Document [original]

Spectrally Efficient Bidirectional

Decode-and-Forward Relaying for

Wireless Networks

von Diplom-Ingenieur

Tobias Josef Oechtering aus Rheine

von der Fakultät IV - Elektrotechnik und Informatik

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften

– Dr.-Ing. –

genehmigte Dissertation

Promotionsausschuss

Vorsitzender: Prof. Dr.-Ing. habil. Gerhard Mönich

1. Berichter: Prof. Dr.-Ing. Dr. rer. nat. Holger Boche,

2. Berichter: Prof. Dr.-Ing. Dr.-Ing. E. h. Joachim Hagenauer

Tag der wissenschaftlichen Aussprache: 09.10.2007

Berlin, 2007

D83

Zusammenfassung

Relaiskonzepte werden in Zunkuft in drahtlosen Netzwerken eine zentrale Rolle spielen.

In dieser Dissertation wird ein Netzwerk mit drei Stationen betrachtet, in welchem zwei

Stationen mittels einer Relaisstation miteinander kommunizieren. Wir untersuchen ein

Zweiphasenprotokoll ohne Rückkopplung, in dem die Relaisstation die Nachrichten für

beide Richtungen gleichzeitig dekodiert und weiterleitet. Damit kann der spektrale Ver-

lust, verursacht durch halbduplexe Stationen in unidirektionalen Protokollen, kompensiert

werden.

Im ersten Abschnitt geht es um schichtübergreifende Konzepte für die bidirektionale Re-

laiskommunikation basierend auf dem Prinzip der Superpositionskodierung. Die Betrach-

tungen werden für die festgelegte und optimale Zeitaufteilung der Phasen durchgeführt. Eine

ausführliche Untersuchung der Kombinatorik der erreichbaren Ratenregion für Stationen mit

einer Antenne erlaubt es uns die Ratenpaare, bei denen die Summe der gewichteten Raten

maximiert wird, explizit zu bestimmen. Diese Untersuchungen werden am Beispiel der pa-

ritätischen und der optimalen Zeitaufteilung ausgeführt.

Die optimalen Ratenpaare werden im Weiteren für den Entwurf einer durchsatzoptimalen

Ressourcenallokationsstrategie und für ein Relaisauswahlkriterium in einem Netzwerk mit

NRelaisstationen verwendet. Dabei ist es vorteilhaft, die abwechselnde Nutzung ver-

schiedener Relaisstationen zu erlauben. Außerdem stellen wir fest, dass die Summe eines

jeden Ratenpaares auf dem Rand der ergodischen Ratenregion für unabhängige und identisch

rayleighverteilte Schwundkanäle asymptotisch mit log(log(N)) wächst.

Als nächstes betrachten wir den Fall, dass die Relaisstation eine Nachricht zusätzlich zur

bidirektionalen Relaiskommunikation an beide Stationen übeträgt. Es zeigt sich, dass es

immer optimal ist die Relaisnachricht zuerst zu dekodieren. Die gemeinsame Ressourcenal-

lokation verbessert die Gesamteffizienz und ermöglicht einen neuen Ratenabtausch. Darüber

hinaus charakterisieren und diskutieren wir das gemeinsame Summenratenmaximum beider

Routingaufgaben.

Anschließend untersuchen wir die erreichbaren Ratenregionen der bidirektionalen Re-

laiskommunikation zwischen Stationen mit mehreren Antennen. Wir spezifizieren die op-

timale Sendestrategie und zeigen, dass die Summe eines jeden Ratenpaares auf dem Rand

der erreichbaren Ratenregion linear gemäß der räumlichen Dimension des Vektorkanals und

der Zeitaufteilung der Phasen ansteigt.

Zusammenfassung

Im zweiten Abschnitt beweisen wir eine optimale Kanalkodierungsstrategie für den bidirek-

tionalen Broadcastkanal mit einem endlichen Alphabet. Dabei werden erreichbare Raten

bezüglich der maximalen Fehlerwahrscheinlichkeit betrachtet. Beim Kodierungssatz folgen

wir der Philosophie der Netzwerkkodierung, die besagt, dass sich Informationsflüsse nicht

wie Flüssigkeiten verhalten.

Im abschließenden Resümee geben wir einen Ausblick auf zukünftige Forschungsarbeiten

und präsentieren Beispiele wie bidirektionale Relaiskommunikation in drahtlose Netzwerke

integriert werden kann.

Abstract

Relaying concepts will play a central role in future wireless networks. In this thesis we

consider a three-node network where two nodes communicate with each other by the support

of a relay node. We study a two-phase decode-and-forward bidirectional relaying protocol

without feedback. Bidirectional relaying has the ability to compensate the spectral loss due

to the half-duplex constraint of nodes in wireless communications.

In the first part we study cross-layer design aspects of bidirectional relaying using super-

position encoding at the relay node. For the two phases we consider the fixed and optimal

time division case. For single-antenna nodes an intensive study of the combinatorial struc-

ture of the achievable rate region allows us to characterize the rate pairs which maximize the

weighted rate sum for the equal and fixed time division case in closed form. These are used

for the design of a throughput optimal resource allocation policy based on the backpressure

strategy and to derive a relay selection criterion for routing in network with Nrelay nodes.

It shows that it is beneficial to allow time-sharing between the usage of relay nodes. We see

that the sum of any rate pair on the boundary of the ergodic rate region for independent and

identical distributed Rayleigh fading channels grows asymptotically with log(log(N)).

Then we add a relay multicast to the bidirectional relay communication. The joint resource

allocation of two routing tasks improves the overall efficiency and enables new rate trade-

offs. It shows that it is always optimal to decode the relay message first. Furthermore,

we characterize and discuss the total sum-rate maximum of both routing tasks. After that

we study the achievable rate region of bidirectional relaying between nodes equipped with

multiple antennas. Therefore, we specify the optimal transmit strategy and show that the

achievable rate region scales linearly with respect to the spatial degrees of the vector channels

and time division.

In the second part we find an optimal channel coding strategy for the bidirectional broadcast

channel considering finite size alphabets. Thereby, we consider achievable rates with respect

to the maximal probability of error. For the coding theorem we follow the philosophy of

network coding and regard information flows not as “fluids”.

In the final conclusion we give an outlook on future research work and show how the bidi-

rectional relaying protocol can be integrated in wireless networks.

Contents

1 Introduction 1

1.1 Trends and Motivations – Related Literature . . . . . . . . . . . . . . . . . 1

1.2 Contribution and Outline of the Thesis . . . . . . . . . . . . . . . . . . . . 10

2 Bidirectional Relay Communication using Superposition Encoding 15

2.1 Introduction.................................. 15

2.2 Achievable Rate Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Gaussian Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Multiple Access Phase . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Broadcast Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.4 Bidirectional Achievable Rate Region . . . . . . . . . . . . . . . . 26

2.2.5 Achievable Rate Regions with Power Scaling . . . . . . . . . . . . 42

2.3 Throughput Optimal Resource Allocation . . . . . . . . . . . . . . . . . . 51

2.3.1 Stability Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.3.2 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . 58

2.4 RelaySelection................................ 61

2.4.1 Relay Selection Criterion . . . . . . . . . . . . . . . . . . . . . . . 63

2.4.2 Scaling Law of the Ergodic Rate Region . . . . . . . . . . . . . . . 67

2.5 Piggyback a Common Relay Message . . . . . . . . . . . . . . . . . . . . 75

2.5.1 Broadcast Phase with Relay Multicast . . . . . . . . . . . . . . . . 78

2.5.2 Total Sum-Rate Maximum . . . . . . . . . . . . . . . . . . . . . . 80

2.5.3 Combinatorial Discussion and Working Examples . . . . . . . . . 85

2.6 Extension to Multi-Antenna Bidirectional Relaying . . . . . . . . . . . . . 103

2.6.1 MIMO Multiple Access Phase . . . . . . . . . . . . . . . . . . . . 107

2.6.2 MIMO Broadcast Phase . . . . . . . . . . . . . . . . . . . . . . . 112

2.6.3 MIMO Bidirectional Achievable Rate Region . . . . . . . . . . . . 117

2.7 Discussion................................... 120

2.8 Appendix:Proofs............................... 125

vii

Contents

3 Optimal Coding Strategy for the Bidirectional Broadcast Channel 181

3.1 Introduction.................................. 181

3.1.1 Two-Phase Bidirectional Relay Channel . . . . . . . . . . . . . . . 182

3.1.2 Capacity Region of the Multiple Access Phase . . . . . . . . . . . 184

3.2 Capacity Region of the Broadcast Phase . . . . . . . . . . . . . . . . . . . 184

3.2.1 Proof of Achievability . . . . . . . . . . . . . . . . . . . . . . . . 186

3.2.2 Proof of weak converse . . . . . . . . . . . . . . . . . . . . . . . . 192

3.2.3 Cardinality of set U......................... 194

3.3 Achievable Bidirectional Rate Region . . . . . . . . . . . . . . . . . . . . 195

3.4 Example with Binary Channels . . . . . . . . . . . . . . . . . . . . . . . . 196

3.5 Discussion and Further Results . . . . . . . . . . . . . . . . . . . . . . . . 198

4 Conclusion and Future Work 201

References 207

viii

List of Figures

2.1 Bidirectional relaying between single-antenna nodes . . . . . . . . . . . . 15

2.2 Bidirectional achievable rate regions . . . . . . . . . . . . . . . . . . . . . 28

2.3 Contour plot of sum-rate maximum . . . . . . . . . . . . . . . . . . . . . 35

2.4 Equivalent characterization of RBRopt .................... 39

2.5 Weighted rate sum optimal rate pairs . . . . . . . . . . . . . . . . . . . . . 41

2.6 Achievable rate region ˜

RBRopt (with scaled powers) . . . . . . . . . . . . 50

2.7 Queueingmodel................................ 52

2.8 Rate and stability regions of different policies . . . . . . . . . . . . . . . . 59

2.9 Queue length evolutions of different policies . . . . . . . . . . . . . . . . . 60

2.10 Achievable rate region of relay selection with time-sharing . . . . . . . . . 66

2.11 Growth of the ergodic rate region with relay selection . . . . . . . . . . . . 72

2.12 Upper and lower bounds on scaling law . . . . . . . . . . . . . . . . . . . 73

2.13 Cases where R⋆

BC(γR)intersects the boundary of the MAC capacity region 86

2.14 Rate trade-offs for constant total sum-rate . . . . . . . . . . . . . . . . . . 90

2.15 Characteristic angles of αCMAC and (1 −α)RBC .............. 95

2.16 Total sum-rate optimal rate pairs with respect to the time division . . . . . . 98

2.17 Total sum-rate maximum with respect to the time division . . . . . . . . . . 101

2.18 Piggyback achievable rate region for equal time division . . . . . . . . . . 102

2.19 Piggyback achievable rate region for optimal time division . . . . . . . . . 102

2.20 Bidirectional relaying between multiple antenna nodes . . . . . . . . . . . 103

2.21 MIMO achievable rate region . . . . . . . . . . . . . . . . . . . . . . . . . 118

2.22 Combinatorial discussion for the proof of Theorem 2.5 . . . . . . . . . . . 127

2.23 Combinatorial discussion for the proof of Lemma 2.46 . . . . . . . . . . . 170

3.1 Two-phase decode-and-forward bidirectional relay channel without feedback 183

3.2 Achievable rate regions for binary symmetric channels . . . . . . . . . . . 198

4.1 Cellular coverage extension using bidirectional relaying . . . . . . . . . . . 202

4.2 Multi-hop communication using bidirectional relaying . . . . . . . . . . . 202

4.3 Coding principle of bidirectional relaying can improve cellular downlink . . 204

Abbreviations, Notation, and Symbols

Abbreviations

AWGN additional white Gaussian noise

iid independent and identical distributed

pdf probability density function

cdf cumulative distribution function

eq equal time division

opt optimal time division

MAC multiple access channel

BC broadcast channel

BR bidirectional relaying

SISO single-input single-output

MIMO multiple-input multiple-output

TDMA time division multiple access

FDMA frequency-division multiple access

CDMA code-division multiple access

RS relay selection

RSTS relay selection with time-sharing between usage of relay nodes

Notation

set of all natural numbers {1,2,3,...}

set of real numbers

+set of non-negative real numbers

set of complex numbers

[a, b]closed interval of real numbers where endpoints ato bare included

(a, b)open interval of real numbers where endpoints ato bare not included

[a, b),(a, b]half-closed intervals of real number where either aor bis included

lhs := rhs the value of the right hand side (rhs) is assigned to the left hand side (lhs)

lhs =: rhs the value of the left hand side (lhs) is assigned to the right hand side (rhs)

αRscaled set αR:= {αR:R∈ R}with α∈

and set R ⊂

Abbreviations, Notation, and Symbols

f:D → C function ffrom domain Dto codomain C

x7→ f(x)mapping of xto f(x)

co(A)convex hull of set A

cl(A)closure of set A

int(A)interior of set A

|A| cardinality of set A

∃x∈ A there exists an element xin set A

∅empty set

Echaracteristic function which is 1if event Eis true and 0else

[x]+abbreviation for max(0, x)

log logarithm to the base two

ln natural logarithm

exp exponential function

∧,∨logical conjunction (AND), logical disjunction (OR)

n=1,WN

n=1 numerated logical conjunctions and disjunctions

||x||1L1-norm of vector x

HH,HThermitian resp. transpose of the matrix or vector H

rank(H)rank of the matrix H

tr(H),det(H)trace and determinant of the matrix H

Iidentity matrix where the dimension follows from the context

diag(a1, a2,...,an)diagonal matrix with diagonal elements a1, a2, . . . , an

X∼ CN(m, σ2)Xis complex Gaussian distributed with pdf fX(x) = 1

πσ2e−|z−m|2

σ2

{X}expectation of the random variable X

{E}probability of event E

Xtime-average of random process X

Θ(g(n)) f(n)is big-theta of g(n)if 0<lim inf

n→∞

f(n)

g(n)≤lim sup

n→∞

f(n)

g(n)<∞.

og(x)f(x)is small-o of g(x)if lim

x→∞

g(x)

f(x)= 0.

Symbols

In Chapter 2 we use boldface letters to denote vectors and matrices and calligraphic letters

denote sets. In Chapter 3 we use the information theoretic typical notation where Xn

1denotes

the sequence X11, X12,...,X1n. We often choose a generic notation for the symbols which

gets at some point quite extensive. The index kenumerates the nodes 1,2, and R respectively,

while the index nenumerates the relay nodes in Section 2.4. The superscripts ∗and ⋆are

used for optimal values. The subscripts eq or opt specify the corresponding rate, rate pair,

or rate region for the equal or optimal time division case. The following tables list the most

important symbols.

Abbreviations, Notation, and Symbols

Pkmean power constraint at node k∈ {1,2,R}

σ2variance of Gaussian noise

γksignal-to-noise ratio Pk/σ2

h1flat fading SISO channel coefficient between node 1 and the relay node

h1,n flat fading SISO channel coefficient between node 1 and the nth relay node

h2flat fading SISO channel coefficient between node 2 and the relay node

h2,n flat fading SISO channel coefficient between node 2 and the nth relay node

H1flat fading MIMO channel matrix between node 1 and the relay node

H2flat fading MIMO channel matrix between node 2 and the relay node

αfraction of time in the MAC phase, i.e. fraction 1−αof the time in BC phase

α∗optimal αto support a certain rate pair in MAC phase

Bset of feasible power distribution vectors [β1, β2]

β1proportion of PRspend for forwarding message m1to node 2

β2proportion of PRspend for forwarding message m2to node 1

βrelay power distribution with β1=βand β2= 1 −β

β∗optimal βto support a certain rate pair in MAC phase

β⋆optimal relay power distribution, cf. Proposition 2.2

γmin

Rminimal necessary γRto support a certain rate pair in CMAC in the BC phase

γMAC

Rminimal necessary γRto support any rate pair in CMAC in the BC phase

γΣMAC

Rminimal necessary γRto support a rate pair [R1, R2]∈ RMAC

Σin the BC phase

γ†

Rcharacteristic signal-to-noise ratio for the broadcast phase, cf. Corollary 2.8

γΣ1

Rminimal necessary γRto support the vertex νΣ1 ∈ CMAC in the BC phase

γ2Σ

Rminimal necessary γRto support the vertex ν2Σ ∈ CMAC in the BC phase

ˆγcharacteristic signal-to-noise ratio γR, cf. Lemma 2.46

νΣ1 vertex between R−→

1R and RMAC

Σof multiple access rate region, cf. (2.5a)

ν2Σ vertex between R−→

2R and RMAC

Σof multiple access rate region, cf. (2.5b)

R1achievable rate form node 1 via relay node to node 2

R2achievable rate from node 2 via relay node to node 1

RΣbidirectional sum-rate R1+R2=RΣ

RRadditional relay multicast rate

Rmax

Rmaximal achievable additional relay rate RR

R†characteristic rate, cf. Lemma 2.46

Rtot total sum-rate Rtot =R1+R2+RR

R∗

tot total sum-rate maximum of piggyback rate region RPiggy

R−→

1R individual rate constraint from node 1 to the relay node in MAC phase

R−→

1R,n R−→

1R for the nth relay node

R−→

2R individual rate constraint from node 1 to the relay node in MAC phase

R−→

2R,n R−→

2R for the nth relay node

RMAC

Σsum-rate constraint of node 1 and 2 to the relay node in the MAC phase

RMAC

Σ,nRMAC

Σfor the nth relay node

xii

Abbreviations, Notation, and Symbols

R2Σ

1maximal achievable rate R1at vertex ν2Σ, cf. (2.5b)

RΣ1

2maximal achievable rate R2at vertex νΣ1, cf. (2.5a)

R−→

R1 individual rate constraint from the relay node to node 2 in BC phase

R−→

R1,n R−→

R1 for the nth relay node

R−→

R2 individual rate constraint from the relay node to node 2 in BC phase

R−→

R2,n R−→

R2 for the nth relay node

R∗

−→

1R maximal unidirectional rate R1achievable in CMIMO

MAC , cf. (2.6.1)

R∗

−→

2R maximal unidirectional rate R2achievable in CMIMO

MAC

R⋆

−→

R1 sum-rate optimal rate R1in BC phase, cf. Proposition 2.2

R⋆

−→

R2 sum-rate optimal rate R2in BC phase, cf. Proposition 2.2

Ropt

1total sum-rate optimal rate R1, cf. Theorem (2.48)

Ropt

2total sum-rate optimal rate R2, cf. Theorem (2.48)

R①proportion of rate R1interchanged for relay rate RR, cf. Corollary 2.49

R②proportion of rate R2interchanged for relay rate RR, cf. Corollary 2.49

R⋆

1BC sum-rate optimal rate R1, cf. Proposition 2.47

R⋆

2BC sum-rate optimal rate R2, cf. Proposition 2.47

RBC

Σsum-rate maximum of broadcast phase, cf. Proposition 2.2

RΣbidirectional sum-rate optimal rate pair, cf. Corollary 2.8

RBC rate pair of parametrized boundary of RBC

Ropt rate pair of parametrized boundary of RBRopt, cf. Corollary 2.15

R⋆

BC BC sum-rate optimal rate pair, cf. Proposition 2.47

RMIMO

MAC weighted rate sum optimal rate pair of CMIMO

MAC , cf. (2.75)

RMIMO

BC weighted rate sum optimal rate pair of RMIMO

BC , cf. (2.79)

R∗

eq weighted rate sum optimal rate pair for α=1

/2, cf. Theorem 2.10

R∗

eq,n R∗

eq for the nth relay node

R∗

opt weighted rate sum optimal rate pair of RBRopt, cf. Theorem 2.16

R∗

opt,n R∗

opt for the nth relay node

R∗

1eq maximal unidirectional rate R1achievable in RBReq, cf. (2.22a)

R∗

1eq,n R∗

1eq for the nth relay node

R∗

2eq maximal unidirectional rate R2achievable in RBRopt, cf. (2.22b)

R∗

2eq,n R∗

2eq for the nth relay node

R∗

keq maximal unidirectional rate Rk,k= 1,2, achievable in RBReq

R∗

keq,n R∗

keq for the nth relay node

R∗

kopt maximal unidirectional rate Rk,k= 1,2, achievable in RBRopt

R∗

kopt,n R∗

kopt for the nth relay node

R∗

kRSeq maximal unidirectional rate Rk,k= 1,2, achievable in RRSTSeq

R∗

kRSopt maximal unidirectional rate Rk,k= 1,2, achievable in RRSTSopt

R∗

kRSeq maximal unidirectional ergodic rate Rk,k= 1,2achievable in RRSTSeq

R∗

kRSopt maximal unidirectional ergodic rate Rk,k= 1,2achievable in RRSTSopt

xiii

Abbreviations, Notation, and Symbols

CBC capacity region in the BC phase

CMAC capacity region in the MAC phase

CMAC,n capacity region of the nth relay node in the MAC phase

CMIMO

MAC capacity region of MIMO MAC

R−→

1R set of rate pairs in CMAC with R1=R−→

R−→

2R set of rate pairs in CMAC with R2=R−→

RMAC

Σset of rate pairs in CMAC with R1+R2=RMAC

RBC achievable rate region in the BC phase

RMIMO

BC achievable rate region of MIMO BC

RBC,n achievable rate region of the nth relay node in the BC phase

R1set of transformed rate pairs for rate constraint R−→

1R, Theorem 2.12

R2set of transformed rate pairs for rate constraint R−→

2R, Theorem 2.12

RΣset of transformed rate pairs for rate constraint RMAC

Σ, Theorem 2.12

RBR bidirectional achievable rate region

RMIMO

BR MIMO bidirectional achievable rate region

RBR ergodic bidirectional rate region

RBReq bidirectional achievable rate region with equal time division

RBReq,n RBReq achievable with the nth relay node

RBRopt bidirectional achievable rate region with optimal time division

RBRopt,n RBRopt achievable with the nth relay node

RMIMO

BRopt MIMO bidirectional achievable rate region with optimal time division

RRSeq achievable rate region with relay selection for equal time division

RRSopt achievable rate region with relay selection for optimal time division

RRSTSeq RRSeq with time-sharing between the usage of relay nodes

RRSTSopt RRSopt with time-sharing between the usage of relay nodes

RRSTSeq ergodic achievable rate region of RRSTSeq

RRSTSopt ergodic achievable rate region of RRSTSopt

RPiggy achievable rate region with additional relay multicast

xiv

1 Introduction

1.1 Trends and Motivations – Related Literature

Most of these days phenomenal technological advances base beside others on the continuous

progress in integrated circuit design which is described by the popular Moore’s law. In

1965 Moore [Moo65] observed that with continuous decreasing packages size the relative

manufacturing cost per component exponentially decreases while additionally the number of

components per integrated circuit exponentially increases. This trend has been observed until

now which has made the development of portable computation and communication devices

with an affordable price possible. With this we have an ever-growing number of wireless

applications and users – the technological vision of the WWRF1expects “7 trillion wireless

devices serving 7 billion people by 2017”.

The broad consumer wireless telecommunications started with generations of cellular voice

networks with a rapidly increasing number of subscribers at the turn of the millennium.

With a growing demand for new high data services such as mobile Internet with video appli-

cations or interactive gaming the development of latest generations show a shift to cellular

data networks which also support the voice application. However, the network bandwidth

in favorable frequency bands is finite since the nowadays technically usable low frequencies

are limited. Future systems will operate at higher frequencies, but there the wireless com-

munication is more susceptible to radio propagation conditions. This will even worsen the

coverage problem service providers already have. Obviously, the providers are interested

to offer satisfactory services anywhere at any time, which is an engineering challenge es-

pecially where the direct link does not have the desired quality, e.g. due to shadowing or

distance. In a single point-to-point connection one could simply increase the transmit power,

which results in a higher interference level in the network and a higher energy consumption.

Both is unattractive, because a lower interference level often allows to accommodate more

users in the network and a reduced energy consumption results either in a longer battery

life or a smaller battery size, which is a crucial point for miniaturization. As a consequence

in traditional cellular network architectures the provider can only install more base-stations

to cover the whole area, but a higher density of base-stations leads to a notable increase in

1WWRF-Wireless World Research Forum - http://www.wireless-world-initiative.org

1 Introduction

infrastructure costs. Katz discusses in [Kat94] many fundamental design issues of wireless

networks at that time which are still important these days.

On the other hand, wireless networks without any wired infrastructure are called ad hoc net-

works. Such networks utilize the broadcast nature of the wireless channel so that in principle

any node can communicate to any other. But due to the attenuation of a wireless channel a

reliable transmission is typically only possible to some neighboring nodes. Therefore, for

a successful transmission to a node that is not in its vicinity the nodes need to cooperate to

forward each others information, which is called a multi-hop communication. Ad hoc net-

works have some appealing properties because of this relay communication technique. First,

we see that any additional node added to the network increases the overall connectivity. Fur-

ther, if a node disappears due to bad channel conditions or power supply insufficiency other

nodes in its vicinity have the possibility to take over its routing responsibilities. This node

redundancy and its adaptability makes ad hoc networks highly robust. And since we do not

require a wired backbone, it is said that an ad hoc network is easily deployed. It follows

that there are many applications where an ad hoc topology is more appropriate than cellular

topology, e.g. sensor networks, home environments, or networks of automobiles.

Accordingly, the integration of multi-hop capability in traditional cellular systems has the

potential to improve the connectivity by closing internal gaps of a cell without installing

more base-stations. It could be also used as a “last mile” technology to extend the periphery

of a wireless cell. Thereby, the relay nodes may be mobile or fixed, but they do not have

access to the wired backbone network of the base-stations or access points. Theoretical and

measurement based radio propagation models indicate that the average received signal power

typically falls off with the n-th power of the distance between transmitter and receiver with

a path-loss coefficient nbetween 2 (free space) and 6 (obstructed in building) [Rap96, Table

4.2]. Since multi-hop transmission splits a long distance into several shorter distances, it

follows that the cumulative path loss is lower than the path loss of the direct transmission.

This leads to an additional reduction in power and therefore energy consumption and/or a

lower power level throughout the network.

That multi-hop or relay strategies are promising technological advancements of future wire-

less networks indicate for instance that for all network sizes there are IEEE 802 LAN/MAN

standardization working groups2which are focused on the developing of mesh-enabled stan-

dards:

• For wireless personal area networks (WPAN) by the Task Group 5 of IEEE 802.15

standard.

• For wireless local area networks (WLAN) in the extension IEEE 802.11s.

2The IEEE 802 LAN/MAN Standards Committee develops Local Area Network standards and Metropolitan

Area Network standards. http://grouper.ieee.org/groups/802/

1.1 Trends and Motivations – Related Literature

• For wireless metropolitan area networks (WMAN) in the amendment IEEE 802.16j –

mobile multi-hop relay. Thereby, one allows every subscriber station to function as a

backhaul, forwarding traffic until it finally arrives at the intended destination.

• Such meshed network principles will be indispensable for a future wireless network,

which should realize broadband wireless access across the country (Mobile Broadband

Wireless Access) and should be someday specified in a IEEE 802.20 standard.

These standardizations are guided by industrial player forums like WiMAX for WMAN,

Wi-Fi for WLAN, ZigBee for low data rate WPAN which try to promote conformance and

interoperability of the devices. In addition to this, many companies are developing various

proprietary mesh network solutions. It is expected that relaying strategies should close the

gap between WLAN-type and cellular systems. For more details we refer to [LZKS06].

For the design and development of communication networks engineers successfully used

the concept of layered architecture to split the enormous complexity of modern communica-

tion networks communication. This allows the separate design of a protocol for each layer.

Among them the ISO-OSI3reference model is still the most prominent. The IEEE 802 stan-

dards for wireless networks specify the two lower layers. However, an efficient design of a

multi-hop communication represents such a significant technical challenge that a cross-layer

design is imperative. This can be seen by a brief inspection of the three (media) layers at the

bottom of the stack. The Physical Layer is concerned with signal transmission aspects. The

Data Link Layer is concerned with the transfers of data between nodes. In IEEE 802.16, the

Data Link Layer is divided in a Logical Link Control Sublayer and Media Access Control

Sublayer. The Logical Link Control is concerned with the flow control, which means that

it multiplexes the service provided to the higher layer and is responsible for acknowledg-

ment and error recovery. The Media Access Control provides addressing and channel access

control mechanisms which means that it is responsible for the spectral resource allocation.

The Network Layer is responsible for the end-to-end transmission, which means that it is

responsible for routing the data from the source to the destination. Obviously, for an effi-

cient resource allocation for a multi-hop communication we have to route our messages via

nodes which have sufficient good channels. Otherwise, if higher layers fix the route, on the

Physical Layer the nodes may be coerced to use higher transmit powers to offer the desired

service to the Data Link Layer. This causes a higher energy consumption and a higher inter-

ference for other nodes which will make the medium access inefficient as well. Therefore,

for an efficient future wireless system a cross-layer design with an adaptive power control

and resource allocation will be necessary. A more detailed discussion about these motivat-

ing thoughts about adaptive power control, energy efficient cross-layer design aspects, and

3Open System Interconnection (OSI) protocol stack, which was developed as a reference model by the Inter-

national Standards Organizations (ISO)[Zim80]. From top to bottom the ISO-OSI reference model defines

seven layers in a stack: Application Layer, Presentation Layer, Session Layer, Transport Layer, Network

Layer, Data Link Layer, and Physical Layer.

1 Introduction

efficient methods of allocating network resources can be found in [Bam98], [GW02], and

[SRK03].

Wireless Communication: Cooperative Diversity

Wireless communications deals with the problem of reliable transmission of information

despite of impairments introduced by the wireless transmission. Propagation impairments

are usually collected in a time-varying fading channel model that collects phenomena like

multi-path, Doppler-shift, path-loss, and shadowing. Additionally, noise and interference

limit the communication. Diversity techniques reduce the probability that a transmission fails

due to a channel which is in a deep fade. We distinguish between time, frequency, and spatial

diversity according to the dimension where the robustness against fading is realized.

In a cellular network other users may receive the signal from the communication between

a user and a base-station due to the broadcast nature of the wireless channel. Then we will

gain spatial diversity if such a user additionally forwards the signal over an independent path

to the destination. In [SEA98] and [SEA03a, SEA03b] Sendonaris, Erkip, and Aazhang

introduced this concept where two users benefit by relaying their signals for each other and

called it user cooperative diversity. Basically, the diversity gain is achieved by implementing

an additional multi-hop communication.

The contribution [LTW01, LTW04] of Laneman, Tse, and Wornell is the next influential

work in the context of cooperative diversity in wireless networks. They studied the out-

age behavior of several cooperative diversity protocols named according to the cooperative

processing strategy: decode-and-forward, amplify-and-forward, selection, and incremental

relaying. It shows that except for the decode-and-forward strategy all strategies achieve the

full diversity order in the high SNR regime. Thereby, the reduced diversity order for the

decode-and-forward protocol is caused by the definition of the outage event. In a subsequent

work [LW03] Laneman and Wornell proposed distributed space-time codes, which also can

achieve the full spatial diversity given by the number of cooperating terminals.

Those works have sparked a growing interest in the design and analysis of cooperative and

therefore relay transmission protocols. Most of them focused on incorporation of vari-

ous types of channel codes, so called coded cooperation, e.g. [HZF04], [HN06], [SE04],

[LSS03], and [MMYZ06] to mention only a few. [NHH04] and [SHMX06] provide more

elaborate introductions.

Nearly all cooperative protocols assume half-duplex nodes, which means that a node cannot

transmit and receive at the same time using the same frequency. This is due to the fact that in

wireless communications it is technically difficult and often impossible to isolate sufficiently

the received signal from the transmit signal where the transmitter and receiver of the node use

the same bandwidth at the same time. For that reason the protocols require a medium access

1.1 Trends and Motivations – Related Literature

strategy in time or frequency domain which allocates exclusive resources for the reception

of the cooperative user signal and the transmission of its own and cooperative users signals,

which can be seen as time or frequency division duplex at the relay node. Accordingly,

Laneman et al. point out that for the benefits from user cooperative diversity we have a loss

in spectral efficiency due to half-duplex operation, which makes a diversity-vs-multiplexing

trade-off discussion necessary [LTW04, AGS05, YE06]. This concept was introduced by

Zheng and Tse in [ZT03] for wireless communication of nodes with multiple antennas and

can be applied if one considers the cooperative users as a distributed multi-antenna system.

The basic idea behind cooperative/multi-hop communication is that a cooperative node relays

the signal or message. Accordingly, to find the optimal respectively a good cooperative

strategy we should look at the results of information theory on the relay channel.

Information Theory: Relay Channel

Shannon stated the channel capacity theorem in 1948 in his seminal work “A Mathematical

Theory of Communication” [Sha48], which is usually seen as the birth of information the-

ory. In 1961 Shannon introduced in [Sha61] the two-way channel where two nodes want to

communicate as effectively as possible with each other. Therein, he obtained the capacity

region for the average error for the restricted two-way channel, this means that a feedback

between the encoders of the two nodes is not allowed. Nowadays this work is regarded as the

first work on multi-user information theory. Compared to the knowledge of the single-user

case we are still in the beginning of understanding multi-user information theory.

Recently, the relay channel is experiencing a revival because multi-hop communication for

meshed network architectures has the potential for a substantial coverage extension. The

relay channel problem was introduced by van der Meulen in [vdM71] in the early seventies.

A few years later, Cover and El Gamal established in [CG79] the capacities when the channel

is degraded, reversely degraded and when feedback is added from both receivers to the sender

and relay node. For the general relay channel they derived an upper bound to the capacity

and an achievable rate. Later, in [GA82] the capacity of the semideterminitic relay channel,

where the output of the relay is a deterministic function of the input of both senders, is found.

However, the capacity of the general relay channel is still unknown.

Since then one tried to get closer by the study of channel models with a simplifying assump-

tion. Zhang derived in [Zha88] a converse for a relay channel where he assumed a noiseless

channel with a certain rate between the relay and the destination. Cover and Kim showed in

[CK06] that this upper bound is actually achievable using alternatively a novel “hash-and-

forward” or “compress-and-forward” strategy. In [GH05] El Gamal and Hassanpour study

another simplified relay channel where the transmitted relay symbols are allowed to depend

additionally on the non-causally present received symbol. It shows that with “instantaneous

1 Introduction

relaying” in a Gaussian channel an amplify-and-forward strategy achieves the cut-set bound

under certain conditions.

Nonetheless, for the lower bound Cover and El Gamal established in [CG79] a coding strat-

egy which gives us an achievable rate. An equivalent but more regular coding scheme was

found by Williams in [Wil82]. Then in [SG00, GK03, XK04] the authors extend these results

to the multiple relay channel. Thereby, in [XK04] an even more practical coding strategy is

presented. In [KGG05] Kramer, Gastpar, and Gupta extend many cooperative strategies to

relay networks with many terminals, antennas, and sources and compare the performances

of the strategies under wireless communication aspects.

All works on the relay channel we considered so far assume full-duplex nodes. El Gamal

noted4that nobody had wireless relay networks in mind when they started working on the

relay channel. Accordingly, in the last years some work on the relay channel is done which

considers wireless communication aspects. Firstly, in [KSA03] Khojastepour, Sabharwal,

and Aazhang transfered some results in a straightforward manner to the relay channel with

half-duplex nodes, which they called cheap relay nodes. They separate the transmission and

reception at the relay node in the time domain. Then El Gamal and Zahedi establish in

[GZ05] the capacity of the relay channels with orthogonal channels in the frequency domain

from the sender to the relay and from the sender and relay to the destination. In [LLG06]

Lai, Liu, and El Gamal consider cooperation strategies for a three-node wireless network. In

particular they propose a cooperation strategy for the relay channel with a noisy feedback

combining the decode-and-forward and compress-and-forward strategy.

In [DK04] Dawy and Kamoun determine relay regions where it is beneficial to use a relay

node. Therefore, they compare the power allocation for a multi-hop transmission with and

without cooperation as well as a single-hop transmission where the channel model includes

the path loss between nodes. Liang and Veeravalli derive in [LV05] the optimal distribu-

tion of the orthogonal resources in frequency domain in a Gaussian relay channel. Finally,

in [HMZ05] Host-Madsen and Zhang study the ergodic capacity of the relay channel in a

Rayleigh distributed fading environment. All authors recognized that in the case of half-

duplex nodes we need to allocate additional resources in time or frequency. This means that

the relay communication is only spectrally more efficient than the direct communication if

the spectral gain using the relay node compensates the “costs” of the additional resource

allocation.

4El Gamal noted this at his talk on the MSRI Workshop: Mathematics of Relaying and Cooperation in Com-

munication Networks, University of California, Berkeley, April 2006.

1.1 Trends and Motivations – Related Literature

Bidirectional Relaying and Network Coding

Cooperative protocols establish a separate route to realize diversity. Therefore, the cooper-

ating users have to split their resources which results in a loss in spectral efficiency. From

the half-duplex relay channel we know that this is only spectrally more efficient if the re-

lay route results in a sufficient large spectral gain. For instance in [SE05] we can find an

example in the context of cooperative space-time codes where a user with a sufficient weak

channel can even worsen the performance of the other user. Thereby, they even did not take

into account the extra allocated spectral resource for the cooperative protocol. Accordingly,

for a user with a good direct link it is inefficient to cooperate with a user which has a weak

direct link. On the other hand the user with the weak direct channel benefits a lot from co-

operation. It follows that in coverage problems only one user will benefit from a cooperative

protocol, which means that for the coverage problem a simple relay or multi-hop communi-

cation protocol is more suitable. Rankov and Wittneben must have had similar thoughts and

motivations when they had the ingenious idea to propose the bidirectional relaying5concept

in [RW05a, RW05b, RW07].

In more detail, in [RW05a] two spectrally efficient amplify-and-forward protocols for uni-

directional and bidirectional relaying are studied. The unidirectional protocol bases on a

strategy where two relay nodes alternately forward messages from the source to the destina-

tion which we similarly proposed in [OS04]. They show that for a sufficiently bad inter-relay

channel the strategy achieves roughly the performance of a full-duplex amplify-and-forward

protocol. For the bidirectional amplify-and-forward protocol two nodes, namely node 1 and

node 2, want to communicate through the support of a common half-duplex relay node. In

the first time slot nodes 1 and 2 transmit their signal to the relay. The relay node scales

the received signal according to its transmit power constraint and retransmits the signal to

nodes 1 and 2 in the following time slot. Since nodes 1 and 2 know their own transmitted

signals they can subtract their contribution before decoding. Thereby, each unidirectional

link still suffers from the additional resource allocation due to the half-duplex node, but the

sum-rate is significantly increased since the relay transmission supports two communication

links at the same time. Note that for the bidirectional protocol between nodes 1 and 2 is

no direct link because both nodes simultaneously transmit in the first phase and receive in

the second phase. A similar strategy for a satellite communication is proposed in the patent

[DMM97].

In [RW05b] Rankov and Wittneben extend the previous spectrally efficient protocols to the

case where the relay nodes support the communication by a decode-and-forward strategy.

The unidirectional protocol with multiple relay nodes leads to an interference channel prob-

lem [CT91] so that this protocol works well for a weak and strong inter-relay channel. The

bidirectional decode-and-forward results in a multiple access channel when nodes 1 and 2

5The bidirectional relaying protocol is also known as the two-way relay channel.

1 Introduction

transmit their information for each other to the relay node. After the relay node successfully

decoded the messages it re-encodes them using superposition encoding and forwards the new

codeword to nodes 1 and 2. Again, both nodes use the knowledge about their own message

to improve their decoding capabilities. Thereby, it is important to notice that they do not

allow cooperation between the encoders of nodes 1 and 2 by feedback, which may improve

the performance. Nonetheless, their final sum-rate performance comparison of all proposed

half-duplex spectrally efficient protocols shows that the bidirectional decode-and-forward

protocol is the most efficient.

We see that if bidirectional communication between the nodes 1 and 2 is desired and the

direct link is sufficiently bad, the bidirectional protocol “trades” the direct links in two half-

duplex relay channels (same routing problem) for increased spectral efficiency. For that

reason, we think that this is conceptually a wise approach for coverage problems where we

have the problem of a weak direct link. In Chapter 2 we study cross-layer design aspects of

this bidirectional decode-and-forward protocol for half-duplex nodes.

In [RW06] Rankov and Wittneben study the achievable rate region for the restricted full-

duplex case. In all works they follow an analogy which Shannon phrased at his Kyoto lec-

ture6in 1985 as follows “A basic idea in information theory is that information can be

treated very much like a physical quantity, such as mass or energy”. However, in the work

“Network information flow” from Ahlswede, Cai, Li, and Yeung [ACLY00] it is shown that

“it is in general not optimal to regard the information to be multicast as a “fluid” which can

simply be routed or replicated. Rather, by employing coding at the nodes, which [they] refer

to as network coding, bandwidth can in general be saved”. This seminal work establishes a

new direction in multiterminal source coding and is closely related to graph theory. The main

result is a Max-flow Min-cut Theorem for the information flow in a multi-terminal source

coding problem with one information source and error-free links. It says that the maximum

rate that a sender can multicast to a set of receivers is given by the minimum cut between

source and receivers. They also present a simple example from Yeung [Yeu95] which shows

that a multi-source problem is not a trivial extension. Finally they conclude that “coding

by superposition is not optimal in general.” These works have caused a paradigm shift and

have stimulated a new flourishing research area in information theory. The textbook [Yeu02,

Chap. 11 and 15] and the tutorial [Yeu05] are good introductions into this rapidly growing

research field and provide more extensive reference lists than we can give.

Ahlswede et al. do not restrict in their work the encoding functions at the terminals. Li,

Yeung, and Cai prove in [LYC03] that linear coding over a certain base field suffices for

a multicast problem. In [KM03], Koetter and Medard present an algebraic framework for

the design of linear network codes in a finite field. Today, linear encoding functions based

on the addition in finite fields, especially XOR-Operation in

2, have found formidable

prominence in the design of network codes. This may go back to the famous “butterfly

6Ahlswede cited this in his Shannon Lecture 2006 [Ahl06].

1.1 Trends and Motivations – Related Literature

network” [Yeu05, Ex. 1.1], [ACLY00, Fig.7] which shows with a striking simplicity the

potential of network coding. With a specific grouping of the terminals the “butterfly network”

describes a bidirectional communication network [Yeu05, Ex. 1.3] which suggests the relay

node to perform an XOR-operation. Then it is interesting to see that the relay node together

with the error-free channels to nodes 1 and 2 describes exactly the XOR-channel of Shannons

example for the two-way channel [Sha61, Fig. 4]. Without considering channel coding

aspects we find this XOR encoding example for bidirectional relaying in [Yeu05, Ex. 1.4], in

a multi-hop context with a simple acknowledgment protocol in [WCK05], and with a cyclic

redundancy check at the relay node in [LJS05]. Popovski and Yomo propose in [PY06] a

Denoise–and–Forward protocol, where the relay tries to eliminate the noise in the received

sum signal by estimation.

The proposed strategies based on network coding principles also show that bidirectional

communication can increase the achievable sum-rate. The channel coding strategy of Rankov

and Wittneben treats information as a physical entity, which is in a multi-terminal source cod-

ing problem in general not optimal. For that reason, in Chapter 3 we study bidirectional re-

laying with classical channel coding arguments following the philosophy of network coding.

We see that the bidirectional relaying with decode-and-forward is a channel where super-

position encoding is indeed not optimal. Unfortunately, we obtained these results after our

cross-layer design studies presented in Chapter 2, where we assume separated information

flows using superposition encoding.

There is lots of work ongoing to transfer the network coding idea to wireless network prob-

lems. The easiest way would be to separate channel-coding and network-coding, but this is in

general not optimal. In [EMH+03] it is shown that separation holds for some networks, but

they also provide networks where separation fails. Likewise in [RK06] it is shown for a relay

network with deterministic channels that channels separation does not hold in general. For

that reason, the design of joint network-channel codes is indispensable. In [HSOB05] Hausl,

Schreckenbach, Oikonomidis, and Bauch introduce a joint network-channel code based on a

distributed Low-Density Parity-Check code for the two-way relay channel, where encoders

and decoders are designed such that all the network coded parity bits send by the relay node

are useful for the decoding at both users. Then in [HH06] Hausl and Hagenauer extend the

concept to a turbo network code for the two-way relay channel. Finally in [HD06] we can

see for the multiple access relay channel that the joint network-channel coding approach

performs better than the separate network-channel coding or a distributed turbo code ap-

proach.

1 Introduction

1.2 Contribution and Outline of the Thesis

In Chapter 2 we study bidirectional relaying based on superposition encoding as proposed

in [RW05b], which means that we consider separated information flows. We study differ-

ent cross-layer design aspects to improve the efficiency and realize synergetic benefits. As

performance metric we consider the achievable rates assuming discrete-time memoryless

Gaussian channels. For that reason, we briefly review some information theoretic arguments

in between to introduce the achievable rates as performance metric.

• In Section 2.2 we intensively examine the achievable bidirectional rate region of a

fixed and optimal time division approach where the nodes are equipped with a single-

antenna element. For a complete understanding of the combinatorial properties we

discuss for different desired rate-pairs the minimal necessary signal-to-noise ratios

in the broadcast phase. Furthermore, we specify the sum-rate optimal rate pairs for

different rate regions. This allows us to characterize in closed form the achievable

rate pairs for the equal and optimal time division case where the weighted rate sum is

maximized, so called Pareto optimal rate pairs7. In the last subsection we characterize

the bidirectional achievable rate region for the optimal time division with scaled mean

power constraints, which corresponds to a different interpretation of the constraint for

a code word concerning its code word length. Parts of the results are published in

[OB06b, OB07a], and should be published in [OB06e] and [OB07b].

• In Section 2.3 we introduce and study a queueing model and consider a throughput

optimal resource allocation policy based on the maximum differential backlog algo-

rithm developed by Tassiulas and Ephrimedes [TE92]. We prove the stability region

of bidirectional relaying using standard arguments for the proof as done in [NMR03],

which base on a well-developed Lyapunov drift analysis of [MT93]. Additionally,

we present some numerical simulation results. Parts of the results are published in

[OB06d, OB07a], and should be published in [OB06e].

• In Section 2.4 we consider the routing problem of finding the best relay node in a

network where Nnodes are willing to support the bidirectional communication be-

tween the nodes 1 and 2. We propose to do relay selection based on the achievable

rate region using the Pareto optimal rate pairs. Finally, we prove that for independent

and identical Rayleigh distributed fading channels the sum of the ergodic achievable

rate pairs on the boundary of the equal and optimal time division cases asymptotically

scale with Θ(log(log(N))). Parts of the results are published in [OB07a, OB07e], and

should be published in [OB06a].

7A feasible point of a vector optimization problem is Pareto optimal if and only if there is no feasible point

which is better in all entries of the vector-valued objective [BV04, Sect. 4.7.3.]. Accordingly, in our context

it means that rate pairs on the boundary in the first quadrant are Pareto optimal since we cannot increase one

rate without decreasing the other.

1.2 Contribution and Outline of the Thesis

• In Section 2.5 we add a multicast of the relay node to the bidirectional relaying pro-

tocol. We examine the joint resource allocation for both routing tasks. Furthermore,

we characterize the optimal decoding order and the total sum-rate optimum for any

fixed time division. Thereby, we identify a rate trade-off property which characterizes

the interchange of bidirectional rate with relay multicast rate while the total sum-rate

remains constant. After that we provide an extensive combinatorial discussion regard-

ing the trade-off, a desired relay rate, and the total sum-rate maximum with respect to

the time division. Parts of the results are published in [OB07a, OB07d], and should be

published in [OB06c].

• In Section 2.6 we consider bidirectional relaying between nodes equipped with mul-

tiple antennas. Therefore, we briefly review the information theoretic concepts for

vector-valued processing and examine the optimal resource allocation assuming per-

fect channel knowledge at the transmitters as well. The look at the high power behavior

shows a linear growth of the sum of any rate pair on the boundary of the achievable

rate region with respect to the spatial degree of the vector channels and time division.

Parts of the results are published in [OB07c].

• In Section 2.7 we sum up the main results of this chapter and discuss some connections

between them.

In Chapter 3 we derive an optimal coding strategy for the two-phase bidirectional decode-

and-forward relaying protocol without feedback. This means that we consider the two-phase

protocol where we require that the relay node has to decode the messages in the first phase

and forwards the re-encoded messages in the succeeding broadcast phase. Thereby, we do

not allow cooperation between the encoders of nodes 1 and 2. We derive the capacity region

with respect to the maximal error probability by proving a coding theorem and a weak con-

verse. For the illustration of the results we present an example based on the binary erasure

multiple access channel and the binary symmetric broadcast channel. After that we give a

brief discussion and refer to further results regarding a strong converse and practical coding

aspects. Parts of the results are published in [OSBB07, BOSB07, SOS07] and should be

published in [OBSB07].

Finally, we conclude the thesis in chapter 4 where we give an outlook on future research

directions and present examples how the bidirectional relaying protocol can be integrated in

wireless networks. A complete publication and reference list finalizes the thesis.

Further results which are not part of the Thesis

During my work at the Technical University of Berlin we obtained further interesting results

which are not part of this Thesis.

1 Introduction

• In [OB03a] and [OB03b] we characterize the optimal power allocation for an amplify-

and-forward cooperative system with and without a direct link between source and

destination respectively. The single-antenna transmit nodes form a distributed antenna

array, which results in a vector channel to the multi-antenna destination node. We

assume a sum-power constraint between all transmitters. In [OB03a] we show that

the asymptotic power distribution differs to the equal power distribution of a classical

MIMO channel. Furthermore, we see that the capacity saturates in the high power

regime due to the noise amplification. On the other hand, in [OB03b] we find a new

behavior for a scenario where a cooperative station has a larger channel gain than of

the direct link. If both channels are highly correlated it shows that it is optimal to use

the cooperative node only in the low power regime. This means that with increasing

sum-power it is optimal to turn off the cooperative relay node, which is again due to

the additional noise of the relay node.

• In [OS04, OB05, OSB06] we propose a cooperative transmission scheme which cir-

cumvents the spectral loss due to the pre-log factor from the half-duplex restriction.

Therefore, we need at least two cooperating nodes which alternatively either transmit

or receive so that at any time the signal from the source is retransmitted after a linear

processing by at least one relay node. It follows that we prevent the pre-log factor

by allocating additional resource in space. But the retransmission scheme could also

cause an unstable system. Accordingly, we find the condition for the linear processing

which ensures system stability. Moreover, we present a necessary and sufficient con-

dition so that the relay network has a finite impulse response, which limits the impact

of the noise added by the relay nodes. We end up with a system equation which is

equivalent to a transmission over a frequency-selective channel with additive colored

noise. For the evaluation of the relay network we study the frame error performance

of a proposed suboptimal Viterbi-decoder, a maximum likelihood sphere-decoder, the

performance of a detector based on the semidefinite relaxation method and the di-

versity order8. Furthermore, we derived the capacity with equal power distribution

between the nodes, which can be solved using Jensens formula [Rud66] and charac-

terize channel conditions where the relay network will always have a better pair-wise

error probability.

• In coauthored works with Aydin Sezgin we obtain results on the field of Space-Time

Block Codes. In [SO04a] we propose a transmit strategy which bases on regular block

Markov encoding. In [SO04b, SO04d, SO04c] we find and utilize spectral properties

of quasi-orthogonal Space-Time Block Codes.

8These discussions were made in the context of the supervised diploma theses: “Analysis of Sphere Decoding

in linear cooperative wireless Relay Networks" of Benjamin Schubert and “Diversity order analysis of com-

plexity efficient decoding techniques in linear cooperative wireless relay networks" of Rafael Wyrembelski.

1.2 Contribution and Outline of the Thesis

• In the coauthored work with Eduard Jorswieck [JOB05], we study the impact of chan-

nel correlation on diversity combiners in a flat fading single-input multiple-output sys-

tem using majorization theory [MO79].

2 Bidirectional Relay Communication

using Superposition Encoding

2.1 Introduction

We consider a three-node network where two nodes communicate with each other using the

support of a half-duplex relay node. The inherent spectral loss of unidirectional protocols can

be significantly reduced through bidirectional relaying. In this chapter we study some cross-

layer design aspects of the bidirectional relaying decode-and-forward protocol which was

proposed by Rankov and Wittneben in [RW05b, RW07]. Therefore, we assume discrete-

time memoryless Gaussian channels between the nodes. At the relay node we apply the

superposition encoding technique so that we consider separated information flows.

System Model

The bidirectional two-hop communication is established in two phases, namely the multi-

ple access (MAC) phase and the broadcast (BC) phase, c.f. Figure 2.1. The protocol starts

with the multiple access phase where node 1 transmits a message with rate R1and node 2

transmits a message with rate R2to the relay node simultaneously. The relay node decodes

both messages. In the succeeding BC phase the relay broadcasts the re-encoded messages

where it uses superposition encoding technique. Since for each receiving node in the BC

phase one message originates from itself, each receiving node performs interference cancel-

lation before decoding the unknown message. This results in two separated interference-free

transmissions essentially.

R 21

(a) MAC phase

R 21

(b) BC phase

Figure 2.1: A three-node network, where each node is equipped with one antenna element.

2 Bidirectional Relay Communication using Superposition Encoding

In this thesis we do not allow cooperation between the two phases. This means that we

do not allow any feedback which could be used for cooperation between the encoders. For

Shannon’s two-way channel [Sha61] this is known as the restricted two-way channel. Fur-

thermore, we assume that the noise and information sources at nodes 1 and 2 are independent

and all nodes are perfectly synchronized.

Throughout the chapter we consider a baseband discrete-time system. We assume a multi-

plicative channel model which includes the physical channel, the shaping pulse at the trans-

mitter, and the whitening matched filter at the receiver. Unless otherwise stated, the channel

is considered to be a single-input single-output (SISO) linear time-invariant system, which

can be seen as a snapshot of a time-varying flat fading channel.

Let h1∈Cdenote the channel gain between node 1 and the relay node and h2∈Cdenotes

the channel gain between node 2 and the relay node. For notational simplicity we assume

reciprocal channels. Then after symbol-rate sampling the system equation for time instants

min the MAC phase is given by

yR[m] = h1x1[m] + h2x2[m] + nR[m],

where yR[m]denotes the received signal at the relay node, x1[m]and x2[m]denote the

transmit signals of nodes 1 and 2, and nR[m]denotes the additive noise. Similarly, after

symbol-rate sampling the system equation for time instants min the BC phase is given by

yk[m] = hkxR[m] + nk[m], k = 1,2,

where yk[m]denotes the received signal at node k= 1,2,xR[m]denotes the transmit signal

of the relay node, and nk[m]denotes the additive noise at node k= 1,2. We assume that

the noise at each antenna is independent additive white Gaussian noise with zero mean and

power σ2. Due to the central limit theorem, the Gaussian noise assumption is reasonable for

a wide class of practical channel models, e.g. thermal noise is modeled by a Gaussian distri-

bution. For each node we have a mean transmit power constraint Pk,k∈ {1,2,R}, so that

we can define the signal-to-noise ratios γk:= Pk/σ2,k∈ {1,2,R}. Throughout the thesis

we consider normalized powers. Without further notification, we assume that |h1|,|h2|,and

Pk, k ∈ {1,2, R}are strictly positive because otherwise bidirectional communication would

not be possible.

In this thesis we consider an orthogonal resource allocation for the two phases in the time

domain. Conceptually the results can be easily transfered to an orthogonal resource allo-

cation in the frequency domain, but then we have to consider the capacity of bandlimited

channels. We assume that all links are reciprocal. In practical systems this condition is ful-

filled if the round-trip time of both phases is shorter than the coherence time of the channel.

Similarly, if we allocate orthogonal resources in frequency domain the condition is fulfilled

if the frequency separation is smaller than the coherence bandwidth. However, in this thesis

2.1 Introduction

we look at the rates which are achievable for a given channel realization. Therefore, channel

reciprocity and equal noise powers are not necessary assumptions since with an appropriate

substitution of the individual transmit power constraints we can include the general case.1

Finally, we want to remark that in most sections we assume a time-invariant channel model.

For the cases where we consider a time-variant channel model we are interested in the time

averages of the achievable rates. Therefore, we assume that the channel remains constant

long enough so that we can achieve the information theoretic rate. An optimization of the

resource allocation over the time-variance of the channels is out of the scope of the thesis.

Outline of this Chapter

In Section 2.2 we examine the achievable bidirectional rate region for a fixed and optimized

time division between the two phases. For a complete understanding of the combinatorial

properties we first study minimal necessary signal-noise ratios γRto support a certain rate

pair in the MAC phase and characterize the sum-rate optimal rate pairs of the rate region

of each phase and of the bidirectional protocol. After that, we characterize for any weight

vector the corresponding rate pair on the boundary of the rate region that maximizes the

weighted rate sum. Rate pairs on the boundary are Pareto optimal since they can be seen

as solutions of a multi-objective optimization problem where we want to maximize each

rate individually. We use these Pareto optimal rate pairs in the following two sections. In

Section 2.3 we discuss a throughput optimal resource allocation policy based on a back-

pressure strategy. In Section 2.4 we consider the problem of relay selection in a network

based on the achievable rate region. Furthermore, we look at the asymptotic growth of the

ergodic rate region assuming independent and identical Rayleigh distributed channels.

In Section 2.5 we examine the joint resource allocation for the bidirectional relaying and

an additional multicast of the relay node. We characterize the optimal decoding order and

the total sum-rate maximum. Furthermore, we characterize a rate trade-off property where

the total sum-rate remains constant. After that we present a combinatorial discussion of the

results. In Section 2.6 we consider the achievable rate region of bidirectional relaying be-

tween nodes equipped with multiple antennas. Therefore, we briefly review the information

theoretic concepts for vector-valued processing and present the optimal resource allocation

assuming perfect channel knowledge at the transmitters. Finally, we identify the linear scal-

ing law of the achievable rates in the high power regime.

For a clear presentation we illustrate many results by examples or numerical simulations. In

1Let hkl and σ2

kwith k, l ∈ {1,2,R}denote the channel coefficient of the channel from node kto node land

the noise power at node k. Then with the substitution |hRk|2

σ2

=|hk|2

σ2, and ˜

Pk=Pk

|hkR|2σ2

|hRk|2σ2

, for k= 1,2,

we can transfer our results to individual channel coefficients and noise powers.

2 Bidirectional Relay Communication using Superposition Encoding

Section 2.7 we give a comprehensive discussion about the cross-layer design aspects consid-

ered in this chapter.

2.2 Achievable Rate Region of Bidirectional Relaying

Rankov and Wittneben only looked at the sum-rate performance assuming an equal time

division between the MAC and the BC phase. Since bidirectional communication is charac-

terized by a two-dimensional rate vector, for cross-layer design aspects we need to know the

whole achievable rate region in detail. Moreover, we notice that we can obviously enlarge

the achievable rate region if we optimize the time division between the phases.

In this chapter we regard information as a “fluid”, which means that multiple messages are

encoded by using the superposition encoding technique. Then the decoding of multiple mes-

sages is done successively. This means that when we decode the codeword of a message

we regard the codewords of the succeeding unknown messages as additional noise. After

the successful decoding of the message the interference of the corresponding codeword is

canceled, which reduces the noise level for the succeeding messages and therefore improves

the decoding capability. This decoding technique is known as successive interference can-

cellation. For the encoding of multiple messages it follows that we can apply the encoding

principles of the single-user point-to-point communication where the encoding has to factor

in the position in decoding sequence. For that reason we first briefly look at the results for

the simple single-input single-output (SISO) Gaussian channel. These can be found in most

textbooks on information theory or good books on wireless communications, e.g. [Gal68],

[CT91], or [TV05]2. Then we will briefly discuss the results regarding the capacity region

of the Gaussian multiple access channel for completeness. For the achievable rate region of

the broadcast phase we use the results of the Gaussian channel. In the rest of the section we

study the resulting achievable rate region of the bidirectional relaying protocol.

2.2.1 Gaussian Channel

We start with a simple single-input single-output discrete-time memoryless AWGN channel.

We assume complex signaling, which is motivated by the baseband processing. Furthermore,

we include a time-invariant multiplicative channel coefficient in this discussion. This results

in a linear channel model with a multiplicative and additive distortion of the transmit signal

2We have argued as for an additive channel model like the AWGN channel since this is the most interesting

channel for wireless communication. But the basic arguments also apply for more general settings, where we

translate noise or interference with uncertainty so that interference cancellation corresponds to a reduction in

uncertainty.

2.2 Achievable Rate Region

with a channel coefficient h, which is perfectly known at the receiver, and a complex Gaus-

sian noise random variable N∼ CN(0, σ2)respectively. We consider a continuous-valued

input and output where the complex random variables Xand Ywith support sets SYand SX

represent single letters of our transmit and receive signals. For the transmit signal we require

a transmit power constraint so that the second moment of the random variable is bounded

by some constant P, i.e.

{|X|2} ≤ P. We assume that the noise is independent of the

transmit signal. Then the single letter input-output relation can be expressed as follows

Y=h X +N.

A single letter description will be sufficient since we consider a memoryless channel.

In the seminal work [Sha48] Shannon studied the problem of how best to encode the in-

formation a sender wants to transmit. For that goal he introduced the concept of mutual

information which measures the amount of information one random variable reveals about

another. Twenty years before, Hartley introduced in communications the concept of entropy

as a measure of uncertainty about a random variable from thermodynamics. Then the mutual

information characterizes the reduction in uncertainty. It therefore measures the quantity of

transmitted information.

Since we consider a channel with continuous alphabets, we get the mutual information in

terms of differential entropies3[CT91, Chapter 9],

I(X;Y) := h(Y)−h(Y|X) = h(Y)−h(N).

The differential entropy h(Y)of a continuous random variables Xdefined on the support set

SYwith the density fY(y)is defined as

h(X) := −ZSY

fY(y) log fY(y)dy.

Since the logarithm is to the base two, we measure the entropy in bits. The conditional

differential entropy h(Y|X)of the random variables (Y, X)defined on the support set SY×

SXwith the joint and conditional densities fY,X(y, x)and fY|X(y|x)is defined as

h(Y|X) := −ZSY×SX

fY,X (y, x) log fY|X(y|x)d(y, x) = h(X, Y )−h(Y).(2.1)

3In the classical Gaussian channel there is no channel coefficient h. Therefore, for our channel model some

authors write I(X;Y, h)to indicate the channel knowledge at the receiver. Since we assume perfect channel

knowledge at the receivers throughout the thesis, we skip this entry to keep the notation simple. Furthermore,

often real-valued signaling is considered. Conceptually there is no difference between complex and real

processing since we can consider each use of a complex AWGN channel as two uses of a real AWGN

channel.

2 Bidirectional Relay Communication using Superposition Encoding

Then the differential entropy of a complex Gaussian random variable Zwith density φ(z) =

πσ2exp(−|z−µ|2

σ2)is given by

h(Z) = −ZSZ

φ(z) log(φ(z))d z =−ZSZ

φ(z)

ln(2)ln( 1

πσ2)−|z−µ|2

σ2d z

= log(πσ2) + 1

ln(2) = log(e πσ2).(2.2)

Therewith, we get the differential entropy of the noise as h(N) = log(e πσ2).

The next important quantity is the relative entropy D(f||g) := Rflog f/g (with

0 log(0/0) := 0) between two densities fand gwhich are finite only if the support set

of fis contained in the support set of g. From Jensen’s Inequality4we know that the relative

entropy between two densities fand gis always non-negative, D(f||g)≥0[CT91, Theo-

rem 9.6.1]. It follows that for any complex random variable Zdefined on the support set SZ

with density f(z)and E{|z|2}=σ2we have h(Z)≤log(e πσ2). This can be easily seen

by the following: Let φ(z)be the density of a complex Gaussian random variable according

to CN(0, σ2), then

0≤D(f||φ) = ZSZ

flog(f/φ) = ZSZ

flog(f)−ZSZ

flog(φ)

=−h(Z)−ZSZ

f(z)

ln(2)ln 1

πσ2−|z|2

σ2d z =−h(Z) + log(e πσ2).

It follows that the complex normal distribution CN(0, σ2)maximizes the entropy over all

complex distributions with the same second moment.

Since the noise Nis independent of the input X, for the second moment of the output we

have

{|Y|2}=

{|h X +N|2}=|h|2

{|X|2}+

{|N|2} ≤ |h|2P+σ2.

Then it follows from the previous that the differential entropy of the output is maximized

with h(Y) = log(e π(|h|2P+σ2)) if the input Xis distributed according to complex normal

distribution CN(0, P). This means that Gaussian distributed codebooks are optimal.

The information capacity Cof the Gaussian channel is defined as the maximum mutual in-

formation between the input and output over all input distributions that satisfy the power

4Jensen’s Inequality can be formulated in different notations. In the notation of probability theory let Xbe a

random variable with support SX⊆

and let f : SX→

be a convex function, then we have

{f(X)} ≥

{X}).

2.2 Achievable Rate Region

constraint [CT91, Chapter 10]. With the previous considerations we can express the infor-

mation capacity as follows

C:= max

fX(x):

{|X|2}≤PI(X;Y) = max

fX(x):

{|X|2}≤Ph(Y)−log(e πσ2)

= log eπ(|h|2P+σ2)−log(e πσ2) = log 1 + |h|2P

σ2.

(2.3)

Then the result of Shannon’s work says that one can construct a sequence of block codes

that satisfies the power constraint, has a rate smaller but arbitrary close to the information

capacity, and has a maximum probability of error which tends to zero with increasing block

code length. Therefore, the information capacity is also the supremum of the achievable

rates of the channel which means that the operational and information capacity are equal.

For more details we refer to any information theory text book that covers the channel coding

theorem for the Gaussian channel, e.g. [CT91].

2.2.2 Multiple Access Phase

In the multiple access phase of the bidirectional relaying protocol nodes 1 and 2 simultane-

ously transmit independent messages m1and m2with rates R1and R2to the relay node.

Thereby, the message m1from node 1 is intended for node 2 and vice versa for message

m2. For the MAC phase we apply the capacity achieving coding strategy. The capacity

region for the discrete memoryless multiple access channel was independently derived by

Ahlswede [Ahl71a] and Liao [Lia72] using finite set alphabets and is now part of any in-

formation theory textbook that contains multiuser information theory, [Wol78], [CK81], and

[CT91]5. The result of the capacity region can be modified to apply to continuous input and

output alphabets with an additional mean input power constraints, which was independently

done by Wyner in [Wyn74] and Cover in [Cov75]. From wireless communication point of

view the multiple access channel with Gaussian channels is most important.

Since the channel is memoryless, it is sufficient to consider single-letters only. Then the

input-output relation of the discrete-time memoryless Gaussian multiple access channel is

given by

YR=h1X1+h2X2+NR.

The continuous random variable YRdenotes the output at the relay node and the continuous

random variables X1and X2denote the inputs from the nodes 1 and 2, which are all complex

since we consider baseband signaling. The complex coefficients h1and h2model the time-

invariant channel gains between the nodes. The received signal at the relay node is distorted

5We reproduce the theorem in Section 3.1.2 where we present a coding theorem for the BC phase of the

bidirectional relay channel using finite set alphabets. Here, we only present the result for the Gaussian

channel which we consider in this chapter.

2 Bidirectional Relay Communication using Superposition Encoding

by complex additive white Gaussian noise NR∼ CN(0, σ2). Then the multiple access

capacity region CMAC is given by the convex hull of the set of rate pairs satisfying

R1≤I(X1;YR|X2),

R2≤I(X2;YR|X1),

R1+R2≤I(X1, X2;YR),

for some input distribution fXk(xk)which satisfies the input power constraints

{|Xk|2} ≤

Pk,k= 1,2. With the same arguments as in Section 2.2.1 it can easily be shown that com-

plex Gaussian input distributions with Xk∼ CN(0, Pk),k= 1,2, maximize I(X1;YR|X2),

I(X2;YR|X1), and I(X1, X2;YR). Since in the scalar case all mutual informations are si-

multaneously maximized, it follows that the rate constraints are tight so that the capacity

region of the MAC phase is given by

CMAC := n[R1, R2]∈

+:R1≤R−→

1R, R2≤R−→

2R, R1+R2≤RMAC

Σo

with the individual and sum-rate constraints

R−→

1R := log 1 + γ1|h1|2,(2.4a)

R−→

2R := log 1 + γ2|h2|2,(2.4b)

RMAC

Σ:= log 1 + γ1|h1|2+γ2|h2|2,(2.4c)

and signal-to-noise ratios γ1=P1

σ2and γ2=P2

σ2.

Similarly to the Gaussian channel it is possible to construct a sequence of block codes that

satisfies the power constraints and achieves a rate pair [R1, R2]∈ CMAC arbitrary close

while the maximum probability of error tends to zero with increasing block length [CT91].

For that reason we assume error-free decoding in the MAC phase if [R1, R2]∈ CMAC6for

the following cross-layer design.

Since CMAC is a pentagon, it can be completely described by five vertices. Thereby, the

vertices where the individual rate constraints intersect with the sum-rate constraint,

νΣ1 := R−→

1R, RΣ1

2with RΣ1

2:= RMAC

Σ−R−→

1R = log 1 + γ2|h2|2

1 + γ1|h1|2,(2.5a)

ν2Σ := R2Σ

1, R−→

2Rwith R2Σ

1:= RMAC

Σ−R−→

2R = log 1 + γ1|h1|2

1 + γ2|h2|2,(2.5b)

are most interesting for the combinatoric as seen in [TH98]. To achieve the vertices, we have

to apply successive interference cancellation. This means that in the first decoding step the

6If we can make the error of the MAC and of the BC phase arbitrary small, it follows from the union bound

that we can make the error of the bidirectional relay channel arbitrary small.

2.2 Achievable Rate Region

codeword of the second message to decode is regarded as additional noise. After decoding

the first message the interference of the corresponding codeword is canceled so that we can

decode the second message without interference. Then each vertex corresponds to a specific

decoding order. To achieve νΣ1 we have to decode the message m2before m1and vice versa

for ν2Σ.

In the following we often consider the boundary of the rate region and therefore define the

interesting sections of the boundary as follows

R−→

1R := n[R−→

1R, R2]∈

+: 0 ≤R2≤RΣ1

2o,(2.6a)

R−→

2R := n[R1, R−→

2R]∈

+: 0 ≤R1≤R2Σ

1o,(2.6b)

RMAC

Σ:= n[R1, RMAC

Σ−R1]∈

+:R2Σ

1≤R1≤R−→

1Ro.(2.6c)

The set RMAC

Σspecifies the dominant face of the pentagon and characterizes the sum-rate

optimal rate pairs in the MAC phase. We can achieve the rates on the dominant face by

time-sharing between νΣ1 and ν2Σ or by a rate splitting technique according to [RU95].

2.2.3 Broadcast Phase

In the succeeding BC phase the relay forwards the previously received message m1to node

2 and message m2to node 1. In this chapter we follow the superposition encoding strategy

as proposed in [RW05b]. Therefore, the messages m1and m2are separately encoded as for

the point-to-point Gaussian channel. Since we consider a memoryless channel, it is again

sufficient to consider single-letters only. Then the input random variable of the relay node

using superposition encoding is given by

XR=W1+W2,

where the random variables W1and W2correspond to the codewords of the messages m1

for node 2 and m2for node 1. Since the messages m1and m2are independent, the random

variables W1and W2are independent as well. From this we get the output at nodes 1 and 2

as follows

Yk=hkXR+N1=hkW1+hkW2+Nk, k = 1,2.

