Document [original]

Power-Aware Spatial Multiplexing

with Unilateral Antenna

Cooperation

vorgelegt von

Diplom-Ingenieur

Martin Schubert

aus Haltern in Westfalen

Von der Fakultät IV - Elektrotechnik und Informatik

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurswissenschaften

- Dr.-Ing. -

genehmigte Dissertation

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

2. Berichter: Prof. Dr. techn. Helmut Bölcskei (ETH Zürich)

Tag der wissenschaftlichen Aussprache: 11. Dezember 2002

Berlin 2003

D 83

Zusammenfassung

In der Arbeit wird eine leistungsoptimierte Lösung für das Problem des räumlichen

Multiplexings von unabhängigen Datenströmen in einem Mehrantennen-System vor-

gestellt, unter der Annahme, daß entweder die Sendeantennen oder die Empfangsan-

tennen nicht kooperieren und die Gesamtleistung beschränkt ist.

Kooperation wird durch lineare räumliche Filterung (Beamforming) erzielt, die

gemeinsam mit den Sendeleistungen optimiert wird. Grundlage der Optimierung ist

Kanalkenntnis an der kooperierenden Seite, welche als bekannt vorausgesetzt wird.

Das Optimierungsziel ist die Gewährleistung individueller Signal-zu-Interferenz-plus-

Rausch-Verhältnisse (signal-to-interference-plus-noise ratio, SINR) bei gleichzeitiger

Minimierung der Gesamtleistung. Dieses Problem tritt z.B. in einem zellularen Mo-

bilfunksystem auf, wo eine Basisstation über ein Antennenarray gleichzeitig mit meh-

reren unabhängigen Mobilstationen kommuniziert, welche jeweils mit einer Einzelan-

tenne ausgerüstet sind. Die Signalübertragung erfolgt im Downlink über kooperie-

rende Sendeantennen und im Uplink über kooperierende Empfangsantennen.

Ein erster Schritt zur Lösung dieses Optimierungsproblems ist die Annahme un-

veränderlicher Beamformer. Dies ermöglicht eine Charakterisierung des globalen Op-

timums. Es zeigt sich, daß eine fundamentale Dualität zwischen räumlichem Mul-

tiplexing in Abwärts- und Aufwärtsverbindung besteht. In beiden Kanälen können

die gleichen SINR-Werte erreicht werden, wobei jeweils die gleiche Beschränkung der

Gesamtleistung zugrundegelegt wird. Eine Lösung für das Downlink-Problem, wel-

ches eine komplizierte analytische Struktur aufweist, kann demnach durch Lösen des

„glatteren“ Uplink-Problems gefunden werden.

Ein zweiter wichtiger Schritt zur Lösung der allgemeinen Aufgabenstellung ist die

Untersuchung von Mehrnutzer-Beamforming unter Vernachlässigung des Rauschens,

wie von Gerlach and Paulraj [10] vorgeschlagen. Dies führt zur Minimierung eines

nicht-differenzierbaren `∞Funktionals. Ein wichtiger Schritt zur Lösung dieser Auf-

gabenstellung ist die Formulierung eines äquivalenten Eigenwert-Optimierungsprob-

lems und der Beweis, daß jedes lokale Optimum auch ein globales Optimum ist. Wei-

terhin wird ein interessanter Zusammenhang zwischen dem optimalen `∞Problem

und einem in [10] vorgeschlagenen `1-Ansatz aufgezeigt. Die analytischen Ergebnis-

se dieser Untersuchungen führen schließlich zu einer iterativen Lösung mit streng

monotonem Konvergenzverhalten. Das globale Optimum wird bereits nach einigen

wenigen Iterationsschritten erreicht.

Aufbauend auf den Erkenntnissen aus der Betrachtung des rauschfreien Falls,

wird eine Lösung des allgemeinen Problems hergeleitet. Der neue Algorithmus hat

dieselben exzellenten Konvergenzeigenschaften wie die oben beschriebene Lösung und

minimiert zudem die Gesamtleistung unter Berücksichtigung des Empfangsrauschens.

Mittels individueller Zielvorgaben lassen sich so beliebige SINR-Werte erreichen, so-

lange das Problem lösbar ist. Notwendige und hinreichende Bedingungen für die

Lösbarkeit werden angegeben. Eine geschlossene Lösung für den 2-Nutzer-Fall wird

hergeleitet.

Schließlich wird gezeigt, wie die gefundene Lösung auf das Problem der Raten-

Kontrolle im Gauß’schen Broadcast-Kanal mit vektorwertigem Eingang und skalarem

Ausgang angewendet werden kann. Mit dem obigen Dualitätsresultat folgt, daß es

eine Übereinstimmung erzielbarer Kapazitäten in Broadcast und Multiple-Access-

Kanal gibt, solange eine Kombination aus linearer Filterung und Decision-Feedback-

Entzerrung bzw. -Vorverzerrung betrachtet wird. Dies zeigt, daß die bekannte Ka-

pazitätsregion des Multiple-Access-Kanals eine direkte Entsprechung im Broadcast

Channel hat, einschließlich der maximalen Summenkapazität. Die Dreiecks-Struktur

des Kanals, hervorgerufen durch die Decision-Feedback-Strategie, ermöglicht eine auf

Rücksubstitution basierende Lösung, welche eine effiziente Steuerung individueller

Datenraten erlaubt. Auch für diese Aufgabenstellung wird eine geschlossene Lösung

für den 2-Nutzer-Fall hergeleitet.

III

Abstract

This thesis offers a power aware solution to the problem of spatial multiplexing

of independent data streams in a multi-antenna system, assuming that either the

transmit antennas or the receive antennas are not allowed to cooperate and a sum

power constraint is imposed.

Cooperation is achieved by using spatial linear filters (beamformers), which are

optimized jointly with the power allocation. To this end, the spatial channel char-

acteristics must be known at the cooperating side. The design goal is to achieve

individual signal-to-interference-plus-noise ratios (SINR) with minimal power con-

sumption. This problem occurs, e.g., in a cellular wireless communication system,

where a multi-antenna base station communicates simultaneously with several de-

centralized mobile terminals, each equipped with a single antenna. Downlink and

uplink correspond to cooperating transmit and receive antennas, respectively.

As a first approach to this optimization problem, the beamformers are assumed to

be fixed. This allows for a characterization of the global optimum, which will prove

useful for the following analysis. It turns out that there is a fundamental duality

between uplink and downlink spatial multiplexing. In particular, the same SINR

values can be achieved in both channels under the same power constraint. Hence, a

solution of the downlink problem, which has a complicated analytical structure, can

be found by solving the “smoother” uplink problem instead.

A second important step towards a general solution is the investigation of joint

beamforming in the absence of noise, as proposed by Gerlach and Paulraj [10]. This

problem consists of minimizing a non-differentiable `∞objective function. A key

step in finding a global solution is to formulate an equivalent eigenvalue optimization

problem and to show that each local minimum is a global minimum. Also, there is an

interesting relation between optimal `∞optimization and the alternative `1objective

proposed by [10]. The analytical results of this study lead to the development of

globally convergent algorithms with strictly monotone iteration sequences. Typically,

only a few iteration steps are required.

The insight gained from studying interference balancing in the absence of noise

is used to derive a power aware strategy for SINR balancing with minimal power

consumption. This new algorithm has the same excellent convergence behavior as

the one derived for the noiseless case. By choosing individual target thresholds,

arbitrary SINR levels can be achieved as long as the problem is feasible. Necessary

and sufficient conditions for feasibility are provided and a closed form solution is

derived for the 2-user case.

Finally, it is shown how the solution can be applied to the problem of control-

ling rate tuples in a Gaussian broadcast channel with vector inputs and independent

scalar outputs. By exploiting the above duality result, it is shown that there is a

one-to-one correspondence between capacities achievable in broadcast and multiple

access channel, as long as a combination of linear filtering and decision feedback

equalization is considered. Hence, the well known multiple access capacity region

has a direct counterpart in the broadcast channel, including the point of maximum

throughput. Exploiting the triangular channel structure imposed by decision feed-

back equalization, a rate balancing technique based on back-substitution is derived.

This approach allows to control individual rate tuples with minimal power consump-

tion and high computational efficiency. A closed form solution is given for the 2-user

case.

Contents

1. Introduction 1

1.1. ProblemStatement............................ 1

1.2. RelatedWorks............................... 2

1.3. Outline................................... 4

2. Duality Between Downlink and Uplink Multiplexing 6

2.1. Narrowband Signal Model . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1. Propagation Channel . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2. Cooperating Transmit Antennas (Downlink) . . . . . . . . . . 8

2.1.3. Cooperating Receive Antennas (Uplink) . . . . . . . . . . . . 9

2.1.4. Crosstalk ............................. 10

2.2. Uplink Power Allocation under a Sum Power Constraint . . . . . . . 12

2.2.1. Maximizing the Balanced SINR Margin . . . . . . . . . . . . . 13

2.2.2. Minimizing the Sum Power . . . . . . . . . . . . . . . . . . . . 14

2.3. Uplink Feasibility in the Absence of Power Constraints . . . . . . . . 16

2.3.1. Feasibility of Coupled Users . . . . . . . . . . . . . . . . . . . 16

2.3.2. General Feasibility . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4. The Dual Downlink Problem . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.1. Downlink Power Allocation . . . . . . . . . . . . . . . . . . . 18

2.4.2. Feasibility Conditions for the Downlink . . . . . . . . . . . . . 19

2.4.3. WeakDuality........................... 20

2.4.4. StrongDuality .......................... 21

2.5. Discussion................................. 23

3. Joint Beamforming in the Absence of Noise 24

3.1. Optimal Interference Balancing based on `∞-Norm Minimization . . . 24

3.1.1. Eigenvalue Minimization . . . . . . . . . . . . . . . . . . . . . 25

3.1.2. Characterization of the Global Minimizer . . . . . . . . . . . . 26

3.1.3. Subspace Characterization of the Optimal Beamforming Solu-

tions................................ 28

3.2. A Globally Convergent Algorithm . . . . . . . . . . . . . . . . . . . . 29

3.2.1. Monotony and Global Convergence . . . . . . . . . . . . . . . 29

Contents

3.2.2. Stopping Criterion . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3. Alternative Strategy based on `1-Norm Minimization . . . . . . . . . 32

3.3.1. Characterization of the `1Minimizer .............. 33

3.3.2. Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . 34

3.3.3. Closed-Form Solution for the 2-User Scenario . . . . . . . . . 36

3.3.4. Numerical Counterexample for K > 2.............. 38

3.3.5. Optimal Algorithm Based on `1-Optimization . . . . . . . . . 39

3.4. Discussion................................. 41

4. Power-Aware Spatial Multiplexing under a Sum Power Constraint 43

4.1. Joint Beamforming and Power Allocation . . . . . . . . . . . . . . . . 43

4.1.1. SINR Balancing in the Downlink . . . . . . . . . . . . . . . . 43

4.1.2. The Dual Uplink Problem . . . . . . . . . . . . . . . . . . . . 44

4.1.3. Problem Reformulation . . . . . . . . . . . . . . . . . . . . . . 45

4.1.4. Necessary and Sufficient Condition for Global Optimality . . . 47

4.2. Global Optimization with Alternating Variables . . . . . . . . . . . . 48

4.2.1. Algorithm Summary . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.2. Monotony and Global Convergence . . . . . . . . . . . . . . . 48

4.2.3. Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3. Minimizing Excess Transmission Power . . . . . . . . . . . . . . . . . 51

4.3.1. Iterative Power Minimization . . . . . . . . . . . . . . . . . . 51

4.3.2. Closed-Form Solution for the 2-User Scenario . . . . . . . . . 53

4.4. Discussion................................. 54

5. Rate Balancing for Gaussian Broadcast and Multiple Access Channels 55

5.1. Duality between Broadcast and Multiple Access Channel . . . . . . . 55

5.1.1. Multiple Access Channel with Successive Decoding . . . . . . 55

5.1.2. Broadcast Channel with ‘Dirty Paper’ Precoding . . . . . . . 57

5.1.3. Duality .............................. 58

5.2. Rate Balancing under a Sum Power Constraint . . . . . . . . . . . . . 58

5.2.1. Joint Beamforming and Power Control via Back-Substitution . 58

5.2.2. Closed Form Solution for the 2-User Case . . . . . . . . . . . 60

5.2.3. Maximizing the Sum Rate . . . . . . . . . . . . . . . . . . . . 62

5.3. Discussion................................. 64

6. Conclusions 66

6.1. Summary and Discussion of the Main Results . . . . . . . . . . . . . 66

6.2. FutureWork................................ 68

Proofs 70

Notation and Symbols 81

VII

Contents

Publication List 83

References 84

VIII

1. Introduction

The spatial structure of the wireless propagation channel can be exploited by using

multiple transmit and receive antennas. By adding the spatial dimension to the classi-

cal system resources bandwidth and power, significant gains in spectral efficiency can

be achieved. This has been demonstrated by evaluating the information-theoretical

capacities [11, 12]. The potential gains depend on the spatio-temporal characteristics

of the propagation channel, the degree of cooperation among the antennas, as well as

the channel information available at transmitter and/or receiver. For an overview,

see e.g. [13].

The general problem of spatial multiplexing consists in finding a transmit covari-

ance matrix and a detector, which are jointly optimal with respect to a chosen design

goal. If both transmit and receive antennas cooperate, then the throughput maxi-

mizing strategy is known to be the water-filling solution based on the singular-value

decomposition of the channel matrix [11, 14].

In many wireless systems, however, maximizing the throughput is not the primary

design goal. Additional system constraints should be considered, like bandwidth-

limited channel feedback or the limited ability to resolve fast fading. So it may

be necessary to trade off spectral efficiency against diversity by sending multiple

replicas of the same signal in order to make reception more reliable. For multi-

antenna systems, this leads to the concept of space-time coding [15–17].

Another important constraint is the degree of antenna cooperation. Cooperation

may be impractical, e.g., for antennas belonging to different decentralized users. In

this case the waterfilling power allocation is generally not optimal. A new strategy is

required, which jointly optimizes the power allocation and the spatial transmission

strategy. In general, this leads to the relatively new field of multiuser MIMO (multiple

input, multiple output) transmission, for which only partial results have appeared so

far. The challenging goal is to adaptively allocate individual data rates in a multiuser

environment. These rates should be achieved with minimal transmission power.

1.1. Problem Statement

This thesis deals with a special case of multi-antenna transmission. Unilateral an-

tenna cooperation is considered, which means that cooperation is allowed at either

1. Introduction

transmitter or receiver. In the following, these scenarios are referred to as downlink

and uplink, respectively. This is in analogy to a cellular wireless communication sys-

tem with a multi-antenna base station and several single-antenna mobile terminals.

Due to the different geographical positions of the mobiles, mutual co-operation is

often impractical. Note, that studying this special scenario is a first step towards the

general case, where also the mobiles are allowed to employ multiple antennas.

The spatial structure of the multiuser channel has to be known at the base station

in order to facilitate meaningful antenna cooperation. The benefit of cooperation is

to enable spatially directed transmission between the base station and the Kdecen-

tralized mobile terminals. Thereby, mutual interference is reduced and the overall

system performance is enhanced. Such spatial multiplexing provides additional de-

grees of freedom through which communication can take place.

Spatial multiplexing always goes hand in hand with power control, which has to

ensure reliable communication by counterbalancing the joint effects of signal prop-

agation and residual interference. Hence, a reasonable design goal is to achieve a

certain link quality for each channel with minimal transmission power. If this prob-

lem is infeasible, the requirements must be relaxed by the resource management,

e.g., by switching off one or more channels. Such a power aware transmission scheme

exploits the available system resources in an optimal way.

A reasonable and often used model is the transmission of Gaussian distributed

signals. Then, the performance measure for each of the Kchannels is the signal-to-

interference ratio (SINR). Considering individual target thresholds γ1,...,γK>0,

the design goal can be formulated as

SINRi≥γi,1≤i≤K . (1.1)

It is desirable to achieve these targets with minimum total transmission power. This

is a joint optimization problem, since all users are typically coupled by the corporate

effects of channel propagation, power adjustment, and spatial filtering. One difficulty

in understanding such systems stems from the intertwining of the effects of all of the

interferers in the system.

1.2. Related Works

•In the context of a synchronous CDMA system with random spreading se-

quences, a similar multi-user problem was studied by Tse and Hanly [18]. The

temporally spread system considered in this work is the direct equivalent to

the spatially spread system considered here. Tse and Hanly carried out an

asymptotical analysis, which shows the limiting effect of mutual interference

for several linear receiver types. They provided an upper bound on the achiev-

able system performance depending on the number of antennas Mand the

1. Introduction

number of users K. However, this provides no complete answer to the ques-

tion how SINR requirements γ1...γKcan be controlled under a sum power

constraint.

•In the context of classical beamforming (for an overview see e.g., [19–23] and

the references therein), the problem of jointly separating several signal sources

in space was addressed. For this purpose, a fixed power allocation is mostly

assumed. The interaction of several users in a power controlled network with

limited transmission power, however, is less well understood.

•Power control strategies for achieving the design goal (1.1) were already stud-

ied [24–28] (assuming fixed beamforming, i.e., each transmitter-receiver pair has

a fixed link gain). An overview is given, e.g. in [29]. This is a special case of

the above problem formulation. The general case that is studied here, how-

ever, is more complicated in that the optimal transmission power can only be

found when knowing the optimal spatial pre-processor. The pre-processor, in

turn, depends on the co-channel interference and therefore on the transmission

powers. Thus, both quantities must be optimized jointly.

•In the context of the Gaussian Broadcast Channel (GBC), a similar prob-

lem was recently studied in [30–34], where strategies for obtaining the sum

capacity and an achievable region were investigated. Most of these results are

based on an information theoretical duality between the BC and the Gaussian

Multiple Access Channel (GMAC), which allows to transfer the well known

principle of decision feedback multiuser equalization (see e.g. [35]) to the GBC.

Yet, an efficient and general strategy for achieving arbitrary target rates within

the resulting achievable region is not known.

•Another line of work is the development of downlink beamforming tech-

niques [10, 36–43], where a joint optimization of beamforming and power allo-

cation is studied. Three different approaches are known:

–In [10, 43] the problem of interference balancing was studied in the absence

of noise. This provides an upper bound on the jointly achievable SINR

margin. This approach will be discussed in detail in Chapter 3.

–Another approach is to minimize the total transmission power while ful-

filling SINR requirements (1.1). This was first proposed in [36] and later

in [37–40]. The disadvantage of this strategy is that it neglects the impor-

tant aspect of feasibility. Algorithms designed under the assumption that

a solution exists may yield unpredictable results in case that the problem

is infeasible.

–A conceptually different approach to the above power minimization prob-

lem appeared in [41, 42], where it was proposed to embed the beamforming

1. Introduction

problem in a semidefinite optimization program. Thus, recently developed

semidefinite programming techniques can be applied to solve the problem

and infeasible scenarios can be handled. However, the optimization is

performed over matrices with more degrees of freedom than the original

beamforming vectors. Hence, the solution comes at the cost of relatively

high computational complexity.

1.3. Outline

Chapter 2. In order to keep this thesis self-contained, some basic concepts of nar-

rowband space processing are reviewed and discussed in Section 2.1.

In Section 2.2, the two seemingly different beamforming concepts of power

minimization [36–42] and interference balancing [10, 28, 43] are analyzed and

compared. It turns out that both techniques are equivalent, as long as the

problem is feasible, i.e., certain SINR thresholds γ1...γKcan be achieved. The

problem of feasibility is studied in Section 2.3, where necessary and sufficient

conditions are given.

An interesting duality between uplink and downlink channels was observed

throughout the last decade in seemingly different contexts, e.g. [27, 33, 39]. In

Section 2.4 it is tried to generalize these results and to provide necessary and

sufficient conditions under which duality holds true.

Chapter 3. In this chapter, the problem of interference balancing by jointly adjust-

ing beamformers and transmission powers is studied in the absence of noise.

As a starting point serves the work of Gerlach and Paulraj [10] who have an-

alyzed this problem and proposed to replace the highly non-smooth `∞-norm

objective function of the original problem by a smoother `1-norm objective.

This leads to an algorithm which was conjectured to solve the original `∞

problem [10]. Another algorithm based on eigenvalue optimization was pro-

posed by Montalbano and Slock [43]. The global convergence of neither of

these algorithms could be proven so far.

In this chapter, both problems are analyzed and a convergence analysis is

carried out. As will turn out, the eigenvalue strategy is globally convergent,

whereas the `1-based strategy only approximates the optimum. These results

provide useful insight into the analytical structure of the general problem ad-

dressed later in Chapter 4, where SINR levels are balanced under the assump-

tion of noise.

Chapter 4. In this chapter the general problem of controlling SINR levels under a

sum power constraint is addressed. Exploiting the duality result, a solution for

1. Introduction

the complicated downlink problem is found by solving the “smoother” uplink

problem instead. This yields a global optimizer for both uplink and downlink.

A key problem that has to be solved in this context, is to expand the interference

balancing technique derived in Chapter 3, such that receiver noise is taken into

account.

Chapter 5. In this chapter, a multiuser Gaussian channel with coding for known

interference is considered. For this special scenario, controlling user capacities

is equivalent to controlling SINR levels. Hence, the solution found in Chapter 4

immediately carries over to the rate balancing problem. The solution can even

be simplified by exploiting the triangular structure of the channel that is im-

posed by the successive coding strategy. It turns out, that for this special case

no iterative optimization is required. The problem can be solved efficiently by

an algorithm based on back-substitution.

