Document [original]

FAKULTÄT FÜR

ELEKTROTECHNIK,

INFORMATIK UND

MATHEMATIK

Network-Coded Cooperation in Wireless

Networks: Theoretical Analysis and

Performance Evaluation

Zur Erlangung des akademischen Grades

DOKTORINGENIEUR (Dr.-Ing.)

der Fakultät für Elektrotechnik, Informatik und Mathematik

der Universität Paderborn

vorgelegte Dissertation

von

M.Sc. Dereje Hailemariam Woldegebreal

Paderborn

Referent: Prof. Dr.-Ing. Reinhold Häb-Umbach

Korreferent: Prof. Dr. rer. nat. Holger Karl

Tag der mündlichen Prüfung: 13.04.2010

Paderborn, den 23.04.2010

Diss. EIM-E/266

Acknowledgments

“Knowledge of what is possible is the beginning of happiness”

George Santayana

I would not have realized my PhD study dream without the financial support of the German

Academic Exchange Service (DAAD). I sincerely thank DAAD first. My deepest gratitude

goes to Prof. Dr. Holger Karl, head of the Computer Networks Research group, for his

support and trust from day one of my research. I enjoyed it very much to work under his

guidance and benefited a lot from his encouragements and meticulous way of thinking. It is

also an honor for me to be his second doctoral graduate.

Special thanks to Prof. Dr.-Ing. Reinhold H¨

ab-Umbach for being my examiner and going

through the thesis. I thank Prof. Dr. Marco Platzner for being supportive when I needed him.

I am so grateful for the constructive discussions and the work we have done together with

Stefan Valentin and Tobias Volkhausen. I also want to thank colleagues at the research group

for their assistance and creating a good working environment. In this regard, the credit goes

to Miss Tanja Langen and Hans-Joachim Kraus. I am indebted to my office mate Matthias

Andree for the good office atmosphere and being resourceful. I am happy for the chat and

friendly discussion with Hermann Simon Lichte, Christian Dannewitz, and Rana Azeem

M. Khan; Azeem, I will always remember those teatime talks. I should also acknowledge

Bernard Bauer from Paderborn Center for Parallel Computing for being collaborative when

he was needed. I appreciate the contribution from my compatriots Mekdes G. Girma and Dr.

Yohannes A. Demessie; their help usually comes in handy. I am thankful to my family for

their contribution in one or the other way, especially to Amarech and my mother. Finally,

I would like to thank my wife Kidist and my daughter Yanet for their unconditional love

and support. At times, the going was tough and I prevailed because of my wife; I love you

Kidist.

Abstract

In today’s wireless networks, there is an increasing demand for high service quality, data

rates, and network coverage. However, when addressing these demands, noise, interference,

fading, power constraints, and bandwidth limitation are some of the fundamental challenges.

Spatial diversity is one way to deal with these challenges and is achieved by sending and re-

ceiving a signal using multiple transmit and/or multiple receive antennas. The use of multiple

transmit and receive antennas in spatial diversity results in a technique called Multiple-Input

Multiple-Output (MIMO). In practice, however, one shortcoming of MIMO is that installing

multiple antennas per wireless node may not be feasible because of limitations in power,

cost, and/or device size.

When nodes are limited in the number of antennas, distributed nodes in the network can be

engaged to emulate MIMO. This technique of gaining spatial diversity is called cooperative

transmission. Information-theoretic studies have shown substantial capacity improvements

as compared to traditional point-to-point wireless networks. In recent years, network-coded

cooperation was proposed as one protocol to realize cooperation in wireless networks. Most

of the previous work done in this area considers error-free inter-user channels; however, this

is usually not the case in wireless networks.

This thesis investigates the performance of two types of network-coded cooperation pro-

tocols under a more practical scenario of erroneous wireless channels, transmissions using

orthogonal channels, and energy constraints. Specifically, we provide the analytical tools to

compute the error rate bounds of these two network-coded cooperation protocols, study their

outage behavior, and show that these protocols can achieve full diversity. We then investi-

gate the coverage area by using network-coded cooperation and study the effect of network

topology on outage performance. In large networks where a source has potential partners in

its surrounding to choose from, metrics that provide insight on how to select a partner are

required. One option would be to select a partner that minimizes the total energy spent in

the network. With energy minimization in mind, we finally analyze the energy consumption

of network-coded cooperation considering transmission, reception, and processing energy at

all cooperating nodes.

iii

Zusammenfassung

In heutigen drahtlosen Netzen w¨

achst die Anforderung an guter Servicequalit¨

at, hohen Daten-

raten und umfassender Netzabdeckung st¨

andig. Gleichzeitig gibt es jedoch Bandbreiten- und

Sendeleistungsbeschr¨

ankungen sowie Interferenz und Fading auf den drahtlosen Kan¨

alen,

die das Erreichen dieser Anforderungen erschweren.

Kooperative ¨

Ubertragung ist ein neues Paradigma in der drahtlosen Kommunikation um

Kanal-Fading zu handhaben. Bei der kooperativen ¨

Ubertragung werden verteilte Knoten in

einem Netzwerk gruppiert und emulieren so Antennendiversit¨

at. Informationstheoretische

Studien haben gezeigt, dass sich die Kapazit¨

at im Vergleich zu herk¨

ommlicher drahtloser

Punkt-zu-Punkt- ¨

Ubertragung verbessern l¨

asst. In den vergangenen Jahren wurde “network-

coded” Kooperation vorgeschlagen und untersucht; ein Verfahren, bei dem Network Coding

in der kooperativen ¨

Ubertragung eingesetzt wird. In der Vergangenheit werden in der Lit-

eratur gr¨

oßtenteils fehlerfreie Kan¨

ale zwischen den Nutzern angenommen. Dies ist jedoch

ublicherweise in drahtlosen Netzen nicht der Fall.