Since the messages m1and m2originate from nodes 1 and 2 respectively, at each receiving

node one message and therefore one codeword is known. This a priori knowledge allows the

receiving nodes to subtract the interference caused by the codeword of its own message. With

this interference cancellation technique we essentially have an interference-free reception at

each receiving node, which results in two equivalent single-user AWGN channels. It follows

2 Bidirectional Relay Communication using Superposition Encoding

that the achievable rates in the BC phase using superposition encoding, i.e. XR=W1+W2,

have to fulfill the constraints

R2≤I(XR;Y1|W1) = I(W2;h1W2+N1),

R1≤I(XR;Y2|W2) = I(W1;h2W1+N2),

for some input distribution fWk(wk),k= 1,2, satisfying the power constraint

{|XR|2}=

{|W1|2}+

{|W2|2} ≤ PR.

Obviously, the mutual informations are only coupled by the relay input power constraint,

which means that we have to distribute the power PRon both codewords. Let β1and β2

denote the proportion of relay transmit power PRspend for the codewords W1and W2

respectively. Then the simplex

B:= {[β1, β2]∈[0,1] ×[0,1] : β1+β2≤1}

characterizes the set of feasible relay power distributions that satisfy the relay transmit

power constraint. For a given feasible relay power distribution [β1, β2]∈ B we know

from Section 2.2.1 that complex Gaussian distributed inputs W1∼ CN(0, β1PR)and

W2∼ CN(0, β2PR)maximize the mutual informations. It follows that with the super-

position encoding in the BC phase we can achieve any rate pair within the rate region

RBC :=n[R1, R2]∈

+:R1≤R−→

R2(β1), R2≤R−→

R1(β2),with [β1, β2]∈ Bo

with rate constraints

R−→

R2 :[0,1] →

+, β17→ log 1 + γRβ1|h2|2,(2.7a)

R−→

R1 :[0,1] →

+, β27→ log 1 + γRβ2|h1|2,(2.7b)

and signal-to-noise ratio γR=PR

σ2.

In next proposition we show that in the BC phase we do not need the convex hull operation

as in the MAC phase since RBC is already convex, which means that in the BC phase time-

sharing is not necessary.

Proposition 2.1. RBC is convex.

Proof. The proof can be found in Appendix 2.8.1.

Similarly to the Gaussian channel it is possible to construct a sequence of block codes that

satisfies the power constraint and achieves a rate pair [R1, R2]∈ RBC arbitrary close while

the maximum probability of error tends to zero with increasing block length. For that reason

2.2 Achievable Rate Region

we assume error-free decoding in the BC phase if [R1, R2]∈ RBC for the following cross-

layer design.

For any rate pair on the boundary of the BC region we obviously have β1+β2= 1. Accord-

ingly, the boundary of RBC can be parametrized by the rate pair function

RBC : [0,1] →

+, β 7→ hR−→

R2(β), R−→

R1(1 −β)i,(2.8)

where we have β1=βand β2= 1 −β. This allows us to characterize the sum-rate optimal

rate pair in the BC phase.

Proposition 2.2. The maximum sum-rate RBC

Σ:= max

[R1,R2]∈RBC

R1+R1of the broadcast

rate region RBC is given by

RBC

Σ=









log(1 + γR|h1|2),if β⋆<0,(2.9a)

log 1

4(1 + γR|h1|2+|h1|2

|h2|2)(1 + γR|h2|2+|h2|2

|h1|2),if 0≤β⋆≤1,(2.9b)

log(1 + γR|h2|2),if β⋆>1,(2.9c)

with β⋆:= 1

2+1

2γR1

|h1|2−1

|h2|2and is attained at the rate pair

[R⋆

−→

R2, R⋆

−→

R1] := 









[0, R−→

R1(1)],if β⋆<0,(2.10a)

[R−→

R2(β⋆), R−→

R1(1 −β⋆)],if 0≤β⋆≤1,(2.10b)

[R−→

R2(1),0],if β⋆>1,(2.10c)

with R−→

R2(β⋆) = log 1

2(1 + γR|h2|2+|h2|2

|h1|2)and R−→

R1(β⋆) = log 1

2(1 + γR|h1|2+|h1|2

|h2|2).

Proof. The proof can be found in Appendix 2.8.2.

Furthermore, for any fixed relay power distribution β∈[0,1] with β1=βand β2= 1 −β

in the BC phase we can achieve any rate pair in the rate region

RBC(β) := n[R1, R2] : R1≤R−→

R2(β), R2≤R−→

R1(1 −β)o(2.11)

so that obviously RBC =Sβ∈[0,1] RBC(β)holds. Finally, we want to remark that RBC is

larger than the rate region of the degraded broadcast channel, where for the encoding and

decoding for one receiving node the own message is regarded as interference.

2 Bidirectional Relay Communication using Superposition Encoding

2.2.4 Bidirectional Achievable Rate Region

For the bidirectional relaying protocol we are in the MAC and BC phases for fraction of

time only so that we have to scale the achievable rate pairs according to the time division.

This means that for a time division parameter α∈[0,1] we can achieve rate pairs Rwithin

the rate regions αCMAC in the MAC phase and (1 −α)RBC(β)in the BC phase with a

relay power distribution β∈[0,1]. For a successful bidirectional relay transmission of the

message m1with rate R1from node 1 to node 2 and message m2with rate R2from node 2

to node 1 the rate pair R= [R1, R2]has to be within the scaled MAC rate region αCMAC

as well as within the scaled BC rate region (1 −α)RBC(β). This means that for any given

time division parameter α∈[0,1] and relay power distribution β∈[0,1] the achievable rate

region of the bidirectional relaying RBR(α, β)is given by the intersection

RBR(α, β) := αCMAC ∩(1 −α)RBC(β).

In the following we will call R1and R2unidirectional rates.

Since this applies to any relay power distribution β∈[0,1], the achievable rate region of

the bidirectional relaying with the optimal relay power distribution for a fixed time division

parameter α∈[0,1] is given by the union over all possible relay power distributions

RBR(α) := S

β∈[0,1]RBR(α, β) = αCMAC ∩(1 −α)S

β∈[0,1]RBC(β)

=αCMAC ∩(1 −α)RBC.

(2.12)

The set RBR(α)is convex since the intersection of convex sets, αCMAC and (1 −α)RBC, is

itself convex.

We are now interested which relay power PRrespectively signal-to-noise ratio γRis neces-

sary to support a rate pair [RM

1, RM

2]∈ CMAC for a given time division parameter α∈[0,1].

From (2.7a) and (2.7b) we can conclude

αRM

1= (1 −α) log(1 + β1γR|h2|2)⇒β1=2α

1−αRM

1−1

|h2|2γR

αRM

2= (1 −α) log(1 + β2γR|h1|2)⇒β2=2α

1−αRM

2−1

|h1|2γR

It can be easily seen by contradiction that for the minimum relay power we have 1 = β1+

β2. From this we get the minimum γRwhich is necessary to support a MAC rate pair

[RM

1, RM

2]∈ CMAC for a time division parameter αas follows

γmin

R(RM

1, RM

2, α) := 2α

1−αRM

1−1

|h2|2+2α

1−αRM

2−1

|h1|2.(2.13)

2.2 Achievable Rate Region

This allows us to characterize the minimum relay power PRrespectively signal-to-noise

ratio γRwhich is necessary to support any rate pair in MAC rate region CMAC in the next

proposition.

Proposition 2.3. Given a MAC rate region CMAC and a time division parameter αwhere

γΣ1

R(α) := γmin

R(R−→

1R, RΣ1

2, α)and γ2Σ

R(α) := γmin

R(R2Σ

1, R−→

2R, α)denote the minimum γR

necessary to support the vertices νΣ1 and ν2Σ. Then the minimum γRwhich is necessary to

support any rate pair [RM

1, RM

2]∈ CMAC is given by

γMAC

R(α) := max γΣ1

R(α), γ2Σ

R(α).

In the case of equal time division, α=1

/2, we have

γΣ1

R(1

/2) = γ1|h1|2

|h2|2+γ2|h2|2

(1 + γ1|h1|2)|h1|2,

γ2Σ

R(1

/2) = γ2|h2|2

|h1|2+γ1|h1|2

(1 + γ2|h2|2)|h2|2.

Proof. Since RBC is a convex set which increases with increasing γR, any rate pair

[RM

1, RM

2]∈ CMAC can be supported if γRis that large that both vertices of αCMAC are

in (1 −α)RBC.

For the equal time division case we get for the vertex νΣ1 = [R−→

1R, RMAC

Σ−R−→

1R] = [log(1+

γ1|h1|2),log(1 + γ2|h2|2

1+γ1|h2|2)] the minimum relay power γΣ1

R(1

/2)and for the vertex ν2Σ =

[RMAC

Σ−R−→

2R, R−→

2R] = [log(1 + γ1|h1|2

1+γ2|h1|2),log(1 + γ2|h2|2)] the minimum relay power

γ2Σ

R(1

/2)using (2.13).

Using γR=γMAC

Rallows to support all possible rate pairs from the MAC phase, but in

general it is the minimal necessary γRfor one vertex only. This leads to the idea that the

relay node can use the remaining power to sent an own message additionally. In Section 2.5

we study a scenario where the relay node adds a multicast communication to the bidirectional

relay communication.

Remark 2.4. For the equal time division case, α=1

/2, we have α

1−α= 1 so that we can

explicitly calculate γΣ1

R(1

/2)and γ2Σ

R(1

/2)in terms of |h1|2,|h2|2, γ1,and γ2. For the same

reason some following closed-form results can be obtained for α=1

/2only. For other

time division parameter such an explicit solution is not possible, but the behavior is similar.

Therefore, we often consider the equal time division case exemplarily.

2 Bidirectional Relay Communication using Superposition Encoding

0 0.2 0.4 0.6

0.2

0.4

0.6

rate R1 [bit/channel use]

rate R2 [bit/channel use]

|h1|2 =0.6

|h2|2 =1

γ1 =1.3

γ2 =1

γR =1.3

(a) Bidirectional achievable rate region with

equal time division RBR(1

/2)(solid line), the

scaled achievable rate region of MAC phase

/2CMAC(dashed line), and the scaled achiev-

able rate region of BC phase 1

/2RBC (dotted

line)

0 0.2 0.4 0.6

0.2

0.4

0.6

rate R1 [bit/channel use]

rate R2 [bit/channel use]

|h1|2 =0.6

|h2|2 =1

γ1 =1.3

γ2 =1

γR =1.3

(b) Bidirectional achievable rate region with

optimal time division RBRopt (solid line)

and RBR(0.4, β) = 0.4CMAC (solid line),

RBR(0.436, β)(dashed line), RBR(1

/2, β)

(dotted line), RBR(0.56, β)(dashed-dotted

line), and RBR(0.6, β) = 0.4RBC (solid line).

Figure 2.2: Bidirectional achievable rate region of an example scenario

The next natural extension is to look at the achievable rate region where we additionally

allow to optimize the time division between the MAC and the BC phase. This rate region is

obviously given by the union over all possible time division parameters αas follows

RBRopt := S

α∈[0,1]RBR(α) = S

α∈[0,1]

(αCMAC ∩(1 −α)RBC).(2.14)

As a union of convex sets the rate region RBRopt need not be convex. Convexity can be

achieved if we allow time-sharing between bidirectional rate pairs with different time divi-

sion parameters, but from the following Corollary 2.13 we will see that RBRopt is already

convex. This means that for any rate pair in RBRopt there exists a time division parameters

αfor which this rate pair is achievable.

In Figure 2.2 we depicted the achievable rate regions of an example. In the left figure

we see the bidirectional achievable rate region RBR(1

/2)for the equal time division case,

which is given by the intersection of the scaled achievable rate regions of the MAC and

BC phases. In the right figure we show the achievable rate region RBRopt of the optimal

time division case. Additionally, the rate regions RBR(α)for some fixed time division pa-

rameters αare depicted. The bullets (•) mark the rate pairs where rate pairs of RBR(α)

achieve the boundary of RBRopt. We see that we have RBR(α) = αCMAC ⊂ RBRopt or

RBR(α) = (1−α)RBC ⊂ RBRopt if the time division parameter is too small or too large re-

2.2 Achievable Rate Region

spectively. Furthermore, we see that for the time division parameter α= 0.436 two rate pairs

achieve the boundary. Note that the marked rate pairs on the boundary RBRopt correspond

to intersection points of the boundaries of the scaled MAC and BC regions. Lemma 2.11 ex-

plains this observation and leads to an equivalent characterization of the bidirectional achiev-

able rate region RBRopt in Theorem 2.12. The simple description of the boundary allows us

to identify for any given weight vector the rate pair which maximize the weighted rate sum

in Theorem 2.16.

In the following we study the properties of the bidirectional achievable rate region for the

fixed and optimal time division. For the fixed time division we focus on the equal time

division case7. For other fixed time division cases the behavior is similar, but often it cannot

be calculated explicitly. After that, we identify the optimal time division and relay power

distribution for an optimal rate allocation. From this we can deduce an equivalent description

of RBRopt. The previous studies allow us to characterize in details the boundaries of the

bidirectional achievable rate regions. In particular we are able to specify in closed form the

rate pairs which maximize the weighted rate sum for any weight vector.

For notationally simplicity we will use the abbreviations

RBReq := RBR(1

/2)and RBReq(β) := RBR(1

/2, β).

Equal time division

With the first theorem and its corollaries we explore the combinatorial structure of the achiev-

able rate region RBReq. In Theorem 2.5 we derive the maximal bidirectional sum-rate RΣ(β)

for any relay power distribution β∈[0,1]. As a direct consequence of the theorem we can

characterize the achievable rate region RBR(β)in Corollary 2.6. But we can also use the

result of Theorem 2.5 to characterize the minimal relay power which is necessary to support

in the BC phase a rate pair [RM

1, RM

2]∈ CMAC which achieves the maximal sum-rate in the

MAC phase, i.e. RM

1+RM

2=RMAC

Σ. Finally in Corollary 2.8 we identify the sum-rate

optimal rate pair of RBReq. For the equal time division case the whole discussion leads to

explicit expressions, but the combinatoric of RBR(α)is the same for any α∈[0,1], cf.

Remark 2.4.

In Theorem 2.19 we use the explicit knowledge about the combinatoric to characterize in

closed form the rate pairs on the boundary of RBReq which maximize the weighted rate

sum, so called Paretto optimal rate pairs. The Pareto optimal rate pairs play a crucial role for

the throughput optimal resource allocation policy in Section 2.3 and for the relay selection

problem in Section 2.4.

7In the following the subscript eq or opt specify the corresponding rate, rate pair, or rate region for the equal

time division or the optimal time division case.

2 Bidirectional Relay Communication using Superposition Encoding

Theorem 2.5. For bidirectional relaying with equal time division the maximal sum-rate

RΣ(β) := max

R∈RBReq(β)R1+R2of the bidirectional rate region RBReq(β)for a given relay

power distribution parameter β∈[0,1] is given by

RΣ(β) = 1

2min RMAC

Σ, R−→

1R +R−→

R1(1 −β), R−→

R2(β) + R−→

2R, R−→

R2(β) + R−→

R1(1 −β)

2









R−→

1R +R−→

R1(1 −β),if β∈ I1,(2.15a)

R−→

R2(β) + R−→

2R,if β∈ I2,(2.15b)

R−→

R2(β) + R−→

R1(1 −β),if β∈ IBC,(2.15c)

RMAC

Σ,if β∈ IΣ,(2.15d)

with sets

I1:= [βΣ1,1] ∩[βB1, β1B],

I2:= [0, β2Σ]∩[βB2, β2B],

IΣ:= (β2Σ, βΣ1)∩((βBΣ, βΣB),if β⋆2≥∆β,

∅,else,

IBC := [0,1] \(I1∪I2∪IΣ),

and characteristic values8

βΣ1 := 1 −γ2|h2|2

γR|h1|2(1 + γ1|h1|2), βB1 := γ1|h1|2

γR|h2|2,

β1B := 1 + 1

γR|h1|2, β2Σ := γ1|h1|2

γR|h2|2(1 + γ2|h2|2),

βB2 := −1

γR|h2|2, β2B := 1 −γ2|h2|2

γR|h1|2,

βBΣ := β⋆−qβ⋆2−∆β, βΣB := β⋆+qβ⋆2−∆β,

β⋆=1

2+1

2γR1

|h1|2−1

|h2|2,and ∆β:= 1

γ2

Rγ2

|h1|2−γR−γ1

|h2|2.

Proof. The proof can be found in Appendix 2.8.3.

The sets I1,I2,IΣ, and IBC characterize the active restriction for a given power distribution

β. If β∈ I1, then the unidirectional rate R1is limited by 1

/2R−→

1R and R2is limited by

/2R−→

R1(1 −β). Accordingly, if β∈ I2then the unidirectional rate R2is limited by 1

/2R−→

8The indices of the points of intersections of f1,f2,fΣ, and fBC, which are used in the proof, indicate the

minimal function for smaller βby the first index and for larger βby the second index.

2.2 Achievable Rate Region

and R1is limited by 1

/2R−→

R2(β). For β∈ IΣthe MAC sum-rate restriction limits the sum-

rate R1+R2≤1

/2RMAC

Σ. If β∈ IBC, the unidirectional rates R1and R2are limited by

/2R−→

R2(β)and 1

/2R−→

R1(1 −β), i.e. the broadcast is the limiting phase. Finally, RBReq is not

restricted by a certain constraint if the corresponding set is empty, e.g. if IBC =∅the whole

MAC region can be supported in the BC phase, i.e. RBReq =1

/2CMAC.

Thus, as a direct consequence of the theorem, the following corollary characterizes the

achievable rate region for a given relay power distribution parameter β∈[0,1].

Corollary 2.6. For bidirectional relaying with equal time division the achievable bidirec-

tional rate region RBReq(β)for a given relay power distribution parameter β∈[0,1] is

given by

RBReq(β) = 1











R∈

+:R1≤R−→

1R, R2≤R−→

R1(1 −β),if β∈ I1,

R∈

+:R1≤R−→

R2(β), R2≤R−→

2R,if β∈ I2,

R∈

+:R1≤R−→

R2(β), R2≤R−→

R1(1 −β),if β∈ IBC,

(R∈

+:R1≤R−→

R2(β), R2≤R−→

R1(1 −β),

R1+R2≤RMAC

Σ),if β∈ IΣ.

For β∈ I1∪I2∪IBC the rate region RBReq(β)is a rectangular, which is characterized by

two individual rate constraints. For those relay power distributions the upper right vertex of

the rate region characterizes a boundary rate pair of RBReq. If β∈ IΣ, the MAC sum-rate

constraint has additionally to be satisfied. For those relay power distributions the sum-rate

constraint of the MAC phase restricts the achievable rate region RBReq.

In the next corollary we characterize the minimal relay power respectively signal-to-noise

ratio which is necessary to support a rate pair [RM

1, RM

2]∈ CMAC that achieves the maximal

MAC sum-rate, i.e. RM

1+RM

2=RMAC

Σ.

Corollary 2.7. For bidirectional relaying with equal time division the minimal signal-to-

noise ratio to support in the BC phase a rate pair [RM

1, RM

2]∈ CMAC that achieves the

maximal possible sum-rate RMAC

Σis given by

γΣMAC

R= max γ⋆

R,min{ˆγΣ1

R, γΣ1

R(1

/2)},min{ˆγ2Σ

R, γ2Σ

R(1

/2)}(2.16)

with γ2Σ

R(1

/2)and γΣ1

R(1

/2)according to Proposition 2.3. and

γ⋆

R:= 2s1 + γ1|h1|2+γ2|h2|2

|h1|2|h2|2−1

|h1|2−1

|h2|2,

ˆγΣ1

R:= 2γ2|h2|2

(1 + γ1|h1|2)|h1|2+1

|h1|2−1

|h2|2,

ˆγ2Σ

R:= 2γ1|h1|2

(1 + γ2|h2|2)|h2|2+1

|h2|2−1

|h1|2.

2 Bidirectional Relay Communication using Superposition Encoding

Proof. The proof can be found in Appendix 2.8.4.

The case where γΣMAC

Ris equal to min{ˆγΣ1

R, γΣ1

R(1

/2)}or min{ˆγ2Σ

R, γ2Σ

R(1

/2)}corresponds

to the case when the vertex νΣ1 or ν2Σ is the rate pair where the sum-rate RMAC

Σis achieved

with the minimal γR. This is only the case when either the sum-rate optimal rate pair of the

broadcast phase [R⋆

−→

R2, R⋆

−→

R1]coincides with the vertex itself or it cannot be supported in the

MAC phase, i.e. 1

/2[R⋆

−→

R2, R⋆

−→

R1]/∈ RBReq.

If γR≥γΣMAC

R, the maximal achievable bidirectional sum-rate is limited by the MAC

sum-rate constraint. Accordingly, the maximal achievable bidirectional sum-rate is equal

to 1

/2RMAC

Σ. The set of sum-rate optimal bidirectional rate pairs is given by

R∗

Σeq :=n[R1, R2]∈1

2RMAC

Σ:∃β∈ IΣso that R1≤1

2R−→

R2(β), R2≤1

2R−→

R1(β)o

=n[R1,1

2RMAC

Σ−R1] : 1

2R−→

R2(βmin

Sum)≤R1≤1

2R−→

R2(βmax

Sum)o(2.17)

with βmin

Sum := minβ∈IΣβand βmax

Sum := maxβ∈IΣβ. In the next corollary we characterize

the rate pairs where the maximal achievable sum-rate is attained if we have γR< γΣMAC

Corollary 2.8. For bidirectional relaying with equal time division and γR< γΣMAC

Rthe rate

pair which achieves the bidirectional sum-rate maximum is given by

RΣ:= arg max

[R1,R2]∈RBReq

R1+R2=











RBC(0),if γR≤min{γ†

R, γ2B

R},(2.18a)

RBC(β2B),if γR> γ2B

R∧γR≥γ⋆1

R,(2.18b)

RBC(β⋆),if |γ†

R|< γR<min{γ⋆1

R, γ2⋆

R},(2.18c)

RBC(βB1),if γR> γB1

R∧γR≥γ2⋆

R,(2.18d)

RBC(1),if γR≤min{−γ†

R, γB1

R}(2.18e)

with characteristic parameters

γ†

R:= 1

|h2|2−1

|h1|2, γB1

R: = γ1|h1|2

|h2|2, γ2B

R:= γ2|h2|2

|h1|2,

γ⋆1

R:= 2γ2|h2|2

|h1|2+1

|h1|2−1

|h2|2,and γ2⋆

R:= 2γ1|h1|2

|h2|2+1

|h2|2−1

|h1|2.

Proof. The proof can be found in Appendix 2.8.5.

In the following remark we just restate the interesting observation where it is sum-rate opti-

mal not to apply the bidirectional protocol.

2.2 Achievable Rate Region

Remark 2.9. If we have γR≤min{−γ†

R, γB1

R}, it is sum-rate optimal to communicate only

from node 1 to node 2. Therefore, for −γ†

R≥0it is necessary that we have |h2|2≥ |h1|2.

Similarly, if we have γR≤min{γ†

R, γ2B

R}, it is sum-rate optimal to communicate only from

node 2 to node 1. For this case it is necessary that we have |h1|2≥ |h2|2.

The knowledge about the combinatoric allows us to characterize the boundary of the achiev-

able rate region RBReq directly. Let R∗

eq(q)denote the rate pair where the weighted rate

sum with weight vector q= [q1, q2]∈

+\{0}is maximized,

R∗

eq(q) = arg max

R∈RBReq

q1R1+q2R2.(2.19)

The rate pair R∗

eq(q)plays a crucial role for the throughput optimal resource allocation

policy in Section 2.3 and for the relay selection problem in Section 2.4. For a convex set like

RBReq the rate pair R∗

eq(q)characterizes the rate pair on the boundary of RBReq where the

tangential hyperplane with normal vector qintersects the boundary of RBReq, i.e. the rate

pairs with the maximal weighted rate sum characterize the boundary of RBReq. This gives

us an equivalent description of the achievable rate region using the convex hull operator

RBReq =coR∗

eq(q) : q= [q1, q2]∈

+\{0}∪0.

In the next theorem we give a closed form solution of the optimization problem given in

equation (2.19).

Theorem 2.10. For bidirectional relaying with equal time division the weighted rate sum

maximum R∗

eq(q)for a weight vector q= [q1, q2]∈

+\{0}is given by

R∗

eq(q) = 1











RBC(1),if βB1 >1∧q2≤q1ϑ(1),(2.20a)

RBC(0),if β2B <0∧q2≥q1ϑ(0),(2.20b)

νΣ1,if βΣ1 ≥βB1 ∧q2≤q1,(2.20c)

ν2Σ,if β2Σ ≤β2B ∧q2≥q1,(2.20d)

RBC(βB1),if βΣ1 < βB1 ≤1∧q2≤q1ϑ(βB1),(2.20e)

RBC(β2B),if 0≤β2B < β2Σ ∧q2≥q1ϑ(β2B),(2.20f)

RBC(βBΣ),if β⋆2>∆β∧β2Σ < βBΣ < βΣ1

∧q1ϑ(βBΣ)≤q2≤q1,(2.20g)

RBC(βΣB),if β⋆2>∆β∧β2Σ < βΣB < βΣ1

∧q1ϑ(βΣB)≥q2≥q1,(2.20h)

RBC(βBC),else, (2.20i)

2 Bidirectional Relay Communication using Superposition Encoding

with

ϑ: [0,1] →

, β 7→ |h2|2(1 + (1 −β)γR|h1|2)

|h1|2(1 + βγR|h2|2),and (2.21a)

βBC :

+\{0} →

,q7→ q1|h2|2(1 + γR|h1|2)−q2|h1|2

|h2|2|h1|2γR(q1+q2).(2.21b)

Proof. The proof can be found in Appendix 2.8.6.

The function ϑ(β)denotes the arctan of the angle of the normal vector of the rate pair on the

boundary RBC(β)for the relay power distribution β∈[0,1]. Further, βBC(q)is the relay

power distribution where the angle of the normal vector of the rate pair on the boundary of

RBC has the same angle as the weight vector q. Since the optimization problem depends on

the direction of the normal vector only, the weighted rate sum maximum is attained at the

same rate pair for any weight vector with the same ratio q1/q2. This means that only the

angle of the weight vector is important. The cases (2.20a) to (2.20h) denote the rate pairs

where on the boundary two restrictions apply simultaneously. Such rate pairs correspond

to intersection points which have multiple tangents so that they are optimal for a range of

weight vector angles. In contrast, the rate pair of the case (2.20i) is optimal for one weight

vector angle only.

In more detail, the cases (2.20a) and (2.20b) characterize the intersection of the boundary of

2RBC with the R1-axis and R2-axis respectively. The individual MAC rate constraint R−→

limits the achievable rate region RBReq in the cases (2.20e) and (2.20c), while for the latter

we additionally have νΣ1 ∈ RBC. Similarly, R−→

2R limits RBReq in the cases (2.20f) and

(2.20d), while for the latter we additionally have ν2Σ ∈ RBC. We have the cases (2.20g)

and (2.20h) if the MAC sum-rate constraint RMAC

Σlimits RBReq and the boundary of RBC

intersects the section between the vertices νΣ1 and ν2Σ.

The theorem also covers some special cases. If we choose q1=q2= 1, we have the

sum-rate maximum maxR∈RBReq R1+R2case, which we also studied in Corollary 2.8 for

γR≤γΣMAC

R. From this the question arises if the sum-rate maximum is Schur-convex or

Schur-concave with respect to the receive signal-to-noise ratios [γ1|h1|2, γ2|h2|2]. But from

the examples of Figure 2.3 we see that there is no such behavior. If the sum-rate maximum

is determined by the individual rate constraints of the MAC phase, R−→

1R or R−→

2R, i.e. at βB1

or β2B, the sum-rate increases with more balanced receive signal-to-noise ratios, which is a

Schur-concave behavior. But if the maximum sum-rate is restricted by the broadcast phase

or MAC sum-rate constraint, i.e. at βBC or βSum, and the receive signal-to-noise ratios are

relatively balanced, we see an oppositional behavior. Nevertheless, this shows that more or

less balanced receive signal-to-noise ratios result in large sum-rates.

2.2 Achievable Rate Region

0 0.5 1 1.5 2 2.5 3

0.5

1.5

2.5

|h1|2γ1

|h2|2γ2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

βBC

βB1

β2B

γ1 =1

γ2 =1

γR =0.5

(a) Symmetric MAC power

0 0.5 1 1.5 2 2.5 3

0.5

1.5

2.5

|h1|2γ1

|h2|2γ2

0.2

0.4

0.6

0.8

1.2

βBC

βSum

βB1

β2B

γ1 =0.4

γ2 =1.1

γR =0.9

(b) Asymmetric MAC power

Figure 2.3: Contour plot of sum-rate maximum with optimal relay power distribution regions

(βSum denotes the case where the MAC sum-rate constraint limits the maximal

sum-rate). The cases of the sum-rate optimal relay power distribution are due to

the combinatoric of the boundary of RBReq.

On the other hand, if we choose either q1>0, q2= 0 or q1= 0, q2>0, we get the maximal

unidirectional rates R∗

1eq and R∗

2eq, i.e. we maximize R1or R2only. This is obviously

achieved if we choose β1= 1 or β2= 1 respectively:

R∗

1eq := max

[R1,R2]∈RBReq

R1=1

2log 1 + min{γ1|h1|2, γR|h2|2},(2.22a)

R∗

2eq := max

[R1,R2]∈RBReq

R2=1

2log 1 + min{γ2|h2|2, γR|h1|2}.(2.22b)

If we maximize R1, the rate R2need not be zero. In more detail, if R−→

1R < R−→

R2(1), all

rate pairs in R∗

1eq := {[R∗

1eq, R2]:0≤R2≤1

2R−→

R1(1 −β1B)}achieve the maximal

unidirectional rate R∗

1eq. Similarly, if we maximize R2, the rate R1need not be zero. If

R−→

2R < R−→

R1(1), all rate pairs in R∗

2eq := {[R1, R∗

2eq]:0≤R1≤1

2R−→

R2(βB2)}achieve the

maximal unidirectional rate R∗

2eq.

Optimal time division

We now study bidirectional relaying with optimal time division between the phases. There-

fore, we identify the optimal time division and relay power distribution for an optimal rate

allocation, which allows us to characterize the boundary of the bidirectional achievable rate

2 Bidirectional Relay Communication using Superposition Encoding

region RBRopt. For that goal, we first specify the optimal time division and relay power

distribution for a fixed operating rate pair in the MAC phase. After that we use this to derive

an equivalent description of the set RBRopt.

In the next lemma, we characterize for any operating rate pair [RM

1, RM

2]∈ CMAC in the

MAC phase the corresponding optimal rate pair in the BC phase and the optimal time division

α∗. It shows that the optimal rate pair is on the boundary of (1−α∗)RBC. Since for any rate

pair on the boundary of RBC we have β1+β2= 1, we have to find the optimal relay power

distribution β∗∈[0,1] with β1=β∗and β2= 1 −β∗.

Lemma 2.11. For an arbitrary but fixed rate pair [RM

1, RM

2]∈ CMAC the feasible set of

time division parameters, where operating at the rate pair [RM

1, RM

2]in the MAC phase is

possible, is given by

A=α∈[0,1] : there exists a relay power distribution [β1, β2]∈ B with

αRM

1≤(1 −α)R−→

R2(β1), αRM

2≤(1 −α)R−→

R1(β2)

with simplex B={[β1, β2]∈[0,1] ×[0,1] : β1+β2≤1}. For a time division parameter

α∈ A we achieve the bidirectional rate pair [R1, R2] = α[RM

1, RM

2]. Then the optimal

time division parameter α∗:= max

α∈A αis uniquely characterized by the equations

α∗RM

1= (1 −α∗)R−→

R2(β∗),(2.23a)

α∗RM

2= (1 −α∗)R−→

R1(1 −β∗),(2.23b)

which also characterizes the optimal relay power distribution [β∗

1, β∗

2]∈ B with β∗

1=β∗

and β∗

2= 1 −β∗.

Proof. The proof can be found in Appendix 2.8.7.

The equations (2.23a) and (2.23b) characterize the optimal time division and relay power

distribution for any rate pair [RM

1, RM

2]∈ CMAC. Obviously, β⋆= 1 is optimal if RM

2is

equal to zero. For any positive rate RM

2the ratio

=R−→

R2(β∗)

R−→

R1(1 −β∗)(2.24)

implicitly defines the optimal relay power distribution β∗∈[0,1) with β∗

1=β∗and β∗

1−β∗so that [β∗

1, β∗

2]∈ B. With the optimal relay power distribution we can calculate the

optimal time division coefficient

α∗=R−→

R2(β∗)

1+R−→

R2(β∗)=R−→

R1(1 −β∗)

2+R−→

R1(1 −β∗).

2.2 Achievable Rate Region

If we do not have RM

1= 0 and RM

2= 0 at the same time, the system of equations (2.23a)

and (2.23b) has only one solution. It follows that the optimal parameters are unique.

Equation (2.24) shows that for any [RM

1, RM

2]∈ CMAC with the same ratio we have the

same optimal relay power distribution. Furthermore, the resulting bidirectional rate pair

[R1, R2]∈ RBRopt has the same ratio as the rate pair [RM

1, RM

2]∈ CMAC. This means that

all lie on the same radial line. Then it is clear that for the optimal time division the bound-

aries of the scaled MAC and BC regions intersect, which we can observe in Figure 2.2 (b).

Furthermore, it follows that the largest bidirectional rates with a certain ratio are achieved by

the rate pairs on the boundaries of the MAC and of the BC region with the same ratio. This

holds for any ratio and therefore for any rate pair on the boundary. With this we can find an

equivalent characterization of RBRopt by transforming the rate pairs of sum and individual

rate constraints of the MAC region using Lemma 2.11.

Theorem 2.12. The bidirectional achievable rate region with optimal time division RBRopt

is given by

RBRopt =R1∩R2∩RΣ(2.25)

with rate regions

R1:= R∈

+:R1≤R11(β), R2≤R12(β), β ∈[0,1],

R2:= R∈

+:R1≤R21(β), R2≤R22(β), β ∈[0,1],

RΣ:= R∈

+:R1≤RΣ1(β), R2≤RΣ2(β), β ∈[0,1],

rate constraints R11, R12, R21, R22, RΣ1, RΣ2 : [0,1] →

+with

R11 :β7→ 1−α⋆

1(β)R−→

R2(β) = R−→

1RR−→

R2(β)

R−→

1R +R−→

R2(β),(2.26a)

R12 :β7→ 1−α⋆

1(β)R−→

R1(1 −β) = R−→

1RR−→

R1(1 −β)

R−→

1R +R−→

R2(β),(2.26b)

R21 :β7→ 1−α⋆

2(β)R−→

R2(β) = R−→

2RR−→

R2(β)

R−→

2R +R−→

R1(1 −β),(2.26c)

R22 :β7→ 1−α⋆

2(β)R−→

R1(1 −β) = R−→

2RR−→

R1(1 −β)

R−→

2R +R−→

R1(1 −β),(2.26d)

RΣ1 :β7→ 1−α⋆

Σ(β)R−→

R2(β) = RMAC

ΣR−→

R2(β)

RMAC

Σ+R−→

R1(1 −β) + R−→

R2(β),(2.26e)

RΣ2 :β7→ 1−α⋆

Σ(β)R−→

R1(1 −β) = RMAC

ΣR−→

R1(1 −β)

RMAC

Σ+R−→

R1(1 −β) + R−→

R2(β),(2.26f)

2 Bidirectional Relay Communication using Superposition Encoding

and optimal time division parameters

α∗

1: [0,1] →

+, β 7→ R−→

R2(β)

R−→

1R +R−→

R2(β),(2.27a)

α∗

2: [0,1] →

+, β 7→ R−→

R1(1 −β)

R−→

2R +R−→

R1(1 −β),(2.27b)

α∗

Σ: [0,1] →

+, β 7→ R−→

R1(1 −β) + R−→

R2(β)

RMAC

Σ+R−→

R1(1 −β) + R−→

R2(β).(2.27c)

Proof. The proof can be found in Appendix 2.8.8.

From the inspection of the equivalent characterization of the rate region we find that the

information flows show a similar behavior as Kirchhoff’s first law (the principle of con-

servation of electric charge). Here, the conservation of the information flows is due to the

superposition encoding approach. In Figure 2.4 (a) we illustrate the regions RΣ,1

/2RBC,

and 1

/2RMAC. We see that the intersection point of 1

/2RΣand 1

/2RBC represents the bound-

ary rate pair on RΣfor the equal time division case. It is easy to conceive that for other time

division parameters αthe intersection point moves on the boundary according to the scaled

rate region (1 −α)RBC and sum-rate constraint αRΣof αRMAC. Similar arguments apply

to R1and R2. Then in Figure 2.4 (b) we illustrate the construction of RBRopt using the

intersection of the sets R1,R2, and RΣaccording to Theorem 2.12.

For the throughput optimal resource allocation policy in Section 2.3 and for the relay selec-

tion problem in Section 2.4 we need to identify the angle of the normal vector of any Pareto

optimal rate pair on the boundary of RBRopt. In the next corollary we characterize the bound-

aries of R1,R2, and RΣ. Since the boundaries on the axis are obvious, we parametrized the

boundaries in the first quadrant only. This also characterizes for each boundary rate pair on

RBRopt the optimal relay power distribution with β⋆

1=βand β⋆

2= 1−β. The parametriza-

tion of the boundary allows us to identify properties of the normal vectors on the boundaries

from which we can conclude that R1,R2, and RΣare convex. Then it follows that RBRopt

is convex as well. Hence, the next corollary is essential for the discussion of the weighted

rate sum maximization problem considered in Theorem 2.16.

Corollary 2.13. We can parametrize the boundaries of R1,R2,RΣby the rate pair func-

tions

R1: [0,1] →

+, β 7→ [R11(β), R12(β)],

R2: [0,1] →

+, β 7→ [R21(β), R22(β)],

RΣ: [0,1] →

+, β 7→ [RΣ1(β), RΣ2(β)],

2.2 Achievable Rate Region

0 0.2 0.4 0.6

0.2

0.4

0.6

rate R1 [bit/channel use]

rate R2 [bit/channel use]

|h1|2 =0.9

|h2|2 =0.3

γ1 =1

γ2 =1.3

γR =1.7

(a) Achievable rate regions RΣ(solid line),

/2RBC (dotted line), 1

/2CMAC (dashed line),

and MAC sum-rate constraint 1

/2RMAC

(dashed-dotted line).

0 0.2 0.4 0.6

0.4

0.8

1.2

rate R1 [bit/channel use]

rate R2 [bit/channel use]

|h1|2 =0.9

|h2|2 =0.3

γ1 =1

γ2 =1.3

γR =1.7

(b) Achievable rate region RBRopt (solid line)

after intersection of R1(dashed line), RΣ

(dashed-dotted line), and R2(dotted line).

Figure 2.4: Equivalent characterization of the bidirectional achievable rate region RBRopt

for the optimal time division case using R1,R2, and RΣ.

respectively. Therewith, the angles of the normal vectors for any boundary rate pair of R1,

R2, and RΣare given by the strictly decreasing functions

ϕ1: [0,1] →[0, π/2], β 7→ arctan −dR11(β)dβ

dR12(β)dβ,

ϕ2: [0,1] →[0, π/2], β 7→ arctan −dR21(β)dβ

dR22(β)dβ,

ϕΣ: [0,1] →[0, π/2], β 7→ arctan −dRΣ1(β)dβ

dRΣ2(β)dβ,

respectively. It follows that RBRopt is convex.

Proof. The proof can be found in Appendix 2.8.9.

Since ϕ1,ϕ2, and ϕΣare continuous and strictly decreasing for each function, there exists

an inverse function ϕ−1

1: [ϕ1(1), ϕ1(0)] →[0,1],ϕ−1

2: [ϕ2(1), ϕ2(0)] →[0,1], and

ϕ−1

Σ: [ϕΣ(1), ϕΣ(0)] →[0,1] respectively. Unfortunately, ϕ−1

1,ϕ−2

2, and ϕ−1

Σhave no

explicit representation.

2 Bidirectional Relay Communication using Superposition Encoding

For a complete characterization of the boundary of RBRopt we need to understand the com-

binatoric of the intersection (2.25). In the next proposition we see that also the combinatoric

of the MAC region carries over to RBRopt.

Proposition 2.14. In the first quadrant there is exactly one intersection between the bound-

aries of R1and R2,R1and RΣ, and R2and RΣ. In particular, for each intersection

there exists exactly one relay power distributions β12, β1Σ, βΣ2 ∈[0,1] where we have

R1(β12) = R2(β12)/∈ RBRopt,R1(β1Σ) = RΣ(β1Σ)∈ RBRopt, and R2(βΣ2) =

RΣ(βΣ2)∈ RBRopt, where the last two rate pairs are the transformed vertices ν1Σ and

νΣ2 respectively. Furthermore, we have

R11(1) ≤RΣ1(1) ≤R21(1),(2.28a)

R12(0) ≤RΣ2(0) ≤R22(0).(2.28b)

Proof. The proof can be found in Appendix 2.8.10.

With the equivalent description of the rate region and the knowledge about the combinatoric

it is finally easy to characterize the boundary of RBRopt.

Corollary 2.15. The section-wise defined rate pair function Ropt : [0,1] →

+with

Ropt :β7→ 









R2(β),for βΣ2 ≥β≥0,

RΣ(β),for β1Σ > β > βΣ2,

R1(β),for 1≥β≥β1Σ.

(2.29)

characterizes the boundary of the bidirectional achievable rate region RBRopt for the opti-

mal time division case.

Proof. The proof can be found in Appendix 2.8.11.

We are now ready to characterize in closed form the rate pair where the weighted rate sum

is maximized. For that goal we make use of all previously introduced functions and char-

acteristic parameters for the rate region RBRopt in the next theorem. Remember that for an

optimal relay power distribution we have β∗

1= 1 −β∗

2=β∗.

Theorem 2.16. Let q= [q1, q2]∈

+\ {0}denote a weight vector with nonnegative

elements and angle θq:= arccos q1

√q2

1+q2

2, then the rate pair where the weighted rate sum

is maximized is given as

R∗

opt(q) = arg max

R∈RBRopt

q1R1+q2R2=Ropt(β∗(q)) (2.30)

2.2 Achievable Rate Region

0 0.2 0.4 0.6

0.2

0.4

0.6

rate R1 [bit/channel use]

rate R2 [bit/channel use]

|h1|2 =0.6

|h2|2 =0.9

γ1 =1

γ2 =1

γR =1.7

(a) Equal time division: RBReq (solid line),

/2CMAC and 1

/2RBC (dotted lines)

0 0.2 0.4 0.6

0.2

0.4

0.6

rate R1 [bit/channel use]

rate R2 [bit/channel use]

|h1|2 =0.6

|h2|2 =0.9

γ1 =1

γ2 =1

γR =1.7

(b) Optimal time division: RBRopt (solid line),

R1,R2, and RΣ(dotted lines)

Figure 2.5: Bidirectional achievable rate regions with Pareto optimal rate pairs.

with the optimal power distribution parameter β∗(q)defined by

β∗:

+→[0,1],q7→











1,if θq< ϕ1(1),

ϕ−1

1(θq),if ϕ1(1) ≤θq≤ϕ1(β1Σ),

β1Σ,if ϕ1(β1Σ)< θq< ϕΣ(β1Σ),

ϕ−1

Σ(θq),if ϕΣ(β1Σ)≤θq≤ϕΣ(βΣ2),

βΣ2,if ϕΣ(βΣ2)< θq< ϕ2(βΣ2),

ϕ−1

2(θq),if ϕ2(βΣ2)≤θq≤ϕ2(0),

0,if ϕ2(0) < θq.

Then the corresponding optimal time division parameter α∗(q)is given by

α∗:

+→[0,1],q7→ 









α1(β∗(q)),if 1≥β∗(q)≥β1Σ,

αΣ(β∗(q)),if β1Σ > β∗(q)> βΣ2,

α2(β∗(q)),if βΣ2 ≥β∗(q)≥0.

Proof. The proof can be found in Appendix 2.8.12.

In Figure 2.5 we depicted the bidirectional rate regions with equal (a) and optimal (b) time

division. The marked boundary rate pairs (bullets) denote the weighted rate sum optimal rate

pairs for the corresponding vector (arrows). We see that the intersection points are optimal

2 Bidirectional Relay Communication using Superposition Encoding

for a range of weight vectors, while the rate pairs between intersection points correspond to

exactly one orthonormal vector.

As for the equal time division the theorem covers some special cases. If we choose q1=q2=

1, we get the sum-rate optimal rate pair. On the other hand, we maximize the unidirectional

rate R1if we choose q1>0and q2= 0 so that we have θq= 0. Obviously, β∗=

1is the optimal power distribution with the corresponding optimal rate pair Ropt(1) =

R1(1) = [R11(1), R12(1)]. This means that the maximal unidirectional rate for the optimal

time division case is given by

R∗

1opt := max

[R1,R2]∈RBRopt

R1=R11(1) = R−→

1RR−→

R2(1)

R−→

1R +R−→

R2(1).(2.31a)

Similarly, if we choose q1= 0 and q2>0, we have θq=π/2so that β∗= 0 is optimal.

From the corresponding optimal rate pair Ropt(0) = R2(0) = [R21(0), R22(0)] we get the

other maximal unidirectional rate as follows

R∗

2opt := max

[R1,R2]∈RBRopt

R2=R22(0) = R−→

2RR−→

R1(1)

R−→

2R +R−→

R1(1).(2.31b)

Since we have R12(1) = R22(0) = 0, for the optimal time division the maximal unidirec-

tional rate pairs are always on the R1-axis and R2-axis.

In the next section we will derive the achievable rate region with an optimal time-division but

with a different interpretation of the block length of a code word which result in a different

individual power constraint which depends on the time-division parameter.

2.2.5 Achievable Rate Regions with Power Scaling

In this section we briefly want to discuss a different interpretation of the mean power con-

straint assumed in the remaining sections. A more detailed discussion on the fundamentals

of the following discussion can be found in [Gal68], but it is also addressed in other text-

books like [Ash65] or [CT91]. In this chapter we consider Gaussian channels, which are

continuous-alphabet channels. In order to have useful results it is necessary to consider an

input constraint, which is usually characterized by a real-valued function f(x)on the input

letters.

The most common and physically meaningful constraint is an energy constraint per transmit-

ted symbol, i.e. f(x) = |x|2. It is common to consider an equivalent power constraint, where

one has normalized the energy constraint by the time duration of a symbol. This means that

we are usually faced with a mean power constraint P.

2.2 Achievable Rate Region

But we still have to interpret the constraint per input symbol from a coding point of view.

Let xm= [xm,1, xm,2,...,xm,N ]denote a code word of length N, then it is common and

reasonable to insist that each code word xmsatisfies the condition

n=1

f(xm,n) =

n=1 |xm,n|2≤NP. (2.32)

Using the random coding arguments we consider the input as a random variable X. Accord-

ingly, it is usual to bound the expected value of f(X)so that (2.32) results in the commonly

used mean power constraint

{|X|2} ≤ P.

The difference between the results obtained in this section and the remaining ones of this

chapter is the interpretation of the definition of a code word. Since the relay node cannot

transmit and receive at the same time using the same frequency we require orthogonal chan-

nels for the transmission and reception. As a consequence the communication is performed

in two phases. In this work we consider orthogonal channels in the time domain, but the

concepts and the results of this section also hold for orthogonal channels in the frequency

domain where we split the total bandwidth into two channels. The time division between the

two phases mean that in the first NMAC time slots we are in the MAC phase. In the following

NBC time slots we are in the BC phase.

In the remaining of this chapter we have defined input power constraints per transmitted

symbol for each phase individually, which means that the code word in the MAC phase

has the length NMAC and the codeword in the BC phase has the length NBC. However, in

this section we will assume that the code words in the MAC and BC phase have the length

NBC +NMAC. To be conform to the time division between the phases we have to assume

that in the MAC phase the last NBC letters of the code words are equal to zero and in the

BC phase the first NMAC letters of the code words are equal to zero. Accordingly, for a code

word in the MAC phase we have

NMAC+NBC

n=1 |xm,n|2=

NMAC

n=1 |xm,n|2≤(NMAC +NBC)P.

which results in the scaled power constraint

{|X|2} ≤ NMAC+NBC

NMAC P. Similarly, for a

codeword in the BC phase we have

NMAC+NBC

n=1 |xm,n|2=

NMAC+NBC

n=NMAC+1 |xm,n|2≤(NMAC +NBC)P.

which results in the scaled power constraint

{|X|2} ≤ NMAC+NBC

NBC P. We see that we scale

the power constraints with respect to the time division. Let αdenote the ratio of NMAC

2 Bidirectional Relay Communication using Superposition Encoding

to NMAC +NBC =Nas Ngets large. Then we end up with the individual mean power

constraints

P1:= P1

α,˜

P2:= P2

α,and ˜

PR:= PR

1−α(2.33)

of the nodes 1 and 2 in the MAC phase and the relay node in the BC phase. For that reason

we consider this as bidirectional relaying with scaled powers. From (2.33) we see that this

model has the property that for each phase the mean transmit power constraint gets arbitrary

large if the time period of the phase is correspondingly short. However, the provided energy

for each node is independent of the duration of the phases, i.e. the energy remains constant

for different time division parameters.

This interpretation of the code word length is often assumed for the multiple access chan-

nel with orthogonal access (TDMA or FDMA) [CT91] and for Gaussian orthogonal relay

channels in [LV05]. In the following we derive the achievable rate region of bidirectional re-

laying with optimal time division and scaled powers. Fortunately, we can principally follow

the same ideas as before, although the proofs get more involved because of the non-linear

dependence of the time division parameter on the achievable rate constraints. This also ham-

pers closed form solutions. Accordingly, in the next sections we specify the achievable rate

region of the MAC phase, the BC phase, and the achievable rate region obtained with optimal

time division between the two phases.

Since all concepts of the previous section can be transfered, the symbols will have the same

meaning but in a different context. For that reason we only added a tilde to the previously

defined symbols to distinguish the variables and results of this section from the results of the

remaining chapter. To avoid distracting case studies we exclude for notational convenience

the degenerate case where one channel or power is equal to zero. However, there is no

mathematical difficulty to extend the discussion for these cases.

Multiple Access Phase

In the multiple access phase node 1 and node 2 transmit their messages to the relay node.

We assume that for the fraction α∈(0,1] we are in the MAC phase, which specifies the

effective block length of the code words. Then we assume that the relay node can decode

the message of node 1 with rate R1and the message of node 2 with rate R2if the rate pair is

within the capacity region

CMAC(α) := n[R1, R2]∈

+:R1≤˜

R−→

1R(α), R2≤˜

R−→

2R(α), R1+R2≤˜

RMAC

Σ(α)o

2.2 Achievable Rate Region

with the individual and sum-rate constraints

R−→

1R : (0,1] →

+, α 7→ αlog 1 + γ1|h1|2

α,(2.34a)

R−→

2R : (0,1] →

+, α 7→ αlog 1 + γ2|h2|2

α,(2.34b)

RMAC

Σ: (0,1] →

+, α 7→ αlog 1 + γ1|h1|2+γ2|h2|2

α,(2.34c)

and the continuously continuations ˜

R−→

kR(0) := 0,k= 1,2, and ˜

RMAC

Σ(0) := 0. The

difference to the rate constraints (2.4a), (2.4b), and (2.4c) of the previous model is that here

the powers, respectively signal-to-noise ratios γ1and γ2, are divided by the time division

factor α. For α= 0 the protocol degenerates and there is no multiple access phase so that

we have ˜

CMAC(0) = ∅. It follows that for α= 0 no bidirectional communication will be

possible. Therefore, we exclude the case α= 0 to avoid distracting case studies caused by

divisions by α.

For any fixed time division parameter α∈(0,1] the capacity region ˜

CMAC(α)describes a

pentagon, which is obviously convex. In the next lemma we prove monotony and concavity

properties of the rate constraints, which we will extensively use in the proofs for the results

on the bidirectional achievable rate region.

Lemma 2.17. The individual and sum-rate constraint functions ˜

R−→

1R(α),˜

R−→

2R(α), and

RMAC

Σ(α)are strictly increasing and concave for α∈(0,1].

Proof. The proof can be found in Appendix 2.8.13.

Broadcast Phase

In the broadcast phase the relay node re-encodes the decoded messages from the MAC phase.

Then the relay node scales the code word of the message of node 1 with rate R1and the code

word of the message of node 2 with rate R2according to the relay power constraint. Again

the parameter β∈[0,1] denotes the fraction of the relay power spent for the message of node

1. Then the relay node transmits a superposition of the scaled codewords. Node 1 and node

2 subtract the interference caused by the code word of its own message before decoding

the unknown message. Therefore, we can assume that node 2 and node 1 can decode the

messages if the rate pair [R1, R2]is within the achievable rate region

RBC(α) := n[R1, R2]∈

+:there exists β∈[0,1] such that

R1≤˜

R−→

R2(α, β), R2≤˜

R−→

R1(α, β)o

2 Bidirectional Relay Communication using Superposition Encoding

with rate constraints

R−→

R2 : [0,1) ×[0,1] →

+,[α, β]7→ (1 −α) log 1 + γRβ|h2|2

1−α,(2.35a)

R−→

R1 : [0,1) ×[0,1] →

+,[α, β]7→ (1 −α) log 1 + γR(1 −β)|h1|2

1−α,(2.35b)

and with the continuously continuation ˜

R−→

Rk(1, β) := 0,k= 1,2, for all β∈[0,1]. The

difference to rate constraints (2.7a) and (2.7b) of the previous model is that here the power,

respectively signal-to-noise ratio γR, is divided by the time division factor (1−α). For α= 1

the protocol degenerates and there is no broadcast phase so that we have ˜

RBC(1) = ∅. It

follows that for α= 1 no bidirectional communication will be possible so that we exclude

the case α= 1 to avoid distracting case studies caused by divisions by 1−α.

We will now introduce some notation which is needed for the following studies. First, we

define the following vector valued function

RBC : [0,1) ×[0,1] →

+,[α, β]7→ h˜

R−→

R2(α, β),˜

R−→

R1(α, β)i

which parametrizes the boundary of ˜

RBC(α)for any time division parameter α∈[0,1) by

the power distribution β∈[0,1]. Obviously, we can express any rate pair ˜

RBC(α, β)6=0

in its polar coordinates. Accordingly, the angle is given by

˜ϕBC : [0,1) ×[0,1] →[0,π

2],[α, β]7→ 





arctan ˜

R−→

R1(α,β)

R−→

R2(α,β),if β6= 0

2,if β= 0

and the radius can be calculated by

RBC : [0,1) ×[0,1] →

+,[α, β]7→ ˜

RBC(α, β)1

with ˜

RBC(α, β)1=q˜

R−→

R1(α, β)2+˜

R−→

R2(α, β)2so that we can write

RBC(α, β) = ˜

RBC(α, β)cos ˜ϕBC(α, β),sin ˜ϕBC(α, β).

In the proofs for the results on the bidirectional achievable rate region we often use the polar

representation of a rate pair. Additionally, we extensively use the monotony and concavity

properties of the rate constraints, which we will prove in the next lemma.

Lemma 2.18. The rate functions ˜

R−→

R2(α, β)and ˜

R−→

R1(α, β)are concave for [α, β]∈[0,1)×

[0,1] and strictly decreasing for α∈[0,1).

Proof. The proof can be found in Appendix 2.8.14.

2.2 Achievable Rate Region

The concavity of the rate constraints allows us to derive convexity of the rate region in the

next proposition.

Proposition 2.19. For any α∈[0,1) the set of rate pairs ˜

RBC(α)is convex.

Proof. The proof can be found in Appendix 2.8.15.

Proposition 2.20. For any arbitrary but fixed time division parameter α∈[0,1) the function

of the angle ˜ϕBC(α, β)is continuously and strictly decreasing for β∈[0,1]. It follows that

there exists an inverse mapping ˜

βBC : [0,1) ×[0,π

2]→[0,1] which is implicitly defined by

˜ϕBC(α, ˜

βBC(α, ϕ)) = ϕ.

Proof. For any non-degenerate fixed time factor αit is obvious that the rates ˜

R−→

R1(α, β)

and ˜

R−→

R2(α, β)are continuous and strictly decreasing and strictly increasing for β∈[0,1].

Accordingly the ratio ˜

R−→

R1(α,β)

R−→

R2(α,β)is continuous and strictly decreasing. Since the trigonometric

function arctan is continuous and strictly increasing the angle ˜ϕBC(α, β)is continuous and

strictly decreasing for β∈[0,1]. Then it follows that for fixed α∈[0,1) there exists an

inverse mapping ˜ϕBC(α, ˜

βBC(α, ϕ)).

With this we can specify the radius ˜

RBC(α, β)with respect to the angle ϕby a new defined

function as follows

RBC : [0,1) ×[0,π

2]→

+,[α, ϕ]7→ ˜

RBC(α, ˜

βBC(α, ϕ)).(2.36)

Similarly, since ˜

RBC(α)is convex for any α∈[0,1) we get an unique parametrization of

the boundary of ˜

RBC(α)in the first quadrant with respect to the angle ϕ∈[0,π

2]according

RBC : [0,1) ×[0,π

2]→

+,[α, ϕ]7→ h˜

R−→

R2(α, ˜

βBC(α, ϕ)),˜

R−→

R1(α, ˜

βBC(α, ϕ))i.

with the equality ˜

RBC(α, ϕ) = ˜

RBC(α, ˜ϕBC(α, ϕ)).

Bidirectional Achievable Rate Region

In this section we study the bidirectional achievable rate region using superposition encoding

and scaled powers. As in the case of non-scaled powers we first prove an equivalent charac-

terization of the achievable rate region which then allows us to prove its convexity. Besides

we can show that the combinatoric of the MAC rate region transfers for the scaled powers as

2 Bidirectional Relay Communication using Superposition Encoding

well. This shows the conceptual parallelism of both interpretations of the power constraints.

However, the proofs here are more involved due to the non-linear dependence of the time

division parameter. This is also the reason that no closed-form solution of the weighted rate

sum maximum can be obtained. But with the equivalent characterization of the boundary,

including its combinatoric, and the knowledge of the convexity this can be easily obtained

algorithmically.

For a successful bidirectional transmission the rate pair [R1, R2]has to be achievable in the

MAC and BC phase simultaneously. It follows that the bidirectional achievable rate region

for a fixed time division parameter α∈[0,1] is given by the intersection of the rate regions

of the MAC and BC phase, i.e.

RBR(α) := ˜

CMAC(α)∩˜

RBC(α).(2.37)

Since ˜

CMAC(α)and ˜

RBC(α)are convex, it follows that the intersection ˜

RBR(α)is convex

as well. Notice, that for α= 0 or α= 1 we have ˜

RBC(α) = ∅.

The next natural extension is to look at the achievable rate region where we additionally

allow to optimize the time division between the MAC and the BC phase. This rate region is

obviously given by the union over all possible time division parameters αas follows

RBRopt := S

α∈[0,1]

RBR(α) = S

α∈(0,1)

RBR(α) = S

α∈(0,1)

CMAC(α)∩˜

RBC(α)(2.38)

where we used the fact that ˜

RBC(α) = ∅for α= 0 or α= 1. As a union of convex sets the

rate region ˜

RBRopt need not be convex. Convexity can be achieved if we allow time-sharing

between bidirectional rate pairs with different time division parameters, but in Theorem 2.26

we show that ˜

RBRopt is already convex. To this end we first prove in the next theorem an

equivalent representation of the achievable rate region ˜

RBRopt.

Theorem 2.21. The bidirectional achievable rate region with optimal time division ˜

RBRopt

is given by

RBRopt =˜

R1∩˜

R2∩˜

RΣ(2.39)

with rate regions

R1:= R∈

+:∃ϕ∈[0,π

2]such that R1≤˜

R1(ϕ) cos(ϕ), R2≤˜

R1(ϕ) sin(ϕ),

R2:= R∈

+:∃ϕ∈[0,π

2]such that R1≤˜

R2(ϕ) cos(ϕ), R2≤˜

R2(ϕ) sin(ϕ),

RΣ:= R∈

+:∃ϕ∈[0,π

2]such that R1≤˜

RΣ(ϕ) cos(ϕ), R2≤˜

RΣ(ϕ) sin(ϕ),

2.2 Achievable Rate Region

R1: [0,π

2]→

+, ϕ 7→ max

α∈[0,1] min ˜

RBC(α, ϕ),˜

R−→

1R(α)

cos(ϕ),(2.40a)

R2: [0,π

2]→

+, ϕ 7→ max

α∈[0,1] min ˜

RBC(α, ϕ),˜

R−→

2R(α)

sin(ϕ),(2.40b)

RΣ: [0,π

2]→

+, ϕ 7→ max

α∈[0,1] min n˜

RBC(α, ϕ),˜

RMAC

Σ(α)

cos(ϕ)+sin(ϕ)o.(2.40c)

Before we refer on the proof of the theorem we will state two simple but important observa-

tions in the following remark and lemma.

Remark 2.22. The rate regions ˜

R1,˜

R2, and ˜

RΣare characterized by the boundaries in

the first quadrant in polar coordinates. This means that ˜

R1(ϕ),˜

R2(ϕ),˜

RΣ(ϕ)specify the

radius of the rate vectors on the boundaries of the rate regions for each ϕ∈[0,π

2].

Lemma 2.23. For any ϕ∈[0,π

2]the maximizing time division parameter αfor the radii

given by (2.40a),(2.40b), and (2.40c) are uniquely characterized by an αwhere the first and

second arguments of the minima are equal. This allows us to define the optimal time division

parameters for any ϕ∈[0,π

2]as follows

˜α∗

1: [0,π

2]→[0,1], ϕ 7→ arg max

α∈[0,1]

min ˜

RBC(α, ϕ),˜

R−→

1R(α)

cos(ϕ),(2.41a)

˜α∗

2: [0,π

2]→[0,1], ϕ 7→ arg max

α∈[0,1]

min ˜

RBC(α, ϕ),˜

R−→

2R(α)

sin(ϕ),(2.41b)

˜α∗

Σ: [0,π

2]→[0,1], ϕ 7→ arg max

α∈[0,1]

min n˜

RBC(α, ϕ),˜

RMAC

Σ(α)

cos(ϕ)+sin(ϕ)o.(2.41c)

Proof. The proof of the Lemma 2.23 and the Theorem 2.21 can be found in Appendices

2.8.16 and 2.8.17 respectively.

In Figure 2.6 (a) we depicted the intersection of the achievable rate regions ˜

R1,˜

R2, and

RΣof an example. We see that for this example the rate regions are convex and that the

combinatoric of the MAC rate region transfers as well. In the following we will prove that

this holds in general. In addition, in Figure 2.6 (b) we depicted the achievable rate regions

RBRopt and ˜

RBRopt with the same power constraint values. However, we want to emphasize

that it is hardly possible to draw consequences from the comparison since different power

constraint models are assumed.

Theorem 2.24. In the first quadrant there is exactly one intersection between the bound-

aries ˜

R1,˜

R2, and ˜

RΣ. The rate pairs of the intersections are transformed vertices of a

corresponding MAC rate region. Let ˜ϕ12,˜ϕΣ1, and ˜ϕ2Σ denote the angles of the intersection

2 Bidirectional Relay Communication using Superposition Encoding

0 0.3 0.6 0.9

0.3

0.6

0.9

1.2

rate R1 [bit/channel use]

rate R2 [bit/channel use]

|h1|2 =0.7

|h2|2 =0.5

γ1 =1.3

γ2 =1

γR =1.7

(a) Equivalent characterization of ˜

RBRopt as

intersection of ˜

R1(dashed line), ˜

R2(dashed-

dotted line), and ˜

RΣ(dotted line).

0 0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

rate R1 [bit/channel use]

rate R2 [bit/channel use]

|h1|2 =0.7

|h2|2 =0.5

γ1 =1.3

γ2 =1

γR =1.7

(b) Comparison between rate region with

scaled powers ˜

RBRopt (solid line) and non-

scaled powers RBRopt (dashed line).

Figure 2.6: Achievable rate region ˜

RBRopt (with scaled powers).

between the boundaries of ˜

R1and ˜

R2,˜

RΣand ˜

R1, and ˜

R2and ˜

RΣrespectively. Then we

have

RΣ( ˜ϕ12)≤˜

R1( ˜ϕ12) = ˜

R2( ˜ϕ12),

R2( ˜ϕΣ1)≥˜

RΣ( ˜ϕΣ1) = ˜

R1( ˜ϕΣ1),

R1( ˜ϕ2Σ)≥˜

R2( ˜ϕ2Σ) = ˜

RΣ( ˜ϕ2Σ).

Furthermore, we have

R1(0) <˜

RΣ(0) <˜

R2(0) (2.42a)

R1(π

2)>˜

RΣ(π

2)>˜

R2(π

2)(2.42b)

Proof. The proof can be found in Appendix 2.8.18.

With the equivalent description of the rate region in Theorem 2.21 and the knowledge about

the combinatoric we can finally characterize the boundary of ˜

RBRopt.

Corollary 2.25. The section-wise defined rate pair function ˜

Ropt : [0,π

2]→

+with

Ropt :ϕ7→ [cos(ϕ),sin(ϕ)] 









R1(ϕ),if ϕ∈[ ˜ϕΣ1,0]

RΣ(ϕ),if ϕ∈( ˜ϕ2Σ,˜ϕΣ1)

R2(ϕ),if ϕ∈[π

2,˜ϕ2Σ]

2.3 Throughput Optimal Resource Allocation

characterizes the boundary of the bidirectional achievable rate region ˜

RBRopt for the opti-

mal time division and power scaling.

Proof. The proof can be found in Appendix 2.8.19.

The next theorem proves that ˜

RBRopt is convex, which is an important property to solve

optimization problems.

Theorem 2.26. The rate regions ˜

R1,˜

R2, and ˜

RΣare convex. It follows that ˜

RBRopt is

convex as well.

Proof. The proof can be found in Appendix 2.8.20.

Since the rate constraints depend non-linearly on the time division parameter αclosed-form

solutions are in general not possible. This includes the weighted rate sum maximum problem

as discussed in the previous sections. But since we know that ˜

RBRopt is convex, we know

that for a given weight vector [q1, q2]∈

+the weighted rate sum maximum is on the

boundary of ˜

RBRopt. Then with the characterization of the boundary given in Corollary 2.25

we can easily implement an iterative algorithm which finds the optimum. For that reason for

the remaining sections of this chapter we will always consider the previous model without

scaled power constraints. However, there is no difficulty in transferring the following results

to the case of bidirectional relaying with optimal time division and power scaling.

2.3 Throughput Optimal Resource Allocation

In this section we are interested in a cross-layer design of the Data Link Layer and the Phys-

ical Layer, which includes the Medium Access Sublayer. We are interested in an efficient

power control and channel allocation strategy with respect to the higher layer traffic. Ac-

cordingly, we are concerned with an adaptive resource allocation policy which takes the

exogenous stochastic arrivals at nodes 1 and 2 from the higher layer into account. To this

end we look at time slotted queueing processes with ergodic arrival processes where the

information theoretic bidirectional achievable rates of the previous section serve as service

rates9.

9In other sections of the thesis we consider achievable rates measured in bits per channel use, which describe

the spectral efficiencies in [bits/s/Hz] for a given wireless system. In this section we consider service rates

which are measured in [bits/s]. For notational and conceptual simplicity and without loss of generality we use

the previous derived rates, which means that the system we consider has normalized band-limited channels

with 1Hz bandwidth.

2 Bidirectional Relay Communication using Superposition Encoding

R ,R ,R , R ,

ρh1h2

β,α

1α

Controller

Figure 2.7: Service rate allocation by a centralized controller according to a rate allocation

policy based on the current queue and channel state for bidirectional relaying

with ergodic arrival processes and queues with infinite buffer size at nodes 1 and

In Figure 2.7 we schematically depicted the considered queueing model. At random time

instants packets arrive at nodes 1 and 2 for the other node, which are stored in queueing

buffers with infinite buffer size until they are served. It is important to notice that the relay

node has no internal queue, which means that we do not allow that any message will be stored

at the relay node. Accordingly, any message received in the MAC phase has immediately to

be forwarded in the BC phase.

We assume independent time-variant flat fading channels, where the channels stay constant

for the duration of a time slot. Thereby, the duration of the time slot is long enough so that

the assumption of a reliable transmission per time slot is reasonable. From time slot to time

slot the channel coefficients change independently. The evolution of each channel coefficient

is modeled by an ergodic and stationary random process.

At the beginning of each time slot the centralized controller adjusts the resource allocation

and therefore the service rates with respect to the bidirectional rate region according to the

rate allocation policy. Therefore, we assume that the centralized controller has access to the

current queue and channel state. For the rate allocation the controller decides for the coding

rates, which implies the power distribution factor βat the relay node and, if it is allowed, the

time division parameter α. The maximum throughput policy for the bidirectional relaying

protocol is an adaptation of the maximum differential backlog algorithm developed by Tas-

siulas and Ephrimedes in the landmark paper [TE92]. Thereby, we do not need the dynamic

routing capability of the differential backlog strategy since we have fixed routes.

As we saw in Section 2.2 the bidirectional achievable rate region depends on the time division

between the two phases. For the following maximum throughput policy we need to know the

weighted rate sum pair of the achievable rate region. Conceptually, it makes no difference if

we consider the fixed or optimal time division. Therefore, let RBR denotes the bidirectional

achievable rate region which stands for the bidirectional rate region RBR(α)or RBRopt

respectively. From the following discussion of the maximum throughput policy we see the

2.3 Throughput Optimal Resource Allocation

importance of the complete knowledge of the achievable rate region and the Pareto optimal

rate pairs.

The design goal for a throughput optimal strategy in our queueing system is stability. The

stability is proved by the well-developed theory of drift analysis using a quadratic Lyapunov

function on the buffer levels [MT93, KM95, MLMN00, MMAW99, LMNM01]. It shows

that the policy establishes stability for all arrival rate vectors within the maximal stability

region, which is equal to the ergodic rate region. Our stability analysis is adapted from

the cross-layer design for a satellite broadcast scenario from Neely, Modiano, and Rhors in

[NMR03]. In [NMR05] they extend their policy to joint routing and power allocation for

wireless networks, which means that their cross-layer design approach includes the routing

problem of the Network Layer. These interesting techniques and results are also presented

more a more comprehensive fashion in [Nee03] and in a more tutorial fashion in [GNT06].

The work [TG95] from Telatar and Gallager is one of first works which combines an informa-

tion theoretic model with a queueing model. After that, several cross-layer design problems

have been studied. In [BJH03, BW06, BW07] Boche et al. study channel aware scheduling

for the MIMO multiple access channel and present an optimization-theoretic analysis of the

cross-layer design as well as an iterative optimization method. Recently, in [YB05] the sta-

bility analysis for the parallel Gaussian relay channel is presented. For more references of

related works we refer to reference lists of the previous mentioned works.