Chapter 6. An overall assessment is given and open problems for future research

are pointed out.

A summary of the notation and symbols used in this work is given in the appendix

on page 81.

2. Duality Between Downlink and

Uplink Multiplexing

In this chapter we focus on finding the optimal power allocation that ensures SINR

target thresholds γ1...γKunder a sum power constraint. To this end, it is assumed

that all beamformers are kept fixed, which has the advantage that the power alloca-

tion can be studied isolated from the effect of spatial processing. Hence the results

can also be seen in the light of classical power control and resource management (see

[29] and the references therein). The insight gained from this approach will prove

useful for the following chapters, where the mutual interaction between beamforming

and power allocation will be studied.

2.1. Narrowband Signal Model

We start with a brief introduction in the signal model and some basic concepts of

spatial signal processing.

2.1.1. Propagation Channel

The wireless propagation channel between transmitter and receiver is characterized

by multipath propagation, which spreads the transmitted signal in three dimensions:

time, frequency and space. In the following, some fundamental properties of wireless

signal propagation are discussed and the vector channel model used in this thesis is

introduced. Note, that this is a special case of the general MIMO channel (see e.g.

[44] for an overview).

Time dispersion is caused by reflections around transmitter and/or receiver, which

cause many delayed replicas of the signal. The channel bandwidth can be

defined as the reciprocal maximum excess delay. If the channel bandwidth is

smaller than the signal bandwidth, then the signal is distorted and the channel

is called “frequency selective”. If the signal is narrowband with respect to the

channel bandwidth, the channel is called “frequency flat”. This means that the

channel impulse response has a multipath delay spread which is much smaller

than the reciprocal bandwidth of the transmitted signal.

2. Duality Between Downlink and Uplink Multiplexing

In the following we assume a frequency flat channel. However, most concepts

proposed here immediately carry over to frequency selective channels, e.g. when

assuming independent spatial processing for each channel tap. In principle,

each frequency selective channel can be turned into a flat channel by temporal

equalization.

Frequency dispersion is caused when scattered signals experience a Doppler shift

due to motion. The Doppler increases with the velocity of the transmitter

and/or the receiver. Doppler shifts can even occur when both transmitter

and receiver are static, caused by moving objects in the surrounding. The

superposition of the scattered signals fades in time, due to different angular

frequencies caused by the Doppler shifts.

Space dispersion is caused by multipath reflections with different propagation an-

gles as seen from the array (cf. Fig 2.1). This causes coherence loss in the

spatial domain, depending on the angle spread as well as the distance between

the antenna elements.

In the following it is assumed that the signal is narrowband with respect to the

array aperture, i.e., the propagation time of the signal across the array is much

less than the reciprocal bandwidth of the transmitted signal. No restriction is

made, however, concerning the angle spread or the geometry of the array.

In this thesis, the channel between the ith mobile and the base station antenna array

is modeled as the superposition of Qscattered multipath replicas (see Fig. 2.1). The

multipath channel is characterized by Doppler shifts ˆωiq,1≤q≤Q, complex path

attenuations βiq, and propagation angles θiq. The path attenuation βiq models the

joint effects of path attenuation and phase shifts. The angle θiq is typically measured

with respect to array broadside. From the narrowband assumption follows that the

time delays at the Mbase station antennas can be modeled as phase shifts collected

in an array response vector a(θ)∈CM. The phase shifts are measured with respect

to an arbitrary reference point, which is commonly chosen to be the first antenna

element. Assuming a carrier frequency ωc, the time-varying channel of the ith mobile

is characterized by a vector function

hi(t) =

q=1

βiq a(θiq) ej(ωc+ˆωiq)t,1≤i≤K . (2.1)

base station array

θ1

β1

scatterer 2

scatterer 1

Figure 2.1.: Schematic illustration of multipath propagation

2. Duality Between Downlink and Uplink Multiplexing

The spatial covariance matrices are given by

Ri=E{hi(t)hH

i(t)},1≤i≤K , (2.2)

where the operator E{·} denotes the ensemble average over the time-fluctuating fad-

ing channel. The variation of hi(t)in time and space is illustrated in Fig. 2.2.

Based on the knowledge of the spatial correlation properties (2.2) at the base

station, all Kcommunication links can be jointly optimized. While the representation

(2.2) is a statistical measure for the propagation channel, it is general enough to

incorporate also the case of perfect channel information. Coherent spatial processing

is possible if rank{Ri}= 1. This is fulfilled, e.g., when the channel does not change

within the observation window, in which case perfect channel information is available.

In general, however, the channel may be rapidly time-variant and the total number

of transmission paths per user easily exceeds the number of antenna elements. Then,

Riwill have full rank.

Note, that most techniques derived in this thesis are general enough to hold for

channels of arbitrary rank, except for the results in Chapter 5. Here, a block fading

channel model is assumed. The channel gains are approximated as piecewise constant

over many symbol intervals, which is fulfilled if the Doppler spread is much less than

the bandwidth of the transmitted signal. This model reflects the need for perfect

channel information, which is required for the techniques described in Chapter 5.

2.1.2. Cooperating Transmit Antennas (Downlink)

Consider a signal vector sdef

= [S1,...,SK]Tto be transmitted from the base station

to the Kmobiles. The components of scan be modeled as wide-sense stationary

stochastic processes which are mutually and temporally uncorrelated, with zero mean.

The variances E{|Si|2},1≤i≤Kare stacked in a vector p= [p1...pK]T. A sum

power constraint

kpk1≤Pmax (2.3)

is imposed. A matrix U∈CM×Kis used to map the signal vector son the M

transmit antennas. The ith data stream is spread over the antenna array by the ith

column of U, which is denoted by ui∈CMand constrained to fulfill

kuik2= 1,1≤i≤K . (2.4)

The base station antennas are uncoordinated if the operator U= [u1,...,uK]has a

diagonal structure. This, is the case, e.g., for the Bell Laboratory Layered Space-Time

(BLAST) architecture [45], which assumes that no channel information is available

at the transmitter. Large performance gains are possible, however, by allowing the

antennas to cooperate.

Cooperation requires that channel information is available prior to transmission.

Then, the vectors uican be interpreted as beamformers which can be adjusted so as

2. Duality Between Downlink and Uplink Multiplexing

space time

field strength (logarithmic)

Figure 2.2.: Space-time fading caused by angle/frequency dispersion

to focus energy in space. This approach has a positive impact on the SINR measured

at the antenna output of the ith mobile:

SINRDL

i(U,p) = piuH

iRiui

k=1

k6=i

pkuH

kRiuk+σ2

,1≤i≤K , (2.5)

where σ2

iis the variance of the additive white Gaussian noise. The stochastic noise

processes are assumed to be mutually uncorrelated, i.e., the vector

σ= [σ2

1,...,σ2

K]T(2.6)

satisfies E{σσH}=I.

In order to illustrate the impact of antenna cooperation on the SINR, think of a

single user channel h1=βa(θ1), as illustrated in Fig. 2.3, for which maximum ratio

combining u1=h1/kh1k2is the optimal transmit strategy. It can easily be verified

that SINR1=p1β2ka(θ1)k2/σ2

1, which is an improvement by a factor ka(θ1)k2=M

as compared to single antenna transmission. The SINR linearly increases with the

number of antenna elements. Note, that the same does not necessarily hold for a

multi-user channel, which is limited by the effects of mutual interference, as will be

shown in the remainder of this chapter.

2.1.3. Cooperating Receive Antennas (Uplink)

Now, assume an uplink scenario, with Mcooperating receive antennas at the base

station. The Knon-cooperating mobiles are equipped with single antennas. Inde-

pendent data streams with powers q1...qKare multiplexed from the mobiles to the

2. Duality Between Downlink and Uplink Multiplexing

path gain β

common

SINR1

noise σ2

Mcooperating

transmit antennas

power p1

array response

a(θ1)

Figure 2.3.: SNR improvement by coordinated multi-antenna transmission for a single

user scenario

base station. As in the downlink, the transmission powers are stacked in a vector

q= [q1,...,qK]T. The same beamforming matrix U= [u1...uK]that has been

used for downlink transmission is now used for reception and can be interpreted as

a linear multiuser receiver. The uplink SINR is given by

SINRUL

i(ui,q) = qiuH

iRiui

iK

k=1

k6=i

qkRk+σ2

iIui

,1≤i≤K . (2.7)

Observe, that the uplink SINR levels (2.7) are coupled by the transmission powers

q, but not by the beamformers U. This special structure makes the uplink SINR

more tractable than the downlink SINR expressions (2.5), which are intertwined in

a complicated way. This is the reason why most studies in the literature so far have

focused on uplink processing. One key result of this chapter will be to show that

albeit their seemingly different structure, there exists a fundamental duality between

multiuser processing in uplink and downlink, thus both links can be optimized jointly.

Again, a first motivating example is provided by the single user scenario depicted

in Fig. 2.3. By reversing the role of transmitter and receiver, the problem immediately

carries over to the uplink. It can easily be verified that the same gain Mis achievable

for both links. Does this observation also hold true for a multiuser scenario? This

problem will be studied in the remainder of this chapter. To this end, we assume

that the uplink transmission powers are constrained in the same way as the downlink

powers, i.e.,

kqk1≤Pmax .(2.8)

2.1.4. Crosstalk

The SINR values in uplink and downlink are mutually coupled by cross-talk, which

can be collected in a K×Kmatrix

[Ψ(U)]ik =uH

kRiuk, k 6=i

0, k =i.(2.9)

In general, this matrix is non-negative and asymmetric.

2. Duality Between Downlink and Uplink Multiplexing

If the spatial covariance matrices are full rank, then uH

kRiuk>0always holds.

Then, it is impossible to completely cancel out the interference by using a zero-forcing

strategy. This situation occurs, e.g., when Riis obtained by averaging over the time-

fluctuating fading channel and the channel has several spatially resolved propagation

paths, as discussed in Section 2.1.1. Then, the matrix Ψis strictly positive and all

users are coupled by interference.

Another, more extreme assumption is perfect channel information. If, in addition,

K≤M, then all covariance matrices have rank one and interference can be canceled

out completely. This results in Ψ=0, thus the users are no longer coupled by

interference.

However, complete interference cancellation is not necessarily optimal when a

power constraint is imposed. In the uplink, linear zeroforcing mostly comes at the

cost of noise enhancement, whereas in the downlink a power enhancement is caused.

Hence, zeroforcing only performs well in the high SNR regime. whereas the aim of this

thesis is to find a power aware transmission strategy that shows good performance

over the whole SNR range. Therefore, the design objective can be seen as to find the

optimal tradeoff between residual interference and signal-to-noise ratio (SNR).

Now, let’s make the notion of “cross-talk coupling” more precise:

Coupled users. A set of users is called coupled by cross-talk if an increase in one

user’s transmission power has direct or indirect impact on the interference ex-

perienced by the other users. This is obviously fulfilled if the link gains between

all users are positive. If, however, there is no direct connection between two

users A and B, they may still be connected via a third party C. Suppose that

A causes interference to C. Then C has to increase its power level to maintain

the same SINR. This in turn may increase the interference level at user B.

An appropriate mathematical model is provided by the theory of directed

graphs (see e.g. [46]). Suppose that each user is represented by a node. The

nodes are connected by directed edges which are given by the positive entries

of the link gain matrix Ψ. If Ψij >0then there is a connection from node

ito node j. A system of nodes is called strongly connected if for each pair of

nodes (Ni, Nj)there is a sequence of directed edges leading from Nito Nj. In

the following, a set of users will be called coupled by cross-talk if and only if

its directed graph is strongly connected. An example scenario is illustrated in

Fig. 2.4.

Uncoupled users. A user is defined to be uncoupled in terms of cross-talk if he

does not belong to a set of coupled users. There are two possible scenarios for

which this is fulfilled:

1. the user receives no crosstalk from other users,

2. the user causes no cross-talk to other users.

2. Duality Between Downlink and Uplink Multiplexing

Ψ=

base station beamformers







0Ψ12 Ψ13 Ψ14

0 0 Ψ23 Ψ24

000Ψ34

Ψ41 000







mobiles

⇒

1 2

Ψ12

Ψ23

Ψ34

Ψ24

Ψ13

Ψ41

Ψ14

Figure 2.4.: All users are coupled by cross-talk. The directed graph is strongly con-

nected

In both cases the transmission power of the user can be increased without the

penalty of increasing its own interference level by the control mechanism of the

multiuser power control. Then, arbitrary levels SINRi≥γican be achieved

(provided that the transmission power is unconstrained). An example scenario

is illustrated in Fig. 2.5.

Ψ=

base station beamformers







0Ψ12 Ψ13 Ψ14

0 0 Ψ23 Ψ24

0 0 0 Ψ34

0 0 0 0







mobiles

⇒

1 2

Ψ14

Ψ12

Ψ23

Ψ34

Ψ24

Ψ13

Figure 2.5.: All users are uncoupled. Such a matrix structure is imposed, e.g., by

coding for known interference (see Chapter 5 for a detailed study of this scenario)

Note, that even when the users are completely uncoupled in terms of interference,

i.e., Ψ=0, their SINR levels are still connected via the sum power constraint. The

absence of interference does not imply independent AWGN channels, as it is the case

for independent power constraints. That is, increasing one users power immediately

forces one or more other users to decrease their power in order for the sum power

constraint to be fulfilled. This aspect will be considered in the next section.

2.2. Uplink Power Allocation under a Sum Power

Constraint

Let’s begin studying the multiuser uplink scenario for a fixed beamforming matrix

U= [˜u1,...,˜uK]. The goal is to find the power allocation that jointly achieves

SINRUL

i≥γi,1≤i≤Kunder a sum power constraint (2.8). This goal may be

infeasible, thus one important aspect of the problem is to find necessary and sufficient

conditions for feasibility.

2. Duality Between Downlink and Uplink Multiplexing

2.2.1. Maximizing the Balanced SINR Margin

The SINR requirements are fulfilled if and only if min1≤i≤KSINRUL

i/γi≥1. A

straightforward strategy to achieve this design goal is to solve the following opti-

mization problem:

CUL(˜

U, Pmax) = max

qmin

1≤i≤K

SINRUL

i(˜ui,q)

γi

,(2.10)

subject to kqk1≤Pmax and q∈RK

Observe that SINRUL

i≥γiis feasible for i= 1,...,Kif and only if CUL(˜

U, Pmax)≥1.

The function CUL is characterized by the following two lemmas:

Lemma 1. Let ˜qbe a global maximizer of the optimization problem (2.10), then

CUL(˜

U, Pmax) = SINRUL

i(˜ui,˜q)

γi

,1≤i≤K , (2.11)

Pmax =k˜qk1.(2.12)

Proof. Assume that there exists an index i0such that SINRUL

i0/γi0>min

1≤k≤KSINRUL

k/γk.

From (2.7) it can be observed that each SINRUL

iis strictly monotonically increasing

in qiand monotonically decreasing in qkfor k6=i. Thus, qi0can be reduced without

decreasing the minimum level. This leads to a new power vector ˆqwith kˆqk1< Pmax.

This power saving can be redistributed among all users by using a scaled power vec-

tor αˆq, where the scalar α > 1is chosen such that kαˆqk1=Pmax. It can be verified

that SINRUL

i(αˆq)>SINRUL

i(ˆq)for all i= 1,...,K. Thus, it would be possible to

obtain a point with the cost function larger than the global maximum, which is a

contradiction. 

Lemma 2. The function CUL(˜

U, Pmax)is strictly monotonically increasing in Pmax.

Proof. This follows from SINRUL

i(˜ui, αq)>SINRUL

i(˜ui,q), with 1≤i≤Kand

α > 1.

The monotony result stated in Lemma 2 is illustrated in Fig. 2.6. It can be

observed that there is a one-to-one relationship between each balanced SINR margin

and a total transmission power. Each point is characterized by the set of equations

(2.11) and (2.12), which can be rewritten using matrix notation. To this end we

define

D= diagγ1

1R1u1

,..., γK

KRKuK,(2.13)

Λ(U, Pmax) = DΨT(U)Dσ

Pmax

KDΨT(U)1

Pmax

KDσ.(2.14)

2. Duality Between Downlink and Uplink Multiplexing

total transmission power Pmax

feasible

infeasible

CUL(˜

U, Pmax)

Figure 2.6.: Balanced SINR margin CUL(˜

U, Pmax)vs. total transmission power Pmax

It can be verified that (2.11) and (2.12) together form an eigensystem

Λ(˜

U, Pmax)˜q

1=1

CUL(˜

U, Pmax)˜q

1.(2.15)

The balanced level CUL(˜

U, Pmax)is a reciprocal eigenvalue of the non-negative ex-

tended coupling matrix Λ. However, not all eigenvalues represent physically mean-

ingful balanced levels. In particular, q∈RK

+and CUL(˜

U, Pmax)∈R+must be

fulfilled.

Lemma 3. For any non-negative real matrix BK×K≥0with spectral radius ρ(B),

there exists a vector q≥0and λ(B)

max =ρ(B)such that Bq =λ(B)

maxq.

Proof. See e.g. [46, p. 670]. 

It was shown by Yang and Xu [28] that if the non-negative matrix has the spe-

cial structure (2.14), then the non-negative eigenvector associated with the maximal

eigenvalue, denoted by λmax(Λ), is strictly positive and the solution is unique. The

uniqueness of λmax(Λ)also follows immediately from Lemma 2, which rules out the

existence of two different balanced levels with the same sum power. For each eigen-

value the power constraint k˜qk1=Pmax can be ensured by scaling the eigenvector

such that the last component equals one. If there are two eigenvalues with strictly

positive eigenvectors, then we know from Lemma 2 that they must have the same

value. Hence, the solution of the SINR balancing problem (2.10) is given by

CUL˜

U, Pmax=1

λmaxΛ(˜

U, Pmax).(2.16)

2.2.2. Minimizing the Sum Power

In the previous section it was shown how to take full advantage of the available total

transmission power Pmax in a fair (balanced) fashion. From a network operator’s

perspective, however, it is desirable to multiplex as many channels as possible under

2. Duality Between Downlink and Uplink Multiplexing

the given SINR constraints (1.1). Thus, another optimization strategy is to mini-

mize the total transmission power while maintaining SINRUL

i≥γifor i= 1,...,K.

Mathematically, this can be expressed as

min

q>0kqk1subject to SINRUL

i(˜ui,q)≥γi,1≤i≤K . (2.17)

If problem (2.17) is feasible, then it leads to a balanced solution with active con-

straints. This is a direct consequence of Lemma 2. Thus, (2.17) can be seen as a

special case of the balancing problem (2.10), where Pmax is chosen such that the bal-

anced optimum equals one (cf. Fig. 2.6 on p. 14). In this case, the global optimum

is characterized by the set of equations (2.11) with CUL˜

U, Pmax= 1. This can be

rewritten as

I−DΨT(˜

U)˜q=Dσ .(2.18)

where ˜qis the optimizer of (2.17).

Note, that CUL˜

U, Pmax= 1 is not always feasible. This can be best illustrated

by considering a 2-user scenario with fixed link gains αik

def

=˜uH

iRk˜ui,i, k ∈[1,2].

With (2.7) the constraints SINRUL

i≥γican be expressed as

q2≥γ2α12

α22

q1+γ2σ2

α22

,(2.19)

q2≤α11

γ1α21

q1−σ2

α21

.(2.20)

These inequalities describe half spaces, as illustrated in Fig. 2.7. If these half spaces

intersect in the positive quadrant, both targets γ1and γ2can be supported and the

solution is feasible. The power minimum is characterized by the intersection point.

It can be easily verified that a feasible solution exists if and only if the slope of (2.20)

is larger than the one of (2.19). This leads to the following necessary and sufficient

condition for feasibility:

α12α21γ1γ2< α11α22 .(2.21)

In the next section, necessary and sufficient conditions for an arbitrary number of

users are derived.

power minimum

γ2σ2

α22

−σ2

α21

feasibility

region

Figure 2.7.: Geometric interpretation of the power minimization problem for fixed

beamformers

2. Duality Between Downlink and Uplink Multiplexing

2.3. Uplink Feasibility in the Absence of Power

Constraints

In this section, conditions are derived under which the power minimization problem

(2.17) is feasible in the absence of power constraints. A scenario is called feasible

if there exist transmission powers such that SINR target thresholds γ1...γKare

achieved. The achievable balanced SINR margin (2.10) is tightly upper bounded by

BUL

Inf (˜

U)def

= sup

q:q>0min

1≤i≤K

SINRUL

i(˜ui,q)

γi.(2.22)

Since CUL(˜

U, Pmax)is strictly monotonically increasing in the sum power kqk1, the

bound BUL

Inf is asymptotically approached in the high power regime, where crosstalk

dominates over noise (see the illustration in Fig. 2.6).