Diese Doktorarbeit untersucht die Performanceleistung zweier Typen von network-coded

Kooperationsverfahren in einem realit¨

atsnahen Szenario mit Energiebeschr¨

ankung bei fehler-

behafteter drahtloser ¨

Ubertragung ¨

uber orthogonale Kan¨

ale. Konkret entwickeln wir ein

Framework, um die Outage Wahrscheinlichkeit der zwei network-coded Kooperationspro-

tokolle zu berechnen, ihr Outage Verhalten zu untersuchen und um zu zeigen, dass diese

Protokolle Diversit¨

at ausnutzen k¨

onnen. Wir untersuchen, wie sich aufgrund von network-

coded Kooperation die abgedeckte Fl¨

ache erweitert und betrachten den Effekt der Netzw-

erktopologie auf die Outage Performance. Abschließend analysieren wir den Energiever-

brauch eines der network-coded Kooperationsverfahren unter Ber¨

ucksichtigung des indi-

viduellen Energieverbrauchs der Sende-, Empfang- und Verarbeitungsoperationen an allen

kooperierenden Knoten.

Contents

1. Introduction 1

1.1. Review of Cooperative Transmission Protocols . . . . . . . . . . . . . . . 4

1.2. Network-Coded Cooperation . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1. Review of network-coded cooperation . . . . . . . . . . . . . . . . 10

1.2.2. Literature survey on network-coded cooperation . . . . . . . . . . 13

1.3. Thesis Motivation and Contributions . . . . . . . . . . . . . . . . . . . . . 17

1.3.1. Thesis motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.2. Thesis contributions . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4. Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2. Introduction to Network Coding 21

2.1. Network Coding in Error-Free Networks . . . . . . . . . . . . . . . . . . 21

2.2. Linear Network Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1. Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.2. Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3. Network Coding and Channel Coding . . . . . . . . . . . . . . . . . . . . 27

2.3.1. Separate network-channel coding . . . . . . . . . . . . . . . . . . 28

2.3.2. Joint network-channel coding . . . . . . . . . . . . . . . . . . . . 29

2.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3. Wireless Channels and Networks 31

3.1. System Model of the Point-to-Point Transmission . . . . . . . . . . . . . . 31

3.1.1. Forward error correction with channel coding . . . . . . . . . . . . 32

3.1.2. Modulation and demodulation . . . . . . . . . . . . . . . . . . . . 33

3.2. Wireless Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.1. Noise and interference . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.2. Fading channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3. Information Theory – Fading Channel Capacity . . . . . . . . . . . . . . . 41

3.3.1. Capacity of Additive White Gaussian Noise Channel . . . . . . . . 42

3.3.2. Capacity of flat and slow fading channels . . . . . . . . . . . . . . 43

3.3.3. Channel state information and channel capacity . . . . . . . . . . 44

vii

Contents

3.3.4. Ergodic fading channels . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.5. Non-ergodic fading channels and outage probability . . . . . . . . 46

3.3.6. Capacity vs. combining schemes . . . . . . . . . . . . . . . . . . 47

3.4. Network-Coded Cooperation . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.1. System model for network-coded cooperation . . . . . . . . . . . 48

3.4.2. Equivalent channel model . . . . . . . . . . . . . . . . . . . . . . 49

3.5. Cross-Layer Design in Cooperative Wireless Network . . . . . . . . . . . . 50

3.5.1. Existing wireless network architecture . . . . . . . . . . . . . . . 51

3.5.2. Cooperative wireless network architecture . . . . . . . . . . . . . 52

3.5.3. Destination node receiver . . . . . . . . . . . . . . . . . . . . . . . 53

3.5.4. Partner node receiver . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4. Outage Behavior of Network-Coded Cooperation 55

4.1. System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2. Outage Probability Computation . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1. Network-coded cooperation . . . . . . . . . . . . . . . . . . . . . 59

4.3. Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 73

4.3.1. Basic assumptions and parameters . . . . . . . . . . . . . . . . . . 73

4.3.2. List of investigated protocols . . . . . . . . . . . . . . . . . . . . 74

4.3.3. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3.4. Conclusion and remarks . . . . . . . . . . . . . . . . . . . . . . . 78

4.4. Diversity-Multiplexing Tradeoff . . . . . . . . . . . . . . . . . . . . . . . 79

4.5. Coverage Area Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5. Energy Efficiency in Wireless Sensor Networks 87

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.1.2. Literature survey . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2. Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.1. WSN Transceiver circuits . . . . . . . . . . . . . . . . . . . . . . 91

5.2.2. Packet structure in point-to-point transmission . . . . . . . . . . . 92

5.3. Energy Consumption of Point-to-Point Transmission . . . . . . . . . . . . 92

5.3.1. Power amplifier calibration . . . . . . . . . . . . . . . . . . . . . 93

5.3.2. Energy consumption formulation . . . . . . . . . . . . . . . . . . 94

5.3.3. Energy efficiency formulation . . . . . . . . . . . . . . . . . . . . 96

5.4. General Assumptions in Network-Coded Cooperation . . . . . . . . . . . . 96

5.5. Energy Consumption in Network-Coded Cooperation . . . . . . . . . . . . 99

viii

Contents

5.5.1. Good inter-user channel . . . . . . . . . . . . . . . . . . . . . . . 99

5.5.2. Bad inter-user channel . . . . . . . . . . . . . . . . . . . . . . . . 101

5.5.3. Average energy consumption . . . . . . . . . . . . . . . . . . . . . 102

5.5.4. Average outage probability . . . . . . . . . . . . . . . . . . . . . . 103

5.5.5. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.5.6. Formulation as an optimization problem . . . . . . . . . . . . . . 106