2.3.1 Stability Region

For this cross-layer design approach we first introduce the assumptions of the considered

queueing model. Then we present the ergodic achievable rate region, which is equal to the

maximal stability region; and finally, we discuss a maximum throughput policy. We consider

the scenario depicted in Figure 2.7, where at nodes 1 and 2 packets for the other node arrive at

random time instants. The packets are stored in queueing buffers with infinite size until they

are served. For simplicity we assume that we can continuously split the data. The service

of transmission to the other node is provided by bidirectional relaying protocol. Without

loss of generality, we consider a normalized system with band-limited channels with 1Hz

bandwidth. Accordingly, the possible service rates are given by the bidirectional achievable

rate region with rates in [bits/s] and are chosen by a centralized controller according to the

rate allocation policy. Thereby, we assume that the controller does not optimize the resource

allocation over time.

For the rest of this section we assume a block-fading channel model, where the flat-fading

channel gains are assumed to be constant during a time period T. This allows us to consider

frames in a time slotted system model where the n-th slot denotes the time period [(n−

1)T, nT ]. We assume that the frame duration Tis long enough that we can transmit a

2 Bidirectional Relay Communication using Superposition Encoding

codeword with a sufficient block length, which means that assuming information theoretic

rates as service rates is a reasonable simplification. We model the evolution of the channel

from nodes 1 and 2 to the relay node by an independent stationary and ergodic process

h[n] = h1[n], h2[n]with finite state space H=H1× H2and steady-state distribution

πh=πh1πh2. Accordingly, let RBRh[n]denote the bidirectional achievable rate region

for the channel state h[n]. The network controller adjusts the rates and therefore the resource

allocation at the beginning of each frame.

Further, we assume independent ergodic arrival processes A[n] = A1[n], A2[n]]at nodes

1 and 2 with bounded first and second moment. This includes homogeneous Poisson arrival

processes with negative exponential distributed inter-arrival times. Let λ= [λ1, λ2]denote

the average arrival rates at nodes 1 and 2. Then

{A}=λTspecifies the mean number of

packet arrivals per time slot. Furthermore, we assume independent random packet lengths

Zi,i= 1,2, with bounded first and second moments at each node. Thus, let Bi[n],i= 1,2,

denote the processes of number of bits arriving in time slot nat nodes 1 and 2. Then the

bit arrival rate at node iis given by ρi=λi

{Zi},i= 1,2, in [bits/s]. Note that we have

{B2

i}<∞,i= 1,2, due to the previous assumptions.

In general, the bits of the data packets cannot be sent immediately after their arrival. There-

fore, at each node the arriving bits are stored in an internal queue with infinite buffer storage

space until they are transmitted. The controller observes the queue lengths at the end of

each time slot. Therefore, let Q[n] = Q1[n], Q2[n]represent the processes of number of

remaining bits in the queues after the n-th time slot, which is obviously the same as in the

beginning of the n+ 1-th time slot..

Finally, we assume a centralized controller which decides at the beginning of each time slot

for the service rates R1[n+ 1], R2[n+ 1]∈ RBR(h[n+ 1]) according to a rate allocation

policy based on the current queue state Q[n]and channel state h[n+ 1] of the new time slot.

Thereto, the controller adjusts the time division and relay power distribution parameters.

Hence, the slot-to-slot dynamics of the queue lengths are given by the equation

Qi[n] = max{Qi[n−1] −Ri[n]T, 0}+Bi[n], i = 1,2.(2.43)

Since the arrival processes are memoryless and the service rates R[n]depend on the queue

state Q(n−1) and the memoryless channel process h[n]only, the process Q[n]has Markov

property. Therefore, Q[n]describes a discrete-time Markov chain with state space

In the following we are interested in a maximum throughput policy. The throughput is de-

fined as the mean number of bits transmitted from node 1 to 2 or vice versa in a unit of time.

From queueing theory we know that for stability the utilization of a queueing system cannot

be larger than one. This can be easily seen from the following. The utilization of a queueing

system specifies the fraction of time in which a server is busy and is defined by the ratio of

the mean bit arrival rate to the mean service rate. If the average number of bits that arrive

2.3 Throughput Optimal Resource Allocation

is larger than the mean service rate, the queueing system is obviously unstable. Therefore,

the maximum throughput is obviously upper bounded by the mean service rates. Since the

information theoretic rates model the service rates, the upper bound is given by the ergodic

bidirectional rate region, which we consider next.

Ergodic Bidirectional Rate Region

Let the overbar in R= [R1, R2]and RBR denote an ergodic rate pair and the ergodic

bidirectional rate region respectively. Since RBR(h)is convex for any channel state h∈H,

the ergodic rate region RBR is convex as well. It follows that the ergodic rate region can

be characterized by the ergodic rate pairs on its boundary. Due to linearity the ergodic

boundary rate pair for the normal vector q∈

+\ {0}is given by the mean of rate pairs

which maximize the weighted rate sum with weight vector q, i.e.

R⋆(q) = arg max

R∈RBR

q1R1+q2R2

h{arg max

R∈RBR(h)

q1R1+q2R2}

h∈H

πh(h) arg max

R∈RBR(h)

q1R1+q2R2,

which can be calculated using Theorem 2.16. Let R⋆

1(q)and R⋆

2(q)denote the components

of the rate pair R⋆(q). With this we get a characterization of the bidirectional ergodic rate

region as follows

RBR = conR∈

+:∃q∈

+\{0}with R1≤R⋆

1(q), R2≤R⋆

2(q)o.(2.44)

Next, we study a resource allocation and scheduling policy for which queue stability is guar-

anteed for all arrival rate vectors ρ= [ρ1, ρ2]within the ergodic rate region RBR.

Maximum Throughput Policy

First we present a rate allocation policy derived from the maximum differential backlog

algorithm presented in [TE93]. The policy achieves the maximum throughput which we will

prove using the well-developed theory of drift analysis using a quadratic Lyapunov function

as done in [MT93, KM95, LMNM01, NMR03], et al.

Let fi(M) := lim sup

t→∞

tRt

[Qi(τ)>M]dτ,i= 1,2, denote the overflow functions as a mea-

sure of the fraction of time that the queue length Qi,i= 1,2, is above a certain value M.

2 Bidirectional Relay Communication using Superposition Encoding

This allows us to define the meaning of system stability from the notion of weak stability for

discrete-time Markov chains10 as follows.

Definition 2.27 ([NMR03]).For a given rate allocation policy a queueing system is stable

if for all i= 1,2we have fi(M)→0as M→ ∞.

With this, we can define the stability region of a policy as the set of bit arrival rate vectors

ρsuch that for any vector in the interior of the stability region system stability is achieved

[TE93]. Accordingly, a policy dominates another policy if the stability region of the one

contains the other. Further, the stability region of a system is the set of bit arrival rate vectors

ρsuch that for any vector in the interior at least one resource allocation policy exists which

achieves system stability. Conversely, no stabilizing policy exists whenever ρis outside. If ρ

lies on the boundary, the system may or may not be stable. A policy that dominates any other

policy is an optimal policy. Since the stability region of any policy is a subset of the maxi-

mum throughput region, a policy whose stability region is equal to the maximum throughput

region is optimal and is called a maximum throughput policy. The above definitions are

adopted from [TE93]11.

From the maximum differential backlog algorithm [TE93] we can deduce the following max-

imum throughput policy which determines the rate and resource allocation and basically tries

to equalize the queue length at nodes 1 and 2. It is noteworthy that the policy does not require

any knowledge about the arrival rates or channel statistics.

Maximum Throughput Policy 2.28. At the beginning of the n+ 1-th time slot the cen-

tralized network controller observes the current queue states [q1, q2] := Q[n]and the new

channel states h:= h[n+ 1], and adjusts the relay power distribution and, if allowed,

the time division parameter on the physical layer so that we achieve the rate pair which

maximizes the weighted rate sum for the weight vector qin RBR(h),

R[n+ 1] = arg max

[R1,R2]∈RBR(h)

q1R1+q2R2.

This policy is a maximum throughput policy if the stability region of the proposed policy is

equal to the ergodic bidirectional rate region. To show this we consider a positive quadratic

Lyapunov function on the buffer levels and show that for any arrival rate vector within the er-

godic rate region, the Lyapunov function has a negative drift whenever the mean queue length

10A Markov chain Q[n],n∈

, is weakly stable if for every ε > 0there exists B∈

such

that lim

n→∞

{||Q[n]|| > B}< ε holds almost surely and strongly stable if it is weakly stable and

lim sup

n→∞

{||Q[n]||} <∞holds almost surely. If the Markov chain is positive recurrent, then it is weakly

stable.

11Some authors call the maximum throughput policy a stability optimal policy to emphasis that this strategy

stabilizes any arrival rate vector in the stability region of the system.

2.3 Throughput Optimal Resource Allocation

is large. This allows us to deduce that the discrete-time Markov chain Q[n]is aperiodic,

irreducible, and positive recurrent, which means that there exists a unique steady-state distri-

bution (equilibrium) and the Markov chain is ergodic [MT93, KM95, LMNM01, NMR03].

Note that if the Markov chain Q[n]has a steady-state distribution it is positive recurrent and

therefore stable.

Theorem 2.29 ([NMR03]).Let be given the Lyapunov function L(q) = P2

i=1 q2

i. If there

exists a compact region Λ⊆

2and ǫ > 0such that

LQ[n+ 1]Q[n] = q<∞,∀q∈

LQ[n+ 1]−LQ[n]Q[n] = q<−ǫ,∀q/∈Λ,

3. whenever Q[n] = q∈Λ, there exists m∈

, m < ∞, such that the probability

(Q[n+m] = 0)>0,

then there exists a steady-state distribution for the queue state Qand hence the system is

stable.

The first two conditions ensure that the mean recurrence time to the Λregion is finite. The

third condition ensures that the zero state is reached infinitely often with finite mean recur-

rence times and therefore the Markov chain reduces to a single ergodic class. It is a necessary

modification for queueing systems with uncountably infinite state space.

But an even stronger drift can be achieved which proves the stability-in-the-mean of the

queues and is called the strong stability of the Markov chain and implies weak stability.

In [NMR03], a corresponding stronger drift condition is presented. Thereto, the authors

generalize results from [LMNM01, KM96] to a Markov chain with uncountably infinite state

space. The fundamental idea is that the drift gets larger in magnitude as the queue lengths

increase, then the mean queue length is bounded.

Corollary 2.30 ([NMR03]).If condition three of the Theorem 2.29 holds and positive values

υand ζexist such that

LQ[n+ 1]−LQ[n]Q[n] = q< υ −ζ

i=1

qi,

then there exists a steady-state distribution with bounded first moments

{Qi}<∞such

that ζP2

i=1

{Qi}< υ.

Note that if the corollary holds then for any ǫ > 0, the negative drift condition of Theorem

2.29 is satisfied whenever we have P2

i=1 qi>(υ+ǫ)/ζ, i.e. the sum of the queue lengths

is sufficiently large. This means that we would have the compact region Λ = {q∈

i=1 qi≤υ+ǫ

ζ}.

2 Bidirectional Relay Communication using Superposition Encoding

The policy here proposed is equivalent to the dynamic power allocation policy in [NMR03].

It is therefore possible to adapt the proof in [NMR03] with the following constant

υ:= T2max

h∈H,R∈RBR(h)R2

1+R2

2+

i=1

{B2

i}.

Since the arrival rate vector ρis assumed to be strictly in the interior of the ergodic rate

region, there exists a ˜

ζ > 0so that ˜

ζ[11] + ρ∈intRBR also holds. Then the proof works

analog with ζ= 2T˜

ζ. In Appendix 2.8.21 we reproduce the proof for completeness.

The assumption of finite channel states and the power constraints ensure that the second mo-

ment of the service rates is bounded. The following remark extracts the necessary condition

which is necessary for the previous proof.

Remark 2.31. The finite channel state condition is used to bound in (2.107) the term

P2

k=1 R2

k[n]|Q[n] = q. We can weaken this condition if ˜υ < ∞exists so that

nmax

R∈RBR(h)

k=1

ko≤˜υ.

Then the proof works with υ:= ˜υT 2+

i=1

{B2

i}since

P2

k=1 R2

k[n]|Q[n] = q≤

max

R∈RBR(h)P2

k=1 R2

kfor any q∈

Finally, let

{Di}denote the average bit delay at node i. With Little’s Theorem12 we have

{Qi}=ρi

{Di}. Therefore, we have υ/ζ > P2

i=1

{Qi}=P2

i=1

{Di}ρiusing the

boundedness of the first moment according to Corollary 2.30. This means that the bound

grows asymptotically like 1/ζ as the arrival rate vector ρis pushed towards the boundary of

the ergodic rate region. A similar discussion is given in [NMR03].

2.3.2 Numerical Simulation

To clarify the previous results, we illustrate and discuss numerical simulation results of a

simple example in this section. For the simulations we assumed independent homogeneous

Poisson arrival processes. In order to indicate the performance gain, we present a comparison

between the proposed bidirectional relaying using the optimal relay power distribution (OpR)

with optimal time division (OpT) and equal time division (EqT), as well as Round-Robin

12Little’s Theorem states that for a queueing system with almost sure finite inter-renewal interval the expected

number of customers in a queueing system is equal to the expected time each customer waits in the system

times the rate of arrival [Gal96, Sect. 3.6].

2.3 Throughput Optimal Resource Allocation

0 0.2 0.4

0.2

0.4

rate R1 [bit/s]

rate R2 [bit/s]

|h1|2 =0.9

|h2|2 =0.3

γ1 =1

γ2 =1.3

γR =1.7

(a) Achievable Rate Regions

0 0.2 0.4

0.2

0.4

ergodic rate R1 [bit/s]

ergodic rate R2 [bit/s]

|h1|2 ~ U({0.3,0.6,0.9,1.2})

|h2|2 ~ U({0.3,0.6,0.9,1.2})

γ1 =1

γ2 =1.3

γR =1.7

(b) Ergodic Rate Regions

Figure 2.8: The solid, dashed, dotted, and dashed-dotted line denote OpR OpT, OpR EqT,

RR OpT, and RR EqT respectively with optimal/equal time division OpT/EqT

and bidirectional relaying with the optimal relay power distribution OpR and

round-robin RR for a normalized system with 1Hz bandwidth.

(RR) scheduling with optimal and equal time division (OpT/EqT). For the Round-Robin

strategy we only change the scheduling strategy so that for any communication between two

nodes the time slot is subdivided into two exclusive time intervals. Thereby, we still do

not allow any node to cooperate across the time intervals. For each protocol we apply the

throughput optimal resource allocation policy.

In Figure 2.8 (a) the achievable rate regions for a certain channel state are depicted. The

dotted line depicts 0.5RMAC and 0.5RBC so that its intersection denotes the correspond-

ing achievable rate region with equal time division RBIR(0.5). Furthermore, we present the

achievable rate regions for the corresponding round robin strategies. On the boundary of

RBRopt some orthogonal vectors which characterize the optimal rate pair for a weight vector

with the same angle are depicted, cf. Theorem 2.16. We see that several orthonormal vectors

with different angles belong to a rate pair of an intersection point, while a rate pair between

the intersection points corresponds to exactly one orthonormal vector. Similar arguments

apply for the other rate regions. For the round robin strategies it is optimal to select the in-

tersection points for any non-negative weight vector. In particular for the equal time division

case the upper right vertex is always optimal, which means that the bidirectional relaying

separates in two independent unidirectional relaying protocols.

Figure 2.8 (b) depicts the corresponding ergodic rate region assuming identically uniformly

distributed channel processes. Since it is hardly possible to observe, we want to note that the

2 Bidirectional Relay Communication using Superposition Encoding

0.1 0.2 0.3 0.4 0.5

101

102

103

abs. of bit arrival rate vector | ρ| in 1/s

empirical mean queue backlog [bit]

OpR OpT

OpR EqT

RR OpT

RR EqT

(a) Average queue length evolution

2 4 6 8 10

x 104

0,2

0,4

0,6

time slot n

Q1(n) [106 bit]

OpR OpT

OpR EqT

2 4 6 8 10

x 104

0,2

0,4

0,6

time slot n

Q2(n) [106 bit]

RR OpT

RR EqT

(b) Temporal queue length evolution

Figure 2.9: Comparison of the queue length evolutions of different resource allocation poli-

cies where OpT and EqT specify the optimal and equal time division case and

OpR and RR denote a bidirectional relaying with an optimal relay power distri-

bution and a round-robin strategy respectively.

ergodic rate region of the RR-OpT strategy is slightly bulged. For the throughput optimal

resource allocation policy the stability region is given by the ergodic rate region. For that

reason we also plotted the arrival rate vectors ρof the simulated queue results in this figure.

We simulated arrival rates along the dotted radial line with an angle ∠= 1.0081.

Figure 2.9 (a) shows the average queue length evolutions along the radial line after 106time

slots with T= 1s. The vertical dotted lines denote the crossing of the stability region

boundaries, which correspond to the bullets on the radial line in Figure 2.8 (b). It can be

clearly seen that the average queue length strongly grows if ρapproaches the boundary of

the stability region.

In Figure 2.9 (b) we present the temporal queue length evolutions for the different protocols

with an arrival rate ρ= [0.16,0.35] bit/s, which is marked in Figure 3 (b) by the cross (×).

Since the vertical axes are scaled in 106bits, the random fluctuation of the queue lengths

are indistinguishable. As expected, it shows that the Round-Robin strategies cannot support

the bit-load. Thereby it is interesting to observe that for the equal time division case (RR-

EqT) the queue at node 1 remains stable while queue at node 2 overflows. The separated

queue evolutions can be explained with the separation of the bidirectional protocol in two

independent unidirectional protocols. At any time slot the rate allocation policy selects for

any channel and queue state the upper right vertex of the rectangular, as mentioned for Figure

2.8 (a). This results in the ergodic rate pair [0.178,0.205] bit/s, which is the upper right vertex

2.4 Relay Selection

of the corresponding ergodic rate region in Figure 2.8 (b) so that that queue evolution at node

1 will be stable and queue at node 2 will be unstable.

2.4 Relay Selection

In this section we consider the problem of relay selection in a network where Nrelay nodes

are willing to assist the bidirectional relay communication between the nodes 1 and 2. Ac-

cordingly, we look for the “best route” which is a problem of the Network Layer. Since

we propose to select a relay node based on the achievable rate region, which includes the

optimal resource allocation and channel state information, this is again a cross-layer design

approach.

The problem of relay selection is closely related to the question if the support of a relay

node is beneficial. Already in [LTW04] Laneman, Tse, and Wornell propose a selective user

strategy where the cooperative user decides based on the channel state if it either supports

the other user or retransmits its own data. In [OS04, OSB06] we derive a sufficient condi-

tion based on the pairwise error probability when it is advantageous to cooperate using an

amplify-and-forward strategy in a linear relay network. Dawy and Kamoun show in [DK04]

that the power consumption in multi-hop communication depends on the route selection and

derive a relay region where using the relay is better than the direct transmission. Generally,

we first have to decide if the support of a certain relay node is beneficial, which obviously

depends on the performance metric, the applied protocol, and the available information to

the controller. Then the controller decides for the “best relay node”.

In a network with Nrelays and independent time-variant fading channels there is the po-

tential for a high spatial diversity order. To exploit this Laneman and Wornell extend in

[LW03] the cooperation idea to larger networks where they introduced the distributed space-

time coding concept so that all relay nodes can participate for cooperation. They show that

distributed space-time coding achieves full spatial diversity in terms of the outage probabil-

ity. Other works follow this concept on distributed space-time coding or beamforming, e.g.

[NBK04, SO04a, LBC+05, DSG+03]. Thereby it is important to notice that the performance

improvement of multiple relay node diversity protocols is often boosted by an increasing sum

power, which means that each participating relay node adds it transmit power to the commu-

nication. Unfortunately, this often obscures the solely gain of the proposed diversity concept.

In [LBC+05] the authors compare different decode-and-forward schemes where one or mul-

tiple relay nodes are employed. They point out that the “effective coding gain” needs to be

considered and propose a distributed power allocation between the relay nodes. The same

conclusion is drawn in [ZAL06] for amplify-and-forward relay networks where the optimal

distributed power allocation is obtained by an extended water-filling solution. Furthermore,

2 Bidirectional Relay Communication using Superposition Encoding

they show that a selection scheme where only one relay node is chosen achieves full diversity

order.

There are obvious analogies to classical receiver combining techniques as equal gain, max-

imum ratio, and selection combining [HM05]. Distributed beamforming and space-time

coding concepts are comparable to equal and maximum ratio combining concepts with an

equal and optimal power distribution among the relay nodes. Those strategies rely on coher-

ent addition at the receiver so that the relay nodes need to be phase synchronized. Therefore,

they are very sensitive to the channel state information which results for a practical system

in a non-negligible technical challenge. This problem is not so serious for relay selection

strategies. Relay selection is comparable to the selection combining concept. Moreover, if

all nodes participate and we do not optimize the power allocation, relay selection strategies

are more power efficient.

Recently, some advanced studies on relay selection for unidirectional cooperative protocols

are done. They study the relay selection problem with respect to the complexity, channel

state information, and energy consumption. Furthermore, distributed solutions are proposed.

In more detail, Lin, Erkip, and Stefanov consider in [LES06] coded user cooperation with

the pairwise error probability as performance metric. They introduce the concept of a user

cooperation decision parameter based on the user cooperation gain, which is used to define

user cooperative regions. Hunter and Nosratinia propose in [HN04] various distributed part-

ner allocation protocols where a certain number of cooperating users is selected according to

a random strategy, a strategy based on the receive SNR, and a strategy with a fixed priority

list of nodes in its neighborhood. Relay selection with respect to the energy consumption,

where the cost of acquiring the CSI is factored in, is considered in [MMMZ06]. Moreover,

in [BKRL06] opportunistic relay selection is presented, which is a distributed method based

on channel measurements at the relay nodes.

However, to the best of our knowledge, relay selection for bidirectional relaying has not

been considered yet. In this thesis we assume a centralized decision for a relay node based

on the achievable rate regions of all relay nodes. Since bidirectional communication is char-

acterized by two rates, the decision for a “best relay node” is a vector optimization problem.

Therefore, it is possible that we have several Pareto optima which may correspond to differ-

ent relay nodes. We will see in the next section that we can further improve the performance

if we allow time-sharing between the usage of different relay nodes. The decision for one

Pareto optimum and its corresponding relay node may depend on other design aspects like

the throughput optimal resource allocation policy of the previous section.

If we consider time-variant block-fading channels with sufficient block-length so that we can

apply relay selection based on the achievable rate region for each channel state, the average

achievable rate region grows with the number of relay nodes N. For that goal we con-

sider independent stationary and ergodic block-fading channel processes so that the average

achievable rates are given by the ergodic rate region. Thereby, we do not optimize the power

2.4 Relay Selection

allocation over the fading states. In Section 2.4.2 we show that in the iid Rayleigh fading

case the growth of both maximal unidirectional achievable rates for bidirectional relaying

with relay selection and equal time division is asymptotically equal to ln(ln(N))

2 ln(2) . We will see

that the scaling law can be used to asymptotically upper and lower bound the sum of any rate

pair on the ergodic achievable rate regions for the equal and optimal time division case since

we can upper and lower bound the sum-rate using the maximal unidirectional ergodic rates

of the equal time division case.

2.4.1 Relay Selection Criterion

Here, we improve the performance by choosing the “best relay node” in a network where N

relay nodes with possibly different relay power constraints offer support for the bidirectional

relay communication between the nodes 1 and 2. We will see that the “best relay node” need

not be optimal for all boundary rate pairs. Accordingly, the relay selection criterion has to be

subtle enough to identify the optimal relay node for each boundary rate pair. For this purpose

we generically extend for the n-th relay node the notation of the already introduced variables

by an additional subscript index, e.g. h1,n denotes the channel between node 1 and the n-th

relay node, etc.

We first argue the reasoning of the relay selection criterion by means of the equal time di-

vision case. After that we briefly discuss the corresponding formulas for the optimal time

division case, where exactly the same arguments apply. It shows that these concepts can be

easily generalized or applied to relay selection problems in other networks.

Equal Time Division

With the first proposition we want to raise awareness of a relay selection criterion based on

the two-dimensional rate pair R= [R1, R2].

Proposition 2.32. We consider a network with two relay nodes, N= 2, and signal-to-noise

ratios γ1=γ2and γR,1=γR,2. Iff we have |h1,1| ≤ |h1,2|and |h2,1| ≤ |h2,2|, then

RBReq,1⊆ RBReq,2with equality iff we have |h1,1|=|h1,2|and |h2,1|=|h2,2|.

Proof. Since R−→

1R,1≤R−→

1R,2,R−→

2R,1≤R−→

2R,2,RΣ,1≤RΣ,2,R−→

R2,1(β1)≤R−→

R2,2(β1),

and R−→

R1,1(β2)≤R−→

R2,2(β2)for any β1, β2∈[0,1] it follows that CMAC,1⊆ CMAC,2and

RBC,1⊆ RBC,2and therefore RBReq,1⊆ RBReq,2. Equality follows immediately.

From the proposition we see that we have a “good channel state” if the corresponding rate

region contains the other; but if one channel gain is larger and the other one is smaller, none

2 Bidirectional Relay Communication using Superposition Encoding

of both regions contains the other. Some rate pairs can be achieved with certain channel states

respectively relay nodes only. This implies that there need not be one relay node which is the

best for the whole two-dimensional achievable rate region. We conclude that for a reasonable

relay selection criterion we need to look at each achievable rate pair individually.

First it is easy to see that in a network of Nrelay nodes with arbitrary channels and indi-

vidual transmit power constraints we can achieve the union of all individual achievable rate

regions,

RRSeq :=

n=1RBReq,n (2.45)

by selecting the corresponding relay node that achieves a certain rate pair. Since the union

of convex regions need not be convex, the rate region using relay selection RRSeq need not

be convex. This is obtained if we additionally allow time-sharing between the usage of the

relay nodes. This means that one has to switch between two relay nodes. The time-sharing

method is exactly described by the convex hull operator so that the rate region using relay

selection with time-sharing is given by

RRSTSeq := coRRSeq.(2.46)

Since a convex set can be characterized by the convex hull of the rate pairs which maximize

the weighted rate sum, the set RRSTSeq can be expressed as

RRSTSeq =conarg max

R∈RRSeq

RqT:q∈

+\{0}o∪0.

Let R∗

eq,n(q)denote the rate pair of the n-th relay node which maximizes the weighted rate

sum for the weight vector qaccording to Theorem 2.10. Obviously, for any weight vector

q∈

+\{0}we have

max

R∈RRSeq

RqT= max

n∈{1,2,...,N}R∗

eq,n(q)qT.

From the previous considerations we can conclude on the following relay selection criterion

for bidirectional relaying with equal time division.

Relay Selection Criterion 2.33. For any weight vector q∈

+\ {0}we have to select

the relay node whose boundary rate pair R∗

eq,n(q)maximizes the weighted rate sum, which

means that we have to do relay selection for each boundary rate pair individually. If there

are multiple relay nodes which all achieve the weighted rate sum maximum, we have to

apply time-sharing between the usage of the relay nodes with the corresponding rate pairs

to achieve all rate pairs on the boundary of RRSTSeq.

2.4 Relay Selection

Accordingly, let

Neq(q) := nn∈ {1,2,...,N}:R∗

eq,n(q)qT= max

n∈{1,2,...,N}R∗

eq,n(q)qTo(2.47)

denote the set of optimal relay nodes for a weight vector q∈

+\{0}with respect to the

achievable rate region RRSTSeq. If Neq(q)is a singleton, then there is only one optimal relay

node. Otherwise we have to apply time-sharing between the usage of the relay nodes with the

corresponding rate pairs to achieve all rate pairs on the boundary of RRSTSeq. Accordingly,

the set

R∗

RSTSeq(q) := co{R∗

eq,n(q) : n∈ Neq(q)}(2.48)

denotes the set of all optimal rate pairs for the weight vector q. Note that if Neq(q)is not a

singleton, than time-sharing between two relay nodes will be sufficient. This is because the

elements of RRSTSeq are two-dimensional rate pairs so that time-sharing between the relay

node which has the largest R1and the relay node which has the largest R2can achieve any

rate pair in RRSTSeq.

For weight vectors qwhich characterize the sum-rate maximum (q1=q2), the maximal

unidirectional rate R1(q1>0and q2= 0), and the maximal unidirectional rate R2(q1= 0

and q2>0) it could be possible that multiple rate pairs on the boundary of RBReq,n with

n∈ Neq(q)achieve the weighted rate sum maximum, cf. R∗

Σeq,n,R∗

1eq,n, and R∗

2eq,n. If

this is the case we have take for R∗

eq,n(q)in (2.48) any rate pair in R∗

Σeq,n,R∗

1eq,n, or R∗

2eq,n

respectively.

Figure 2.10 (a) illustrates the rate region using relay selection in a network with N= 3 relay

nodes and equal time division. Note that some rate pairs can be achieved by time-sharing

between two rate pairs of different relay nodes only. This means that one has to switch

between two relay nodes.

Optimal Time Division

For the optimal time division case the same arguments apply as for the equal time division

case. Some rate pairs can be achieved by certain relay nodes only. Therefore, the achiev-

able rate region using relay selection is given by the union of all individual bidirectional

achievable rate regions

RRSopt :=

n=1RBRopt,n.

2 Bidirectional Relay Communication using Superposition Encoding

0 0.2 0.4 0.6

0.2

0.4

0.6

rate R1 [bit/channel use]

rate R2 [bit/channel use]

γ1 =0.7

γ2 =1

γR =1.3

|h1|2, |h2|2

0.8, 0.8

1, 0.6

0.6, 1

(a) Achievable rate regions for equal time di-

vision: RRSTSeq (solid line) and RBReq,n for

n= 1,2,3(dashed line).

0 0.2 0.4 0.6

0.2

0.4

0.6

rate R1 [bit/channel use]

rate R2 [bit/channel use]

γ1 =1

γ2 =0.7

|h1|2, |h2|2, γR

0.5, 1.4, 1.5

0.8, 0.9, 0.8

1.1, 0.6, 1.3

(b) Achievable rate regions for optimal time di-

vision: RRSTSopt (solid line) and RBRopt,n for

n= 1,2,3(dashed line).

Figure 2.10: Achievable rate regions of relay selection with time-sharing for N= 3 relay

nodes. Note that some rate pairs can be only achieved by time-sharing between

two rate pairs of different relay nodes.

If we additionally allow time-sharing between the usage of the relay nodes, then the achiev-

able rate region is convex and can be expressed as

RRSTSopt :=coRRSopt(2.49)

=conarg max

R∈RRSopt

RqT:q∈

+\{0}o∪0.

Let R∗

opt,n(q)denote the rate pair of the n-th relay node which maximizes the weighted rate

sum for the weight vector q= [q1, q2]according to Theorem 2.16. Then for any weight

vector q∈

+\{0}we have

max

R∈RRSopt

RqT= max

n∈{1,2,...,N}R∗

opt,n(q)qT.

Again, we propose to do relay selection for each boundary rate pair individually.

Relay Selection Criterion 2.34. For any weight vector q∈

+\ {0}we have to select

the relay node whose boundary rate pair R∗

opt,n(q)maximizes the weighted rate sum. If

there are multiple relay nodes which all achieve the weighted rate sum maximum, we have to

apply time-sharing between the usage of the relay nodes with the corresponding rate pairs

to achieve all rate pairs on the boundary of RRSTSopt.

2.4 Relay Selection

Similarly, we can define for any weight vector q∈

+\{0}the set

Nopt(q) := nn∈ {1,2,...,N}:R∗

opt,n(q)qT= max

n∈{1,2,...,N}R∗

opt,n(q)qTo,(2.50)

which characterizes the optimal relay nodes with respect to the achievable rate region

RRSTSopt. As before, there is only one optimal relay node if Nopt(q)is a singleton. Oth-

erwise we have to apply time-sharing between the usage of the relay nodes with the corre-

sponding rate pairs to achieve all rate pairs on the boundary of RRSTSopt. Accordingly, the

set

R∗

RSTSopt(q) := co{R∗

opt,n(q) : n∈ Nopt(q)}

denotes the set of all optimal rate pairs for the weight vector q. Since the elements of

RRSTSopt are two-dimensional, there exist two relay nodes so that time-sharing between

those two relay nodes will be sufficient to achieve any rate pair in R∗

RSTSopt(q).

Figure 2.10 (b) illustrates the rate region using relay selection with time-sharing for a sce-

nario with N= 3 relay nodes and optimal time division. Thereby note that some rate pairs

can be achieved by time-sharing between two rate pairs of different relay nodes only.

In the next section, we study the scaling of the diversity gain in a time-variant fading con-

text.

2.4.2 Scaling Law of the Ergodic Rate Region

We now consider a network with Nrelay nodes in the presence of time-variant fading.

Therefore, we assume independent stationary and ergodic block-fading channel processes

{hk,n[m]}m,k= 1,2,n= 1,2,...,N. The channel remains constant over blocks con-

sisting of Msymbol periods and changes independently from block to block. Thereby, we

assume a block length Mso that the error-free coding assumption is reasonable. We do not

optimize the power allocation over time-varying channel states, which we regard as future

work. Accordingly, we are interested in the time average of the achievable rates obtained at

each fading instant, which is given by the ergodic achievable rate region. We first study the

ergodic rate region for the equal time division case. After that we will extend the results for

the ergodic rate region with the optimal time division between the phases.

Equal Time Division

Let RRSTSeq(h)denote the achievable rate region using relay selection with time-sharing for

the channel state haccording to (2.46). Then let R∗

RSTSeq(q,h)denote the set of optimal

achievable rates for a weight vector qand channel state h. We characterize the ergodic rate

2 Bidirectional Relay Communication using Superposition Encoding

region RRSTSeq by its boundary. To this end, we choose for any weight vector qand channel

state ha rate pair in R∗

RSTSeq(q,h). However, if R∗

RSTSeq(q,h)is not a singleton we have

to consider any possible rate allocation for this weight vector and channel state. We sum

this up with a definition of a rate allocation policy. For a weight vector qlet Ξeq(q)denote

a rate allocation policy R(h)hthat decides for any channel realization hfor a rate pair

R(h)∈ R∗

RSTSeq(q,h). Then

R∗

RSTSeq(q) := nR∈

+:there exists a policy Ξeq(q)with R=

R(h)o

denotes the set of ergodic rates we can achieve for a weight vector q∈

+\{0}. Therewith

we can express the ergodic rate region as follows

RRSTSeq := con[R1, R2]∈

+:there exists q∈

+\{0}with

R1≤R1, R2≤R2,[R1, R2]∈ R∗

RSTSeq(q)o.

In the following let R∗

RSTSeq(q)∈ RRSTSeq denote a rate pair on the boundary that achieves

the weighted rate sum maximum with weight vector q∈

+\ {0}, this means we have

R∗

RSTSeq(q)qT= maxR∈RRSTSeq RqT. Due to the time-sharing operation (convex hull)

the rate pair need not be unique.

We now study how the ergodic rate region using relay selection increases with an increasing

number of relay nodes. To this end we look at the maximal unidirectional ergodic rates,

which allows us to upper and lower bound the sum of any ergodic rate pair on the boundary

of RRSTSeq. Then for iid Rayleigh fading channels we can conclude from a scaling law for

the maximal unidirectional ergodic rates how the rates of rate pairs on the boundary of the

ergodic rate region increase with increasing number of relay nodes.

According to (2.22a) and (2.22b) let R∗

1eq,n := 1

/2min{R−→

1R,n, R−→

R2,n(1)}and R∗

2eq,n :=

/2min{R−→

2R,n, R−→

R1,n(1)}denote the maximal unidirectional rate of the n-th relay node.

Therewith, we get the maximal unidirectional rates using relay selection as follows

R∗

kRSeq := max

n∈{1,2,...,N}R∗

keq,n, k = 1,2.

We first state a technical lemma where we do some algebra for random variables. After that,

we will use the lemma to characterize the statistics of the maximal unidirectional rates using

relay selection. The lemma is stated for arbitrary real channel distributions but most of the

succeeding results are obtained for the Rayleigh distributed fading channels.

2.4 Relay Selection

Lemma 2.35. Let Xnand Yn,n= 1,2,...,N, be pairwise independent non-negative real

random variables with probability density functions (pdf) fXn(xn)and fYn(yn). Further,

let FXn(xn)and FYn(yn)denote the corresponding cumulative distribution functions (cdf).

Then the random variable Zn:= min{anXn, bnYn}with positive weights anand bnhas the

pdf

fZn(zn) = 1

anfxn(zn

an)(1 −Fyn(zn

bn)) + 1

bnfyn(zn

bn)(1 −Fxn(zn

an))

and the cdf

FZn(zn) = 1 −(1 −Fxn(zn

an))(1 −Fyn(zn

bn)).

The maximum Z:= max{Z1, Z2,...,ZN}has the pdf

fZ(z) =

n=1

fZn(z)

j=1,j6=n

FZj(z).

And finally,

fR(R) = 2 ln(2)22RfZ(22R−1)

specifies the pdf of the random variable R=1

2log[1 + Z]as a function of the random

variable Z.

Proof. The proof can be found in Appendix 2.8.22.

Remark 2.36. It is interesting to observe that Z= min{aX, bY }in the case of exponential

distributions Fx(x) = 1 −e−x/σ2

xand Fy(y) = 1 −e−y/σ2

yis again exponential distributed

with Fz(z) = 1 −e−z/λ and λ=1

aσ2

x+1

bσ2

Before deriving the asymptote of the growth of the sum of the ergodic rates on the boundary

of the ergodic rate region using relay selection, we first look at the probability that the relay

selection can be drawn by Proposition 2.32. In other words, we are interested in the proba-

bility that there exists a relay node ηwhose rate region RBReq,η contains the regions of all

others, i.e.

(∃η:RRSeq =RBReq,η). Such a relay node exists iff we have Neq(q) = {η}

for all q. The intuition says that with an increasing number of relay nodes it will be less

probable.

Proposition 2.37. For bidirectional relaying with relay selection and equal time division let

f|hk,n|(|hk,n|)and F|hk,n|(|hk,n|)denote the pdf and cdf of the absolute value of the channel

hk,n defined on the support set S|hk,n|. In a network with independent channels where each

relay node has the same power constraint γR, the probability that there exists a relay node

2 Bidirectional Relay Communication using Superposition Encoding

η∈ {1,2,...,N}whose rate region contains the rate regions of all others is given by

(∃η:RRSeq =RBR,η) =

η=1

RBReq,η ⊇

[

n=1 RBReq,n

η=1

k=1 ZS|hk,n|

n=1, n6=η

F|hk,n|(|hk,η|)f|hk,η|(|hk,η|)d|hk,η|.

In the case of iid Rayleigh fading hk,n ∼ CN(0, σ2

k)we have

(∃η:RRSeq =RBReq,η) = 1

Proof. The proof can be found in Appendix 2.8.23.

In the case of iid Rayleigh fading the intuition is proved. We see that with an increasing

number of relay nodes the Proposition 2.32 is more often not applicable. In the next theo-

rem we characterize the distributions of the maximal unidirectional rates in a network with

independent Rayleigh distributed fading channels.

Theorem 2.38. For bidirectional relaying with relay selection and equal time division, we

assume independently distributed channel coefficients hk,n ∼ CN(0, σ2

k,n)for all k= 1,2

and n= 1,2,...,N. Then the probability density function of the maximal unidirectional

rate using relay selection R∗

kRSeq,k= 1,2, is given by

f(R∗

kRSeq) = 2 ln(2)

n=1

λk,n exp (1 −22R∗

kRSeq )λk,n22R∗

kRSeq

m=1,m6=n1−exp (1 −22R∗

kRSeq )λk,m

with λ1,n =1

γ1σ2

1,n

γRσ2

2,n

and λ2,n =1

γ2σ2

2,n

γRσ2

1,n

Proof. The proof can be found in Appendix 2.8.24.

The theorem allows us to calculate the ergodic maximal unidirectional rates.

2.4 Relay Selection

Corollary 2.39. The ergodic maximal unidirectional rate R∗

kRSeq =

R∗

kRSeq,k= 1,2,

for bidirectional relaying with relay selection and equal time division in the case of indepen-

dent Rayleigh distributed fading is given by

R∗

kRSeq =

n=1

λk,n

2 ln(2)hexp(λk,n)E1(λk,n)

λk,n

N−1

m=1

(−1)mX

L⊆Jn,|L|=m

exp(λk,n +P

l∈L

λk,l)E1(λk,n +P

l∈L

λk,l)

λk,n +P

l∈L

λk,l i

(2.51)

with the exponential integral E1(x) = R∞

e−t

tdt. In the last sum we have to sum over all

subsets Lof the index set Jn={1,2,...,N}\{n}with cardinality |L| =m. If we have

λk,n =λkfor all n= 1,2,...N, then (2.51) results in

R∗

kRSeq =N

N−1

n=0 N−1

nexp((n+ 1)λk)E1((n+ 1)λk)

(−1)n(n+ 1)2 ln(2) .(2.52)

Proof. The proof can be found in Appendix 2.8.25.

In Figure 2.11 (a) we illustrate the enlargement of the ergodic rate region due to relay selec-

tion for the equal time division case. Thereby, we assume equal relay transmit powers and

iid Rayleigh fading. We see that the gain decreases with increasing number of relays. In the

following we are interested in the scaling law of this growth. For its derivation, we need the

analytical expression for the maximal unidirectional ergodic rate of the previous corollary.

Since RRSTSeq is convex, we can upper and lower bound the sum-rate of any boundary rate

pair R∗

RSTSeq(q)of the ergodic rate region as follows

min{R∗

1RSeq, R∗

2RSeq} ≤ R∗

RSTSeq(q)1≤2

k=1

R∗

kRSeq.(2.53)

The first inequality holds because RRSTSeq is convex and the sum of any rate pair from the

time-sharing between the rate pairs [R∗

1RSeq,0] and [0,R∗

2RSeq]already fulfills the condition.

The second inequality follows from the fact that R∗

1RSeq and R∗

2RSeq are both the maximal

unidirectional ergodic rates so that the sum of any rate pair on the boundary of RRSTSeq has

to be smaller.

We now derive a scaling law for the growth of the sum-rate of any rate pair on the boundary

of the ergodic rate region RRSTSeq with increasing number of relay nodes in an iid Rayleigh

fading scenario. To this end, we present in the next theorem an upper and lower bound on

the maximal unidirectional ergodic rate R∗

kRSeq,k= 1,2, which are asymptotically tight.

2 Bidirectional Relay Communication using Superposition Encoding

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ergodic rate R1

ergodic rate R2

γ1 =0.5

γ2 =1.5

γR =1.3

#R↑

(a) RRSTSeq

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

ergodic rate R1

ergodic rate R2

γ1 =0.5

γ2 =1.5

γR =1.3

#R↑

(b) RRSTSopt

Figure 2.11: Growth of the ergodic rate region with increasing number of relay nodes N=

1,2,...,12 for the equal and optimal time division case. In the left plot we

depicted the maximal unidirectional ergodic rates according to Corollary 2.39

for N= 12.

Theorem 2.40. For bidirectional relaying with relay selection and equal time division we

assume independently identical distributed channel coefficients hk,n ∼ CN(0, σ2), equal

signal-to-noise ratios γR,n =γR, and coefficients λk=γk+γR

σ2γkγRfor all k= 1,2and n=

1,2,...,N. Then we can upper and lower bound the maximal unidirectional ergodic rate

R∗

kRSeq,k= 1,2, as follows

R∗

kRSeq ≥1−e−a

2log 1 + 1

λk

ln N

a,

R∗

kRSeq ≤e−b+b

2log 1 + 1

λk

ln(N

b)+b

2log 1 + 1

λk+ ln N

b

with arbitrary a, b ∈(0, N). The asymptotic upper and lower bound meet if we choose

a→ ∞ and b→0when N→ ∞. Therefore, the growth of the maximal unidirectional

ergodic rate is asymptotically equal 1

2log(ln(N)).

Proof. The proof can be found in Appendix 2.8.26.

Note that the asymptote on the growth of the maximal ergodic unidirectional rate is indepen-

dent of λkand therefore independent of the the channel gain variance σ2and signal-to-noise

ratios γRand γk.

2.4 Relay Selection

100101102103104105

0.5

1.5

2.5

Number of Relay Nodes N

Maximal Ergodic Rate R1

Simulation

Lower Bound

Upper Bound

Figure 2.12: Growth of the maximal unidirectional ergodic rate R∗

1RSeq using relay selection

with an increasing number of relay nodes with upper and lower bounds with

optimized (in gray) and fixed coefficients (in black) with a=b= 1/2,γR=

1.3,γ1= 1, and σ2= 1.

Remark 2.41. Since the derivation applies to any AWGN achievable rate calculation of

a system where a selection combining technique between Nindependently exponentially

distributed SNRs is utilized, the scaling law of the growth of the ergodic rate of such a

system has always a log(ln(N)) asymptote.

Since we get the coefficients aand bfrom the analysis in the proof, they have no physical

meaning. Therefore, we can optimize the bounds for any fixed number of relay nodes Nby

optimizing the coefficients aand b. In Figure 2.12 we plotted the growth of the maximal

unidirectional ergodic rate R∗

1RSeq using relay selection with the number of relay nodes N

(Monte Carlo Simulation) and the upper and lower bounds with optimized and fixed coeffi-

cients. The bounds are loose but they have the same growth rates. Furthermore, we see that

the bounds with the optimized coefficients (“slowly”) converge.

Finally, as a direct consequence of the previous theorem and the inequalities in (2.53) we can

conclude on the asymptotic growth of the ergodic rate region RRSTSeq.

Corollary 2.42. For bidirectional relaying using relay selection with time-sharing and equal

time division in an iid Rayleigh fading scenario with γR,n =γRthe sum of any ergodic

rate pair on the boundary of the ergodic rate region RRSTSeq grows with Θ(log(log(N))).

In more detail, for any q∈

\ {0}we can asymptotically lower and upper bound the

asymptotic growth of the sum-rate of any boundary rate pair R∗

RSTSeq(q)as follows

lim inf

N→∞ R∗

RSTSeq(q)1

log(ln(N)) ≥1

2,lim sup

N→∞ R∗

RSTSeq(q)1

log(ln(N)) ≤1.

2 Bidirectional Relay Communication using Superposition Encoding

The difference between the asymptotic lower and upper bound is exactly the difference be-

tween a unidirectional protocol and the bidirectional protocol with an improved spectral effi-

ciency. As we see in Figure 2.10, the increase in spectral efficiency depends on the rate ratio

which vanishes in the case of maximal unidirectional ergodic rates. Finally, we want to re-

mark that the maximal unidirectional ergodic rates evolve according to the same asymptotic

scaling law as found in [DSG+03] using distributed beamforming approach with a two-phase

separation constraint.

Optimal Time Division

Let RRSTSopt(h)denote the bidirectional achievable rate region using relay selection with

time-sharing and optimal time division between the phases for the channel state haccording

to (2.49). Then let R∗

RSTSopt(q,h)denote the set of optimal achievable rates for a weight

vector qand channel state h. As for the equal time division case, we characterize the ergodic

rate region by the rate pairs on its boundary. Therefore, we choose for any channel state ha

rate pair in R∗

RSTSopt(q,h). However, if R∗

RSTSopt(q,h)is not a singleton, we have to take

any possible rate allocation into account. Again, we sum this up with the definition of a rate

allocation policy. For a weight vector qlet Ξopt(q)denote a rate allocation policy R(h)h

that decides for any channel realization hfor a rate pair R(h)∈ R∗

RSTSopt(q,h). Then

R∗

RSTSopt(q) := nR∈

+:there exists a policy Ξopt(q)with R=

R(h)o

denotes the set of ergodic rates we can achieve for a weight vector q∈

+\{0}. Therewith,

we can express the ergodic rate region as follows

RRSTSopt := con[R1, R2]∈

+:there exists q∈

+\{0}with

R1≤R1, R2≤R2,[R1, R2]∈ R∗

RSTSopt(q)o.

In the following let R∗

RSTSopt(q)∈ RRSTSopt denote a rate pair on the boundary that

achieves the weighted rate sum maximum with weight vector q∈

+\ {0}, this means

we have R∗

RSTSopt(q)qT= maxR∈RRSTSopt RqT. Due to the time-sharing operation (con-

vex hull) the rate pair need not be unique.

Clearly, the ergodic rate region RRSTSopt of relay selection with time-sharing and optimal

time division grows as well with the number of relay nodes. In Figure 2.10 (b) we illustrate

the enlargement for iid Rayleigh fading channels and equal relay powers. We see a similar

behavior as for the equal time division case. For the derivation of the growth we can make

use of the asymptotic result of the maximal unidirectional ergodic rates for the equal time

division case. To this end, we upper and lower bound in the next proposition the sum of any

rate pair on the boundary of RRSTSopt in terms of the ergodic maximal unidirectional rates

for the equal time division case.

2.5 Piggyback a Common Relay Message

Proposition 2.43. For bidirectional relaying using relay selection with time-sharing and

optimal time division we can upper and lower bound the sum-rate of any boundary rate pair

R∗

RSTSopt(q)of the ergodic rate region RRSTSopt as follows

min R∗

1RSeq,R∗

2RSeq≤R∗

RSTSopt(q)1≤2

k=1

R∗

kRSeq.(2.54)

Proof. The proof can be found in Appendix 2.43.

With the upper and lower bound we can directly apply the asymptotic result for the ergodic

maximal unidirectional rates R∗

kRSeq of Theorem 2.40 as we did for Corollary 2.42.

For bidirectional relaying using relay selection with time-sharing and equal time division

in an iid Rayleigh fading scenario with γR,n =γRthe sum of any ergodic rate pair on the

boundary of the ergodic rate region RRSTSeq grows with Θ(log(log(N))). In more detail, in

the next corollary we can lower and upper bound the asymptotic growth of the sum-rate of

any boundary rate pair R∗

RSTSeq(q)for any q∈

\{0}.

Corollary 2.44. For bidirectional relaying using relay selection with time-sharing and equal

time division in an iid Rayleigh fading scenario with γR,n =γRthe sum of any ergodic rate

pair on the boundary of the ergodic rate region RRSTSopt grows with Θ(log(log(N))). In

more detail, for any q∈

+\{0}we can asymptotically lower and upper bound the growth

of the sum-rate of any boundary rate pair R∗

RSTSopt(q)as follows

lim inf

N→∞ R∗

RSTSopt(q)1

log(ln(N)) ≥1

2,lim sup

N→∞ R∗

RSTSopt(q)1

log(ln(N)) ≤2.

It is interesting to see that although the ergodic rate region of the optimal time division case

is always larger than for the equal time division case, the sum of any ergodic rate pair has

the same asymptotic scaling law.

2.5 Piggyback a Common Relay Message

Nowadays, there is a trend to offer multiple wireless services at a single device, which is

known as the convergence of wireless services. Up to now this is realized by the coexistence

of multiple wireless transceiver chains each equipped with an own antenna. This requires a

careful radio segmentation with adequate transmit spectrum masks that allow a sufficient iso-

lation of the receivers to the transmitters with respect to the receivers sensitivity levels. With

a convergence on the Data Link Layer and Physical Layer we can overcome such practical

problems. The joint design on the Data Link Layer and Physical Layer allows the reduction

2 Bidirectional Relay Communication using Superposition Encoding

of the number of transceiver chains in the device. Moreover, the reduction of RF front-ends

is interesting for further miniaturization of the devices. In addition it allows a more efficient

joint resource allocation, which results in a reduction of the energy consumption.

Accordingly, in this section we add a relay multicast communication to the bidirectional

relaying protocol. This means that we assume that the Network Layer assigns two routing

tasks to Data Link Layer. Accordingly, we are interested in an efficient joint resource allo-

cation design for both routing tasks across the Physical and Data Link Layer. We consider

again superposition encoding, which means that the relay node encodes its own message and

superimposes the codeword of the relay multicast message on the bidirectional broadcast

transmit signal. In accordance to this we say that the relay messages is piggybacked on the

bidirectional relaying.

In the seminal work [GK00] Gupta and Kumar introduce the concept of transport capacity,

which is a measure that factors in the distance between the source and destination pairs in

a wireless network. The mean feat is to use this measure to derive upper and lower bounds

of scaling laws which characterize the amount of information a network can transport in the

limit of large number of nodes. A weak point of the work is that they considered concurrent

transmissions as interference so that in [GK03, XK04] Gupta, Xie, and Kumar consider

interference as information and therewith improve the scaling laws by studying large relay

networks. In this context they raised the fundamental question “How should nodes cooperate

in maximizing information transfer in a wireless network?” In [OLT07] Özgür, Lévêque, and

Tse achieve a linear scaling law with a hierarchical cooperation scheme using distributed

MIMO communication. Recently, in [XK07] Xie and Kumar extend their coding scheme to

multi-source, multi-destination, multi-relay networks where they group nodes along multi-

relay routes. Thereby, they merge routes if they intersect at any relay node so that they can

apply a joint backward decoding scheme. They point out that the coding scheme depends on

the topology and note that such a coding scheme excludes two-way multi-relay networks.

The afore mentioned works study information theoretic coding strategies for large relay net-

works. Since it is a hard problem to find the optimal coding strategy for an arbitrary net-

work, for most networks the optimal communication strategy is not known. However, the

promising benefits of already proposed relaying concepts for wireless networks encourage

researcher to study routing, media access, and power control for such problems. Accord-

ingly, there is recently a vast growing literature on power control problems for wireless cel-

lular and multi-hop networks which considers the joint optimization of routing, scheduling,

and power control problems, which allows the development of efficient centralized and dis-

tributed resource allocation algorithms. In [Bam98] Bambos points out that general power

control design goals for wireless networks are to prolong battery life, mitigate interference,

and maintain link quality. For the design of resource allocation policies usually a network

flow model based on flow conservation is assumed. This means that separated information

flows are considered. Then the optimization is done using utility functions per user, which

2.5 Piggyback a Common Relay Message

are often assumed to be continuous, differentiable, concave, and increasing, e.g. the capacity

function of a Gaussian channel. Then the main goal is to maximize the sum of the user util-

ity functions, which is called the network utility maximization problem. In [XJB04], Xiao,

Johansson, and Boyd introduce a generic formulation of an optimization problem for the si-

multaneous routing and power allocation problem for wireless networks, which allows them

to derive efficient solution methods based on convex optimization theory. Algorithms which

compute distributed or centralized power control solution with respect to routing and/or link

scheduling for multi-hop networks can be found for instance, in [TE02, CS03, Chi04]. While

in [NMR05] Neely, Modiano, and Rohrs study a dynamic routing and power allocation pol-

icy for wireless networks with time-varying channels based on the maximum differential

backlog strategy of Tassiulas and Ephremides [TE93]. For an introduction in the theory and

algorithms of resource allocation in wireless networks with a more comprehensive literature

survey on this topic we refer on the textbook [SWB06].

However, our goal in this section is not to find another algorithm which solves efficiently a

power or rate allocation problem. We are interested in finding some properties of the achiev-

able rate region, which then could be used for improving the efficiency of a resource alloca-

tion policy according to some higher layer strategies. Moreover, we can identify the optimal

decoding order at nodes 1 and 2 in the BC phase. Since we consider two simple routing

tasks and single-antenna nodes, some of the following results can be astonishingly obtained

in closed form. But we see from the derivations that this is already a tedious combinato-

rial problem so that we suppose that for joint optimization of more or more complicated13

routing tasks the resource allocation problems can only be solved algorithmically.

In the following section we extend the bidirectional broadcast by an additional relay message.

We characterize the optimal decoding order and the total sum-rate maximum which can

be achieved with both routing tasks. Moreover, we identify bidirectional rate pairs which

result in the same total sum-rate so that we know how to interchange additional relay rate

with bidirectional rate. The explicit characterization leads to combinatorial problems which

we discuss for the equal time division and for |h1|>|h2|. This allows us to characterize

for a desired relay rate the bidirectional rate pair which results in the largest bidirectional

sum-rate. Throughout this section we illustrate the main results and the combinatoric with

representative examples for a clear presentation of the arguments.

Most closed form results are obtained for the equal time division case, cf. Remark 2.4.

However, the behavior will be similar for other fixed time division cases. According to this

we study in the latter part of the section the maximal sum-rates with respect to the time

division parameter. In the end we present for two examples the achievable rate regions for

13Already for the routing task where the relay node adds individual messages for nodes 1 and 2 to the bidi-

rectional relaying protocol we have 3! = 6 possible decoding orders at nodes 1 and 2, which results in 36

different cases to consider. For comparison, for the here considered common relay message we have 2! = 2

decoding orders each node, which results in 4different cases where one is always optimal, c.f. Theorem 2.45.

2 Bidirectional Relay Communication using Superposition Encoding

the equal and optimal time division case. For comparison we also depicted the achievable

rate region with a simple energy-equivalent TDMA approach which realizes the same routing

tasks. This illustrates the efficiency and the new established rate trade-off possibilities gained

from the joint resource allocation approach.

2.5.1 Broadcast Phase with Relay Multicast

In this section, we extend the bidirectional relaying protocol by an additional relay multicast

communication. This means that the relay node wants to transmit an independent common

message to nodes 1 and 2. The additional relay multicast and the bidirectional broadcast are

simultaneously performed in the BC phase of the bidirectional relay protocol. For the study

of the problem it is enough to consider a network with one relay node only and time-invariant

channels. In this subsection we assume an arbitrary but fixed time division parameter α∈

[0,1].

As described in Section 2.2.3, in the BC phase the relay node forwards the previously re-

ceived message m1to node 2 and message m2to node 1. In addition to this, the relay node

additionally encodes a relay message mRfor nodes 1 and 2 with rate RR. For the trans-

mission, we again follow the superposition encoding strategy. Since we have a memoryless

Gaussian channel, it is sufficient to consider single-letters only. Accordingly, the random

input variable of the relay node can be expressed as

XR=W1+W2+WR

where the random variables W1,W2, and WRcorrespond to the codewords of messages m1,

m2, and mRrespectively. Since the messages are assumed to be independent, we consider

independent random variables. From this we get the output at node k,k= 1,2, as follows

Yk=hkXR+N1=hkW1+hkW2+hkWR+Nk, k = 1,2,

where Nkdenotes the independent additive complex Gaussian distributed noise at node k,

k= 1,2. Both nodes receive the signal of their own bidirectional message as interfer-

ence. Therefore, before decoding the unknown messages each node subtracts the interfer-

ence caused by its own message. This means that node 1 subtracts h1X1and node 2 sub-

tracts h2X2so that for each node it remains to decode the bidirectional message from the

other node and the common relay message. Accordingly, the non-negative achievable rates

for the unknown messages have to fulfill the constraints

R2+RR≤(1 −α)I(XR;Y1|W1) = (1 −α)I(W2;h1W2+h1WR+N1),(2.55a)

R1+RR≤(1 −α)I(XR;Y2|W2) = (1 −α)I(W1;h2W1+h2WR+N2),(2.55b)

RR≤min (1 −α)I(XR;Y1|W1)−R2

|{z }

=:RR@1

,(1 −α)I(XR;Y2|W2)−R1

| {z }

=:RR@2 ,(2.55c)

2.5 Piggyback a Common Relay Message

for some input distributions fWk(wk),k∈ {1,2,R}, satisfying the power constraint

{|XR|2}=

{|W1|2}+

{|W2|2}+

{|WR|2} ≤ PR. Thereby, the rates RR@1 and

RR@2 denote the achievable relay rates at nodes 1 and 2.

Again, the mutual informations are coupled by the relay power distribution. This time we

have to distribute the relay transmit power PRamong three messages. Therefore, as before let

β1and β2denote the proportion of relay transmit power PRspend for the codewords W1and

W2and additionally let βRdenote the proportion of relay transmit power PRspend for the

codeword WR. Now, the relay transmit power constraint requires that β1+β2+βR≤1holds.

For a feasible relay power distribution we know from Section 2.2.1 that complex Gaussian

distributed inputs W1∼ CN(0, β1PR),W2∼ CN(0, β2PR), and WR∼ CN(0, βRPR)

maximize the mutual informations so that we have

I(XR;Y1|W1) = log 1 + γR|h1|2(β2+βR),

I(XR;Y2|W2) = log 1 + γR|h2|2(β1+βR).

In the following we look at the achievable rates R1,R2, and RRmore explicitly. Therefore,

in the next theorem we identify an always optimal decoding order at nodes 1 and 2. On

that score, we first look at the case where each node decodes the additional relay message

first. After a successful decoding of the relay message and before decoding of the unknown

bidirectional message each node subtracts the interference caused by the additional relay

message. Effectively, this means that the bidirectional relay communication is essentially

interference-free. For that reason, for the bidirectional communication the rate constraints

(2.7a) and (2.7b) apply so that for a desired bidirectional rate pair [R1, R2]∈ RBR(α)⊆

(1 −α)RBC we need at least

β1=2R1

1−α−1

|h2|2γR

and β2=2R2

1−α−1

|h1|2γR

.(2.56)

With this the power fraction βR= 1 −β1−β2remains at most for the multicast communi-

cation. Accordingly, for a feasible bidirectional rate pair [R1, R2]∈ RBR(α)the maximal

achievable rate of the additional relay multicast at nodes 1 and 2 is given by

RR@1(R1, R2) = (1 −α) log 1 + βR|h1|2γR

1 + β2|h1|2γR,(2.57a)

RR@2(R1, R2) = (1 −α) log 1 + βR|h2|2γR

1 + β1|h2|2γR.(2.57b)

Since for a multicast both nodes should be able to decode the relay message, the maximal

achievable relay rate is given by the minimum of (2.57a) and (2.57b). In the next theorem we

show that this is the optimal decoding order and power distribution if we desire to achieve a

certain feasible bidirectional rate pair.

2 Bidirectional Relay Communication using Superposition Encoding

Theorem 2.45. For a given rate pair [R1, R2]∈ RBR(α)the maximal achievable additional

relay rate RRis achieved if nodes 1 and 2 decode the additional relay message first. Then

the maximal achievable additional relay rate is given by

RR(R1, R2) = min RR@1(R1, R2), RR@2(R1, R2).(2.58)

Proof. The proof can be found in Appendix 2.8.28.

Since it is optimal for any desired bidirectional rate pair that at each node the relay message

is decoded first, then its interference is canceled, and finally the unknown bidirectional mes-

sage is decoded without any interference of the relay message we say that the common relay

message is piggybacked on the bidirectional relay communication. Moreover, this implies

that potentially existing resources at the relay node are used to add a multicast communica-

tion on top of the bidirectional relaying.

The achievable rate region of the bidirectional relaying protocol with an additional relay

multicast for a fixed time division parameter α∈[0,1] can be expressed as

RPiggy(α) := [R1, R2, RR]∈

+: [R1, R2]∈ RBR(α), RR≤RR(R1, R2).

Since this applies for any time division parameter, for the optimal time division case we can

achieve

Ropt

Piggy := S

α∈[0,1]RPiggy(α).

We depicted the achievable rate region RPiggy(1

/2)and Ropt

Piggy for two examples in Fig-

ure 2.18 and Figure 2.19 in the end of this section. But first we want to identify more

properties of the achievable rate region RPiggy(α). In particular in the next subsection we

characterize the sum-rate optimal rate tuple which achieves the total sum-rate maximum.

2.5.2 Total Sum-Rate Maximum

Let RΣ=R1+R2denote the bidirectional sum-rate and let Rtot =R1+R2+RRspecify

the total sum-rate, which includes the additional relay rate. The optimization problem for

the total sum-rate maximum for a time division parameter αis given by

R∗

tot := max

[R1,R2,RR]∈RPiggy(α)R1+R2+RR

= max

[R1,R2]∈RBR(α)R1+R2+RR(R1, R2).(2.59)

For the characterization of the total sum-rate maximum R∗

tot in Theorem 2.48 we need the

following lemma and proposition. In Lemma 2.46 we first characterize the bidirectional

2.5 Piggyback a Common Relay Message

rate pair [R1, R2]with a fixed bidirectional sum-rate RΣ=R1+R2which maximizes

the additional relay rate RR(R1, R2). Then with Proposition 2.47 we explicitly identify the

optimal rate pair. In the next subsection we present an explicit discussion and some helpful

illustrations of the following results for the equal time division case.

Lemma 2.46. Given a feasible bidirectional sum-rate RΣ, which means that at least one

feasible rate pair [R1, R2]∈ RBR(α)with R1+R2=RΣexists, then there are R⊲

1and R⊳

so that for all R1with R⊲

1≤R1≤R⊳

1we have [R1, RΣ−R1]∈ RBR(α). Furthermore, the

maximal achievable additional relay rate for the desired bidirectional sum-rate RΣis given

Rmax

R(RΣ) := max

[R1,R2]∈RBR(α),

R1+R2=RΣ

RR(R1, R2)

=









(1 −α) log |h1h2|ˆγ−2

RΣ

2(1−α)−1

2RΣ,if R⊲

1≤R⋆

1≤R⊳

1,(2.60a)

(1 −α) log |h2|2(ˆγ−1

|h1|22

RΣ−R⊲

1−α)−R⊲

1,if R⋆

1< R⊲

1,(2.60b)

(1 −α) log |h1|2(ˆγ−1

|h2|22

R⊳

1−α)−RΣ+R⊳

1,if R⋆

1> R⊳

1(2.60c)

with R⋆

1:= 1

2RΣ−1−α

2R†,R†:= log |h1|2

|h2|2, and ˆγ:= γR+1

|h1|2+1

|h2|2. The additional

relay rate maximum Rmax

R(RΣ)is attained at the bidirectional rate pair

[R1(RΣ), R2(RΣ)] = 









[R⋆

1, RΣ−R⋆

1],if R⊲

1≤R⋆

1≤R⊳

[R⊲

1, RΣ−R⊲

1],if R⋆

1< R⊲

[R⊳

1, RΣ−R⊳

1],if R⋆

1> R⊳

(2.61)

Proof. The proof can be found in Appendix 2.8.29.

The lemma characterizes for any bidirectional sum-rate RΣthe maximal achievable addi-

tional relay rate Rmax

R(RΣ)and the corresponding rate pair [R1(RΣ), R2(RΣ)] ∈ RBR(α)

with R1(RΣ) + R2(RΣ) = RΣ. This allows us to rewrite the total sum-rate maximum

problem (2.59) as follows

R∗

tot = max

RΣ=R1+R2

[R1,R2]∈RBR(α)

Rmax

R(RΣ) + RΣ.(2.62)

This means that for the total sum-rate optimum R∗

tot we have to find the optimal bidirectional

sum-rate R∗

Σwhich maximizes the function Rtot(RΣ) := Rmax

R(RΣ)+RΣfor feasible bidi-

rectional sum-rates. For the discussion of Rtot(RΣ)we need to know how R⋆

1of Lemma 2.46

is related to the bidirectional rate region RBR(α).

2 Bidirectional Relay Communication using Superposition Encoding

Let RBC(γR)denote the achievable broadcast rate region for a given signal-to-noise ra-

tio γR. Then let RBC

Σ(γR),R⋆

−→

R2(γR), and R⋆

−→

R1(γR)denote the maximum sum-rate and

the corresponding optimal rates of RBC(γR)according to Proposition 2.2. The function

RBC

Σ(γR)is obviously bijective on the domain of non-negative γRand the range of non-

negative bidirectional sum-rates. Thus, there exists a unique inverse function γBC

R(RΣ)with

RBC

Σ(γBC

R(RΣ)) = RΣfor all feasible bidirectional sum-rates. Accordingly, we can define

R⋆

−→

R2(γR)and R⋆

−→

R1(γR)in terms of RΣas follows

R⋆

−→

R2(RΣ) := R⋆

−→

R2(γBC

R(RΣ)) and R⋆

−→

R1(RΣ) := R⋆

−→

R1(γBC

R(RΣ))

with R⋆

−→

R2(RΣ) + R⋆

−→

R1(RΣ) = RΣas well as the corresponding broadcast rate region

RBC(RΣ) := RBC(γBC

R(RΣ)).

For a bidirectional rate pair [R1, R2]∈ RBR(α) = αCMAC ∩(1 −α)RBC with sum-rate

R1+R2=RΣwe have [R1

1−α,R2

1−α]∈ RBC so that the relay signal-to-noise ratio γRhas to

be larger or equal to γBC

R(RΣ

1−α). The next proposition shows that R⋆

1of Lemma 2.46 for a

feasible bidirectional sum-rate RΣspecifies the first sum-rate optimal rate (1−α)R⋆

−→

R2RΣ

1−α

of a corresponding scaled BC rate region (1 −α)RBCRΣ

1−αaccording to (2.10b) if RΣ

(respectively γR) is large enough, which means that we have β⋆∈[0,1].

Proposition 2.47. The sum-rate optimal rate pair with sum-rate RΣof the broadcast rate

region (1 −α)RBC(RΣ

1−α)is given by the vector-valued function

R⋆

BC :

+→

+, RΣ7→ 









[R⋆

1(RΣ), R⋆

2(RΣ)],if RΣ≥(1 −α)|R†|,(2.63a)

[RΣ,0],if RΣ<−(1 −α)R†,(2.63b)

[0, RΣ],if RΣ<(1 −α)R†,(2.63c)

with R⋆

1(RΣ) = 1

2RΣ−1−α

2R†,R⋆

2(RΣ) := 1

2RΣ+1−α

2R†, and R†= log |h1|2

|h2|2.

Proof. The proof can be found in Appendix 2.8.30.

The function RBC(RΣ)of Proposition 2.47 defines for any bidirectional sum-rate RΣthe

unique sum-rate optimal rate pair of the rate region (1−α)RBC(RΣ

1−α)of the BC phase. This

allows us to identify for a desired bidirectional sum-rate RΣthe rate pair where the maximal

additional relay rate Rmax

R(RΣ)is attained, cf. (2.61) of Lemma 2.46, explicity. With this we

can characterize the total sum-rate maximum for a desired bidirectional sum-rate, Rtot(RΣ).

Let us assume that the sum-rate RΣis feasible, which means that there exists at least one

bidirectional rate pair with the desired sum-rate so that the intersection of RBR(α)with the

graph GfRΣof the function fRΣ: [0, RΣ]→[0, RΣ]with R17→ RΣ−R1is non-empty.

Then, we have to distinguish between the following cases:

2.5 Piggyback a Common Relay Message

1. The maximal additional relay rate Rmax

R(RΣ)is attained at [R⋆

1(RΣ), R⋆

2(RΣ)]

if the intersection point between fRΣ(R1)and R⋆

BC(RΣ)is achievable, i.e.