2.3.1. Feasibility of Coupled Users

Let’s start by assuming that all users are coupled by crosstalk, as defined in Sec-

tion 2.1.4. Then, the asymptotic limit BUL

Inf can be found by dropping the noise term

and balancing the signal-to-interference ratios

SIRUL

i(ui,q) = qiuH

iRiui

iK

k=1

k6=i

qkRkui

,1≤i≤K . (2.23)

Observe that each SIR is invariant with respect to a scaling of q, thus without loss

of generality we can assume kqk1= 1. The limit BUL

Inf (˜

U)is obtained by solving the

following max-min balancing problem

BUL

Inf (˜

U) = max

qmin

1≤i≤K

SIRUL

i(˜ui,q)

γisubject to kqk1= 1 .(2.24)

Since all SIR are coupled, (2.24) has a solution ˜qthat balances all quantities SIRUL

i(˜ui,˜q)

at a level BUL

Inf (˜

U). Using matrix notation, this can be expressed as

DΨT(˜

U)·˜q=1

BUL

Inf ·˜q.(2.25)

It can be observed that BUL

Inf (˜

U)is a reciprocal eigenvalue of the coupling matrix

DΨT(˜

U). Since coupled users are assumed, the directed graph associated with the

non-negative matrix ΨT(˜

U)is strongly connected (see Section 2.1.4). This implies

that DΨT(˜

U)is irreducible [46]. From the Perron-Frobenius Theorem it can be

2. Duality Between Downlink and Uplink Multiplexing

concluded that DΨT(˜

U)has a simple, real and strictly positive maximal eigenvalue

λmaxDΨT(˜

U), which equals the spectral radius. The eigenvector associated with

the maximal eigenvalue is strictly positive. Moreover, all other eigenvectors have at

least one negative component. Hence, the maximal eigenvalue is the only valid one.

It follows that

BUL

Inf (˜

U) = 1

λmaxDΨT(˜

U).(2.26)

To conclude, the power minimization problem (2.17) is feasible for a set of coupled

users if and only if λmaxDΨT(˜

U)<1. For K= 2 users, condition (2.26) reduces

to λmax(DΨT) = √α12α21γ1γ2<√α11α22. This is in good agreement with the

feasibility condition (2.21) derived for the 2-user case in Section 2.2.2.

Note, that the problem of achieving SINR targets was already studied by Zander

[25–27, 29] in the context of classical power control. The result (2.26) was derived in

[26], under the implicit assumption that Ψis irreducible and the Perron-Frobenius

Theorem can be applied. This is always fulfilled in the context of classical power con-

trol. However, when additional spatial filtering is used, some users may be decoupled

by nullsteering beamforming, as discussed in Section 2.1.4.

2.3.2. General Feasibility

Now, consider a mixed scenario with some uncoupled users and (possibly) several

subsets of coupled users. The subsets are mutually uncoupled, so each of them can

be regarded as a separate system with an individual performance bound. This leads

to the following observations:

•If all user are uncoupled then λmax(DΨT)=0, which means that the system

is not limited by crosstalk. An example has already been given in Fig. 2.5 on

page 12.

•Suppose that there is one subsystem consisting of two or more coupled users,

characterized by an irreducible coupling matrix Ψ1. All other users are un-

coupled. Then, the performance is limited by the coupled subsystem, i.e.,

λmax(DΨT) = λmax(DΨT

1).

•Suppose that there are Lsubsystems with irreducible coupling matrices Ψ1,...,ΨL.

The systems are mutually uncoupled. Then, the performance is limited by

λmax(DΨT) = maxλmax(DΨT

1),...,λmax(DΨT

L).

The results are further specified by the following theorems:

Theorem 1. Suppose that the power minimization problem (2.17) is feasible for given

noise variances ˜σ2

i>0,1≤i≤K, that is, there exists a power vector ˜q>0such

that SINRUL

i(˜ui,˜q,˜σ2

i)≥γi. Then (2.17) is feasible for any σ2

i>0,1≤i≤K.

2. Duality Between Downlink and Uplink Multiplexing

Proof. See the appendix on page 72. 

Theorem 2. For any noise vector σ>0, problem (2.17) has a feasible solution

˜q>0if and only if λmaxDΨT(˜

U)<1.

Proof. See the appendix on page 73. 

These results hold for both coupled and uncoupled users. It can be concluded

that asymptotic feasibility depends on the one hand on the target thresholds γ1...γK

(included in D), on the other hand on the channel properties and spatial filters

(included in the cross-talk matrix Ψ).

2.4. The Dual Downlink Problem

Now, let’s turn our attention to the downlink channel, where Mcooperating trans-

mit antennas are used to transmit independent data streams to Knon-cooperating

receive antennas. As for the uplink, a sum power constraint is imposed and a fixed

beamforming matrix ˜

Uis assumed. Many arguments used for the uplink analysis

obviously hold for the downlink as well, thus some proofs can be omitted.

2.4.1. Downlink Power Allocation

A necessary and sufficient condition for the achievability of individual SINR thresh-

olds γ1...γKunder a sum power constraint kpk1≤Pmax is given by CDL(˜

U, Pmax)≥

1, where

CDL(˜

U, Pmax) = max

pmin

1≤i≤K

SINRDL

i(˜

U,p)

γi

,(2.27)

subject to kpk1≤Pmax and p∈RK

First, it can be shown that the optimum of (2.27) is balanced:

Lemma 4. Let ˜pbe a global maximizer of the optimization problem (2.27), then

CDL(˜

U, Pmax) = SINRDL

i(˜

U,˜p)

γi

,1≤i≤K , (2.28)

Pmax =k˜pk1.(2.29)

Proof. This can be shown in analogy to Lemma 1. 

Lemma 5. The function CDL(˜

U, Pmax)is monotonically increasing in Pmax.

Proof. This follows from SINRDL

i(˜

U, αp)>SINRDL

i(˜

U,p), with 1≤i≤Kand

α > 1.

2. Duality Between Downlink and Uplink Multiplexing

Hence, the function CDL(˜

U, Pmax)has the same properties as the corresponding

uplink function CUL(˜

U, Pmax)discussed in Section 2.2. Defining

Υ(U, Pmax) = DΨ(U)Dσ

Pmax

KDΨ(U)1

Pmax

KDσ,(2.30)

the set of equations (2.28) and (2.29) can be written as

Υ(˜

U, Pmax)˜p

1=1

CDL(˜

U, Pmax)˜p

1.(2.31)

Using the same monotony argument as in Section 2.2, it can be shown that

CDL˜

U, Pmax=1

λmaxΥ(˜

U, Pmax),(2.32)

where λmaxΥ(Pmax)is the maximal eigenvalue of the extended coupling matrix Υ.

In analogy to the uplink, the power minimum is achieved for CDL˜

U, Pmax= 1.

If the SINR requirements are feasible, then a power allocation that minimizes the

total transmission power is found by solving

min

p>0kpk1subject to SINRDL

i(˜

U,p)≥γi,1≤i≤K . (2.33)

Problem (2.33) is called feasible if there exists a solution ˜p>0that fulfills the

constraints in (2.33). Moreover, since SINRDL

iis strictly monotonically increasing

in piand monotonically decreasing in pk,k6=i, this optimum is characterized by

SINRDL

i(˜

U,˜p) = γi. Using matrix notation, this point can be found by solving the

following set of equations

I−DΨ(˜

U)˜p=Dσ .(2.34)

As in the uplink case, an interesting question is: under which conditions is (2.33)

feasible in the asymptotic limit for Pmax → ∞? This will be answered in the next

section.

2.4.2. Feasibility Conditions for the Downlink

Comparing (2.34) and (2.18), it can be observed that the downlink users are coupled

by cross-talk matrix Ψ, whereas the uplink users are coupled by the transpose ΨT.

Hence, the same arguments that have been used in Section 2.3 to derive a feasibil-

ity bound for the uplink, immediately carry over to the downlink. The results are

summarized by the following theorems:

2. Duality Between Downlink and Uplink Multiplexing

Theorem 3. Suppose that the power minimization problem (2.33) is feasible for given

noise variances ˜σ2

i>0,1≤i≤K, that is, there exists a power vector ˜p>0such

that SINRDL

i(˜

U,˜p,˜σ2

i)≥γi. Then (2.33) is feasible for any σ2

i>0,1≤i≤K.

Proof. This can be shown by using the same technique as for Theorem 1. 

Theorem 4. For any noise vector σ>0, problem (2.33) has a feasible solution

˜p>0if and only if λmaxDΨ(˜

U)<1.

Proof. This can be shown by using the same technique as for Theorem 2. 

2.4.3. Weak Duality

Now we compare uplink and downlink performance in terms of their feasibility range.

To this end we need the following lemma, which will prove very useful throughout

the remainder of this thesis.

Lemma 6. The maximal eigenvalue of any non-negative real matrix BK×Kcan be

expressed as

λmax(B) = max

x>0min

y>0

xTBy

xTy= min

x>0max

y>0

xTBy

xTy.(2.35)

Proof. See [47, p. 28]. 

With Lemma 6 the following result can be proven:

Lemma 7. Let ΨK×Kbe non-negative real and Das defined in (2.13). Then

λmax(DΨ) = λmax(DΨT).(2.36)

Proof. Using Lemma 6 and λmax(B) = λmax(BT), we have

λmax(DΨT) = λmax(ΨD)

= min

y>0max

x>0

xTΨDy

xTy.(2.37)

The matrix Dis regular, thus we can substitute x1

def

=D−1xand y1

def

=Dy. Hence,

(2.37) can be rewritten as

λmax(DΨT) = min

y1>0max

x1>0

1DΨy1

1y1

=λmax(DΨ).(2.38)



2. Duality Between Downlink and Uplink Multiplexing

Lemma 7 shows that the asymptotic feasibility bounds for uplink and downlink

are identical. We refer to this result as weak duality:

Theorem 5. The uplink problem (2.17) is feasible if and only if the downlink problem

(2.33) is feasible. Then both problems are feasible for arbitrary noise vectors σ>0.

Proof. The theorem follows directly from Theorem 2, Theorem 4, and Lemma 7. 

An alternative proof for Theorem 5 was provided by Zander [27].

2.4.4. Strong Duality

Next, λmaxDΨ(˜

U)<1is taken for granted, i.e., both uplink and downlink targets

are feasible. Let Adef

=D−1−Ψ(˜

U)−1. It was already shown that the power mini-

mizing uplink and downlink allocation can be found by solving the sets of equations

(2.18) and (2.34), respectively. Hence, there exist positive downlink and uplink power

vectors

p=Aσ ,(downlink) (2.39)

q=ATσ.(uplink) (2.40)

We now whish to know under what conditions does the total uplink power equal the

total downlink power, i.e.,

kpk1=

k=1

pk=

k=1

qk=kqk1.(2.41)

If (2.41) holds together with weak duality, i.e., the same SINR levels are achievable

in uplink and downlink, then we say that strong duality holds.

Theorem 6. Let σ=ν·1Kwith ν∈R+. Then strong duality is fulfilled for any

non-negative A.

Proof. This is an immediate consequence of

kqk1=1T

Kq=ν·1T

KAT1K=ν·1T

KA1K=1T

Kp=kpk1.(2.42)

The result was published by Boche and Schubert in [6] and was independently derived

by Tse and Viswanath in [34]. 

Now, we are interested in a more general characterization of the noise vectors

that guarantee strong duality.

2. Duality Between Downlink and Uplink Multiplexing

Lemma 8. Let σ(1) >0and σ(2) >0be noise vectors. The resulting downlink and

uplink power vectors follow directly from (2.39) and (2.40). They are denoted by p(1),

p(2),q(1), and q(2). Also, consider a noise vector σ=ασ(1)+βσ(2), where the positive

scalars αand βsatisfy α+β= 1. The resulting power vectors are denoted by pand

q. Then, kpk1=kqk1holds for all α, β ≥0if and only if kp(l)k1=kq(l)k1, l = 1,2.

Proof. Assume kpk1=kqk1. Substituting σ=ασ(1) +βσ(2) in (2.39) and (2.40)

we have p=αp(1) +βp(2) and q=αq(1) +βq(2), respectively. From the assumption

follows kp(l)k1=kq(l)k1, l = 1,2. To prove the reverse direction, assume kp(l)k1=

kq(l)k1, l = 1,2. This implies kpk1=αkp(1)

kk1+βkp(2)

kk1=kqk1.

Lemma 9. Suppose that p=Aσ and q=ATσfulfill (2.41), where σ>0is an

arbitrary positive noise vector and Ais non-negative. Then (2.41) is fulfilled for all

noise vectors σ≥0.

Proof. Let ˆσbe an arbitrary noise vector with [ˆσ]l= 0 for some index l. Let

ˆp=Aˆσand ˆq=ATˆσ. Now, consider the convergent sequence ˆσ(n)=ˆσ+1

n1Kwith

[ˆσ(n)]l>0,1≤l≤K. Since the linear operator Ais continuous, the sequences

ˆp(n)=Aˆσ(n)and ˆq(n)=ATˆσ(n)are also convergent. With the assumption that

(2.41) holds for any σ>0, we have kˆp(n)k1=kˆq(n)k1for all n. Hence,

lim

n→∞ kˆp(n)k1=kˆpk1= lim

n→∞ kˆq(n)k1=kˆqk1.

From this we can conclude that (2.41) holds for ˆσ≥0.

Lemma 10. Given a matrix A, the relation (2.41) is fulfilled for all σ>0if and

only if it is fulfilled for

[e(k)]l=(1, l =k

0, l 6=k,1≤k≤K . (2.43)

Proof. If (2.41) is fulfilled for all σ>0, then it follows from Lemma 9 that it is also

fulfilled for vectors (2.43). On the other hand, if (2.41) is fulfilled for (2.43), we know

from Lemma 8 that it is fulfilled for all linear combinations, i.e., for all σ>0.

Using the above results, it can be shown that strong duality holds for any noise

vector if and only if the matrix Ahas a specific structure specified by the following

theorem (the result was published by Boche and Schubert in [1]).

Theorem 7. Relation (2.41) holds for all σ>0if and only if

l=1

Akl =

l=1

Alk ,1≤k≤K . (2.44)

Proof. This follows directly from Lemma 10 with p(k)=Ae(k)=

l=1

Alk and q(k)=

ATe(k)=

l=1

Akl, where e(k)is defined as in (2.43). 

2. Duality Between Downlink and Uplink Multiplexing

2.5. Discussion

In this chapter, a fundamental duality between the downlink point-to-multipoint

channel and the uplink multipoint-to-point channel has been found. If one of the

conditions formulated by Theorem 6 and Theorem 7 hold true, then identical SINR

levels can be achieved in both links with the same total transmission power.

This duality relation is extremely useful, as will be shown in the following. In

particular, each SINR-based signaling strategy that is designed for the uplink imme-

diately carries over to the downlink. One merely has to ensure that the condition in

Theorem 6 is fulfilled, i.e., all receivers have the same noise level.

For the downlink, this is easily achieved by scaling the SINR (2.5) such that

σ=1K. Then, the optimization has to be performed with respect to scaled matrices

R1/σ2

1,...,RK/σ2

K. Such a scaling has no impact on the downlink SINR. However,

when applying the same matrices to the uplink problem, one has

SINRUL

i(ui,p) = piuH

σ2

iK

k=1

k6=i

pkRk

σ2

+Iui

,1≤i≤K . (2.45)

This scaled SINR may in general be different from the true uplink SINR (2.7). Thus,

strong duality only holds between the scaled channels. But by solving the scaled

“virtual” uplink problem, one always gets a solution for the “true” downlink problem.

If one is interested in the uplink problem, the condition σ=1Kcan likewise be

ensured. While the downlink channel is invariant to a scaling of the link gains, the

uplink channel is invariant to a scaling of the beamforming weights. By constraining

σ2

iuH

iui= 1, it would in principle be possible to achieve target SINR’s in the uplink

by considering the dual “virtual” downlink problem instead. However, this is less

interesting from a practical point of view since the downlink has a more difficult

analytical structure.

Finally, note that the duality found in this chapter stands in an interesting rela-

tionship to other results in the literature:

•In [39, 40] an iterative algorithm for joint beamforming and power control was

proposed which uses virtual uplink power control in each iteration step.

•In [48] it was shown that the capacity of the MIMO channel remains the same

when reversing the role of transmitter and receiver

•In [33] duality was found in an information theoretical context, between the

Gaussian broadcast channel and the multiple access channel.

The generalization of these independent observations is still an open problem. Partial

results will be presented later in Chapter 5.

3. Joint Beamforming in the

Absence of Noise

In the previous chapter it was shown how the achievable SINR margin is limited by

the effects of mutual interference. Assuming coupled users, an upper bound on the

system performance can be found by balancing the signal-to-interference ratios (SIR)

in the absence of noise.

In this chapter the problem of SIR balancing is studied with additional optimiza-

tion over the beamforming filters. This problem was first addressed by Gerlach and

Paulraj [10] in the context of downlink beamforming. Algorithms have been pre-

sented in [10, 43]. However, no proof of global convergence was found so far. One

goal of this chapter is to analyze both optimization strategies and to carry out a

convergence analysis.

By studying the joint beamforming problem in the absence of noise, necessary and

sufficient conditions for feasibility are obtained, as discussed in Section 2.3. Moreover,

it is expected that the results provide deeper insight into the analytical structure of

the general problem of controlling SINR levels.

3.1. Optimal Interference Balancing based on

`∞-Norm Minimization

We start with the downlink SIR balancing problem, as formulated by Gerlach and

Paulraj [10]. For given channel matrices R1,...,RK, one is interested in the beam-

forming matrix Uand power allocation p, which jointly balance the values SIRDL

i/γi,

1≤i≤K, as high as possible. Each SIR is a function of pand U, i.e.,

SIRDL

i(p,U) = piuH

iRiui

k=1

k6=i

pkuH

kRiuk

,1≤i≤K . (3.1)

This problem can be expressed mathematically as

BDL

Inf = max

U,pmin

1≤i≤K

SIRDL

i(U,p)

γisubject to kuik2= 1,1≤i≤K ,

kpk1= 1 .

(3.2)

3. Joint Beamforming in the Absence of Noise

Note, that this problem is only defined as long as each user receives interference, in

which case the optimum is characterized by balanced relative SIR levels. Thus, min-

max operations are interchangeable and one can equivalently balance the reciprocal

quantities γi/SIRDL

i. Hence, problem (3.2) is solved by minimizing the `∞-norm of

the vector

ξ(U,p) = γ1

SIRDL

1(U,p),..., γK

SIRDL

K(U,p)T

.(3.3)

By optimizing over both Uand p, the optimally balanced level BDL

Inf is found:

BDL

Inf

= min

U,pkξ(U,p)k∞.(3.4)

Note, that (3.4) has a non-differentiable objective function which in general may have

many local minima. Thus, there is no obvious way for finding the global optimum.

Fortunately, the analytical results of Chapter 2 provide a characterization of the

global optimum, which leads to a form that is better suited for optimization. This

will be shown in the following section.

3.1.1. Eigenvalue Minimization

In Section 2.3.1 it has been shown that for any fixed beamforming matrix ˜

Uthe

balanced optimum is given by the inverse maximal eigenvalue λmaxDΨ(˜

U). Hence,

the global optimum of the `∞optimization problem (3.4) is found by solving an

eigenvalue optimization problem

BDL

Inf =1

min

UλmaxDΨ(U)subject to kuik2= 1,1≤i≤K . (3.5)

Note, that this problem no longer depends on p. Once an optimizer Uopt is found,

the associated popt is given as the dominant right-hand eigenvector of the matrix

DΨ(Uopt), as shown in Section 2.3.1.

Another straightforward observation is that the eigenvalue optimization problem

(3.5) not only provides a solution of the `∞balancing problem (3.2), but also provides

a necessary and sufficient criterion for the feasibility of SINR ≥γi,1≤i≤K. It has

been shown in Section 2.3.2, that this does not require the assumption of coupled

users, which can therefore be dropped as long as the eigenvalue optimization problem

(3.5) is concerned. The maximal eigenvalue is always defined, even if the users are

completely decoupled. If (3.5) has a solution BDL

Inf = 0, then SINR ≥γiis always

feasible, but (3.4) is not defined.

While the problem formulation (3.5) has less degrees of freedom than the original

problem (3.2), it is still very unlikely to allow for efficient algorithmic solution. As

the objective function may generally be non-convex, global optimization may be

prevented by local minima.

3. Joint Beamforming in the Absence of Noise

In this situation one can make use of the weak duality formulated in Section 2.4.3.

From this result it is known that instead of balancing the downlink quantities SIRDL

i/γi

one can equivalently balance the uplink quantities SIRUL

i/γi, where SIRUL

iis defined

in (2.23). The following balanced level can be achieved by optimizing over both U

and q:

BUL

Inf = max

U,qmin

1≤i≤K

SIRUL

i(ui,q)

γisubject to kuik2= 1,1≤i≤K ,

kqk1= 1 .

(3.6)

Note, that SIRUL

i(ui,q)is scalable in q, thus kqk1= 1 can be assumed without

loss of generality. As in the downlink, the max-min balancing problem (3.6) can be

equivalently stated as an eigenvalue optimization problem

BUL

Inf =1

min

UλmaxDΨT(U),subject to kuik2= 1,1≤i≤K . (3.7)

From Lemma 7 in Section 2.4.3 it is known that BDL

Inf =BUL

Inf for any beamforming

matrix, including the global minimizers of (3.5) and (3.7). Hence, any beamformer

that solves the uplink problem (3.7) also solves the downlink problem (3.5) and

vice versa. Note that this one-to-one correspondence pertains to the beamforming

solution, but not to the transmission powers. The optimal power allocation in uplink

and downlink may in general be different. Having found a global minimizer Uopt,

the power allocations are found by solving the following two eigenvalue problems (see

Section 2.3):

DΨT(Uopt)·qopt =1

BUL

Inf ·qopt ,(uplink) (3.8)

DΨ(Uopt)·popt =1

BDL

Inf ·popt .(downlink) (3.9)

3.1.2. Characterization of the Global Minimizer

Next, the global optimum of (3.7) is characterized. With Lemma 6 we have

λmaxDΨT(U)= max

x>0min

y>0

xTDΨT(U)y

xTy= min

x>0max

y>0

xTDΨT(U)y

xTy.(3.10)

Using (3.10), the eigenvalue minimization problem (3.7) can be rewritten as

BUL

Inf

= min

U:kuik2=1 min

q>0

f(U,q),with ˆ

f(U,q) = max

x>0

xTDΨT(U)q

xTq.(3.11)

Here, xis an auxiliary variable which has no physical meaning. The vector qcan

be interpreted as the uplink power vector. Now, the idea is to find an alternating

optimization scheme that keeps one of the variables Uand qfixed while minimizing

with respect to the other.