5.6. Energy Efficiency Formulation . . . . . . . . . . . . . . . . . . . . . . . . 108

5.6.1. Good inter-user channel . . . . . . . . . . . . . . . . . . . . . . . 109

5.6.2. Bad inter-user channel . . . . . . . . . . . . . . . . . . . . . . . . 109

5.6.3. Average energy efficiency . . . . . . . . . . . . . . . . . . . . . . 109

5.6.4. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6. Incremental Redundancy Network-Coded Cooperation 113

6.1. System Model, Joint Network-Channel Coding . . . . . . . . . . . . . . . 114

6.1.1. System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.1.2. Joint network-channel encoding . . . . . . . . . . . . . . . . . . . 116

6.1.3. Joint network-channel decoding . . . . . . . . . . . . . . . . . . . 119

6.2. Outage Behavior of Incremental Redundancy Network-Coded Cooperation 120

6.2.1. Inter-user transmission . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2.2. Uplink transmission . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.3. Result and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7. Conclusion 131

7.1. Contribution of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.1.1. Conventional network-coded cooperation . . . . . . . . . . . . . . 131

7.1.2. Energy consumption of network-coded cooperation . . . . . . . . . 132

7.1.3. Incremental redundancy network-coded cooperation . . . . . . . . 132

7.2. Recommendations for Future Research . . . . . . . . . . . . . . . . . . . 133

A. Outage Probability Approximation 135

A.1. Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

A.2. Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Contents

List of Figures

1.1. Cooperative transmission in a mobile communication deployment. . . . . . 3

1.2. Relay-channel system in the uplink of the mobile communication system. . 4

1.3. Three-node cooperative transmission in the mobile communication system. 6

1.4. Classification of cooperative transmission protocols. . . . . . . . . . . . . . 11

1.5. Network-coded cooperation in mobile communication system. . . . . . . . 12

1.6. Classification of network-coded cooperation schemes. . . . . . . . . . . . . 16

2.1. Canonical butterfly topology that explains network coding. . . . . . . . . . 22

2.2. Linear network coding in the butterfly network. . . . . . . . . . . . . . . . 26

2.3. Local encoding over the Galois field 𝔽2. . . . . . . . . . . . . . . . . . . . 26

2.4. Separate network-channel coding. . . . . . . . . . . . . . . . . . . . . . . 28

3.1. System model for the point-to-point transmission. . . . . . . . . . . . . . 32

3.2. Normalized received power, in dB, vs. distance. . . . . . . . . . . . . . . . 36

3.3. Cooperative transmission in the uplink of the mobile communication system. 38

3.4. Channel model with small-scale fading coefficient 𝛼and noise samples z. . 41

3.5. Channel capacity for various modulated input symbols. . . . . . . . . . . . 44

3.6. Network-coded cooperation in the uplink of the mobile communication. . . 49

3.7. Existing wireless network protocol stack. . . . . . . . . . . . . . . . . . . 52

3.8. Cooperative transmission physical layer. . . . . . . . . . . . . . . . . . . 53

4.1. System diagram of network-coded cooperation. . . . . . . . . . . . . . . . 56

4.2. Timing diagram of network-coded cooperation. . . . . . . . . . . . . . . . 57

4.3. Both users decode each others’ codeword correctly. . . . . . . . . . . . . . 61

4.4. Neither user decods its partner’s codeword correctly. . . . . . . . . . . . . 64

4.5. The partner correctly decodes the source’s codeword, but not vice versa. . . 66

4.6. The source correctly decodes the partner’s codeword, but not vice versa. . . 67

4.7. Outage results of point-to-point and network-coded cooperation. . . . . . . 70

4.8. Block diagram showing the compared protocols. . . . . . . . . . . . . . . . 73

4.9. Outage probability vs. Γ𝑠,𝑑 of static protocols. . . . . . . . . . . . . . . . . 74

4.10. Outage probability vs. Γ𝑠,𝑑 of adaptive and repetition coding protocols. . . . 75

List of Figures

4.11. Outage probability vs. Γ𝑠,𝑑 of adaptive protocols for Γ𝑠,𝑝 = 0 dB. . . . . . . 76

4.12. Outage probability vs. Γ𝑠,𝑑 of adaptive protocols for Γ𝑠,𝑝 = 10 dB. . . . . . 77

4.13. Outage probability vs. Γ𝑠,𝑑 of adaptive protocols for Γ𝑠,𝑝 =30 dB. . . . . . 78

4.14. Outage probability of adaptive protocols. . . . . . . . . . . . . . . . . . . . 79

4.15. Outage probability vs. Γ𝑠,𝑑 of adaptive protocols. . . . . . . . . . . . . . . 80

4.16. Diversity-multiplexing tradeoff of the network-coded cooperation. . . . . . 82

4.17. Outage probability contours of static protocols. . . . . . . . . . . . . . . . 83

4.18. Intra-cooperation gain contours. . . . . . . . . . . . . . . . . . . . . . . . 84

4.19. Intra-cooperative gain contours. . . . . . . . . . . . . . . . . . . . . . . . 85

5.1. Outage probability of network-coded cooperation. . . . . . . . . . . . . . . 88

5.2. Network-coded cooperation in sensor network. . . . . . . . . . . . . . . . 89

5.3. Block diagram of a Wireless Sensor Network (WSN) transceiver circuit. . . 92

5.4. The link-layer packet format. . . . . . . . . . . . . . . . . . . . . . . . . 92

5.5. Timing diagram of the cooperative and point-to-point transmissions. . . . . 97

5.6. Summary of parameters used in energy allocation. . . . . . . . . . . . . . 98

5.7. Possible states when inter-user channels are good quality. . . . . . . . . . . 100

5.8. Possible states when inter-user channels are bad quality. . . . . . . . . . . . 101

5.9. Transmit power contours for various locations of the partner node. . . . . . 105

5.10. Average energy consumed, in mill Joule, per information bit. . . . . . . . . 106

5.11. Average consumed energy for system-level outage probability of 0.0001. . . 107

5.12. Energy efficiency versus code rate. . . . . . . . . . . . . . . . . . . . . . 110

5.13. Energy efficiency versus source-partner separation. . . . . . . . . . . . . . 111