[R⋆

1(RΣ), R⋆

2(RΣ)] ∈ RBR(α). Then with (2.60a) the total sum-rate can be expressed

Rtot(RΣ) = Rmax

R(RΣ)+RΣ= (1−α) log 2

RΣ

2(1−α)|h1h2|ˆγ−2

RΣ

2(1−α).(2.64a)

2. Otherwise, if we have [R⋆

1(RΣ), R⋆

2(RΣ)] /∈ RBR(α)for a feasible bidirectional sum-

rate RΣ, we have to characterize R⊲

1and R⊳

1from the intersection of RBR(α)with the

graph GfRΣ.

a) If RΣ<min{(1 −α)R†, αR−→

2R}, we have R⊲

1= 0 and R⋆

1(RΣ)< R⊲

1. There-

fore, the maximal additional relay rate Rmax

R(RΣ)is attained at R⋆

BC(RΣ) =

[0, RΣ]∈ RBR(α). Then with (2.60b) the total sum-rate can be expressed as

Rtot(RΣ) = (1 −α) log |h2|2

|h1|22

RΣ

1−αˆγ|h1|2−2

RΣ

1−α.(2.64b)

b) If RΣ>min{αR−→

2R,2αR−→

2R −(1 −α)R†} ⇔ RΣ> αR−→

2R ∧R⋆

1(RΣ)<

RΣ−αR−→

2R, the line fRΣ(R1)intersects the individual MAC rate constraint

αR−→

2R so that we have R⊲

1=RΣ−αR−→

2R and R⋆

1(RΣ)< R⊲

1. Accordingly,

the maximal additional relay rate Rmax

R(RΣ)is attained at [R⊲

1, RΣ−R⊲

1] =

[RΣ−αR−→

2R, αR−→

2R]∈ RBR(α). Then with (2.60b) the total sum-rate can be

expressed as

Rtot(RΣ) = (1 −α) log |h2|2

|h1|22α

1−αR−→

2R ˆγ|h1|2−2α

1−αR−→

2R .(2.64c)

c) If RΣ<min{−(1 −α)R†, αR−→

1R}, we have R⊳

1=RΣand R⋆

1(RΣ)>

R⊳

1. Therefore, the maximum additional relay rate Rmax

R(RΣ)is attained at

R⋆

BC(RΣ) = [RΣ,0] ∈ RBR(α). Then with (2.60c) the total sum-rate can

be expressed as

Rtot(RΣ) = (1 −α) log |h1|2

|h2|22

RΣ

1−αˆγ|h2|2−2

RΣ

1−α.(2.64d)

d) If RΣ>min{αR−→

1R,2αR−→

1R + (1 −α)R†} ⇔ RΣ> αR−→

1R ∧R⋆

2(RΣ)<

RΣ−αR−→

1R, the line fRΣ(R1)intersects the individual MAC rate constraint αR−→

so that we have R⊳

1=RΣ−αR−→

1R and R⋆

1(RΣ)> R⊳

1. Accordingly, the maximal

additional relay rate Rmax

R(RΣ)is attained at [R⊳

1, RΣ−R⊳

1] = [αR−→

1R, RΣ−

αR−→

1R]∈ RBR(α). Then with (2.60c) the total sum-rate can be expressed as

Rtot(RΣ) = (1 −α) log |h1|2

|h2|22α

1−αR−→

1R ˆγ|h2|2−2α

1−αR−→

1R .(2.64e)

2 Bidirectional Relay Communication using Superposition Encoding

Note that if we have R†<0⇔ |h1|2<|h2|2, the case 2. a) is not possible. If the other

cases occur depend on the scenario specific parameters. Similarly, we can a priori exclude

the case 2.c) if we have R†>0⇔|h1|2<|h2|2.

We have now specified the total sum-rate for any feasible bidirectional sum-rate. There-

with, we can define section-wise a function Rtot(RΣ)on the range [0, Rmax

Σ], where Rmax

denotes the maximal feasible bidirectional sum-rate. This allows us to characterize in the

next theorem the bidirectional rate pair [Ropt

1, Ropt

2]where the total sum-rate maximum R∗

tot

according to (2.62) is achieved.

Theorem 2.48. For a given time division parameter α∈[0,1] the total sum-rate maximum

R∗

tot of the rate region RPiggy(α)is attained at the bidirectional rate pair [Ropt

1, Ropt

2]where

the function R⋆

BC(RΣ)intersects the boundary of the bidirectional rate region RBR(α).

Proof. The proof can be found in Appendix 2.8.31.

From the proof of the theorem and from the (2.64c) and (2.64e) we see that for the cases

2. b) and 2. d) the total sum-rate remains constant for a range of large bidirectional sum-

rates, Ropt

1+Ropt

2≤RΣ≤Rmax

Σ. This means that we can interchange additional relay

rate for bidirectional sum-rate without losing total sum-rate optimality. The next corollary

records this observation and states that such an interchange between additional relay rate and

bidirectional rate exists not only for the total sum-rate maximum.

Corollary 2.49. For any sum-rate RΣ≤Ropt

1+Ropt

2with the rate pair [R⋆

1, R⋆

2] :=

[R⋆

1(RΣ), R⋆

2(RΣ)] = [1

2RΣ−1−α

2R†,1

2RΣ+1−α

2R†]the rate tuples

[R1, R2, RR] = [R⋆

1+R①, R⋆

2, RR(R⋆

1+R①, R⋆

2)]

with R①≥0and [R⋆

1+R①, R⋆

2]∈ RBR(α)and

[R1, R2, RR] = [R⋆

1, R⋆

2+R②, RR(R⋆

1, R⋆

2+R②)]

with R②≥0and [R⋆

1, R⋆

2+R②]∈ RBR(α)achieve the same total sum-rate.

Proof. The proof can be found in Appendix 2.8.32.

The corollary characterizes the bidirectional rate pairs where the total sum-rate remains con-

stant. For a bidirectional sum-rate RΣ≤Ropt

1+Ropt

2there exists the rate pair [R⋆

1, R⋆

which determines the rate pairs where the total sum-rate remains constant. Then the corollary

states that we can either increase R1or R2. For a constant total sum-rate this means that if

we increase one bidirectional rate, we equally decrease the relay multicast rate. Thereby, the

interchange range follows from the condition that the bidirectional rate pairs [R⋆

1+R①, R⋆

2.5 Piggyback a Common Relay Message

or [R⋆

1, R⋆

2+R②]have to be within RBR(α)with appropriate R①, R②≥0. However, if

we increase both bidirectional rates we know from Theorem 2.48 that we increase the total

sum-rate. At the total sum-rate optimum R∗

tot there may no interchange or interchange with

only one bidirectional rate possible. For a better understanding we briefly discuss the low

and high bidirectional sum-rate cases next.

For low sum-rates RΣwhere we have R⋆

1(RΣ)+R⋆

2(RΣ)<|R†|we have a negative R⋆

1(RΣ)

iff |h1|2<|h2|2and a negative R⋆

2(RΣ)iff |h2|2<|h1|2. If R⋆

1(RΣ)<0we can only

interchange relay rate in the direction R1since for any rate R②≥0the rate pair [R⋆

1, R⋆

R②]/∈ RBR(α). Furthermore, for a rate pair within the bidirectional rate region we need

R①≥R⋆

1. Similarly, we can interchange relay rate in the direction R2only if R⋆

2(RΣ)is

negative.

If we have [R⋆

1(Rmax

Σ), R⋆

2(Rmax

Σ)] ∈ RBR(α), the rate pair achieves the bidirectional sum-

rate maximum R⋆

1(Rmax

Σ) + R⋆

2(Rmax

Σ) = Rmax

Σ, which means that the rate pair is also bidi-

rectional sum-rate optimal. We see that this is the case iff the intersection point of the func-

tion RBC(RΣ)with the boundary of RBR(α)coincides with the intersection of RBC(RΣ)

with the boundary of (1 −α)RBC or the sum-rate constraint of the MAC region αRMAC.

Since the rate pair characterizes a sum-rate optimum of a BC rate region it follows that in

this case no interchange without loosing total sum-rate optimality is possible. In the next

subsection we elaborate the combinatorial discussion on the previous results.

2.5.3 Combinatorial Discussion and Working Examples

In this section we present some combinatorial discussions of the achievable rate region using

the previous results. We start with an explicit study of the equal time division case. After

that we fix the additional relay rate and look for achievable bidirectional rate pairs. Then we

discuss the total sum-rate maximum with respect to the time division parameter α. Finally,

we compare the achievable rate regions with the achievable rate region of a simple TDMA

approach which realizes the same routing tasks.

Equal Time Division

For the equal time division case we can explicitly express the total sum-rate optimal rate pairs

and describe the possible rate trade-offs. The explicit characterization of the combinatorial

results leads to simple arithmetical problems. Because of the symmetry we do the discussion

for case |h1|2≥ |h2|2. We first specify the combinatorial cases for which we characterize the

total sum-rate optimality ranges. If we interchange the indices 1 and 2 an analog discussion

gives the results for the case |h1|2≤ |h2|2.

2 Bidirectional Relay Communication using Superposition Encoding

0 0.5 1

0.2

0.4

0.6

0.8

rate R1 [bit/channel use]

rate R2 [bit/channel use]

γ1 =1.5

γ2 =0.4

0 0.5 1

0.2

0.4

0.6

0.8

rate R1 [bit/channel use]

rate R2 [bit/channel use]

γ1 =2

γ2 =1.3

0 0.5 1

0.2

0.4

0.6

0.8

rate R1 [bit/channel use]

rate R2 [bit/channel use]

γ1 =0.4

γ2 =3.8

0 0.5 1

0.2

0.4

0.6

0.8

rate R1 [bit/channel use]

rate R2 [bit/channel use]

γ1 =1

γ2 =2.5

Figure 2.13: For |h1|2≥ |h2|2we can distinguish between four possible cases where

R⋆

BC(γR)(solid line) intersects the boundary of 1

/2CMAC (dashed line). Ad-

ditionally, we depicted broadcast rate regions of selected relay powers (dashed-

dotted line). The cases are specified by the intersection points ,,♦, and ◦,

where the function R⋆

BC(γR)intersects the two individual rate constraints and

the sum-rate constraint of the MAC region. For the intersection of the first in-

dividual rate constraint we additionally have to differentiate whether in the BC

phase the unidirectional communication is sum-rate optimal. This is the case

if the rate pair is not achievable in the MAC phase, i.e. /∈1

/2CMAC. Fi-

nally, the rate pairs ▽and △denote the vertices of the MAC region 1

/2CMAC

respectively.

2.5 Piggyback a Common Relay Message

For |h1|2≥ |h2|2we have R†≥0so that there are four cases where and how the function

R⋆

BC(RΣ)intersects the boundary of the MAC rate region 1

2CMAC. The four cases depend

on the channel realizations and power constraints of nodes 1 and 2. In Figure 2.13 we depict

the cases for four representative examples with some characteristic rate pairs. In accordance

to that we can characterize the four cases as follow:

UL (upper left): The function R⋆

BC(RΣ)intersects the individual rate constraint 1

2R−→

2R for

the case (2.63c) ⇔1

2R−→

2R <1

2R†⇔γ2>|h1|2

|h2|2(1

|h2|2−1

|h1|2).

UR (upper right): The function R⋆

BC(RΣ)intersects the individual rate constraint 1

2R−→

for the case (2.63a) ⇔1

2R−→

2R ≥1

2R†∧1

2R−→

2R −1

2R†≤1

2R2Σ

1=1

2(RMAC

Σ−R−→

2R).

LL (lower left): The function R⋆

BC(RΣ)intersects the individual rate constraint 1

2R−→

1R for

the case (2.63a) ⇔1

2R−→

2R ≥1

2R†∧1

2R−→

1R +1

2R†≤1

2RΣ1

2=1

2(RMAC

Σ−R−→

1R).

LR (lower right): The function R⋆

BC(RΣ)intersects the sum-rate constraint 1

2RMAC

Σfor

the case (2.63a) ⇔1

2R−→

2R ≥1

2R†∧RMAC

Σ<min{2R−→

2R −R†,2R−→

1R +R†}.

For the identification of the rate pair where the total sum-rate optimum R∗

tot is achieved ac-

cording Theorem 2.48 we need to know the intersection point between R⋆

BC(RΣ)and the

boundary of RBReq. If the intersection point with the boundary of the MAC rate region

2CMAC is within 1

2RBC it coincides with the intersection point with the boundary of RBReq.

Otherwise, we find the intersection point on the boundary of the rate region 1

2RBC, which is

given by [1

2R⋆

−→

R2,1

2R⋆

−→

R1]according to Proposition 2.2. As a consequence, we have to charac-

terize for each of the previous cases the condition that the intersection point is within the BC

rate region. This is the case iff

UL: 1

2R−→

R1(1) ≥1

2R−→

2R ⇔γR≥|h2|2

|h1|2γ2,

UR: 1

2R−→

R1(1 −β⋆)≥1

2R−→

2R ⇔γR≥γ2R

Rwith γ2R

R=γ1|h1|2

(1+γ2|h2|2)|h2|2+γ2|h2|2

|h1|2,

LL: 1

2R−→

R2(β⋆)≥1

2R−→

1R ⇔γR≥γ1R

Rwith γ1R

R=γ2|h2|2

(1+γ1|h1|2)|h1|2+γ1|h1|2

|h2|2,

LR: 1

2RBC

Σ≥1

2RMAC

Σ⇔γR≥γΣ

Rwith γΣ

R= 2q1+γ1|h1|2+γ2|h2|2

|h1|2|h2|2−1

|h1|2−1

|h2|2,

where we solved the conditions for γR. To sum up the discussion on the intersection point

for the equal time division case with |h1|2≥ |h2|2we now explicitly characterize the total

sum-rate optimal rate pair according to Theorem 2.48 using the previous case definitions as

2 Bidirectional Relay Communication using Superposition Encoding

follows

[Ropt

1, Ropt

2] =











[0,1

2R−→

R1(1)],UL with γR<|h2|2

|h1|2γ2,(2.65a)

[0,1

2R−→

2R],UL with γR≥|h2|2

|h1|2γ2,(2.65b)

2R−→

2R −1

2R†,1

2R−→

2R],UR with γR≥γ12

R,(2.65c)

2R−→

1R,1

2R−→

2R +1

2R†],LL with γR≥γ21

R,(2.65d)

2RMAC

Σ−1

4R†,1

2RMAC

Σ+1

4R†],LR with γR≥γΣ

R,(2.65e)

2R−→

R2(β⋆),1

2R−→

R1(β⋆)],else. (2.65f)

Then the total sum-rate optimal relay rate RRresults from (2.58) accordingly. Since UL is

the case where [R⋆

1(RΣ), R⋆

2(RΣ)] /∈1

2CMAC for any RΣ, for the cases (2.65a) and (2.65b)

we have a total sum-rate optimal bidirectional rate pair with R1= 0. This means that for the

case UL there exists a unidirectional rate allocation which is total sum-rate optimal.

According to Corollary 2.49 in some cases we can increase one bidirectional rate by de-

creasing the relay rate. Therefore, we need that [Ropt

1+R①, Ropt

2]for any R①>0or

[Ropt

1, Ropt

2+R②]for any R②>0is within RBReq. This is impossible if the intersection

point is on the boundary of the BC rate region, which corresponds to the cases (2.65a) and

(2.65f), or the intersection is specified by the sum-rate constraint 1

/2RMAC

Σof the MAC rate

region 1

/2CMAC, which corresponds to the case (2.65e). In all these cases we cannot further

increase the bidirectional sum-rate, which obviously implies that we cannot increase one

bidirectional rate. As a consequence it follows that in the cases (2.65a), (2.65e), and (2.65f)

the total sum-rate rate pair is unique so that we can say that no interchange is possible without

loosing total sum-rate optimality.

Moreover, for the cases (2.65a) and (2.65f) the intersection point is on the boundary of the

BC rate region so that we have β1+β2= 1 and it follows that the additional relay rate RRis

equal to zero. This means that if γRis small enough the intersection point with the boundary

of RBReq coincides with the intersection of the boundary of the BC rate region 1

2RBC. For

that reason we can say that at low signal-to-noise ratios γRit is total sum-rate optimal not to

have an additional relay multicast communication. Furthermore, if the signal-to-noise ratio

γRis that small that the case (2.65a) applies, we can only achieve the total sum-rate with

rates R1= 0 and RR= 0. This means in this case it is also necessary for total sum-rate

optimality to have a relay communication in one direction only.

In the case (2.65b) the unidirectional relay communication is total sum-rate optimal as well,

however we have a positive additional relay rate RR. From Corollary 2.49 we know that for

any interchange rate R①≥0with [R①,1

2R−→

2R]∈ RBReq we can interchange relay rate RR=

RR(R①,1

2R−→

2R)with bidirectional rate R1=R①without loosing total sum-rate optimality,

which means that we have R①+1

2R−→

2R +RR(R①,1

2R−→

2R) = R∗

tot =const. Obviously, we

2.5 Piggyback a Common Relay Message

need that RR(R①,1

2R−→

2R)≥0is satisfied. The maximum relay rate for the total sum-rate

optimum is attained at the bidirectional rate pair of the intersection point [0,1

2R−→

2R]. Since we

have RR(R①,1

2R−→

2R) = RR(0,1

2R−→

2R)−R①it follows that we can increase R①at most up

to the maximum relay rate RR(0,1

2R−→

2R). Furthermore, since [R①,1

2R−→

2R]has to be feasible,

we can increase R1at most up to maximum rate 1

2R2Σ

1, if ν2Σ ∈ RBC. To sum this up,

for the case (2.65b) the total sum-rate optimality range with an interchange rate R①can be

expressed as

[R1, R2,RR] = [R①,1

2R−→

2R, RR(R①,1

2R−→

2R)]

R①∈0,min{RR(0,1

2R−→

2R),1

2R2Σ

1}.(2.66)

In the case (2.65c) with R①≥0we can interchange relay rate RR=RR(1

2R−→

2R −1

2R†+

R①,1

2R−→

2R)with bidirectional rate R1=1

2R−→

2R −1

2R†+R①as well. As before we need a

non-negative relay rate RR(1

2R−→

2R −1

2R†+R①,1

2R−→

2R)≥0and R1≤1

2R2Σ

1. Then with

similar arguments it follows that for the case (2.65c) the total sum-rate optimality range with

an interchange rate R①can be expressed as

[R1, R2, RR] = 1

2R−→

2R −1

2R†+R①,1

2R−→

2R, RR(1

2R−→

2R −1

2R†+R①,1

2R−→

2R)

R①∈0,min{RR(1

2R−→

2R −1

2R†,1

2R−→

2R),1

2R2Σ

1−1

2R−→

2R +1

2R†}.(2.67)

Finally, in the case (2.65d) with R②≥0we can interchange relay rate RR=

RR(1

2R−→

1R,1

2R−→

1R +1

2R†+R②)with bidirectional rate R2=1

2R−→

1R +1

2R†+R②. For

this case we need again RR(1

2R−→

1R,1

2R−→

1R +1

2R†+R②)≥0and R2≤1

2RΣ1

2. Similarly as

before, it follows that for the case (2.65d) the total sum-rate optimality range with arbitrary

interchange rate R②can be expressed as

[R1, R2, RR] = 1

2R−→

1R,1

2R−→

1R +1

2R†+R②, RR(1

2R−→

1R,1

2R−→

1R +1

2R†+R②)

R②∈0,min{RR(1

2R−→

1R,1

2R−→

1R +1

2R†),1

2RΣ1

2−1

2R−→

2R −1

2R†}.(2.68)

In Figure 2.14 we illustrate the total sum-rate Rtot(R1, R2)with respect to the bidirectional

rate pairs [R1, R2]∈ RBReq for the four different cases. The result of Corollary 2.49 is

shown by some contour lines. It shows that the total sum-rate optimum is attained at the

intersection of R⋆

BC(RΣ)with the boundary of RBReq. Furthermore, we see that if the inter-

section point lies on one individual rate constraint of the MAC region, there is an optimality

range where we can interchange relay rate with bidirectional rate in one direction without

loosing total sum-rate optimality.

2 Bidirectional Relay Communication using Superposition Encoding

0.5

0.1

0.2

0.4

0.6

γ1 =1.5

γ2 =0.4

γR =1.4

Rtot

0.5

γ1 =2

γ2 =1.3

γR =3.3

Rtot

0.1

0.2

0.5

γ1 =0.4

γ2 =3.8

γR =3

Rtot

0.5

0.2

0.4

0.6

γ1 =1

γ2 =2.5

γR =1.7

Rtot

Figure 2.14: For |h1|2≥ |h2|2and equal time division, α=1

/2, we see the total sum-rate

Rtot(R1, R2)for [R1, R2]∈ RBReq for the four possible cases where R⋆

BC(γR)

intersects the boundary of 1

2CMAC. The total sum-rate contour lines illustrate

the interchange property between the additional relay rate and one bidirectional

rate. Moreover, we see that the total sum-rate optimum is unique in the lower

right figure only. In the upper figures we can interchange RRwith R1and in the

lower left figure we can interchange RRwith R2without loosing total sum-rate

optimality.

2.5 Piggyback a Common Relay Message

Desired Relay Rate

The explicit knowledge about the optimal rate pairs and the optimality ranges allows us to

change the point of view and characterize for a desired relay rate RRthe bidirectional rate

pair [R1(RR), R2(RR)] ∈ RBR(α)which maximizes the bidirectional sum-rate,

RΣ(RR) := max

[R1,R2]∈RBR(α)

RR(R1,R2)=RR

R1+R2.

This means we have RΣ(RR) = R1(RR) + R2(RR). Again we present the results for

the case |h1|2≥ |h2|2. For the case |h2|2≥ |h1|2the optimal bidirectional rate pair in a

permuted order, i.e. [R2(RR), R1(RR)], follows immediately by interchanging the indices 1

and 2. However, this time we do the combinatorial discussion for an arbitrary but fixed time

division parameter α∈[0,1].

With increasing bidirectional rates the parameter βRdecreases. It follows that the relay

rate RRdecreases with increasing bidirectional rates so that the desired relay rate is obvi-

ously bounded by the maximal possible relay rate Rmax

R:= RR(0,0). At high relay rates,

where we have RR≥RR(Ropt

1, Ropt

2), the optimal bidirectional rate pairs are specified

by the function R⋆

BC(RΣ). For lower desired relay rates in some cases we can addition-

ally apply Corollary 2.49 to increase the bidirectional rate sum-rate of the intersection point

[Ropt

1, Ropt

2]while the total sum-rate remains constant. In the following we discuss the cases

in more detail.

First we look at the case (2.63c) where the BC sum-rate optimal rate pair is on the R2-axis.

This means that we consider relay rates larger than RR(0,min{(1 −α)R†, αR−→

2R}). Since

we have R1(RR) = 0 for such relay rates, it follows that RΣ=R2(RR)≤min{(1 −

α)R†, αR−→

2R}. We get R2(RR)if we solve (2.60b) with R⊲

1= 0 for RΣ. This gives us

[R1(RR), R2(RR)] = h0,(1 −α) log |h1|2(ˆγ−21

1−αRR

|h2|2)i.

Next we consider relay rates for the case (2.63a) where the optimal bidirectional rate pair

[R⋆

1(RΣ), R⋆

2(RΣ)] is achievable, i.e. [R⋆

1(RΣ), R⋆

2(RΣ)] ∈ RBR(α). This means we con-

sider relay rates RR∈[RR(Ropt

1, Ropt

2), RR(0,(1 −α)R†)). This interval may be empty if

the BC sum-rate optimal rate pair is on the R2-axis, i.e. Ropt

1= 0. If the interval in non-

empty it follows from (2.63a) that [R1(RR), R2(RR)] = [1

2RΣ−1−α

2R†,1

2RΣ+1−α

2R†]

where we get the sum-rate RΣif we solve (2.60a) for RΣ. This gives us

[R1(RR), R2(RR)] = h(1 −α) log |h2|2ˆγ

2(1−α)+1 ,(1 −α) log |h1|2ˆγ

2(1−α)+1 i.

2 Bidirectional Relay Communication using Superposition Encoding

For a large signal-to-noise ratio γRwe have (1 −α)[R⋆

−→

R2, R⋆

−→

R1]/∈ RBR(α)so that

[Ropt

1, Ropt

2]is on the boundary of αCMAC. In some cases we can apply Corollary 2.49

to increase the bidirectional sum-rate for a certain range of relay rate until the MAC sum-

rate constraint restricts the interchange of relay rate with bidirectional rate. According to

Corollary 2.49 an interchange is possible with one bidirectional rate only. For the following

case study we again distinguish between the cases where R⋆

BC(RΣ)intersects the boundary

of αCMAC.

For the case (2.63c), where R⋆

BC(RΣ)intersects the boundary of RBR(α)at the rate pair

[Ropt

1, Ropt

2] = [0, αR−→

2R], we have αR−→

2R <(1−α)R†. By the way if this holds, the previous

considered case does not occur, i.e. [R⋆

1(RΣ), R⋆

2(RΣ)] /∈ RBR(α)for all RΣ. In this case

we are looking for relay rates which are achievable with bidirectional rate pairs [R1, αR−→

2R]∈

RBR(α)where the total sum-rate remains constant according to the interchange property of

Corollary 2.49. The largest relay rate is achieved at R1= 0 and the smallest relay rate is

either 0or it is attained at the vertex αν2Σ. This means that we consider relay rates RR∈

[max{0, RR(0, αR−→

2R)−αR2Σ

1}, RR(0, αR−→

2R)). Then the corresponding bidirectional rate

pair with the largest bidirectional sum-rate for those relay rates RRis given by

[R1(RR), R2(RR)] = [RR(0, αR−→

2R)−RR, αR−→

2R].

If we have RR(0, αR−→

2R)> αR2Σ

1, the interchange of relay rate and bidirectional rate is

limited by the sum-rate constraint of the MAC phase. Therefore, for smaller relay rates

RR< RR(0, αR−→

2R)−αR2Σ

1at most a maximal bidirectional sum-rate αRMAC

Σis achievable,

e.g. by the vertex αν2Σ.

Next we consider the case αR−→

2R ≥(1 −α)R†so that the function R⋆

BC(RΣ)given by

(2.63a) intersects the boundary of RBR(α)at the rate pair [Ropt

1, Ropt

2] = [αR−→

2R −(1 −

α)R†, αR−→

2R]. In this case we look for non-negative relay rates which are achievable with

bidirectional rate pairs given by [Ropt

1, Ropt

2]and the vertex αν2Σ where the total sum-rate

remains constant according to the interchange property of Corollary 2.49. This means that we

consider relay rates RR∈[max{0, RR(Ropt

1, Ropt

2)−(αR2Σ

1−Ropt

1)}, RR(Ropt

1, Ropt

2)).

Then the corresponding bidirectional rate pair with the largest bidirectional sum-rate for

those relay rates RRis given by

[R1(RR), R2(RR)] = [Ropt

1+ (RR(Ropt

1, Ropt

2)−RR), Ropt

2].

If we have RR(Ropt

1, Ropt

2)> αR2Σ

1−Ropt

1, the interchange of relay rate and bidirectional

rate is again limited by the sum-rate constraint of the MAC phase. It similarly follows that for

smaller relay rates RR< RR(Ropt

1, Ropt

2)−(αR2Σ

1−Ropt

1)at most a maximal bidirectional

sum-rate αRMAC

Σis achievable, e.g. by the vertex αν2Σ.

Similar arguments apply for the case where the function R⋆

BC(RΣ)given by (2.63a) inter-

sects the boundary of RBR(α)at a rate pair [Ropt

1, Ropt

2] = [αR−→

1R, αR−→

1R + (1 −α)R†]. In

2.5 Piggyback a Common Relay Message

this case we look for non-negative relay rates which are achievable with bidirectional rate

pairs [Ropt

1, Ropt

2]and the vertex ανΣ1 where the total sum-rate remains constant accord-

ing to the interchange property of Corollary 2.49. This means that we consider relay rates

RR∈[max{0, RR(Ropt

1, Ropt

2)−(αRΣ1

2−Ropt

2)}, RR(Ropt

1, Ropt

2)). Then the correspond-

ing bidirectional rate pair with the largest bidirectional sum-rate for those relay rates RRis

given by

[R1(RR), R2(RR)] = [Ropt

1, Ropt

2+ (RR(Ropt

1, Ropt

2)−RR)].

If we have RR(Ropt

1, Ropt

2)> αRΣ1

2−Ropt

2, the interchange of relay rate and bidirectional

rate is again limited by the sum-rate constraint of the MAC phase. It similarly follows that for

smaller relay rates RR< RR(Ropt

1, Ropt

2)−(αRΣ1

2−Ropt

2)at most a maximal bidirectional

sum-rate αRMAC

Σis achievable, e.g. by the vertex ανΣ1.

Finally, if R⋆

BC(RΣ)intersects the boundary of RBR(α)defined by the MAC sum-rate con-

straint, i.e. we have [Ropt

1, Ropt

2] = [R⋆

1(RMAC

Σ), R⋆

2(RMAC

Σ)], there is no interchange of re-

lay rate and bidirectional rate possible. It follows that for relay rates RR< RR(Ropt

1, Ropt

bidirectional rate pairs with a sum-rate at most αRMAC

Σare achievable, e.g. by the rate pair

[R⋆

1(RMAC

Σ), R⋆

2(RMAC

Σ)].

Total Sum-Rate Maximum Discussion

Theorem 2.48 characterizes for any time division parameter α∈[0,1] the rate-pair where

the total sum-rate maximum is attained. We are now interested how the total sum-rate max-

imum depends on the time division parameter α. For that goal, in this subsection we need

to extend the notation of some symbols by an additional argument to specify the time divi-

sion. Accordingly, let the function R⋆

BC(RΣ, α)denote the sum-rate optimal rate pair of a

broadcast region (1 −α)RBC(RΣ

1−α)with sum-rate equal RΣaccording to Proposition 2.47.

Similarly, let R∗

tot(α)and [Ropt

1(α), Ropt

2(α)] denote the total sum-rate maximum and the

corresponding rate pair where the maximum is attained for a time division parameter αac-

cording to Theorem 2.48. Since the discussion of the total sum-rate with respect to the time

division parameter αleads again to a case study, we restrict our discussion without loss of

generality to the case where we have |h1|2≥ |h2|2.

First we distinguish between the cases β⋆<0and β⋆≥0according to Proposition

2.2. If β⋆<0we always have [R⋆

−→

R2, R⋆

−→

R1] = [0, R−→

R1(1)] from which it follows that

the function R⋆

BC(RΣ, α)always intersects the boundary of RBR(α)on the R2-axis, i.e.

[Ropt

1(α), Ropt

2(α)] = [0,min{αR−→

2R,(1 −α)R−→

R1(1)}]. This means that it is always sum-

rate optimal to have a relay communication only in one direction. If we have a time di-

vision parameter α≥α0:= R−→

R1(1)

R−→

2R+R−→

R1(1) , we have αR−→

2R ≥(1 −α)R−→

R1(1) which

means that the total sum-rate maximum achieving rate pair is on the boundary of the BC

2 Bidirectional Relay Communication using Superposition Encoding

rate region (1 −α)RBC. This is equivalent to the fact that the broadcast sum-rate op-

timal rate pair (1 −α)[R⋆

−→

R2, R⋆

−→

R1]is within αCMAC. Accordingly, if α < α0we have

(1 −α)[R⋆

−→

R2, R⋆

−→

R1]/∈αCMAC so that the total sum-rate optimal rate pair is given by

[Ropt

1(α), Ropt

2(α)] = [0, αR−→

2R].

On the other hand, if β⋆≥0holds, we have [R⋆

−→

R2, R⋆

−→

R1] = [R−→

R2(β⋆), R−→

R1(1 −β⋆)] so

that it is sum-rate optimal to communicate in both directions. With an increasing time di-

vision parameter α∈[0,1] we enlarge the MAC rate region αCMAC and reduce the BC

rate region (1 −α)RBC. Accordingly, for some αthe broadcast sum-rate optimal rate

pair (1 −α)[R−→

R2(β⋆), R−→

R1(1 −β⋆)] is within αCMAC. As before, we denote the cor-

responding time division parameter by α0. For the explicit characterization we need to

distinguish between the boundary sections of αCMAC where we find a rate pair which is

equal to the sum-rate optimal rate pair of (1 −α)RBC. For α∈[0,1] the rate pairs

(1 −α)[R−→

R2(β⋆), R−→

R1(1 −β⋆)] specify a line from the origin in the first quadrant with

an angle

φBC := arctan R−→

R1(1 −β⋆)

R−→

R2(β⋆)= arctan log(1

2ˆγ|h1|2)

log(1

2ˆγ|h2|2).(2.69a)

Accordingly, for α∈[0,1] the vertices ανΣ1 = [αR−→

1R, α(RMAC

Σ−R−→

1R)] and αν2Σ =

[α(RMAC

Σ−R−→

2R), αR−→

2R]specify lines from the origin in the first quadrant with angles

φΣ1 : = arctan RMAC

Σ−R−→

R−→

,(2.69b)

φ2Σ : = arctan R−→

RMAC

Σ−R−→

(2.69c)

respectively. The growing MAC rate regions αCMAC and shrinking BC rate regions (1 −

α)RBC with increasing αand the corresponding lines of the vertices and broadcast sum-rate

optimal rate pairs are illustrated in Figure 2.15. Since the angles φΣ1, φ2Σ,and φBC do not

depend on the time division parameter, we can use the angles to identify the section of the

MAC boundary where we find the broadcast sum-rate optimal rate pair for α0.

Thus, if we have φBC ≥φ2Σ there exists a time division parameter α0where the sum-rate

optimal rate pair (1 −α0)[R⋆

−→

R2, R⋆

−→

R1]of the BC rate region (1 −α0)RBC is on the section

of the boundary of α0CMAC which is characterized by the individual rate constraint R−→

2R, i.e.

(1 −α0)[R⋆

−→

R2, R⋆

−→

R1]∈ {α0[R1, R−→

2R]∈

+: 0 ≤R1≤RMAC

Σ−R−→

2R}. Accordingly, if

we solve (1 −α0)R⋆

−→

R1 =α0R−→

2R for α0we get the time division parameter α0=R⋆

−→

R⋆

−→

R1+R−→

for the case φBC ≥φ2Σ.

Similarly, if we have φBC ≤φΣ1 there exists a time division parameter α0where the sum-

rate optimal rate pair (1−α0)[R⋆

−→

R2, R⋆

−→

R1]is on the section of the boundary of αCMAC which is

2.5 Piggyback a Common Relay Message

0.3 0.6 0.9 1.2 1.5

0.3

0.6

0.9

1.2

1.5

rate R1 [bit/channel use]

rate R2 [bit/channel use]

α =0.2

α =0.4

α =0.6

α =0.8

α =1

φΣ1

φ2Σ

|h1|2 =1

|h2|2 =0.7

γ1 =1

γ2 =2.5

0.3 0.6 0.9 1.2 1.5

0.3

0.6

0.9

1.2

1.5

rate R1 [bit/channel use]

rate R2 [bit/channel use]

φBC

α =0.8

α =0.6

α =0.4

α =0.2

α =0

|h1|2 =1

|h2|2 =0.7

γR =1.7

Figure 2.15: The left and right figures show the increasing MAC rate regions αCMAC and

decreasing BC rate regions (1 −α)RBC for increasing α. Furthermore, we de-

picted the lines characterized by the vertices and the broadcast sum-rate optimal

rate pairs. In the right figure the function R⋆

BC(RΣ, α)for α= 0,0.2,0.4,0.6,

and 0.8(dashed lines) are additionally depicted.

characterized by the individual rate constraint R−→

1R. Accordingly, if we solve (1−α0)R⋆

−→

R2 =

α0R−→

1R for α0we get α0=R⋆

−→

R⋆

−→

R2+R−→

for the case φBC ≤φΣ1.

Finally, if we have φ2Σ > φBC > φΣ1 there exists a time division parameter α0where

the sum-rate optimal rate pair of (1 −α0)RBC is also sum-rate optimal with respect to the

MAC rate region α0CMAC, i.e. the rate pair is on the section of the boundary of α0CMAC

which is characterized by the sum-rate constraint RMAC

Σ. Accordingly, if we solve (1 −

α0)(R⋆

−→

R2 +R⋆

−→

R1) = (1 −α0)RBC

Σ=α0RMAC

Σfor α0we get α0=RBC

RBC

Σ+RMAC

for the case

φ2Σ > φBC > φΣ1.

In the next proposition we summarize the previous considerations and characterize the total

sum-rate optimal rate pair according to Theorem 2.48. Since φBC is greater than π

2for

β⋆<0, we can merge the case β⋆<0with the case β⋆≥0and φBC > φ2Σ according to

the definition of [R⋆

−→

R2, R⋆

−→

R1]in Proposition 2.2.

Proposition 2.50. The sum-rate optimal rate pair (1−α)[R⋆

−→

R2, R⋆

−→

R1]of the broadcast region

(1 −α)RBC is within the scaled MAC rate region αCMAC and therefore within RBR(α)if

2 Bidirectional Relay Communication using Superposition Encoding

we have a time division parameter α∈ ABC := [α0,1] with

α0:= 









R⋆

−→

R⋆

−→

R1+R−→

,if φBC ≥φ2Σ,

RBC

Σ+RMAC

,if φ2Σ > φBC > φΣ1,

R⋆

−→

R⋆

−→

R2+R−→

,if φΣ1 ≥φBC.

(2.70)

Then the BC sum-rate optimal pair is also sum-rate optimal with respect to the total sum-rate

R∗

tot(α). This means for α≥α0that we have

[Ropt

1(α), Ropt

2(α)] = (1 −α)[R⋆

−→

R2, R⋆

−→

R1](2.71)

with total sum-rate R∗

tot(α) = (1 −α)RBC

Σ= (1 −α)(R⋆

−→

R1 +R⋆

−→

R2).

The total sum-rate R∗

tot(α)follows from the fact that for α≥α0the total sum-rate optimal

rate pair is equal to the sum-rate optimal rate pair of (1 −α)RBC so that βRand therefore

the additional relay rate RRare equal to zero.

The set ABC = [α0,1] denotes the set of time division parameters αwhere the sum-rate

optimal rate pair of the BC phase (1 −α)[R⋆

−→

R2, R⋆

−→

R1]is achievable in the MAC phase. Ac-

cordingly, for α∈ A0:= [0, α0)the rate pair (1 −α)[R⋆

−→

R1, R⋆

−→

R2]is not achievable in the

MAC phase. Then it follows from Theorem 2.48 that the total sum-rate optimal rate pair

is attained at the rate pair where R⋆

BC(RΣ, α)intersects the boundary of RBR(α). As in

Section 2.5.3 we can distinguish between four possible intersections for |h1|2≥ |h2|2. For

the following characterization of the cases we first ignore the feasibility of the time division

parameters. In the proposition after it we take the feasibility into account.

1. For small time division parameters α < α1:= R†

R†+R−→

the function R⋆

BC(RΣ, α)

intersects the boundary of αCMAC on the R2-axis, this means at the rate pair [0, αR−→

2R].

Accordingly, α1characterizes the time division parameter where we have α1R−→

2R =

(1 −α1)R†.

2. For time division parameters α1≤α < α2:= R†

2R−→

2R−RMAC

Σ+R†the function

R⋆

BC(RΣ, α)intersects the boundary of αCMAC characterized by the individual rate

constraint αR−→

2R, this means at a rate pair in αR−→

2R ={α[R1, R−→

2R] : 0 ≤R1< R2Σ

1}.

Thereby, α2characterizes the time division parameter where R⋆

BC(RΣ, α)intersects

the boundary of αCMAC in its vertex αν2Σ. Accordingly, we get α2if we solve

R⋆

1(α2RMAC

Σ) = α2

2RMAC

Σ−1−α2

2R†=α2(RMAC

Σ−R−→

2R)for α2.

2.5 Piggyback a Common Relay Message

3. For time division parameters α2≤α≤α3:= R†

RMAC

Σ−2R−→

1R+R†the function

R⋆

BC(RΣ, α)intersects the boundary of αCMAC characterized by the sum-rate con-

straint αRMAC

Σ, this means at a rate pair in αRMAC

Σ={α[R1, RMAC

Σ−R1] :

R2Σ

1≤R1≤R−→

1R}. Thereby, α3characterizes the time division parameter where

R⋆

BC(RΣ, α)intersects the boundary of αCMAC in its vertex ανΣ1. Accordingly, we

get α3if we solve R⋆

1(α3RMAC

Σ) = α3

2RMAC

Σ−1−α3

2R†=α3R−→

1R for α3.

4. Finally, for large time division parameters α > α3the function R⋆

BC(RΣ, α)intersects

the boundary of αCMAC characterized by the individual rate constraint αR−→

1R, this

means at a rate pair in αR−→

1R ={α[R−→

1R, R2]:0≤R2< RΣ1

2}.

Since for |h1|2≥ |h2|2we have R†≥0. Then the parameters α2or α3are negative if we

have R−→

2R +R†< R2Σ

1or RΣ1

2< R−→

1R −R†respectively. Such time division parameters are

obviously not feasible. The feasible time division parameters are given by the set A0. Then

in the following proposition we show that the feasible cases are again specified by the angles

φBC,φΣ1, and φ2Σ.

Proposition 2.51. With A0:= [0, α0)let A1:= [0, α1)∩ A0,A2:= [α1, α2)∩ A0,

A3:= [α2, α3)∩ A0, and A4:= A0\A1∪ A2∪ A3denote the sets of feasible time

division parameters of the previous cases. Then we have

A16=∅,A2,A3,A4=∅,if φBC >π

A1,A26=∅,A3,A4=∅,if π

2≥φBC > φ2Σ,

A1,A2,A36=∅,A4=∅,if φ2Σ ≥φBC > φΣ1,

A1,A2,A3,A46=∅,if φΣ1 ≥φBC.

For a time division parameter α∈ A0the total sum-rate optimal rate pair

[Ropt

1(α), Ropt

2(α)] on the boundary of the scaled MAC region αCMAC is given by

[Ropt

1(α), Ropt

2(α)] = 









[0, αR−→

2R],if α∈ A1,

[αR−→

2R −(1 −α)R†, αR−→

2R],if α∈ A2,

[α

2RMAC

Σ−1−α

2R†,α

2RMAC

Σ+1−α

2R†],if α∈ A3,

[αR−→

1R, αR−→

2R + (1 −α)R†],if α∈ A4,

so that we have the total sum-rate maximum

R∗

tot(α) = 









(1 −α) log |h2|2

|h1|22α

1−αR−→

2R ˆγ|h1|2−2α

1−αR−→

2R ,if α∈ A1∪A2,

(1 −α) log 2α

2(1−α)RMAC

Σ|h1h2|ˆγ−2α

2(1−α)RMAC

Σ,if α∈ A3,

(1 −α) log |h1|2

|h2|22α

1−αR−→

1R ˆγ|h2|2−2α

1−αR−→

1R ,if α∈ A4.

2 Bidirectional Relay Communication using Superposition Encoding

0.3

rate R1 [bit/channel use]

rate R2 [bit/channel use]

|h1|2 =1

|h2|2 =0.5

γ1 =1

γ2 =1.5

γR =0.3

0.3

rate R1 [bit/channel use]

rate R2 [bit/channel use]

|h1|2 =1

|h2|2 =0.7

γ1 =2

γ2 =1.8

γR =0.7

0.3 0.6

0.3

0.6

rate R1 [bit/channel use]

rate R2 [bit/channel use]

|h1|2 =1

|h2|2 =0.7

γ1 =1.3

γ2 =2

γR =2.5

0.3 0.6

0.3

0.6

rate R1 [bit/channel use]

rate R2 [bit/channel use]

|h1|2 =1

|h2|2 =0.8

γ1 =0.7

γ2 =2.5

γR =3.5

Figure 2.16: The figures show parametrized curves of the total sum-rate optimal rate pairs

[Ropt

1(α), Ropt

2(α)] for α∈[0,1] according to the four different cases for

|h1|2≥ |h2|2. For the upper left figure we have φBC >π

2, for the up-

per right figure we have π

2≥φBC > φ2Σ, for the lower left figure we have

φ2Σ ≥φBC > φΣ1, and for the lower right figure we have φΣ1 ≥φBC. Fur-

thermore, we depicted in parts αCMAC,αRBC, and R⋆

BC(RΣ, α)for α= 0.2

(solid), α= 0.4(dashed-dotted), α= 0.6(dashed), and α= 0.8(dotted)

with gray lines. Additionally, we marked some characteristic optimal rate pairs

specified by (:α= 0.2), (:α= 0.4), (△:α= 0.6), (▽:α= 0.8), (◦:

α=α0), (:α=α1), (♦:α=α2), and (✩:α=α3).

2.5 Piggyback a Common Relay Message

Proof. The proof can be found in Appendix 2.8.33.

The parametrized curves in Figure 2.16 illustrate the optimal rate pairs [Ropt

1(α), Ropt

2(α)]

for time division parameters α∈[0,1] for the four possible cases for |h1|2≥ |h2|2. For

comparison between the cases the optimal rate pairs and in parts the broadcast and multiple

access rate regions for time division parameters α= 0.2,0.4,0.6, and 0.8are depicted.

Furthermore, the optimal rate pairs of the characteristic time division parameters α0,α1,α2,

and α3are marked. In more detail, in the upper left we have the case where R†> R−→

2R so

that we have ABC ∪A1= [0,1]. It follows that we have only two sections. For α≥α0we

have [Ropt

1(α), Ropt

2(α)] = [0,(1−α)R−→

R1(1)] ∈ RBR(α)according to Proposition 2.50 and

for α < α0we have [Ropt

1(α), Ropt

2(α)] = [0, αR−→

2R]∈ RBR(α)according to Proposition

2.51. In the upper right, we have A26=∅so that we additionally have the case α1≤α < α0

where the optimal rate pair is given by [Ropt

1(α), Ropt

2(α)] = [αR−→

2R −(1 −α)R†, αR−→

2R]

according to Proposition 2.51. Similarly in the lower left and lower right we first additionally

have A36=∅and then A46=∅as well.

The Propositions 2.50 and 2.51 characterize for each time division parameter the correspond-

ing total sum-rate R∗

tot(α). Since the parametrized curves [Ropt

1(α), Ropt

2(α)] are continuous

for each case, the section-wise defined function R∗

tot(α)is continuous as well. For α∈ ABC

we know from Proposition 2.50 that with increasing αthe total sum-rate Rtot(α)decreases

linearly. This means that for this section the total sum-rate is always maximized at α0. But

for time division parameters α∈ A0a closed form discussion of the total sum-rate is no

longer possible, c.f. Remark 2.4. Moreover, from numerical examples there is unfortunately

no clear behavior observable. In Figure 2.17 we depicted some representative numerical ex-

amples which should give some idea how the total sum-rate behaves with respect to the time

division parameter α. In the upper left we see that the total sum-rate maximum is attained

at α0. In the upper right the total sum-rate is attained at some αwithin the interval (0, α0)

and in the lower left and right it is optimal with respect to the total sum-rate to have a relay

multicast only.

From Theorem 2.48 we know that for a given time division parameter αthe total sum-rate is

maximized at the bidirectional rate pair R⋆

BC(RΣ, α)∈ RBR(α)of the largest possible bidi-

rectional sum-rate RΣ. This means that we first allocate relay power to support an efficient

bidirectional rate pair then the remaining transmit power can be used for the additional relay

communication. But since we a priori fixed the time division parameter we do not factor in

the costs of transmitting the messages to the relay node in the previous MAC phase. In the

lower left and right figure, we see that the resource costs (large fraction of time) of the MAC

phase are higher than the spectral efficiency of the bidirectional broadcast phase so that it

is total sum-rate optimal to have no bidirectional communication. However, since it was

the aim to have two routing tasks, it is questionable if such a solution is desired. Neverthe-

less, it shows that for a spectrally efficient bidirectional relaying we need enough resources

2 Bidirectional Relay Communication using Superposition Encoding

(transmit power at nodes 1 and 2) in the MAC phase as well.

Comparison with a TDMA protocol

To demonstrate the efficiency of the proposed protocol we compare the achievable rate re-

gions of the piggyback on bidirectional relaying protocol with equal time division with a

straightforward Round Robin TDMA approach. Thereby, we assume a TDMA protocol with

five time slots of equal duration. For a fair comparison, we adapt the power constraints in the

TDMA protocol so that both protocols consume the same amount of energy. Accordingly,

in the first time slot node 1 transmits its message to the relay node with transmit power 5

2P1.

Similarly, in the second time slot node 2 transmits its message to the relay node with transmit

power 5

2P2. In the third and fourth time slot the relay node separately forwards the messages

with power 5

6PR. With the same power the relay message is transmitted to nodes 1 and 2 in

the last time slot. This give us the achievable rates for a comparable TDMA protocol

RTDMA

1=1

5log 1 + min{5

2γ1|h1|2,5

6γR|h2|2},

RTDMA

2=1

5log 1 + min{5

2γ2|h2|2,5

6γR|h1|2},

RTDMA

R=1

5log 1 + 5

6γRmin{|h1|2,|h2|2}.

Figure 2.18 illustrates for two scenarios the dominance of the bidirectional relaying protocol

with an additional relay multicast for the equal time division case. For both cases we see that

the achievable rate region RPiggy(1

/2)is significantly larger in any direction than the achiev-

able rate region of the TDMA protocol (cuboid), which means that the entire achievable rate

region of the TDMA protocol (cuboid) is within the piggyback rate region. Accordingly,

the piggyback approach achieves higher weighted total sum-rates, which is important for

service adapted network operations. Moreover, it illustrates that the joint resource allocation

allows new advantageous rate trade-offs so that we conclude that the synergy from the joint

resource allocation of two routing tasks, multicast and bidirectional relaying, improves the

performance significantly.

Finally, for completeness we present the achievable rate regions of bidirectional relaying

with an additional relay multicast and optimal time division Ropt

Piggy of two representative

examples in Figure 2.19. Since the optimal decoding order holds for any fixed time division

parameter, it is also optimal to decode the relay message first for the optimal time division.

Unfortunately, a closed form discussion as for the fixed time division case is not possible, cf.

Remark 2.4, but we can use the equivalent characterization of the bidirectional achievable

rate region RBRopt according to Theorem 2.12 to illustrate the rate region Ropt

Piggy.

100

2.5 Piggyback a Common Relay Message

0 1

0.1

0.2

0.3

0.4

rate Rtot [bit/channel use]

α0

|h1|2 =1

|h2|2 =0.5

γ1 =1

γ2 =1.5

γR =0.3

0 1

0.1

0.2

0.3

0.4

0.5

0.6

rate Rtot [bit/channel use]

α1α0

|h1|2 =1

|h2|2 =0.7

γ1 =2

γ2 =1.8

γR =0.7

0 1

0.5

1.5

rate Rtot [bit/channel use]

α1α2α0

|h1|2 =1

|h2|2 =0.7

γ1 =1.3

γ2 =2

γR =2.5

0 1

0.5

1.5

rate Rtot [bit/channel use]

α1α2α3α0

|h1|2 =1

|h2|2 =0.8

γ1 =0.7

γ2 =2.5

γR =3.5

Figure 2.17: The figures show the total sum-rate maximum R∗

tot(α)with respect to the time

division parameter α∈[0,1] for the four cases for |h1|2≥ |h2|2. For the

upper left figure we have φBC >π

2, for the upper right figure we have π

2≥

φBC > φ2Σ, for the lower left figure we have φ2Σ ≥φBC > φΣ1, and for

the lower right figure we have φΣ1 ≥φBC. (∗) specifies the maximum for

each case. Additionally, we marked the total sum-rates of some characteristic

time division parameters (◦:α=α0), (:α=α1), (♦:α=α2), and (✩:

α=α3).

101

2 Bidirectional Relay Communication using Superposition Encoding

0.2

0.4

0.6

00.2 0.4 0.6

0.1

0.2

0.3

0.4

0.5

0.6

γ1 =0.8

γ2 =1.2

γR =1.3

|h1|2 =1

|h2|2 =0.8

0.2

0.4

0.6

00.2 0.4 0.6

0.1

0.2

0.3

0.4

0.5

γ1 =1

γ2 =0.9

γR =1.3

|h1|2 =1

|h2|2 =0.7

Figure 2.18: Comparison of achievable rate region RPiggy(1

/2)of piggyback a common relay

message on bidirectional relaying with equal time division and a TDMA proto-

col realizing the same routing task (cuboid) with the same energy consumption.

In the left figure we have at the bullet (•) a unique total sum-rate optimum; in

the right figure we can interchange rate R1with RRalong the bold line between

the the bullets without loosing total sum-rate optimality. On the contour lines

for equal additional relay rates the bidirectional sum-rate optimum is marked

by a cross (×).

0.2

0.4

0.6

00.2 0.4 0.6

0.2

0.4

0.6

0.8

γ1 =0.8

γ2 =1.2

γR =1.3

|h1|2 =1

|h2|2 =0.8

0.2

0.4

0.6

00.2 0.4 0.6

0.2

0.4

0.6

0.8

γ1 =1

γ2 =0.9

γR =1.3

|h1|2 =1

|h2|2 =0.7

Figure 2.19: Achievable rate region Ropt

Piggy of piggyback a common message on bidirec-

tional relaying with optimal time division with the same system parameters

as in Figure 2.18. The contour lines denote achievable rate pairs for fixed

βR=1

15 ,2

15 ,...,1. The contrasting colors characterize the three sections ac-

cording to Therorem 2.12.

102

2.6 Extension to Multi-Antenna Bidirectional Relaying

In this section we extend the bidirectional relaying protocol with superposition encoding

to nodes equipped with multiple antennas. The leads to a multiple-input multiple-output

(MIMO) wireless system as depicted in Figure 2.20. The Gaussian vector channel was stud-

ied first by Tsybakov in [Tsy65] in 1965. The work of Teletar, Foschini and Gans have shown

in [Tel99] and [FG98] that multiple-antenna systems have the ability to reach higher trans-

mission rates than single-antenna systems. The basic idea is to exploit the spatial dimension

offered by the MIMO channel. With an appropriate processing at the transmitter and receiver

we obtain multiple subchannels, also called the channel eigenmodes, which allow data mul-

tiplexing on several substreams and lead therefore to a linear increase in capacity. On the

other hand the spatial degrees of freedom can be utilized to transmit the data signal over

multiple fading paths to increase the robustness of the transmission through diversity. Both

works have sparked an active and flourishing research area. For a comprehensive discussion

on recent results on MIMO wireless communications with large reference lists we refer to

the book [BCC+07] and the overview paper [GJJV03].

In [WZHM05] Wang, Zhang, and Host-Madsen can calculate upper and lower bounds for

the Gaussian MIMO relay channel with a full-duplex relay node by simplifying the opti-

mization over the joint distribution of the source and relay node input. Then they present

sufficient conditions when the upper and lower bound on the ergodic capacity meet in the

case of Rayleigh fading. But the spatial MIMO gains can be also obtained using multiple

relay nodes. Wittneben and Rankov propose in [WR03] the relay assisted MIMO link where

amplify-and-forward relays ensure the rich scattering requirement of MIMO systems. In

[BNOP06] Bölcskei, Nabar, Oyman, and Paulraj derive scaling laws of the network capacity

with different processing strategies at the relay nodes depending on its channel state infor-

mation. The source and destination are equipped with Mantennas so that the capacity scales

linearly with M/2in the limit of the number of half-duplex relay nodes. Moreover, there

are some other interesting works which address other important MIMO relaying aspects like

coverage extension in cellular networks [HKW06], linear processing for a multiuser MIMO

system [TCHC06], etc.

NRN2

R 21

(a) MAC phase

NRN2

R 21

(b) BC phase

Figure 2.20: A three-node network, where the relay node, node 1, and node 2 are equipped

with NR,N1, and N2antennas respectively.

103

2 Bidirectional Relay Communication using Superposition Encoding

As before, we consider a two-phase bidirectional decode-and-forward relaying protocol

without feedback. In the multiple access phase the nodes 1 and 2 transmit their messages

with rate R1and R2to the relay node. We assume that the relay node decodes both messages

successfully if the rates are within the capacity region of the MAC phase. In the succeeding

broadcast phase the relay separately re-encodes the messages and transmits the superposi-

tion of the codewords. In [OB07c] we have studied the achievable rate region of such a

multi-antenna bidirectional relaying protocol. Recently, in [HKE+07] Hammerström et. al.

compare the sum-rate performance of superposition encoding and XOR precoding at the re-

lay node based on the concept of [WCK05] and [LJS05]. Since the XOR coding approach

is more power efficient, it leads to a better performance. We expect another improvement if

one uses the coding idea of Chapter 3 for the Gaussian MIMO bidirectional relay channel.

The MIMO system model is similar to the SISO case considered in Section 2.1. However,

in this section we require perfect channel knowledge available at each node. As before,

we assume a perfectly synchronized three-node network as depicted in Figure 2.20. The

relay node, nodes 1 and 2 are equipped with NR,N1, and N2antennas respectively. Let

H1∈

NR×N1and H2∈

NR×N2characterize the discrete-time time-invariant multi-

plicative MIMO channels between the relay node and nodes 1 and 2 respectively. For sim-

plicity we assume reciprocal channels. The rank r1= rank(H1)≤min{N1, NR}and

r2= rank(H2)≤min{NR, N2}denote the number of spatial degree of the channels,

i.e. each non-zero eigenmode of the channel can support a data stream. The mean transmit

powers of each node are restricted by power constraints Pk,k∈ {1,2, R}, respectively.

Furthermore, the reception at each antenna of every node is distorted by independent addi-

tive white Gaussian noise n1,n2,and nRwith equal covariance matrices σ2I=ρ−1I. A

generalization to individual and more general noise covariance matrices is straightforward.

Then after symbol-rate sampling the system equation in the MAC phase for time instants m

is given by

yR[m] = H1x1[m] + H2x2[m] + nR[m],

where yR[m]∈

NRdenotes the received signal vector at the relay node, x1[m]∈

and x2[m]∈

N2denote the transmit signal vector of nodes 1 and 2, and nR[m]∈

denotes the additive noise vector. Similarly, after symbol-rate sampling the system equation

in the BC phase for time instants mis given by

yk[m] = HT

kxR[m] + nk[m], k = 1,2,

where yk[m]∈

Nkdenotes the received signal vector at node k,k= 1,2,xR[m]∈

denotes the transmit signal vector of the relay node, and nk[m]∈

Nkdenotes the additive

noise vector at node k,k= 1,2.

In the following we first briefly review the optimal transmit strategies of the Gaussian MIMO

channel with CSI at the transmitter and receiver. In the next sections we summarize the

104

2.6 Extension to Multi-Antenna Bidirectional Relaying

known capacity results on the Gaussian MIMO multiple access channel and present the opti-

mal transmit strategy for the Gaussian MIMO channels in the BC phase using superposition

encoding. Due to the complicated structure of the rate regions in the MIMO case the com-

binatoric cannot be given in closed form. Nevertheless, with the results presented here it

is possible to characterize the optimal transmit strategies for any rate pair using standard

methods of convex optimization. Furthermore, we will see that the achievable rate in the

high power regime scales linearly with the minimum of the fraction of time weighted spatial

degree of both MIMO channels.

MIMO Gaussian Channel

We first look at a Nt×Nrmultiple-input multiple-output (MIMO) wireless point-to-point

connection where we consider a linear time-invariant multiplicative channel with additive

white Gaussian noise. We assume that the transmitter and receiver have perfect channel

knowledge for encoding and decoding. In the following, we briefly reproduce the deriva-

tion of the information capacity of the MIMO Gaussian channel as done in [Tel99, TV05].

Therefore, the vector-valued linear input-output relation can be expressed as follows

Y=HX +N

where H∈

Nr×Ntdenotes the channel matrix and the random vectors X∈

Nt×1,

Y∈

Nr×1, and N∼ CN(0, σ2INr)denote the input, the output, and the complex Gaus-

sian noise of the channel respectively. As in the scalar case we assume that the noise is

independent of the input. Furthermore, we require that the input has to fulfill a mean trans-

mit power constraint

{XHX} ≤ P.

Similar to the scalar case we can define the mutual information for the vector-valued channel

as follows

I(X;Y) := h(Y)−h(Y|X) = h(Y)−h(N)

where h(Y)and h(Y|X)denote the differential entropy and conditional differential en-

tropy. The differential entropy for a continuous random vector Ydefined on the support SY

with density fY(y)is defined as

h(Y) := −ZSY

fY(y) log fY(y)dy.

The conditional differential entropy for continuous random vectors (Y,X)defined on SY×

SXwith joint and conditional densities fY,X(y,x)and fY|X(x|y)is defined as

h(Y|X) := −ZSY×SX

fY,X(y,x) log fY|X(y|x)d(y,x).

105

2 Bidirectional Relay Communication using Superposition Encoding

Then Lemma 2 in [Tel99] shows that a circularly symmetric complex Gaussian14 distribu-

tion X∼ CN(0,Q)maximizes the differential entropy with h(X) = log det(πeQ)of

complex random vectors with zero mean and covariance matrix Q. Since the noise vector

Nis independent of the the input vector Xwe have

{Y Y H}=H

{XXH}HH+

{NNH}=HQHH+σ2INr,

where Q:=

{XXH}. It follows that a circularly symmetric complex Gaussian distributed

input vector X∼ CN(0,Q)maximizes the entropy h(Y)as well.

The information capacity Cis again defined as the maximal mutual information over all input

distributions. With a circularly symmetric complex Gaussian input vector X∼ CN(0,Q)

we have

C= max

fX(x):tr(Q)≤PI(X;Y) = max

fX(x):tr(Q)≤Ph(Y)−log det(πeINrσ2)

= max

tr(Q)≤Plog det πe(HQHH+INrσ2)−log det(πeINrσ2)

= max

tr(Q)≤Plog det(INr+1

σ2HQHH).

Therefore, it remains to find the optimal covariance matrix Q∗which satisfies the power

constraint tr(Q∗)≤P.

This convex optimization problem of finding the optimal covariance matrix can be explicitly

solved using the Lagrangian method [Tel99, BV04]. This leads to the so called water-filling

solution. In the following we briefly describe the procedure to calculate the optimal transmit

covariance matrix Q∗according to the water-filling solution. Let

HHH=VΣVH,with Σ:= diag(λ1, λ2, . . . , λNt)

and unitary matrix Vbe the eigenvalue decomposition of HHHwith eigenvalues sorted in

decreasing order, i.e. λ1≥ ··· ≥ λNt≥0. Then

Q∗=Vdiag(ξ1, ξ2,...,ξNt)VH

with eigenvalues

ξn:= (max ν−σ2

λn,0,if λn6= 0,

0,if λn= 0,

14The probability density function of a complex valued circularly symmetric (or spherically invariant) Gaussian

random vector X∼ CN(µ,Q)of dimension Nwith mean µand covariance matrix Qis given by

fX(x) = 1

πNdet(Q)e−(x−µ)HQ−1(x−µ),(2.72)

where additionally

{(X−µ)(X−µ)T}=0∈

N×Nholds. This circularity condition ensures that

uncorrelated jointly Gaussian random variables are also independent[KSH00, App 3.C],[Doo53, Sec. II.3].

106

2.6 Extension to Multi-Antenna Bidirectional Relaying

where the so called water-level νfollows from the transmit power constraint

n=1

ξn= tr(Q∗),

which is obviously fulfilled with equality for the optimal transmit covariance matrix. For the

water-filling procedure we introduce the short notation Q(HHH, P)so that for the MIMO

Gaussian channel the optimal transmit covariance is given by

Q∗=Q(HHH, P).

With the optimal transmit covariance matrix Q∗the information capacity for the MIMO

channel is given as follows

C= log det(INr+1

σ2HQ∗HH) =

n=1

log 1 + λnξn

σ2.

It depends on the rank of the channel matrix Hand the transmit power constraint how many

eigenmodes of the channel are used. In the so called low-power regime beamforming is

optimal. This means that only the strongest eigenmode of the channel is used. Furthermore,

the case where all eigenmodes of the channel are used is called the high-power regime.

2.6.1 MIMO Multiple Access Phase

In the first phase, nodes 1 and 2 transmit their information for each other to the relay node.

The encoding and the decoding are performed as in the classical discrete-time Gaussian

MIMO MAC channel. In principle the Gaussian MIMO MAC follows from the classical

result from Ahlswede [Ahl71a] and Liao [Lia72], but many researchers have explicitly stud-

ied various interesting aspects [CV93, YRBC01, BJ02, VJG03, BW06] to mention only a

few. Since we consider memoryless channels, it is sufficient to consider single letters only.

Therefore, the vector-valued linear input-output relation can be expressed as follows

YR=H1X1+H2X2+NR

where the continuous random vector YR∈

NRdenotes the output at the relay node,

the continuous random vector Xk∈

NRdenotes the input from node k,k= 1,2, and

NR∈

NRdenotes the complex additive white Gaussian noise distributed according to

CN(0,Iσ2)at the relay node.

107

2 Bidirectional Relay Communication using Superposition Encoding

Again, we assume uncorrelated sources. Then the multiple access capacity region CMIMO

MAC is

given by the convex hull of the set of rate pairs which satisfy the following individual and

sum-rate constraints

R1≤I(X1;YR|X2) = h(YR|X2)−h(YR|X1,X2) = h(H1X1+NR)−h(NR),

R2≤I(X2;YR|X1) = h(YR|X1)−h(YR|X1,X2) = h(H2X2+NR)−h(NR),

R1+R2≤I(X1,X2;YR) = h(YR)−h(YR|X1,X2) = h(YR)−h(NR)

for some vector input distributions fX1(x1)and fX2(x2)which satisfy the power constraint

{XH

kXk} ≤ Pk,k= 1,2.

Since we assume that the inputs X1,X2and noise NRare pairwise independent, the co-

variance matrix of the output YRcan be expressed as

{YRYH

R}=H1

{X1XH

|{z }

:=Q1

1+H2

{X2XH

|{z }}

:=Q2

{NRNH

|{z }

=σ2INR

with tr(Qk)≤Pk,k= 1,2, to fulfill the power constraints. Since a circularly symmet-

ric complex Gaussian vector distributed according to CN(0,H1Q1HH

1+H2Q2HH

σ2INR)maximizes the differential entropy of a random vector Ywith covariance matrix

H1Q1HH

1+H2Q2HH

2+σ2INRwe have

h(H1X1+NR)≤log det πe(H1Q1HH

1+σ2INR),

h(H2X2+NR)≤log det πe(H2Q2HH

2+σ2INR),

h(Y)≤log det πe(H1Q1HH

1+H2Q2HH

2+σ2INR)

with equalities if we have circularly symmetric complex Gaussian inputs Xk∼ CN(0,Qk),

k= 1,2. It follows that the optimal input for given covariance matrix is distributed ac-

cording to zero mean circularly symmetric complex Gaussian distribution. With h(NR) =

log det πeσ2INRand the optimal input distributions Xk∼ CN(0,Qk),k= 1,2, the

individual and sum-rate constraints can be expressed as

R−→

1R(Q1) := log det INR+1

σ2H1Q1HH

1,(2.73a)

R−→

2R(Q2) := log det INR+1

σ2H2Q2HH

2,(2.73b)

RMAC

Σ(Q1,Q2) := log det INR+1

σ2H1Q1HH

1+1

σ2H2Q2HH

2.(2.73c)

Accordingly, for covariance matrices Q1and Q2we can achieve rate pairs within the re-

gion

CMIMO

MAC (Q1,Q2) := n[R1, R2]∈

+:R1+R2≤RMAC

Σ(Q1,Q2),

R1≤R−→

1R(Q1), R2≤R−→

2R(Q2)o.

108

2.6 Extension to Multi-Antenna Bidirectional Relaying

For each pair of covariance matrices the achievable region is described by a pentagon. As

in the scalar case, the encoding methods to achieve the vertices can be deduced from the

single-user MIMO Gaussian channel. For the decoding the relay node applies successive

interference cancellation so that each vertex corresponds to a certain decoding order. In

more detail, we can achieve the vertex νΣ1(Q1,Q2) := [R−→

1R(Q1), RΣ1

2(Q1,Q2)] with

RΣ1

2(Q1,Q2) :=RMAC

Σ(Q1,Q2)−R−→

1R(Q1)

= log det σ2INR+H1Q1HH

1+H2Q2HH

2

det σ2INR+H1Q1HH

1,(2.74)

if node 1 encodes as in single-user Gaussian MIMO channel to achieve the rate R−→

1R(Q1).

It therefore uses a code with codewords distributed according to a circularly symmetric

complex Gaussian distribution CN(0,Q1). Node 2 encodes as in the single-user Gaussian

MIMO channel to achieve the rate RΣ1

2(Q1,Q2). The relay node first decodes the message

of node 2 with high probability while it considers the interference of node 1 as additional

Gaussian noise with covariance H1Q1HH

1. After the message of node 2 is determined, the

relay can cancel the interference H2X2from the receive vector YRso that it can decode

the message of node 1 with high probability.

Similarly, we can achieve the vertex ν2Σ(Q1,Q2) := [R2Σ

1(Q1,Q2), R−→

2R(Q2)] with

R2Σ

1(Q1,Q2) :=RMAC

Σ(Q1,Q2)−R−→

2R(Q2)

= log det σ2INR+H1Q1HH

1+H2Q2HH

2

det σ2INR+H2Q2HH

2

by interchanging the decoding order and the corresponding encoding at nodes 1 and 2. In

accordance, let πk,k= 1,2, denote the decoding order where node kis decoded last.

However, in contrast to the scalar case there are in general no input distributions so that

the sum-rate and individual rate constraints are tight simultaneously. This means that the

pair of covariance matrices which maximize the individual rate constraints in general do not

maximize the sum-rate constraint as well. Therefore, the capacity region of the Gaussian

MIMO-MAC is given by the convex hull of the union over all circularly symmetric complex

Gaussian inputs that satisfy the power constraints, tr(Qk)≤Pk,

CMIMO

MAC := co 

[

tr(Qk)≤Pk,k=1,2CMIMO

MAC (Q1,Q2)

.

This means that the capacity region CMIMO

MAC is the convex hull of the union over all pentagons

corresponding to a pair of covariance matrices. In general we will have two curved sections

109

2 Bidirectional Relay Communication using Superposition Encoding

on the boundary, each corresponds to a certain decoding order. Thereby, each rate pair on

the curved section is achieved by an individual set of covariance matrices. The sum-rate

optimal section between the curved parts can only be achieved with time-sharing. Finally,

we will have sections where we achieve the single-user capacities. It can be easily seen by

contradiction that for any rate pair on the boundary the power constraints are satisfied with

equality. In [WB04] the discussion of the boundary is presented for an arbitrary number

of users, we briefly look at the sections of the boundary for our two user multiple access

channel.

Since the boundary of a convex set can be characterized by rate pairs that maximize the

weighted rate sum we consider the optimization problem

RMIMO

MAC (q) = arg max

[R1,R2]∈CMIMO

MAC

q1R1+q2R2(2.75)

for a weight vector q∈

+\ {0}. There are multiple solutions if we have q1=q2,

q1= 0, q2>0, or q1>0, q2= 0, which gets clearer from the following.

First, we consider weight vectors qwith q1>0and q2= 0. This means, we want to

maximize the unidirectional rate R1which then corresponds to the single-user capacity.