3. Joint Beamforming in the Absence of Noise

Theorem 8. For a fixed beamforming matrix ˆ

Uthe objective function ˆ

f(ˆ

U,q)is

minimized by the dominant eigenvalue of DΨT(ˆ

U).

Proof. From (3.10) we know that the minimum of the objective function is min

q>0

f(ˆ

U,q) =

λmaxDΨT(ˆ

U). Let ˆqbe a solution of

DΨT(ˆ

U)·ˆq=λmaxDΨT(ˆ

U)·ˆq.(3.12)

Then the following relation holds for any x>0:

xTDΨT(ˆ

U)ˆq

xTˆq=λmaxDΨT(ˆ

U).

This can be shown by left-hand multiplying both sides of (3.12) with xT/xTˆq. Hence,

f(ˆ

U,ˆq) = λmaxDΨT(ˆ

U).

Theorem 9. For a fixed power allocation ˜qthe objective function ˆ

f(U,˜q)is mini-

mized by ˜

U= [˜u1,...,˜uK], where ˜uiis the dominant eigenvector of the generalized

eigenproblem

Riui=λmaxQi(˜q)ui(3.13)

with Qi(q) =

k=1

k6=i

[q]kRk.(3.14)

Proof. There exists an x0>0such that

f(U,˜q) = max

x>0

xTDΨT(U)˜q

xT˜q=xT

0DΨT(U)˜q

0˜q

i=1

[x0]iγi

iQi(˜q)ui

iRiui

.(3.15)

Observe that the minimization with respect to U= [u1,...,uK]leads to a set of K

decoupled problems. The optimizers can equivalently be found by maximizing the

reciprocal value, i.e.,

˜ui= arg max

iRiui

iQi(˜q)ui

subject to kuik2= 1 ,1≤i≤K . (3.16)

Hence, the ith beamforming vector ˜uiis given by the dominant generalized eigenvector

of the matrix pair Ri,Qi(˜q).

3. Joint Beamforming in the Absence of Noise

Note, that (3.16) maximizes the ratio between the desired signal power of the ith

user and the interference received by all other users, which provides an additional

interpretation. Interestingly, the SIR requirements contained in Dplay no role for

the beamforming optimization. However, they are used for each power allocation

step, thus they indirectly influence the optimal beamforming solution.

With (3.10) and Theorem 9 we are able to prove the following necessary and

sufficient condition for global optimality:

Theorem 10. A beamforming matrix ¯

Usolves the eigenvalue minimization problem

(3.7) if and only if for all x>0

xTDΨT(¯

U)¯q

xT¯q= min

xTDΨT(U)¯q

xT¯q,(3.17)

where ¯qfulfills DΨT(¯

U)·¯q=λmaxDΨT(¯

U)·¯q.(3.18)

Proof. See the appendix on page 75. 

It can be concluded from Theorem 10 that the beamformers Uopt and transmission

powers qopt that solve the uplink SIR balancing problem (3.6) are always linked in

the special way described by (3.17) and (3.18). Hence, the optimum is completely

determined if either qopt and Uopt is known. This will be exploited in the next section.

3.1.3. Subspace Characterization of the Optimal

Beamforming Solutions

Suppose that the power allocation qopt solves the SIR balancing problem (3.6). The

optimal beamforming solution that is associated with qopt need not be unique. The

set of possible solutions is denoted as M∞. From Theorem 10 we know that an

optimal beamformer Uopt fulfills

DΨT(Uopt)·qopt =λopt

max ·qopt ,(3.19)

where λopt

max

def

= 1/BUL

Inf . The set of equations (3.19) can be rewritten as

(uopt

i)HΩi(λopt

max,qopt)uopt

i= 0 ,1≤i≤K , (3.20)

where the functions Ωiare defined as

Ωi(λ, q) = λ qiRi−γi

k=1

k6=i

qkRk,1≤i≤K . (3.21)

It can be observed from (3.20) that a solution Uopt ∈ M∞exists as long as the

matrices Ωi(λopt

max,qopt)are singular. This provides a new criterion for feasibility.

Unlike the criterion used in Section 2.3.1, this criterion is not based on a given

set of beamforming weights. Instead, feasibility is characterized for a given power

allocation.

3. Joint Beamforming in the Absence of Noise

Theorem 11. The constraints SINRUL

i≥γi,1≤i≤K, are feasible if and only if

there exist λ < 1and ˜q>0such that the matrices Ωi(λ, ˜q)are singular. Then, each

˜uichosen from the nullspace of Ωi(λ, ˜q)fulfills SIRUL

i(˜ui,˜q)> γi.

Proof. See the appendix on page 76. 

From Theorem 11 some useful properties can be derived:

Corollary 1. An alternative strategy for solving the eigenvalue optimization problem

(3.6) is as follows. Let Ωi(λ, q)be singular matrices as defined in (3.21), then

BUL

Inf

= min

q,λ λsubject to singular matrices Ωi(λ, q),1≤i≤K , (3.22)

which provides the smallest positive λfor which representation (3.20) holds true.

Corollary 2. Let (Uopt,qopt)be a minimizer of (3.6), then Uopt is unique if and

only if the matrix Ωi(1/BUL

Inf ,qopt)has a one-dimensional null space.

Corollary 3. If the balancing problem (3.6) has two distinct solutions UI= [uI

1,...,uI

and UII = [uII

1,...,uII

K], then ˜uk=µuI

k+ (1 −µ)uII

kwith 0<µ<1is also a solu-

tion. All solutions are associated with the same power allocation.

3.2. A Globally Convergent Algorithm

The analytical results in Section 3.1.2 motivate an alternating optimization strategy

for problem (3.6). By keeping either Uor qfixed and minimizing ˆ

f(U,q)with

respect to the other variable, one can expect to approach a minimum. The required

optimization steps are provided by Theorems 8 and 9. This motivates Algorithm 1,

which was first described in [43], but without proof of convergence. A complete

convergence analysis will be provided in the next section. The result was published

by Schubert and Boche in [5].

3.2.1. Monotony and Global Convergence

Next, monotony and global convergence of Algorithm 1 will be proven. This is

perhaps surprising, since the objective function is non-convex and may have local

minima. Similar strategies based on alternating variables are known to have very

poor convergence properties and may even get stuck before reaching a minimum.

However, the following results show that this is not the case for Algorithm 1, which

exhibits excellent convergence behavior. A key step in the proof is Theorem 10, which

shows that each local minimum of the objective ˆ

f(U,q)is a global minimum. Hence,

it remains to prove convergence.

3. Joint Beamforming in the Absence of Noise

Algorithm 1 Solution of the eigenvalue minimization problem (3.7)

1: n⇐0

2: q(0) ⇐[1,...,1]T

3: λmax(0) ⇐0

4: repeat

5: n⇐n+ 1

6: u(n)

i⇐arg max

iRiui

iQi(q(n−1))ui

,subject to kuik2= 1,1≤i≤K

7: λmax(n),q(n)⇐solution of DΨT(U(n))·q(n)=λmax(n)·q(n)

8: until |λmax(n)−λmax(n−1)| ≤ 

Theorem 12. The iteration sequence λmax(n)provided by Algorithm 1 is monoton-

ically decreasing and converges towards the global optimum of the eigenvalue mini-

mization problem (3.7).

Proof. Define

λn(x)def

=xTDΨT(U(n))q(n−1)

xTq(n−1) ,x>0.(3.23)

By the same argumentation as in the proof of Theorem 9, it can be shown that U(n)

is a minimizer of xTDΨT(U)q(n−1) =

i=1

[x]iγi

iQi(q(n−1))ui

iRiuifor any x>0. Thus

λn(x)≤xTDΨT(U(n−1))q(n−1)

xTq(n−1) =λmax(n−1) .(3.24)

This follows directly from DΨT(U(n))·q(n)=λmax(n)·q(n). Combining (3.10) and

(3.23), we have

λmax(n) = max

x>0min

y>0

xTDΨT(U(n))y

xTy

≤max

x>0

xTDΨT(U(n))q(n−1)

xTq(n−1) = max

x>0

λn(x).(3.25)

Likewise, it can be shown that λmax(n)is lower bounded:

λmax(n) = min

x>0max

y>0

xTDΨT(U(n))y

xTy

≥min

x>0

xTDΨT(U(n))q(n−1)

xTq(n−1) = min

x>0

λn(x).(3.26)

Combining (3.24), (3.25), and (3.26) we have

min

x>0

λn(x)≤λmax(n)≤max

x>0

λn(x)≤λmax(n−1) .(3.27)

3. Joint Beamforming in the Absence of Noise

Hence, the iteration sequence λmax(n)is monotonically decreasing and there exists a

limit value λ∞def

= limn→∞ λmax(n). Using the fact that the quantities q(n)and U(n)

are both bounded, there must exist limit values q∞and U∞such that

lim

n→∞ kU(n)−U∞kF= 0

lim

n→∞ kq(n)−q∞k1= 0 .

Since Ψ(U)is continuous, we have

lim

n→∞ kDΨT(U(n))−DΨT(U∞)kF= 0

and therefore

lim

n→∞ DΨT(U(n))q(n)=DΨT(U∞)q∞

lim

n→∞ λmax(n)q(n)=λ∞q∞.

Consequently, λ∞is a solution of DΨT(U∞)q∞=λ∞q∞and

U∞= arg min

xTDΨT(U)q∞

xTq∞.

From Theorem 10 we know that U∞fulfills the condition for global optimality. Thus

it solves the `∞-norm minimization problem (3.4). 

3.2.2. Stopping Criterion

The above results show that the iteration sequence is strictly monotonically decreas-

ing as long as the global optimum is not found. A stopping criterion is needed

that allows to control the desired accuracy by defining a threshold . Obviously,

λmax(n−1) −λmax(n)≤is such a criterion. Necessary and sufficient conditions for

the optimality of a solution U(n)are provided in the following.

Firstly, a solution U(n)solves the eigenvalue minimization problem (3.7) if and

only if λmax(n) = λmax(n+ 1), which follows directly from Theorem 10. This crite-

rion, however, requires the computation of U(n+1), which means that an additional

iteration step has to be carried out. It is therefore desirable to assess the optimality

of U(n)directly, without the need of additional eigenvalue decompositions.

Theorem 13. U(n)is a global minimizer of the eigenvalue minimization problem

(3.7) if and only if ˜

λn(x) = λmax(n)for any x>0.

Proof. See the appendix on page 77. 

3. Joint Beamforming in the Absence of Noise

Theorem 13 implies that the global optimum is achieved if and only if maxx˜

λn(x) =

minx˜

λn(x). If this is fulfilled then λmax(n) = λmax(n+ 1). The converse also holds,

thus both conditions are equivalent.

Lemma 11. Let x,b, and cbe positive, K-dimensional vectors, then the following

two equalities hold:

max

xTb

xTc= max

1≤k≤K

[b]k

[c]k

(3.28)

min

xTb

xTc= min

1≤k≤K

[b]k

[c]k

.(3.29)

Proof. See the appendix on page 70. 

Applying Lemma 11 to the function ˜

λn(x), as defined in (3.23), and using the

monotony result (3.27), we have

min

1≤k≤Kγk

SIRUL

k(q(n−1),u(n)

k)≤λmax(n)≤max

1≤k≤Kγk

SIRUL

k(q(n−1),u(n)

k)≤λmax(n−1).

Combining Lemma 11 with Theorem 13, it becomes clear how the optimality of a

solution U(n)is related to the SIR balancing problem (3.6). The global optimum

is achieved if and only if all relative SIR are balanced at the same level after the

beamforming step. Note, that the upper bound for λmax(n)is always smaller than

λmax(n−1). This explains the rapid convergence of the algorithm. Typically, only a

few iteration steps are required. The convergence behavior is illustrated in Fig. 3.1.

step 1 step 2 step 3

λmax(n)

min

1≤k≤Kγk

SIRUL

k(q(n−1),u(n)

k)

max

1≤k≤Kγk

SIRUL

k(q(n−1),u(n)

k)

Figure 3.1.: Typical convergence behavior of the proposed Algorithm 1

3.3. Alternative Strategy based on `1-Norm

Minimization

Gerlach and Paulraj [10] proposed a different strategy for solving the downlink SIR

balancing problem (3.2). Instead of minimizing the non-differentiable objective func-

tion kξ(U,p)k∞defined in (3.3), they proposed to minimize the `1-norm kξ(U,p)k1

3. Joint Beamforming in the Absence of Noise

instead. In general, this approach has the advantage that one has a smoother objec-

tive

F1(U,p) = 1

Kkξ(U,p)k1=1

i=1

piuH

iGi(p)ui,(3.30)

where Gi(p) =

k=1

k6=i

γkRk

.(3.31)

The following optimization problem has been considered in [10]:

B1= min

U,pF1(U,p)subject to uiRiui= 1,1≤i≤K ,

kpk1= 1 .(3.32)

This will now be compared with the `∞approach

BDL

Inf = max

U,pmin

1≤i≤K

SIRDL

i(U,p)

γisubject to uiRiui= 1,1≤i≤K ,

kpk1= 1 .

(3.33)

Note, that both (3.32) and (3.33) pose quadratic constraints on the beamforming

vectors ui, whereas the `∞approach (3.2) studied in Section 3.1 uses a different

constraint kuik2= 1. This, however, will not affect the level on which the relative

SIR are be balanced, but only the transmission powers and the scaling of the vectors

ui. Quadratic constraints will be assumed exclusively in the remainder of this section

in order to compare the `∞and `1based strategies proposed in [10].

In the following we denote M1the set of beamforming matrices Uthat solve

(3.32). Also (ˆ

U,ˆp)is a global minimizer of (3.32), thus B1=F1(ˆ

U,ˆp). The set of

optimizers of the `∞problem (3.33) is denoted by M∞.

It was conjectured in [10] that each ˆ

U∈ M1solves the original SIR balancing

problem (3.33), i.e., M1⊆ M∞. Now, an interesting question is, whether or not this

is fulfilled. This was not ultimately answered in [10]. In order to find an answer, we

start by characterizing the optimum of the `1problem. The `∞problem was already

studied in Section 3.1.

3.3.1. Characterization of the `1Minimizer

We start by assuming a fixed beamforming matrix U. The power vector pthat

minimizes the cost function F1(U,p)is characterized by the following lemma:

Lemma 12. For any fixed Uthe optimization problem minpF1(U,p)has a unique

solution ˜p= [˜p1,...,˜pK]Twith k˜pk1= 1. This solution is characterized by

˜pi

k=1

k6=i

˜pkuH

kγiRiuk

˜piuH

iK

k=1

k6=i

γkRk

˜pkui

.(3.34)

3. Joint Beamforming in the Absence of Noise

Proof. See the appendix on page 71. 

Next, assume that pis the given parameter and that F1(U,p)is minimized with

respect to U. From (3.30) it can be observed that this leads to a set of Kdecoupled

problems

min

iGi(p)ui,s.t. uH

iRiui= 1,1≤i≤K . (3.35)

The optimizers are given as the dominant generalized eigenvectors of the matrix

pairs (Ri,Gi(p)). With these results we are able to characterize the optimal solution

(ˆ

U,ˆp)that solves (3.32):

Theorem 14. The global minimizer (ˆ

U,ˆp)of the cost function F1(U,p)is charac-

terized by the following two equations:

Riˆui=λmaxRi,Gi(ˆp)·Gi(ˆp)ˆui,1≤i≤K , (3.36)

ˆpi

k=1

k6=i

ˆpkˆuH

kγiRiˆuk

ˆpiˆuH

iGi(ˆp)ˆui

,1≤i≤K . (3.37)

Proof. This can be shown by combining (3.35) and Lemma 12. 

3.3.2. Optimality Conditions

Next, necessary and sufficient conditions will be derived for the optimality of the `1

solution (ˆ

U,ˆp)with respect to the original `∞problem (3.33). Then, a necessary and

sufficient condition will be derived for the optimality of the beamforming solution ˆ

alone.

The optimality of ˆ

Uwas conjectured in [10], where it was proposed to solve (3.32),

then to solve an eigenproblem

ΓΨ(ˆ

U)pg=λmaxΓΨ(ˆ

U)pg,(3.38)

where Γ= diag{γ1,...,γK}. Note, that the power allocation pgbalances the quan-

tities SIRDL

i/γiat the same level.

Now, consider the `∞solution (Uopt,popt), which also balances all SIRDL

i/γiat the

same level, thus 1/BDL

Inf =λopt

max =F1(Uopt,popt). This optimal level can be compared

with the `1optimum B1=F1(ˆ

U,ˆp)by using the following relation:

B1=1

Kmin

U,p

i=1

γi

SIRDL

i(U,p)

≤1

Kmin

U,p

i=1

max

1≤k≤K

γk

SIRDL

k(U,p)=λopt

max .(3.39)

3. Joint Beamforming in the Absence of Noise

It can be concluded that

B1≤λopt

max ≤λmaxΓΨ(ˆ

U)

F1(ˆ

U,ˆp)≤F1(Uopt,popt)≤F1(ˆ

U,pg).

(3.40)

Using this result the following theorem can be proven:

Theorem 15. The global minimizer (ˆ

U,ˆp)of the cost function F1(U,p)solves the

`∞-problem (3.33), i.e., F1(ˆ

U,ˆp) = F1(Uopt,popt), if and only if one of the following

two equivalent conditions holds true:

ΓΨ(ˆ

U)·ˆp=λmaxΓΨ(ˆ

U)·ˆp,(3.41)

ˆp−1T·ΓΨ(ˆ

U) = ˆp−1T·λmaxΓΨ(ˆ

U).(3.42)

where ˆp−1def

= [ˆp−1

1,...ˆp−1

K]T.

Proof. See the appendix on page 78. 

From Theorem 15 we can conclude that the `1solution solves the `∞problem

if and only if the resulting levels SIRDL

i(ˆ

U,ˆp)/γiare balanced. It was shown in

Section 3.1.1 that this corresponds to the case where ˆpis the right-hand eigenvector

of ΓΨ(ˆ

U). Now, Theorem 15 shows that this is equivalently fulfilled when ˆp−1is the

left-hand eigenvector. Comparison with (3.8) reveals that ˆp−1is the optimal power

allocation of the dual uplink problem. This provides additional insight in how uplink

and downlink transmission powers are related.

Theorem 15 can also be expressed in terms of the quantities B1and λopt

max, which

leads to the following corollary:

Corollary 4. The global minimizer (ˆ

U,ˆp)of the cost function F1(U,p)solves the

`∞-problem (3.33) if and only if B1=λopt

max.

Proof. From (3.39) it can be concluded that B1=λopt

max if and only if the quantities

SIRDL

i(ˆ

U,ˆp)/γiare balanced. Then ˆpis the right-hand eigenvector of ΓΨ(ˆ

U). From

Theorem 15 follows that (ˆ

U,ˆp)solves (3.33). In the reverse direction assume that

(ˆ

U,ˆp)solves the `∞problem (3.33). Theorem 15 implies that ˆpsolves (3.41), from

which we can conclude that λopt

max =B1.

Further insight is gained by multiplying both sides of (3.38) with (ˆp−1)T, i.e.,

(ˆp−1)TλmaxΓΨ(ˆ

U)pg= (ˆp−1)TΓΨ(ˆ

U)pg

i=1

iˆuH

iGi(ˆp)ˆui.(3.43)

3. Joint Beamforming in the Absence of Noise

Multiplying (3.30) by Kand subtracting the result from both sides of (3.43) we

obtain

K

i=1

ˆpiλmaxΓΨ(ˆ

U)−K·B1=

i=1

(pg

i−ˆpi)ˆuH

iG(ˆp)ˆui.(3.44)

It can be observed from (3.44) that for small pg

i−ˆpi, we can approximate

λmaxΓΨ(ˆ

U)≈B1.(3.45)

This provides additional interpretation for Theorem 15 and Corollary 4.

Next, a necessary and sufficient condition for the optimality of the balanced `1

solution (ˆ

U,pg)with respect to the `∞problem (3.33) is provided:

Theorem 16. Let ˆ

U∈ M1. Furthermore, let qgand pgdenote the dominant

left- and right-hand eigenvectors of the matrix ΓΨ(ˆ

U), respectively. Let (qg)−1def

[(qg

1)−1,...,(qg

K)−1]T. The solution (ˆ

U,pg)solves the `∞-problem (3.33) if and only

min

UF1(U,(qg)−1) = F1(ˆ

U,(qg)−1).(3.46)

Proof. See the appendix on page 78. 

If ˆ

Ufulfills the condition in Theorem 16, then the right side of inequality (3.40)

is satisfied with equality. The same, however, need not be true for the left side. That

is, ˆ

Umay be an optimizer of the `∞problem although the quantities SIRi(ˆ

U,ˆp)/γi

are not balanced.

Up to this point, necessary and sufficient conditions have been given for the

optimality of the `1solution (ˆ

U,ˆp)and the balanced `1solution (ˆ

U,pg)with respect

to the `∞optimization problem (3.33). Now we are interested in finding out whether

these problems are actually equivalent or not.