6.1. System diagram of incremental network-coded cooperation. . . . . . . . . 115

6.2. Codewords sent by the source (shaded) and partner in Case 1. . . . . . . . 122

6.3. Codewords sent by the source (shaded) and partner in Case 2. . . . . . . . 123

6.4. Codewords sent by the source (shaded) and partner in Case 3. . . . . . . . 125

6.5. Codewords sent by the source (shaded) and partner in Case 4. . . . . . . . 126

6.6. Outage probability vs. uplink SNR Γ𝑠,𝑑 when Γ𝑠,𝑝 =0, 10, and 30 dB. . . . 127

6.7. Three dimensional plot of the outage probability, 𝛼, and 𝛽. . . . . . . . . . 128

6.8. Three dimensional plot of the outage probability, 𝛼, and 𝛽. . . . . . . . . . 129

xii

List of Abbreviations

LNA Low Noise Amplifier

ACK Acknowledgment

NACK Negative Acknowledgment

BER Bit Error Rate

BPSK Binary Phase-Shift Keying

CRC Cyclic Redundancy Check

LLR Log-likelihood Ratio

CSI Channel-State Information

FEC Forward Error Correction

LNC Linear Network Coding

MAC Medium-Access Control Layer

OSI Open Systems Interconnection

PER Packet Error Rate

PHY Physical Layer

SNR Signal-to-Noise Ratio

TDMA Time-Division Multiple Access

WSN Wireless Sensor Network

BS Base Station

MS Mobile Station

PPT Point-to-Point Transmission

xiii

List of Figures

ASK Amplitude Shift Keying

PSK Phase Shift Keying

8-PSK 8 Phase Shift Keying

FSK Frequency Shift Keying

ARQ Automatic Repeat Request

ADC Analog-to-Digital Converter

AWGN Additive White Gaussian Noise

CDMA Code-Division-Multiple-Access

DAC Digital-to-Analog Converter

DSP Digital Signal Processor

EGC Equal-Gain Combining

CC Code Combining

FDMA Frequency Division Multiple Access

ISI Inter-symbol Interference

MIMO Multiple-Input Multiple-Output

SISO Single-Input Single-Output

MRC Maximum-Ratio Combining

SC Selection Combining

QoS Quality of Service

QPSK Quadrature Phase Shift Keying

TCP Transmission Control Protocol

xiv

1. Introduction

In today’s wireless networks, there is an increasing demand for service quality, high data

rates, network coverage, and lesser processing time. The scarcity of two fundamental re-

sources for communications, namely, energy and bandwidth, is a serious challenge to fulfill

these demands [1]. Moreover, wireless channels feature fading, shadowing, interference, and

other impairments that make the channel unpredictable.

Signal fading is the most severe among these impairments. In a wireless channel, random

scattering from reflectors with different attenuation coefficients results in multiple copies of

a transmitted signal arriving (and interfering) at a receiver with different gains, phase shifts,

and delays. These multiple signal replicas can add together in a constructive or destructive

way, amplifying or attenuating the received signal’s amplitude. Destructive interference re-

sults in fading which causes temporary failure of communication as the amplitude of the

received signal may be low to the extent that the receiver may not be able to distinguish it

from thermal noise [2, 3].

Diversity is one technique to combat fading in a wireless channel. Diversity can be defined

as the technique by which multiple copies of a signal are delivered to the receiver via inde-

pendently fading channels. If one or more copies are highly degraded due to severe fading,

then the receiver can still receive the signal from the other received copies. Diversity gain

is used to quantify the gain from diversity. It is related to the number of independent chan-

nels over which the signal is being received. Independent channels can be generated in three

physical domains: time, frequency, and space. Time diversity is achieved by transmitting

the same signal via different time slots; these slots should be well separated to ensure that

the channels at these slots are uncorrelated. Drawbacks of time diversity include loss in the

system signal rate and an increase in transmission delay. Frequency diversity is achieved

by transmitting the same data on different frequency bands. This diversity is inefficient in

bandwidth utilization. Spatial diversity, on the other hand, is achieved by sending and re-

ceiving a signal using multiple transmit and/or multiple receive antennas that are physically

separated from one another. The received copies of the signal are likely to be uncorrelated as

they propagate along spatially separated paths. The use of spatial diversity has gained lots of

interest in recent years as it does not increase the bandwidth or the transmission delay [1].

1. Introduction

The use of multiple transmit and receive antennas in spatial diversity results in a Multiple-

Input Multiple-Output (MIMO) system. In MIMO, the transmitter has the capability to trans-

mit different signals from each antenna and the receiver can observe different signals from

each antenna as an input. The signal present at each receive antenna is the combination of

signals from the transmit antennas, after each has traveled through possibly different fading

channels. Statistically independent channels between any pair of transmit and receive anten-

nas are obtained by placing the antenna pairs a few wavelengths apart from each other.

Depending on whether multiple antennas are used for transmission or reception, the diversity

from MIMO can further be classified as transmit antenna diversity or receive antenna diver-

sity, or both. In transmit antenna diversity, multiple transmit antennas are deployed at the

transmitter to send multiple copies of a signal. If we take a mobile communication system

as an example, transmit antenna diversity is feasible in a downlink transmission (i.e., from

Base Station (BS) to Mobile Station (MS)) because multiple antennas can easily be deployed

at the BS. On the other hand, in receive antenna diversity, multiple receive antennas are de-

ployed at the receiver to receive multiple copies of the transmitted signal. This diversity can

be used to improve the uplink (i.e., MS to BS) transmission of the mobile communication

system.