Therefore, we have to maximize (2.73a) according to the water-filling procedure given in

the previous section which gives us

R∗

−→

1R := max

R∈CMAC

R1= max

Q1: tr(Q1)≤P1

R−→

1R(Q1) = R−→

1R (Q∗

using the water-filling solution Q∗

1:= Q(HH

1H1, P1). To achieve this we have to apply the

decoding order π1which allows node 2 to transmit with rates RΣ1

2(Q∗

1,Q2). Then with the

Cholesky-decomposition C1CH

1:= (INR+1

σ2H1Q∗

1HH

1)−1we can define an equivalent

channel ˜

H2:= C1H2so that we can rewrite (2.74) as follows

RΣ1

2(Q∗

1,Q2) = log det INR+1

σ2˜

H2Q2˜

2.

Then the water-filling solution Q(˜

2˜

H2, P2)maximizes the rate RΣ1

2(Q∗

1,Q2). Both to-

gether gives us the first characteristic rate pair and section where node 1 achieves its single-

user capacity

E1:= hR∗

−→

1R , RΣ1

2Q∗

1,Q(˜

2˜

H2, P2)i,

E1:= n[R∗

−→

1R, R2] : 0 ≤R2≤RΣ1

2Q∗

1,Q(˜

2˜

H2, P2)o.

The same procedure applies to weight vectors qwith q1= 0 and q2>0so that we get

the corresponding characteristic rate pair and section where node 2 achieves its single-user

110

2.6 Extension to Multi-Antenna Bidirectional Relaying

capacity

E2:= hRΣ1

2Q∗

2,Q(˜

1˜

H1, P1), R∗

−→

2Ri,

E2:= n[R1, R∗

−→

2R]:0≤R1≤R2Σ

1Q(˜

1˜

H1, P1),Q∗

2o

with the maximal unidirectional rate R∗

−→

2R := R−→

2R (Q∗

2)achieved with the water-filling so-

lutions Q∗

2:= Q(HH

2H2, P2)and Q(˜

1˜

H1, P1)with the equivalent channel ˜

H1:=

C2H1using the Cholesky-decomposition C2CH

2:= (INR+1

σ2H2Q∗

2HH

2)−1.

For q1≥q2it is shown in [BW06] that the decoding order π1is optimal. For that reason we

can rewrite (2.75) as follows

RMIMO

MAC (q) = arg max

[R1,R2]∈CMIMO

MAC

q1R−→

1R(Q1) + q2RΣ1

2(Q1,Q2)

= arg max

[R1,R2]∈CMIMO

MAC

(q1−q2)R−→

1R(Q1) + q2RMAC

Σ(Q1,Q2).

With q1≥q2the objective is a sum of concave functions and therefore the optimization

problem is convex [BW06]. Similarly, it is shown that the decoding order π2is optimal for

weight vectors with q2≥q1.

In the following we look at the Lagrangian function of the optimization problem for the case

q1≥q2,

L(Q1,Q2,Ψ1,Ψ2, µ1, µ2) = −(q1−q2)R−→

1R(Q1)−q2RMAC

Σ(Q1,Q2)

−2

k=1

µkPk−tr(Qk)−2

k=1

tr(QkΨk).

Similarly to [WB04] the optimal transmit strategies Q1and Q2for a rate pair RMIMO

MAC (q)

are uniquely characterized by the Karush-Kuhn-Tucker (KKT) conditions

µ1IN1+Ψ1=HH

1(q2−q1)σ2INR+H1Q1HH

1−1

−q2σ2INR+H1Q1HH

1+H2Q2HH

2−1H1,(2.76a)

µ2IN2+Ψ2=−q2HH

2σ2INR+H1Q1HH

1+H2Q2HH

2−1H2,(2.76b)

tr(QkΨk) = 0, µk(Pk−tr(Qk)) = 0, k = 1,2,(2.76c)

Ψk0Nk, µk≥0, k = 1,2,(2.76d)

Qk0Nk, Pk≥tr(Qk), k = 1,2,(2.76e)

with complementary slackness, dual, and primal conditions (2.76c), (2.76d), and (2.76e)

respectively. Since the optimization problem is convex, efficient algorithms, like the interior-

point method, exist to calculate the optimal covariance matrices [BV04]. This allows us

111

2 Bidirectional Relay Communication using Superposition Encoding

to specify the curved section on the boundary corresponding to the decoding order π1as

follows

D1:= RMIMO

MAC (q) : q∈

+, q1≥q2>0.

Furthermore, we denote by D1the sum-rate maximum RMIMO

MAC ([1,1]) ∈ D1.

Similarly, in the case q1≤q2the decoding order π2is optimal. Then the Lagrange function

and KKT conditions follow by interchanging the indices 1 and 2 so that the set

D2:= RMIMO

MAC (q) : q∈

+, q1≤q2>0

denotes the curved section on the boundary corresponding to the decoding order π2. Accord-

ingly, let D2=RMIMO

MAC ([1,1]) ∈ D2denote its sum-rate maximum.

The last section on the boundary is given by the connecting line between the sum-rate max-

ima D1and D2,

T:= R:R=τD1+ (1 −τ)D2, τ ∈[0,1]

and can be reached by time-sharing between the corresponding strategies of D1and D2

only. If one is interested in sum-rate optimal rate pairs only, one can also use an iterative

water-filling algorithm presented in [YRBC01].

In Figure 2.21 we depicted the characteristic sections and rate pairs of the capacity region

CMIMO

MAC for an example where each node is equipped with two antennas.

2.6.2 MIMO Broadcast Phase

As in the scalar case in the BC phase the relay forwards the messages m1and m2which it has

received in the previous MIMO MAC phase. Since we consider in this chapter superposition

encoding, the messages are separately encoded and afterwards the superposition of both

is transmitted. Again, since we assume a memoryless channel, it is sufficient to consider

vector-valued single-letters only. To this end, let W1∈

NRdenote a single-letters of the

codeword for message m1for node 2 and W2∈

NRa single-letter of the codeword m2

for node 1 so that the input of the relay node in the broadcast channel is given by the sum

XR=W1+W2.

We assume that the messages m1and m2are independent so that the random vectors W1

and W2are independent as well. From this we get the vector-valued output at node k,

k= 1,2, as follows

Yk=HH

kXR+Nk=HH

kW1+HH

kW2+Nk, k = 1,2.

112

2.6 Extension to Multi-Antenna Bidirectional Relaying

As in the scalar case, the receiving nodes perform interference cancellation so that we essen-

tially have two interference-free MIMO Gaussian channels between the relay node and the

nodes 1 and 2. It follows that the achievable rates for transmitting the unknown messages

have to fulfill the constraints

R2≤I(XR;Y1|W1) = I(W2;HH

1W2+N1),

R1≤I(XR;Y2|W2) = I(W1;HH

2W1+N2)

for some input distributions fW1(w1)and fW2(w2)which satisfy the power constraint

{XH

RXR}=

{WH

1W1}+

{WH

2W2} ≤ PR.

Obviously, the mutual informations are only coupled by the relay power distribution. As in

the scalar case, we denote by β1and β2the proportion of relay transmit power PRspend for

the codewords W1and W2respectively. Then the simplex

B={[β1, β2]∈[0,1] ×[0,1] : β1+β2≤1}

characterizes the set of feasible relay power distributions that satisfy the relay transmit power

constraint.

For any feasible relay power distribution [β1, β2]∈ B we have two separated MIMO Gaus-

sian channels so that circularly symmetric complex Gaussian distributed inputs

W1∼ CN(0,QR,1(β1)) with QR,1(β1) := Q(H2HH

2, β1PR),

W2∼ CN(0,QR,2(β2)) with QR,2(β2) := Q(H1HH

1, β2PR)

maximize the mutual informations. Thereby, the optimal covariance matrices QR,1(β1)and

QR,2(β2)are given by the water-filling solution as described for the Gaussian MIMO chan-

nel in Section 2.6. Therefore, in the MIMO BC phase we can achieve rate pairs within the

rate region

RMIMO

BC =n[R1, R2]∈

+:R1≤R−→

R2(β1), R2≤R−→

R1(β2),[β1, β2]∈ Bo

with rate constraints

R−→

R2(β1) := log det IN2+1

σ2HH

2QR,1(β1)H2,(2.77)

R−→

R1(β2) := log det IN1+1

σ2HH

1QR,2(β2)H1(2.78)

for the MIMO broadcast channel. It can be easily seen by contradiction that for any rate pair

on the boundary of the achievable rate region we have β1+β2= 1.

From the water-filling solution we see that with increasing power it is optimal to increase the

number of the used eigenmodes if the power exceeds certain thresholds. In the low-power

113

2 Bidirectional Relay Communication using Superposition Encoding

regime we use the strongest eigenmode of the channel only, which is known as beamform-

ing. In the high-power regime it is optimal to use all eigenmodes. The maximal number of

eigenmodes is given by the rank of the channel matrix. In the next proposition we specify

the threshold values which also characterize the so-called beamforming optimality range and

high-power regime. To this end, let

HkHH

k=VkΣkVH

k,with Σk:= diag(λk,1, λk,2,...,λk,NR)

with unitary matrix Vk∈

NR×Rbe the eigenvalue decomposition of HkHH

k∈

NR×R,

k= 1,2with eigenvalues sorted in decreasing order, i.e. λk,1≥ ··· ≥ λk,NR≥0. Further-

more, we define the coefficients

Lk,m :=

i=1

λk,i

, k = 1,2

for m= 1,2,...,r2for k= 1 and m= 1,2,...,r1for k= 2 respectively and Lk,0:= 0

for k= 1,2.

Proposition 2.52. For k= 1 with r2= rank(H2)maximal possible eigenmodes it is

optimal to use











r2eigenmodes, if β1PR> σ2r2−1

λ2,r2−L2,r2−1,

1eigenmode, if β1PR≤σ21

λ2,2−L2,1, or

meigenmodes, if σ2m

λ2,m −L2,m−1< β1PR≤σ2m

λ2,m+1 −L2,m.

Similarly, for k= 2 with r1= rank(H1)maximal possible eigenmodes it is optimal to use











r1eigenmodes, if β2PR> σ2r1−1

λ1,r1−L1,r1−1,

1eigenmode, if β2PR≤σ21

λ1,2−L1,1, or

meigenmodes, if σ2n

λ1,m −L1,m−1< β2PR≤σ2m

λ1,m+1 −L1,m.

Proof. The proof can be found in Appendix 2.8.34.

Next, we are interested in the weighted rate sum maximum

RMIMO

BC (q) := arg max

[R1,R2]∈RMIMO

q1R1+q2R2.(2.79)

Since RMIMO

BC is convex, the strategy is similar to the previous derivations where we look for

the rate pair on the boundary whose normal vector has the same angle than the weight vector

114

2.6 Extension to Multi-Antenna Bidirectional Relaying

q. Therefore, we have to identify for each rate pair on the boundary RMIMO

BC the number of

used eigenmodes of the channels Hk,k= 1,2.

Proposition 2.52 characterizes the number of eigenmodes with respect to the power β1PR

and β2PR. We will look at a parametrization of the boundary in terms of β=β1= 1 −β2.

Therefore, we solve the power threshold values for β1and β2so that we can express the

threshold values in terms of the power fraction as follows

β1,m := 1

γRm

λ2,m+1 −L2,m, m = 1,2,...,r2−1,

β2,m := 1

γRm

λ1,m+1 −L1,m, m = 1,2,...,r1−1.

Additionally, we define the value βk,0:= 0,k= 1,2, as the value where no eigenmode is

used. In order to identify the number of eigenmodes used for a given relay power fraction β1

or β2we define the intervals

B1,m := (β1,m−1, β1,m], m = 1,2,...,r2−1,

B2,m := (β2,m−1, β2,m], m = 1,2,...,r1−1

with B1,r2:= (β1,r2−1,∞),B2,r1:= (β1,r1−1,∞), and Bk,0:= {0},k= 1,2. Therewith,

we can define the following indicator functions

m1: [0,1] → {0,1,2,...,r2}, β17→ mwhere we have β1∈ B1,m,

m2: [0,1] → {0,1,2,...,r1}, β27→ mwhere we have β2∈ B2,m,

which specify the numbers of used eigenmodes for a power fraction β1or β2. Obviously, for

β1= 0 we have R−→

R2(β1) = 0 and for β2= 0 we have R−→

R1(β2) = 0. For β1, β2>0the

characterization of the used eigenmodes allows us to state in the next proposition a closed

form solution of the rates R−→

R2(β1)and R−→

R1(β2), which we use for the derivation of the

weighted rate sum optimal rate pair.

Proposition 2.53. The rates R−→

R2(β1)or R−→

R1(β2)are zero if β1or β2are equal to zero. For

β1, β2>0we have

R−→

R2(β1) =

m1(β1)

n=1

log λ2,n

m1(β1)(β1γR+L2,m1(β1)),

R−→

R1(β2) =

m2(β2)

n=1

log λ1,n

m2(β2)(β2γR+L1,m2(β2)).

Proof. The proof can be found in Appendix 2.8.35.

115

2 Bidirectional Relay Communication using Superposition Encoding

Then

M1:= β1,m :β1,m ≤1for m= 0,1,2,...,r2−1and

M2:= β2,m :β2,m ≤1for m= 0,1,2, . . . , r1−1

denote the numbers of maximal used eigenmodes. Next we define M:= M1+M2.

Then let [˜

β1,˜

β2,...˜

βM]denote the sorted vector of [β1,0, β1,1, . . . , β1,M1−1,1−β2,0,1−

β2,1,...,β2,M2−1]in increasing order, i.e. we have 0 = ˜

β1≤˜

β2≤ ··· ≤ ˜

βM= 1.

We are now prepared to characterize in closed form the rate pair which maximizes the

weighted rate sum according to (2.79).

Theorem 2.54. Let q= [q1, q2]∈

+\{0}denote a weight vector with nonnegative ele-

ments and angle θq:= arccos q1

√q2

1+q2

2, then the rate pair RMIMO

BC (q)where the weighted

rate sum is maximized is given by

RMIMO

BC (q) = 









[0, R2,BC([0,1])],if θq≥ϕ1,

[R1,BC(q), R2,BC(q)],if ϕ1> θq> ϕM,

[R1,BC([1,0]),0],if ϕM≥θq,

with characteristic angles where the number of allocated eigenmodes changes

ϕn:= 









arctan γR+L1,M1

M1L2,1,if n= 1,

arctan m1(˜

βn)“(1−˜

βn)γR+L1,m2(1−˜

βn)”

m2(1−˜

βn)“˜

βnγR+L2,m1(˜

βn)”,if 1< n < M,

arctan M2L1,1

γR+L2,M2,if n=M,

which are given for a weight vector qby

η1(q) := 









0,if θq≥ϕ1,

m1(˜

βn+1),if ϕn> θq≥ϕn+1,

M1,if ϕM> θq,

η2(q) := 









M2,if θq> ϕ1,

m2(˜

βn),if ϕn≥θq> ϕn+1,

0,if ϕM≥θq,

116

2.6 Extension to Multi-Antenna Bidirectional Relaying

so that the rates are given by

R1,BC(q) :=

η1(q)

i=1

log γR+L1,η2(q)+L2,η1(q)

q1η1(q) + q2η2(q)q1λ2,i,

R2,BC(q) :=

η2(q)

i=1

log γR+L1,η2(q)+L2,η1(q)

q1η1(q) + q2η2(q)q2λ1,i.

Proof. The proof can be found in Appendix 2.8.36.

With this theorem we can identify for any non-negative weight vector q∈

+\ {0}the

Pareto optimal rate pair on the boundary of the BC phase. In Figure 2.21 we depicted the

characteristic sections and rate pairs of the achievable rate region RMIMO

BC for an example

where each node is equipped with two antennas.

2.6.3 MIMO Bidirectional Achievable Rate Region

As in the scalar case we use the MAC and BC phases for a fraction of time only. Let the

time division parameter α∈[0,1] denote the fraction of time in the MAC phase so that

1−αcharacterizes the fraction of time in the BC phase. Therefore, we have to scale the

rate regions according to the time fraction. This means for a successful bidirectional relay

transmission of a message m1with rate R1from node 1 to node 2 and message m2with rate

R2from node 2 to node 1 the rate pair R= [R1, R2]has to be within αCMIMO

MAC as well as

within (1 −α)RMIMO

BC . This means that for any given time division parameter α∈[0,1] the

bidirectional achievable rate pairs are given by the intersection of the scaled rate regions

RMIMO

BR (α) := αCMIMO

MAC ∩(1 −α)RMIMO

BC ,

which is convex since the intersection of convex sets is itself convex. Accordingly, the

boundary is characterized by the rate pairs which maximize the weighted rate sum for weight

vectors q∈

+\{0},

RMIMO

BR (α, q) := arg max

[R1,R2]∈RMIMO

BR (α)

q1R1+q2R2.

Because of the difficult MIMO MAC capacity region the optimization problem cannot be

solved in closed form as in the SISO case. However, for the solution we can distinguish

117

2 Bidirectional Relay Communication using Superposition Encoding

0 0.2 0.4 0.6 0.8 1 1.2

0.2

0.4

0.6

0.8

1.2

rate R1 [bit/channel use]

rate R2 [bit/channel use]

γ1 =1

γ2 =1

γR =1.2

MAC

ε1

ε2

H1=

»1−0.5j1−0.5j

−0.5 1.3–,

H2=

»0.4−0.4j1.2

1.6−0.2j–

with λ1,1= 3.16,λ1,2= 1.28,

λ2,1= 3.04, and λ2,2= 1.32.

Figure 2.21: MIMO achievable rate regions RMIMO

BR (1

/2)with N1=N2=NR= 2.

between the following cases. First we consider the cases where the optimal rate pair is also

the optimal rate pair of the MAC or BC phase. Accordingly, we can conclude as follows

αRMIMO

MAC (q)∈ RMIMO

BR (α)⇒RMIMO

BR (α, q) = αRMIMO

MAC (q),

(1 −α)RMIMO

BC (q)∈ RMIMO

BR (α)⇒RMIMO

BR (α, q) = (1 −α)RMIMO

BC (q).

If both cases do not apply the boundaries have to intersect at least once. From the previous

we know that a rate pair where the boundaries intersect is optimal for a range of weight

vector angles. In the case where αRMIMO

MAC (q),(1−α)RMIMO

BC (q)/∈ RMIMO

BR (α)the optimal

rate pair RMIMO

BR (α, q)is given by a rate pair where the boundaries intersect. Since multiple

intersections may possible it remains to determine the intersection point which is optimal for

the weight vector q. Unfortunately the characterization of the intersection points is much

more involved than in the SISO case and cannot be solved in closed form.

As in the scalar case we can define the rate region and the weighted rate sum optimal rate

pair for the optimal time division case as follows

RMIMO

BRopt := co S

α∈[0,1]αCMIMO

MAC ∩(1 −α)RMIMO

BC !,

RMIMO

BRopt(q) := arg max

[R1,R2]∈RMIMO

BRopt

q1R1+q2R2,

which we use to study the high-power behavior in the following.

118

2.6 Extension to Multi-Antenna Bidirectional Relaying

High Power Behavior

We can deduce the high power behavior from the asymptotic behavior of the maximal unidi-

rectional rates for a fixed time division parameter α,

R∗MIMO

1(α) := min nαR∗

−→

1R,(1 −α)R−→

R2(1)o,

R∗MIMO

2(α) := min nαR∗

−→

2R,(1 −α)R−→

R1(1)o,

and for the optimal time division,

R∗MIMO

1opt := R∗

−→

1RR−→

R2(1)

R∗

−→

1R +R−→

R2(1),and R∗MIMO

2opt := R∗

−→

2RR−→

R1(1)

R∗

−→

2R +R−→

R1(1),

where R∗

−→

1R and R∗

−→

2R denote the maximal unidirectional rates in the MAC phase as defined

in Section 2.6.1 and R−→

R2(1) and R−→

R1(1) denote the maximal unidirectional rates in the BC

phase as defined in Section 2.6.2.

In the high power regime it is asymptotically optimal to allocate equal proportions of power

for each non-zero eigenmode. This means at high powers P1, P2, and PRwe can approxi-

mate R∗

−→

kRand R−→

Rk(1),k= 1,2, by

R∗

−→

kR≈

n=1

log 1 + Pk

λk,n

σ2≈

n=1

log Pk

λk,n

σ2,

R−→

Rk(1) ≈

n=1

log 1 + PR

λk,n

σ2≈

n=1

log PR

λk,n

σ2.

For conceptual clarity we set P1=P2=PR=P. Then in the high power regime the rates

scale linearly with the number of eigenmodes of the corresponding MIMO channel

R∗

−→

kR≈rklog(P) + ck,and R∗

−→

Rk(1) ≈rklog(P) + dk,

with constants ckand dkfor k= 1,2.

Therewith, we get the asymptotic of the maximal unidirectional rates as follows15

R∗MIMO

1= min{αr1,(1 −α)r2}log(P) + olog(P),

R∗MIMO

2= min{αr2,(1 −α)r1}log(P) + olog(P),

R∗MIMO

kopt =r1r2

r1+r2

log(P) + olog(P)for k= 1,2.

15The little-o notation collects asymptotically insignificant terms of an expression. This means, if g(x)∈

o(f(x)) it follows that we have lim

x→∞

g(x)

f(x)= 0.

119

2 Bidirectional Relay Communication using Superposition Encoding

This means that the maximal unidirectional rates for the fixed time division case scale with

the minimum of the fraction of time weighted number of eigenmodes and for the optimal

time division case scale with r1r2

r1+r2.

Since the rate regions RMIMO

BR (α),α∈[0,1], and RMIMO

BRopt are convex, for the sum of any

rate pair on the boundaries we have

min{R∗MIMO

1, R∗MIMO

2} ≤ RMIMO

BR (α, q)1≤R∗MIMO

1+R∗MIMO

2,

min{R∗MIMO

1opt , R∗MIMO

2opt } ≤ RMIMO

BRopt (q)1≤R∗MIMO

1opt +R∗MIMO

2opt .

It follows that the asymptotics of the maximal unidirectional rates give us a lower and upper

bound on the asymptotics of the sum-rate of the rate pairs on the boundary which we sum up

in the following proposition.

Proposition 2.55. Let r1= rank(H1)and r2= rank(H2)denote the spatial degree

offered by the channel between the relay node and the nodes 1 and 2 respectively. Then

the asymptotic scaling of the sum of any rate pair on the boundary of the achievable rate

regions RMIMO

BR (α)and RMIMO

BRopt with increasing power Pcan be upper and lower bounded

as follows

RMIMO

BR (α, q)1≤(s1+s2) log(P) + olog(P),

RMIMO

BR (α, q)1≥min{s1, s2}log(P) + olog(P),

RMIMO

BRopt(q)1≤2r1r2

r1+r2

log(P) + olog(P),

RMIMO

BRopt(q)1≥r1r2

r1+r2

log(P) + olog(P).

with s1:= min{αr1,(1 −α)r2}and s2:= min{αr2,(1 −α)r1}.

We see that as for the point-to-point MIMO channel the spatial degree results in a pre-log

factor in the high power regime, which depends on the spatial degree of both MIMO channels

as well as on the time division between the phases.

2.7 Discussion

We considered various cross-layer design aspects for a two-phase bidirectional decode-and-

forward relaying protocol using superposition encoding in the broadcast phase which was

proposed first by Rankov and Wittneben in [RW05b, RW07]. In their work they propose

half-duplex relaying protocols with increased spectral efficiency and show by comparing

120

2.7 Discussion

the maximal achievable ergodic sum-rates of the proposed bidirectional protocols and the

achievable ergodic rates of the proposed unidirectional protocols that the bidirectional ap-

proaches can “recover a large portion of the half-duplex loss.” After that they did not carry

on research to explore further properties of their bidirectional protocols.

On that score in Section 2.2 we have extensively studied the bidirectional decode-and-

forward protocol for a three-node network assuming single-antenna nodes. First, it is im-

portant to realize that a bidirectional relaying protocol is characterized by two achievable

rates. This means that we have to examine a two-dimensional achievable rate region. Then

it easily follows that the achievable rate region is given by the intersection of the scaled rate

regions of the multiple access and broadcast phases. In this chapter, the relay node applies

superposition encoding technique to forward the messages in the broadcast phase as pro-

posed by Rankov and Wittneben. The characterization of the intersection of achievable rate

regions usually results in a combinatorial problem. Theorem 2.5 and its corollaries reveal the

combinatorial structure of the bidirectional achievable rate region for the equal time division

case. In particular it allows us to characterize the sum-rate maximum in Corollary 2.8, which

Rankov and Wittneben characterize with some simplifying assumptions only. Since the in-

tersection of convex sets is itself convex, we know from convex theory that the boundary of

the achievable rate region is characterized by the rate pairs which maximize the weighted

rate sum. Theorem 2.10 characterizes in closed form the optimal rate pairs for any positive

weight vector using the knowledge of the combinatorics.

Another simple extension is to relax their equal time division assumption from which it

is natural to ask for the optimal time division. The achievable rate region of the optimal

time division case is given by the union over all time division parameters and therefore

need not be convex. However, in Corollary 2.13 we have concluded that the region is indeed

convex using the equivalent description of the achievable rate region proved in Theorem 2.12.

From Proposition 2.14 we see that the combinatoric of the MAC capacity region transfers

to the bidirectional rate region. Again, in Theorem 2.16 we characterize the boundary of

the achievable rate region with the optimal time division by the rate pairs that maximize the

weighted rate sum. Since the boundary rate pairs of a convex set are Pareto optimal in the

sense that we cannot increase one rate without decreasing the other, its characterization is

crucial for the succeeding cross-layer designs.

For the two phase protocol there are two possibilities to interpret the mean power input con-

straint of a Gaussian continuous-alphabet channel with respect to a code word. We can either

consider each phase separately, which means that we average over all actually transmitted

symbols of each phase, or we average over both phases so that we scale the mean power

constraints according to the inverse of the time division parameter. While for the equal time

division case the results for both models can be obtained by an appropriate substitution, for

the optimal time division case the non-linear dependence on the time division case makes

a reconsideration for the second model necessary. Accordingly, in Theorem 2.39 we find

121

2 Bidirectional Relay Communication using Superposition Encoding

an equivalent characterization of the achievable rate region. Furthermore, in Theorem 2.24

we show that the combinatoric of the MAC capacity region transfers to bidirectional achiev-

able rate region as well. And finally, in Theorem 2.26 we establish the convexity of the rate

region. The non-linear dependence of the rate constraints on the time division hinders the

derivation of closed-form results. However, the obtained results makes efficient algorithmic

optimization solutions feasible.

In Section 2.3 we use the results for a cross-layer design across the Data Link Layer and

Physical Layer, where we are interested in an efficient resource allocation with respect to

the traffic generated at higher layers. Therefore, we assume stationary and ergodic time-

variant block-fading channel processes. At nodes 1 and 2 we consider queues with infinite

buffer length and a centralized controller which slot-wise adapts the service rates according

to the maximum weighted rate sum with weights equal to the buffer length. Since this pol-

icy is based on the maximum differential backlog algorithm of Tassiulas and Ephrimedes

[TE93], we can use ”standard” Lyapunov drift techniques to characterize the arrival rates

under which the policy guarantees queue stability. In Corollary 2.30 we adapt the proof from

[NMR03] to show that the stability region is equal to the bidirectional ergodic rate region by

proving a negative drift of a quadratic Lyapunov function on the buffer levels whenever the

mean number of unfinished work is large. Moreover, it follows that the mean delay behaves

asymptotically to the inverse of the distance of the bit arrival rate vector to the boundary of

the stability region. The following numerical simulations confirm the stability results and

illustrate the efficiency of the bidirectional relay protocol for the equal and optimal time

division case compared to classical Round-Robin strategies.

In a network where Nrelay nodes are willing to support the bidirectional communication

it is a fundamental problem to identify the best relay node. But again, since bidirectional

communication is characterized by two rates, this problem is a vector optimization problem

so that relay selection for a bidirectional protocol is more involved than for unidirectional

relaying protocols. In Section 2.4 we derive a relay selection criterion based on the achiev-

able rate region of each relay node. From the Propositions 2.32 and 2.37 we can conclude

that the probability that the achievable rate region of one relay node contains the rate region

of all others decreases with an increasing number of relay nodes N, which means that there

is more often no over-all best relay node. For that reason we propose to select the optimal

relay node individually for any Pareto optimal rate pair on the boundary of the achievable

rate region. Moreover, from Figure 2.10 we see that if two relay nodes achieve the same

weighted rate sum for a certain weight vector, we can enlarge the achievable rate region if

we allow time-sharing between the usage of the corresponding relay nodes.

For iid Rayleigh fading channels and for the equal time division case in Theorem 2.40 we

show that the growth of the maximal unidirectional ergodic rates is asymptotically equal to

/2log(ln(N)), which is the same scaling law achieved with the separation based relaying

approach in [DSG+03]. Since the sum-rate of any ergodic rate pair on the boundary of the

122

2.7 Discussion

ergodic rate region for the equal and optimal time division cases can be upper and lower

bounded by the maximal unidirectional rates of the equal time division case, in the Corol-

laries 2.42 and 2.44 we conclude that the ergodic sum-rate of any boundary rate pair scales

with Θlog(log(N))as well.

Relay selection is obviously a routing problem, which belongs to the Network Layer. Then

for relay selection in a wireless system it remains to decide for a weight vector. We can

again look at this problem from a cross-layer design perspective. This means that if we

use the buffer levels as the weight vector we tie together relay selection with the maximum

throughput policy of Section 2.3. It follows that we end up with a load adaptive routing

protocol where the routing policy takes the resource allocation, the channel states as well as

the queue states into account.

In Section 2.5 we consider the next cross-layer design approach where we study the joint

resource allocation of two routing tasks. For this purpose, we add in the broadcast phase

a multicast communication from the relay node to the nodes 1 and 2 to the bidirectional

relaying protocol. The goal is to realize synergistic efficiency and enable rate trade-offs by

converging two routing schemes. Moreover, if the relay does not fully exploit its transmit

power constraint for the bidirectional communication, the relay can add the relay multicast

communication without worsening the bidirectional communication. For that reason and in

dependence to piggyback communication on higher layer protocols, we call this approach

piggyback on bidirectional communication. We see this as the first step in integrating bidi-

rectional relay communication in a wireless network.

In Theorem 2.45 we show that for the receiving decode-and-forward nodes 1 and 2 it is

always optimal to decode the relay message first. This fixed decoding order simplifies fol-

lowing case studies, but the explicit characterization is still exhaustive. Then Theorem 2.48

specifies the total sum-rate maximum of the bidirectional rates and the additional relay mul-

ticast rate. We see that the total sum-rate behavior is dominated by the function which char-

acterizes the rate pairs where the sum-rate maximum of a broadcast rate region is attained

for a certain relay power. In Corollary 2.49 we identify a rate trade-off between additional

relay rate and one bidirectional rate where the total sum-rate remains constant. In the follow-

ing combinatorial discussion we look at some examples and work out some further aspects,

which are interesting for the design and operation of service adapted network protocols. In

particular we change the point of view and optimize the bidirectional communication while

we desire a certain relay rate for a multicast application or we discuss the total sum-rate

maximum with respect to the time division parameter. Most of the closed form results can

be obtained for the equal time division case only, but the behavior for any fixed time division

will be similar.

In Section 2.6 we extend the bidirectional decode-and-forward relaying protocol using super-

position encoding to vector-valued processing. For this purpose, we assume that each node is

equipped with multiple antennas and that each transmitter has perfect channel knowledge as

123

2 Bidirectional Relay Communication using Superposition Encoding

well. Accordingly, we present the optimal transmit strategies at each node for both phases.

In the BC phase the optimal transmit strategy is given by two point-to-point water-filling so-

lutions which are only coupled by the relay power distribution. This allows us to characterize

the rate pair which maximizes the weighted rate sum in Theorem 2.54. A combinatorial dis-

cussion of the achievable rate region as in the SISO case is not possible in the MIMO case

because of the complicated structure of the MIMO-MAC capacity region. However, from

the maximal unidirectional rates in the high power regime we find a scaling law for the max-

imal unidirectional rates, which depends on spatial degrees of the MIMO channels as well

as on the time division between the phases. In Proposition 2.55 we use the scaling law for

the maximal unidirectional rates to upper and lower bound the sum-rate of any rate pair on

the boundary of the achievable rate regions for the fixed and optimal time division case.

Nevertheless, one can directly apply the ideas of the previous sections to the MIMO case,

which means that for an optimal throughput in a system the rates should be allocated accord-

ing to the queue states, the relay selection should be performed with respect to the achievable

rate region, and the resource efficiency can be improved if we consider the joint resource al-

location of multiple routing tasks.

124

2.8 Appendix: Proofs

2.8.1 Proof of Proposition 2.2.3

Let [R1i, R2i]∈ RBC,i= 1,2denote two arbitrary achievable rate pairs. It follows that

there exist feasible relay power distributions [β1i, β2i]∈ B with R1i=R−→

R2(β1i)and R2i=

R−→

R1(β2i)for i= 1,2. From the concavity of the logarithm we can conclude that for any

t∈[0,1] we have

tR11 + (1 −t)R12 =tR−→

R2(β11) + (1 −t)R−→

R2(β12)≤R−→

R2(β1(t)),

tR21 + (1 −t)R22 =tR−→

R1(β21) + (1 −t)R−→

R1(β22)≤R−→

R1(β2(t))

with βk(t) := tβk1+ (1 −t)βk2,k= 1,2. Since [β1i, β2i]∈ B, we have β1i+β2i≤1for

i= 1,2so that for any t∈[0,1] we have

k=1

βk(t) = tβ11 + (1 −t)β12 +tβ21 + (1 −t)β22 ≤t+ (1 −t) = 1.

This means that for any t∈[0,1] we have [β1(t), β2(t)] ∈ B and therefore

[R−→

R2(β1(t)), R−→

R1(β2(t))] ∈ RBC. It follows that any convex combination of achievable

rate pairs is achievable, which proves that RBC is convex.

2.8.2 Proof of Proposition 2.2

The sum-rate maximum is obviously attained on the boundary of RBC. The boundary is

parametrized by (2.8). Therefore, we consider the sum-rate

R−→

R2(β) + R−→

R1(1 −β) = log (1 −β|h2|2γR)(1 −(1 −β)|h1|2γR)

|{z }

:=fBC(β),

which is maximized for the same βas the concave parabola fBC(β). The concave parabola

f(β)is maximized at its vertex

[β⋆,fBC(β⋆)] = h1

2+1

2γR1

|h1|2−1

|h2|2,1

4(1 + γR|h1|2+|h1|2

|h2|2)(1 + γR|h2|2+|h2|2

|h1|2)i.

This characterizes the maximum sum-rate if β⋆is a feasible relay power distribution, i.e.

β⋆∈[0,1]. If we have β⋆<0, the function fBC(β)is strictly decreasing on [0,1] so that

the maximum sum-rate is attained at RBC(0). On the other hand, if we have β⋆>1, the

function fBC(β)is strictly increasing on [0,1] so that the maximum sum-rate is attained at

RBC(1).

2 Bidirectional Relay Communication using Superposition Encoding

2.8.3 Proof of Theorem 2.5

For a given relay power distribution β=β1= 1 −β2∈[0,1] the maximal sum-rate has to

satisfy

RΣ(β) = max

R∈RBReq(β)R1+R2<1

/2RMAC

Σ,

because of the MAC sum-rate constraint (2.4c). Furthermore, the unidirectional rate R1has

to satisfy the individual rate constraint (2.4a) of the MAC phase and (2.7a) of the BC phase

and similarly the unidirectional rate R2has to satisfy (2.4b) and (2.7b) so that we have

R1≤1

/2min{R−→

1R, R−→

R2(β)}and R2≤1

/2min{R−→

2R, R−→

R1(1 −β)}.

The combination of all constraints gives us the maximum sum-rate

RΣ(β) = 1

/2min min{R−→

1R, R−→

R2(β)}+ min{R−→

2R, R−→

R1(1 −β)}, RMAC

Σ

/2min R−→

1R +R−→

R1(1 −β), R−→

R2(β) + R−→

2R, R−→

R2(β) + R−→

R1(1 −β),

R−→

1R +R−→

2R, RMAC

Σ

/2min R−→

1R +R−→

R1(1 −β), R−→

R2(β) + R−→

2R, R−→

R2(β) + R−→

R1(1 −β), RMAC

Σ

(2.80)

using R−→

1R +R−→

2R = log (1+γ1|h1|2)(1+γ2|h2|2)≥log(1+γ1|h1|2+γ2|h2|2) = RMAC

in the last equality.

For the next equality we have to examine the combinatorics of (2.80). This becomes clearer

if we discuss the geometry of the arguments of the logarithms of the rates. Accordingly, we

define the following functions

fΣ(β) := 2RMAC

Σ= 1 + γ1|h1|2+γ2|h2|2,

f1(β) := 2R−→

1R+R−→

R1(1−β)= (1 + γ1|h1|2)(1 + (1 −β)γR|h1|2),

f2(β) := 2R−→

2R+R−→

2R(β)= (1 + γ2|h2|2)(1 + βγR|h2|2),

fBC(β) := 2R−→

R1(1−β)+R−→

R2(β)= (1 + (1 −β)γR|h1|2)(1 + βγR|h2|2),

where we easily see that fΣ(β)is a constant function, f1(β)is a linearly decreasing function,

f2(β)is a linearly increasing function, and fBC(β)is a concave parabola, which we already

know from the Proof of Proposition 2.2. To solve (2.80) we will study the function

min{f1(β),f2(β),fΣ(β),fBC(β)}(2.81)

for β∈[0,1] in the following. In Figure 2.22 we depicted two representative examples.

The intersection points characterize the combinatorics of the rate region and identify the

active restriction for each feasible power distribution β∈[0,1]. In the following derivation

126

2.8 Appendix: Proofs

1.6

1.8

2.2

f1,f2,fΣ,fBC(β)

β⋆

β21 βB1

β2B

01β

|h1|2 =0.4

|h2|2 =0.5

γ1 =1.4

γ2 =0.9

γR =1.3

f1(β)→

f2(β)→

fΣ(β)↑

fBC(β)↓

1.6

1.8

2.2

f1,f2,fΣ,fBC(β)

βΣ1

β2Σ

βBΣ βΣB

0 1 β

|h1|2 =0.8

|h2|2 =0.7

γ1 =0.4

γ2 =1

γR =1.2

f1(β)→← f2(β)

fΣ(β)↑

↓ fBC(β)

Figure 2.22: Exemplary plots for the discussion of f1(β),f2(β),fΣ(β), and fBC(β)for β⋆2<

∆βin the left figure and β⋆2≥∆βin the right figure.

of the combinatorics we first neglect the feasibility, but later we will take the feasibility into

account. With simple calculations of the intersection points between f1(β),fΣ(β),f2(β), and

fBC(β)it is easily seen that we have

f1(β)≤fΣ(β)for β∈[βΣ1,∞)and f1(β)≥fΣ(β)for β∈(−∞, βΣ1],(2.82a)

f2(β)≤fΣ(β)for β∈(−∞, β2Σ]and f2(β)≥fΣ(β)for β∈[β2Σ,∞),(2.82b)

f1(β)≤fBC(β)for β∈[βB1, β1B]and f1(β)≥fBC(β)for β∈(−∞, βB1]∪[β1B,∞),

(2.82c)

f2(β)≤fBC(β)for β∈[βB2, β2B]and f2(β)≥fBC(β)for β∈(−∞, βB2]∪[β2B,∞),

(2.82d)

f2(β)≤f1(β)for β∈(−∞, β21]and f2(β)≥f1(β)for β∈[β21,∞)(2.82e)

with βΣ1,β2Σ,βB1,β1B,βB2, and β2B as given in the theorem and β21 :=

(1+γ1|h1|2)(1+γR|h1|2)−(1+γ2|h2|2)

(1+γ1|h1|2)γR|h1|2+(1+γ2|h2|2)γR|h2|2. Furthermore, we have

f1(β21) = f2(β21) = (1+γ1|h1|2)(1+γ2|h2|2)(γR|h1|2+γR|h2|2+γ2

R|h1|2|h2|2)

(1+γ1|h1|2)γR|h1|2+(1+γ2|h2|2)γR|h2|2>0.(2.83)

It follows that we have βB2 < β21 < β1B since we have f1(β1B) = f2(βB2) = 0 and the

linear functions f1(β)and f2(β)are strictly decreasing and increasing.

If we multiply the left and the right hand side of f1(β21) = (1+γ1|h1|2)(1+(1−β21)γR|h1|2)

to the left and right hand side of f1(β21) = f2(β21) = (1 + γ2|h2|2)(1 + β21γR|h2|2)we

127

2 Bidirectional Relay Communication using Superposition Encoding

get

f1(β21)2= (1 + γ2|h2|2)(1 + γ1|h1|2)

|{z }

:=c21

(1 + βγR|h2|2)(1 + (1 −β21)γR|h1|2)

| {z }

=fBC(β21)

(2.84)

with c21 = (1+γ2|h2|2)(1+γ1|h1|2)>(1+γ1|h1|2+γ2|h2|2) = fΣ(β)for all β. Therewith,

we will prove

f1(β21) = f2(β21)>min{fΣ(β21),fBC(β21)}(2.85)

next. First, we consider the case fΣ(β21)≤fBC(β21)so that we can conclude

fΣ(β21)2<fΣ(β21)c21 ≤fBC(β21)c21 = f1(β21)2

using (2.84). Since we know from (2.83) that f1(β21)>0and fΣ(β)>0for all β, it follows

that we have f1(β21) = f2(β21)>fΣ(β21)so that (2.85) is fulfilled for this case. Otherwise,

if fΣ(β21)>fBC(β21)we have fBC(β21)<c21 since fΣ(β21)<c21 holds. Then from (2.84)

it follows that

fBC(β21)2<c21fBC(β21) = f1(β21)2

so that f1(β21) = f2(β21)>fBC(β21)since we have f1(β21),fBC(β21)>0. This proves the

other case so that (2.85) holds.

We are now ready to derive I1. From (2.82a), (2.82c), and (2.82e) we have

f1(β)≤min{fΣ(β),fBC(β),f2(β)}for β∈[βΣ1,∞)∩[βB1, β1B]∩[β21,∞).

We now claim that [βΣ1,∞)∩[βB1, β1B]∩[β21,∞) = [βΣ1,∞)∩[βB1, β1B]holds. To see

this we first look at the case that the set on the left hand side is empty, which is equivalent

to the condition that max{βΣ1, βB1, β21}> β1B holds. Since we have βB1, β21 < β1B, it

follows that βΣ1 > β1B has to hold so that the set on the right hand side is empty as well.

On the other hand, if we have βΣ1 ≤β1B both sets are non-empty. Then both sets are equal

if we have max{βΣ1, βB1, β21}= max{βΣ1, βB1}, which is equivalent to the condition that

β21 ≤max{βΣ1, βB1}holds. We will prove the last condition by contradiction. Therefore,

let us assume that β21 >max{βΣ1, βB1}is true. This is equivalent to the condition that

β21 > βΣ1 and β21 > βB1 holds simultaneously. Since we have β21 < β1B, it follows from

(2.82a) that f1(β21)<fΣ(β21)and from (2.82c) that f1(β21)<fBC(β21). Both together

gives us f1(β21)<min{fΣ(β21),fBC(β21)}which is a contradiction to (2.85). Accordingly,

the set I1= [βΣ1,1] ∩[βB1, β1B]denotes the set of feasible β∈[0,1] where the minimum

of (2.81) is equal to f1(β).

Similarly, from (2.82b), (2.82d), and (2.82e) we have

f2(β)≤min{fΣ(β),fBC(β),f1(β)}for β∈(−∞, β2Σ]∩[βB2, β2B]∩(−∞, β21].

128

2.8 Appendix: Proofs

We now claim that (−∞, β2Σ]∩[βB2, β2B]∩(−∞, β21] = (−∞, β2Σ]∩[βB2, β2B]holds. As

before, we first look at the case that the set on the left hand side is empty, which is equivalent

to the condition that min{β2Σ, β2B, β21}< βB2 holds. Since we have β2B, β21 > βB2, it

follows that β2Σ < βB2 has to hold so that the right hand side is empty as well. On the

other hand, if we have β2Σ ≥βB2 both sets are non-empty. Then both sets are equal if we

have min{β2Σ, β2B, β21}= min{β2Σ, β2B}, which is equivalent to the condition that β21 ≥

min{β2Σ, β2B}holds. Again, we will prove the last condition by contradiction. Therefore,

let us assume that β21 <min{βΣ1, βB1}is true. This is equivalent to the condition that

β21 < βΣ1 and β21 < βB1 holds simultaneously. Since we have β21 > βB2, if follows from

(2.82b) that f2(β21)<fΣ(β21)and from (2.82c) that f2(β21)<fBC(β21). Both together

gives us f2(β21)<min{fΣ(β21),fBC(β21)}which is a contradiction to (2.85). Accordingly,

the set I2= [0, β2Σ]∩[βB2, β2B]denotes the set of feasible β∈[0,1] where the minimum

of (2.81) is equal to f2(β).

The MAC sum-rate constraint is only active if the concave parabola fBC(β)intersects fΣ(β).

This is the case if and only if we have β⋆2≥∆β. Then the corresponding intersection

points are characterized by βBΣ and βΣB as in the theorem so that we have fΣ(β)≤fBC(β)

if β∈[βBΣ, βΣB]. Furthermore, from (2.82a) and (2.82b) we know that

fΣ(β)<min{f1(β),f2(β)}for β∈(−∞, βΣ1)∩(β2Σ,∞)

so that for feasible β∈ IΣas in the theorem the minimum of (2.81) is equal to fΣ(β).

Finally, for all other feasible power distributions β∈ IBC = [0,1] \(I1∪I2∪IΣ)we have

fBC(β)≤min{f1(β),f2(β),fΣ(β)}and therefore the BC phase is more restrictive than any

restriction of the MAC phase.

2.8.4 Proof of Corollary 2.7

For the proof of the corollary we use the definitions of the proof of Theorem 2.5. Accord-

ingly, the right plot in Figure 2.22 is a helpful illustration.

A rate pair [RM

1, RM

2]∈ CMAC with the sum-rate RMAC

Σcan be supported in the BC phase

iff the set IΣis non-empty. If one solves (β⋆)2= ∆βfor γRwe get γ⋆

Ras given in the

corollary. It follows that for IΣ6=∅we have to have (β⋆)2≥∆βso that it is necessary that

γR≥γ⋆

Since RBC increases with increasing γRfor the minimal γRthe set IΣhas to be a singleton,

this means that for γΣMAC

Rwe have max{β2Σ, βBΣ}= min{βΣ1, βΣB}. This is equivalent

to the conditions

β2Σ ≤βΣB ∧βBΣ ≤βΣ1 ∧βBΣ ≤βΣB ∧β2Σ ≤βΣ1,(2.86)

129

2 Bidirectional Relay Communication using Superposition Encoding

where we have at least one equality. Since we require γR≥γ⋆

Rthe condition βBΣ ≤βΣB is

already fulfilled.

Next, we consider the case βBΣ ≤βΣ1, which is equivalent to the condition β⋆−βΣ1 ≤

pβ⋆2−∆β. Since we already require γR≥γ⋆

Rwe have β⋆2−∆β≥0so that the condition

is fulfilled if β⋆−βΣ1 ≤0holds, which is easily seen the case if we have γR≥ˆγΣ1

R. On the

other hand, if β⋆> βΣ1 we have βBΣ ≤βΣ1 if we have γR≥γΣ1

R(1

/2). Both together gives

us that βBΣ ≤βΣ1 is fulfilled if we have γR≥min{ˆγΣ1

R, γΣ1

R(1

/2)}.

Similarly, the case β2Σ ≤βΣB is equivalent to the condition β2Σ −β⋆≤pβ⋆2−∆β. Since

we have β⋆2−∆β≥0the condition is fulfilled if β2Σ−β⋆≤0holds, which is the case if we

have γR≥ˆγ2Σ

R. On the hand, if β⋆> β2Σ we have β2Σ ≤βΣB if we have γR≥γ2Σ

R(1

/2).

Both together gives us that β2Σ ≤βΣB is fulfilled if we have γR≥min{ˆγ2Σ

R, γ2Σ

R(1

/2)}.

Finally, we prove by contradiction that the following holds

β2Σ ≤βΣB ∧βBΣ ≤βΣ1 ∧βBΣ ≤βΣB ⇒β2Σ ≤βΣ1.

Therefore, let us assume that we have β2Σ > βΣ1. Then using the first and second inequality

gives us βΣB ≥β2Σ > βΣ1 ≥βBΣ which is a contradiction to the inequality βBΣ ≤βΣB.

Accordingly, the minimum necessary relay power has to satisfy the first three conditions of

(2.86) and therefore is given by (2.16).

2.8.5 Proof of Corollary 2.8

For the proof of the corollary we use the definitions of the proof of Theorem 2.5. Accord-

ingly, the left plot in Figure 2.22 is a helpful illustration. We prove the corollary by further

investigation of the combinatorial structure of the geometry of f1,f2, and fBC. Therefore, let

us define the function

r(β) := min{f1(β),f2(β),fBC(β)}.

Since we assume γR< γΣMAC

R, we have r(β)<fΣ(β)for all β∈[0,1]. Remember

that f1(β)and f2(β)are strictly decreasing and increasing. Furthermore, since fBC(β)is a

concave parabola which has its vertex at β⋆, it is strictly increasing for β < β⋆and strictly

decreasing for β > β⋆. Therefore, the sum-rate maximum is given by the combinatoric of

the vertex of the parabola at β⋆and the intersection points at βB1 and β2B.

Since we have IΣ=∅,β1B >1, and βB1 <0, the set of feasible power distributions where

fBC(β)<r(β)is given by

IB= [0,1] \I1∪I2= [0,1] ∩(β2B, βB1).

130

2.8 Appendix: Proofs

Since fBC(β)is a concave parabola, we have fBC(β⋆) = r(β⋆)≥r(β)for all β∈[0,1].

Accordingly, if we have

β⋆∈ IB⇔β⋆≥0∧β⋆≤1∧β⋆> β2B ∧β⋆< βB1,(2.87)

then the sum-rate maximum is given by 1

2RBC(β⋆). If we solve the conditions in (2.87) for

γRwe get

β⋆=1

2+1

2γR1

|h1|2−1

|h2|2≥0⇔γR≥γ†

R=1

|h2|2−1

|h1|2,

β⋆=1

2+1

2γR1

|h1|2−1

|h2|2≤1⇔γR≥ −γ†

R=1

|h1|2−1

|h2|2,

β⋆> β2B = 1 −γ2|h2|2

γR|h1|2⇔γR< γ⋆1

R=2γ2|h2|2

|h1|2+1

|h1|2−1

|h2|2,

β⋆< βB1 =γ1|h1|2

γR|h2|2⇔γR< γ2⋆

R=2γ1|h1|2

|h2|2+1

|h2|2−1

|h1|2.

Therefore, we have β⋆∈ IBwith the sum-rate optimum 1

2RBC(β⋆)if we have |γ†

R|< γR<

min{γ⋆1

R, γ2⋆

R}, which proves the case (2.18c).

If we have β⋆≤max{β2B,0}, then fBC(β)is decreasing for all β∈ IB. Since f2(β)and

f1(β)are strictly increasing and decreasing, the rate pair 1

2RBC(max{β2B,0})is sum-rate

optimal. Then the sum-rate optimal rate pair is on the R2-axis if we additionally have

β2B ≤0⇔γR≤γ2B

R=γ2|h2|2

|h1|2.

Then it follows that for 1

2RBC(0) to be sum-rate optimal we require β2B ≤0and β⋆≤0,

which is equivalent to γR≤min{γ†

R, γ2B

R}. This proves the case (2.18a). Accordingly, if

we have β2B >0and β⋆≤β2B the rate pair 1

2RBC(β2B)is sum-rate optimal. This is case

if γR> γ2B

Rand γR≥γ⋆1

R, which proves the case (2.18b).

Similarly, if we have β⋆≥min{βB1,1}, then fBC(β)is increasing for all β∈ IB. Again,

since f2(β)and f1(β)are strictly increasing and decreasing the rate pair 1

2RBC(min{βB1,1})

is sum-rate optimal. Then the sum-rate optimal rate pair is on the R1-axis if we additionally

have

βB1 ≥1⇔γR≤γB1

R=γ1|h1|2

|h2|2.

Then it follows that for 1

2RBC(1) to be sum-rate optimal we require βB1 ≥1and β⋆≥1,

which is equivalent to γR≤min{−γ†

R, γB1

R}. This proves the case (2.18e). Accordingly, if

we have βB1 <1and β⋆≥the rate pair 1

2RBC(βB1)is sum-rate optimal. This is case if

γR> γB1

Rand γR≥γ2⋆

R, which proves the case (2.18d).

131

2 Bidirectional Relay Communication using Superposition Encoding

2.8.6 Proof of Theorem 2.10

We first separate a simple calculation in the following lemma, which also justifies ϑ(β).

Lemma 2.56. The normal angle of the supporting hyperplane of the rate region RBC

for a relay power distribution β1= 1 −β2=βis given by ϕ(β) = tan ϑ(β) =

tan |h2|2(1+(1−β)γR|h1|2)

|h1|2(1+βγR|h2|2).

Proof of Lemma 2.56. The normal vector of the parametrized boundary [R−→

R2(β), R−→

R1(1 −

β)] is given by [−R′

−→

R1(β),R′

−→

R2(β)]

||[R−→

R2(β),R−→

R1(1−β)]|| with R′

−→

R1(β) = d R−→

R1(1−β)

d β =−γR|h1|2

1+γR(1−β)|h1|2and

R′

−→

R2(β) = d R−→

R2(β)

d β =γR|h2|2

1+γRβ|h2|2. Thus the angle ϕ(β)is given by tan R′

−→

R2(β)

−R′

−→

R1(β).

Proof of Theorem 2.10. The weighted rate sum maximum in Theorem 2.10 is given by the

boundary rate pair of RBReq where the angle of the normal vector of the supporting hyper-

plane is equal to the angle of the weight vector, i.e. tan q2/q1. Furthermore, two restrictions

apply simultaneously at boundary intersection rate pairs. Those rate pairs are also optimal

for weight vectors with an angle inbetween the angles of the normal vectors of the restric-

tions at this rate pair. Since for all weight vectors with non-negative elements, i.e. angles

inbetween [0, π/2], the tan is a strictly increasing function. For that reason, we prefer to

characterize the different cases by the argument of the tan. From Theorem 2.5 and its corol-

laries we know the combinatorial structure of the rate region and its boundary. Therefore, in

the following we determine for any weight vector q∈

+\{0}the corresponding boundary

rate pair.

First we characterize all intersection rate pairs. If I1=∅ ⇔ βB1 >1, the MAC single user

bound 1

2R−→

1R is not achievable in the BC phase, therefore the boundary rate pair 1

2RBC(1) is

optimal for all weight vectors ϑ(1) > q2/q1. Similarly, (2.20b) follows from I2=∅. The

MAC single user bound 1

2R−→

1R is active for some weight vectors qif I16=∅ ⇔ βB1 ≤1.

Next we have to distinguish if γRis large enough so that the MAC vertex 1

2νΣ1 can be

reached, i.e. βΣ1 ≥βB1. Then 1

2νΣ1 is optimal for all weight vectors with q2/q1≤1. If

βΣ1 < βB1 ≤1the vertex νΣ1 cannot be supported but the MAC single user bound 1

2R−→

is active for β > βB1. Therefore, the intersection rate pair 1

2RBC(βB1)is optimal for weight

vectors q2/q1≤ϑ(βB1). Similarly, if β2Σ ≤β2B the vertex 1

2ν2Σ is optimal for q2/q1≥1

and the case (2.20f) if not.

Finally, two possible intersection rate pairs of the MAC sum-rate 1

2RΣwith RBC are left.

For both it is necessary that β⋆2>∆β. If β2Σ < βBΣ < βΣ1, i.e. the MAC vertex

2ν2Σ cannot be supported, the intersection rate pair 1

2RBC(βBΣ)is optimal for all weight

vectors with ϑ(βBΣ)≤q2/q1≤1. On the other hand, the MAC vertex 1

2νΣ1 cannot be

132

2.8 Appendix: Proofs

supported if β2Σ < βΣB < βΣ1. Therefore 1

2RBC(βΣB)is optimal for weight vectors with

ϑ(βΣB)≥q2/q1≥1.

For all other weight vectors RBReq is bounded by the rate region of the BC phase. Therefore,

the optimal rate vector 1

2RBC(βBC)is given by the boundary rate pair with the corresponding

normal vector, where the optimal power distribution βBC follows if one solves ϑ(βBC) =

q2/q1for βBC.

2.8.7 Proof of Lemma 2.11

Since for any α∈ A a power distribution [β1, β2]∈ B with αRM

1≤(1 −α)R−→

R2(β1)and

αRM

2≤(1 −α)R−→

R1(β2)exists, it is possible for the relay node to broadcast the messages

from nodes 1 and 2 with rates αRM

1and αRM

2. Accordingly, for any α∈ A we achieve

the bidirectional rate pair [R1, R2] = α[RM

1, RM

2]so that the largest α∗∈ Amaximizes the

bidirectional rate pair in each component. Next we prove by contradiction that α∗is uniquely

characterized by (2.23a) and (2.23b). Therefore, we need that R−→

R2(β1)and R−→

R1(β2)are

continuous and increasing in β1and β2.

Let us assume that for the largest element α∗of the set Aat least one equality is not fulfilled.

Since α∗∈ A, a relay power distribution [β1, β2]∈ B exists so that either α∗RM

1<(1 −

α∗)R−→

R2(β1)or α∗RM

2<(1 −α∗)R−→

R1(β2).

If α∗RM

1<(1−α∗)R−→

R2(β1), we can find a ˜

β1< β1where still α∗RM

1<(1−α∗)R−→

R2(˜

β1)

holds. With ˜

β2= 1 −˜

β1> β2we get α∗RM

2<(1 −α∗)R−→

R1(˜

β2). Since in both cases we

have a strict inequality, we can find an ˜α > α∗while both inequalities are still fulfilled, this

means ˜α∈ A. This contradicts the assumption that α∗is the largest element in A.

Accordingly, if α∗RM

2<(1−α∗)R−→

R1(β2), we can find a ˜

β2< β2and a ˜

β1= 1−˜

β2> β1so

that we have α∗RM

1<(1 −α∗)R−→

R2(˜

β1)and α∗RM

2<(1 −α∗)R−→

R1(˜

β2). This allows us to

increase αwhile both inequalities are still fulfilled, what again contradicts the assumption.

Finally, we assume that for the largest element α∗of the set Awe have β1+β2<1.

Therefore, we can find ˜

β1> β1and ˜

β2> β2with [˜

β1,˜

β2]∈ A so that we have α∗RM

(1 −α∗)R−→

R2(˜

β1)and α∗RM

2<(1 −α∗)R−→

R1(˜

β2). This allows us to increase αwhile both

inequalities are still fulfilled, which contradicts the assumption β1+β2<1.

2.8.8 Proof of Theorem 2.12

The equivalent description follows from a transformation of rate pairs [RM

1, RM

2]on the

boundary of the MAC rate region using Lemma 2.11. We get the sets R1,R2, and RΣwith

133

2 Bidirectional Relay Communication using Superposition Encoding

time division parameters α⋆

1(β),α⋆

2(β), and α⋆

Σ(β)if we solve (2.23a) and (2.23b) for α⋆

with RM

1=R−→

1R,RM

2=R−→

2R, and RM

1+RM

2=RMAC

Σrespectively.

For an arbitrary rate pair R∈ R1∩R2∩RΣwe will show in the following that R∈ RBRopt

holds. Since Rhas to be within R1there exists ˜

β1∈[0,1] so that R1≤R11(˜

β1) =

1−α⋆

1(˜

β1)R−→

R2(˜

β1)and R2≤R12(˜

β1) = 1−α⋆

1(˜

β1)R−→

R1(1−˜

β1)hold. This means that

we have R∈1−α⋆

1(˜

β1)RBC. Similarly, from R∈ R2and R∈ RΣwe know that there

exist ˜

β2,˜

βΣ∈[0,1] so that we have R∈1−α⋆

2(˜

β2)RBC and R∈1−α⋆

Σ(˜

βΣ)RBC

respectively. With ˜α= max{α⋆

1(˜

β1), α⋆

2(˜

β2), α⋆

Σ(˜

βΣ)}we still have R∈(1 −˜α)RBC.

Furthermore, we see from

R1≤R11(˜

β1) = α⋆

1(˜

β1)R−→

1R ≤˜αR−→

1R,

R2≤R22(˜

β2) = α⋆

2(˜

β2)R−→

2R ≤˜αR−→

2R,

R1+R2≤RΣ1(˜

βΣ) + RΣ2(˜

βΣ) = α⋆

Σ(˜

βΣ)RMAC

Σ≤˜αRMAC

that we have R∈˜αCMAC. It follows that R∈ RBRopt(˜α)⊆ RBRopt so that we conclude

that R1∩R2∩RΣ⊆ RBRopt.

For the reverse inclusion we show that for any rate pair R∈ RBRopt we also have R∈ R1∩

R2∩RΣ. For any R∈ RBRopt there exists an ˜α∈[0,1] so that R∈˜αCMAC ∩(1−˜α)RBC

holds. From R∈˜αCMAC it follows that R1≤˜αR−→

1R,R2≤˜αR−→

2R, and R1+R2≤˜αRMAC

Σ,

which is equivalent to (1 −˜α)≥R−→

1R−R1

R−→

,(1 −˜α)≥R−→

2R−R2

R−→

, and (1 −˜α)≥RMAC

Σ−R1−R2

RMAC

respectively. Likewise, from (1 −˜α)RBC we know that there exists a ˜

β∈[0,1] where we

have R1≤(1−˜α)R−→

R2(˜

β)and R2≤(1−˜α)R−→

R1(1−˜

β). If we solve (1−˜α)≥R−→

1R−R1

R−→

and

R1≤(1 −˜α)R−→

R2(˜

β)for R1we get R1≤R−→

1RR−→

R2(˜

β)

R−→

1R+R−→

R2(˜

β)=1−α⋆

1(˜

β)R−→

R2(˜

β) = R11(˜

β).

We can use the previous inequalities to bound R2as follows

R2≤(1−˜α)R−→

R1(1−˜

β) = R−→

1R −R1

R−→

R1(1−˜

β)≤1−R11(˜

β)

R−→

1R R−→

R1(1−˜

β) = R12(˜

β).

It follows that R∈ R1. If we use (1 −˜α)≥R−→

2R−R2

R−→

and (1 −˜α)≥RMAC

Σ−R1−R2

RMAC

can similarly conclude with R1≤(1 −˜α)R−→

R2(˜

β)and R2≤(1 −˜α)R−→

R1(1 −˜

β)that

R1≤R21(˜

β)and R2≤R22(˜

β)so that R∈ R2and that R1≤RΣ1(˜

β)and R2≤RΣ2(˜

β)

so that R∈ RΣrespectively. This means that we have R∈ R1∩R2∩RΣfrom which we

conclude that RBRopt ⊆ R1∩R2∩RΣholds. Hence, we proved the equality RBRopt =

R1∩R2∩RΣ.

134

2.8 Appendix: Proofs

2.8.9 Proof of Corollary 2.13

For the proof of the results we basically look at the derivatives of some rate functions. For

notational simplicity we omit the arguments; but we note the subtle particularities, which

however do not change the main results.

For a specific but arbitrary β∈[0,1], let R−→

R1,R−→

R2,R′

−→

R1,R′

−→

R2,R′′

−→

R1, and R′′

−→

R2 denote R−→

R1(β)

and R−→

R2(1 −β)and its first and second derivatives with respect to β. It can be easily seen

that we have R−→

R1,R−→

R2,R′

−→

R2 >0and R′

−→

R1,R′′

−→

R1,R′′

−→

R2 <0for all β∈[0,1] except for

β= 0, where we have R−→

R2 = 0, and for β= 1, where we have R−→

R1 = 0. Later, we will still

achieve strict monotony since we never have equality for both simultaneously.

From this we get the inequalities for the first derivatives

dR11(β)

dβ=R2

−→

1RR′

−→

(R−→

1R +R−→

R2)2≥0,

dR21(β)

dβ=R−→

2R(R′

−→

R2(R−→

2R +R−→

R1)−R−→

R2R′

−→

R1)

(R−→

2R +R−→

R1)2>0,

dRΣ1(β)

dβ=RMAC

Σ(R′

−→

R2(RMAC

Σ+R−→

R1)−R−→

R2R′

−→

R1))

(RMAC

Σ+R−→

R1 +R−→

R2)2>0.

Therefore, R11(β)is increasing and R21(β)and RΣ1(β)are strictly increasing with β. Ac-

cordingly, from

dR12(β)

dβ=R−→

1R(R′

−→

R1(R−→

1R +R−→

R2)−R−→

R1R′

−→

R2)

(R−→

1R +R−→

R2)2<0,

dR22(β)

dβ=R2

−→

2RR′

−→

(R−→

2R +R−→

R1)2≤0,

dRΣ2(β)

dβ=RMAC

Σ(R′

−→

R1(RMAC

Σ+R−→

R2)−R−→

R1R′

−→

R2))

(RMAC

Σ+R−→

R1 +R−→

R2)2<0

we see that R22(β)is decreasing and R12(β)and RΣ2(β)are strictly decreasing with β.

Therefore, the functions R1(β),R2(β), and RΣ(β)are parameterizations of the boundaries

of R1,R2, and RΣin the first quadrant.

The normalized normal vector for the parametrized curve R1(β)is given by −

dR12(β)

dβ,dR11(β)

dβ||R1(β)|| with the angle ϕ1(β) = arctan −q1(β)where q1(β) =

dR11(β)

dβdR12(β)

dβ. Since both components of the normal vector are nonnegative, the function

135

2 Bidirectional Relay Communication using Superposition Encoding

ϕ1has a range [0, π/2]. Since R′′

−→

R2R′

−→

R1 −R′′

−→

R1R′

−→

R2 >0for all β∈[0,1] the derivative

dq1(β)

dβ=R−→

1R(R′′

−→

R2R′

−→

R1 −R′′

−→

R1R′

−→

R2)(R−→

1R +R−→

R2)

((R′

−→

R1(R−→

1R +R−→

R2)−R−→

R1R′

−→

R2))2

is positive. Since arctan is strictly increasing, it follows that ϕ1(β)is strictly decreasing.

Since R11(β)is increasing and R12(β)and ϕ1(β)are strictly decreasing it follows that R1

is convex.

Similarly, we get the angles ϕ2(β) = arctan −q2(β)and ϕΣ(β) = arctan −qΣ(β)with

q2(β) = dR21(β)

dβdR22(β)

dβand qΣ(β) = dRΣ1(β)

dβdRΣ2(β)

dβfor the normal vectors of rate pairs

on the parametrized curves R2(β)and RΣ(β)respectively. Again, ϕ2(β)and ϕΣ(β)are

strictly decreasing for all βsince the derivatives

dq2(β)

dβ=R−→

2R(R′′

−→

R2R′

−→

R1 −R′′

−→

R1R′

−→

R2)(R−→

2R +R−→

R1)

((R′

−→

R2(R−→

2R +R−→

R1)−R−→

R2R′

−→

R1))2,

dqΣ(β)

dβ=RMAC

Σ(R′′

−→

R2R′

−→

R1 −R′′

−→

R1R′

−→

R2)(RMAC

Σ+R−→

R2 +R−→

R1)

((R′

−→

R1(RMAC

Σ+R−→

R2)−R−→

R1R′

−→

R2))2

are positive. Therefore, R2and RΣare convex.

Finally, RBRopt is convex since the intersection of convex sets is itself convex.

2.8.10 Proof of Proposition 2.14

We have an intersection between R1(β)and RΣ(β)if there exist ˆ

β, ˇ

β∈[0,1] with

[R11(ˆ

β), R12(ˆ

β)] = [RΣ1(ˇ

β), RΣ2(ˇ

β)]. First we consider the case that the intersection point

is not on the axes. Then from

R11(ˆ

β)

R12(ˆ

β)=R−→

R2(ˆ

β)

R−→

R1(1 −ˆ

β)=R−→

R2(ˇ

β)

R−→

R1(1 −ˇ

β)=R11(ˇ

β)

R12(ˇ

β)

and the fact that R−→

R2(β)and R−→

R1(1 −β)are strictly increasing and decreasing in βwe

conclude that for an intersection point we have ˆ

β=ˇ

β. Since α⋆

1(β), α⋆

Σ(β)<1holds

for any β, for an intersection point on the axes we either need R−→

R2(ˆ

β) = R−→

R2(ˇ

β) = 0 or

R−→

R1(1 −ˆ

β) = R−→

R1(1 −ˇ

β) = 0 so that we have either ˆ

β=ˇ

β= 0 or ˆ

β=ˇ

β= 1. It

follows that for an intersection point between R1(β)and RΣ(β)we always have ˆ

β=ˇ

β.

The same arguments apply for an intersection point between R2(β)and RΣ(β)and R1(β)

and R2(β).

136

2.8 Appendix: Proofs

Accordingly, we have an intersection point between R1(β)and RΣ(β)iff there exists a

β1Σ ∈[0,1] with R11(β1Σ) = R1Σ(β1Σ)and R21(β1Σ) = R2Σ(β1Σ). This is fulfilled iff

we have

R−→

1R +R−→

R2(β1Σ)=RMAC

RMAC

Σ+R−→

R1(1 −β1Σ) + R−→

R2(β1Σ),

which is equivalent to the equation R−→

RMAC

Σ−R−→

=R−→

R2(β1Σ)

R−→

R1(1−β1Σ). Since this is exactly equation

(2.24) we see that β1Σ corresponds to the unique optimal relay power distribution of the

MAC rate pair [RM

1, RM

2] = [R−→

1R, RMAC

Σ−R−→

1R] = ν1Σ ∈ CMAC, cf. (2.5a).

With similar arguments we see that the unique intersection between R1(β)and RΣ(β)at

βΣ2 ∈[0,1] corresponds to [RM

1, RM

2] = [RMAC

Σ−R−→

2R, R−→

2R] = νΣ2 ∈ CMAC.

Finally, we have an intersection of R1(β)and R2(β)iff there exists a β12 ∈[0,1] with

R11(β12) = R12(β12)and R11(β12) = R12(β12). This is fulfilled iff we have

R−→

1R +R−→

R2(β12)=R−→

R−→

2R +R−→

R1(1 −β12) + R−→

R2(β12),

which is equivalent to the equation R−→

R−→

=R−→

R2(β12)

R−→

R1(1−β12). Let us assume that the rate pair

R1(β12)∈ RBRopt. From Lemma 2.11 we know that [R−→

1R, R−→

2R]is the corresponding rate

pair in the MAC phase. But since R−→

1R +R−→

2R > RMAC

Σ, this rate pair is not achievable so

that we have R1(β12)/∈ RBRopt.

The inequalities (2.28a) and (2.28b) follow after simple calculations using the relations

R−→

R2(1) ≤1

RMAC

R−→

R2(1) ≤1

R−→

R2(1) and 1

R−→

R1(1) ≤1

RMAC

R−→

R1(1) ≤1

R−→

R1(1) .