3.3.3. Closed-Form Solution for the 2-User Scenario

For K= 2, the `1minimization problem (3.32) reduces to

B1= min

p1,p2,u1,u2

2γ1p2uH

2R1u2

+γ2p1uH

1R2u1

p2s.t. uiRiui= 1,∀i ,

kpk1= 1 .

(3.47)

It can be observed that the minimization with respect to u1and u2does not depend

on the transmission powers. Each summand of the cost function can be minimized

independently. The minimizers ˆu1,ˆu2are obtained by solving the following two

generalized eigenproblems

R2ˆu2=λ(2)

maxR1ˆu2,(3.48)

R2ˆu1=λ(2)

minR1ˆu1,(3.49)

3. Joint Beamforming in the Absence of Noise

where λ(2)

max and λ(2)

min denote the maximal and minimal generalized eigenvalue of the

matrix pair (R2,R1), respectively. Considering the constraint uH

iRiui= 1, we have

λ(2)

max = 1/ˆuH

2R1ˆu2and λ(2)

min =ˆuH

1R2ˆu1, which can be substituted back into (3.47).

This leads to a new cost function

C(p1, p2) = 1

2 γ1p2

p1λ(2)

max

+γ2p1λ(2)

min

p2!.(3.50)

Since SIRDL

i(U,p)is scalable in p, we can set p= [µ, 1], where µdenotes the

ratio between the transmission powers p1and p2. This leads to a 1-dimensional

optimization problem

B1= min

2 γ1

µλ(2)

max

+µγ2λ(2)

min!.(3.51)

The unique minimizer

ˆµ=sγ1

γ2λ(2)

maxλ(2)

min

(3.52)

is obtained from computing the derivative of the cost function. Putting (3.52) into

(3.51) it can be observed that

SIRDL

1([ˆu1,ˆu2],ˆµ

1)

γ1

=SIRDL

2([ˆu1,ˆu2],ˆµ

1)

γ2

λ(2)

max

γ1γ2λ(2)

min

.(3.53)

The SIR levels are balanced. From the results in Section 3.3.2 we know that in this

case the `1solution solves the optimal `∞problem. Thus, for the 2-user case both

problems are indeed equivalent.

An additional interpretation is obtained by rewriting (3.53) as an eigenvalue de-

composition

ΓΨ([ˆu1,ˆu2]) ·ˆµ1/2

ˆµ−1/2=sγ1γ2λ(2)

min

λ(2)

max ·ˆµ1/2

ˆµ−1/2.

Hence, [ˆµ1/2,ˆµ−1/2]Tis a (scaled) eigenvector of the coupling matrix ΓΨ. Likewise,

it can be shown that [ˆµ−1/2,ˆµ1/2]Tis a dominant eigenvector of (ΓΨ)T. From The-

orem 15 it can be concluded, that the closed-form solution (ˆu1,ˆu2,ˆp1,ˆp2) not only

solves the `1-problem (3.32), but also the `∞-problem (3.33). Thus we have

BDL

Inf = 1/B1=qλ(2)

max/(γ1γ2λ(2)

min).(3.54)

3. Joint Beamforming in the Absence of Noise

Now, assume that SIRDL

i≥γiis required for i= 1,2. These requirements can only

be fulfilled if BDL

Inf ≥1holds. This leads to the following condition

λ(2)

max/λ(2)

min ≥γ1γ2.(3.55)

Thus, the feasibility of a certain scenario, characterized by the channel covariance

matrices R1,R2and SIR requirements γ1, γ2, is determined by (3.55). An illustration

of this result is given in Fig. 3.2.

infeasible

γ1(target SIR user 1)

γ2(target SIR user 2)

feasible

Figure 3.2.: 2-user scenario: tradeoff between SIR requirements γ1and γ2

From (3.55) it becomes clear that we must avoid situations where the ratio

λ(2)

max/λ(2)

min becomes small. Clearly, the worst case is given for R1=R2, where

λ(2)

min =λ(1)

min holds. This corresponds to a situation where the users cannot be distin-

guished by spatial filtering.

3.3.4. Numerical Counterexample for K > 2

Now, consider the general case K > 2. Even when assuming the most simple scenario

K= 3 and M= 1, the `∞problem becomes quite complicated and there is no obvious

way to decide whether or not it is equivalent to the `1approach. Fortunately, optimal

numerical solutions are known for the `1problem (see [10]) and the `∞problem (see

Section 3.2 and the corresponding paper [5]). Thus, together with the optimality

conditions derived in Section 3.3.2, it is possible to find a numerical counterexample

that disproves general equivalence. The results for a specific channel scenario are

illustrated in Fig. 3.3.

From Fig. 3.3.a it can be observed that the SIR levels resulting from (ˆ

U,ˆp)are

not balanced. From Theorem 15 it can be concluded that in this case the `1solution

does not solve the `∞optimization problem (3.33). Hence, both problems are not

equivalent.

Is it possible that the balanced `1solution (ˆ

U,pg)is optimal? From Corollary 4

it is known that this holds true as long as `1and `∞balancing achieves the same

level. A counterexample is depicted in Fig. 3.3.b, where F1(Uopt,popt)< F1(ˆ

U,pg)

holds. Hence, ˆ

Uis not always an optimizer of the `∞optimization problem (3.33).

3. Joint Beamforming in the Absence of Noise

-50 -50

-30 -30

-10 -10

10 10

30 30

50 50

-40 -40

-30 -30

-20 -20

-10 -10

10 10

17.1

12 13.2

22.2

12 13.2

11.6

12 [dB]13.2

11.3

12 13.2

MS 2 MS 4 MS 5MS 3MS1

[dB]

MS 2 MS 3 MS 4 MS 5MS1

array gain [dB]

azimuth angle [deg]

array gain [dB]

azimuth angle [deg]

SIRDL

U,pg)

SIRDL

U,ˆp)

a) b)

SIRDL

i(Uopt,popt)

Figure 3.3.: Comparison of a) `1optimization b) `∞optimization strategies in the

absence of noise. The normalized beam patterns are depicted for a 5-user scenario.

The results have been found with the `1-based algorithm proposed in [10] and

the optimal `∞optimization scheme summarized on p. 30. The same simulation

parameters were assumed for both optimization techniques.

While the observations in Fig. 3.3 only hold for a specific scenario, the statistical

analysis in Fig. 3.4 shows, that the same behavior holds for most other scenarios

as well, including the case K= 3. However, the difference becomes smaller as the

number of users increases. It can be concluded that for most scenarios the `1-based

strategy proposed in [10] only approximates the `∞optimum. In the next section, a

modified algorithm will be proposed that always converges to the `∞optimum.

3.3.5. Optimal Algorithm Based on `1-Optimization

From Theorem 16 it is known that the global optimum of the `∞problem (3.33) is

characterized in two different ways: An optimal solution Uopt ∈ M∞can be found by

minimizing the cost function F1U,(qg)−1, which requires knowledge of the optimal

qg. On the other hand, if Uopt is known, then the associated qgis given as the

dominant left-hand eigenvector of the matrix ΓΨ(Uopt). This provides intuition

for a new alternating optimization scheme, which is based on the `1cost function

F1, as proposed by Gerlach and Paulraj in [10], but which uses a different way of

adjusting the power vector in each iteration step. Instead of minimizing F1(U,p)

with respect to p, the power update is done by solving the eigenvalue problem (3.38).

The algorithm is summarized below. It was published by Boche and Schubert in [4].

The optimal beamforming strategy for given q(n−1) is found by evaluating

3. Joint Beamforming in the Absence of Noise

3 4 5 6 7 8

number of co-channel users

optimal `∞balancing (Alg. 1)

balanced `1norm minimization [10]:

average balanced SIR margin

BUL

Inf =SIRDL

1(Uopt,popt) = ···=SIRDL

K(Uopt,popt)

B1=SIRDL

1(ˆ

U,pg) = ···=SIRDL

K(ˆ

U,pg)

Figure 3.4.: Achievable levels B1and BUL

Inf averaged over 1000 Monte Carlo runs.

Random channel matrices are assumed and γ1=···=γK= 1

U(n)= arg min

UF(U,(q(n−1))−1),s.t. uH

iRiui= 1,1≤i≤K

= arg min

i=1

[x]iuH

iK

k=1

k6=i

[q(n−1)]kγkRkui,for any x∈RK

= arg min

xTΓΨ(U)Tq(n−1)

xTq(n−1) .(3.58)

This strategy is similar to the approach chosen for Algorithm 1 on p. 30. However,

it differs in the way the targets γ1...γKaffect the iteration process. In Algorithm 2,

they influence both beamforming and power control steps, while in Algorithm 1 only

the power control step is involved.

Global convergence of Algorithm 2 can be shown by defining a function

λn(x)def

=xTΓΨ(U(n))Tq(n−1)

xTq(n−1) ,where x>0. (3.59)

It follows from (3.58) that

λn(x) = min

xTΓΨ(U)Tq(n−1)

xTq(n−1)

≤xTΓΨ(U(n−1))Tq(n−1)

xTq(n−1) =λmax(n−1)

3. Joint Beamforming in the Absence of Noise

Algorithm 2 Optimal SIR balancing based on `1optimization.

1: n⇐0

2: q(0) ⇐[1,...,1]T

3: λmax(0) ⇐0

4: repeat

5: n⇐n+ 1

6: for given q(n)minimize the `1cost function

U(n)⇐arg min

UF1(U,(q(n−1))−1),s.t. uH

iRiui= 1,1≤i≤K(3.56)

7: for given U(n)solve the eigenproblem

ΓΨ(U(n))T·q(n)=λmax(n)·q(n).(3.57)

8: until λmax(n−1) −λmax(n)≤

for any x>0. Hence,

λmax(n) = max

x>0min

y>0

xTΓΨ(U(n))Ty

xTy

≤max

x>0

xTΓΨ(U(n))Tq(n−1)

xTq(n−1)

= max

x>0

λn(x)≤λmax(n−1) .(3.60)

The sequence λmax(n)is strictly monotonically decreasing and converges to λopt

max. In

analogy to the proof in Section 3.2.1, it can be shown that the optimality condition

(3.46) is fulfilled after convergence. The resulting beamforming solution Uopt is an

optimizer of the `∞problem (3.33).

3.4. Discussion

Joint beamforming and power allocation in the absence of noise consists of minimizing

the `∞norm of the vector (3.3). This objective function is non-differentiable, so at

first sight it seems unlikely that a globally convergent algorithm can be found. The

equivalent eigenvalue optimization problem has a smoother objective function, but

may still have local minima. So a key result of this chapter is to prove that each

local minimum is a global minimum. This allows to develop a rapidly converging

algorithm, which typically requires only a few iteration steps. These results complete

the studies in [43], where the same problem was studied.

An interesting aspect of the solution is, that it need not be unique. Knowing the

optimal allocation, all optimal beamforming solutions can be found by the subspace

3. Joint Beamforming in the Absence of Noise

characterization proposed in Section 3.1.3. This is practically relevant since it allows

to choose the solution that is best suited, e.g. with respect to additional design

criteria, like the peak to average ratio of the beamforming filters.

Finally, the `1-norm based optimization strategy proposed in [10] is studied. By

providing necessary and sufficient conditions for optimality along with counterexam-

ples, it is shown that this approach is in general not equivalent to the original `∞

norm optimization problem. A modified version of the `1strategy is proposed. This

new algorithm has the same convergence properties as the one above and always

converges to the global `∞optimum.

4. Power-Aware Spatial

Multiplexing under a Sum

Power Constraint

Next, the results are extended to the case where each receiver has a certain noise

level and a sum power constraint is imposed. The goal is to achieve SINR thresholds

γ1...γKby jointly optimizing over all beamforming vectors and transmission powers.

The solution builds on the duality result in Chapter 2 and the interference balancing

technique in the absence of noise, which has been derived in Chapter 3. From the

discussion in Section 2.5 it becomes clear that multiuser channels can always be

scaled, so without loss of generality, σ=1Kcan be assumed in the remainder of this

thesis.

4.1. Joint Beamforming and Power Allocation

4.1.1. SINR Balancing in the Downlink

The design goal min1≤i≤KSINRDL

i≥γimotivates the following optimization strategy:

CDL

opt(Pmax) = max

U,pmin

1≤i≤K

SINRDL

i(U,p)

γi

subject to kpk1≤Pmax

kuik2= 1,1≤i≤K .

(4.1)

The achievable level CDL

opt(Pmax)provides a single performance measure for the quality

of several spatially multiplexed channels. Individual targets γ1...γKcan be achieved

if and only if CDL

opt(Pmax)≥1. Each beamforming solution obtained by (4.1) is optimal

in the sense that no other beamforming algorithm can achieve a balanced level larger

than CDL

opt(Pmax)under the same power constraint. Since CDL

opt(Pmax)is monotonically

increasing in Pmax, the SINR levels SINRDL

i=CDL

opt(Pmax)γiare achieved with minimal

transmission power.

4. Power-Aware Spatial Multiplexing under a Sum Power Constraint

In Section 2.2.1 it was shown that for any fixed Uthe maximal balanced level

CDL(U, Pmax)is given by the reciprocal maximal eigenvalue of the extended coupling

matrix Υ(U, Pmax). Hence, the global optimum CDL

opt(Pmax) = maxUCDL(U, Pmax)is

found by solving

CDL

opt(Pmax) = 1

min

UλmaxΥ(U, Pmax).(4.2)

Any Uopt that solves (4.2) is an optimizer of the SINR balancing problem (4.1).

The eigenvalue minimization problem (4.2) has a reduced dimensionality as com-

pared to the original problem (4.1). Yet, it is still very unlikely to allow for an efficient

and global algorithmic solution because the objective is typically non-convex in U

and may have many local minima.

4.1.2. The Dual Uplink Problem

The same SINR balancing problem that was discussed for the downlink in Sec-

tion 4.1.1 can be formulated for the uplink:

CUL

opt(Pmax) = max

U,qmin

1≤i≤K

SINRUL

i(ui,q)

γi

subject to kqk1≤Pmax

kuik2= 1,1≤i≤K .

(4.3)

Again, this leads to a beamforming solution, which is optimal in terms of the design

goal SINRUL

i≥γi. For given Pmax, a maximal balanced level CUL

opt(Pmax)is achieved.

The associated levels SINRUL

i=CUL

opt(Pmax)γiare achieved with minimal power. The

global optimum CUL

opt(Pmax) = maxUCUL(U, Pmax)is found by solving an eigenvalue

optimization problem

CUL

opt(Pmax) = 1

min

UλmaxΛ(U, Pmax).(4.4)

The relationship between the downlink problem (4.2) and the uplink problem (4.4)

is described by the following corollary, which is an immediate consequence of the

duality observed in Section 7.

Corollary 5. Suppose that strong duality, as defined in Section 2.4.4, holds. Then,

the same balanced level CDL

opt(Pmax) = CUL

opt(Pmax)can be achieved in uplink and down-

link under the same sum power constraint kqk1=kpk1=Pmax. This implies that

each beamforming solution which solves the uplink problem (4.4) also solves the down-

link problem (4.2) and vice versa.

4. Power-Aware Spatial Multiplexing under a Sum Power Constraint

As discussed in Section 2.5, the downlink channel can always be scaled such

that σ=1Kholds. Hence, without loss of generality the downlink beamforming

problem (4.2) can be solved by considering the dual uplink problem (4.4) instead.

This approach is motivated by the results of Chapter 3, where it was observed that

the uplink has a “smoother” analytical structure.

Finally, it is important to notice that the optimizer of (4.4) need not be unique.

Let Mdenote the set of optimal solutions, then Uopt ∈ Msolves the uplink balancing

problem (4.3). Hence, Uopt is characterized by CUL

opt =SINRUL

i(Uopt,qopt)/γi, which

can be rewritten as

(uopt

i)H1

CUL

opt

qopt

iRi−γiZi(qopt)uopt

i= 0 ,1≤i≤K , (4.5)

where Ziis the spatial interference+noise covariance matrix

Zi(q) =

k=1

k6=i

[q]kRk+I,1≤i≤K . (4.6)

In analogy to Theorem 11 it can be shown that all vectors ˜uithat are chosen from

the nullspace of 1

CUL

opt qopt

iRi−γiZi(qopt)are elements of M. This important char-

acterization is discussed in more detail in Section 3.1.3, where the same result has

been derived in the absence of noise.

4.1.3. Problem Reformulation

Using (2.35), the maximal eigenvalue of the extended coupling matrix Λcan be

expressed as

λmaxΛ= max

x>0min

y>0

xTΛy

xTy= min

x>0max

y>0

xTΛy

xTy.(4.7)

Introducing a function

λn(U,y) = max

x>0

xTΛ(U, Pmax)y

xTy,(4.8)

problem (4.4) can be rewritten as

CDL

opt(Pmax) = 1

min

Umin

qext>0

λn(U,qext).(4.9)

Here, qext =q

1is the extended uplink power vector, as introduced in (2.15), whereas

the auxiliary variable x>0has no physical meaning. The global optimum is found

by minimizing the cost function ˆ

λn(U,qext)over all Uand qext >0. At first sight,

4. Power-Aware Spatial Multiplexing under a Sum Power Constraint

the problem (4.9) seems to be more complicated than the original problem (4.4), since

it depends on more variables. However, the representation (4.9) allows for an efficient

algorithmic solution. By keeping either Uor qext fixed and minimizing ˆ

λn(U,qext)

with respect to the other variable, the minimum of (4.9) is approached.

For fixed ˜

U, the cost function ˆ

λn(˜

U,qext)is minimized by the vector ˆqext, which

fulfills

Λ(˜

U, Pmax)·ˆqext =λmaxΛ(˜

U, Pmax)·ˆqext with [ˆqext]K+1 = 1 .(4.10)

This can easily be shown by multiplying both sides of (4.10) by xT∈RK+1

+and

dividing by xTˆqext. Combining (4.7) and (4.8) we have

xTΛ(˜

U, Pmax)ˆqext

xTˆqext

=λmaxΛ(˜

U, Pmax)

= min

qext

λn(˜

U,qext).

Having found the optimal power control strategy, we are now interested in finding

the beamformer ˆ

Uthat is optimal for given ˆqext, i.e.,

U= arg min

λn(U,ˆqext).(4.11)

To this end we need the following lemma:

Lemma 13. The cost function ˆ

λn(U,qext)does not depend on the last row of the

extended coupling matrix Λ, i.e.,

max

x>0

xTΛ(U, Pmax)qext

xTqext

= max

x: [x]K+1=0

xTΛ(U, Pmax)qext

xTqext

= max

1≤i≤K

γi

SINRUL

i(ui,q),where q= [qext]1:K.(4.12)

The same relation holds when replacing max by min.

Proof. See the appendix on page 71. 

A beneficial consequence from Lemma 13 is that minimizing the cost function

λn(U,qext)with respect to U(keeping qext =q

1fixed) leads to a set of Kdecoupled

problems

ˆui= arg max

SINRUL

i(ui,q)

= arg max

iRiui

iZi(q)ui

,1≤i≤K . (4.13)

The matrices Zi, as defined in (4.6), are non-singular and symmetric, thus (4.13) is

solved by the dominant generalized eigenvectors of the matrix pairs Ri,Zi(q),1≤

i≤K.

4. Power-Aware Spatial Multiplexing under a Sum Power Constraint

4.1.4. Necessary and Sufficient Condition for Global

Optimality

Using Lemma 13, a characterization of each global optimizer Uopt ∈ Mcan be found.

The result is summarized by the following theorem, which will prove very useful in

Section 4.2, where an iterative solution will be derived.

Theorem 17. A matrix ˆ

U= [ˆu1,...,ˆuK]is optimal, i.e., ˆ

U∈ M if and only if

there exists a ˆqext such that

λn(ˆ

U,ˆqext) = min

λn(U,ˆqext),(4.14)

and Λ(ˆ

U, Pmax)ˆqext =λmaxΛ(ˆ

U, Pmax)ˆqext .(4.15)

Proof. Assume that (4.14) and (4.15) hold. Let x∈RK+1

+be an arbitrary vector.

Left-hand multiplication on both sides of (4.15) with xT/xTˆqext leads to

xTΛ(ˆ

U, Pmax)ˆqext

xTˆqext

=λmaxΛ(ˆ

U, Pmax).(4.16)

The same ˆ

Uthat minimizes ˆ

λn(U,ˆqext) = max

1≤i≤K

γi

SINRUL

i(ui,ˆq)is also a minimizer of

min

1≤i≤K

γi

SINRUL

i(ui,ˆq). It follows from Lemma 13 that

min

x>0

xTΛ(ˆ

U, Pmax)ˆqext

xTˆqext

= min

Umin

x>0

xTΛ(U, Pmax)ˆqext

xTˆqext

.(4.17)

Since (4.16) holds for any x>0, relation (4.17) can be rewritten as

min

Umin

x>0

xTΛ(U, Pmax)ˆqext

xTˆqext

=λmaxΛ(ˆ

U, Pmax).(4.18)

Combining (4.4), (4.7), and (4.18) we have

1/CUL

opt(Pmax) = min

UλmaxΛ(U, Pmax)

= min

Umin

x>0max

y>0

xTΛ(U, Pmax)y

xTy

≥min

Umin

x>0

xTΛ(U, Pmax)ˆqext

xTˆqext

=λmaxΛ(ˆ

U, Pmax).(4.19)

Inequality (4.19) can only be satisfied with equality, since 1/CDL

opt(Pmax)is the global

minimum. From this we can conclude that ˆ

U∈ M holds.