The gains of MIMO in terms of increasing channel capacity, higher throughput, improved

error performance, and better energy efficiency are well established by now. In practice,

however, one limitation of MIMO is that installing multiple antennas on wireless nodes (e.g.,

MS in a mobile communication system) may not be feasible because of limitations in power,

cost, and/or size. To achieve full diversity gain in MIMO, there must be sufficient separation

between the antenna elements at the transmitter and receiver sides. If not sufficiently sepa-

rated, the fades of the channels between different antenna pairs will be correlated, thereby

reducing the diversity gain. In the mobile communication system example above, transmit

diversity in the uplink transmission may not be feasible, i.e., the uplink has some bottleneck

in current mobile communication systems.

To overcome the above mentioned drawback of MIMO, distributed wireless nodes (active

terminals or fixed relays) can be engaged in a cooperative fashion to emulate antenna diver-

sity. This mode of gaining transmit diversity is called cooperative transmission (also called

virtual antenna arrays [4], cooperative diversity [5], or user cooperation [6, 7]). In cooper-

ative transmission, nodes can share their time, frequency, and/or other resources to form a

distributed or virtual MIMO.

Figure 1.1 shows cooperation in a mobile communication deployment. Nodes in one shaded

region are partnered to cooperatively transmit each other’s message to the BS, i.e., the MSs

form (virtual) transmit antenna diversity for the uplink transmission. The lower part of the

V-MIMO

First phase

Second phase





BS Antenna Array

Cooperative transmisison











Figure 1.1.: Cooperative transmission in a mobile communication deployment. The lower

part of the figure shows two mobile stations, called source, S, and partner, P,

cooperatively sending their messages to a base station, D. Solid and dashed lines

show the transmission of the source’s message, 𝑖𝑠, and partner’s message, 𝑖𝑝,

respectively, and V-MIMO stands for virtual MIMO.

figure shows a three-node cooperation scenario where two mobile stations, called source, S,

and partner, P, cooperate while communicating with the base station, D. Each node forwards

the same message received from its partner (this is called repetition coding). The antenna

elements in the cooperative schemes are widely separated and connected through wireless

links, unlike the physical cabling in MIMO [8]; this necessitates cooperation to benefit from

the broadcast nature of the wireless channel.

Because of the spatial diversity gain, transmission failures are reduced in cooperative trans-

missions which leads to an increase of aggregate throughput. On the other hand, when users

share their resources, e.g., time, the effective transmission rate of an individual user will re-

duce; hence, it may not be intuitive at a glance whether the loss in data rate is compensated

for the spatial diversity gain. The performance of cooperative transmission is influenced by a

number of factors. To mention but two: type of cooperative protocol implemented and qual-

ity of channels among cooperating nodes (also called inter-user channels). If the inter-user

1. Introduction

First phase

Second phase



Figure 1.2.: Relay-channel system in the uplink of the mobile communication system, where

S, R, and D stands for the source, relay, and destination, respectively, and 𝑖𝑠is

the message of the source.

channels are error free, then cooperative transmission behaves like conventional MIMO. In

contrast, the performance of cooperative transmission is worse if the quality of the inter-user

channels is bad. Performance analysis, conditioned on the quality of inter-user channels,

is usually followed in the study of cooperative transmission protocols (e.g., [5, 9, 10]) and

is also adopted in this thesis. In the next section, the working principle of the three-node

cooperation and various protocols to implement cooperation are revised.

1.1. Review of Cooperative Transmission Protocols

In this section, various cooperative protocols for the three-node cooperation are discussed.

One way to view this cooperation is as an extension of the classical relay-channel system,

which consists of a source, a relay, and a destination [11]. Figure 1.2 shows a typical relay-

channel system in the uplink of the mobile communication system. The source broadcasts

its message1to the relay and destination in a first phase; the relay forwards the message

it has received to the destination in a second phase. The destination recovers the source’s

information bits based on only the message received from the relay. Usually, the relay is

located in the path between the source and destination to split a longer path into shorter

segments so that the effect of overall path loss is reduced. Even more so, if energy and

propagation environment constraints preclude Point-to-Point Transmission (PPT), relaying

emerges as the only option to provide connectivity [9]. However, spatial diversity is ignored

in the relay-channel system.

In the case of cooperative transmission, the relay node, after receiving the source’s message,

resends a processed version of this message to the destination. Consider the solid lines in

the first phase of Figure 1.3 that indicate transmission of the source’s message. Because of

1In the course of this thesis, we use ‘message’, ‘codeword’, and sometimes ‘packet’ interchangeably to refer

to a group of channel-coded bits.

1.1. Review of Cooperative Transmission Protocols

the broadcast nature of the wireless channel, the same message sent by the source is likely

to be received by the destination as well. The destination combines the two copies received

from the source and relay. This way, spatial diversity is exploited as the two messages

are received from potentially uncorrelated channels (contrary to the relay-channel system).

Moreover, depending on the location of the relay, cooperative transmission also benefits from

the path-loss reduction as in the relay-channel system [9].