2.8.11 Proof of Corollary 2.15

Since RBRopt is given by the intersection (2.25), the boundary is characterized by the rate

pairs on the boundaries of R1,R2, and RΣwhich are most restrictive. For β= 0 we have

R−→

R2(0) = 0 ⇒R11(0) = R21(0) = RΣ1(0) = 0. From (2.28b) we see that the rate pair

R2(0) is most restrictive for rate pairs on the ordinate [0, R2]. For small β > 0the boundary

R2(β)is still most restrictive until RΣ(β)intersects the boundary at βΣ2. Until the next

intersection, the boundary RΣ(β)is most restrictive, i.e. for β1Σ < β < βΣ2. And finally,

for the last section R1(β)with β≥β1Σ is most restrictive.

137

2 Bidirectional Relay Communication using Superposition Encoding

2.8.12 Proof of Theorem 2.16

For a convex set it is well-known that the weighted rate sum is attained at the boundary rate

pair where the direction of the normal vector is equal to the direction of the weight vector.

This can be easily seen by the following geometrical interpretation. Consider a hyperplane

2with a normal vector qwhich intersects the feasible set RBRopt. Due to orthogonality,

any rate pair of this intersection leads to the same weighted rate sum. In order to maximize

the weighted rate sum we have to shift the hyperplane in the direction of the normal vector

as far as possible until the hyperplane is finally tangential to the boundary of RBRopt. Such a

tangential hyperplane is called a supporting hyperplane. The rate pair where the supporting

hyperplane with normal vector qintersects the boundary is the rate pair with the largest

weighted rate sum for this weight vector.

Accordingly, the procedure to find the optimum is obvious. First we have to find the opti-

mal relay power distribution β∗corresponding to the weight vector q. Therefore, we have

to find the rate pair on the boundary where the angle of the normal vector is equal to the

angle of the weight vector, i.e. θq. Since the boundary is defined section-wise, we have

to distinguish between the following cases: If θq∈[ϕ1(1), ϕ1(β1Σ)], the optimal rate pair

is on the boundary of R1. Therefore, for those angles we can calculate the optimal relay

power distribution using the inverse function ϕ−1

1. Similarly, if θq∈[ϕΣ(β1Σ), ϕΣ(βΣ2)] or

θq∈[ϕ2(βΣ2), ϕ2(0)] the optimal rate pair is on the boundary of RΣor R2respectively. For

weight vectors with an angle θq∈[0, ϕ1(1)),(ϕ1(β1Σ), ϕΣ(β1Σ)),(ϕΣ(βΣ2), ϕ2(βΣ2)), or

(ϕ2(0), π/2), the rate pairs R1(1),RΣ(β1Σ),RΣ(βΣ2), or R2(0) at the intersections of

R1(β),RΣ(β), and R2(β)with β∗= 1,β1Σ,βΣ2, or 0respectively are optimal. This char-

acterizes the optimal rate pair on the boundary R∗

opt(β∗(q)) and the corresponding optimal

time division parameter α∗(q)for any weight vector q∈

+\{0}.

2.8.13 Proof of Lemma 2.17

From [AS64, 4.1.33] we have x

1+x<ln(1 + x)for x > −1, x 6= 0. Therewith, we get for a

constant a > 0the inequality ln(1 + a/x)>a/x

1+a/x ,for x > 0. We can use this inequality

138

2.8 Appendix: Proofs

to see that the first derivatives

d˜

R−→

1R(α)

dα=1

ln(2) ln 1 + γ1|h1|2

α−

γ1|h1|2

1 + γ1|h1|2

α!>0

d˜

R−→

2R(α)

dα=1

ln(2) ln 1 + γ2|h2|2

α−

γ2|h2|2

1 + γ2|h2|2

α!>0

d˜

RMAC

Σ(α)

dα=1

ln(2) ln 1 + γ1|h1|2+γ2|h2|2

α−

γ1|h1|2+γ2|h2|2

1 + γ1|h1|2+γ2|h2|2

α!>0

are positive so that ˜

R−→

1R(α),˜

R−→

2R(α), and ˜

RMAC

Σ(α)are strictly increasing for α∈(0,1).

The second derivatives of ˜

R−→

1R(α),˜

R−→

2R(α), and ˜

RMAC

Σ(α)are given by

d2˜

R−→

1R(α)

dα2=−γ1|h1|22

α31 + γ1|h1|2

α2ln(2) ≤0

d2˜

R−→

2R(α)

dα2=−γ2|h2|22

α31 + γ2|h2|2

α2ln(2) ≤0

d2˜

RMAC

Σ(α)

dα2=−γ1|h1|2+γ2|h2|22

α31 + γ1|h1|2+γ2|h2|2

α2ln(2) ≤0.

Since ˜

R−→

1R(α),˜

R−→

2R(α), and ˜

RMAC

Σ(α)are non-positive for all α∈(0,1) it follows that they

are concave.

2.8.14 Proof of Lemma 2.18

From [AS64, 4.1.33] we have x

1+x<ln(1 + x)for x > −1, x 6= 0. Therewith, we get

for a constant a > 0the inequality ln(1 + a

1−x)>

1−x

1+ a

1−x,for x∈[0,1). We can use this

inequality to see that for an arbitrary but fixed βthe first partial derivatives with respect to

∂˜

R−→

R2(α, β)

∂α =1

ln(2) γR|h2|2β

1−α

1 + γR|h2|2β

1−α−ln 1 + γR|h2|2β

1−α!<0

∂˜

R−→

R1(α, β)

∂α =1

ln(2) γR|h1|2(1−β)

1−α

1 + γR|h1|2(1−β)

1−α−ln 1 + γR|h1|2(1 −β)

1−α!<0

139

2 Bidirectional Relay Communication using Superposition Encoding

are negative so that ˜

R−→

R2(α, β)is strictly decreasing for α∈(0,1]. The Hessian matrix of

R−→

R2 for α∈(0,1] and β∈[0,1] can be easily calculated as follows

H(˜

R−→

R2) = 



∂2˜

R−→

∂α ∂α

∂2˜

R−→

∂α ∂β

∂2˜

R−→

∂β ∂α

∂2˜

R−→

∂β ∂β 

=−γR|h2|22

1 + γRβ|h2|2

1−α2ln(2) "β2

(1−α)3

(1−α)2

(1−α)21

(1−α)#.

From Q2

i=1 λi(˜

R−→

R2) = det H(˜

R−→

R2)= 0 it follows that at least one eigenvalue

λi(˜

R−→

R2),i= 1,2is zero. Furthermore, from P2

i=1 λi(˜

R−→

R2) = tr H(˜

R−→

R2)=

−(γR|h2|2)2(β2+(1−α)2)

1+ γRβ|h2|2

1−α2ln(2)(1−α)3≤0it additionally follows that the other eigenvalue is non-positive.

Therefore, both eigenvalues λi(˜

R−→

R2),i= 1,2, are non-positive so that the rate function

R−→

R2(α, β)is concave on [0,1) ×[0,1].

Similarly, the Hessian matrix of ˜

R−→

R1 for α∈[0,1) and β∈[0,1] is given by

H(˜

R−→

R1) = 



∂2˜

R−→

∂α ∂α

∂2˜

R−→

∂α ∂β

∂2˜

R−→

∂β ∂α

∂2˜

R−→

∂β ∂β .

=−γR|h1|22

1 + γR(1−β)|h1|2

1−α2ln(2) "(1−β)2

(1−α)3

1−β

(1−α)2

1−β

(1−α)21

(1−α)#.

Again from Q2

i=1 λi(˜

R−→

R1) = det H(˜

R−→

R1)= 0 and P2

i=1 λi(˜

R−→

R1) = tr H(˜

R−→

R1)=

−(γR|h2|2)2(β2+(1−α)2)

1+ γRβ|h2|2

1−α2ln(2)(1−α)3≤0it follows that both eigenvalues λi(˜

R−→

R1),i= 1,2, are non-

positive so that the rate function ˜

R−→

R1(α, β)is concave on [0,1) ×[0,1] as well.

2.8.15 Proof of Proposition 2.19

For any arbitrary but fixed time-division parameter α∈[0,1) and rate pairs R(1),R(2) ∈

RBC(α)we have to show that for any t∈[0,1] we have R(t) := tR(1) + (1 −t)R(2) ∈

RBC(α). Then ˜

RBC(α)is convex.

Let Ri(t) := tR(1)

i+ (1 −t)R(2)

i,i= 1,2, denote the components of R(t). For each rate

pair R(k),k= 1,2, there exists a β(k)∈[0,1] such that we have R(k)

1≤˜

R−→

R2(α, β(k))and

R(k)

2≤˜

R−→

R1(α, β(k)). For t∈[0,1] we define β(t) := tβ(1) + (1 −t)β(2). Then we have

R1(t) = tR(1)

1+ (1 −t)R(2)

1≤t˜

R−→

R2(α, β(1)) + (1 −t)˜

R−→

R2(α, β(2))≤˜

R−→

R2(α, β(t))

R2(t) = tR(1)

2+ (1 −t)R(2)

2≤t˜

R−→

R1(α, β(1)) + (1 −t)˜

R−→

R1(α, β(2))≤˜

R−→

R1(α, β(t))

140

2.8 Appendix: Proofs

where the last inequalities follow from concavity of ˜

R−→

R2(α, β)and ˜

R−→

R1(α, β)for β∈[0,1].

This means that for all t∈[0,1] we have β(t)∈[0,1] and R1(t)≤˜

R−→

R2(α, β(t)) and

R1(t)≤˜

R−→

R2(α, β(t)) so that R(t)∈˜

RBC(α)follows. For α= 0 we have ˜

RBC(0) = ∅,

which is convex by definition.

2.8.16 Proof of Lemma 2.23

Since ˜

R−→

R1(α, β)and ˜

R−→

R2(α, β)are continuous, positive, and strictly decreasing for α∈

[0,1) and a fixed βthe absolute value of the rate pair ˜

RBC(α, β)is strictly decreas-

ing as well. Likewise, ˜

R−→

1R(α),˜

R−→

2R(α), and ˜

RMAC

Σ(α)are continuous, positive, and

strictly increasing for α∈(0,1]. Furthermore, for the continuous continuation we have

R−→

R1(1, β) = ˜

R−→

R2(1, β) = ˜

R−→

1R(0) = ˜

R−→

2R(0) = ˜

RMAC

Σ(0) = 0 for all β∈[0,1]. Since

both arguments of the minima are continuous and one is strictly increasing while the other is

strictly decreasing it follows that for the maximizing time-division parameter both arguments

are equal and the maximizing time-division parameter is unique.

2.8.17 Proof of Theorem 2.21

First, we will prove the inclusion ˜

R1∩˜

R2∩˜

RΣ⊆˜

RBRopt. To this end we have to show

that for any arbitrary rate pair R∈˜

R1∩˜

R2∩˜

RΣit follows that R∈˜

RBRopt. Let ϕRand

RRdenote the angle and the radius of the rate pair R, i.e. the rate pair in polar coordinates

is given by R= [RRcos(ϕR), RRsin(ϕR)]. If R∈˜

R1∩˜

R2∩˜

RΣwe have R∈˜

R1,

R∈˜

R2, and R∈˜

RΣso that we have

RR≤min n˜

R1(ϕR),˜

R2(ϕR),˜

RΣ(ϕR)o.

From Lemma 2.23 we know that for each ˜

R1(ϕR),˜

R2(ϕR),˜

RΣ(ϕR)there exist unique

maximizing time-division parameters ˜α∗

1(ϕR),˜α∗

2(ϕR), and ˜α∗

Σ(ϕR)∈[0,1] where we

have

RBC(˜α∗

1(ϕR), ϕR) =

R−→

1R(˜α∗

1(ϕR))

cos(ϕR)

RBC(˜α∗

2(ϕR), ϕR) =

R−→

2R(˜α∗

2(ϕR))

sin(ϕR)

RBC(˜α∗

Σ(ϕR), ϕR) = ˜

RMAC

Σ(˜α∗

Σ(ϕR))

cos(ϕR) + sin(ϕR)

respectively. Let ˜α∗:= max{˜α∗

1(ϕR),˜α∗

2(ϕR),˜α∗

Σ(ϕR)}denote the maximum, which cor-

responds to the shortest broadcast phase. With the power distribution ˜

β∗:= ˜

βBC(˜α∗, ϕR),

141

2 Bidirectional Relay Communication using Superposition Encoding

which is the power distribution of the rate pair on the boundary of ˜

RBC(˜α∗)with angle ϕR,

we have RR≤˜

RBC(˜α∗,˜

β∗)so that R∈˜

RBC(˜α∗)follows. Moreover, since ˜

R−→

1R(α),

R−→

2R(α), and ˜

RMAC

Σ(α)are increasing for α∈[0,1] we have for the individual rate con-

straints

R1=RRcos(ϕR)≤˜

R1(ϕR) cos(ϕR) =

R−→

1R(˜α1(ϕR))

cos(ϕR)cos(ϕR)≤˜

R−→

1R(˜α∗)

R2=RRsin(ϕR)≤˜

R2(ϕR) sin(ϕR) =

R−→

2R(˜α2(ϕR))

sin(ϕR)sin(ϕR)≤˜

R−→

2R(˜α∗)

and for the sum-rate constraint

R1+R2=RR(cos(ϕR) + sin(ϕR)) ≤˜

RΣ(ϕR)(cos(ϕR) + sin(ϕR))

=˜

RMAC

Σ(˜αΣ(ϕR))

cos(ϕR) + sin(ϕR)(cos(ϕR) + sin(ϕR)) = ˜

RMAC

Σ(˜αΣ(ϕR)) ≤˜

RMAC

Σ(˜α∗)

so that R∈˜

CMAC(˜α∗). It follows that R∈˜

RBC(˜α∗)∪˜

CMAC(˜α∗)⊆Sα∈[0,1] ˜

RBC(α)∪

CMAC(α) = ˜

RBRopt.

For the reverse inclusion we show that for any R∈˜

RBRopt it follows that R∈˜

R1∩˜

R2∩

RΣ. For any R= [RRcos(ϕR), RRsin(ϕR)] ∈˜

RBRopt there exists an ˜α∗∈[0,1] such

that R∈˜

CMAC(˜α∗)and R∈˜

RBC(˜α∗). This implies that we have

R1=RRcos(ϕR)≤˜

R−→

1R(˜α∗)⇒RR≤

R−→

1R(˜α∗)

cos(ϕR),(2.88a)

R2=RRsin(ϕR)≤˜

R−→

2R(˜α∗)⇒RR≤

R−→

2R(˜α∗)

sin(ϕR),(2.88b)

R1+R2=RRcos(ϕR) + sin(ϕR)≤˜

RMAC

Σ(˜α∗)⇒RR≤˜

RMAC

Σ(˜α∗)

cos(ϕR) + sin(ϕR).

(2.88c)

For a given time-division ˜α∗the rate pair with largest radius with the same angle as the rate

pair Rhas the relay power distribution ˜

β∗:= β(˜α∗, ϕR). This implies ϕR=ϕBC(˜α∗,˜

β∗).

Then it follows from R∈˜

RBC(˜α)that

R1=RRcos(ϕR)≤˜

R−→

R2(˜α∗,˜

β∗),

R2=RRsin(ϕR)≤˜

R−→

R1(˜α∗,˜

β∗).

Furthermore, we have

h˜

R−→

R2(˜α∗,˜

β∗),˜

R−→

R1(˜α∗,˜

β∗)i=h˜

R−→

R2˜α∗, β(˜α∗, ϕR),˜

R−→

R1˜α∗, β(˜α∗, ϕR)i

=˜

RBC(˜α∗, ϕR) = ˜

RBC(˜α∗, ϕR)[cos(ϕR),sin(ϕR)].

142

2.8 Appendix: Proofs

Therefore, we have RR≤˜

RBC(˜α∗, ϕR). This together with (2.88a), (2.88b), and (2.88c)

gives us

RR≤min (˜

RBC(˜α∗, ϕR),

R−→

1R(˜α∗)

cos(ϕR))

≤max

α∈[0,1] min (˜

RBC(α, ϕR),

R−→

1R(α)

cos(ϕR))=˜

R1(ϕR)

RR≤min (˜

RBC(˜α∗, ϕR),

R−→

2R(˜α∗)

sin(ϕR))

≤max

α∈[0,1] min (˜

RBC(α, ϕR),

R−→

2R(α)

sin(ϕR))=˜

R2(ϕR)

RR≤min (˜

RBC(˜α∗, ϕR),˜

RMAC

Σ(˜α∗)

cos(ϕR) + sin(ϕR))

≤max

α∈[0,1] min (˜

RBC(α, ϕR),˜

RMAC

Σ(α)

cos(ϕR) + sin(ϕR))=˜

RΣ(ϕR)

This let us conclude that

R1=RRcos(ϕR)≤˜

R1(ϕR) cos(ϕR)

R2=RRsin(ϕR)≤˜

R1(ϕR) sin(ϕR))⇒R= [R1, R2]∈˜

R1,

R1=RRcos(ϕR)≤˜

R2(ϕR) cos(ϕR)

R2=RRsin(ϕR)≤˜

R2(ϕR) sin(ϕR))⇒R= [R1, R2]∈˜

R2,

R1=RRcos(ϕR)≤˜

RΣ(ϕR) cos(ϕR)

R2=RRsin(ϕR)≤˜

RΣ(ϕR) sin(ϕR))⇒R= [R1, R2]∈˜

RΣ.

Accordingly, we have R∈˜

R1∩˜

R2∩˜

RΣfinally. Therewith, we have proved ˜

RBRopt =

R1∩˜

R2∩˜

RΣ.

143

2 Bidirectional Relay Communication using Superposition Encoding

2.8.18 Proof of Theorem 2.24

We first prove (2.42a) and (2.42b). For ϕ= 0 and ϕ=π

2we have ˜

RBC(α, 0) = ˜

R−→

R2(α, 1)

and ˜

RBC(α, π

2) = ˜

R−→

R1(α, 0) respectively. Therewith, we have

R1(0) = max

α∈[0,1] min ˜

R−→

R2(α, 1),˜

R−→

1R(α),

RΣ(0) = max

α∈[0,1] min ˜

R−→

R2(α, 1),˜

RMAC

Σ(α),

R2(0) = lim

ϕ→0max

α∈[0,1] min (˜

RBC(α, ϕ),

R−→

2R(α)

sin(φ))= max

α∈[0,1]

R−→

R2(α, 1) = ˜

R−→

R2(0,1).

We know that ˜

R−→

R2(1,1) = 0 and ˜

R−→

R2(α, 1) >0for α < 1and ˜

R−→

1R(0) = ˜

RMAC

Σ(0) =

0and ˜

R−→

1R(α),˜

RMAC

Σ(α)>0for α∈(0,1]. It follows that the optimal time-division

parameters ˜α1(0),˜αΣ(0) are in the open interval (0,1). Since ˜

R−→

1R(α)<˜

RMAC

Σ(α)for all

α∈(0,1) it follows that ˜

R1(0) <˜

RΣ(0). Furthermore, since ˜

R−→

R2(α, 1) <˜

R−→

R2(0,1) for

all α∈(0,1) we have ˜

RΣ(0) <˜

R2(0) so that (2.42a) follows. Similarly, for ϕ=π

2we

have

R2(π

2) = max

α∈[0,1] min ˜

R−→

R1(α, 0),˜

R−→

2R(α),

RΣ(π

2) = max

α∈[0,1] min ˜

R−→

R1(α, 0),˜

RMAC

Σ(α),

R1(π

2) = lim

ϕ→π

max

α∈[0,1] min (˜

RBC(α, ϕ),

R−→

1R(α)

cos(φ))= max

α∈[0,1]

R−→

R1(α, 0) = ˜

R−→

R1(0,0).

As before, we can conclude with ˜

R−→

2R(α)≤˜

RMAC

Σ(α)and ˜

R−→

R1(α, 0) ≤˜

R−→

R2(0,0) for all

α∈(0,1) that (2.42b) holds.

Since the boundaries are continuous curves in the first quadrant we see from (2.42a) and

(2.42b) that boundaries of ˜

R1,˜

R2, and ˜

RΣhave to intersect at least once. In the following

we will see that they intersect at one boundary rate pair only.

For an intersection rate pair between the boundaries ˜

R1and ˜

R2there must exist an angle

˜ϕ12 ∈(0,π

2)where the radii are equal, i.e. we have ˜

R1( ˜ϕ12) = ˜

R2( ˜ϕ12). From Lemma 2.23

we additionally know that

R1( ˜ϕ12) = ˜

RBC(α∗

1( ˜ϕ12),˜ϕ12) =

R−→

1R(˜α1( ˜ϕ12))

cos( ˜ϕ12)

R2( ˜ϕ12) = ˜

RBC(α∗

2( ˜ϕ12),˜ϕ12) =

R−→

2R(˜α2( ˜ϕ12))

sin( ˜ϕ12)

144

2.8 Appendix: Proofs

so that we have ˜

RBC(α∗

1( ˜ϕ12),˜ϕ12) = ˜

RBC(α∗

2( ˜ϕ12),˜ϕ12)>0. Since ˜

RBC(α, ϕ)is

continuous and strictly decreasing for α∈[0,1) and a fixed ϕ∈[0,π

2]it follows that

˜α∗

1( ˜ϕ12) = ˜α∗

2( ˜ϕ12) =: ˜α12.

The equality ˜

R−→

1R(˜α12)

cos( ˜ϕ12)=˜

R−→

2R(˜α12)

sin( ˜ϕ12)characterizes the radius of the intersection rate pair of the

individual rate constraints of ˜

CMAC(˜α12). For any αthe rate region ˜

CMAC(α)has only one

intersection rate pair of the individual rate constraints, which angle we denote by ˜ϕ12(α).

Since both rate constraints strictly increase with α, also the radius of the intersection strictly

increases. Since ˜

RBC(α, ϕ)strictly decreases, we can conclude that there is only one time-

division parameter where the intersection rate pair is on the boundary of ˜

RBC(α, ϕ)as well.

It follows that ˜ϕ12 is unique.

From the equality of ˜

R1( ˜ϕ12) = ˜

R2( ˜ϕ12)we can conclude

RMAC

Σ(˜α12)<˜

R−→

1R(˜α12) + ˜

R−→

2R(˜α12) = ˜

RBC(˜α12,˜ϕ12)(sin( ˜ϕ12) + cos( ˜ϕ12)).

Since ˜

RMAC

Σ(α)and ˜

RBC(α, ϕ)are strictly increasing and decreasing for αand fixed ϕ, it

follows that for the optimal time-division parameter we have ˜α∗

Σ( ˜ϕ12)<˜α12 so that

R1( ˜ϕ12) = ˜

R2( ˜ϕ12) = ˜

RBC(˜α12,˜ϕ12)>˜

RBC(˜α∗

Σ( ˜ϕ12),˜ϕ12) = ˜

RΣ( ˜ϕ).

Since ˜

R−→

2R(α, ϕ)and ˜

RBC(α, ϕ)are strictly increasing and decreasing for αand fixed ϕ, it

follows that for the optimal time-division parameter we have ˜α∗

2( ˜ϕΣ1)<˜αΣ1 so that

R1( ˜ϕΣ1) = ˜

RΣ( ˜ϕΣ1) = ˜

RBC(˜αΣ1,˜ϕΣ1)<˜

RBC(˜α∗

2( ˜ϕΣ1),˜ϕΣ1) = ˜

R2( ˜ϕ).

The discussion for the intersections between the boundaries of ˜

RΣand ˜

R1or ˜

R2is similar,

which we will present here for completeness. For an intersection of the boundaries ˜

RΣ

and ˜

R1there must exist an angle ˜ϕΣ1 ∈(0,π

2)where the radii are equal, i.e. we have

RΣ( ˜ϕΣ1) = ˜

R1( ˜ϕΣ1). From Lemma 2.23 we additionally know that

R1( ˜ϕΣ1) = ˜

RBC(α∗

1( ˜ϕΣ1),˜ϕΣ1) =

R−→

1R(˜α1( ˜ϕΣ1))

cos( ˜ϕΣ1)

RΣ( ˜ϕΣ1) = ˜

RBC(α∗

Σ( ˜ϕΣ1),˜ϕΣ1) = ˜

RMAC

Σ(˜αΣ( ˜ϕΣ1))

sin( ˜ϕΣ1) + cos( ˜ϕΣ1)

so that we have ˜

RBC(α∗

1( ˜ϕΣ1),˜ϕΣ1) = ˜

RBC(α∗

Σ( ˜ϕΣ1),˜ϕΣ1)>0. Since ˜

RBC(α, ϕ)is

continuous and strictly decreasing for α∈[0,1) and a fixed ϕ∈[0,π

2]it follows that

˜α∗

Σ( ˜ϕΣ1) = ˜α∗

2( ˜ϕΣ1) =: ˜αΣ1.

The equality ˜

R−→

1R(˜αΣ1)

cos( ˜ϕΣ1)=˜

RMAC

Σ(˜αΣ1)

sin( ˜ϕΣ1)+cos( ˜ϕΣ1)characterizes the radius of the intersection rate

pair of the individual rate constraint and the sum-rate constraint of ˜

CMAC(˜αΣ1). For any

145

2 Bidirectional Relay Communication using Superposition Encoding

αthe rate region ˜

CMAC(α)has only one intersection rate pair, which angle we will denote

by ˜ϕΣ1(α). Since the individual and sum-rate constraints strictly increase with α, also the

radius of the intersection strictly increases. Since ˜

RBC(α, ϕ)strictly decreases, we can con-

clude that there is only one time-division parameter where the intersection rate pair is on the

boundary of ˜

RBC(α, ϕ)as well. It follows that ˜ϕΣ1 is unique.

From the equality of ˜

R1( ˜ϕΣ1) = ˜

RΣ( ˜ϕΣ1)we can conclude

R−→

2R(˜αΣ1)>˜

RMAC

Σ(˜αΣ1)−˜

R−→

1R(˜αΣ1) = ˜

RBC(˜αΣ1,˜ϕΣ1) sin( ˜ϕΣ1).

Since ˜

R−→

2R(α)and ˜

RBC(α, ϕ)are strictly increasing and decreasing for αand fixed ϕ, it

follows that for the optimal time-division parameter we have ˜α∗

2( ˜ϕΣ1)<˜αΣ1 so that

R1( ˜ϕΣ1) = ˜

RΣ( ˜ϕΣ1) = ˜

RBC(˜αΣ1,˜ϕΣ1)<˜

RBC(˜α∗

2( ˜ϕΣ1),˜ϕΣ1) = ˜

R2( ˜ϕ).

Finally, for an intersection of the boundaries ˜

R2and ˜

RΣthere must exist an angle ˜ϕ2Σ ∈

(0,π

2)where the radii are equal, i.e. we have ˜

RΣ( ˜ϕ2Σ) = ˜

R2( ˜ϕ2Σ). From Lemma 2.23 we

additionally know that

R2( ˜ϕ2Σ) = ˜

RBC(α∗

1( ˜ϕ2Σ),˜ϕ2Σ) =

R−→

2R(˜α1( ˜ϕ2Σ))

sin( ˜ϕ2Σ)

RΣ( ˜ϕ2Σ) = ˜

RBC(α∗

Σ( ˜ϕ2Σ),˜ϕ2Σ) = ˜

RMAC

Σ(˜αΣ( ˜ϕ2Σ))

sin( ˜ϕ2Σ) + cos( ˜ϕ2Σ)

so that we have ˜

RBC(α∗

1( ˜ϕ2Σ),˜ϕ2Σ) = ˜

RBC(α∗

Σ( ˜ϕ2Σ),˜ϕ2Σ)>0. Since ˜

RBC(α, ϕ)is

continuous and strictly decreasing for α∈[0,1) and a fixed ϕ∈[0,π

2]it follows that

˜α∗

2( ˜ϕ2Σ) = ˜α∗

Σ( ˜ϕ2Σ) =: ˜α2Σ.

The equality ˜

R−→

2R(˜α2Σ)

sin( ˜ϕ2Σ)=˜

RMAC

Σ(˜α2Σ)

sin( ˜ϕ2Σ)+cos( ˜ϕ2Σ)characterizes the radius of the intersection rate

pair of the individual rate constraint and the sum-rate constraint of ˜

CMAC(˜α2Σ). For any

αthe rate region ˜

CMAC(α)has only one intersection rate pair, which angle we will denote

by ˜ϕ2Σ(α). Since the individual and sum-rate constraints strictly increase with α, also the

radius of the intersection strictly increases. Since ˜

RBC(α, ϕ)strictly decreases, we can con-

clude that there is only one time-division parameter where the intersection rate pair is on the

boundary of ˜

RBC(α, ϕ)as well. It follows that ˜ϕ2Σ is unique.

From the equality of ˜

R2( ˜ϕ2Σ) = ˜

RΣ( ˜ϕ2Σ)we can conclude

R−→

1R(˜α2Σ)>˜

RMAC

Σ(˜α2Σ)−˜

R−→

2R(˜α2Σ) = ˜

RBC(˜α2Σ,˜ϕ2Σ) cos( ˜ϕ2Σ).

Since ˜

R−→

1R(α)and ˜

RBC(α, ϕ)are strictly increasing and decreasing for αand fixed ϕ, it

follows that for the optimal time-division parameter we have ˜α∗

1( ˜ϕ2Σ)<˜α2Σ so that

R2( ˜ϕ2Σ) = ˜

RΣ( ˜ϕ2Σ) = ˜

RBC(˜α2Σ,˜ϕ2Σ)<˜

RBC(˜α∗

1( ˜ϕ2Σ),˜ϕ2Σ) = ˜

R1( ˜ϕ).

This finishes the proof.

146

2.8 Appendix: Proofs

2.8.19 Proof of Corollary 2.25

Since ˜

RBRopt is given by the intersection (2.39), the boundary is characterized by the rate

pairs on the boundaries of ˜

R1,˜

R2, and ˜

RΣwhich are most restrictive. For ϕ= 0 we

know from (2.42a) that radius ˜

R1(0) is most restrictive for rate pairs on the abscissa [R1,0].

This remains the case until the boundary of ˜

RΣintersects the boundary of ˜

R1, i.e. for

ϕ≤˜ϕ1Σ. Then the radius ˜

RΣ(ϕ)is most restrictive until the boundary of ˜

R2intersects the

boundary of ˜

RΣ, i.e. for ϕ∈( ˜ϕ2Σ,˜ϕΣ1). Finally, for ϕ∈[π

2,˜ϕ2Σ]the radius ˜

R2(ϕ)is most

restrictive.

2.8.20 Proof of Theorem 2.26

We first prove that ˜

R1is convex. Therefore, we have to show that for any two rate pairs

R(k)= [R(k)

1, R(k)

2]∈˜

R1,k= 1,2, we have R(t) = [R1(t), R2(t)] := tR(1) + (1 −

t)R(2) ∈˜

R1for all t∈[0,1]. Therefore, let ϕR(k)and RR(k)denote the angle and radius of

the rate vector R(k),k= 1,2.

From R(k)∈˜

R1it follows that RR(k)≤˜

R1(ϕR(k)),k= 1,2. According to Lemma 2.23

there exist time-division parameters ˜α(k)

1:= α∗

1(ϕR(k))with RR(k)≤˜

R−→

1R(˜α(k)

cos(ϕR(k))so that we

have

R(k)

1=RR(k)cos(ϕR(k))≤˜

R−→

1R(˜α(k)

1), k = 1,2.(2.89)

Then from Proposition 2.20 we additionally get the relay distribution factors ˜

β(k)

1:=

βBC(˜α(k)

1, ϕR(k))so that we have RR(k)≤˜

RBC(˜α(k)

1,˜

β(k)

1),k= 1,2, which is equivalent

R(k)

1≤˜

R−→

R2(˜α(k)

1,˜

β(k)

1), R(k)

2≤˜

R−→

R1(˜α(k)

1,˜

β(k)

1).(2.90)

For t∈[0,1] let us define ˜α1(t) := t˜α(1)

1+(1−t)˜α(2)

1and ˜

β1(t) := t˜

β(1)

1+(1−t)˜

β(2)

1. From

Lemma 2.17 and 2.18 we know that ˜

R−→

1R(α),˜

R−→

R2(α, β), and ˜

R−→

R1(α, β)are concave.

Then let ˜ϕ1(t)be the angle of [min{˜

R−→

R2(˜α1(t),˜

β1(t)),˜

R−→

1R(˜α1(t))},˜

R−→

R1(˜α1(t),˜

β1(t))],

which is given by

˜ϕ1(t) := 





arctan ˜

R−→

R1(˜α1(t),˜

β1(t))

min{˜

R−→

R2(˜α1(t),˜

β1(t)),˜

R−→

1R(˜α1(t))},if ˜

β1(t)6= 0

2,if ˜

β1(t) = 0.

Note that if ˜

R−→

R2(˜α1(t),˜

β1(t)) ≤˜

R−→

1R(˜α1(t)) we have ˜ϕ1(t) = ˜ϕBC(˜α1(t),˜

β1(t)), else

we have ˜ϕ1(t)>˜ϕBC(˜α1(t),˜

β1(t)) because the inverse function of the tangent is strictly

increasing.

147

2 Bidirectional Relay Communication using Superposition Encoding

In the following, we show that the boundary rate pair on ˜

R1with angle ˜ϕ1(t)is component-

wise larger than R(t). To this end for the radius for this boundary rate pair we have

R1( ˜ϕ1(t)) = max

α∈[0,1] min (˜

RBC(α, ˜ϕ1(t)),

R−→

1R(α)

cos( ˜ϕ1(t)))

=˜

RBC(˜α∗

1( ˜ϕ1(t)),˜ϕ1(t)) =

R−→

1R(˜α∗

1( ˜ϕ1(t)))

cos( ˜ϕ1(t)) (2.91)

≥min (˜

RBC(˜α1(t),˜ϕ1(t)),

R−→

1R(˜α1(t))

cos( ˜ϕ1(t)) ).(2.92)

We will now show that R1(t)≤˜

R1(˜α1(t)) cos( ˜ϕ1(t)) and R2(t)≤˜

R1( ˜ϕ1(t)) sin( ˜ϕ1(t)).

Therefore, we have to distinguish between two cases of the minimization in (2.92).

First, if ˜

RBC(˜α1(t),˜ϕ1(t)) >˜

R−→

1R(˜α1(t))

cos( ˜ϕ1(t)) , which implies that ˜ϕ1(t)<π

2, we have

cos( ˜ϕ1(t)) ˜

R1( ˜ϕ1(t)) = ˜

R−→

1R(˜α∗

1( ˜ϕ1(t))) ≥˜

R−→

1R(˜α1(t))

≥t˜

R−→

1R(˜α(1)

1) + (1 −t)˜

R−→

1R(˜α(2)

1)≥tR(1)

1+ (1 −t)R(2)

1=R1(t)

where the first equality comes from (2.91) and the inequalities follow from (2.92), from the

concavity of ˜

R−→

1R(α), and from the definition of ˜α(k)

1,k= 1,2, respectively. Further, we can

conclude

sin( ˜ϕ1(t)) ˜

R1( ˜ϕ1(t)) = tan( ˜ϕ1(t)) ˜

R−→

1R(˜α∗

1( ˜ϕ1(t))) ≥tan( ˜ϕ1(t)) ˜

R−→

1R(˜α1(t))

R−→

R1(˜α1(t),˜

β1(t)) ˜

R−→

1R(˜α1(t))

min{˜

R−→

R2(˜α1(t),˜

β1(t)),˜

R−→

1R(˜α1(t))}

≥

R−→

R1(˜α1(t),˜

β1(t)) ˜

R−→

1R(˜α1(t))

R−→

1R(˜α1(t)) =˜

R−→

R1(˜α1(t),˜

β1(t))

≥t˜

R−→

R1(˜α(1)

1,˜

β(1)

1) + (1 −t)˜

R−→

R1(˜α(2)

1,˜

β(2)

1)≥tR(1)

2+ (1 −t)R(2)

2=R2(t)

where the first equality and inequality are consequences of (2.91) and (2.92), the next follow

from the definition of ˜ϕ1(t), from the concavity of ˜

R−→

R1(α, β), and from the definition of

˜α(k)

1,k= 1,2, respectively.

For the case ˜

RBC(˜α1(t),˜ϕ1(t)) ≤˜

R−→

1R(˜α1(t))

cos( ˜ϕ1(t)) , which is equivalent to

R−→

1R(˜α1(t)) ≥˜

RBC(˜α1(t),˜ϕ1(t)) cos( ˜ϕ1(t)) = ˜

R−→

R2(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))),(2.93)

we first prove the following claim.

148

2.8 Appendix: Proofs

Claim 2.57. From (2.93) it follows that ˜

R−→

R2(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))) =

R−→

R2(˜α1(t),˜

β1(t)).

Proof. If ˜ϕ1(t) = π

2then we have ˜

βBC(˜α1(t),π

2) = 0 = ˜

β1(t)so that the claim

holds. For ˜ϕ1(t)<π

2the claim will be proved by contradiction. To this end we as-

sume that from (2.93) it follows that ˜

R−→

R2(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))) 6=˜

R−→

R2(˜α1(t),˜

β1(t)).

Then we have to distinguish between two cases. First, we assume that we have

R−→

R2(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))) >˜

R−→

R2(˜α1(t),˜

β1(t)). Since ˜

R−→

R2(α, β)is strictly increas-

ing for βwe have ˜

βBC(˜α1(t),˜ϕ1(t)) >˜

β1(t). This implies

R−→

R1(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))) <˜

R−→

R1(˜α1(t),˜

β1(t))

since ˜

R−→

R1(α, β)is strictly decreasing for β. Further, with (2.93) we have

R−→

1R(˜α1(t)) ≥˜

R−→

R2(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))) >˜

R−→

R2(˜α1(t),˜

β1(t))

so that we get the contradiction for the first case

tan ˜ϕ1(t) =

R−→

R1(˜α1(t),˜

β1(t))

R−→

R2(˜α1(t),˜

β1(t)) >

R−→

R1(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t)))

R−→

R2(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))) = tan ˜ϕ1(t).

Now we assume ˜

R−→

R2(˜α1(t),˜

β1(t)) >˜

R−→

R2(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))) so that ˜

β1(t)>

βBC(˜α1(t),˜ϕ1(t)) and ˜

R−→

R1(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))) >˜

R−→

R1(˜α1(t),˜

β1(t)) follow imme-

diately. Then we get a contradiction for the second case from the following

tan ˜ϕ1(t) =

R−→

R1(˜α1(t),˜

β1(t))

min{˜

R−→

R2(˜α1(t),˜

β1(t)),˜

R−→

1R(˜α1(t))}

≤

R−→

R1(˜α1(t),˜

β1(t))

min{˜

R−→

R2(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))),˜

R−→

1R(˜α1(t))}

R−→

R1(˜α1(t),˜

β1(t))

R−→

R2(˜α1(t),˜

β1(t)) <

R−→

R1(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t)))

R−→

R2(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))) = tan ˜ϕ1(t)

where we used the fact that for a < b,a, c 6= 0 we have 1

min{b,c}≤1

min{a,c}, (2.93), and the

monotony of ˜

R−→

R2(α, β)and ˜

R−→

R1(α, β)for β.

Note that the claim implies ˜

β1(t) = ˜

βBC(˜α1(t),˜ϕ1(t)). From Claim 2.57 with (2.93)

we conclude that ˜

R−→

1R(˜α1(t)) ≥˜

R−→

R2(˜α1(t),˜

β1(t)) holds so that we have ˜ϕ1(t) =

149

2 Bidirectional Relay Communication using Superposition Encoding

˜ϕBC(˜α1(t),˜

β1(t)). Accordingly, we get

cos( ˜ϕ1(t)) ˜

R1( ˜ϕ1(t)) = cos( ˜ϕ1(t)) ˜

RBC(˜α1(t),˜ϕ1(t)) = ˜

R−→

R2(˜α1(t),˜

β1(t))

≥t˜

R−→

R2(˜α(1)

1,˜

β(1)

1) + (1 −t)˜

R−→

R2(˜α(2)

1,˜

β(2)

1)≥tR(1)

1+ (1 −t)R(2)

1=R1(t)

where we used (2.93), Claim 2.57 and the concavity of ˜

R−→

R2(˜α1(t),˜

β1(t)). For the rate R2(t)

we have

sin( ˜ϕ1(t)) ˜

R1( ˜ϕ1(t)) = sin( ˜ϕ1(t)) ˜

RBC(˜α1(t),˜ϕ1(t)) = tan( ˜ϕ1(t)) ˜

R−→

R2(˜α1(t),˜

β1(t))

R−→

R1(˜α1(t),˜

β1(t)) ˜

R−→

R2(˜α1(t),˜

β1(t))

min{˜

R−→

R2(˜α1(t),˜

β1(t)),˜

R−→

1R(˜α1(t))}

R−→

R1(˜α1(t),˜

β1(t)) ˜

R−→

R2(˜α1(t),˜

β1(t))

min{˜

R−→

R2(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))),˜

R−→

1R(˜α1(t))}

R−→

R1(˜α1(t),˜

β1(t)) ˜

R−→

R2(˜α1(t),˜

β1(t))

R−→

R2(˜α1(t),˜

βBC(˜α1(t),˜ϕ1(t))) =

R−→

R1(˜α1(t),˜

β1(t)) ˜

R−→

R2(˜α1(t),˜

β1(t))

R−→

R2(˜α1(t),˜

β1(t))

=˜

R−→

R1(˜α1(t),˜

β1(t)) ≥t˜

R−→

R1(˜α(1)

1,˜

β(1)

1) + (1 −t)˜

R−→

R1(˜α(2)

1,˜

β(2)

≥tR(1)

2+ (1 −t)R(2)

2=R2(t)

where we have used (2.93) and Claim 2.57, the definition of ˜ϕ1(t), and the concavity of

R−→

R1(˜α1(t),˜

β1(t)). It follows that R(t)∈˜

R1for all t∈[0,1] so that we conclude that ˜

is convex!

The proof of convexity of ˜

R2is similar, but differs at some points so that we will present

it here for completeness. Again we have to show that for any two rate pairs R(k)=

[R(k)

1, R(k)

2]∈˜

R2,k= 1,2, we have R(t) = [R1(t), R2(t)] := tR(1) + (1 −t)R(2) ∈˜

for all t∈[0,1]. Therefore, let ϕR(k)and RR(k)denote the angle and radius of the rate

vector R(k),k= 1,2.

From R(k)∈˜

R2it follows that RR(k)≤˜

R2(ϕR(k)),k= 1,2. According to Lemma 2.23

there exist time-division parameters ˜α(k)

2:= α∗

2(ϕR(k))with RR(k)≤˜

R−→

2R(˜α(k)

sin(ϕR(k))so that we

have

R(k)

2=RR(k)sin(ϕR(k))≤˜

R−→

2R(˜α(k)

2), k = 1,2.(2.94)

Then from Proposition 2.20 we additionally get the relay distribution factors ˜

β(k)

2:=

βBC(˜α(k)

2, ϕR(k))so that we have RR(k)≤˜

RBC(˜α(k)

2,˜

β(k)

2),k= 1,2, which is equivalent

R(k)

1≤˜

R−→

R2(˜α(k)

2,˜

β(k)

2), R(k)

2≤˜

R−→

R1(˜α(k)

2,˜

β(k)

2).(2.95)

150

2.8 Appendix: Proofs

For t∈[0,1] let us define ˜α2(t) := t˜α(1)

2+(1−t)˜α(2)

2and ˜

β2(t) := t˜

β(1)

2+(1−t)˜

β(2)

2. From

Lemma 2.17 and 2.18 we know that ˜

R−→

2R(α),˜

R−→

R2(α, β), and ˜

R−→

R1(α, β)are concave.

Then let ˜ϕ2(t)be the angle of [˜

R−→

R2(˜α2(t),˜

β2(t)),min{˜

R−→

R1(˜α2(t),˜

β2(t)),˜

R−→

2R(˜α2(t))}],

which is given by

˜ϕ2(t) := 





arctan min{˜

R−→

R1(˜α2(t),˜

β2(t)),˜

R−→

2R(˜α2(t))}

R−→

R2(˜α2(t),˜

β2(t)) ,if ˜

β2(t)6= 0

2,if ˜

β2(t) = 0.

Note that if ˜

R−→

R2(˜α2(t),˜

β2(t)) ≤˜

R−→

2R(˜α2(t)) we have ˜ϕ2(t) = ˜ϕBC(˜α2(t),˜

β2(t)), else

we have ˜ϕ2(t)>˜ϕBC(˜α1(t),˜

β1(t)) because the inverse function of the tangent is strictly

increasing.

In the following, we show that the boundary rate pair on ˜

R2with angle ˜ϕ2(t)is component-

wise larger than R(t). To this end for the radius for this boundary rate pair we have

R2( ˜ϕ2(t)) = max

α∈[0,1] min (˜

RBC(α, ˜ϕ2(t)),

R−→

2R(α)

sin( ˜ϕ2(t)))

=˜

RBC(˜α∗

2( ˜ϕ2(t)),˜ϕ2(t)) =

R−→

2R(˜α∗

2( ˜ϕ2(t)))

sin( ˜ϕ2(t)) (2.96)

≥min (˜

RBC(˜α2(t),˜ϕ2(t)),

R−→

2R(˜α2(t))

sin( ˜ϕ2(t)) ).(2.97)

We will now show that R1(t)≤˜

R2( ˜ϕ2(t)) cos( ˜ϕ2(t)) and R2(t)≤˜

R2( ˜ϕ2(t)) sin( ˜ϕ2(t)).

Therefore, we have to distinguish between two cases of the minimization in (2.97).

First, if ˜

RBC(˜α2(t),˜ϕ2(t)) >˜

R−→

2R(˜α2(t))

sin( ˜ϕ2(t)) , which implies that ˜ϕ2(t)>0, we have

cos( ˜ϕ2(t)) ˜

R2( ˜ϕ2(t)) =

R−→

2R(˜α∗

2( ˜ϕ2(t)))

tan( ˜ϕ2(t)) ≥

R−→

2R(˜α2(t))

tan( ˜ϕ2(t))

R−→

R2(˜α2(t),˜

β2(t)) ˜

R−→

2R(˜α2(t))

min{˜

R−→

R1(˜α2(t),˜

β2(t)),˜

R−→

2R(˜α2(t))}

≥

R−→

R2(˜α2(t),˜

β2(t)) ˜

R−→

2R(˜α2(t))

R−→

2R(˜α2(t)) =˜

R−→

R2(˜α2(t),˜

β2(t))

≥t˜

R−→

R2(˜α(1)

2,˜

β(1)

2) + (1 −t)˜

R−→

R2(˜α(2)

2,˜

β(2)

2)≥tR(1)

1+ (1 −t)R(2)

1=R1(t)

where the first equality and inequality are consequences of (2.96) and (2.97), the next follow

from the definition of ˜ϕ2(t), from the concavity of ˜

R−→

R2(α, β), and from the definition of

151

2 Bidirectional Relay Communication using Superposition Encoding

˜α(k)

2,k= 1,2, respectively. Further, we can conclude

sin( ˜ϕ2(t)) ˜

R2( ˜ϕ2(t)) = ˜

R−→

2R(˜α∗

2( ˜ϕ2(t))) ≥˜

R−→

2R(˜α2(t))

≥t˜

R−→

2R(˜α(1)

2) + (1 −t)˜

R−→

2R(˜α(2)

2)≥tR(1)

2+ (1 −t)R(2)

2=R2(t)

where the first equality comes from (2.96) and the inequalities follow from (2.97), from the

concavity of ˜

R−→

2R(α), and from the definition of ˜α(k)

2,k= 1,2, respectively.

For the case ˜

RBC(˜α2(t),˜ϕ2(t)) ≤˜

R−→

2R(˜α2(t))

sin( ˜ϕ2(t)) , which is equivalent to

R−→

2R(˜α2(t)) ≥˜

RBC(˜α2(t),˜ϕ2(t)) sin( ˜ϕ2(t)) = ˜

R−→

R1(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))),(2.98)

we first prove the following claim.

Claim 2.58. From (2.98) it follows that ˜

R−→

R1(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))) =

R−→

R1(˜α2(t),˜

β2(t)).

Proof. If ˜ϕ2(t) = 0 then we have ˜

βBC(˜α2(t),0) = 1 = ˜

β2(t)so that the claim

holds. For ˜ϕ2(t)>0the claim will be proved by contradiction. To this end we as-

sume that from (2.98) it follows that ˜

R−→

R1(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))) 6=˜

R−→

R1(˜α2(t),˜

β2(t)).

Then we have to distinguish between two cases. First, we assume that we have

R−→

R1(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))) >˜

R−→

R1(˜α2(t),˜

β2(t)). Since ˜

R−→

R2(α, β)is strictly increas-

ing for βwe have ˜

βBC(˜α2(t),˜ϕ2(t)) <˜

β2(t). This implies

R−→

R2(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))) <˜

R−→

R2(˜α2(t),˜

β2(t))

since ˜

R−→

R2(α, β)is strictly increasing for β. Further, with (2.98) we have

R−→

2R(˜α2(t)) ≥˜

R−→

R1(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))) >˜

R−→

R1(˜α2(t),˜

β2(t))

so that we get the contradiction for the first case

tan ˜ϕ2(t) =

R−→

R1(˜α2(t),˜

β2(t))

R−→

R2(˜α2(t),˜

β2(t)) <

R−→

R1(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t)))

R−→

R2(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))) = tan ˜ϕ2(t).

Now we assume ˜

R−→

R1(˜α2(t),˜

β2(t)) >˜

R−→

R1(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))) so that ˜

β2(t)<

βBC(˜α2(t),˜ϕ2(t)) and ˜

R−→

R2(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))) >˜

R−→

R2(˜α2(t),˜

β2(t)) follow imme-

152

2.8 Appendix: Proofs

diately. Then we get a contradiction for the second case from the following

tan ˜ϕ2(t) = min{˜

R−→

R1(˜α2(t),˜

β2(t)),˜

R−→

2R(˜α2(t))}

R−→

R2(˜α2(t),˜

β2(t))

≥min{˜

R−→

R1(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))),˜

R−→

2R(˜α2(t))}

R−→

R2(˜α2(t),˜

β2(t))

R−→

R1(˜α2(t),˜

β2(t))

R−→

R2(˜α2(t),˜

β2(t)) >

R−→

R1(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t)))

R−→

R2(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))) = tan ˜ϕ2(t)

where we used the fact that for a < b we have min{b, c} ≥ min{a, c}, (2.98), and the

monotony of ˜

R−→

R1(α, β)and ˜

R−→

R2(α, β)for βand fixed α.

Note that the claim implies ˜

β2(t) = ˜

βBC(˜α2(t),˜ϕ2(t)). From Claim 2.58 with (2.98)

we conclude that ˜

R−→

2R(˜α2(t)) ≥˜

R−→

R2(˜α2(t),˜

β2(t)) holds so that we have ˜ϕ2(t) =

˜ϕBC(˜α2(t),˜

β2(t)). Accordingly, we get

cos( ˜ϕ2(t)) ˜

R2( ˜ϕ2(t)) = cos( ˜ϕ2(t)) ˜

RBC(˜α2(t),˜ϕ2(t))

R−→

R1(˜α2(t),˜

β2(t))

tan( ˜ϕ2(t)) =

R−→

R1(˜α2(t),˜

β2(t)) ˜

R−→

R2(˜α2(t),˜

β2(t))

min{˜

R−→

R1(˜α2(t),˜

β2(t)),˜

R−→

2R(˜α2(t))}

R−→

R1(˜α2(t),˜

β2(t)) ˜

R−→

R2(˜α2(t),˜

β2(t))

min{˜

R−→

R1(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))),˜

R−→

2R(˜α2(t))}

R−→

R1(˜α2(t),˜

β2(t)) ˜

R−→

R2(˜α2(t),˜

β2(t))

R−→

R1(˜α2(t),˜

βBC(˜α2(t),˜ϕ2(t))) =

R−→

R1(˜α2(t),˜

β2(t)) ˜

R−→

R2(˜α2(t),˜

β2(t))

R−→

R1(˜α2(t),˜

β2(t))

=˜

R−→

R2(˜α2(t),˜

β2(t)) ≥t˜

R−→

R2(˜α(1)

2,˜

β(1)

2) + (1 −t)˜

R−→

R2(˜α(2)

2,˜

β(2)

≥tR(1)

1+ (1 −t)R(2)

1=R1(t)

where we have used (2.98) and Claim 2.58, the definition of ˜ϕ2(t), and the concavity of

R−→

R2(˜α2(t),˜

β2(t)). For the rate R2(t)we have

sin( ˜ϕ2(t)) ˜

R2( ˜ϕ2(t)) = sin( ˜ϕ2(t)) ˜

RBC(˜α2(t),˜ϕ2(t)) = ˜

R−→

R1(˜α2(t),˜

β2(t))

≥t˜

R−→

R1(˜α(1)

2,˜

β(1)

2) + (1 −t)˜

R−→

R1(˜α(2)

2,˜

β(2)

2)≥tR(1)

2+ (1 −t)R(2)

2=R2(t)

where we used (2.98), Claim 2.58 and the concavity of ˜

R−→

R1(˜α2(t),˜

β2(t)). It follows that

R(t)∈˜

R2for all t∈[0,1] so that we conclude that ˜

R2is convex!

Finally, we have to prove that ˜

RΣis convex. Therefore, we have to show that for any

two rate pairs R(k)= [R(k)

1, R(k)

2]∈˜

RΣ,k= 1,2, we have R(t) = [R1(t), R2(t)] :=

153

2 Bidirectional Relay Communication using Superposition Encoding

tR(1) + (1 −t)R(2) ∈˜

RΣfor all t∈[0,1]. Therefore, let ϕR(k)and RR(k)denote the angle

and radius of the rate vector R(k),k= 1,2.

From R(k)∈˜

RΣit follows that RR(k)≤˜

RΣ(ϕR(k)),k= 1,2. According to Lemma 2.23

there exist time-division parameters ˜α(k)

Σ:= α∗

Σ(ϕR(k))with RR(k)≤˜

RMAC

Σ(˜α(k)

Σ)

cos(ϕR(k))+sin(ϕR(k))

so that we have

R(k)

1+R(k)

2=RR(k)(cos(ϕR(k)) + sin(ϕR(k))) ≤˜

RMAC

Σ(˜α(k)

Σ), k = 1,2.(2.99)

Then from Proposition 2.20 we additionally get the relay distribution factors ˜

β(k)

Σ:=

βBC(˜α(k)

Σ, ϕR(k))so that we have RR(k)≤˜

RBC(˜α(k)

Σ,˜

β(k)

Σ),k= 1,2, which is equivalent

R(k)

1≤˜

R−→

R2(˜α(k)

Σ,˜

β(k)

Σ), R(k)

2≤˜

R−→

R1(˜α(k)

Σ,˜

β(k)

Σ).(2.100)

For t∈[0,1] let us define ˜αΣ(t) := t˜α(1)

Σ+(1−t)˜α(2)

Σand ˜

βΣ(t) := t˜

β(1)

Σ+(1−t)˜

β(2)

Σ. From

Lemma 2.17 and 2.18 we know that ˜

RMAC

Σ(α),˜

R−→

R2(α, β), and ˜

R−→

R1(α, β)are concave.

We define the rate vector RBC(˜αΣ(t),˜

βΣ(t)) := [ ˜

R−→

R2(˜αΣ(t),˜

βΣ(t)),˜

R−→

R1(˜αΣ(t),˜

βΣ(t))] with

angle ˜ϕBC(˜αΣ(t),˜

βΣ(t)) of the polar coordinates. Furthermore, ϕR(t)specifies the angle of

the rate pair R(t) = [R1(t), R2(t)]. Then the angle ˜ϕΣ(t)of the component-wise dominant

rate pair is given by

˜ϕΣ(t) := (ϕR(t),if ˜

RBC

Σ(˜αΣ(t),˜

βΣ(t)) >˜

RMAC

Σ(˜αΣ(t))

˜ϕBC(˜αΣ(t),˜

βΣ(t)),if ˜

RBC

Σ(˜αΣ(t),˜

βΣ(t)) ≤˜

RMAC

Σ(˜αΣ(t)).(2.101)

with RBC(˜αΣ(t),˜

βΣ(t)) = [ ˜

R−→

R2(˜αΣ(t),˜

βΣ(t)),˜

R−→

R1(˜αΣ(t),˜

βΣ(t))].

In the following, we show that the boundary rate pair on ˜

RΣwith angle ˜ϕΣ(t)is component-

wise larger than R(t). To this end for the radius for this boundary rate pair we have

RΣ( ˜ϕΣ(t)) = max

α∈[0,1] min (˜

RBC(α, ˜ϕΣ(t)),˜

RMAC

Σ(α)

cos( ˜ϕΣ(t)) + sin( ˜ϕΣ(t)))

=˜

RBC(˜α∗

Σ( ˜ϕΣ(t)),˜ϕΣ(t)) = ˜

RMAC

Σ(˜α∗

Σ( ˜ϕΣ(t)))

cos( ˜ϕΣ(t)) + sin( ˜ϕΣ(t)) (2.102)

≥min (˜

RBC(˜αΣ(t),˜ϕΣ(t)),˜

RMAC

Σ(˜αΣ(t))

cos( ˜ϕΣ(t)) + sin( ˜ϕΣ(t))).(2.103)

We will now show that R1(t)≤˜

RΣ( ˜ϕΣ(t)) cos( ˜ϕΣ(t)) and RΣ(t)≤

RΣ( ˜ϕΣ(t)) sin( ˜ϕΣ(t)). Therefore, we have to distinguish between two cases of the

154

2.8 Appendix: Proofs

minimization in (2.103) for the two cases of possible angles ˜ϕΣ(t)of (2.101). To this end

we will use the concavity of ˜

RMAC

Σ(˜αΣ(t)), i.e.

RMAC

Σ(˜αΣ(t)) ≥t˜

RMAC(1)

Σ+ (1 −t)˜

RMAC(2)

Σ≥t(R(1)

1+R(1)

2) + (1 −t)(R(2)

1+R(2)

≥(tR(1)

1+ (1 −t)R(2)

1) + (tR(1)

2+ (1 −t)R(2)

2) = R1(t) + R2(t).

(2.104)

First we consider the case ˜

RBC(˜αΣ(t),˜ϕΣ(t)) >˜

RMAC

Σ(˜αΣ(t))

cos( ˜ϕΣ(t))+sin( ˜ϕΣ(t)) . Then we have to

distinguish between the two cases of (2.103). Since for ˜

RMAC

Σ(˜αΣ(t)) ≥˜

RBC

Σ(˜αΣ(t),˜

βΣ(t))

we have ˜ϕΣ(t) = ˜ϕBC(˜αΣ(t),˜

βΣ(t)) which implies that we have ˜

RBC(˜αΣ(t),˜ϕΣ(t)) =

RBC(˜αΣ(t),˜

βΣ(t)) it follows that this case is not possible. Furthermore, it follows that we

have ˜

RMAC

Σ(˜αΣ(t)) <˜

RBC

Σ(˜αΣ(t),˜

βΣ(t)) with ˜ϕΣ(t) = ϕR(t). Therewith, we get

cos( ˜ϕΣ(t)) ˜

RΣ( ˜ϕΣ(t)) = cos( ˜ϕΣ(t)) ˜

RMAC

Σ(˜α∗

Σ( ˜ϕΣ(t)))

cos( ˜ϕΣ(t)) + sin( ˜ϕΣ(t)) ≥˜

RMAC

Σ(˜αΣ(t)) cos( ˜ϕΣ(t))

cos( ˜ϕΣ(t)) + sin( ˜ϕΣ(t))

=˜

RMAC

Σ(˜αΣ(t))

1 + tan( ˜ϕΣ(t)) =˜

RMAC

Σ(˜αΣ(t))

1 + tan(ϕR(t))

≥R1(t) + R2(t)

1 + tan(ϕR(t))=R1(t) + R2(t)

1 + R2(t)/R1(t)=R1(t)

where the first equality and inequality are consequences of (2.102) and (2.103), then we

used (2.104), the fact ˜ϕΣ(t) = ϕR(t), and finally the definition of ˜ϕΣ(t). Similarly, we can

conclude that

sin( ˜ϕΣ(t)) ˜

RΣ( ˜ϕΣ(t)) = sin( ˜ϕΣ(t)) ˜

RMAC

Σ(˜α∗

Σ( ˜ϕΣ(t)))

cos( ˜ϕΣ(t)) + sin( ˜ϕΣ(t)) ≥˜

RMAC

Σ(˜αΣ(t)) sin( ˜ϕΣ(t))

cos( ˜ϕΣ(t)) + sin( ˜ϕΣ(t))

=˜

RMAC

Σ(˜αΣ(t))

1 + 1/tan( ˜ϕΣ(t)) =˜

RMAC

Σ(˜αΣ(t))

1 + 1/tan(ϕR(t))

≥R1(t) + R2(t)

1 + 1/tan(ϕR(t))=R1(t) + R2(t)

1 + R1(t)/R2(t)=R2(t)

so that it follows that [R1(t), R2(t)] ∈˜

RΣfor this case.

We now consider the case ˜

RBC(˜αΣ(t),˜ϕΣ(t)) ≤˜

RMAC

Σ(˜αΣ(t))

cos( ˜ϕΣ(t))+sin( ˜ϕΣ(t)) . Again we have

to distinguish between the two cases of (2.103). First, we assume that we have

RBC

Σ(˜αΣ(t),˜

βΣ(t)) ≤˜

RMAC

Σ(˜αΣ(t)) so that we have ˜ϕΣ(t) = ˜ϕBC(˜αΣ(t),˜

βΣ(t)) which

implies that we have ˜

βΣ(t) = ˜

βBC(˜αΣ(t),˜ϕΣ(t)) or equivalently ˜

RBC(˜αΣ(t),˜ϕΣ(t)) =

155

2 Bidirectional Relay Communication using Superposition Encoding

RBC(˜αΣ(t),˜

βΣ(t)). Therewith, we have

cos( ˜ϕΣ(t)) ˜

RΣ( ˜ϕΣ(t)) = cos( ˜ϕΣ(t)) ˜

RBC(˜α∗

Σ( ˜ϕΣ(t)),˜ϕΣ(t))

≥cos( ˜ϕΣ(t)) ˜

RBC(˜αΣ(t),˜ϕΣ(t)) = cos( ˜ϕΣ(t)) ˜

RBC(˜αΣ(t),˜

βΣ(t))

=˜

R−→

R2(˜αΣ(t),˜

βΣ(t)) ≥t˜

R−→

R2(˜α(1)

Σ,˜

β(1)

Σ) + (1 −t)˜

R−→

R2(˜α(2)

Σ,˜

β(2)

Σ)

≥tR(1)

1+ (1 −t)R(2)

1=R1(t)

where the first equality and inequality are consequences of (2.102) and (2.103), then we

used the fact ˜

RBC(˜αΣ(t),˜ϕΣ(t)) = ˜

RBC(˜αΣ(t),˜

βΣ(t)) and the concavity of ˜

R−→

R2(α, β).

Similarly, with the concavity of ˜

R−→

R1(α, β)we get

sin( ˜ϕΣ(t)) ˜

RΣ( ˜ϕΣ(t)) = sin( ˜ϕΣ(t)) ˜

RBC(˜α∗

Σ( ˜ϕΣ(t)),˜ϕΣ(t))

≥sin( ˜ϕΣ(t)) ˜

RBC(˜αΣ(t),˜ϕΣ(t)) = sin( ˜ϕΣ(t)) ˜

RBC(˜αΣ(t),˜

βΣ(t))

=˜

R−→

R1(˜αΣ(t),˜

βΣ(t)) ≥t˜

R−→

R1(˜α(1)

Σ,˜

β(1)

Σ) + (1 −t)˜

R−→

R1(˜α(2)

Σ,˜

β(2)

Σ)

≥tR(1)

2+ (1 −t)R(2)

2=R2(t)

so that [R1(t), R2(t)] ∈˜

RΣfor this case as well.

Finally, we assume that we have ˜

RBC(˜αΣ(t),˜ϕΣ(t)) ≤˜

RMAC

Σ(˜αΣ(t))

cos( ˜ϕΣ(t))+sin( ˜ϕΣ(t)) and

RBC

Σ(˜αΣ(t),˜

βΣ(t)) >˜

RMAC

Σ(˜αΣ(t)) which implies that we have ˜ϕΣ(t) = ϕR(t). First note

that ˜

RBC(˜αΣ(t),˜ϕΣ(t)) denotes the radius of the rate pair on the boundary of ˜

RBC(˜αΣ(t))

with angle ˜ϕΣ(t), i.e.

RBC(˜αΣ(t),˜ϕΣ(t)) = max

[R1,R2]∈˜

RBC(˜αΣ(t))

R1sin ˜ϕΣ(t)=R2cos ˜ϕΣ(t)qR2

1+R2

To prove the component-wise dominance of ˜

RBC(˜αΣ(t),˜ϕΣ(t))[cos( ˜ϕΣ(t)),sin( ˜ϕΣ(t))] we

define the downward comprehensive hull16 ˆ

R(t) := dch(RBC(˜αΣ(t),˜

βΣ(t))) of the rate

pair RBC(˜αΣ(t),˜

βΣ(t)). Then, let

Rˆ

R(t)( ˜ϕΣ(t)) := max

[R1,R2]∈ˆ

R(t)

R1sin ˜ϕΣ(t)=R2cos ˜ϕΣ(t)qR2

1+R2

denote the radius of the rate pair on the boundary of the downward comprehensive hull

of the vector RBC(˜αΣ(t),˜

βΣ(t)) = [ ˜

R−→

R2(˜αΣ(t),˜

βΣ(t)),˜

R−→

R1(˜αΣ(t),˜

βΣ(t))] with angle

16A set Ais downward comprehensive in

+, if for any x= [x1, x2]∈ A it follows that for any y= [y1, y2]∈

+with xi≤yi,i= 1,2, we have y∈ A. Then we define the downward comprehensive hull of the vector

x∈

+by the set dch(x) := {y∈

+:yi≤xi, i = 1,2}.

156

2.8 Appendix: Proofs

˜ϕΣ(t). Since we have RBC(˜αΣ(t),˜

βΣ(t)) ∈˜

RBC(˜αΣ(t)), it follows the inclusion ˆ

R(t)⊆

RBC(˜αΣ(t)). Therefrom, we can conclude that Rˆ

R(t)( ˜ϕΣ(t)) ≤˜

RBC(˜αΣ(t),˜ϕΣ(t)) holds.

Furthermore, from the concavity of ˜

R−→

R2(α, β)and ˜

R−→

R1(α, β)we know

R1(t)≤˜

R−→

R2(˜αΣ(t),˜

βΣ(t)), R2(t)≤˜

R−→

R1(˜αΣ(t),˜

βΣ(t)).

It follows that R(t) = [R1(t), R2(t)] ∈ˆ

R(t). Since we have ˜ϕΣ(t) = ϕR(t), we can

conclude that ||R(t)||1=p(R1(t))2+ (R2(t))2≤Rˆ

R(t)( ˜ϕΣ(t)). Altogether, we have

RΣ( ˜ϕΣ(t)) = ˜

RBC(˜α∗

Σ( ˜ϕΣ(t)),˜ϕΣ(t)) ≥˜

RBC(˜αΣ(t),˜ϕΣ(t))

=˜

RBC(˜αΣ(t), ϕR(t))≥Rˆ

R(t)(ϕR(t))≥ ||R(t)||1

so that we have

cos( ˜ϕΣ(t)) ˜

RΣ( ˜ϕΣ(t)) ≥cos(ϕR(t))||R(t)||1=R1

sin( ˜ϕΣ(t)) ˜

RΣ( ˜ϕΣ(t)) ≥sin(ϕR(t))||R(t)||1=R2.

This was the last case where we have shown that [R1(t), R2(t)] ∈˜

RΣfor all t∈[0,1] so

that we can conclude that ˜

RΣis convex as well.

Finally, since the intersection of convex sets is convex it follows that ˜

RBRopt =˜

R1∩˜

R2∩˜

RΣ

is convex as well.

2.8.21 Proof of Corollary 2.30

From the queue evolution equation (2.43) we have

k[n+ 1] ≤Qk[n]−Rk[n]T2+B2

k[n] + 2Bk[n]Qk[n]−Rk[n]T+

≤Qk[n]−Rk[n]T2+B2

k[n] + 2Bk[n]Qk[n]

≤Q2

k[n]−2TQk[n]Rk[n]−Bk[n]

T+Rk[n]T2+B2

k[n]

157

2 Bidirectional Relay Communication using Superposition Encoding

for k= 1,2. Using this inequality the one-step drift gives us

LQ[n+ 1]−LQ[n]Q[n] = q=

2

k=1

k[n+ 1] −Q2

k[n]Q[n] = q

≤

2

k=1 −2TQk[n]Rk[n]−Bk[n]

T+Rk[n]T2+B2

k[n]Q[n] = q

k=1

R2

k[n]|Q[n] = q+

B2

k[n](2.105)

−2T

k=1

qk

Rk[n]|Q[n] = q−

Bk[n]

T

|{z }

=λk(2.106)

≤υ−2T

k=1

qk

Rk[n]|Q[n] = q−

Bk[n]

T,

where we used for the first term of the sum (2.105) the inequality

2

k=1

k[n]|Q[n] = q≤T2max

h∈H,R∈RBR(h)R2

1+R2

2=υ−

B2

k[n].(2.107)

For the second term of the sum (2.106) we have

k=1

Rk[n]|Q[n] = q= max

R∈RBR

k=1

qkRk> λ1+λ2,

since we require [λ1, λ2]to be strictly in the interior of RBR. Accordingly, there exists ˜

ζ > 0

so that the vector [λ1+˜

ζ, λ2+˜

ζ]is still in RBR. Therefore, we get for the second term of

the sum (2.106) the inequality

k=1

qk

Rk[n]|Q[n] = q−λk≥2T˜

k=1

qk.

All together we have

LQ[n+ 1]−LQ[n]Q[n] = q≤υ−ζ

k=1

with ζ= 2T˜

ζ. For any arbitrary ε > 0we can define the compact region Λ := nq∈

+

k=1

qk≤υ+ε

ζo.Whenever q/∈Λwe have 2

k=1

qk>υ+ε

ζand therefore we have a

negative one-step Lyapunov drift

LQ[n+ 1]−LQ[n]Q[n] = q<−ε.

158

2.8 Appendix: Proofs

2.8.22 Proof of Lemma 2.35

The cdf of Znfollows from

FZn(zn) =

min{anxn, bnyn} ≤ zn

= 1 −

min{anxn, bnyn}> zn

= 1 −

anxn> zn

bnyn> zn

= 1 −(1 −FXn(zn/an))(1 −FYn(zn/bn))

and the pdf is given by its derivation. Accordingly, fZ(z)is the derivation of the cdf

FZ(z) =

max

n{zn} ≤ z=Y

z≥zn=Y

FZn(zn),

since we assume that the random variables Znare pairwise independent. Finally, R=

g(Z) = log(1 + Z)/2is the function of the random variable Zwhere g(Z)is strictly in-

creasing and differentiable in Z[PP02, Chap. 5]. For that reason the pdf is given by

fR(R) = fZ(g−1(R))

|g′(g−1(R))|= 2 ln(2)22RfZ(22R−1),

where g′(Z) = 1

2 ln(2)(1+Z)and g−1(R) = 22R−1denote the derivative and inverse of g(Z)

respectively.