To prove the reverse direction, suppose that ˆ

Uis optimal, i.e., ˆ

U∈ M. This

implies λmaxΛ(ˆ

U, Pmax)≤λmaxΛ(U, Pmax)for all U. Let ˆqext be the dominant

4. Power-Aware Spatial Multiplexing under a Sum Power Constraint

eigenvector of Λ(ˆ

U, Pmax)with ˆqext =ˆq

1and kˆqk1=Pmax. With (4.7) and (4.8)

the following inequality is obtained:

λmaxΛ(ˆ

U, Pmax)≤λmaxΛ(U, Pmax)

= min

y>0max

x>0

xTΛ(U, Pmax)y

xTy

≤max

x>0

xTΛ(U, Pmax)ˆqext

xTˆqext

=ˆ

λn(U,ˆqext).(4.20)

This holds for any U. Thus, using (4.16) it follows that

min

λn(U,ˆqext)≥λmaxΛ(ˆ

U, Pmax)

= max

z>0

zTΛ(ˆ

U, Pmax)ˆqext

zTˆqext

=ˆ

λn(ˆ

U,ˆqext).(4.21)

Inequality (4.21) can only be fulfilled with equality, which concludes the proof. 

4.2. Global Optimization with Alternating Variables

The method of alternating variables in its simplest form keeps all but one variable

fixed and seeks a value of this variable so as to minimize the objective function. This is

repeated successively for all variables. In general, this strategy is know to approach

the optimum very slowly and may even get stuck. The variant proposed in this

section, however, exploits the analytical structure of the given problem and therefore

rapidly converges towards the global optimum of the eigenvalue minimization problem

(4.4).

4.2.1. Algorithm Summary

In Section 4.1.4 it was shown how the objective function ˆ

λn(U,qext)is minimized

when either Uor qext is kept fixed. Also, a necessary and sufficient condition for

global optimality is provided by Theorem 17. This motivates the alternating algo-

rithm summarized below. The quantities associated with the nth iteration step are

denoted by the superscript (·)(n).

4.2.2. Monotony and Global Convergence

Theorem 18. The iteration sequence λmax(n)obtained from Algorithm 3 is monoton-

ically decreasing and converges to the global optimum of the eigenvalue minimization

problem (4.4), regardless of the chosen initialization.

4. Power-Aware Spatial Multiplexing under a Sum Power Constraint

Algorithm 3 Algorithmic solution of the eigenvalue minimization problem (4.4)

1: n⇐0

2: q(0)

ext ⇐[0,...,0,1]T

3: repeat

4: n⇐n+ 1

5: for given q(n−1)

ext compute U(n)= [u(n)

1,...,u(n)

K]as the solution of

u(n)

i= arg max

iRiui

iZi(q(n−1)

ext )ui

,1≤i≤K(4.22)

6: for given U(n)compute q(n)

ext as the solution of

Λ(U(n), Pmax)q(n)

ext =λmax(n)q(n)

ext ,with [q(n)

ext]K+1 = 1

7: until λmax(n−1) −λmax(n)< 

Proof. From Lemma 13 follows that for given q(n−1)

ext , the solution U(n)minimizes the

cost function ˆ

λn(U,q(n−1)

ext ), i.e,

λn(U(n),q(n−1)

ext )≤ˆ

λn(U(n−1),q(n−1)

ext ) = λmax(n−1) .(4.23)

Using (4.7) and (4.23) we have

λmax(n) = min

y>0max

x>0

xTΛ(U(n), Pmax)y

xTy

≤ˆ

λn(U(n),q(n−1)

ext )

≤λmax(n−1) .(4.24)

Hence, the iteration sequence λmax(n)is monotonically decreasing. Using (4.7), it

can be shown that the quantity λmax(n)is lower bounded:

λmax(n) = max

y>0min

x>0

xTΛ(U(n), Pmax)y

xTy

≥min

x>0

xTΛ(U(n), Pmax)q(n−1)

ext

xTq(n−1)

ext

.(4.25)

Combining (4.12), (4.24), and (4.25) we have

min

1≤i≤K

γi

SINRUL

i(u(n)

i,q(n−1))≤λmax(n)≤max

1≤i≤K

γi

SINRUL

i(u(n)

i,q(n−1))≤λmax(n−1) .

(4.26)

4. Power-Aware Spatial Multiplexing under a Sum Power Constraint

max

1≤i≤K

SINRUL

i(u(n)

i,q(n−1))

γi

CUL(U(n), Pmax)=1/λmax(n)

min

1≤i≤K

SINRUL

i(u(n)

i,q(n−1))

γi

step 1 step 2 step 3

Figure 4.1.: Typical convergence behavior of Algorithm 3

The monotony result (4.26) implies the existence of a limit λ∞= limn→∞ λmax(n)

(an illustration is given in Fig. 4.1). Using the continuity of Λ(U), as well as the

fact that both kq(n)

extk1and kU(n)kFare bounded, it can be shown in the same way

as in Section 3.2.1, that λ∞is a solution of Λ(U∞)q∞

ext =λ∞q∞

ext. Thus,

min

λn(U,q∞

ext) = ˆ

λn(U∞,q∞

ext).(4.27)

From Theorem 17 we know that (4.27) is necessary and sufficient for the optimality

of U∞.

4.2.3. Stopping Criteria

Next, stopping criteria are derived in analogy to the noiseless case (see Section 3.2.2).

A solution U(n)is a global optimizer of the eigenvalue minimization problem

(4.4) if and only if λmax(n) = λmax(n+ 1). This is an immediate consequence from

Theorem 17. Thus, the algorithm can be stopped as soon as the difference λmax(n)−

λmax(n+ 1) falls below a threshold .

A drawback of this approach is that the quantity U(n+1) must be computed in

order to assess the optimality of U(n), which means that an additional iteration step

must be carried out. A more direct stopping criterion can be found by considering

the ratios SINRUL

i(U(n),q(n−1))/γi, where q= [qext]1:K. If they are balanced at the

same level, the global optimum is reached.

Theorem 19. A solution U(n)is optimal with respect to (4.4), i.e., U(n)∈ M, if

and only if

min

1≤i≤K

SINRUL

i(U(n),q(n−1))

γi

= max

1≤i≤K

SINRUL

i(U(n),q(n−1))

γi

.(4.28)

Proof. See the appendix on page 80. 

4. Power-Aware Spatial Multiplexing under a Sum Power Constraint

Theorem 19 provides an additional interpretation in terms of SINR balancing.

As long as the optimum is not reached, the quantity CUL(U(n), Pmax)=1/λmax(n)

is lower and upper bounded as in (4.26). The iteration process can be stopped as

soon as the difference between the relative SINR levels in (4.28) is smaller than a

threshold .

4.3. Minimizing Excess Transmission Power

If CUL

opt(Pmax)>1, then there are excess degrees of freedom which can be used to

minimize the total transmission power:

min

U,q

i=1

qisubject to SINRUL

i≥γi,1≤i≤K

kuik2= 1,1≤i≤K .

(4.29)

In Section 2.2.2 it was shown that (4.29) can be seen as a special case of the SINR

balancing problem (4.1). Thus, the global optimum (4.29) can be found by extending

Algorithm 3.

4.3.1. Iterative Power Minimization

The first step for solving the power minimization problem (4.29) is to make sure

that the constraints SINRUL

i≥γican be fulfilled. As discussed in Section 2.2,

this is exactly the case when there exists a beamforming solution U(n)such that

CUL(U(n), Pmax)≥1(the function CUL is defined by (2.10) and illustrated in Fig. 2.6).

The problem is infeasible if and only if Algorithm 3 yields a solution CUL(U(n), Pmax)<

1for n→ ∞. Such a situation should be avoided by proper countermeasures taken

by the dynamic resource management.

As soon as it turns out that the problem is feasible, the global power minimum

(4.29) can be found by changing the power control policy. Instead of computing

the power allocation which maximizes the balanced ratio SINRUL

i/γifor given Pmax,

we are now interested in finding the allocation that minimizes the total tranmis-

sion power while satisfying SINRUL

i/γi= 1. This problem was already discussed in

Section 2.2.2. The solution is found by solving the set of linear equations (2.18).

These steps are repeated until convergence. The algorithm is summarized below and

illustrated in Fig. 4.2.

Theorem 20. Suppose that the power minimization problem (4.29) is feasible, then

Algorithm 4 approximates the global power minimum up to a desired accuracy.

Proof. Suppose that (4.29) is feasible. In Section 2.2.2 it was shown that for given

U(n)the allocation q(n)=I−DΨT(U(n))−1Dσ achieves the SINR targets with

4. Power-Aware Spatial Multiplexing under a Sum Power Constraint

Algorithm 4 Algorithmic solution of the power minimization problem (4.29)

1: initialize: n⇐0,q(0) ⇐[0,...,0]T,C(0) ⇐0

2: repeat

3: n⇐n+ 1

4: U(n)= [u(n)

1,...,u(n)

K]⇐arg max

iRiui

iZi(q(n−1))ui,1≤i≤K

5: if C(n)<1then

6: P(n)

sum ⇐Pmax

7: solve Λ(U(n), P(n)

sum)q(n)

1=λmax(n)q(n)

1

8: C(n)⇐1/λmax(n)

9: else

10: q(n)⇐I−DΨT(U(n))−1

Dσ

11: P(n)

sum ⇐ kq(n)k1

12: C(n)⇐1

13: end if

14: until max

1≤i≤KSINRUL

i(u(n)

i,q(n−1))−min

1≤i≤KSINRUL

i(u(n)

i,q(n−1))< 

minimal transmission power. Hence, P(n)

sum ≤P(n−1)

sum , as illustrated in Fig 4.2. The

algorithm converges towards a solution P(∞)

sum. That is, for a given sum power P(∞)

sum, the

beamforming step achieves no further improvement. With the results in Section 4.2

it can be concluded that

CUL(U(∞), P(∞)

sum) = max

UCUL(U, P(∞)

sum).(4.30)

Now, assume that there exists a beamforming solution ˜

Uwhich fulfills the SINR

constraints with a minimal power level ˜

P < P(∞)

sum. Since CUL is strictly monotonically

increasing in the total power, this implies CUL(˜

U, P(∞)

sum)> CUL(U(∞), P(∞)

sum), which

is a contradiction to (4.30). 

global

power minimum

initialization (infeasible)

feasible

sum power

CUL(U, Psum)

U(1)

U(2)

U(3)

U(4)

U(∞)

Pmax

P(3)

sum

P(4)

sum

P(∞)

sum

Figure 4.2.: Schematic illustration of Algorithm 4 (iterative power minimization)

4. Power-Aware Spatial Multiplexing under a Sum Power Constraint

If the power minimization problem (4.29) is not feasible, then convergence is

achieved with C(∞)<1. Then, the algorithm does not differ from the SINR balancing

algorithm in Table 3. As discussed before, C(∞)≥1is a necessary and sufficient

condition for feasibility required by the dynamic resource allocation.

4.3.2. Closed-Form Solution for the 2-User Scenario

Consider a 2-user scenario with perfect channel information Ri=hihH

i,i= 1,2,

and equal noise levels σ2

1=σ2

2= 1. The goal is to balance the quantities CUL

SINRUL

i(ui, q1, q2)/γi,i= 1,2, with minimal transmission power. The first user

achieves a level

CUL

1=SINRUL

1(u1, q1, q2)

γ1

=q1|uH

1h1|2

γ1(q2|uH

1h2|2+ 1) .(4.31)

This is maximized by the normalized MMSE solution

uopt

1=q2h2hH

2+I−1h1

kq2h2hH

2+I−1h1k2

.(4.32)

Putting (4.32) in (4.31) we have

CUL

1=q1hH

1q2h2hH

2+I−1h1

γ1

Using the Sherman-Morrison inversion lemma (see e.g. [46]) this can be rewritten as

CUL

1=q1

γ1hH

1h1−q2|hH

1h2|2

1 + q2hH

2h2.(4.33)

Likewise, the level of the second user is given by

CUL

2=q2

γ2hH

2h2−q1|hH

2h1|2

1 + q1hH

1h1.(4.34)

Now, assume that both users are balanced at the same level CUL

1=CUL

2= 1. Solving

(4.33) for q1, then putting q1in (4.34), then solving for q2yields

q2=−a2+pa2

2+ 4b2c2

2c2

(4.35)

where a2=kh2k2kh1k2(1 + γ1−γ2−γ1γ2) + γ2|hH

1h2|2−γ1|hH

2h1|2

b2=γ2kh1k2(1 + γ1)

c2=kh2k4kh1k2(1 + γ1)−kh2k2|hH

1h2|2−γ1kh2k2|hH

2h1|2.

4. Power-Aware Spatial Multiplexing under a Sum Power Constraint

Likewise, we can derive a closed form expression for user 1:

q1=−a1+pa2

1+ 4b1c1

2c1

(4.36)

where a1=kh1k2kh2k2(1 + γ2−γ1−γ2γ1) + γ1|hH

2h1|2−γ2|hH

1h2|2

b1=γ1kh2k2(1 + γ2)

c1=kh1k4kh2k2(1 + γ2)−kh1k2|hH

2h1|2−γ2kh1k2|hH

1h2|2.

The power levels (4.35) and (4.36) achieve the targets γ1and γ2with minimum total

transmission power Pmax =q1+q2. The optimal beamformer uopt

1is obtained from

(4.32). Likewise, uopt

2can be computed. With the duality stated in Theorem 6 it

is known that uopt

1and uopt

2are optimal for both uplink and downlink. Finally, the

optimal downlink powers can be found by the power control strategy discussed in

Section 2.2.1.

4.4. Discussion

In this chapter the problem of joint beamforming and power control under a sum

power constraint has been solved. A generic solution has been presented, which

shares the available power resource in a fair manner, taking into account individual

SINR targets. This technique can be used in two different ways:

1. by maximizing the balanced SINR margin CUL(U, Pmax)while maintaining a

sum power constraint kqk1≤Pmax,

2. by minimizing the total power kqk1while ensuring target SINR’s γ1...γK.

Both design goals are closely linked. If the latter is infeasible, this can be detected by

the necessary and sufficient condition provided by Algorithm 3. Infeasible scenarios

must be handled by proper resource management.

Given an optimal beamforming solution, an optimal power allocation follows di-

rectly from (4.15). On the other hand, an optimal beamforming solution is obtained

from (4.14) if an optimal power allocation is known. This fundamental relationship

will prove useful in the next chapter in order to find the maximum throughput of the

Gaussian multiuser channel.

The solution derived in this chapter is general enough to hold for spatial covariance

matrices of arbitrary rank. This is important, since perfect spatial channel knowledge

is rarely available, due to channel fluctuations in time. Incoherent beamforming

can be performed based on knowledge of the spatial covariance matrices that are

obtained by averaging over several fading cycles (see Section 2.1.1). Such a “long-

term” approach is robust due to the slowly changing geometry of the multipath

directions. The rapid convergence behavior of the algorithm makes this approach

suitable for real time applications.

5. Rate Balancing for Gaussian

Broadcast and Multiple Access

Channels

In this chapter we study the SINR balancing problem in an information theoretical

context with i.i.d. Gaussian signaling under a sum power constraint. Assuming

perfect channel knowledge at the base station, the multiplexed data streams can be

decoupled by a combination of linear equalization and successive coding for known

interference. This leads to a triangularized channel structure, which is a special

case of the general scenario considered so far. As will be shown in this chapter,

the triangular structure allows for an efficient solution based on back substitution.

By defining threshold targets, rate tuples can be controlled according to individual

capacity needs. Such a strategy is required when strict delay constraints are imposed

and throughput has to be traded off against fairness.

The results presented in this chapter were published by Boche and Schubert in

[1, 2].

5.1. Duality between Broadcast and Multiple

Access Channel

5.1.1. Multiple Access Channel with Successive Decoding

We start by considering a Multiple Access Channel (MAC) with Kscalar inputs

and a vector output. The Kindependent transmitters use i.i.d. Gaussian random

codebooks. The Mreceive antennas are jointly processed. From the discussion in

Section 2.5 it is clear that a noise vector σ=1Kcan be assumed without loss of

generality.

An optimal receiver structure for the MAC has been found in [35], where it was

shown that the vertices of the capacity region can be achieved by using a combination

of successive decoding and linear MMSE receivers uMMSE

i, i = 1,...,K. Assuming a

5. Rate Balancing for Gaussian Broadcast and Multiple Access Channels

decoding order K . . . 1, the following signal is obtained at the ith cancellation step:

yi(t) = si(t)uH

ihi+

i−1

k=1

sk(t)(uMMSE

i)Hhk+ni(t),1≤i≤K . (5.1)

This leads to “coded” SINR’s

SINRUL,SD

i(uMMSE

i, q1...qi) = qi|(uMMSE

i)Hhi|2

(uMMSE

i)Hˆ

Zi(q1...qi−1)uMMSE

=qihH

iˆ

Z−1

ihi.(5.2)

where uMMSE

i=ˆ

Z−1

ihi/kˆ

Z−1

ihik2and

Zi(q1...qi−1) = I+

i−1

k=1

qkhkhH

k.(5.3)

There is a one-to-one correspondence between (5.2) and the maximal spectral effi-

ciency

RUL

i= log2|qihihH

i+ˆ

Zi|−log2|ˆ

Zi|

= log2|I+qiˆ

Z−1/2

ihihH

iˆ

Z−1/2

= log2(1 + qihH

iˆ

Z−1

ihi)

= log2(1 + SINRUL,SD

i(uMMSE

i, q1...qi)) .(5.4)

For simplicity we will also refer to Rias the rate in the following.

Hence, data rates can be controlled by jointly adjusting transmission powers and

MMSE beamformers. By defining target thresholds γ1...γK, the SINR balancing

strategy developed in Chapter 4 can be used to achieve rate tuples within the capacity

region, including the maximal sum capacity. Provided that γ1...γKare achievable

under the given power constraint, the optimal power allocation and the optimal

MMSE filters can be found by solving the power minimization problem

min

q,U

i=1

qisubject to SINRUL,SD

i(ui, q1,...,qi)≥γi,1≤i≤K . (5.5)

Note, that each optimizer (Uopt,qopt)of the joint power minimization problem (5.5)

is characterized by Theorem 17 (with Lemma 13) in two different ways:

1. Given qopt, the optimal beamformer Uopt is found by Kindependent SINR

maximization problems.

2. Given Uopt, the optimal power allocation qopt can be found by the power control

strategies described in Section 2.2.

Unlike the general problem treated in Chapter 4, problem (5.5) always feasible in

the absence of power constraints. This is due to the triangular channel structure

imposed by the successive decoding (see the discussion in Section 2.3 and the example

in Fig. 2.5).

5. Rate Balancing for Gaussian Broadcast and Multiple Access Channels

5.1.2. Broadcast Channel with ‘Dirty Paper’ Precoding

Next, consider a Gaussian broadcast channel (BC) with Mcooperating transmit

antennas and Knon-cooperating receivers. This channel belongs to the class of

non-degraded broadcast channels for which the general capacity region is not known

[49].

A partial solution was recently found by Caire and Shamai [30] for the special

case of two users and two transmit antennas. The result shows that throughput-

wise optimal transmission is possible by using a combination of linear pre-filtering

and coding for non-causally known interference at the transmitter. Such a precoder

triangularizes the channel in a similar way as the decision-feedback equalizer does.

The channel capacity is the same as if the non-causal interference would not exist.

No further power enhancement is caused as long as optimal precoding is assumed.

The existence of such a an optimal precoder was first shown by Costa [50] for a

Gaussian channel with Gaussian interference. More recently, this was expanded to

the case of non-Gaussian interference [51]. Also, strategies for achieving this capacity

were proposed [32, 51]. The principle is very much reminiscent of classical Tomlin-

son/Harashima precoding [52, 53], which can also be expanded to vector broadcast

channels [54].

In this work we make use of these results, i.e., we assume that perfect channel

state information is available at the transmitter and an optimal precoding scheme

exists. We refer to this optimal strategy as “Dirty Paper” precoding after Costa’s

famous title “Writing on Dirty Paper” [50].

Assume a precoding order 1...K. Having chosen a codeword for the first user,

the interference experienced by the users 2...K is known non-causally at the receiver

and can be taken into account when precoding the following users. Repeating this

strategy for all users, a channel with a triangular structure is created (see Fig. 2.5).

The Kreceiving mobiles are non-cooperating and thus independently decoded.

Multi-user interference appears as independent, zero-mean and non-Gaussian addi-

tive noise. It was shown in [55] that in this case the capacity of the ith user is given

RDL

i= log2(1 + SINRDL,CO

i),1≤i≤K , (5.6)

where

SINRDL,CO

i(U,p) = pi|uH

ihi|2

k=i+1

pk|uH

khi|2+ 1

,1≤i≤K . (5.7)

Again, by jointly adjusting the beamformers U= [u1...uK]and transmission power

p= [p1...pK]T, individual capacities can be controlled. Individual targets γ1...γK

can be achieved by solving the optimization problem

min

p,U

i=1

pisubject to SINRDL,CO

i(U,p)≥γi.(5.8)

5. Rate Balancing for Gaussian Broadcast and Multiple Access Channels

As for the MAC, this problem is always feasible in the absence of power constraints,

which is a consequence of the triangular channel structure (see Section 2.3).

5.1.3. Duality

Suppose that both MAC and BC have the same sum power constraint. From Theo-

rem 7 follows that certain SINR targets γ1...γKcan be achieved in the uplink if and

only if the same targets can be achieved in the downlink. The optimization problems

(5.5) and (5.8) have the same global optimum, which is achieved by the same uplink

MMSE beamforming solution, but with different power allocations.