The cooperative transmission explained above assumes the presence of a “dedicated” relay

node. In a case that both the source and relay have messages of their own, the two nodes

can be partnered such that they cyclically interchange their roles, i.e., in a next cycle, the

source becomes the relay and vice versa. This partnering of nodes is for mutual benefit

and hereafter we call the relay node partner node. As an example, consider the network

in Figure 1.3 where the source and partner send their messages 𝑖𝑠and 𝑖𝑝, respectively, to

the destination, D. In the first phase, the source and partner, using orthogonal channels,

transmit their messages to the destination as well as exchange each other’s message (because

of the broadcast nature of the channel). The orthogonality could be in time as in Time-

Division Multiple Access (TDMA) or frequency as in Frequency Division Multiple Access

(FDMA). In the following discussions, we assume TDMA-based channel sharing by the

source and partner as it is frequently used; however, all discussions hold for FDMA-based

counterparts.

In the second phase, the source and partner forward “processed” versions of each others’

messages to the destination. In repetition-coding-based cooperation shown in Figure 1.3,

the partner forwards the very message received from the source. The destination combines

identical messages received from the source and partner (e.g., by using Maximum-Ratio

Combining (MRC), Equal-Gain Combining (EGC), or Selection Combining (SC)) and de-

cides if the message is correctly received. On the other hand, if the source and partner

forward different versions of the received messages (e.g., in coded cooperation where nodes

forward incrementally redundant symbols [10]), then the destination uses Code Combining

(CC) to form stronger messages (in coding sense) from messages/symbols received in the

two phases, decodes, and makes decision on the correct reception of messages.

In the following, three salient contributions in the study of cooperative transmission are

revised. Sendonaris et al. were the first to introduce cooperative transmission and used the

name user cooperation [6, 7]. In their two-part paper, a three-node cooperative transmission

was developed for a Code-Division-Multiple-Access (CDMA) system (orthogonal codes to

avoid multiple-access interference) in which the source and partner operate in half-duplex

mode (i.e., cannot transmit and receive at the same time). Assuming knowledge of channel

phases at the source and partner, increased data rates for the cooperating users have been

demonstrated.

1. Introduction

First phase

Second phase







Figure 1.3.: Cooperative transmission in the uplink transmission of the mobile communica-

tion system; it is based on repetition coding. Solid and dashed lines show the

source’s message, 𝑖𝑠, and partner’s message, 𝑖𝑝.

Laneman et al. introduced cooperative diversity for the same three-node cooperation [5].

Several cooperative protocols were proposed and their outage behavior was analyzed. In this

work, more practical considerations such as half-duplex and orthogonality constraint based

on TDMA (i.e., a node cannot transmit its own and partner’s data simultaneously at the same

frequency) were considered.

Hunter et al. proposed coded cooperation and analyzed its outage behavior [10]. In coded

cooperation, error control coding was incorporated into cooperation such that the source

and partner cooperate by transmitting incrementally redundant symbols to each other. In

this protocol, the source encodes 𝑘of its source bits into a codeword of 𝑁symbols, and

the 𝑁symbols are further partitioned into a weaker codeword of 𝑁1symbols and 𝑁2=

𝑁−𝑁1incremental symbols. The source transmits the weaker codeword in the first phase,

and if the partner successfully decodes the 𝑘bits from this codeword, then it generates and

transmits the source’s 𝑁2remaining symbols. If decoding fails, then the partner will transmit

additional 𝑁2symbols of its own message. At the destination, the symbols received from the

source and partner are code combined to form a codeword of 𝑁symbols, and this codeword

is further decoded to generate information bits of the source [10, 12].

In general, the cooperative protocols mentioned in the above prominent (and plenty of follow-

up) literature can be broadly categorized based on various parameters, to mention but few:

relaying/forwarding strategy, level of adaptiveness to decoding error, and type of coding used

in the second phase. Considering the relaying strategy at the relaying node (in the following,

we consider the transmission of the source’s message only such that the partner will be the

relaying node; the same argument holds for the partner’s message), some of the common

relaying strategies are:

∙Amplify-and-forward: In this relaying strategy, the partner simply forwards an ampli-

fied version of the received message (works in the analog domain) [5]. Because of the

noise added at the partner and as there is no error-checking mechanism, the forwarded

message is a noisy version of the original message from the source. In spite of the

1.1. Review of Cooperative Transmission Protocols

noise propagation, it was shown that amplify-and-forward can achieve full diversity

gain, which is equal to two for one relay.

∙Decode-and-forward: In this relaying protocol, the partner decodes the received mes-

sage, re-encodes it (using the same codebook as in repetition coding [13] or a different

codebook as in coded cooperation [10]) and then forwards it to the destination [5, 9].

Decode-and-forward requires correct decoding of the message at the partner for the

forwarded message to be usable at the destination; otherwise the forwarded message

leads to error propagation and further decoding error at the destination. The perfor-

mance of this protocol is limited by the worst link of either source-partner or source-

destination, as the protocol will not benefit from either the relayed transmission or the

direct transmission [14, 15]. Comparing the above two forwarding strategies, decode-

and-forward protocol performs better when the two inter-user channels are of high

quality (where chance of correctly decoding is high); on the other hand, amplify-and-

forward performs better when the inter-user channels are of poor quality.

∙Compress-and-forward: In this protocol, the partner first samples, quantizes, and com-

presses (in order to reduce redundancy) the received message. Second, it encodes the

compressed message into a new message (as if they were information bits) and for-

wards it to the destination [16, 17]. The destination jointly processes the observa-

tions from the source and partner. The compression at the partner is realized using

Wyner-Ziv source coding [18, 19]. As this protocol works in semi-analog and semi-

digital domain, it incorporates benefits of both the amplify-and-forward and decode-

and-forward protocols.

In the decode-and-forward-based protocol, when the inter-user channels are bad, the part-

ner is more likely to forward an erroneous message. Moreover, when the source-destination

channel is not very bad (or even better than the partner-destination channel), a high per-

centage of messages transmitted by the source are likely to be received correctly by the

destination; in this case transmissions from the partner are a waste. To overcome these draw-

backs, relaying protocols can be designed to adapt to decoding results at the partner and/or

destination. This leads to further classification of protocols as static and adaptive.