2.8.23 Proof of Proposition 2.37

For independent channel realizations the achievable rate region of the η-th relay node con-

tains the regions of all others iff we have |hk,η| ≥ maxn=1,...,N {|hk,n|} for k= 1,2simul-

taneously. With

|hk,η| ≥ |hk,n||hk,η|=F|hk,n|(|hk,η|)it follows that

nRBReq,η ⊇[

nRBReq,no=

k=1

n|hk,η| ≥ max

n=1,...,N{|hk,n|}o

k=1 Z

n|hk,η| ≥ max

n=1,...,N{|hk,n|}|hk,η|of|hk,η|(|hk,η|)d|hk,η|

k=1 ZN

n=1

n6=η

n|hk,η| ≥ |hk,n||hk,η|of|hk,η |(|hk,η|)d|hk,η|.

159

2 Bidirectional Relay Communication using Superposition Encoding

Then the probability that there exists one relay node η∈ {1, . . . , N}which rate region

contains the others is given by the disjoint disjunction

∃η:RRSeq =RBReq,η=

η=1

(RBReq,η ⊇[

nRBReq,n)o

η=1

nRBReq,η ⊇[

nRBReq,no.

In the case of iid Rayleigh fading we have

n|hk,η| ≥ max

n=1,...,N{|hk,n|}o

∞

Z01−exp −|hk,η|2

σ2

kN−12|hk,η|

σ2

exp −|hk,η|2

σ2

kd|hk,η|

σ2

N−1

n=0 N−1

n(−1)n

∞

exp −|hk,η|2(n+ 1)

σ2

kd|hk,η|2

N−1

n=0 N−1

n(−1)n

n+ 1 =1

Finally, this gives us

η=1 RBReq,η ⊇[

nRBReq,no=

η=1

k=1

n|hk,η| ≥ max

n=1,...,N |hk,n|o=1

2.8.24 Proof of Theorem 2.38

Since |hk,n|is Rayleigh distributed, |hk,n|2is exponential distributed with pdf

f|hk,n|2(|hk,n|2) = 1

σ2

k,n

exp −|hk,n|2

σ2

k,n for k= 1,2and n= 1,...,N. According to

(2.22a) and (2.22b) the maximal unidirectional relay rates are given by

R∗

1RSeq =1

2log 1 + max

n{min{γ1|h1,n|2, γR|h2,n|2}},

R∗

2RSeq =1

2log 1 + max

n{min{γ2|h2,n|2, γR|h1,n|2}}.

160

2.8 Appendix: Proofs

Therefore, we can apply Lemma 2.35 to derive the pdf of R∗

kRSeq,k= 1,2. For a given k

we consider Xn:= |h1,n|2and Yn:= |h2,n|2. To be in terms of Lemma 2.35, we define

Zn:= min{γ1Xn, γRYn}. Then according to Remark 2.36 the random variable Znis again

exponential distributed with pdf fZn(zn) = λk,n exp(−znλk,n). with λ1,n =1

γ1σ2

1,n

γRσ2

2,n

and λ2,n =1

γ2σ2

2,n

γRσ2

1,n

respectively. We consider next the random variable

Z:= maxn{Zn}for which we can calculate the pdf as follows

fZ(z) =

n=1

fZn(z)

j=1,j6=n

FZj(z) =

n=1

λk,n e−zλk,n

m=1,m6=n

(1 −e−zλk,m )(2.108)

according to Lemma 2.35. Finally, we can again apply Lemma 2.35 to calculate the pdf

of the random variable R∗

kRSeq = log(1 + Z)as stated in the theorem which finishes the

proof.

2.8.25 Proof of Corollary 2.39

In a scenario with independent Rayleigh distributed fading channels and for a given kwe

can define a random variable Zas in the proof of Theorem 2.38 with a pdf according to

(2.108)

fz(z) =

n=1

λk,n e−zλk,n

m=1,m6=n

(1 −e−zλk,m )

n=1

λk,n e−zλk,n 1 +

N−1

m=1

(−1)mX

L⊆Jn,|L|=m

exp(−zX

l∈L

λk,l)

n=1

λk,nexp(−zλk,n) +

N−1

m=1

(−1)mX

L⊆Jn,|L|=m

exp −z(λk,n +X

l∈L

λk,l).

Then the ergodic unidirectional rate is given by

R∗

kRSeq =1

2Z∞

log(1 + z)fz(z)dz =

n=1

λk,n

2 ln(2)hexp(λk,n)E1(λk,n)

λk,n

N−1

m=1

(−1)mX

L⊆Jn,|L|=m

exp(λk,n +P

l∈L

λk,l)E1(λk,n +P

l∈L

λk,l)

λk,n +P

l∈L

λk,l i

using the identity R∞

0log(1 + t) exp(−ta)dt =eaE1(a)

ln(2)afor a≥0.

161

2 Bidirectional Relay Communication using Superposition Encoding

In the case of iid Rayleigh fading, we have λk,n =λkfor all nso that we get the pdf

fz(z) = Nλke−λkz(1 −e−λkz)N−1=Nλk

N−1

n=0 N−1

n(−1)nexp(−(n+ 1)λkz)

(2.109)

using the binomial theorem (x+y)n=Pn

k=0 n

kxn−kyk. Then (2.52) follows using again

the identity R∞

0log(1 + t) exp(−ta)dt =eaE1(a)

ln(2)afor a≥0.

2.8.26 Proof of Theorem 2.40

In the case of iid Rayleigh fading, we have fz(z) = Nλke−λkz(1 −e−λkz)N−1according

to (2.109). By substitution of τ= 1 −e−λkzwe get

R∗

kRSeq =1

2Z∞

log(1 + z)fz(z)dz

2 ln(2) Z1

ln 1−1/λkln(1 −τ)τN−1dτ.

Therewith, we get for any x∈(0,1)

R∗

kRSeq ≥N

2 ln(2) Z1

ln 1−1/λkln(1 −τ)τN−1dτ

≥N

2 ln(2) ln 1−1/λkln(1 −x)Z1

τN−1dτ

2 ln(2) ln 1−1/λkln(1 −x)(1 −xN),

where we used in the first inequality that the integrand is positive and in the second that

ln 1−1/λkln(1 −τ)is increasing with τ. If we set x= 1 −a/N with a∈(0, N)we

have

R∗

kRSeq ≥1

2 ln(2) ln 1−1/λkln(a/N)(1 −(1 −a/N)N).(2.110)

From [AS64, 4.2.36] we have the inequality ewy

w+y<1 + w

yyfor w, y > 0. Therefrom, we

get with z:= wy

w+y>0the following inequality

ez= e wy

w+y<1 + w

yy=y

y+w−y=1−w

y+w−y=1−z

y−y

so that we have e−z>1−z

yyfinally. With this inequality we can finally bound (2.110) so

that we have

R∗

kRSeq >1

2 ln(2) ln 1−1/λkln(a/N)(1 −e−a),

162

2.8 Appendix: Proofs

which is the proposed lower bound.

For the upper bound, accordingly for any x∈(0,1) we have

R∗

kRSeq ≤ln(1 −1/λ ln(1 −x))

2 ln(2) NZx

τN−1dτ +N

2 ln(2) Z1

ln 1−1/λkln(1 −τ)dτ,

where we used the fact that τN−1takes it maximum at one. If we set x= 1 −b/N with

b∈(0, N)we get for the first integral

NZx

τN−1dτ =xN= (1 −b

N)N<e−b

and for the second integral (integration by parts technique)

NZ1

1−b/N

ln 1−1/λkln(1−τ)dτ =1

bln 1+1/λkln(N/b)+NeλkE1(λk+ln(N/b)).

Let λk+ ln(N/b) =: ξ, then

NeλkE1(λk+ ln(N/b)) = beξE1(ξ)< b ln(1 + 1/(λk+ ln(N/b)))

using the inequality eξE1(ξ)<ln(1 + 1/ξ)[AS64, 5.1.20]. Combining all terms leads to

the upper bound.

For the asymptotic lower and upper bound we have

lim inf

N→∞

R∗

kRSeq

2log(ln(N)) ≥lim inf

N→∞

log(1 + 1

λkln(N

a))1−e−a

2log(ln(N)) = 1 −e−a>0,

lim sup

N→∞

R∗

kRSeq

2log(ln(N)) ≤lim sup

N→∞

log(1 + 1

λkln(N

b))e−b+b

2+b

2log(1 + 1

λk+ln( N

b))

2log(ln(N))

= e−b+b < ∞,

which meet if we choose a→ ∞ and b→0, i.e. lim

N→∞

R∗

kRSeq

2log(ln(N)) = 1. This means that

R∗

kRSeq is asymptotically equal to 1

2log(ln(N)).

Since any ergodic sum-rate on the boundary of the ergodic rate region is larger than

mink{R∗

kRSeq}and smaller than the sum P2

k=1 R∗

kRSeq the ergodic rate region grows asymp-

totically with Θlog(log(N))as concluded in Corollary 2.42.

163

2 Bidirectional Relay Communication using Superposition Encoding

2.8.27 Proof of Proposition 2.43

The maximal unidirectional rates of the n-th relay node for the optimal time division case

are given by

R∗

1opt,n := R−→

1R,nR−→

R2,n

R−→

1R,n +R−→

R2,n

,and R∗

2opt,n := R−→

2R,nR−→

R1,n

R−→

2R,n +R−→

R1,n

according to (2.31a) and (2.31b). Then the maximal unidirectional rate using relay selection

is given by

R∗

kRSopt := max

n∈{1,2,...,N}R∗

kopt,n k= 1,2.(2.111)

With the maximal unidirectional rates of the equal time division case R∗

1eq,n =

/2min{R−→

1R,n, R−→

R2,n}and R∗

2eq,n =1

/2min{R−→

2R,n, R−→

R1,n}for the n-th relay node ac-

cording to (2.22a) and (2.22b) we get

R∗

keq,n ≤R∗

kopt,n ≤2R∗

keq,n (2.112)

using the inequalities 1

/2min{x, y} ≤ xy

x+y≤min{x, y}for x, y > 0, which can be easily

seen from the following

/2min{x, y}=1

2 max{1

x,1

y}≤1

x+1

y≤1

max{1

x,1

y}= min{x, y}.

Since (2.112) holds for any n, we get R∗

kRSeq ≤R∗

kRSopt ≤2R∗

kRSeq,k= 1,2, for (2.111).

Furthermore, since the inequalities hold for any channel realization, we get an upper and

lower bound for the maximal unidirectional ergodic rates as follows

R∗

kRSeq ≤R∗

kRSopt ≤2R∗

kRSeq, k = 1,2.(2.113)

Similar to (2.53), since RRSTSopt is convex, we can bound the sum of any ergodic rate pair

R∗

RSTSopt(q)on the boundary of the ergodic rate region RRSTSopt as follows

min R∗

1RSopt, R∗

2RSopt≤R∗

RSTSopt(q)1≤2

k=1

R∗

kRSopt.(2.114)

Then (2.54) obviously follows from (2.113) and (2.114).

164

2.8 Appendix: Proofs

2.8.28 Proof of Theorem 2.45

Since nodes 1 and 2 cancel the interference caused by their own message before decoding

the unknown messages, for the proof we have to distinguish between four possible decoding

orders at nodes 1 and 2. Therefore, let the superscript R2 and R1 denote the decoding order of

the unknown messages where the relay message at nodes 1 and 2 is decoded first and accord-

ingly 2R and 1R where the relay messages is decoded last. For decoding the first unknown

message the received signal of the second unknown message is regarded as interference.

After successful decoding the nodes apply interference cancellation so that the nodes are

interference-free in the second decoding step. Since [R1, R2]∈ RBR(α), there exists a fea-

sible power distribution for each decoding order. Since the relay rate strictly increases with

βRfor any decoding order, we have βR= 1 −β1+β2.

We start with the decoding order R2,R1 where the relay message is decoded first, i.e. the

bidirectional messages are decoded last. Since the second decoding step is interference-

free we achieve the bidirectional rate R1= (1 −α)R−→

R2(β1)and R2= (1 −α)R−→

R1(β2).

Accordingly, we need for the bidirectional relay communication the relay power distribution

parameters

βR2,R1

1:= r1q2and βR2,R1

2:= r2q1

with r1:= 2 R1

1−α−1≥0,r2:= 2 R2

1−α−1≥0,q1:= 1

γR|h1|2>0, and q2:= 1

γR|h2|2>0

to achieve a certain rate pair [R1, R2]∈ RBR(α). In the first decoding step at node 1 the

message m2is unknown. Therefore, node 1 can successfully decode a relay message mR

with a rate at most

RR2,R1

R@1 := (1 −α) log 1 + βR2,R1

R|h1|2γR

1 + βR2,R1

2|h1|2γR= (1 −α) log 1 + 1−r1q2−r2q1

q1+r2q1

with βR2,R1

R= 1 −βR2,R1

1−βR2,R1

2. Similarly, the message m1is unknown at node 2 so

that node 2 can successfully decode a relay message mRwith a rate at most

RR2,R1

R@2 := (1 −α) log 1 + βR2,R1

R|h2|2γR

1 + βR2,R1

1|h2|2γR= (1 −α) log 1 + 1−r1q2−r2q1

q2+r1q2.

Since both nodes have to decode the relay message only the minimum of both rates can be

achieved, this means that with the decoding order R2,R1 the maximal achievable additional

relay rate is given by

RR2,R1

R:= (1 −α) log 1 + 1−r1q2−r2q1

max{q1(1 + r2), q2(1 + r1)}.(2.115)

For the decoding order 2R,1R nodes 1 and 2 decode the relay message last. Since [R1, R2]∈

RBR(α)there exists a feasible relay power distribution. Since the relay message mRis

165

2 Bidirectional Relay Communication using Superposition Encoding

unknown at nodes 1 and 2 in the first decoding step, the received signal of the relay message

is regarded as interference. Therefore, for the desired bidirectional rate R1at node 2 we

have

R1= (1 −α) log 1 + β2R,1R

1|h2|2γR

1 + β2R,1R

R|h2|2γR= (1 −α) log 1 + β2R,1R

q2+β2R,1R

R

⇒r1= 2 R1

1−α−1 = β2R,1R

q2+βR

=1−β2R,1R

R−β2

q2+β2R,1R

(2.116)

⇒β2R,1R

2= 1 −q2r1−β2R,1R

R(1 + r1),(2.117)

where we used β2R,1R

1= 1 −β2R,1R

2−β2R,1R

Rin the last equation of (2.116). Accordingly,

for the bidirectional rate R2at node 1 we have

R2= (1 −α) log 1 + β2R,1R

2|h1|2γR

1 + β2R,1R

R|h1|2γR= (1 −α) log 1 + β2R,1R

q1+β2R,1R

R

⇒r2= 2 R2

1−α−1 = β2R,1R

q1+β2R,1R

=1−q2r1−β2R,1R

R(1 + r1)

q1+β2R,1R

(2.118)

⇒β2R,1R

R=1−r1q2−r2q1

1 + r1+r2

,(2.119)

where we used (2.117) in the last equation of (2.118). At both nodes the relay message mR

is decoded in the second decoding step, which is interference-free due to the interference

cancellation. Therefore, node 1 can decode the relay message if the rate is at most

R2R,1R

R@1 := (1 −α) log(1 + β2R,1R

RγR|h1|2).

Similarly, node 2 can decode the relay message if the rate is at most

R2R,1R

R@2 := (1 −α) log(1 + β2R,1R

RγR|h2|2).

Since both nodes should decode the relay message the largest possible relay rate for the

decoding order 2R,1R is given by

R2R,1R

R:= min R2R,1R

R@1 , R2R,1R

R@2 = (1 −α) log(1 + β2R,1R

Rmin{γR|h1|2, γR|h2|2})

=(1 −α) log 1 + β2R,1R

Rmin{1

q1,1

q2}= (1 −α) log 1 + β2R,1R

max{q1, q2}

=(1 −α) log 1 + 1−r1q2−r2q1

max{q1(1 + r1+r2), q2(1 + r1+r2)},(2.120)

where we used in the last equation (2.119).

166

2.8 Appendix: Proofs

Next, we study the decoding order 2R,R1 where at node 1 the relay message is decoded last

and at node 2 the relay message is decoded first. Therefore, for the interference-free second

decoding step at node 2 we have

R1= (1 −α)R−→

R2(β2R,R1

1)⇒β2R,R1

1=r1q2.(2.121)

For the decoding of message m2at node 1 we regard the received signal of the relay message

as the interference. Thus we have

R2= (1 −α) log 1 + β2R,R1

2|h1|2γR

1 + β2R,R1

R|h1|2γR= (1 −α) log 1 + β2R,R1

q1+β2R,R1

R

⇒r2=β2R,R1

q1+β2R,1R

=1−β2R,R1

1−β2R,R1

q1+β2R,1R

=1−r1q2−β2R,R1

q1+β2R,1R

(2.122)

⇒β2R,R1

R=1−r1q2−r2q1

1 + r2

where we used β2R,R1

1= 1 −β2R,R1

2−β2R,R1

Rand (2.121) in (2.122). At node 1 the relay

message is decoded in the second interference-free decoding step so that the rate

R2R,R1

R@1 := (1 −α) log(1 + β2R,R1

RγR|h1|2) = (1 −α) log 1 + 1−r1q2−r2q1

q1(1 + r2)

is achievable. In the first decoding step at node 2 the message m1is unknown, therefore at

node 2 we can achieve the relay rate

R2R,R1

R@2 := (1−α) log 1+ βR2,R1

R|h2|2γR

1 + βR2,R1

1|h2|2γR= (1−α) log 1+ 1−r1q2−r2q1

q2(1 + r1)(1 + r2).

Since the message mRshould be decoded at both nodes, for the decoding order 2R,R1 we

have the maximal relay rate

R2R,R1

R:= min R2R,R1

R@1 , R2R,R1

R@2 

=(1 −α) log 1 + 1−r1q2−r2q1

max{q1(1 + r2), q2(1 + r1)(1 + r2)}.(2.123)

Finally, we look at the decoding order R2,1R, which is similar to the previous one. Form

the bidirectional achievable rates R1and R2we have

R2= (1 −α)R−→

R1(βR2,1R

2)⇒βR2,1R

2=r2q1,

R1= (1 −α) log 1 + βR2,1R

1|h2|2γR

1 + βR2,1R

R|h2|2γR= (1 −α) log 1 + βR2,1R

q2+βR2,1R

R

⇒r1=βR2,1R

q2+βR2,R1

=1−βR2,1R

2−βR2,1R

q2+βR2,R1

=1−r2q1−βR2,1R

q2+βR2,R1

⇒βR2,1R

R=1−r1q2−r2q1

1 + r1

167

2 Bidirectional Relay Communication using Superposition Encoding

Accordingly, at nodes 1 and 2 we can achieve the relay rates

RR2,1R

R@1 := (1 −α) log 1 + β2R,1R

R|h1|2γR

1 + β2R,1R

2|h1|2γR= (1 −α) log 1 + 1−r1q2−r2q1

q1(1 + r1)(1 + r2),

RR2,1R

R@2 := (1 −α) log(1 + βR2,1R

RγR|h2|2) = (1 −α) log 1 + 1−r1q2−r2q1

q2(1 + r1).

This gives us the maximal achievable relay rate for the decoding order R2,1R

RR2,1R

R:= min RR2,1R

R@1 , RR2,1R

R@2 

=(1 −α) log 1 + 1−r1q2−r2q1

max{q1(1 + r1)(1 + r2), q2(1 + r1)}.(2.124)

Since q1, q2, r1, and r2are non-negative we have

max{q1(1 + r2), q2(1 + r1)} ≤ min max{q1(1 + r1+r2), q2(1 + r1+r2)},

max{q1(1 + r2), q2(1 + r1)(1 + r2)},

max{q1(1 + r1)(1 + r2), q2(1 + r1)}.

These are the differing denominators from (2.115), (2.120), (2.123), and (2.124) so that it

directly follows that

RR(R1, R2) := max RR2,R1

R, R2R,1R

R, R2R,R1

R, RR2,1R

R=RR2,R1

This means that the maximal achievable additional relay rate is always achieved with the

decoding order where the relay message is decoded first.

2.8.29 Proof of Lemma 2.46

The graph GfRΣof the function fRΣ: [0, RΣ]→[0, RΣ]with R17→ RΣ−R1denotes

the set of non-negative rate pairs with a sum-rate equal to RΣ. Then the intersection of

GfRΣ∩RBR(α)characterizes the set of rate pairs in RBR(α)with the desired sum-rate RΣ

and is non-empty since we assume that a rate pair [R1, R2]∈ RBR(α)with the desired

sum-rate RΣ=R1+R2exists. Since RBR(α)is convex there exist R⊲

1and R⊳

1so that

GfRΣ∩RBR(α) = [R1, fRΣ(R1)] : R⊲

1≤R1≤R⊳

1.

This means that the interval [R⊲

1, R⊳

1]characterizes all bidirectional rate pair in RBR(α)with

a sum-rate equal to RΣ. For any R1∈[R⊲

1, R⊳

1]we can achieve the relay rate RR(R1, RΣ−

R1)according to Theorem 2.45. From (2.56) we have

β1=2R1

1−α−1

|h2|2γR

and β2=2

RΣ−R1

1−α−1

|h1|2γR

168

2.8 Appendix: Proofs

and therefore βR= 1 −β1−β2= 1 −2

1−α−1

|h2|2γR−2

RΣ−R1

1−α−1

|h1|2γR. This allows us to write

RR(R1, RΣ−R1)as follows

RR(R1, RΣ−R1)

= min n(1 −α) log 1 + βR|h1|2γR

1 + β2|h1|2γR,(1 −α) log 1 + βR|h2|2γR

1 + β1|h2|2γRo

= (1 −α) log 1 + 1−2R1

1−α−1

|h2|2γR−2

RΣ−R1

1−α−1

|h1|2γRmin n|h1|2γR

RΣ−R1

1−α

,|h2|2γR

2R1

1−αo

= (1 −α) log min n|h1|2

RΣ

1−α

|{z}

:=ξ

γR+1

|h1|2+1

|h2|2

|{z }

:=ˆγ2R1

1−α

|{z}

:=x−2R1

1−α)2

|h2|2

|{z }

=x2/ζ ,

|h2|2

|{z}

:=ζγR+1

|h1|2+1

|h2|2

|{z }

=ˆγ1

2R1

1−α

|{z}

=1/x

−2

RΣ

1−α

|h1|2

|{z}

=1/ξ 1

2R1

1−α2

|{z }

=1/x2

o

= (1 −α) log min nξˆγx −x2

ζ

|{z }

:=p(x)

, ζˆγ1

x−1

ξx2

|{z }

:=q(x)o

= (1 −α) log min p(x), q(x)=: RR(x),

where the new defined function RR(x)has the domain [xmin, xmax]with xmin := 2

R⊲

1−α

and xmax := 2

R⊳

1−α. The rate RR(x)is non-negative for any x∈[xmin, xmax], since

we assume that a feasible bidirectional rate pair with the desired sum-rate exists. Thus

min{p(x), q(x)} ≥ 1for any x∈[xmin, xmax].

In Figure 2.23 we illustrate the functions p(x)and q(x)to make the following discussion

clear. Since ξand ζare non-negative p(x) = ξ/ζ(ˆγζ −x)xis a concave parabola with roots

0,ˆγζ and vertex [ˆγζ/2,ˆγ2ξζ/4]. Similarly, q(1/x) = ζ/ξ(ˆγξ −x)xis a concave parabola

for the inverse argument with roots 0,ˆγξ and vertex [ˆγξ/2,ˆγ2ξζ/4]. Therefore, q(x)has a

negative pole at 0, a root at 1/ˆγξ, and a vertex at [2/ˆγξ, ˆγ2ξζ/4]. Furthermore, q(x)is strictly

increasing on the interval (0,2/ˆγξ)and strictly decreasing on the interval (2/ˆγξ, ∞).

Since p(x), q(x)≤ˆγ2ξζ/4for any x∈(0,∞)and min{p(x), q(1/x)} ≥ 1for any x∈

[xmin, xmax]6=∅, we have 1≤ˆγ2ξζ/4and therefore ˆγζ/2≥2/ˆγξ. This means that vertex

of q(x)is always to the left of the vertex of p(x). From a simple calculation it follows that

169

2 Bidirectional Relay Communication using Superposition Encoding

ˆγ2ξ ζ

ˆγ√ξ ζ −1

xpq

xqp1 xqp2

ˆγζ ˆγξ

← p(x)

↑ q(x)

← q(1/x)

p(x), q(1/x), q(x)

Figure 2.23: Discussion of p(x),q(1/x),q(x)with vertexes [ˆγζ

2,ˆγ2ξζ

4],[ˆγξ

2,ˆγ2ξζ

4],[2

ξˆγ,ˆγ2ξζ

respectively (not explicitly depicted).

the graphs of p(x)and q(x)intersect at the following intersection points

[xqp1, p(xqp1)] = [ˆγζ/2−p(ˆγζ/2)2−ζ/ξ, 1],

[xqp2, p(xqp2)] = [ˆγζ/2 + p(ˆγζ/2)2−ζ/ξ, 1],

[xpq, p(xpq)] = [pζ/ξ, ˆγpξζ −1].

Obviously, the first two points coincide if we have (ˆγζ/2)2=ζ/ξ which is equivalent to

ˆγζ/2 = 2/ˆγξ, i.e. the vertexes of p(x)and q(x)coincide as well. From the intersection

points and the vertexes it follows that p(x), q(x)≥1for x∈[xqp1, xqp2]only and therefore

we have [xmin, xmax]⊆[xqp1, xqp2]because rates are always non-negative. Finally it follows

that for x∈[xmin, xmax]we have

min p(x), q(x)=(q(x),if x≤xpq,

p(x),if x > xpq.

Moreover, min p(x), q(x)is increasing for x∈[xqp1, xpq]and decreasing for x∈

[xpq, xqp2].

Therefore, if xpg ∈[xmin, xmax]⇔R⊲

1≤R⋆

1≤R⊳

1the maximal additional relay rate RR(x)

is attained at xpg ⇔at the rate pair [R⋆

1, RΣ−R⋆

1]with

RR(RΣ) = RR(xpg) = (1 −α) log p(xpg)= (1 −α) log ˆγpξζ −1

= (1 −α) log |h1h2|ˆγ−2

RΣ

2(1−α)−1

2RΣ,

170

2.8 Appendix: Proofs

otherwise if xpg < xmin ⇔R⋆

1< R⊲

1the maximal additional relay rate RR(x)is attained at

xmin ⇔at the rate pair [R⊳

1, RΣ−R⊳

1]with

RR(RΣ) = RR(xmin) = (1 −α) log q(xmin)= (1 −α) log ζˆγ1

xmin −1

ξx2

min 

= (1 −α) log |h2|2(ˆγ−1

|h1|22

RΣ−R⊲

1−α)−R⊲

or if xpg > xmax ⇔R⋆

1> R⊲

1the maximal additional relay rate RR(x)is attained at xmax

⇔at the rate pair [R⊲

1, RΣ−R⊲

1]with

RR(RΣ) = RR(xmax) = (1 −α) log p(xmax)(1 −α) log ξˆγxmax −x2

max

ζ

= (1 −α) log |h1|2(ˆγ−1

|h2|22

R⊳

1−α)−RΣ+R⊳

This finishes the proof.

2.8.30 Proof of Proposition 2.47

For a given time division parameter αwe have the broadcast rate region (1−α)RBCRΣ

1−α=

(1 −α)RBCγBC

R(RΣ

1−α)with the sum-rate maximum (1 −α)RΣ

1−α=RΣ. It follows that

for the broadcast region (1 −α)RBC(γR)with γR=γBC

RRΣ

1−αthe sum-rate maximum

is attained at the rate pair (1 −α)R⋆

−→

R2(γR), R⋆

−→

R1(γR)according to Proposition 2.2 with a

maximum sum-rate RΣ. Therefore, we have to distinguish for β⋆between three cases with

respect to γRrespectively RΣ.

If β⋆∈[0,1], we have

(1 −α)R⋆

−→

R1(γR) = (1 −α)R−→

R1(1 −β⋆)

= (1 −α) log 1

2(1 + γR|h1|2+|h1|2

|h2|2)

= (1 −α) log 1

2(|h2|2

|h1|2+γR|h2|2+ 1)|h1|2

|h2|2

= (1 −α) log 1

2(|h2|2

|h1|2+γR|h2|2+ 1)+ (1 −α) log |h1|2

|h2|2

= (1 −α)R−→

R2(β⋆) + (1 −α)R†

= (1 −α)R⋆

−→

R2(γR) + (1 −α)R†.

From the definitions of R⋆

1(RΣ)and R⋆

2(RΣ)it follows that

R⋆

1(RΣ) + R⋆

2(RΣ) =RΣ= (1 −α)R−→

R2(β⋆) + (1 −α)R−→

R1(β⋆)

R⋆

1(RΣ)−R⋆

2(RΣ) = −(1−α)R†= (1 −α)R−→

R2(β⋆)−(1 −α)R−→

R1(β⋆).

171

2 Bidirectional Relay Communication using Superposition Encoding

If we solve the equation system for R⋆

1(RΣ)and R⋆

2(RΣ)we have R⋆

1(RΣ) = (1 −

α)R−→

R2(β⋆)and R⋆

2(RΣ) = (1 −α)R−→

R1(1 −β⋆), which means that [R⋆

1(RΣ), R⋆

2(RΣ)]

characterizes the sum-rate optimal rate pair of (1 −α)RBCRΣ

1−α.

For the characterization of the case β⋆∈[0,1] we consider the now simple equivalences

β⋆≥0⇔R−→

R2(β⋆)≥0⇔RΣ≥(1 −α)R†,

β⋆≤1⇔R−→

R1(1 −β⋆)≥0⇔RΣ≥ −(1 −α)R†.

It follows that the case β⋆∈[0,1] is equivalent to RΣ≥(1 −α)|R†|.

Finally, it follows that if RΣ<(1 −α)R†we have β⋆<0so that the sum-rate optimal rate

pair is on the R2-axis with [0, RΣ]and if RΣ<−(1 −α)R†we have β⋆>1so that the

sum-rate optimal rate pair is on the R1-axis with [0, RΣ].

2.8.31 Proof of Theorem 2.48

We prove the theorem for the case |h1|2≥ |h2|2. The case |h2|2≥ |h1|2follows accordingly

by interchanging the indices one and two. To identify the bidirectional rate pair where the

total sum-rate maximum is attained we consider for any feasible bidirectional sum-rate RΣ∈

[0, Rmax

Σ]the corresponding total sum-rate Rtot(RΣ). Thereby, let γB:= γBC

R(RΣ)≤γR

denote the necessary relay signal-to-noise ratio for a bidirectional sum-rate RΣ≤Rmax

Σ.

Effectively, this means that we have a bidirectional relay communication with reduced relay

power and a relay power distribution parameter β=γR

γBβ1= 1 −γR

γBβ2. Accordingly, we

can define γP:= γR−γP≥0which denotes the relay signal-to-noise ratio of the piggyback

communication. In the following we will discuss the different cases of Rtot(RΣ).

At low sum-rates RΣ, where we have RΣ<min{αR−→

2R,(1 −α)R†}, we have R⋆

1(RΣ)<0

so that the bidirectional sum-rate maximum is attained at [0, RΣ] = [0,(1 −α) log(1 +

γB|h1|2)] with γB=γBC

R(RΣ). For those bidirectional sum-rates the total sum-rate is given

by (2.64b). Since the argument of the logarithm is a concave parabola in 2

RΣ

1−α, which is

maximized at its vertex, the function Rtot(RΣ)is maximized at (1−α) log 1

2|h1|2ˆγ. From

R⋆

1(RΣ)<0it follows that β⋆=1

2−1

2γB1

|h2|2−1

|h1|2<0⇔γB+1

|h1|2<1

|h2|2.

Therefore, we have

(1 −α) log 1

2|h1|2ˆγ= (1 −α) log |h1|2

2γR+1

|h1|2+1

|h2|2

>(1 −α) log |h1|2

2γB+1

|h1|2+1

|h2|2

>(1 −α) log 1 + γB|h1|2=RΣ.

172

2.8 Appendix: Proofs

Since the vertex of the parabola is attained at a rate larger than RΣit follows that Rtot(RΣ)

is strictly increasing on [0, RΣ]. This implies that for this section the largest total sum-rate is

attained at the largest possible bidirectional sum-rate RΣ.

At higher sum-rates RΣ, where we have RΣ≥R†, we possibly have [R⋆

1(RΣ), R⋆

2(RΣ)] ∈

RBR(α)so that (2.64a) applies for Rtot(RΣ). Now, the argument of the logarithm is a

concave parabola in 2

RΣ

2(1−α)which maximizes at R⋆

Σ:= 2(1 −α) log 1

2|h1h2|ˆγ. It follows

that the function Rtot(RΣ)is increasing on the interval [|R†|, R⋆

Σ]. For the vertex at R⋆

Σwe

have

Rtot(R⋆

Σ) = R⋆

Σ+Rmax

R(R⋆

Σ) = (1−α)(log |h1h2|ˆγ−1

2|h1h2|ˆγ+log[1

2|h1h2|ˆγ]) = R⋆

Σ,

so that we obviously have Rmax

R(R⋆

Σ) = 0. If we have [R⋆

1(R⋆

Σ), R⋆

2(R⋆

Σ)] ∈ RBR(α)it

follows that the largest total sum-rate is attained at R⋆

Σ, which is also the largest feasible

bidirectional sum-rate at all. Otherwise, if we have [R⋆

1(R⋆

Σ), R⋆

2(R⋆

Σ)] /∈ RBR(α), the

largest total sum-rate Rtot(RΣ)for this section is attained at the largest bidirectional sum-

rate RΣwhere we have [R⋆

1(RΣ), R⋆

2(RΣ)] ∈ RBR(α). Again it follows that the largest

feasible bidirectional sum-rate for this section maximizes Rtot(RΣ).

If we further increase the sum-rate RΣwe have R⋆

BC(RΣ)/∈ RBR(α). This means that the

total sum-rate Rtot(RΣ)is given by (2.64c) if R⋆

1(RΣ)< R⊲

1=RΣ−αR−→

2R or (2.64d) if

R⋆

1(RΣ)> R⊳

1=RΣ−αR−→

1R. In both cases the total sum-rate is independent of RΣand

therefore constant.

Since the section-wise defined function Rtot(RΣ)is continuous and increasing for rate pairs

RBC(RΣ)∈ RBR(α)and constant for rate pairs RBC(RΣ)/∈ RBR(α)we achieve the max-

imum total sum-rate R∗

tot at the largest sum-rate RΣwhere we have R⋆

BC(RΣ)∈ RBR(α),

which is given by the intersection point of RBC(RΣ)with the boundary of RBR(α).

2.8.32 Proof of Corollary 2.49

In the following we prove the interchange of RRwith R1only. The interchange with R2

follows accordingly.

If we can prove the equality RR(R⋆

1+R①, R⋆

2) = Rmax

R(R⋆

1+R⋆

2)−R①for non-negative

R①, it would follow from

Rtot(R⋆

1+R⋆

2) = R⋆

1+R⋆

2+Rmax

R(R⋆

1+R⋆

=R⋆

1+R①+R⋆

2+Rmax

R(R⋆

1+R①, R⋆

2)(2.125)

that the total sum-rate remains constant. From Lemma 2.46 we know that Rmax

R(R⋆

1+R⋆

2) =

RR(R⋆

1, R⋆

2)with R⋆

1=1

2RΣ−1−α

2R†,R⋆

2=1

2RΣ+1−α

2R†, and R†= log |h1|2

|h2|2.

173

2 Bidirectional Relay Communication using Superposition Encoding

Therewith, it follows that

1−αRR@1(R⋆

1+R①,R⋆

2)−1 = βR|h1|2γR

1 + β2|h1|2γR

=1−2

R⋆

1+R①

1−α−1

|h2|2γR−2

R⋆

1−α−1

|h1|2γR|h1|2γR

1 + 2

R⋆

1−α−1

|h1|2γR|h1|2γR

=|h1|2γR−|h1|2

|h2|221

2(1−α)RΣ−1

2R†+1

1−αR①−1−21

2(1−α)RΣ+1

2R†−1

2(1−α)RΣ+1

2R†·2−R†

2−R†

=|h2|2γR−21

2(1−α)RΣ−1

2R†+1

1−αR①−1−|h2|2

|h1|221

2(1−α)RΣ+1

2R†−1

2(1−α)RΣ−1

2R†·21

1−αR①

= 2 1

1−αR①1−2

R⋆

1+R①

1−α−1

|h2|2γR−2

R⋆

1−α−1

|h1|2γR|h2|2γR

1 + 2

R⋆

1+R①

1−α−1

|h2|2γR|h2|2γR

= 2 1

1−αR①βR|h2|2γR

1 + β1|h2|2γR

= 2 1

1−αR①21

1−αRR@2(R⋆

1+R①,R⋆

2)−1),(2.126)

where we used βR= 1 −β1−β2and (2.56). Similarly, we can follow

1−αRR@2(R⋆

1+R①,R⋆

2)= 1 + 1−2

R⋆

1+R①

1−α−1

|h2|2γR−2

R⋆

1−α−1

|h2|2γR|h2|2γR

1 + 2

R⋆

1+R①

1−α−1

|h2|2γR|h2|2γR

1 + 1−2

R⋆

1−α−1

|h2|2γR|h2|2γR

R⋆

1+R①

1−α

= 2−R①

1−α·

1 + 1−2

R⋆

1−α−1

|h2|2γR|h2|2γR

1 + 2

R⋆

1−α−1

|h2|2γR|h2|2γR

= 2−R①

1−α·



1 + 1−2

R⋆

1−α−1

|h2|2γR−2

R⋆

1−α−1

|h2|2γR|h2|2γR

1 + 2

R⋆

1−α−1

|h2|2γR|h2|2γR







= 2−R①

1−α21

1−αRR@2(R⋆

1,R⋆

2).(2.127)

Then for R①= 0 it follows from (2.126) that

RR@1(R⋆

1, R⋆

2) = RR@2(R⋆

1, R⋆

2) = RR(R⋆

1, R⋆

2).(2.128)

Furthermore, for R①≥0we have RR@1(R⋆

1+R①, R⋆

2)≥RR@2(R⋆

1+R①, R⋆

2)so that we

174

2.8 Appendix: Proofs

can conclude

RR(R⋆

1+R①, R⋆

2) = min{RR@1(R⋆

1+R①, R⋆

2), RR@2(R⋆

1+R①, R⋆

2)}

=RR@2(R⋆

1+R①, R⋆

(2.127)

=RR@2(R⋆

1, R⋆

2)−R①

(2.128)

=RR(R⋆

1, R⋆

2)−R①.

This means that (2.125) holds and therefore the total sum-rate remains constant for R①≥0.

Similar arguments apply for R②≥0.

2.8.33 Proof of Proposition 2.51

For φBC >π

2we have R−→

R2(β⋆)<0as well as β⋆<0. This is equivalent to γR<

|h2|2−1

|h1|2from which it follows that R⋆

−→

R1 =R−→

R1(1) = log(1 + γR|h1|2)<log(1 +

|h2|2−1

|h1|2)|h1|2) = R†. Therefore, we have α1=1

R−→

2R/R†+1 >1

R−→

2R/R⋆

−→

R1+1 =α0so that

we have A1= [0, α0)and A2=A3=A4=∅.

For π

2≥φBC we have β⋆≥0⇒γR≥1

|h2|2−1

|h1|2⇒R−→

R1(1) ≥R†⇒α0≥α1so that

we additionally have A1= [0, α1).

Further, for π

2≥φBC > φ2Σ we have R−→

RMAC

Σ−R−→

<R⋆

−→

R⋆

−→

=R⋆

−→

R⋆

−→

R1−R†where we used

R⋆

−→

R2 =R−→

R2(β⋆) = R−→

R1(β⋆)−R†=R⋆

−→

R1 −R†, which is equivalent to 2R−→

2RR⋆

−→

R1 −

R⋆

−→

R1RMAC

Σ< R⋆

−→

2RR†. If we add on both sides R−→

R1R†and divide both sides by (R⋆

−→

R1 +

R−→

2R)(2R−→

2R −RMAC

Σ+R†)we get α0=R⋆

−→

R⋆

−→

R1+R−→

<R†

2R−→

2R−RMAC

Σ+R†=α2so that we have

A2= [α1, α0)and A3=A4=∅.

On the other hand, if we have φ2Σ ≥φBC we have α0≥α2using the same arguments with

the opposite relations. It follows that A2= [α1, α2).

Then, for φ2Σ ≥φBC > φΣ1 we have RMAC

Σ−R−→

R−→

<R⋆

−→

R⋆

−→

from which we get 2(R⋆

−→

R1 +

R⋆

−→

R2)R−→

1R >2R⋆

−→

R2RMAC

Σ= (R⋆

−→

R1 +R⋆

−→

R2 −R†)RMAC

Σ⇔(R⋆

−→

R1 +R⋆

−→

R2)(RMAC

Σ−2R−→

1R +

R†)< R†(R⋆

−→

R1 +R⋆

−→

R2 +RMAC

Σ). If we divide both side with (R⋆

−→

R1 +R⋆

−→

R2 +RMAC

Σ)(RMAC

Σ−

2R−→

1R +R†)we get α0=R⋆

−→

R⋆

−→

R1+R−→

<R†

RMAC

Σ−2R−→

1R+R†=α3so that we have A3= [α2, α0)

and A4=∅.

Finally, if we have φBC ≥φ2Σ we have α0≥α3so that A3= [α2, α3)and A4=

[α3, α0).

175

2 Bidirectional Relay Communication using Superposition Encoding

Then the total sum-rate optimal rate pairs follow immediately. If α∈ A2we have to solve

R⋆

2(RΣ) = 1

2RΣ+1−α

2R†=αR−→

2R for RΣand plug in the solution RΣ= 2αR−→

2R −(1 −

α)R†in R⋆

1(RΣ), which gives us R1=αR−→

2R −(1 −α)R†. Similarly, if α∈ A4we

have to solve R⋆

1(RΣ) = αR−→

1R for RΣand plug in the solution in R⋆

2(RΣ), which gives us

R2=αR−→

1R + (1 −α)R†. Finally, for α∈ A3we require R⋆

1(RΣ) + R⋆

2(RΣ) = αRMAC

so that we get RΣ=αRMAC

Σ. Then R⋆

BC(αRMAC

Σ, α)specifies the optimal rate pair.

With the optimal bidirectional rate pairs [Ropt

1(α), Ropt

2(α)] we get the maximal total sum-

rate R∗

tot(α) = Ropt

1(α) + Ropt

2(α) + Rmax

R(Ropt

1(α) + Ropt

2(α)) using Lemma 2.46. In

more detail, for time division parameters αA1we have R⋆

1=R−→

R2(β⋆)< R⊲

1= 0 so that

the maximum relay rate is given by (2.60b). Accordingly, we get the total sum-rate

Rtot(α) = αR−→

2R + (1 −α) log |h2|2(ˆγ−1

|h1|22α

1−αR−→

2R )

= (1 −α) log |h2|2

|h1|22α

1−αR−→

2R ˆγ|h1|2−2α

1−αR−→

2R for α∈ A1.

If α∈ A2∪ A3∪ A4we have R⋆

1=Ropt

1(α). Since [Ropt

1(α), Ropt

2(α)] ∈ RBR(α)we

have R⊲

1≤R⋆

1≤R⊳

1so that the case (2.60a) applies. This gives us in the case α∈ A2with

RΣ= 2αR−→

2R −(1 −α)R†

Rtot(α) = αR−→

2R −1−α

2R†+ (1 −α) log |h1h2|ˆγ−2α

1−αR−→

2R−1

2R†

= (1 −α) log 2α

1−αR−→

2R−1

2R†|h1h2|ˆγ−2α

1−αR−→

2R−1

2R†

= (1 −α) log |h2|2

|h1|22α

1−αR−→

2R ˆγ|h1|2−2α

1−αR−→

2R ,

where we used the equality 2−1

2R†=|h2|

|h1|. For α∈ A3we have to plug in RΣ=αRMAC

Σin

(2.60a) which gives us

Rtot(α) = 1

2RMAC

Σ+ (1 −α) log |h1h2|ˆγ−2

RMAC

2(1−α)

= (1 −α) log 2α

2(1−α)RMAC

Σ|h1h2|ˆγ−2α

2(1−α)RMAC

Σ.

Similarly, for α∈ A4we have to plug in RΣ= 2αR−→

1R +(1−α)R†in (2.60a) which finally

gives us

Rtot(α) = αR−→

1R +1−α

2R†+ (1 −α) log |h1h2|ˆγ−2α

1−αR−→

1R+1

2R†

= (1 −α) log |h1|2

|h2|22α

1−αR−→

1R ˆγ|h2|2−2α

1−αR−→

1R .

This finishes the proof.

176

2.8 Appendix: Proofs

2.8.34 Proof of Proposition 2.52

For k= 1 we use all eigenmodes if for all λ2,n 6= 0 we have ξ1,n 6= 0. Therefore, we need

a water-level ν1>σ2

λ2,r2. The condition follows if we look at the power constraint in the

high-power regime constraint

β1PR=

n=1 ν1−σ2

λ2,n =r2ν1−σ2L2,r2>σ2r2

λ2,r2−σ2L2,r2=σ2r2−1

λ2,r2−L2,r2−1.

Similarly, beamforming is optimal if we have ξ2,2= 0. Therefore, we need a water-level

ν1≤σ2

λ2,2. Again the condition follows from the power constraint

β1PR=ν1−σ2

λ2,1≤σ2

λ2,2−σ2L2,1.

Finally, it is optimal to use meigenmodes if we have ξ2,m 6= 0 and ξ2,m+1 = 0. Therefore,

we need a water-level within σ2

λ2,m < ν1≤σ2

λ2,m+1 . From the power constraint we get

mσ2

λ2,m −σ2L2,m < β1PR=

n=1 ν1−σ2

λ2,n ≤mσ2

λ2,m+1 −σ2L2,m.

The case k= 2 follows accordingly.

2.8.35 Proof of Proposition 2.53

From (2.77) we have

R−→

R2(β1) = log det IN2+ρHH

2QR,1H2= log

n=1 1 + λ2,nξ2,n

σ2

m1(β1)

n=1

log 1 + λ2,n

σ2(ν1−σ2

λ2,n

)=

m1(β1)

n=1

log λ2,nν1

σ2

m1(β1)

n=1

log λ2,n

β1γR+L2,m1(β1)

m1(β1),

where we used in the last equality the water-level ν1=β1PR+σ2L2,m1(β1)

m1(β1)which follows from

the power constraint β1PR=Pr2

n=1 ξ2,n =Pm1(β1)

n=1 ν1−σ2

λ2,n =m1(β1)ν1−σ2L2,m1(β1).

The rate R−→

R1(β2)follows accordingly.

177

2 Bidirectional Relay Communication using Superposition Encoding

2.8.36 Proof of Theorem 2.54

For a rate pair on the boundary of RMIMO

BC we have β1+β2= 1. Therefore, the rate pair

[R−→

R2(β), R−→

R1(1 −β)],β∈[0,1] characterizes a parametrization of the boundary. Using the

representation of R−→

R2(β)for β > 0given in Proposition 2.53 we can calculate the derivation

as follows

dR−→

R2(β)

dβ=

m1(β)

n=1

γR

βγR+L2,m1(β)

ln(2) =m1(β)γR

βγR+L2,m1(β)

ln(2).

Since R−→

R2(β1)is continuous in β1= 0 we get the right hand side derivation

dR−→

R2(β)

dββ=0 = lim

β↓0

m1(β)γR

βγR+L2,m1(β)

ln(2) =γR

L2,1

ln(2).

Similarly, for R−→

R1(1 −β)for β < 1we can easily calculate the derivation as follows

dR−→

R1(1 −β)

dβ=

m2(1−β)

n=1

−γR

(1 −β)γR+L1,m2(1−β)

ln(2) =−m2(1 −β)γR

(1 −β)γR+L1,m2(1−β)

ln(2)

as well as the left hand side derivation at β= 1

dR−→

R1(1 −β)

dββ=1 = lim

β↑1

m2(1 −β)γR

(1 −β)γR+L1,m2(1−β)

ln(2) =γR

L1,1

ln(2).

With this we get the angle of the normal vector ϕBC(β)of the parametrized boundary

[R−→

R2(β), R−→

R1(1 −β)],β∈[0,1], from

tan ϕBC(β) = dR−→

R2(β)dβ

−dR−→

R1(1 −β)dβ=









γR+L1,M2

M2L2,1,if β= 0,

m1(β)((1−β)γR+L1,m2(1−β))

m2(1−β)(βγR+L2,m1(β)),if 0< β < 1,

M1L1,1

γR+L2,M ,if β= 1.

Obviously, θqdenotes the angle of the weight vector q. Then we know from previous dis-

cussion, e.g. in the proof of Theorem 2.10 in Appendix 2.8.6, that the boundary rate pair

[R−→

R2(β), R−→

R1(1 −β)] with a normal angle ϕBC(β)is the weighted rate sum optimal rate

pair for weight vectors qwhere we have θq=ϕBC(β). Since RMIMO

BC is convex it follows

that ϕBC(β)decreases with increasing β.

178

2.8 Appendix: Proofs

For weight vectors qwith an angle θq≥ϕBC(0) = ϕBC(˜

β1) = ϕ1the intersection of the

boundary [R−→

R2(β), R−→

R1(1 −β)] with the R2-axis is optimal, this means we have β= 0.

Thus, for θq≥ϕ1the weighted rate sum is attained at the rate pair

[R−→

R2(0), R−→

R1(1)] = h0,

n=1

log λ1,n

(γR+L1,M2)i

according to Proposition 2.53 with m2(1) = M2.

Similarly, for weight vectors qwith an angle θq≤ϕBC(0) = ϕBC(˜

βM) = ϕMthe intersec-

tion of the boundary [R−→

R2(β), R−→

R1(1 −β)] with the R1-axis is optimal. Thus, for θq≤ϕM

the weighted rate sum is attained at the rate pair

[R−→

R2(1), R−→

R1(0)] = hM1

n=1

log λ2,n

(γR+L2,M1),0i

according to Proposition 2.53 with m1(1) = M1.

For the characterization of the rate pair for a weight vector qwith ϕM< θq< ϕ1we need

to know the number of used eigenmodes for this weight vector. The functions m1(β1)and

m2(β2)characterize the number of eigenmodes for a given relay power fraction. Further-

more, we know that the number of used eigenmodes change at relay power fractions ˜

βn,

n= 1,2,...,M so that for all β∈(˜

βn,˜

βn+1]we have m1(β) = m1(˜

βn+1)and simi-

larly for all β∈[˜

βn,˜

βn+1)we have m2(β) = m2(˜

βn). Since ϕBC(β)is decreasing for β

we can characterize this property also in term of the angles. Therefore, let ϕn=ϕBC(˜

βn)

denote the angle of the characteristic power fraction ˜

βn,n= 1,2,...,M. Accordingly,

for weight vectors qwith angle θq∈[ϕn+1, ϕn)we use m1(˜

βn+1) = η1(q)eigenmodes

of the channel H2and similarly for weight vectors qwith angle θq∈(ϕn+1, ϕn]we use

m2(˜

βn) = η2(q)eigenmodes of the channel H1. And of course, for q≥ϕ1we use

η1(q) = 0 and η2(q) = M2as well as for q≤ϕMwe have η1(q) = M1and η2(q) = 0

eigenmodes of H1and H2respectively.

We are now able to characterize for a weight vector qwith ϕM< θq< ϕ1the rate pair on

the boundary RMIMO

BC explicitly. Therefore, we solve tan ϕBC(β) = tan θq=q1

q2for βwith

0< β < 1which gives us the corresponding optimal power fraction β(q)as follows

β(q) = q1η1(q)γR+L1,η2(q)−q2η2(q)L2,η1(q)

γR(q2η2(q) + q1η1(q)) .

Then a simple calculation shows that R−→

R2(q)and R−→

R1(q)are given by R−→

R2(β(q)) and

R−→

R1(1 −β(q)) using the representation of the rates given by Proposition 2.53. Finally,

we finish the proof with the observation that the identities R1,BC([1,0]) = R−→

R2(1) and

R2,BC([0,1]) = R−→

R1(1) hold because we defined Lk,0= 0,k= 1,2.

179

3 Optimal Coding Strategy for the

Bidirectional Broadcast Channel

3.1 Introduction

We still consider a three-node network where two nodes want to communicate with each

other using the support of the third node as a relay, which we call the bidirectional relay

channel. Between the nodes we assume discrete memoryless channels with finite size input

and output alphabets. In this chapter we present an information theoretic optimal chan-

nel coding approach for the broadcast phase based on the philosophy of network coding

[ACLY00] which suggests that information should not be treated as a fluid.

In classical information theory often full-duplex nodes are assumed. However, in wireless

communication this assumption is hard to fulfill. Therefore, we assume half-duplex nodes

which means that a node contributes either an input or an output to the channel. A natu-

ral consequence of this assumption is that the relay communication is performed in phases.

Often the relay communication should be integrated in existing infrastructures and most pro-

tocol proposals base usually on orthogonal components which require exclusive resources

for each link. As a consequence they suffer from an inherent loss in spectral efficiency. As

we know from the previous, this loss can be significantly reduced if bidirectional relay com-

munication is desired, since bidirectional communication can be efficiently performed in two

phases. First, we have the multiple access phase where nodes 1 and 2 transmit their infor-

mation to the relay node. In the succeeding broadcast phase, the relay node simultaneously

forwards the information to its destinations.

Network coding is originally a distributed source coding problem. The bidirectional relaying

protocols proposed in [WCK05, LJS05, FBW06, PY06, HKE+07] apply an XOR operation

on the decoded data at the relay node so that they assume a source (network) and channel

coding separation. The separation of network and channel coding is in general suboptimal

[EMH+03, RK06]. In this thesis we consider a decode-and-forward protocol and find the

optimal channel coding strategy for the bidirectional broadcast channel which factors in

the distributed knowledge of the sources. But the decode-and-forward approach causes an

operational separation between the phases, which means that we require that the relay node

successfully decodes the messages of nodes 1 and 2 in the first phase. Furthermore, we do

181

3 Optimal Coding Strategy for the Bidirectional Broadcast Channel

not allow any feedback between the phases so that the encoders of nodes 1 and 2 cannot

cooperate. In the first phase we have, due to the operational separation, the classical multiple

access channel where the optimal coding strategy and capacity region are known. It remains

to find the optimal coding strategy for the succeeding broadcast phase, which we call the

bidirectional broadcast channel. It is important to notice that the proposed coding strategy

is only optimal for the bidirectional relay channel under the restriction of the operational

separation.

While for single-user channels it is of no importance whether we use vanishing average or

maximal probabilities of error in the definition of the achievable rate, in a multi-terminal

system the average and maximal error capacity region can be different, even in the case of

asymptotically vanishing errors as shown by Dueck in [Due78]. For this reason, in this chap-

ter we will pay attention to the consideration of the maximal and average error probabilities

and the relation between them.

In the following two subsections we more precisely introduce the considered two-phase bidi-

rectional relaying model and after that we briefly restate for completeness the multiple access

channel capacity region. In Section 3.2 we prove for the bidirectional broadcast channel an

optimal coding theorem and a weak converse for the maximum error probability. The proof

shows that the capacity region is independent of whether we use asymptotically vanishing

average or maximum probabilities of error. After we found the capacity region of the bidi-

rectional broadcast channel we got aware of the work of Tuncel [Tun06] where a joint source

and channel coding approach for a broadcast channel problem is studied. It seems that one

can deduce the achievable rate region with respect to the average error from this work, how-

ever for the bidirectional broadcast channel we do not need the Slepian-Wolf source coding

part since the relay node knows which side information is known at the nodes 1 and 2. For

that reason we can prove the achievable rate region for the average error probability with

classical channel coding arguments only. Moreover, this allows us to derive the capacity

region with respect to the maximum probability of error.

After we obtained the capacity region for each phase we optimize the time division be-

tween the MAC and the BC phases. This gives us the largest achievable rate region for the

discrete memoryless bidirectional relay channel using finite set alphabets with the simplifi-

cations of the operational separation of the two phases. Finally, we illustrate the bidirectional

achievable rate region by means of a binary channel example and finish this chapter with a

discussion as well as an outlook on further results.

3.1.1 Two-Phase Bidirectional Relay Channel

We consider a three-node network with two independent information sources W1and W2at

nodes 1 and 2. The random variables Wk,k= 1,2, are uniformly distributed on the message

182

3.1 Introduction

Node 1 Node 2Relay

Source 1 Source 2

Node 1 Node 2Relay

Source 1 Source 2

1MAC BC

W W

X1X2Y1Y2

Figure 3.1: Multiple access (MAC) and broadcast (BC) phase of a two-phase decode-and-

forward bidirectional relay channel without feedback.

set Wk:= {1,2,...,M(n)

k},k= 1,2. For the bidirectional channel we want the messages

w1∈ W1and w2∈ W2to be known at node 2 and node 1, respectively. For this goal,

the nodes 1 and 2 use the support of the relay node. Note, that we have no direct channel

between the nodes 1 and 2 due to the half-duplex assumption and the two-phase protocol.

We simplify the problem by assuming an a priori operational separation of the communi-

cation into two phases. In this thesis we consider the decode-and-forward approach, which

means that we require the relay node to decode the messages from nodes 1 and 2. Further-

more, we do not allow cooperation between the encoders at nodes 1 and 2. Otherwise, a

transmitted symbol could depend on previously received symbols. For a two-way channel

this is known as a restricted two-way channel. With this simplification we end up with a

multiple access phase, where the nodes 1 and 2 transmit their messages w1and w2to the

relay node, and a broadcast phase, where the relay node forwards the messages to the nodes

2 and 1, respectively. Then the operational separation allows us to look at the two phases

separately. After that we will briefly consider the optimal time division between the two

phases.

In the multiple access phase (MAC) we have a classical multiple access channel, where the

optimal coding strategy and capacity region CMAC is known [Ahl71a, Lia72]. We will restate

the capacity region in the next subsection. As before, let R−→

1R and R−→

2R denote the achievable

rates between the nodes 1 and 2 and the relay node in the MAC phase.

For the broadcast phase (BC) we assume that the relay node has successfully decoded the

messages w1and w2in the multiple access phase. From the union bound1we know that the

error probability of the two-phase protocol is at most the sum of the error probabilities of

each phase. Therefore, an error-free MAC phase is reasonable if we assume rates within the

MAC capacity region and a sufficient coding length. From this we end up with a broadcast

channel where the message w1is known at node 1 and the relay node and the message w2

1Let

˘E1¯and

˘E2¯denote the probabilities of event E1and E2, then the union bound upper bounds the

probability that at least one of the events happens as follows

˘E1∪E2¯≤

˘E1¯+

˘E2¯.

183

3 Optimal Coding Strategy for the Bidirectional Broadcast Channel

is known at node 2 and the relay node as depicted in Figure 3.1. Thereby, let x1,x2, and x

denote the input and y1,y2, and ythe output symbols of node 1, node 2, and the relay node,

respectively. Furthermore, let R−→

R1 and R−→

R2 denote the achievable rates between the relay

node and the nodes 1 and 2 in the BC phase.

The mission of the relay node is to broadcast a message to the nodes 1 and 2 which allows

them to recover the unknown messages. This means that node 1 wants to recover message w2

and node 2 wants to recover message w1. In Section 3.2 we present an information theoretic

optimal coding strategy, which gives us the capacity region of the bidirectional broadcast

channel.

3.1.2 Capacity Region of the Multiple Access Phase

In this subsection, we restate the capacity region of the multiple access channel, which was

independently found by Ahlswede [Ahl71a] and Liao [Lia72] and is nowadays part of any

textbook on multiuser information theory [Wol78, CK81, CT91].

Definition 3.1. A discrete memoryless multiple access channel is the family {p(n):Xn

1×

2→ Yn}n∈Nwith finite input alphabets Xk,k= 1,2, and a finite output alphabet Ywhere

the probability transition functions are given by p(n)(yn|xn

1, xn

2) := Qn

i=1 p(yi|xi1, xi2)for

a given probability transition function {p(y|x1, x2)}x1∈X1,x2∈X2,y∈Y .

Theorem 3.2. The capacity region CMAC with respect to a vanishing average and maximum

probability of error of the memoryless multiple access channel is the set of all rate pairs

[R−→

1R, R−→

2R]satisfying

R−→

1R ≤I(X1;Y|X2, U),

R−→

2R ≤I(X2;Y|X1, U),and

R−→

1R +R−→

2R ≤I(X1, X2;Y|U)

for random variables [U, X1, X2, Y ]with values in U×X1×X2×Y and the joint distribution

{q(u)q1(x1|u)q2(x2|u)p(y|x1, x2)}u∈U,x1∈X1,x2∈X2,y∈Y . Furthermore, the range Uof the

auxiliary random variable Uhas a cardinality bounded by |U| ≤ 2.

3.2 Capacity Region of the Broadcast Phase

In this section we present the capacity region of the bidirectional broadcast channel where

the receiving nodes have perfect knowledge about the message which should be transmitted

to the other node. This means that the independent uniformly distributed information sources

184

3.2 Capacity Region of the Broadcast Phase

W1and W2at the relay node are also known at node 2 and node 1 respectively. We prove the

capacity region by classical channel coding principles. To this end we first introduce some

standard notation.

Definition 3.3. Let Xand Yk,k= 1,2, be finite sets, which denote the input alphabet of

the relay node and the output alphabet of nodes 1 and 2. A discrete memoryless broadcast

channel is defined by a family {p(n):Xn→ Yn

1×Yn

2}n∈Nof probability transition functions

given by p(n)(yn

1, yn

2|xn) := Qn

i=1 p(yi1, yi2|xi)for a probability transition function p:

X → Y1×Y2, i.e. {p(y1, y2|x)}x∈X,y1∈Y1,y2∈Y2is a stochastic matrix.

In what follows we will suppress the super-index nin the definition of the n-th extension of

the channel p, i.e. we will write simply pinstead of p(n). This should cause no confusion

since it will always be clear from the context which block length is under consideration. In

addition, we will use the abbreviation V:= W1×W2, where W1and W2denote the message

sets.

Definition 3.4. A(M(n)

1, M(n)

2, n)-code for the bidirectional broadcast channel consists of

one encoder at the relay node

xn:V → Xn

and decoders at nodes 1 and 2

g1:Yn

1×W1→ W2∪{0},

g2:Yn

2×W2→ W1∪{0}.

The element 0in the definition of the decoders is included for convenience only and plays

the role of an erasure symbol.

From the definition we see that we allow the decoders at nodes 1 and 2 to utilize the knowl-

edge about its own message. When the relay node sends the message v= [w1, w2], the

receiver of node 1 is in error if g1(Yn

1, w1)6=w2. The probability of this event is denoted

λ1(v) :=

g1(Yn

1, w1)6=w2|xn(v)has been sent.

Accordingly, we denote the probability that the receiver of node 2 is in error by

λ2(v) :=

g2(Yn

2, w2)6=w1|xn(v)has been sent.

Hereby, Yn

1and Yn

2denote the random outputs at nodes 1and 2given that the sequence

xn(v)has been sent down the channel. This allows us to introduce the notation for the

maximum and average probabilities of error for the k-th node

λ(n)

k:= max

v∈V λk(v),and µ(n)

k:= 1

|V| P

v∈V

λk(v).

185

3 Optimal Coding Strategy for the Bidirectional Broadcast Channel

Definition 3.5. A rate pair [R−→

R2, R−→

R1]is said to be achievable for the bidirectional broad-

cast channel if for any δ > 0there is an n(δ)∈Nand a sequence of (M(n)

1, M(n)

2, n)-codes

such that for all n≥n(δ)we have log M(n)

n≥R−→

R2 −δand log M(n)

n≥R−→

R1 −δwhile the

maximum probabilities of error λ(n)

1, λ(n)

2→0when n→ ∞. The set of all achievable rate

pairs is the capacity region of the bidirectional broadcast channel and is denoted by CBC.

Remark 3.6. Achievable rate pairs and a capacity region can be similarly defined for aver-

age probability of error µ(n)

1, µ(n)

2→0when n→ ∞.

We are now ready to present the main result of this chapter.

Theorem 3.7. The capacity region CBC of the bidirectional memoryless broadcast channel

is the set of all rate pairs [R−→

R2, R−→

R1]satisfying

R−→

R2 ≤I(X;Y2|U),

R−→

R1 ≤I(X;Y1|U)(3.1)

for random variables [U, X, Y1, Y2]with values in U×X ×Y1×Y2and the joint probability

distribution {q1(u)q2(x|u)p(y1, y2|x)}u∈U,x∈X,y1∈Y1,y2∈Y2. The cardinality of the range of

Ucan be bounded by |U| ≤ 2.

The theorem is proved in the following three subsections. In the first subsection we prove the

achievability, i.e. a coding theorem. We prove a weak converse with respect to the maximum

probability of error in the second subsection. Then the theorem is proved with the third

subsection where we show that a cardinality of two is enough for the range of the auxiliary

random variable.

3.2.1 Proof of Achievability

Here, we adapt the random coding proof for the degraded broadcast channel of [Ber73] to

our context. First, we prove the achievability of all rate pairs [R−→

R2, R−→

R1]satisfying

R−→

R2 ≤I(X;Y2),and R−→

R1 ≤I(X;Y1),(3.2)

for some probability function p(x)p(y1, y2|x). Then, we extend this to prove that all points

in the closure of the convex hull of (3.2) are achievable, which we will see is exactly the

region stated in Theorem 3.7.

186

3.2 Capacity Region of the Broadcast Phase

Random codebook generation

For any δ > 0we define for any R−→

Rk,k= 1,2, the rate of the code ¯

R−→

Rk:=

max{1

n⌊n(R−→

Rk−δ

2)⌋,0}.2Then we generate M(n)

1M(n)

2independent codewords Xn(v),

v= [w1, w2], of length nwith M(n)

1:= 2n¯

R−→

R2 and M(n)

2:= 2n¯

R−→

R1 according to

i=1 p(xi).

Encoding

To send the pair v= [w1, w2]with wk∈ Wk,k= 1,2, the relay sends the corresponding

codeword xn(v).

Decoding

The receiving nodes use typical set decoding. First, we characterize the decoding sets. For

the decoder at node k= 1,2let

I(xn;yn

k) := 1

nlog2

p(yn

k|xn)

p(yn

with average mutual information I(X;Yk) :=

xn,yn

kI(xn;yn

k). This gives the decoding

set

S(yn

k) := xn∈ Xn:I(xn;yn

k)≥¯

R−→

Rk+I(X;Yk)

2

and indicator function

d(xn, yn

k) := (1,if xn/∈ S(yn

k),

0,otherwise.

When xn(v)with v= [w1, w2]has been sent, and yn

1and yn

2have been received we say

that the decoder at node kmakes an error if either xn(v)is not in S(yn

k)(occurring with

probability P(1)

e,k (v)) or if at node 1 xn(w1,ˆw2)with ˆw26=w2is in S(yn

1)or at node 2

xn( ˆw1, w2)with ˆw16=w1is in S(yn

2)(occurring with P(2)

e,k (v)). If there is no or more than

one codeword xn(w1,·)∈ S(yn

1)or xn(·, w2)∈ S(yn

2), the decoders map on the erasure

symbol 0.

2We need not consider the trivial cases ¯

R−→

Rk= 0 for any kbecause then the error probability is zero by

definition.

187

3 Optimal Coding Strategy for the Bidirectional Broadcast Channel

Analysis of the probability of error

From the union bound we have λk(v)≤P(1)

e,k (v) + P(2)

e,k (v)with

P(1)

e,k (v) := P

k∈Yn

p(yn

k|xn(v)) d(xn(v), yn

k)for k= 1,2

and

P(2)

e,1(v) := P

1∈Yn

p(yn

1|xn(v))

|W2|

ˆw2=1

ˆw26=w21−d(xn(w1,ˆw2), yn

1),

P(2)

e,2(v) := P

2∈Yn

p(yn

2|xn(v))

|W1|

ˆw1=1

ˆw16=w11−d(xn( ˆw1, w2), yn

2).

For uniformly distributed messages W1and W2we define P(m)

e,k :=

|W1| |W2|Pv∈W1×W2P(m)

e,k (v)for m= 1,2so that µ(n)

k≤P(1)

e,k +P(2)

e,k .

Next, we average over all codebooks and show that

xnµ(n)

k≤

xnP(1)

e,k +P(2)

e,k →0

as n→ ∞ if R−→

Rk≤I(X, Yk),k= 1,2. Recall that ¯

R−→

Rk≤R−→

Rk−δ

2holds so that we

have

xnP(1)

e,k =1

|W1||W2|X

v∈W1×W2

xnP(1)

e,k (v)

for any

fixed vX

k∈Yn

xnp(yn

k|xn(v)) d(xn(v), yn

k)

k∈Yn

xn∈Xn

p(xn)p(yn

k|xn)d(xn, yn

xn,yn

kd(xn, yn

k)=

d(xn, yn

k) = 1

I(xn;yn

k)≤¯

R−→

Rk+I(X;Yk)

2

≤

I(xn;yn

k)≤I(X;Yk)−δ

4−→

n→∞ 0

exponentially fast by the law of large numbers. For the calculation of

xnP(2)

e,k we have

to distinguish between the receiving nodes. We present the analysis for k= 1, the case

k= 2 follows accordingly. Thereby, we use the fact that for v= [w1, w2]6= [w1,ˆw2]the

188

3.2 Capacity Region of the Broadcast Phase

random variables p(yn

1|Xn(v)) and d(Xn(w1,ˆw2), yn

1)are independent for each choice of

1∈ Yn

xnP(2)

e,1=1

|W1||W2|X

v∈W1×W2

xnP(2)

e,1(v)

for any

fixed vX

1∈Yn

xnnp(yn

1|xn(v))

|W2|

ˆw2=1

ˆw26=w21−d(xn(w1,ˆw2), yn

1)o

1∈Yn

|W2|

ˆw2=1

ˆw26=w2

p(yn

1|xn(v))

1−d(xn(w1,ˆw2), yn

1)

1∈Yn

|W2|

ˆw2=1

ˆw26=w2

p(yn

xn1−d(xn(w1,ˆw2), yn

1)

1∈Yn

|W2|

ˆw2=1

ˆw26=w2

p(yn

1)X

xn∈Xn

p(xn)1−d(xn, yn

1)

= (|W2|−1) X

1∈Yn

xn∈S(yn

p(xn)p(yn

1).