For the Gaussian channel studied here, this duality immediately carries over to

a duality in terms of achievable rates. Hence, the MAC capacity region has a direct

counterpart in the BC. This coincides with recent results [33, 56], where a similar

duality was observed by using a different approach. It is still not clear, however,

whether the “dirty paper” achievable region is actually the capacity region of the BC

or not.

In the following, the theoretical framework of SINR balancing is used to solve the

optimization problems (5.5) and (5.8). The general solution for this power minimiza-

tion problem was already given in Chapter 4. An even more efficient approach will

be derived in the next section, by exploiting the special triangular channel structure

imposed by successive decoding and “dirty paper” precoding.

5.2. Rate Balancing under a Sum Power Constraint

Having shown the duality between the rate balancing problems (5.8) and (5.5), we

can now focus on the multiple access channel, which has a less complicated analytical

structure.

5.2.1. Joint Beamforming and Power Control via

Back-Substitution

Consider the power minimization problem (5.5) with a decoding order K . . . 1. It was

already shown in Section 2.2.2 that the global minimizer (Uopt,qopt)fulfills all SINR

constraints with equality, i.e.,

SINRUL,SD

i(uopt

i, qopt

1...qopt

i) = γi,1≤i≤K . (5.9)

It can be exploited that the solution of the power minimization problem is a special

case of the more general SINR balancing problem (see discussion in Section 2.2.2).

Theorem 17 states that for given qopt, the optimal beamforming matrix Uopt can be

found directly by solving Kdecoupled optimization problems

uopt

i= arg max

SINRUL,SD

i(ui, q1...qi),1≤i≤K . (5.10)

5. Rate Balancing for Gaussian Broadcast and Multiple Access Channels

Now, we exploit that the interference of the first user is completely canceled out,

thus

SINRUL,SD

1(u1, q1) = q1|uH

1h1|2.(5.11)

This provides a way to find a globally optimal beamforming solution directly, without

the need of iterative beamforming and power adjustment, as it is required for coupled

user (see Chapter 4). The strategy that maximizes (5.11) is the spatial matched filter

uopt

1=h1/kh1k2. With (5.2), the minimal transmission power qopt

1, that is needed to

achieve SINRUL,SD

1=γ1is given by

qopt

1=γ1

(kh1k2)2.(5.12)

Note, that the transmission power (5.12) is optimal with respect to all users, because

the functions SINRUL,SD

2,...SINRUL,SD

Kare monotonically decreasing in qopt

With qopt

1, the SINR of the 2nd user becomes

SINRUL,SD

2(u2, q1, q2) = q2|uH

2h2|2

2ˆ

Z2(qopt

1)u2

,with ˆ

Z2(qopt

1) = qopt

1h1hH

1+I.(5.13)

This is maximized by the normalized MMSE solution

uopt

Z−1

2(qopt

1)h2

kˆ

Z−1

2(qopt

1)h2k2

.(5.14)

Putting (5.14) in (5.13) leads to

SINRUL,SD

2(u2, q1, q2) = q2hH

2ˆ

Z−1

2(qopt

1)h2.(5.15)

Hence, the minimal transmission power qopt

2needed for the 2nd user to achieve an

SINR level γ2is given by

qopt

2=γ2

2ˆ

Z−1

2(qopt

1)h2

.(5.16)

Using the same monotony argument as above, the solution (5.16) is optimal for all

users in the sense that it causes the least interference. Continuing this strategy,

all optimal beamformers uopt

1...uopt

Kand uplink transmission powers qopt

1...qopt

Kare

found successively. The algorithm is summarized below: The beamforming matrix

Uopt = [uopt

1,...,uopt

K]yielded by Algorithm 5 is optimal for both, uplink and down-

link. However, the same does not hold for the power allocation. Thus, it remains

to find the optimal downlink power allocation popt = [popt

1,...,popt

K]T, which fulfills

SINRDL,CO

i(Uopt,popt) = γi, where SINRDL,CO

iis defined as in (5.7).

It is important to note that a MAC decoding order K . . . 1corresponds to a BC

pre-coding order 1...K. Only then does the duality between both channels hold

5. Rate Balancing for Gaussian Broadcast and Multiple Access Channels

Algorithm 5 Rate balancing for the multiple access channel, decoding order K . . . 1

1: for i= 1 to Kdo

2: ˆ

Zi(qopt

1,...,qopt

i−1)⇐

i−1

k=1

qopt

khkhH

k+I

3: uopt

i⇐

Z−1

ihi

Z−1

ihik2

4: qopt

i⇐γi/(hH

iˆ

Z−1

ihi)

5: end for

true. Under this assumption, the Kth user in both channels sees no interference,

which implies that the same transmission power popt

K=γK/(khKk2)2is required to

fulfill the target γK. Having found popt

K, all other power levels popt

(K−1) ...popt

1can be

determined in descending order of their indices. This leads to Algorithm 6, which is

summarized below:

Algorithm 6 Rate balancing for the broadcast channel, precoding order 1...K

1: optimal beamformers Uopt are obtained by Algorithm 5 with inverse decoding

order K . . . 1

2: for i=Kto 1do

3: popt

i⇐γi

|uH

ihi|2K

k=i+1

pk|uH

khi|2+ 1

4: end for

5.2.2. Closed Form Solution for the 2-User Case

Now, a closed form solution is derived for K= 2. The solution balances individual

targets γ1and γ2with minimal power consumption. The solution is derived for the

MAC. From the duality result discussed in Section 5.1.3 it is known that each MAC

optimizer also solves the equivalent BC problem.

Assume a decoding order 2,1. Thus, the first user sees no interference after

demodulation. By adjusting the transmission power q1, the following relative SINR

value can be achieved:

SINRUL,SD

1(u1, q1)

γ1

=q1|uH

1h1|2

γ1

.(5.17)

The spatial matched filter uopt

1=h1/kh1k2is optimal among all possible beamform-

ing solutions, since it maximizes (5.17). Hence, a target threshold C1is achieved

with minimal transmission power

qopt

1=C1γ1

(kh1k2)2.(5.18)

5. Rate Balancing for Gaussian Broadcast and Multiple Access Channels

Thus, the individual solution uopt

1causes the least interference to all other users.

In analogy to Section 5.2.1, it can be concluded that uopt

1is globally optimal with

respect to the joint optimization problem (5.5).

The second user is interfered by the first user, i.e.,

SINRUL,SD

2(u2, q1, q2)

γ2

=q2|uH

2h2|2

γ2(q1|uH

2h1|2+ 1) .(5.19)

The optimal beamformer is the MMSE solution

uopt

2=q1h1hH

1+I−1h2

kq1h1hH

1+I−1h2k2

.(5.20)

Putting (5.20) in (5.19) and applying the Sherman-Morrison formula (see e.g. [46]),

we have

SINRUL,SD

2(u2, q1, q2)

γ2

=q2hH

2q1h1hH

1+I−1h2

γ2

=q2

γ2hH

2h2−q1|hH

2h1|2

1 + q1hH

1h1.(5.21)

Assume that target thresholds C1and C2are given for (5.17) and (5.21), respectively.

These targets can are jointly achieved with minimal transmission power qopt

1and

qopt

2=γ2C2(1 + C1γ1)

2h2(1 + C1γ1)−C1γ1|hH

2h1|2

1h1

.(5.22)

This can be verified by putting (5.18) in (5.21), setting (5.21) to C2and solving for

q2.

Now, assume that both users are balanced at the same level C=C1=C2and

the transmission powers are constrained to fulfill Pmax =q1+q2. Defining

adef

=kh2k2

kh1k2−|hH

2h1|2

kh1k4,

bdef

=kh2k2

kh1k2+Pmax|hH

2h1|2

kh1k2−Pmaxkh2k2,

and combining (5.18) and (5.22), we obtain a unique positive solution

C=−γ1b−γ2

2(γ2

1a+γ1γ2)

+s(γ1b+γ2)2

4(γ2

1a+γ1γ2)2+Pmaxkh2k2

(γ2

1a+γ1γ2).

(5.23)

5. Rate Balancing for Gaussian Broadcast and Multiple Access Channels

As discussed in Chapter 2, individual SINR targets γ1, γ2can be fulfilled if and only

if C≥1. With this additional constraint, (5.23) can be rewritten as an inequality

that characterizes the achievable SINR region under a given sum power constraint

q1+q2≤Pmax and a coding order 2,1. As shown before, this result immediately

carries over to the problem of balancing data rates Ri= log2(1+γi)in both MAC and

BC. By taking the convex hull of the regions resulting from both coding orders, an

achievable rate region is found. This regions contains all rates that can be achieved

by a combination of linear equalization and successive coding. The duality result

of Chapter 2 implies that these rates can be achieved in both MAC and BC (see

discussion in Section 5.1.3). Two examples are illustrated in Fig. 5.1.

Note, that the region can always be expanded by increasing the total transmis-

sion power. Thus, arbitrary rate tuples can be achieved in the absence of power

constraints. This behavior differs from the one observed for the interference limited

scenarios in Chapter 3 and Chapter 4.

5.2.3. Maximizing the Sum Rate

The uplink/downlink duality can be used to find a strategy that maximizes the

downlink throughput (rate sum) under a given power constraint kpk1≤Pmax.

The maximal throughput of the downlink broadcast channel was characterized

in [31], where it was shown that this point lies on the closure of the “Dirty Paper”

region and can be achieved by a combination of decision feedback precoding and

beamforming. The rate-sum

CDL

sum = max

kpk1≤Pmax

i=1 RDL

i(5.24)

is achieved by optimizing over all beamformers Uand power allocations p.

The optimization problem (5.24) has a complicated analytical structure. For-

tunately, a solution can be found indirectly by exploiting the one-to-one monotonic

relationship between RDL

iand SINRDL,CO

i(U,p). The optimum CDL

sum can be achieved

by controlling the SINR levels (5.7) such that SINRDL,CO

i= ˜γifor the given power

constraint. Here ˜γ1...˜γKdenote the unknown target thresholds that lead to the

maximal throughput (5.24).

The duality theory states that the same targets ˜γ1...˜γKare achievable for BC

and MAC. By using a combination of beamforming and successive interference can-

cellation, we can achieve

˜γi=SINRUL,CO

i=qi|uH

ihi|2

iZi(q1,...,qi−1)ui

,1≤i≤K . (5.25)

In particular, ˜γ1...˜γKalso maximize the uplink rate sum CUL

sum =CDL

sum, which is

5. Rate Balancing for Gaussian Broadcast and Multiple Access Channels

0 2 4 6 8 10 12

user 2 user 1

cell sector

scenario a)

low channel correlation

precoding order 1,2

precoding order 2,1

sum capacity

capacity user 2 [bits/symbol]

capacity user 1 [bits/symbol]

0 2 4 6 8 10 12 14 16

user 2

user 1

cell sector

scenario b)

high channel correlation

sum capacity

precoding order 1,2

precoding

order 2,1

capacity user 2 [bits/symbol]

capacity user 1 [bits/symbol]

Figure 5.1.: “Dirty paper” achievable regions obtained by combined beamforming and

precoding for non-causally known interference for a micro-cell cellular environment.

In scenario a) interference is suppressed by the linear beamformer, thus the users

are nearly decoupled, in which case precoding has only little impact. In scenario

b), linear spatial filtering is ineffective because of the small angular separation of

the mobiles. In this case the impact of precoding is more pronounced. Within

the “Dirty paper” region, the corresponding MAC regions for individual power

constraints are depicted for each power pair.

5. Rate Balancing for Gaussian Broadcast and Multiple Access Channels

given by

CUL

sum = max

kqk1≤Pmax

i=1 RUL

= max

kqk1≤Pmax

log2I+

k=1

qkhkhH

k.(5.26)

The objective in (5.26) is concave in the transmission powers. The optimal power

allocation qopt, with kqoptk1=Pmax, can be found, e.g. by using the interior point

technique proposed in [57]. Knowing qopt, the optimal MAC spatial receivers are

given by

uopt

i=Zi(q1,...,qi−1)−1hi

kZi(q1,...,qi−1)−1hik2

,1≤i≤K . (5.27)

Here, a coding order K . . . 1has been assumed. However, the sum capacity (5.26)

does not depend on the coding order (cf. Fig. 5.1). Additional optimizers are found

for other coding orders.

From the duality result in Section 2.4.4 we know that the solutions (5.27) are

optimal for both uplink and downlink. Thus, it remains to find the optimal downlink

power allocation pthat achieves SINRDL,CO

i= ˜γi. This can be done with Algo-

rithm 6.

5.3. Discussion

In this chapter it was shown how the generic results from Sections 2-4 can be used to

solve existing problems in communication theory. By studying the Shannon capacity

of the multi-user channel, we can find theoretical limits and understand how to

achieve those limits with practical schemes.

Considering Gaussian signaling and coding for known interference, the problem

of controlling SINR levels is equivalent to the one of controlling data rates. By

properly choosing target thresholds γ1...γK, rate tuples within the capacity region

of the Gaussian MAC can be achieved with the algorithm derived in Chapter 4. The

duality result found in Chapter 2 implies that this strategy immediately carries over

to the downlink. Thus, the MAC capacity region has a direct equivalence in the

broadcast channel, where a combination of downlink beamforming and precoding for

non-causally known interference is used. It is still unknown, however, whether this

is the actual BC capacity region or not.

The throughput-wise optimal transmission strategy proposed in Section 5.2.3

achieves the capacity of a fully cooperating MIMO system under a worst case noise

covariance. This follows from the results in [31, 58]. Hence, cooperation between

transmitters (or receivers) need not always be beneficial. Under certain conditions,

5. Rate Balancing for Gaussian Broadcast and Multiple Access Channels

beamforming is the optimal strategy. This is an interesting aspect, which may influ-

ence the design of future multi-user MIMO communication systems.

The choice of the optimal coding order is still an open problem. Here, ‘optimal’

means that certain target rates are achieved with minimum total transmission power.

The error propagation issue is neglected since perfect channel knowledge and optimal

power control is assumed. Numerical simulations indicate that for the MAC it is

almost always a good choice to decode the channels in decreasing order of their

average path gains khik2,1≤i≤K. For the broadcast channel, the inverse order

seems to be preferable.

6. Conclusions

This thesis provides the solution to the problem of spatial multiplexing for a multi-

antenna system, if either the transmit antennas or the receive antennas are not

allowed to cooperate and a sum power constraint is imposed. The solution proposed

here is power-aware, in a sense that each channel is allocated only as much power as

it needs to fulfill individual SINR requirements. This allows to trade-off capacities

between the multiplexed channels and provides necessary and sufficient conditions

for feasibility.

6.1. Summary and Discussion of the Main Results

•In Section 2.4.4 it is shown that there is a fundamental duality between the

downlink point-to-multipoint channel and the uplink multipoint-to-point chan-

nel. This duality holds as long as either all receiver noise levels are equal or

the crosstalk-matrix Ψsatisfies the symmetry condition in Theorem 7. If this

is fulfilled, SINR targets are achievable in the downlink if and only if the same

targets are achievable in the uplink under the same sum power constraint.

This result is very useful in that many results known for the uplink immediately

carry over to the downlink. For a longtime, very few results appeared for down-

link processing due to the more complicated analytical structure. The uplink

has a smoother structure, which allows for the design of efficient algorithms.

This has been exploited in the Chapters 3, 4, and 5, where globally convergent

algorithmic solutions have been derived.

•The problem of SIR balancing in the absence of noise [10, 43] has been solved

in Chapter 3. This not only provides insight into the analytical structure of

the general problem, but also yields a necessary and sufficient condition for the

achievability of certain SINR targets in the high SNR regime.

It is proven that the global optimum can be achieved by an iterative strategy,

as conjectured in [43]. This is perhaps surprising, since the original problem

consists of minimizing an `∞norm objective function that is highly non-smooth

and non-differentiable. The other algorithm proposed in [10], which is based

6. Conclusions

on a smoother `1-norm objective, however, is in general not globally conver-

gent. This could be shown by providing necessary and sufficient conditions for

optimality along with a numerical counter-example. An exception is the 2-user

scenario for which `1-norm minimization and `∞-norm are indeed equivalent.

Finally, it is shown how the algorithm proposed in [10] should be modified

in order to solve the original SIR balancing problem. This new algorithm is

presented in Section 3.3.

•The general problem of controlling SINR levels under a sum power constraint

is solved in Chapter 4. A simple iterative solution is proposed, which rapidly

converges to the global optimum. The first key step in the convergence analysis

is the characterization of the iteration sequence in Section 4.1.4. It turns out

that each local minimum is also a global one. Monotony and global convergence

are proven in Section 18. The algorithm, which is summarized in Section 4.2,

can be applied to systems with both perfect and covariance channel information.

Important aspects of the solution are:

–A necessary and sufficient condition for the achievability of SINR targets

under a given sum power constraint is provided. The targets are achievable

if and only if the global optimum of the iterative solution is greater than

one. This plays an important role for the resource management. Only

if this condition is fulfilled, the power minimization problem studied in

[36–42] has a solution.

–A modified version of the iterative technique is proposed, which mini-

mizes the total power while satisfying SINR constraints (see Section 4.3).

In combination with the necessary and sufficient condition for feasibility

described above, this provides a complete solution of the problem studied

in [36–42]. Minimizing the total power is preferable from an operator’s

perspective, since it maximizes the number of users that can be accom-

modated per cell.

•A power aware rate balancing strategy for Gaussian multiuser channels with

unilateral antenna cooperation is proposed in Chapter 5. The algorithm ex-

ploits the triangular channel structure imposed by decision feedback decod-

ing/precoding. For this specific case, all users are uncoupled and the SINR

control problem can be solved very efficiently by back-substitution. As com-

pared to the algorithm proposed in Chapter 4, the solution is not found iter-

atively, but can be computed directly. By properly choosing targets γ1...γK,

it is possible to achieve arbitrary rate tuples within the MAC capacity region.

With the above duality result, this strategy immediately carries over to the

BC, where rates within the “dirty paper” achievable region can be achieved.

6. Conclusions

Whether or not this achievable region is the actual BC capacity region remains

to be shown.

•Finally, closed-form solutions for the 2-user case are derived for all scenarios

under consideration.

6.2. Future Work

•The generic solution for power aware spatial multiplexing presented in Chap-

ter 4 can be applied to related vector systems with unilateral cooperation, e.g.,

–synchronous CDMA systems with random sequences [18]

–wireline communication systems such as digital subscriber lines (DSL),

where co-channel interference is caused by electromagnetic coupling and

cooperation is only possible at one side [59].

–Bell Laboratory Layered Space-Time (BLAST) architecture [45] with ad-

ditional power control at the transmitter

•While multiplexing aims at optimizing throughput, diversity is often required

in oder to make the transmission more robust. Hence, an optimal trade-off

between multiplexing and diversity is desirable (see, e.g., the work in [60–62]

and the reference therein). The technique proposed here can be seen as a step

in this direction, in a sense that it allows to smoothly trade-off throughput

against robustness by varying the noise parameter.

•If the base station only has covariance channel knowledge, the information

theoretical capacity may be achieved by transmitting several independent data

streams per user. In this case, the beamforming strategy proposed here is

suboptimal. The optimality range depends on the channel properties as well as

the SNR, as was recently shown for a single user scenario in [63, 64]. Optimal

balancing strategies for multiuser scenarios are still unknown.

•Similarly, the general problem of controlling data rates in a multiuser MIMO

channel with minimum total transmission power is still open. A partial result

is the iterative waterfilling solution [65], which maximizes the sum rate under

independent power constraints.

•Multiuser diversity can be exploited by a scheduler, which transmits only to

the user with the best channel at any time. This is optimal in an information

theoretical sense as long as the base station is equipped with a single antenna

[66, 67]. When several antennas are employed, however, the maximal through-

put may be achieved by simultaneously multiplexing severals users in space

6. Conclusions

[30]. The algorithm proposed in Section 4.2 provides a single measure for the

quality of the instantaneous spatially multiplexed channel. This could lead to

the development of new space-time scheduling strategies.

Proofs

P.1. Proof of Lemma 11

Substituting [z]k

def

= [x]k·[c]k,1≤k≤K, we have

max

xTb

xTc= max

k=1

[z]k[b]k

[c]k

k=1

[z]k

.(P.1)

Observe that (P.1) is invariant with respect to a scaling of z, hence the maximization

can be restricted to the set {z:kzk1= 1}, i.e.,

max

xTb

xTc= max

kzk1=1

k=1

[z]k

[b]k

[c]k

Next, we have

k=1

[z]k

[b]k

[c]k≤max

1≤l≤K

[b]l

[c]l

k=1

[z]k

= max

1≤l≤K

[b]l

[c]l

(P.2)

Denote by k0the index that fulfills

[b]k0

[c]k0

= max

1≤k≤K

[b]k

[c]k

Now, we use ˜zwith

[˜z]k=1k=k0

0, k 6=k0.

With ˜zthe inequality (P.2) is always satisfied with equality, hence (3.28) is fulfilled.

Relation (3.29) can be proven in the same way, by replacing the max operator by

min.

Proofs

P.2. Proof of Lemma 12

For arbitrary Uand p>0we have

F1(U,p)≥1

γi

SIRDL

i(U,p),1≤i≤K,

=γi

Kpi

k=1

k6=i

pkuH

kRiuk>0,1≤i≤K .

For fixed pk, k 6=i, we have

lim

pi→0F1(U,p)=+∞,1≤i≤K .