∙Static protocols: In static (or fixed) protocols, the partner always forwards the source’s

message without checking errors [5, 9]. This protocol is easy to implement, but is

prone to error propagation.

∙Adaptive protocol: Here, the partner decides whether to forward or not, depending on

its success of decoding the source’s message. If successful, then it may forward the

same message or a modified version of it. If decoding fails, then the partner has the

1. Introduction

options to switch to amplify-and-forward, transmit its own message, or even remain

silent [9].

In adaptive protocols, the decision at the partner to either forward the received message or

switch to the other options can be made at different levels of granularity. To mention two:

∙Cyclic Redundancy Check (CRC): The received message/packet is completely decoded

and then a CRC is performed. This approach is useful in a packet-based transmission

where the CRC is done on a group of bits, and it insures that the partner forwards only

correctly received packets [15].

∙Threshold-based decoding: Using this approach, the partner forwards if the Signal-to-

Noise Ratio (SNR) of the received signal exceeds a certain threshold [9]. Alternatively,

soft information, which is obtained from the Log-likelihood Ratio (LLR) value of the

received signals, can be used instead of the SNR [20, 21]. In threshold-based decoding,

the choice of an appropriate threshold is not easy, decision is done on signal/bit level

rather than on packet level, and is less reliable than the CRC-based approach because

a message may be received in error even if its SNR (or soft information) is above a

threshold.

Depending on the level of Channel-State Information (CSI) available at the source and part-

ner, adaptive protocols are further categorized as selective or incremental relaying [5].

∙Selective relaying: When the source knows the fading state of the inter-user channels,

then it can reasonably estimate if the partner correctly receives and forwards its mes-

sage to the destination. In the case of decoding failure at the partner, the source can

transmit a copy of its message to the destination instead. One drawback of this relaying

is the high overhead of acquiring CSI, which may overwhelm the gain from coopera-

tion. Alternatively, reciprocity in reception can be assumed, i.e., if the source decodes

the partner’s message, then the source can assume that its message is also likely to be

decoded by the partner.

∙Incremental relaying: Here, the destination feedbacks an acknowledgment to the part-

ner if it was able to receive the source’s message correctly from the first-phase trans-

mission (assuming a feedback channel from the destination to the partner). Accord-

ingly, the partner stops forwarding the source’s message. In this relaying scheme,

nodes cooperate only when the direct link to the destination fails. It was shown that

the protocol has better spectral efficiency as the partner does not need to always trans-

mit (especially when the source-destination channel is good and a high percentage of

the messages transmitted by the source can be received correctly by the destination)

[5].

1.1. Review of Cooperative Transmission Protocols

In the decode-and-forward-based schemes, once the partner receives the source’s message

correctly it can employ the following coding strategies before forwarding the message to the

destination.

∙Repetition coding: The partner uses the same codebook used at the source. At the

destination, either MRC, EGC, or SC is used to combine the two messages received

from the source and partner [12]. The forwarded message helps to accumulate SNR

(i.e., increase the strength of the signal); however, the partner forwards no additional

information that would help decoding.

∙Incremental redundancy coding: The partner uses a different codebook to re-encode

the received message and forwards the resulting message to the destination. At the

destination, this message is used as incrementally redundant information and code

combining is used to combine it with the message received from the source in the

first phase [11, 10]. One drawback of repetition coding and incremental redundancy

coding is that all the resources in the second phase are dedicated to either the source or

partner only. This leads to unfair cooperation especially when the source-partner and

partner-source channels have different quality, where one user may forward for the

other but not vice versa. To overcome this drawback, the following coding approaches

are proposed.

∙The partner combines the source’s message with its own message (e.g., network coding

[22, 13], space-time coding [23, 24, 25], superposition modulation [26], or differen-

tial modulation [27, 28]) and forwards to the destination. Using one of these coding

strategies, resource in the second phase is shared between the two users instead of

being dedicated to one user. At the destination, this message is “combined” with the

messages received in the first phase and then decoded to recover the source’s message.

In addition to the above classification of protocols, other considerations in the study of coop-

eration are: purpose of cooperation, resource allocation, and type of diversity gain. Purpose

of cooperation is related to whether cooperation is beneficial for a single user (called user-

level cooperation and occurs, for example, when the partner is a dedicated relay with no

data of its own) or both users (called system-level cooperation and is for mutual benefit).

Resource allocation refers to how users allocate their resources to cooperation; this includes

total energy (how the total energy is allocated in the two phases of cooperation) and avoiding

multiple access interference (orthogonality constraints and half-duplex transmission). Type

of diversity gain refers to whether the gain is from spatial diversity, temporal diversity, or a

combination of these two gains. Temporal diversity is achieved when the fading statistics of

a channel changes from phase one to phase two, and in this case diversity gain is achieved

by repeating an own message. If the channel state remains constant over the two phases,

1. Introduction

the channel is called block fading. Block fading channel is the most common assumption in

almost all work in the area of cooperation.

The various protocols discussed above are summarized in Figure 1.4. Shaded areas in the

diagram indicate protocols/parameters considered in this thesis. We see that by combining

the various degrees of freedom (e.g., relaying strategy, level of adaptiveness, coding strategy)

one can design a variety of cooperative transmission protocols, trading-off between complex-

ity of a protocol and its performance [9]. Section 1.2 briefly discusses the working principle

of network-coded cooperation protocols, which are the focus of this thesis, and reviews some

of the related work.