Whenever xn∈ S(yn

1), we have I(xn;yn

1) = 1

nlog2

p(yn

1|xn)

p(yn

1)>1

2(¯

R−→

R1 +I(X;Y1)) or

p(yn

1)< p(yn

1|xn)2−n

2(¯

R−→

R1+I(X;Y1)). Consequently,

xnP(2)

e,1<|W2|X

1∈Yn

xn∈S(yn

p(xn)p(yn

1|xn)2−n

2(¯

R−→

R1+I(X;Y1))

≤2n¯

R−→

R1 2−n

2(¯

R−→

R1+I(X;Y1))

= 2n

2(¯

R−→

R1−I(X;Y1))

≤2n

2(R−→

R1−δ

2−I(X;Y1))

≤2−nδ

4−→

n→∞ 0

where the last inequality holds if we have R−→

R1 ≤I(X, Y1). Hence, if R−→

Rk≤I(X, Yk),

k= 1,2, the average probability of error, averaged over codebooks and codewords, gets

arbitrary small for sufficiently large block length n.

189

3 Optimal Coding Strategy for the Bidirectional Broadcast Channel

Code construction with arbitrary small maximum probability of error

If R−→

R1 ≤I(X;Y1)and R−→

R2 ≤I(X;Y2)for any δ > 0and its code rate ¯

R−→

Rk,k= 1,2, we

can choose ε > 0and n∈Nso that we have

xnµ(n)

1+µ(n)

2< ε. Since the average

probabilities of error over the codebooks is small, there exists at least one codebook C⋆with

a small average probabilities of error µ(n)

1+µ(n)

2< ε. This implies that we have µ(n)

1< ε

and µ(n)

2< ε. We define sets

Q:= {v∈ V :λ1(v)<8εand λ2(v)<8ε},

Rk:= {v∈ V :λk(v)≥8ε},for k= 1,2.

Since ε > 1

|V| Pv∈V λk(v)≥|Rk|

|V| 8ε, we can bound the cardinality |Rk|<|V|

8for k= 1,2.

Then from V=Q∪R1∪R2it follows

|Q| ≥ |V|−|R1|−|R2|>3

4|V|.

Now, let Tbe the set of w1having the property that for each w1there are at least 1

2M(n)

choices of w2so that [w1, w2]∈ Q. Therefore, for w1∈ T there are at most M(n)

2choices

w2∈ W2and for w1/∈ T there are less than 1

2M(n)

2choices w2∈ W2such that [w1, w2]∈

Q. Accordingly, we have

|T|M(n)

2+|W1\T |1

2M(n)

2>|Q| >3

4M(n)

1M(n)

so that it follows that |T | >1

2M(n)

1using |W1\T| =M(n)

1−|T|. This means that there

exists an index set Q⋆

1⊂ W1with 1

2M(n)

1indices w1, to each of which we can find an

index set Q⋆

2(w1)⊂ W2with 1

2M(n)

2indices w2so that we have for each w1∈ Q⋆

1and

w2∈ Q⋆

2(w1)a maximum error λk(w1, w2)<8ε,k= 1,2.

It follows that there exist one-to-one mappings Φ : V⋆→ Q⋆,Φ1:W⋆

1→ Q⋆

1,Φw1

W⋆

2→ Q⋆

2(w1)for each w1∈ Q⋆

1with Φ(w1, w2) := [Φ1(w1),Φw1

2(w2)] with sets V⋆:=

W⋆

1×W⋆

2,W⋆

k:= {1,2. . . , 1

2M(n)

k}for k= 1,2,Q⋆:= {[w1, w2]∈ V :w1∈ Q⋆

1, w2∈

Q⋆

2(w1)} ⊂ Q. Accordingly, there exist mappings Ψk:Q⋆→ W⋆

k,k= 1,2, with v=

[Ψ1(Φ(v)),Ψ2(Φ(v))].

This allows us finally to define a (1

2M(n)

1,1

2M(n)

2, n)-code with an encoder ˜xn:V⋆→ Xn

with ˜xn(v) := xn(Φ(v)) and decoders ˜g1:Yn

1×W⋆

1→ W⋆

2and ˜g2:Yn

2×W⋆

2→ W⋆

1with

˜g1(yn

1, w1) := ˜

Ψ2(w1, g1(yn

1, w1)) and ˜g2(yn

2, w2) := ˜

Ψ1(g2(yn

2, w2), w2)where we use the

mappings ˜

Ψk:V → W⋆

kgiven by

Ψk(v) := (Ψk(v),if v∈ Q⋆,

0,if v /∈ Q⋆,

190

3.2 Capacity Region of the Broadcast Phase

for k= 1,2. The idea is that the encoder uses only codewords xn(v)of the code C⋆with an

index v∈ Q⋆, which have a maximum error λk(v)<8ε,k= 1,2. Since the decoders use

the typical set decoder of the code C⋆, they could erroneously find an xn(v)with v∈ V\Q⋆.

In this case, the mapping ˜

Ψkdecides on the erasure symbol 0. It was already a wrong

decision by the decoder gk, since the encoder chooses only codewords xn(v)with v∈ Q⋆.

Therefore, this does not add any error to the decoding. The code has a rate pair

h¯

R−→

R2 −1

n,¯

R−→

R1 −1

ni=h1

n⌊n(R−→

R2 −δ

2)⌋− 1

n,1

n⌊n(R−→

R1 −δ

2)⌋− 1

which is element-wise larger than [R−→

R2 −δ, R−→

R1 −δ]as n > 4

δusing ⌊x⌋ ≥ x−1. This

proves the achievability of any rate pair satisfying the equation (3.2).

Closure of convex hull

Let R(p(x)) denote the set of rates which we can achieve with the input distribution p(x).

For k= 1,2, we can rewrite the right hand side of (3.1) as follows

I(X;Yk|U) =

|U|

u=1

p(u)I(X;Yk|U=u) =

|U|

u=1

p(u)I(X;Yk)p(x|u),

where in I(X;Yk)p(x|u)we choose a specific input distribution p(x|u)according to the

auxiliary random variable U. For the input distribution p(x|u)we know from the first part

of the proof that any rate pair Ru∈ R(p(x|u)) ⊂

2is achievable. Therefore, for any

convex combination Pm

u=1 αuRuwe can regard the weights as a probability mass function

with p(u) := αuand u∈ U := {1,2,...,m}and choose for any uan input distribution

p(x|u)that achieves the rate pair Ru. For that reason, the conditional mutual informations

given by the right hand sides of (3.1) are also achievable rates.

Finally, the set of achievable rate pairs is closed by definition of achievability since for any

sequence of achievable rate pairs with [R1, R2] = limm→∞[R(m)

1, R(m)

2]the limit point itself

is achievable.

In general in multi-terminal systems the average and maximal error capacity region can

be different. Ahlswede has shown for the two-way channel in [Ahl71b] that “one cannot

reduce a code with average errors to a code with maximal errors without an essential loss

in code length or error probability, whereas for one-way channels it is unessential whether

one uses average or maximal errors.” The problem in the two-way channel is to find a

maximal error sub-code with a Cartesian product structure. This problem is equivalent to

191

3 Optimal Coding Strategy for the Bidirectional Broadcast Channel

a combinatorial problem by Zarankiewicz3[Ahl71b] and arises since the transmitter and

receiver have disjoint partial knowledge only. Here, the relay node has full knowledge so

that for the code construction with arbitrarily small maximum probability of error we need

not require a sub-code with Cartesian product structure.

In the next subsection we prove the weak converse for the maximal error. Since Fano’s

inequalities apply for the average error as well, the weak converse for the average error

follows analogously.

3.2.2 Proof of weak converse

We have to show that any given sequence of (M(n)

1, M(n)

2, n)-codes with λ(n)

1, λ(n)

2→

0must satisfy 1

nlog M(n)

1≤I(X;Y2|U) + o(n0)and 1

nlog M(n)

2≤I(X;Y1|U) +

o(n0)for a joint probability distribution q1(u)q2(x|u)p(y1, y2|x). For a fixed block

length nwe define the joint probability distribution p(w1, w2, xn, yn

1, yn

2) := 1

|W1|

|W2|

q2(xn|w1, w2)Qn

i=1 p(y1i, y2i|xi)on W1× W2× Xn× Yn

1× Yn

2where the conditional

distribution q2(xn|w1, w2) = 1 if xnis the codeword corresponding to w1, w2or is equal to

0else. In what follows we consider for k= 1,2uniformly distributed random variables Wk

with values in Wk.

Lemma 3.8. For our context we have the Fano’s inequality

H(W2|Yn

1, W1)≤λ(n)

1log |W2|+ 1 = nε(n)

1(3.3)

with ε(n)

1=log |W2|

nλ(n)

1+1

n→0for n→ ∞as λ(n)

1→0.

Proof. From Yn

1and W1node 1 estimates the index W2from the sent codeword

Xn(W1, W2). We define the event of an error at node 1 as

E1:= (1,if g1(Yn

1, W1)6=W2,

0,if g1(Yn

1, W1) = W2,

so that we have for the mean probability of error µ(n)

E1= 1≤λ(n)

1. From the

chain rule for entropies we have

H(E1, W2|Yn

1, W1) = H(W2|Yn

1, W1) + H(E1|Yn

1, W1, W2)

=H(E1|Yn

1, W1) + H(W2|E, Y n

1, W1).

3For an n×nmatrix which contains zeros and ones only Zarankiewicz [Zar51] posed the combinatorial problem

to find the smallest natural number of ones, kj(n), which ensures that a n×nmatrix contains a j×jminor

which consists entirely of ones. In [Ahl71b] Ahlswede generalizes the problem to find the smallest number

of ones, ki,j (m, n), which ensures that a m×nmatrix contains a i×jsubmatrix entirely of ones.

192

3.2 Capacity Region of the Broadcast Phase

Since E1is a function of W1, W2, and Yn

1, we have H(E1|Yn

1, W1, W2) = 0. Further, since

E1is a binary-valued random variable, we get H(E1|Yn

1, W1)≤H(E1)≤1. So that finally

with the next inequality

H(W2|Yn

1, W1, E1) =

E1= 0H(W2|Yn

1, W1, E1= 0)

E1= 1H(W2|Yn

1, W1, E1= 1)

≤(1 −µ(n)

1)0 + µ(n)

1log(|W2|−1) ≤λ(n)

1log |W2|

we get Fano’s inequality for our context.

Therewith, we can bound the entropy H(W2)as follows

H(W2) = H(W2|W1)

=I(W2;Yn

1|W1) + H(W2|Yn

1, W1)

≤I(W2;Yn

1|W1) + nε(n)

≤I(W1, W2;Yn

1) + nε(n)

≤I(Xn;Yn

1) + nε(n)

≤H(Yn

1)−H(Yn

1|Xn) + nε(n)

where the equations and inequalities follow from the independence of W1and W2, the defi-

nition of mutual information, Lemma 1, the chain rule for mutual information, the positivity

of mutual information, and the data processing inequality. If we divide the inequality by n

we get the rate

nH(W2) = 1

nH(Yn

1)−H(Yn

1|Xn)+ε(n)

≤1

i=1H(Y1i|Yi−1

1)−H(Y1i|Yi−1

1, Xn)+ε(n)

≤1

i=1H(Y1i)−H(Y1i|Xi)+ε(n)

i=1

I(Y1i;Xi) + ε(n)

using the chain rule of mutual information, the memoryless property of the channel, and the

definition of mutual information. A similar derivation for the source rate 1

nH(W1)gives us

the bound 1

nH(W1)≤1

i=1

I(Y2i;Xi) + ε(n)

2with ε(n)

2=log |W1|

nλ(n)

2+1

n→0for n→ ∞

as λ(n)

2→0.

This means that the entropies H(W1)and H(W2)are bounded by averages of the mutual

informations calculated at the empirical distribution in column iof the codebook. Therefore,

193

3 Optimal Coding Strategy for the Bidirectional Broadcast Channel

we can rewrite these inequalities with an auxiliary random variable U, where U=i∈ U =

{1,2,...,n}with probability 1

n. We finish the proof of the converse with the following

inequalities

nH(W2)≤1

i=1

I(Y1i;Xi) + ε(n)

i=1

U=iI(Y1i;Xi|U=i) + ε(n)

=I(Y1U;XU|U) + ε(n)

=I(Y1;X|U) + ε(n)

and 1

nH(W1)≤I(Y2;X|U) + ε(n)

2accordingly where ε(n)

k→0,k= 1,2, when n→ ∞.

Thereby, Yk:= YkU and X:= XUare new random variables whose distributions depend on

Uin the same way as the distributions of Yki and Xidepend on i.

Up to now the auxiliary random variable Uis defined on a set Uwith arbitrary cardinality.

Next, we will show that |U| = 2 is enough.

3.2.3 Cardinality of set U

With Fenchel–Bunt’s extension of Carathéodory’s theorem it follows that any rate pair in the

convex hull coSp(x)R(p(x))=Su∈U R(p(x|u)) is achievable by time-sharing between

two rate pairs from Sp(x)R(p(x)), i.e. |U| = 2 is enough.

Theorem 3.9 ([HUL01, Theorem 1.3.7]).If S ⊂

nhas no more than nconnected com-

ponents (in particular, if Sis connected), then any x∈co(S)can be expressed as a convex

combination of nelements of S.

Since for any x∈ X we have [0,0] ∈ R(p(x)), the set Sp(x)R(p(x)) is connected. There-

fore, any rate pair in CBC =coSp(x)R(p(x))can be expressed as a convex combination

of n= 2 rate pairs of Sp(x)R(p(x)).

This finishes the proof of the capacity region of the bidirectional broadcast channel.

Remark 3.10. Since the coding theorem includes the achievability of rate pairs in terms of

the average probability of error and the proof of the weak converse for the average error

works analogously, CBC is also the capacity region in terms of the average probability of

error.

194

3.3 Achievable Bidirectional Rate Region

Remark 3.11. The characterization of the bidirectional broadcast capacity region for Gaus-

sian channels is analogous. We would have to deal with discrete channels with Gaussian

channel transfer distributions and would have to add an input power constraints but the

arguments are similar to the arguments considered here.

The coding principles of the bidirectional broadcast are similar to the network coding

approach with a bitwise XOR operation on the decoded messages at the relay node

[WCK05, LJS05, FBW06, PY06, HKE+07]. But since network coding is originally a multi-

terminal source coding problem, the achievable rates with network and channel code separa-

tion in the broadcast phase using the XOR coding approach are limited by the worst receiver.

This means that with a network coding approach using the XOR operation on the decoded

data we can achieve in our network the rates

R−→

R2, R−→

R1 ≤min{I(X;Y1), I(X;Y2)}

for some common input distribution p(x). The achievable rates depend on the common input

distribution and both channel transfer distributions. For our coding scheme each achievable

rate depends on the common input distribution and its own channel transfer distribution

only. For each channel we can separately find the optimal input distribution which achieves

the maximal achievable rate for this link (equal to the single link capacity), but the optimal

input distribution for one channel need not be optimal for the other channel.

Accordingly, we see that the network coding approach using XOR on the decoded

messages at the relay node is in general inferior, but it achieves the capacity of the

bidirectional broadcast if and only if for the maximizing input distribution p⋆(x) =

arg maxp(x)max{I(X;Y1), I(X;Y2)}we have I(X;Y1) = I(X;Y2). In the following

we will look at the bidirectional achievable rate region.

3.3 Achievable Bidirectional Rate Region

We will now look at the achievable bidirectional rate region where we use in each phase

an optimal coding strategy. Thereby, we optimize the time division between the MAC phase

with memoryless multiple access channel p(y|x1, x2)and BC phase with memoryless broad-

cast channel p(y1, y2|x). Of course, due to the a priori separation into two phases, this strat-

egy need not be the optimal strategy for the bidirectional relay channel.

Let R1and R2denote the achievable rates for transmitting a message w1from node 1 to node

2 and a message w2from node 2 to node 1 with the support of the relay node. In more detail,

node 1 wants to transmit message w1with rate nR1in nchannel uses of the bidirectional

relay channel to node 2. Simultaneously, node 2 wants to transmit message w2with rate

nR2in nchannel uses to node 1. Then let nMAC and nBC =n−nMAC denote the number

195

3 Optimal Coding Strategy for the Bidirectional Broadcast Channel

of channel uses in the MAC phase and BC phase with the property nMAC

n→α∈[0,1] and

nBC

n→1−αwhen n→ ∞, respectively. As before, we call αthe time division parameter

the between multiple access and broadcast phase. With a sufficient block length n(respec-

tively nMAC and nBC) we can achieve a bidirectional transmission of messages w1and w2

with arbitrary small decoding error if rate pairs [R−→

1R, R−→

2R]∈ CMAC and [R−→

R2, R−→

R1]∈ CBC

exist so that we have

nR1≤min{nMACR−→

1R, nBCR−→

R2},

nR2≤min{nMACR−→

2R, nBCR−→

R1}.

Thus, the achievable rate region of the bidirectional relay channel using the time division

is given by the set of all rate pairs [R1, R2]which are achievable with any time division

parameter α∈[0,1] as n→ ∞. We collect the previous consideration in the following

proposition.

Proposition 3.12. The achievable rate region RBRC of the two-phase bidirectional relay

channel is given by

RBRC =n[R1, R2]∈

2:R1≤min{αR−→

1R,(1 −α)R−→

R2},

R2≤min{αR−→

2R,(1 −α)R−→

R1}with α∈(0,1),

[R−→

1R, R−→

2R]∈ CMAC,and [R−→

R2, R−→

R1]∈ CBCo.

The region CBC is in general larger than the broadcast region using superposition encoding

where we additionally require separated information flows and XOR on the decoded data

at the relay node. It follows that the achievable bidirectional rate region RBRC is larger as

well.

3.4 Example with Binary Channels

Finally, we will briefly look at an example with binary channels. Therefore, we assume

to have a binary erasure multiple access channel [GW75],[CT91, Example 14.3.3] in the

MAC phase and two independent binary symmetric channels [Ash65, Section 3.3] in the BC

phase.

Binary Erasure Multiple Access Channel

We briefly reproduce the definitions and the resulting capacity region of the binary erasure

multiple access channel from [GW75] and [CT91, Example 14.3.3]. Therefore, we have

196

3.4 Example with Binary Channels

binary input alphabets X1=X2={0,1}, an output alphabet YR={0,1, E}, and transition

probabilities

PYR|X1,X2(0|0,0) = 1, PYR|X1,X2(1|1,1) = 1,

PYR|X1,X2(E|0,1) = 1, PYR|X1,X2(E|1,0) = 1

so that we have a deterministic channel. We see that if the output of the channel is the erasure

symbol Ethe relay node cannot uniquely determine the input. With the input distribution

given by

PX1(0) = p1,and PX2(0) = p2,

it can be easily seen that the mutual informations I(X1;YR|X2) = H(X1),I(X2;YR|X1) =

H(X2)and I(X1, X2;YR) = H(YR)are simultaneously maximized when p1=p2=1

/2.

Then the capacity region is given by

CMAC =[R1, R2]∈

+:R1≤1, R2≤1, R1+R2≤1.5(3.4)

which is shown in Figure 3.2.

Remark 3.13. Gaarder and Wolf introduced this binary erasure multiple access channel in

[GW75] to show that a noiseless feedback for the binary erasure multiple access channel

can enlarge the capacity region. Accordingly, we can use this channel with a lossless broad-

cast channel to construct an example which shows that allowing cooperation between the

encoders enlarges the bidirectional achievable rate region.

Binary Symmetric Broadcast Channel

The binary symmetric broadcast channel consists of two independent binary symmetric chan-

nels [Ash65, Section 3.3]. For the binary symmetric broadcast channel we have binary input

and output alphabets X=Y1=Y2={0,1}and transition probabilities

PY1|X(0|0) = PY1|X(1|1) = q1,and PY2|X(0|0) = PY2|X(1|1) = q2,

so that we have a symmetric channel, i.e. H(Yk|X)does not depend on the input distribution.

Accordingly, the problem of maximizing I(Yk|X) = H(Yk)−H(Yk|X),k= 1,2, reduces

to the problem of maximizing H(Yk),k= 1,2. Then it is easy to see that if all input

symbols of a symmetric channel are equally likely, the output symbols are also equally likely.

Therefore, the uniform input distribution maximizes both channels simultaneously so that the

broadcast capacity region for the binary symmetric channel is given by

CBC = [0,1−H(p2)] ×[0,1−H(p1)] (3.5)

with H(pk) = −pklog(pk)−(1 −pk) log(1 −pk),k= 1,2. The capacity region CBC

is shown in Figure 3.2. Obviously, it includes the region [0,1−max{H(p1), H(p2)}]×

[0,1−max{H(p1), H(p2)}]which is achievable using XOR on the decoded data at the

relay node.

197

3 Optimal Coding Strategy for the Bidirectional Broadcast Channel

cMAC

cBC

R1RRR2

R2RRR1 R2

BRC

1/2

1/2 1

1/4

Figure 3.2: The left figure shows the capacity regions CMAC (dotted line) and CBC (dashed

line), the right figure shows the corresponding achievable rate region RBRC

(solid line). The dashed-dotted line exemplarily shows for one angle φthe

achievable rate pair (•) on the boundary of RBRC with the optimal time divi-

sion between the two rate pairs (×) on the boundary of CMAC and CBC.

Achievable Bidirectional Rate Region

In Figure 3.2 we depicted the capacity regions CMAC and CBC and the achievable rate region

RBRC with a symmetric binary erasure multiple access channel, c.f. (3.4), and a binary

symmetric broadcast channel, cf. (3.5). The boundary of the achievable rate region can be

obtained geometrically if one takes for any angle φ∈[0, π/2] half of the arithmetical mean

between the boundary rate pairs of the capacity regions where we have tan φ=R−→

2R/R−→

1R =

R−→

R1/R−→

R2.

3.5 Discussion

The proposed coding scheme follows the network coding idea and treats information flows in

a network not as physical fluids. But since network coding assumes error-free links between

the nodes it does not consider channel coding aspects. Due to the half-duplex assumption

we will have a natural separation of the protocol into two phases. In this thesis we assume

that the encoders of nodes 1 and 2 cannot cooperate and that the relay node has to decode the

messages. Then the optimal coding strategy for the MAC phase is well-known. We prove an

optimal coding strategy for the remaining BC phase, which gives us the capacity region for a

network with the assumed simplifications. It follows that this strategy is superior to the prior

198

3.5 Discussion and Further Results

coding strategies based on superposition encoding or XOR operation on the decoded data at

the relay node.

For a given broadcast input distribution we achieve on each link the corresponding discrete

memoryless channel rate. But it is important to notice that the achievable rates are coupled by

the common input distribution, which means that we cannot optimize the rates on each link

separately. Then it is curious to see that if we transfer the result to scalar Gaussian channels

with a mean power constraint, obviously the Gaussian input distribution will maximize both

links simultaneously. However, for the vector valued Gaussian channel this is no longer

the case. For the optimal input distribution we can apply the same arguments as for the

MIMO MAC in Section 2.6.1. For each MIMO channel there is an optimal input distribution.

This leads to two boundary rate pairs where at each rate pair one rate is maximal. Then

the boundary inbetween the two rate pairs will be curved and can be calculated by convex

optimization methods.

Further Results

Finally, we want briefly mention two further results which we derived in this context. The

the main contribution to the the strong converse result is due to Igor Bjelakovic and to the

practical coding aspects is due to Clemens Schnurr.

Strong Converse

In [BOSB07, OBSB07] we prove the strong converse for the maximum error probability and

show that this implies that [ε1, ε2]-capacity region in terms of average probability of error is

constant for small values of error parameters ε1and ε2.

Let CBC,max(ε1, ε2)denote the set of all achievable rate pairs with maximum errors less

than ε1and ε2as n→ ∞. Then we can prove the strong converse for the memoryless

bidirectional broadcast channel which means that we have CBC =CBC,max(ε1, ε2)for all

ε1, ε2∈(0,1). The proof is based on the blowing-up technique of Ahlswede, Gács, Körner

[AGK76]. Therefore, we "blow-up" the decoding sets and use the Blowing-up Lemma to

bound the error event in a variant of Fano’s inequality. Then we follow the line of the weak

converse.

Similarly, we define CBC,av(ε1, ε2)as the capacity region for the average errors. For suffi-

ciently small ε1, ε2we can construct a sub-code with bounded maximum error probability

from which it follows from the strong converse for the maximum error that the corresponding

rate pair has to be within CBC. As a corollary we have that for the memoryless bidirectional

broadcast channel it holds that CBC =CBC,av(ε1, ε2)for all [ε1, ε2]∈(0,1

2)×(0,1

4)or

[ε1, ε2]∈(0,1

4)×(0,1

2).

199

3 Optimal Coding Strategy for the Bidirectional Broadcast Channel

Practical Coding Aspects

The coding theorem offers a hint how to design a good practical channel code. Accordingly,

in [SOS07] an interesting coset coding strategy for symmetric channels is discussed.

The coding theorem suggests to design for each receiving node and for each side information

one triplet a code, an encoder, and a decoder. Let {E1,w1(w2)}w2∈W2denote the code for

node 1with side information w1∈ W1and similarly let {E2,w2(w1)}w1∈W1denote the code

for node 2with side information w2∈ W2. Then the coding theorem requires that the two

code families are interwoven so that we have E1,w1(w2) = E2,w2(w1)for each message pair

[w1, w2]∈ W1×W2the relay wants to transmit.

Then we make the observation that for symmetric channels ("channel distortion" is input

independent) the cosets of the code perform as good as the original code. This suggests that

we can construct the code families for each user as coset codes by simply shifting the base

codes. If the two single codes base on an Ablelian additive group this leads to a simple

superposition of the codewords in the group, i.e. ψ(w1, w2) = E1(w2) + E2(w1). Then

each code will perform as good as the base code. We also present a simple counter example

which shows that this construction rule does not extend to non-symmetric channels.

200

4 Conclusion

Historically, information theory has played a crucial role in the development of one-way

point-to-point communication. With the development of the wireless networks and the Inter-

net the research focus has recently shifted to multi-terminal problems. A one-hop transmis-

sion over a long distance or of a shadowed node needs high transmit power which causes a

large interference for a wide range and results in a high energy consumption. Relaying pro-

tocols have the potential to enhance the coverage and the throughput in wireless networks by

utilizing the broadcast nature of the wireless medium. For that reason, it is common sense

that relaying concepts will play a central role in future wireless communication systems.

In this thesis we study bidirectional decode-and-forward relaying which has the ability to

reduce the spectral loss due to the half-duplex constraint of wireless nodes. From the half-

duplex constraint the relay communication is naturally separated into two phases where the

relay node either transmits or receives. The two-phase separation of the protocol has the

appealing property that the bidirectional relaying protocol can be easily integrated in a con-

ventional cellular network or extended to multi-hop communication as depicted in Figure 4.1

and Figure 4.2. Accordingly, we think that the study of the two-phase bidirectional decode-

and-forward protocol is of great interest. We consider the optimal and fixed time division

case. The optimal case is interesting on its own, while the fixed may be interesting if the

bidirectional protocol should be integrated in an existing wireless network.

The scarce of radio resources and the limited energy supply of mobile nodes in wireless

networks make efficient resource and power allocation strategies absolutely necessary. It is

widely known that a spectrally efficient wireless network cannot be designed with a layered

architecture where we optimize each layer separately. Furthermore, the performance can be

substantially enhanced if we allow interaction between layers, which leads to the cross-layer

design concept. In Chapter 2 we studied different cross-layer design aspects for bidirectional

relaying based on the achievable rates using the superposition encoding strategy at the relay

node.

From Section 2.3 we see that for the maximal throughput over a bidirectional relaying link,

the optimal resource allocation not just aims for the maximal sum-rate with respect to the

channel states. An optimal resource allocation policy depends on the achievable rate re-

gion and factors in the traffic of the higher layers. This means that a throughput optimal

201

4 Conclusion and Future Work

BS BC

MAC

(a) MAC phase/cellular downlink

MAC

Bi−BC

(b) BC phase/cellular uplink

Figure 4.1: The integration of the bidirectional broadcast channel (Bi-BC) can be used for

coverage extension in a cellular network, where BS denotes the basestation and

Mi,i= 1,2,3,4denote the mobile terminals. Mobile M2works as the relay

node. In the cellular downlink the bidirectional relay channel works in its MAC

phase, while M1may cause interference at other nodes. In the cellular uplink the

bidirectional relay channel works is in its broadcast phase (Bi-BC).

M1M3

Bi−BC MACMAC

M1M3M4M5

MAC Bi−BCBi−BC

Figure 4.2: Bidirectional relaying can be easily extended to multi-hop communication. In

the upper figure, node M1,M3, and M5transmit and M2and M4receive. The

signal of node M1and M5cause interference at nodes M4and M2respectively.

In the lower figure, node M2and M4transmit and M1,M3, and M5receive. The

signal of node M2and M4cause interference at nodes M5and M1respectively.

(Chain topology as proposed in [KRH+06])

202

resource allocation policy depends, of course, on the channel states but also on the load at

each node.

This insight has a direct consequence for the routing problem in a wireless network consid-

ered in Section 2.4. While for unidirectional relay protocols we select the relay node which

results in the largest achievable rate, for bidirectional protocols we have to select the relay

node with respect to the achievable rate pair at which we want to operate. It is therefore a

vector optimization problem. Moreover, this means that if we apply the throughput optimal

policy from Section 2.3 the route we choose depends again on the channel state as well as

on the load, which therefore leads to a load adaptive routing strategy. Furthermore, we see

that the performance can be improved if we allow time-sharing between the usage of relay

nodes.

However, the scaling law of the ergodic rate region for relay nodes with iid Rayleigh fad-

ing channels shows that the spatial diversity gain decreases with increasing number of relay

nodes. Since in real wireless networks the channels of the relay nodes are usually not iden-

tically distributed we conclude that for a practical implementation it will be sufficient to

decide between a small number of preselected relay nodes.

In Section 2.5 we see from the joint resource allocation for the bidirectional relaying and

an additional relay multicast communication that it is always optimal to decode the relay

message first. Although we specified the total sum-rate maximum for this simple problem,

we see that the closed form discussion is tedious and it is probably impossible for more or

more difficult routing problems. This lets us conclude that joint optimization of multiple

routing tasks always improves the overall performance and makes new trade-offs possible,

but it also increases the complexity of the problem.

The previous insights were obtained from the discussion of single-antenna nodes, but all

principles transfer to the multiple antenna case which we studied in Section 2.6. The diffi-

cult optimal transmit strategy for the vector valued processing in the MIMO-MAC prevents

a closed form discussion as in the scalar case. Nevertheless, multiple antennas improve

the achievable rate for each link in the usual manner so that it follows that the achievable

rates linearly scale in the high power regime according to the spatial degrees of the MIMO

channels and the time division between the phases.

In Chapter 3 we find the optimal channel coding strategy with respect to asymptotically

vanishing maximal probability of error for the bidirectional broadcast channel of the bidi-

rectional decode-and-forward protocol without feedback. The important improvement to the

superposition coding approach in Chapter 2 is that we do not treat information as a fluid,

which is in accordance with the philosophy of network coding [ACLY00]. Although we

obtained the results for discrete memoryless channels with finite-size alphabets, the coding

strategy can be easily transfered using standard arguments to continuous Gaussian channels

with an input mean power constraint.

203

4 Conclusion and Future Work

MAC

(a) MAC phase/cellular uplink

M3M4

Bi−BC

(b) BC phase/cellular downlink

Figure 4.3: If two nodes within one cell (here M1and M2) want to communicate with each

other, the integration of the coding idea of the bidirectional broadcast channel

(Bi-BC) can improve the downlink performance. Thereby, the base-station works

as the relay node.

For multi-terminal channel coding problems it generally makes a difference if one considers

achievable rates with respect to the average or maximal probability of error. We follow the

classical strategy and extract a sub-code of essentially the same size with vanishing maxi-

mal error probability from the code which suffices the average error probability condition.

Thereby, we surprisingly do not rely on a Cartesian mapping of the message indexes at the

relay node. This is because of the special distribution of the knowledge about the messages

for the bidirectional broadcast channel. But this will cause a difference for the achievable

rates if one considers channel coding for other network problems, e.g. [Ahl71b].

A more explicit discussion of the obtained results is given in Sections 2.7 and 3.5.

Future Work

Unfortunately, we obtained the optimal coding result in Chapter 3 after we have studied the

cross-layer design aspects presented in Chapter 2. However, all insights, techniques, and

conclusions can be transfered to the optimal coding approach. In particular, all results based

on the maximal unidirectional rates will be the same. It will be future work to characterize

the results explicitly.

In Figure 4.3 we present an idea how bidirectional relaying can be used to improve the

performance in a cellular wireless network. If two nodes in one cell want to communicate

with each other we can use the broadcast coding techniques to improve the broadcast in the

downlink. In the future we will work out this example and similar networks as depicted in

Figure 4.1 and Figure 4.2.

204

It follows that bidirectional relaying can be applied to cellular, ad-hoc, or hybrid networks in

order to enhance the coverage or increase the overall throughput and we are sure that there

are more situations where the bidirectional relaying protocol can be integrated in a meshed

wireless network architecture. Moreover, we see that this will again open up new possibilities

for further improvement. For example in Figure 4.1 the transmission of M1interferes with

the broadcast of the basestation so that node M1causes additional interference at other nodes

in the cell. If those regard the interference as additional information, it should be possible to

improve the overall performance again. This example illustrates the potential improvement

of the performance in a meshed architecture, which makes a careful study necessary. But we

also see that these concepts will increase the system‘s complexity. For that reason it will be

necessary to specify conditions where additional cooperation is still beneficial in a wireless

system.

A more closely related extension to our results is to find the optimal coding strategy if we

drop the assumptions we made in Chapter 3. In particular this means that we do not require

that the relay node has to decode the messages and we allow the encoders at nodes 1 and 2

to cooperate via a feedback.

In some situations in this thesis we considered the averages of the achievable rates with

a short-term (over a frame) average power constraint. Therefore, it would be interesting

to consider adaptive resource allocation policies which take advantage of a time-varying

channel as done in [KH95, GV97, TH98, BPS98]. Next, it would be interesting if we allow

queueing at the relay node. Additional queueing and adaptive resource allocation policies

are closely related to problems concerning the delay.

For an implementation we have to re-evaluate the performance where we take into account

the costs for providing the system state information at a centralized controller and/or other

nodes. In accordance, for adaptive resource allocation policies it would be interesting to find

decentralized decision strategies based on the local information or to develop new strategies

which base on partial knowledge only.

Of course, one has to answer many further questions before considering a practical imple-

mentation. However, we are absolutely convinced that further studies on bidirectional relay-

ing will pay off, because we think that bidirectional relaying is conceptually a wise approach.

It avoids the spectral loss of two separated unidirectional relaying protocols due to the half-

duplex constraint, and in addition to this it can fully exploit the network coding idea. All this

lets us finally conclude that bidirectional relaying is spectrally efficient and has the potential

to enhance the throughput and coverage in wireless networks.

205

References

[ACLY00] Rudolf Ahlswede, Ning Cai, Shuo-Yen Robert Li, and Raymond W. Yeung.

Network information flow. IEEE Trans. Inf. Theory, 46(4):1204–1216, July

2000.

[AGK76] Rudolf Ahlswede, P. Gács, and János Körner. Bounds on Conditional Probabil-

ities with Applications in Multi-User Communication. Z. Wahrscheinlichkeit-

stheorie verw. Gebiete, 34:157–177, 1976.

[AGS05] Kambiz Azarian, Hesham El Gamal, and Philip Schniter. On the achiev-

able diversity–multiplexing tradeoff in half-duplex cooperative channels. IEEE

Transactions on Information Theory, 51(12):4152–4172, December 2005.

[Ahl71a] Rudolf Ahlswede. Multi-way communication channels. In Proc. 2nd Int. Sym-

posium on Information Theory, pages 23–52, Tsahkadsor, Armenian, USSR,

September 1971.

[Ahl71b] Rudolf Ahlswede. On two-way communication channels and a problem by

Zarankiewicz. In Sixth Prague Conf. on Inf. Th., Stat. Dec. Fct’s and Rand.

Proc., pages 23–37. Publ. House Chechosl. Academy of Sc., September 1971.

[Ahl06] Rudolf Ahlswede. Towards a general theory of information transfer. Shan-

non Lecture at ISIT in Seattle 13th July 2006, IEEE Inform. Theory Society

Newsletter, 2006.

[AS64] Milton Abramowitz and Irene A. Stegun. Handbook of Mathematical Func-

tions. U.S. Department of Commerce, National Bureau of Standards, Wash-

ington D.C., 1964.

[Ash65] Robert B. Ash. Information Theory. Interscience Publishers, New York, 1965.

[Bam98] Nicholas Bambos. Toward power-sensitive network architectures in wireless

communications: concepts, issues, and design aspects. IEEE Personal Com-

munications, 5(3):50–59, June 1998.

207

References

[BCC+07] Ezio Biglieri, Robert Calderbank, Anthony Constantinides, Andrea Goldsmith,

Arogyaswami Paulraj, and H. Vincent Poor. MIMO Wireless Communications.

Cambridge University Press, 2007.

[Ber73] Patrick P. Bergmans. Random coding theorem for broadcast channels with

degraded components. IEEE Transactions on Information Theory, 19(2):197–

207, March 1973.

[BJ02] Holger Boche and Eduard Jorswieck. Sum Capacity Optimization of the

MIMO Gaussian MAC. In Proc. 5th Intern. Symp. On Wireless Personal Mul-

timedia Communications (WPMC 2002), volume 1, pages 130–134, Honolulu,

USA, October 2002.

[BJH03] Holger Boche, Eduard Jorswieck, and Thomas Haustein. Channel aware

scheduling for multiple antenna multiple access channels. In Proc. 37th Asilo-

mar Conference on Signals, Systems, and Computers, volume 1, pages 992–

996, Monterey, USA, November 2003.

[BKRL06] Aggelos Bletsas, Ashishi Khisti, David P. Reed, and Andrew Lippman. A

Simple Cooperative Diversity Method Based on Network Path Selection. IEEE

Journal on Selected Areas in Communications, 24(3):659–672, March 2006.

[BNOP06] Helmut Bölcskei, Rohit U. Nabar, Özgür Oyman, and Arogyaswami J. Paulraj.

Capacity scaling laws in MIMO relay networks. IEEE Trans. Wireless Com-

munications, 5(6):1433–1444, June 2006.

[BOSB07] Igor Bjelakovic, Tobias J. Oechtering, Clemens Schnurr, and Holger Boche.

On the strong converse for the broadcast capacity region of two-phase bidi-

rectional relaying. In Proc. IEEE Information Theory Workshop (ITW ’07),

Bergen, Norway, July 2007.

[BPS98] Ezio Biglieri, John Proakis, and Shlomo Shamai (Shitz). Fading channels:

Information-theoretic and communications aspects. IEEE Transactions on In-

formation Theory, 44(6):2619–2692, October 1998.

[BV04] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge

University Press, 2004.

[BW06] Holger Boche and Marcin Wiczanowski. Stability-optimal transmission pol-

icy for the multiple antenna multiple access channel in the geometric view.

EURASIP Signal Processing Journal, Special Issue on Advances in Signal

Processing-assisted Cross-layer Designs, June 2006.

[BW07] Holger Boche and Marcin Wiczanowski. Optimization-theoretic analysis of

the stability-optimal transmission policy for multiple antenna multiple access

channel. IEEE Transactions on Signal Processing, 2007. to appear.

208

References

[CG79] Thomas M. Cover and Abbas El Gamal. Capacity theorems for the relay chan-

nel. IEEE Transactions on Information Theory, 25(5):572–584, Sept. 1979.

[Chi04] Mung Chiang. To layer or not to layer: balancing transport and physical layers

in wireless multihop networks. In Proc. 23rd IEEE Infocom, Hong Kong,

March 2004.

[CK81] Imre Csiszár and János Körner. Information Theory: Coding Theorems for

Discrete Memoryless Systems. Academic Press, New York, 1981.

[CK06] Thomas M. Cover and Young-Han Kim. Capacity of a class of deterministic

relay channels, 2006.

[Cov75] Thomas M. Cover. Some advances in broadcast channels. Advances in Com-

munication Systems, 4:229–260, 1975.

[CS03] Rene L. Cruz and Arvind V. Santhanam. Optimal routing, link scheduling and

power control in multi-hop wireless networks. In Proc. 22nd IEEE Infocom,

pages 702–211, San Francisco, USA, March 2003.

[CT91] Thomas M. Cover and Joy A. Thomas. Elements of Information Theory. John

Wiley & Sons, Inc., New York, 1991.

[CV93] Roger S. Cheng and Sergio Verdu. Gaussian multiaccess channels with ISI:

Capacity region and multiuser water-filling. IEEE Transactions on Information

Theory, 39(3):773–785, May 1993.

[DK04] Zaher Dawy and Hichem Kamoun. The general gaussian relay channel: Anal-

ysis and insights. In Proc. 5th Int. ITG Conf. on Source and Channel Coding,

Erlangen, Germany, January 2004.

[DMM97] Mark D. Dankberg, Mark J. Miller, and Michael G. Mulligan. Self-interference

cancellation for two-party relayed communication. U.S. Patent 5 596 439, Jan.

21, 1997.

[Doo53] Joseph L. Doob. Stochastic Processes. John Wiley, New York, 1953.

[DSG+03] Amir F. Dana, Masoud Sharif, Radhika Gowaikar, Babak Hassibi, and

Michelle Effros. Is broadcast plus multi-access optimal for gaussian wireless

networks. In Proc. Asilomar Conference on Signals, Systems, and Computers,

November 2003.

[Due78] Gunter Dueck. Maximal error capacity regions are smaller than average error

capacity regions for multi-user channels. Problems of Control and Information

Theory, 7(1):11–19, 1978.

209

References

[EMH+03] Michelle Effros, Muriel Médard, Tracey Ho, Siddharth Ray, David Karger, and

Ralf Koetter. Linear network codes: A unified framework for source, channel,

and network coding. In Proc. DIMACS Workshop Networking Information

Theory, Piscataway, NJ, USA, 2003.

[FBW06] Christina Fragouli, Jean-Yves Le Boudec, and Jörg Widmer. Network coding:

An instant primer. SIGCOMM Comput. Commun. Rev., 36(1):63–68, 2006.

[FG98] Gerard. J. Foschini and Michael. J. Gans. On limits of wireless communica-

tion in a fading environment when using multiple antennas. Wireless Personal

Communications, 6:311–335, 1998.

[GA82] Abbas El Gamal and Mohammad Aref. The capacity of the semideterministic

relay channel. IEEE Transactions on Information Theory, 28(3):536, May

1982.

[Gal68] Robert G. Gallager. Information Theory and Reliable Communication. John

Wiley & Sons, Inc., New York, 1968.

[Gal96] Robert G. Gallager. Discrete Stochastic Processes. Kluwer Academic Pub-

lishers, Boston, 1996.

[GH05] Abbas El Gamal and Navid Hassanpour. Relay-without-delay. In IEEE

International Symposiom of Information Theory (ISIT), Adelaide, Australia,

September 2005.

[GJJV03] Andrea Goldsmith, Syed Ali Jafar, Nihar Jindal, and Sriram Vishwanath. Ca-

pacity limits of mimo channels. IEEE Journal on Selected Areas in Communi-

cations, 21(5):684–702, June 2003.

[GK00] Piyush Gupta and P. R. Kumar. The capacity of wireless networks. IEEE

Transactions on Information Theory, 46(2):388–404, March 2000.

[GK03] Piyush Gupta and P. R. Kumar. Towards an information theory of large net-

works: An achievable rate region. IEEE Transactions on Information Theory,

49(8):1877–1894, August 2003.

[GNT06] Leonidas Georgiadis, Michael J. Neely, and Leandros Tassiulas. Resource al-

location and cross-layer control in wireless networks. Foundations and Trends

in Networking, 1(1):1–144, 2006.

[GV97] Andrea J. Goldsmith and Pravin P. Varaiya. Capacity of fading channels

with channel side information. IEEE Transactions on Information Theory,

43(6):1986–1992, November 1997.

210

References

[GW75] N. Thomas Gaarder and Jack K. Wolf. The capacity region of a multiple-access

discrete memoryless channel can increase with feedback. IEEE Transactions

on Information Theory, 21(1):100–102, January 1975.

[GW02] Andrea J. Goldsmith and Stephen B. Wicker. Design challenges for energy-

constrained ad hoc wireless networks. IEEE Wireless Communications Maga-

zine, 9(4):8–27, August 2002.

[GZ05] Abbas El Gamal and Sina Zahedi. Capacity of a class of relay channels with or-

thogonal components. IEEE Transactions on Information Theory, 51(5):1815–

1817, May 2005.

[HD06] Christoph Hausl and Philippe Dupraz. Joint network-channel coding for the

multiple-access relay channel. In Proc. of International Workshop on Wireless

Ad-hoc and Sensor Networks, New York, USA, June 2006.

[HH06] Christoph Hausl and Joachim Hagenauer. Iterative Network and Channel De-

coding for the Two-Way Relay Channel. In Proc. IEEE Intern. Conf. on Com-

munications (ICC ’06), Istanbul, Turkey, 2006.

[HKE+07] Ingmar Hammerström, Marc Kuhn, Celal Esli, Jian Zhao, Armin Wittneben,

and Gerhard Bauch. Mimo two-way relaying with transmit csi at the relay. In

Proc. 8th IEEE Workshop on Signal Processing Advances in Wireless Commu-

nications (SPAWC), Helsinki, Finland, June 2007.

[HKW06] Ingmar Hammerström, Marc Kuhn, and Armin Wittneben. Distributed MIMO

for cellular networks with multihop transmission protocols. In Asilomar Con-

ference on Signals, Systems, and Computers 2006, oct 2006.

[HM05] Simon Haykin and Michael Moher. Modern Wireless Communications. Pren-

tice Hall PTR, Upper Saddle River, New Jersey, 2005.

[HMZ05] Anders Host-Madsen and Junshan Zhang. Capacity bounds and power allo-

cation for wireless relay channels. IEEE Transactions on Information Theory,

51(6):2020–2040, June 2005.

[HN04] Todd Hunter and Aria Nosratinia. Distributed protocols for user cooperation

in multi-user wireless networks. In Proc. IEEE GlobeCom, Dallas, Texas,

November 2004.

[HN06] Todd E. Hunter and Aria Nosratinia. Diversity through coded cooperation.

IEEE Transaction on Wireless Communications, 5(2):283–289, February 2006.

[HSOB05] Christoph Hausl, Frank Schreckenbach, Ioannis Oikonomidis, and Gerhard

Bauch. Iterative network and channel decoding on a tanner graph. In Proc.

43rd Allerton Conference, September 2005.

211

References

[HUL01] Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal. Fundamentals of Convex

Analysis. Springer-Verlag, New York, 2001.

[HZF04] Patrick Herold, Ernesto Zimmermann, and Gerhard Fettweis. On the perfor-

mance of cooperative amplify-and-forward relay networks. In Proc. 5th Int.

ITG Conf. on Source and Channel Coding, Erlangen, Germany, January 2004.

[JOB05] Eduard A. Jorswieck, Tobias J. Oechtering, and Holger Boche. Performance

Analysis of Combining Techniques with Correlated Diversity. In Proc. IEEE

Wireless Communications and Networking Conference (WCNC ’05), volume 2,

pages 849–854, New Orleans, USA, March 2005.

[Kat94] Randy H. Katz. Adaptation and mobility in wireless information systems.

IEEE Personal Communications, 1:6–17, 1994.

[KGG05] Gerhard Kramer, Michael Gastpar, and Piyush Gupta. Cooperative strategies

and capacity for relay networks. IEEE Transactions on Information Theory,

51(9):3037–3063, September 2005.

[KH95] Raymond Knopp and Pierre A. Humblet. Information capacity and power

control in single-cell multiuser communications. In Proc. IEEE International

Conference on Communications (ICC), pages 331–335, Seattle, WA, USA,

June 1995.

[KM95] P. R. Kumar and Sean P. Meyn. Stability of queueing networks and scheduling

policies. IEEE Transactions on Automatic Control, 40(2):251–260, February

1995.

[KM96] P. R. Kumar and Sean P. Meyn. Duality and linear programs for stability and

performance analysis of queueing networks and scheduling policies. IEEE

Transactions on Automatic Control, 41(1):4–17, January 1996.

[KM03] Ralf Koetter and Muriel Médard. An algebraic approach to network coding.

IEEE/ACM Transactions on Networking, 5:782–795, October 2003.

[KRH+06] Sachin Katti, Hariharan Rahul, Wenjun Hu, Dina Katabi, Muriel Médard, and

Jon Crowcroft. Xors in the air: Practical wireless network coding. In Proc.

ACM SIGCOMM 2006, Pisa, Italy, September 2006.

[KSA03] Mohammad Ali Khojastepour, Ashutosh Sabharwal, and Behnaam Aazhang.

On the capacity of ‘cheap´ relay networks. In Proceedings of the 37th Annual

Conference on Information Sciences and Systems (CISS), March 2003.

[KSH00] Thomas Kailath, Ali H. Sayed, and Babak Hassibi. Linear Estimataion. Pren-

tice Hall PTR, Upper Saddle River, New Jersey, 2000.

212

References

[LBC+05] Jianghong Luo, Rick S. Blum, Len J. Cimini, Larry J. Greenstein, and Alexan-

der M. Haimovich. Link-failure probabilities for practical cooperative relay

networks. In Proc. 61th-IEEE Conference on Vehicular Technology, volume 3,

pages 1489–1493, Stockholm, Sweden, 2005.

[LES06] Zinan Lin, Elza Erkip, and Andrej Stefanov. Cooperative regions and partner

choice in coded cooperative systems. IEEE Transactions on Communications,

54:1323–1334, July 2006.

[Lia72] Henry H. J. Liao. Multiple access channels. PhD thesis, Department of Elec-

trical Engineering, University of Hawaii, Honolulu, 1972.

[LJS05] Peter Larsson, Niklas Johansson, and Kai-Erik Sunell. Coded Bi-directional

Relaying. In Proc. 5th Scandinavian Workshop on Ad Hoc Networks (AD-

HOC’05), Stockholm, Sweden, May 2005.

[LLG06] Lifeng Lai, Ke Liu, and Hesham El Gamal. The three-node wireless network:

Achievable rates and cooperation strategies. IEEE Transactions on Informa-

tion Theory, 52(3):805–828, March 2006.

[LMNM01] Emilio Leonardi, Marco Mellia, Fabio Neri, and Marco Ajmone Marsan.

Bounds on average delays and queue size averages and variances in input-

queued cell-based switches. In IEEE INFOCOM 2001, pages 1095–1103, An-

chorage, Alaska, April 2001.

[LSS03] Ruoheng Liu, Predrag Spasojevi´

c, and Emina Soljanin. User cooperation with

punctured turbo codes. In Proc. Allerton Conference on Communication, Con-

trol, and Computing, October 2003.

[LTW01] J. Nicholas Laneman, David N. C. Tse, and Gregory W. Wornell. An efficient

protocol for realizing cooperative diversity in wireless networks. In Proc. IEEE

International Symposium on Information Theory (ISIT’01), June 2001.

[LTW04] J. Nicholas Laneman, David N. C. Tse, and Gregory W. Wornell. Cooperative

diversity in wireless networks: Efficient protocols and outage behavior. IEEE

Transactions on Information Theory, 50(12):3062–3080, Dec. 2004.

[LV05] Yingbin Liang and Venugopal V. Veeravalli. Gaussian orthogonal relay chan-

nel: optimal resource allocation and capacity. IEEE Transactions on Informa-

tion Theory, 51(9):3284–3289, September 2005.

[LW03] J. Nicholas Laneman and Gregory W. Wornell. Distributed space-time coded

protocols for exploiting cooperative diversity in wireless networks. IEEE

Transactions on Information Theory, 49(10):2415–2425, October 2003.

213

References

[LYC03] Shou-Yen Robert Li, Raymond W. Yeung, and Ning Cai. Linear network

coding. IEEE Transactions on Information Theory, 49(2):371–381, February

2003.

[LZKS06] Myung J. Lee, Jianliang Zheng, Young-Bae Ko, and Deepesh Man Shrestha.

Emerging standards for wireless mesh technology. IEEE Wireless Communi-

cations Magazine, 13(4):56–63, April 2006.

[MLMN00] Marco Ajmone Marsan, Emilio Leonardi, Marco Mellia, and Fabio Neri. On

the stability of input-buffer cell switches with speed-up. In IEEE INFOCOM

2000, pages 1604–1613, Tel Aviv, Israel, March 2000.

[MMAW99] Nick McKeown, Adisak Mekkittikul, Venkat Anantharam, and Jean Walrand.

Achieving 100% throughput in an input-queued switch. IEEE Transactions on

Communications, 47:1260–1267, August 1999.

[MMMZ06] Ritesh Madan, Neelesh B. Mehta, Andreas F. Molisch, and Jinyun Zhang.

Energy-Efficient Cooperative Relaying over Fading Channels with Simple Re-

lay Selection. In Proc. IEEE GlobeCom, San Francisco, California, November

2006.

[MMYZ06] Andreas F. Molisch, Neelesh B. Mehta, Jonathan S. Yedidia, and Jinyun

Zhang. Cooperative relay networks using fountain codes. In Proc. IEEE

GlobeCom, San Francisco, California, November 2006.

[MO79] Albert W. Marshall and Ingram Olkin. Inequalities: Theory of Majorization

and Its Applications. Academic Press, New York, 1979.

[Moo65] Gordon E. Moore. Cramming more components onto integrated circuits. Elec-

tronics Magazine, 38(8), April 1965.

[MT93] Sean P. Meyn and Richard L. Tweedie. Markov Chains and Stochastic Stabil-

ity. Springer-Verlag, London, 1993.

[NBK04] Rohit U. Nabar, Helmut Bölcskei, and Felix W. Kneubühler. Fading relay

channels: Performance limits and space-time signal design. IEEE Journal on

Selected Areas in Communications, 22(6):1099–1109, August 2004.

[Nee03] Michael J. Neely. Dynamic Power Allocation and Routing for Satellite and

Wireless Networks with Time Varying Channels. PhD thesis, Department of

Electrical Engineering and Computer Science, Massachusetts Institute of Tech-

nology, Cambridge, MA, USA, 2003.

[NHH04] Aria Nosratinia, Todd E. Hunter, and Ahmadreza Hedayat. Cooperative com-

munication in wireless networks. IEEE Communications Magazine, 42:68–73,

October 2004.

214

References

[NMR03] Michael J. Neely, Eytan Modiano, and Charles E. Rohrs. Power Allocation

and Routing in Multibeam Satellites with Time-Varying Channels. IEEE/ACM

Transactions on Networking, 11(1):138–152, February 2003.

[NMR05] Michael J. Neely, Eytan Modiano, and Charles E. Rohrs. Dynamic Power

Allocation and Routing for Time-Varying Wireless Networks. IEEE Journal

on Selected Areas in Communications, 23(1):89–103, January 2005.

[OB03a] Tobias J. Oechtering and Holger Boche. On the Capacity of a Distributed Mul-

tiantenna System using Cooperative Transmitters. In Proc. 57th IEEE Semian-

nual Vehicular Techn. Conf. (VTC ’03), volume 1, pages 75–79, Jeju, Korea,

April 2003.

[OB03b] Tobias J. Oechtering and Holger Boche. Optimality Range of Transmission

over Different Terminals in Cooperative Multiantenna Systems. In Proc. IEEE

Intern. Conf. on Acoustics Speech and Signal Processing (ICASSP ’03), vol-

ume 4, pages 365–368, Hong Kong, April 2003.

[OB05] Tobias J. Oechtering and Holger Boche. Capacity of a Gaussian FIR Linear

Relay Network. In Proc. WirelessCom 2005, Maui, Hawaii, USA, June 2005.

[OB06a] Tobias J. Oechtering and Holger Boche. Bidirectional Regenerative Half-

duplex Relaying using Relay Selection. IEEE Transactions on Wireless Com-

munications, 2006. accepted.

[OB06b] Tobias J. Oechtering and Holger Boche. Optimal Resource Allocation for

a Bidirectional Regenerative Half-duplex Relaying. In Proc. IEEE Interna-

tional Symposium on Information Theory and its Applications, pages 528 –

533, Seoul, Korea, 2006.

[OB06c] Tobias J. Oechtering and Holger Boche. Piggyback a Common Message on

Half-Duplex Bidirectional Relaying. IEEE Transactions on Wireless Commu-

nications, 2006. revised.

[OB06d] Tobias J. Oechtering and Holger Boche. Stability Region of an Efficient Bidi-

rectional Regenerative Half-duplex Relaying Protocol. In IEEE Information

Theory Workshop (ITW 06) , pages 380–384, Chengdu, China, October 2006.

[OB06e] Tobias J. Oechtering and Holger Boche. Stability Region of an Optimized

Bidirectional Regenerative Half-duplex Relaying Protocol. IEEE Transaction

on Communications, 2006. accepted.

[OB07a] Tobias J. Oechtering and Holger Boche. Bidirectional Relaying using Inter-

ference Cancellation. In Proc. ITG/IEEE International Workshop on Smart

Antennas (WSA ’07), Vienna, Austria, February 2007.

215

References

[OB07b] Tobias J. Oechtering and Holger Boche. Optimal Time-Division for Bidirec-

tional Relaying using Superposition Encoding. IEEE Communications Letter,

2007. submitted.

[OB07c] Tobias J. Oechtering and Holger Boche. Optimal Transmit Strategies in Multi-

Antenna Bidirectional Relaying. In Proc. IEEE Intern. Conf. on Acoustics,

Speech, and Signal Processing (ICASSP ’07), Honolulu, Hawaii, USA, April

2007.

[OB07d] Tobias J. Oechtering and Holger Boche. Piggyback a Common Message on

Bidirectional Relaying. In Proc. Workshop on Wireless Networks - Communi-

cation, Cooperation and Competition (WNC3-’07), 2007.

[OB07e] Tobias J. Oechtering and Holger Boche. Relay Selection in Bidirectional Relay

Communication. In Proc. 8th IEEE Workshop on Signal Processing Advances

in Wireless Communications (SPAWC), Helsinki, Finland, June 2007.

[OBSB07] Tobias J. Oechtering, Igor Bjelakovic, Clemens Schnurr, and Holger

Boche. Broadcast Capacity Region of Two-Phase Bidirectional Relaying.

IEEE Transactions on Information Theory, 2007. submitted, available at

http://arxiv.org/abs/cs.IT/0703078.

[OLT07] Ayfer Özgür, Olivier Lévêque, and David Tse. Hierarchical cooperation

achieves optimal capacity scaling in ad hoc networks. IEEE Transactions on

Information Theory, page to appear, December 2007.

[OS04] Tobias J. Oechtering and Aydin Sezgin. A new Cooperative Transmission

Scheme using the Space-Time Delay Code. In Proc. ITG Workshop on Smart

Antennas, Munich, Germany, March 2004.

[OSB06] Tobias J. Oechtering, Benjamin Schubert, and Holger Boche. FIR Linear Relay

Network with Frequency Selective Channels. In Proc. IEEE Intern. Conf. on

Communications (ICC ’06), Istanbul, Turkey, 2006.

[OSBB07] Tobias J. Oechtering, Clemens Schnurr, Igor Bjelakovic, and Holger Boche.

Achievable Rate Region of a Two Phase Bidirectional Relay Channel. In Pro-

ceedings of the 41st Annual Conference on Information Sciences and Systems,

March 2007.

[PP02] Athanasios Papoulis and S. Unnikrishna Pillai. Probability, Random Variables

and Stochastic Processes. McGraw-Hill Book Company, New York, 4. edition,

2002.

[PY06] Petar Popovski and Hiroyuki Yomo. The anti–packets can increase the achiev-

able throughput of a wireless multi–hop network. In Proc. IEEE International

Conference on Communications (ICC), Istanbul, Turkey, June 2006.

216

References

[Rap96] Theodore S. Rappaport. Wireless Communications. Prentice Hall, Englewood

Cliffs, New Jersey, 1996.

[RK06] Niranjan Ratnakar and Gerhard Kramer. The multicast capacity of determin-

istic relay networks with no interference. IEEE Transactions on Information

Theory, 52(6):2425–2432, June 2006.

[RU95] Bixio Rimoldi and Rüdiger Urbanke. A rate-splitting approach to the gaus-

sian multiple access channel. IEEE Transactions on Information Theory,

42(2):364–375, February 1995.

[Rud66] Walter Rudin. Real and Complex Analysis. McGraw Hill Book Company,

1966.

[RW05a] Boris Rankov and Armin Wittneben. Spectral efficient protocols for nonregen-

erative half-duplex relaying. In Proc. 43rd Allerton Conference, September

2005.

[RW05b] Boris Rankov and Armin Wittneben. Spectral efficient signaling for half-

duplex relay channels. In Proc. Asilomar Conference on Signals, Systems,

and Computers, November 2005.

[RW06] Boris Rankov and Armin Wittneben. Achievable rate regions for the two-way

relay channel. In Proc. IEEE Int. Symposium on Information Theory (ISIT), jul

2006.

[RW07] Boris Rankov and Armin Wittneben. Spectral efficient protocols for half-

duplex fading relay channels. IEEE Journal on Selected Areas in Commu-

nications, feb 2007.

[SE04] Andrej Stefanov and Elza Erkip. Cooperative coding for wireless networks.

IEEE Transactions on Communications, 52:1470–1476, September 2004.

[SE05] Andrej Stefanov and Elza Erkip. Cooperative space–time coding for wireless

networks. IEEE Transactions on Communications, 53:1804–1809, November

2005.

[SEA98] Andrew Sendonaris, Elza Erkip, and Behnaam Aazhang. Increasing uplink

capacity via user cooperation diversity. In Proc. IEEE International Sympo-

sium on Information Theory (ISIT’98), page 156, Cambridge, MA, USA, Aug.

1998.

[SEA03a] Andrew Sendonaris, Elza Erkip, and Behnaam Aazhang. User cooperation

diversity - Part I: System description. IEEE Transactions on Communications,

51:1927–1938, November 2003.

217

References

[SEA03b] Andrew Sendonaris, Elza Erkip, and Behnaam Aazhang. User cooperation

diversity - Part II: Implementation aspects and performance analysis. IEEE

Transactions on Communications, 51:1939–1948, November 2003.

[SG00] Brett Schein and Robert G. Gallager. The gaussian parallel relay network.

In Proc. IEEE International Symposium on Information Theory (ISIT’00),

page 22, Sorrento, Italy, June 2000.

[Sha48] Claude E. Shannon. A mathematical theory of communication. Bell System

Technical Journal, 27:379–423 and 623–656, Jul and Oct 1948.

[Sha61] Claude E. Shannon. Two-way communication channel. In Proc. 4th Berkeley

Symp. Math Stat. and Prob., volume 1, pages 611–644, 1961.

[SHMX06] Vladimir Stankovic, Anders Host-Madsen, and Zixiang Xiong. Cooperative di-

versity for wireless ad hoc networks. IEEE Communications Magazine, 23:37–

49, September 2006.

[SO04a] Aydin Sezgin and Tobias J. Oechtering. A new resource efficient transmission

scheme for Cooperative Systems. In Proc. IEEE Intern. Workshop on Sig-

nal Processing Advances for Wireless Communications (SPAWC ’04), Lisboa,

Portugal, July 2004.

[SO04b] Aydin Sezgin and Tobias J. Oechtering. Antenna Selection with Capacity-

Approaching Space-Time Block Codes. In Proc. IEEE Asilomar Conf. on Sig-

nals, Systems, and Computers, Pacific Grove, CA, USA, November 2004.

[SO04c] Aydin Sezgin and Tobias J. Oechtering. Complete Characterization of the

Equivalent MIMO Channel for Quasi-Orthogonal Space-Time Codes. IEEE

Transactions on Information Theory, 2004, December 2004. submitted.

[SO04d] Aydin Sezgin and Tobias J. Oechtering. On the Outage Probability of Quasi-

Orthogonal Space-Time Codes. In Proc. IEEE Information Theory Workshop

(ITW ’04), San Antonio, Texas, October 2004.

[SOS07] Clemens Schnurr, Tobias J. Oechtering, and Slawomir Sta´

nczak. On coding

for the broadcast phase in the two-way relay channel. In Proceedings of the

41st Annual Conference on Information Sciences and Systems, 2007.

[SRK03] Sanjay Shakkottai, Theodore S. Rappaport, and Peter C. Karlsson. Cross-layer

design for wireless networks. IEEE Communications Magazine, 41(10):74–80,

October 2003.

[SWB06] Slawomir Sta´

nczak, Marcin Wiczanowski, and Holger Boche. Resource Allo-

cation in Wireless Networks - Theory and Algorithms. Lecture Notes in Com-

puter Science (LNCS 4000). Springer-Verlag, 2006.

218

References

[TCHC06] Taiwen Tang, Chan-Byoung Chae, Robert W. Health, and Sunghyun Cho. On

achievable sum rates of a multiuser mimo relay channel. In Proc. IEEE Int.

Symposium on Information Theory (ISIT), jul 2006.

[TE92] Leandros Tassiulas and Anthony Ephremides. Stability properties of con-

strained queueing systems and scheduling systems and scheduling policies for

maximum throughput in multihop radio networks. IEEE Transactions on Au-

tomatic Control, 37(12):1936–1948, 1992.

[TE93] Leandros Tassiulas and Anthony Ephremides. Dynamic server allocation to

parallel queues with randomly varying connectivity. IEEE Transactions on

Information Theory, 39(2):466–478, 1993.

[TE02] Anthony Ephremides Tamer ElBatt. Joint scheduling and power control for

wireless ad-hoc networks. In Proc. 21st IEEE Infocom, New York, NY, USA,

June 2002.

[Tel99] Imre E. Telatar. Capacity of multi-antenna gaussian channels. European Trans-

actions on Telecommunications, 10(6):585–595, Nov./Dec 1999.

[TG95] Imre E. Telatar and Robert G. Gallager. Combining Queueing Theory with In-

formation Theory with Mulitaccess. IEEE Journal on Selected Areas in Com-

munications, 13(6):963–969, August 1995.

[TH98] David N. C. Tse and Stephen V. Hanly. Multiaccess Fading Channels–Part I:

Polymatroid Structure, Optimal Resource Allocation and Throughput Capac-

ities. IEEE Transactions on Information Theory, 44:2796–2815, November

1998.

[Tsy65] Boris S. Tsybakov. The capacity of a memoryless gaussian vector channel.

Problems of Information Transmission, 1(1):18–29, 1965.

[Tun06] Ertem Tuncel. Slepian–wolf coding over broadcast channels. IEEE Trans. Inf.

Theory, 52(4):1469–1482, April 2006.

[TV05] David N. C. Tse and Pramod Viswanath. Fundamentals of Wireless Commu-

nication. Cambridge University Press, June 2005.

[vdM71] Edward C. van der Meulen. Three-terminal communication channels. Adv.

Appl. Prob., 3:120–154, Sept. 1971.

[VJG03] Sriram Vishwanath, Nihar Jindal, and Andrea Goldsmith. Duality, achiev-

able rates, and sum-rate capacity of gaussian mimo broadcast channels. IEEE

Transactions on Information Theory, 49(10):2658–2668, October 2003.

219

References

[WB04] Marcin Wiczanowski and Holger Boche. Optimal Transmit Covariance Ma-

trices for MIMO High Speed Uplink Packet Access. In Proc. IEEE Wireless

Communications and Networking Conference (WCNC 2004), volume 2, pages

771–776, Atlanta, USA, March 2004.

[WCK05] Yunnan Wu, Philip A. Chou, and S. Y. Kung. Information exchange in wire-

less networks with network coding and physical-layer broadcast. In Proc.

39th Conference on Information Sciences and Systems, Baltimore, MD, USA,

March 2005.

[Wil82] Frans M. J. Willems. Informationtheoretical results for the discrete mem-

oryless multiple access channel. Doctor in Wetenschappen Proefschrift,

Katholieke Univeriteit Leuven, Belgium, 1982.

[Wol78] Jacob Wolfowitz. Coding Theorems of Information Theory. Springer-Verlag,

Berlin, 3. edition, 1978.

[WR03] Armin Wittneben and Boris Rankov. Impact of cooperative relays on the ca-

pacity of rank-deficient mimo channels. In in Proc. 12th IST Summit on Mobile

Wireless Communications, Aveiro, Portogal, June 2003.

[Wyn74] Aaron D. Wyner. Recent Results in the Shannon Theory. IEEE Transactions

on Information Theory, 20(1):2–10, January 1974.

[WZHM05] Bo Wang, Junshan Zhang, and Anders Host-Madsen. On the capacity of mimo

relay channels. IEEE Transactions on Information Theory, 51(1):29–43, Jan-

uary 2005.

[XJB04] Lin Xiao, Mikael Johansson, and Stephen Boyd. Simultaneous routing and

resource allocation via dual decomposition. IEEE Transactions on Communi-

cations, 52:1136–1144, July 2004.

[XK04] Liang-Liang Xie and P. R. Kumar. A network information theory for wireless

communication: Scaling law and optimal operation. IEEE Transactions on

Information Theory, 50(5):748–767, May 2004.

[XK07] Liang-Liang Xie and P. R. Kumar. Multi-source, multi-destination, multi-relay

wireless networks. IEEE Transactions on Information Theory, page to appear,

October 2007.

[YB05] Edmund M. Yeh and Randall Berry. Throughput Optimal Control of Coopera-

tive Relay Networks. In IEEE International Symposium of Information Theory

(ISIT), Adelaide, Australia, September 2005.

220

References

[YE06] Melda Yuksel and Elza Erkip. Diversity-multiplexing tradeoff in cooperative

wireless systems. In Proceedings of the 40th Annual Conference on Informa-

tion Sciences and Systems (CISS), Princeton, March 2006.

[Yeu95] Raymond W. Yeung. Multilevel diversity coding with distortion. IEEE Trans-

actions on Information Theory, 41(3):412–422, March 1995.

[Yeu02] Raymond W. Yeung. A First Course in Information Theory. Springer-Verlag,

New York, 2002.

[Yeu05] Raymond W. Yeung. Network Coding Theory Part I/II: Single/Multiple

Source. Foundations and Trends in Communications and Information Theory,

2(4–5):241–381, 2005.

[YRBC01] Wei Yu, Wonjong Rhee, Stephen Boyd, and John M. Cioffi. Iterative water-

filling for gaussian vector multiple access channels. In IEEE International

Symposium on Information Theory (ISIT’01), Washington, DC, USA, June

2001.

[ZAL06] Yi Zhao, Raviraj Adve, and Teng Joon Lim. Improving amplify-and-forward

relay networks: Optimal power allocation versus selection. In Proc. IEEE

International Symposium on Information Theory (ISIT’06), pages 1234–1238,

Seattle, WA, USA, July 2006.

[Zar51] Kazimierz Zarankiewicz. Problem p 101. Colloquium Mathematicum, (2):301,

1951.

[Zha88] Zhen Zhang. Partial converse for a relay channel. IEEE Transactions on Infor-

mation Theory, 34(5):1106–1110, September 1988.

[Zim80] Hubert Zimmermann. OSI Reference Model – The IS0 Model of Architecture

for Open Systems Interconnection. IEEE Transactions on Communications,

28:425–432, April 1980.

[ZT03] Lizhong Zheng and David N. C. Tse. Diversity and multiplexing: A fundamen-

tal tradeoff in multiple-antenna channels. IEEE Transactions on Information

Theory, 49(5):1073–1096, May 2003.

221