Thus, there exists a component-wise positive optimizer ˜p>0with k˜pk1= 1 such

that

min

p:k˜pk1=1 F1(U,p) = inf

pF1(U,p) = F1(U,˜p).

Hence, the function F1, as defined in (3.30), is infinitely often differentiable with

respect to p. The first partial derivative is given by

∂F1(U,p)

∂pl

KuH

lGl(p)ul−1

Kp2

k=1

k6=l

pkuH

kγlRluk1≤l≤K . (P.3)

Setting (P.3) to zero finally leads to (3.34).

P.3. Proof of Lemma 13

From Lemma 11 we know that

λn(U,qext) = max

1≤k≤K+1

kΛ(U)qext

kqext

where ekis defined as a vector whose kth component equals one, while all other

components are zero. The same relation can be formulated for the minimum.

With qext =q

1we have

max

1≤k≤K

kΛ(U)qext

kqext

= max

1≤k≤K

γk

SINRUL

k(uk,q).(P.4)

We also have

K+1Λ(U)qext

K+1qext

Pmax 1T

KDΨT(U)q+1T

KD1K.(P.5)

Proofs

Using

KDΨT(U)q=

i=1

γi

SIRUL

i(ui,q)

and

KD1K=

i=1

γi

SNRUL

i(ui, qi),

where SNRUL

i(ui, qi)def

=qiuH

iRiui/σ2

i, we can conclude that

K+1Λ(U)qext

K+1qext

Pmax

i=1

qiγi1

SIRUL

i(ui,q)+1

SNRUL

i(ui, qi)

Pmax

i=1

γi

SINRUL

i(ui,q).(P.6)

Now we can use

Pmax

i=1

γi

SINRUL

i(ui,q)≤max

1≤k≤K

γk

SINRUL

k(uk,q)1

Pmax

i=1

= max

1≤k≤K

γk

SINRUL

k(uk,q),(P.7)

which shows that (P.5) is never greater than (P.4), i.e.,

λn(U,qext) = max

1≤k≤K+1

kΛ(U)qext

kqext

= max

1≤k≤K

kΛ(U)qext

kqext

P.4. Proof of Theorem 1

For any ν > 0and noise variances σ2

i>0,1≤i≤K, we have

SINRUL

i(˜ui, ν ˜q, σ2

i)≥SINRUL

i˜ui, ν ˜q,max

1≤k≤Kσ2

k,1≤i≤K. (P.8)

The power allocation ˜q>0solves (2.17), i.e., SINRUL

i(˜ui,˜q,˜σ2

i)≥γi. This implies

SINRUL

i˜ui,˜q,min1≤k≤K˜σ2

2> γi,1≤i≤K . (P.9)

From (P.8) and (P.9) it can be concluded, that ˜qcan always be scaled by a factor

ν0= 2 ·max1≤k≤Kσ2

min1≤k≤K˜σ2

Proofs

such that

SINRUL

i(˜ui, ν0˜q, σ2

i)≥SINRUL

i(˜ui, ν0˜q,max

1≤k≤Kσ2

=SINRUL

i(˜ui,˜q,max1≤k≤Kσ2

ν0

)

=SINRUL

i˜ui,˜q,min1≤k≤K˜σ2

2> γi,1≤i≤K ,

holds for arbitrary noise powers σ2

i>0.

P.5. Proof of Theorem 2

Define ˜

Ψdef

=Ψ(˜

U). Assume that λmax(D˜

ΨT)<1and therefore ρ(D˜

ΨT)<1(see

Lemma 3). This implies [46, p. 618]

I−D˜

ΨT−1=

∞

r=0

(D˜

ΨT)r.(P.10)

All terms of the Neuman series (P.10) are non-negative, thus for any σ>0the vector

˜q= (I−D˜

ΨT)−1Dσ is component-wise positive. With the results of Section 2.2.2

it can be concluded that ˜qis an optimizer of the power minimization problem (2.17),

which is therefore feasible.

To prove the reverse direction, assume that (2.17) is feasible. Thus, there exists

a positive solution ˜q>0for any σ>0. It was shown in Section 2.2.2 that this

solution fulfills

Φ˜q=z,where Φdef

=I−D˜

ΨTand zdef

=Dσ. (P.11)

Now, it will be shown that Φis regular. Observe that each zlies in the range of Φ,

i.e., z∈Φ(RK

+). The operator Φis linear and continuous, thus the range Φ(RK

+)is

a closed subspace of RK

+. Let ek,1≤k≤K, be non-negative orthogonal vectors

spanning RK

+. The Cauchy sequences

z(n)

k=ek+1

n1K,1≤k≤K

always have strictly positive components, which follows from [z(n)

k]l≥1/n > 0,1≤

l≤K. Thus, for each z(n)

kthere exists a ˜qsuch that (P.11) is fulfilled. Consequently,

each z(n)

klies in the range of Φ. With

lim

n→∞ kz(n)

k−ekk2

2= lim

n→∞

l=1

n2= 0,1≤k≤K ,

Proofs

it can be concluded that ek∈Φ(RK

+). Consequently, the operator Φmaps RK

+on

+. It follows from Banach’s homomorphism theorem that Φis regular.

It remains to show that the assumption of feasibility implies ρ(D˜

ΨT)<1. It is

known from Theorem 1 that the power minimization problem (2.17) has a strictly

positive solution for any noise vector σ∈RK

+with [σ]l>0, l = 1,...,K. Consider-

ing special noise vectors σk(α)with

[σk(α)]l=(1, l =k

α, l 6=k,1≤k, l ≤K , α > 0

it can be concluded that the power vectors qk= lim

α→0

Φ−1Dσk(α)are strictly positive,

i.e., [qk]l>0, l, k = 1,...,K. This implies that each component of Φ−1is non-

negative.

Next, consider noise vectors 1Kand σ, where kσk1≤1. Since Φ−1is non-

negative, we have

[Φ−1Dσ]m≤[Φ−1D1K]m,1≤m≤K . (P.12)

Using these noise vectors we can define sequences

q(k+1) =Dσ + (D˜

ΨT)q(k),q(0) =Dσ,

q(k+1)

1=D1K+ (D˜

ΨT)q(k)

1,q(0)

1=D1K.

Hence,

q(k+1) =

k+1

r=0

(D˜

ΨT)rDσ ,

q(k+1)

k+1

r=0

(D˜

ΨT)rD1K.

Since the components of D˜

ΨTare non-negative, we have

[q(k+1)]l≥[q(k)]l,1≤l≤K .

That is, the lth component of q(k+1) is monotonically increasing in k. The same holds

for q(k+1)

1. Using (P.10) we have

q(k+1) = (I−D˜

ΨT)−1b(k),

where b(k)def

= (I−(D˜

ΨT)k+2)Dσ. The components of (D˜

ΨT)k+1 are non-negative,

thus [b(k)]l≤[Dσ]lfor 1≤l≤K. Since Φ−1has been shown to be non-negative, it

follows

[q(k+1)]l= [Φ−1b(k)]l≤[Φ−1Dσ]l,1≤l≤K . (P.13)

Proofs

Combining (P.12) and (P.13), it can be concluded that for any σwith kσk1≤1we

have

[q(k+1)]l≤[(I−D˜

ΨT)−1D1K]l,1≤l≤K . (P.14)

Now, it is known form Lemma 3 that D˜

ΨTcan be decomposed as follows:

D˜

ΨTq∗=λ∗

maxq∗=ρ(D˜

ΨT)q∗,(P.15)

where λ∗

max is the maximal eigenvalue of D˜

ΨTequaling the spectral radius ρ(D˜

ΨT).

The associated non-negative eigenvector q∗can be scaled such that q∗≤1. Now

consider a series

q(k+1)

∗=q∗+D˜

ΨTq(k)

∗,q(0)

∗=q∗.(P.16)

Defining ρdef

=ρ(D˜

ΨT)and using (P.15), the series (P.16) can be expressed as

q(k)

∗=

r=0

ρrq∗=(1−ρk+1

1−ρq∗, ρ 6= 1

(k+ 1)q∗, ρ = 1 .

The case ρ= 1 can be ruled out since I−D˜

ΨThas been shown to be regular. For

ρ > 1, the relation (P.14) implies

ρk+1 −1

ρ−1[q∗]l≤[(I−D˜

ΨT)−1D1K]l.(P.17)

There is at least one lfor which [q∗]l6= 0, thus relation (P.17) cannot hold for all

kand ρ > 1(the left side tends to infinity for k→ ∞ and the right side does not

depend on k). This leaves ρ < 1. From Lemma 3 we know that this is equivalent to

λmax(D˜

ΨT)<1.

P.6. Proof of Theorem 10

Assume that ¯

Usolves the eigenvalue minimization problem (3.7). This implies

max

x>0

xTDΨT(U)¯q

xT¯q≥min

y>0max

x>0

xTDΨT(U)y

xTy

=λmaxDΨT(U)←Eq. (3.10)

≥λmaxDΨT(¯

U)

=zTDΨT(¯

U)¯q

zT¯q←Eq. (3.18)

for any Uand z>0. Thus,

min

Umax

x>0

xTDΨT(U)¯q

xT¯q≥zTDΨT(¯

U)¯q

zT¯q.

Proofs

There always exists an optimizer x0>0such that

min

0DΨT(U)¯q

0¯q≥xT

0DΨT(¯

U)¯q

0¯q.

This inequality can only be fulfilled with equality. Moreover, it can be observed from

(3.15) that the minimization with respect to Uis independent of x0. Thus, for all

x>0and Uwe have

min

xTDΨT(U)¯q

0¯q=xTDΨT(¯

U)¯q

xT¯q.

In the reverse direction assume that a certain matrix ¯

Ufulfills (3.17), thus for all

x>0we have

xTDΨT(U)¯q

xT¯q≥xTDΨT(¯

U)¯q

xT¯q.(P.18)

Using (3.10) we have

λmaxDΨT(U)= min

x>0max

y>0

xTDΨT(U)y

xTy

≥min

x>0

xTDΨT(U)¯q

xT¯q.(P.19)

Combining (P.18) and (P.19) we can conclude that

λmaxDΨT(U)≥min

x>0

xTDΨT(¯

U)¯q

xT¯q=λmaxDΨT(¯

U).(P.20)

Hence, ¯

Uis the global minimizer of the function λmaxDΨT(U).

P.7. Proof of Theorem 11

Suppose that SINRUL

i≥γi,1≤i≤K, is feasible. This implies that the eigenvalue

optimization problem (3.7) has a solution BUL

Inf = 1/λopt

max >1. The optimizers

qopt >0and uopt

ifulfill SIRUL

i(uopt

i,qopt)≥SIRUL

i(ui,qopt)for any ui∈CM. Thus,

λopt

max =γi

SIRUL

i(uopt

i,qopt)≤γi

SIRUL

i(ui,qopt),1≤i≤K ,

which can be rewritten as

iΩi(λopt

max,qopt)ui≤0,1≤i≤K . (P.21)

Thus, Ωi(λopt

max,qopt)is negative semidefinite. Inequality (P.21) can only be fulfilled

with equality if uilies in the nullspace of Ωi(λopt

max,qopt).

Proofs

Conversely, assume that there exists a power allocation ˜q>0and λ < 1such

that the matrices Ωi(λ, ˜q),1≤i≤K, are singular. This implies that there exist

non-trivial solutions ˜uk6= [0,...,0]Tsuch that 0 = ˜uH

iΩi(λ, ˜q)˜ui,1≤i≤K. This

can be rewritten as γi/SIRUL

i(˜ui,˜q) = λ. Thus, SIRUL

i(˜ui,˜q)> γi, which implies

feasibility and concludes the proof.

P.8. Proof of Theorem 13

Assume ˜

λn(x) = λmax(n)for any x>0. This implies

λmaxDΨT(U)(a)

= max

y>0min

x>0

xTDΨT(U)y

xTy

≥min

x>0

xTDΨT(U)q(n−1)

xTq(n−1)

≥min

x>0min

xTDΨT(U)q(n−1)

xTq(n−1)

(b)

= min

x>0

λn(x)

=λmax(n),(P.22)

where (a) follows from Lemma 6 and (b) follows from the fact that U(n)minimizes

xTDΨT(U)q(n−1) for an arbitrary x>0(see the proof of Theorem 9). Inequality

(P.22) holds for any U, thus we can conclude that

λmaxDΨT(U(n))=λmax(n)≤min

UλmaxDΨT(U)=λopt

max .(P.23)

Since λopt

max is the global minimum, (P.23) can only be satisfied with equality. Thus,

U(n)∈ MDL.

To prove the reverse direction assume that U(n)solves the eigenvalue minimization

problem (3.7), i.e., U(n)∈ MDL. Since U(n)is computed by Algorithm 1, we have

U(n)= arg min

xTDΨT(U)q(n−1)

xTq(n−1) .

Since U(n)is the global optimizer of (3.7), Theorem 10 implies

DΨT(U(n))q(n−1) =λmax(n)q(n−1) .(P.24)

Combining (3.23) and (P.24) we have

λn(x) = xTDΨT(U(n))q(n−1)

xTq(n−1) =λmax(n),

for any x>0, which concludes the proof.

Proofs

P.9. Proof of Theorem 15

The equivalence of (3.41) and (3.42) for a given `1solution (ˆ

U,ˆp)follows from (3.34).

Now, assume that (3.41) holds true. This implies that the `1solution is balanced,

thus

B1=F1(ˆ

U,ˆp) = γi

SIRDL

i(ˆ

U,ˆp),1≤i≤K

=λmaxΓΨ(ˆ

U)≥λopt

max .(P.25)

From (3.40) we know that this can only be satisfied with equality, thus λopt

max =B1

must hold.

In the reverse direction, assume that (ˆ

U,ˆp)solves the `∞problem (3.33). The

solution is balanced, thus λopt

max =γi/SIRDL

i(ˆ

U,ˆp),∀iand ˆpis the dominant right-

hand eigenvector of ΓΨ(ˆ

U). With Theorem 14, the optimum is characterized by

BDL

Inf =1

λopt

max

=ˆpi

γi

k=1

k6=i

ˆpkˆuH

kRiˆuk

,1≤i≤K(P.26)

ˆpiˆuH

iGi(ˆp)ˆui

,1≤i≤K . (P.27)

Using matrix notation, (P.26) and (P.27) can be rewritten as (3.41) and (3.42),

respectively.

P.10. Proof of Theorem 16

Assume that ˆ

Usolves (3.46) and qgis the dominant left-hand eigenvector of the

matrix ΓΨ(ˆ

U). With (3.30), this implies that for all x>0,

U= arg min

UF1(U,(qg)−1),s.t. uH

iRiui= 1,1≤i≤K

= arg min

i=1

[x]iuH

iK

k=1

k6=i

[qg]kγkRkui,s.t. uH

iRiui= 1,1≤i≤K

= arg min

xTΓΨ(U)Tqg

xTqg,s.t. uH

iRiui= 1,1≤i≤K . (P.28)

Hence, the following relation holds for all Uand x>0:

xTΓΨ(U)Tqg

xTqg≥xTΓΨ(ˆ

U)Tqg

xTqg.(P.29)

Proofs

Using (3.10) we have

λmaxΓΨ(U)T= min

xmax

xTΓΨ(U)Ty

xTy

≥min

xTΓΨ(U)Tqg

xTqg.(P.30)

Combining (P.29) and (P.30) we can conclude that

λmaxΓΨ(U)T≥min

xTΓΨ(ˆ

U)Tqg

xTqg=λmaxΓΨ(ˆ

U)T.(P.31)

Hence, ˆ

Uis the global minimizer of the function λmaxΓΨ(U). From the discussion

in Section 3.1.1 it is known that in this case ˆ

Uand pgsolve the `∞-problem (3.33).

In the reverse direction, assume that ˆ

Uis the `∞-optimizer, which implies

λmaxΓΨ(ˆ

U)≤λmaxΓΨ(U).

Thus, the following inequality holds for any Uand z>0:

max

x>0

xTΓΨ(U)Tqg

xTqg≥min

y>0max

x>0

xTΓΨ(U)Ty

xTy

=λmaxΓΨ(U)T←Eq. (3.10)

≥λmaxΓΨ(ˆ

U)T

=zTΓΨ(ˆ

U)Tqg

zTqg.←Eq. (3.18)

The above inequality holds for any U, thus

min

Umax

x>0

xTΓΨ(U)Tqg

xTqg≥zTΓΨ(ˆ

U)Tqg

zTqg.

There exists an optimizer x0>0such that

min

0ΓΨ(U)Tqg

0qg≥xT

0ΓΨ(ˆ

U)Tqg

0qg.

This inequality can only be fulfilled with equality. Moreover, it can be observed from

(3.15) that the minimization with respect to Uis independent of x0. Thus, for all

x>0and Uwe have

min

xTΓΨT(U)qg

0qg=xTΓΨT(ˆ

U)qg

xTqg.

It can be concluded with (P.28) that ˆ

Usolves (3.46).

Proofs

P.11. Proof of Theorem 19

Assume that (4.28) holds. Combining (4.12) and the monotony result (4.26) we have

min

x>0

xTΛ(U(n), Pmax)q(n−1)

ext

xTq(n−1)

ext

=λmaxΛ(U(n), Pmax).(P.32)

The same arguments that have led to (4.17) can be used to show that U(n)fulfills

min

Umin

x>0

xTΛ(U, Pmax)q(n−1)

ext

xTq(n−1)

ext

= min

x>0

xTΛ(U(n), Pmax)q(n−1)

ext

xTq(n−1)

ext

.(P.33)

Combining (P.32) and (P.33) we have

min

Uλmax(Λ(U, Pmax)) = min

Umin

x>0max

y>0

xTΛ(U, Pmax)y

xTy

≥min

Umin

x>0

xTΛ(U, Pmax)q(n−1)

ext

xTq(n−1)

ext

= min

x>0

xTΛ(U(n), Pmax)q(n−1)

ext

xTq(n−1)

ext

=λmaxΛ(U(n), Pmax).

This can only be satisfied with equality, thus U(n)∈ M.

To prove the reverse direction, assume that U(n)∈ M. From Theorem 17 we

know that U(n)is characterized by

Λ(U(n), Pmax)q(n−1)

ext =λmax(n)q(n−1)

ext .(P.34)

Left-hand multiplication on both sides of (P.34) with xT/(xTq(n−1

ext ),x∈RK

+, leads

λmax(n) = xTΛ(U(n), Pmax)q(n−1)

ext

xTq(n−1)

ext

.(P.35)

Equation (P.35) holds for all x>0, hence

min

x>0

xTΛ(U(n), Pmax)q(n−1)

ext

xTq(n−1)

ext

= max

x>0

xTΛ(U(n), Pmax)q(n−1)

ext

xTq(n−1)

ext

With Lemma 13 this implies (4.28).

Notation and Symbols

E{·} expectation operator

[·]ij component with index i, j of a matrix or a vector

(a)−1if applied to a vector: component-wise inversion

(·)Ttranspose

(·)HHermitian (complex conjugate) transpose

(·)(n)nth iteration step

(·)UL uplink quantity

(·)DL downlink quantity

B1optimum of the `1cost function p. 33

BInf asymptotically achievable upper bound for miniSINRip. 16

C(U, Pmax)balanced SINR optimum for a given beamforming matrix Up. 13,18

Copt(Pmax)balanced SINR optimum optimized over all beamformers

and power allocations

p. 43,44

Dnormalized threshold matrix p. 13

F1`1cost function p. 33

Gi(p) = PK

k6=i

γkRk

pkp. 33

γitarget threshold for the ith user

Γthreshold matrix Γ= diag{γ1,...,γK}p. 34

hichannel vector of the ith user p. 7

Knumber of users (non-cooperating terminals)

λmax(·)maximal eigenvector

λopt

max optimum of the eigenvalue minimization problem in the

absence of noise

p. 28

λncost function for SIR balancing p. 30

λncost function for extended `1balancing p. 40

λncost function for SINR balancing p. 45

Λ(U, Pmax)extended uplink coupling matrix p. 13

Mnumber of base station antennas (cooperating terminals)

Ψ(U)coupling matrix p. 10

Pmax maximal total power

pidownlink power of the ith user

pdownlink power allocation p. 8

Notation and Symbols

qiuplink power of the ith user

quplink power allocation p. 10

qext extended uplink power vector qext =q

1p. 45

Qi(q)interference covariance in the absence of noise p. 27

σ2

inoise variance of the ith user

σnoise vector σ= [σ2

1,...,σ2

K]T

SIR signal-to-interference ratio p. 16,24

SINR signal-to-interference-plus-noise ratio p. 9,10

uibeamforming vector of the ith user (both uplink an

downlink)

Ubeamforming matrix U= [u1,...,uK]p. 8

Υ(U, Pmax)extended downlink coupling matrix p. 19

xauxiliary variable p. 45

ξvector with ratios SIRi/γip. 25

Zi(q)interference-plus-noise spatial covariance matrix p. 45

Zi(q1...qi−1)interference-plus-noise spatial covariance matrix with

successive decoding, order K . . . 1

p. 56

Publication List

[1] H. Boche and M. Schubert, “A general duality theory for uplink and downlink

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Theory (ISIT), Lausanne, Switzerland, p. 452, June 2002.

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and Signal Proc. (ICASSP), Orlando, USA, May 2002.

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[8] H. Boche and M. Schubert, “Analysis of SIR-based downlink beamforming,”

IEICE Trans. Commun., vol. E85-B, pp. 1160–1168, June 2002.

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