1.2. Network-Coded Cooperation

In the previous section, we have presented various cooperative protocols and mentioned that

network coding is one coding strategy for decode-and-forward-based protocols. This thesis

focuses on protocols that use network coding; hereafter, we call the protocols “network-

coded cooperation”, or a variant of it. In this section, we briefly talk about two possible

implementations of network-coded cooperation and present a literature survey on existing

approaches and their limitations.

1.2.1. Review of network-coded cooperation

In network-coded cooperation, the partner, instead of simply forwarding a received mes-

sage, transmits a network-coded version of its own and the source’s messages (assuming

it has correctly received the source’s message). The network-coded message is a weighted

sum (or linear combination) of the received and locally generated (own) messages, where

the weighing coefficients are generated either randomly or deterministically. This coded

message contains information bits of both users, unlike other cooperative schemes where a

message forwarded in the second phase contains information originating from one user only.

At the destination, network-coded message(s) received from the source and/or partner are

used to recover both the source’s and partner’s information bits, i.e., both users benefit from

any one or both of the two coded messages sent by the source and partner. Conceptually,

the network-coded cooperation resembles the distributed space-time cooperation protocols

in [23, 10], where, for example, space-time block codes based upon orthogonal designs are

proposed to implement the coding.

1.2. Network-Coded Cooperation

Cooperative

Transmission

Forwarding

strategy

Amplify-and-forward

Decode-and-forward

Compress-and-forward

Level of adaptiveness

Static (partner always forwards)

Adaptive (partner decides when to forward)

Forwarding decision

Cyclic redundancy check

Threshold-based decoding

Coding strategy

Repetition coding

Incremental redundancy

Network coding

Space-time coding

Superposition/differential modulation

Channel state

information

Selective relaying (source re-transmits its own message)

Incremental relaying (partner forwards based on

feedback from the destination

Combining scheme

Selection combining

Maximum ratio combining

Code combining (for incremental redundancy transmission)

Space-time coding

Purpose of cooperation

User benefit (e.g., when the partner is a

dedicated relay with no data of its own)

System benefit (mutualism)

Diversity

Spatial diversity

Are channels in two phases

uncorrelated?

Yes, temporal diversity

No, no temporal diversity (called block fading channels)

Resource allocation

Total energy

Shared equally in the two phases

Shared differently in the two phases

Orthogonality constraint

Time

Frequency

Code

Duplex Half-duplex

Full-duplex

Figure 1.4.: Classification of cooperative transmission schemes. Shaded areas indicate the

focus in this thesis.

1. Introduction

First phase

Second phase

⊕

⊕ ⊕

⊕ i

⊕

⊕ ⊕

⊕ i

Figure 1.5.: System diagram of network-coded cooperation in the uplink transmission. Solid,

dashed, and dotted lines show the transmission of the source’s message, partner’s

message, and network-coded message, respectively.

One simple implementation of network coding is by modulo-2 summation of received mes-

sages. Figure 1.5 shows that the network-coded cooperation and transmissions in the second

phase are the modulo-2 summed messages 𝑖𝑠⊕𝑖𝑝. Once the destination receives the four

messages, it first combines the two network-coded messages using either SC or MRC. Then

the source’s message, for example, is recovered using one of the two options: either from the

source’s first-phase transmission, provided no error occurs, or through additional network

coding of the partner’s and network-coded messages, i.e., 𝑖𝑠=𝑖𝑝⊕(𝑖𝑠⊕𝑖𝑝), provided both

messages are correctly received. Note that, at the destination, the network-coded message

is considered as an independent message and will be usable if it is received error-free. We

call this cooperation conventional network-coded cooperation to differentiate it from other

network-coding-based cooperation to be discussed in the next paragraph. Intuitively, as in-

formation bits are recovered using either of the two options, a maximum diversity order of

two can be obtained. The actual value of the diversity order depends on other system pa-

rameters such as received SNR and transmission rate. A detailed discussion on the working

principle of this protocol and investigation of its outage behavior are presented in Chapter 4.

Alternatively, when both network and channel codings are used, network coding can be

done on parity symbols of the two users. We coin this cooperation incremental redundancy

network-coded cooperation. In this cooperation protocol, the partner, for example, after

receiving the source’s message in the first phase, generates two groups of parity symbols:

one for its own and the other for the source. Then it network codes these groups of parity

symbols to form network-coded parity symbols and forwards the symbols to the destination

in the second phase. At the destination, the network-coded parity symbols are code combined

with message(s) received in the first phase to form a stronger codeword. This is unlike

conventional network-coded cooperation where the network-coded message is considered as

an independent message. We see that this approach embeds network coding into channel

coding in a way that the redundancy in the network code is used to support the channel code

1.2. Network-Coded Cooperation

for better error protection [29, 30]. A detailed discussion of the working principle, various

coding approaches, and the outage behavior of incremental network-coded cooperation are

discussed in Chapter 6.

1.2.2. Literature survey on network-coded cooperation

So far, we briefly explained the use of network coding in cooperative transmission; next,

we review some of the work in network-coding-based cooperation that are related to our

study. In general, different terminologies are used for various ways how network coding is

implemented. To mention some: joint vs. separate network-channel coding [31], symbol vs.

packet level [32], analog (or continuous) vs. digital [33], physical layer vs. network layer

[33].

The idea of using network coding in a two-way relay channel system is investigated in

[34, 30] and references therein. The two-way relay channel consists of two sources that

want to exchange information with the help of a third dedicated relay. The relay is usually

located in the path between the two sources, and this relay is referred to as a “routing node”

[34]. If no network coding is used, a total of four time slots is required to complete the

exchange of information: two slots for each source to transmit to the relay and another two

slots to forward/route from the relay to each source. If network coding is used instead, the

relay modulo-2 sums the messages received from the two sources and sends the message in

the third time slot, i.e., three slots are sufficient to exchange the information and this im-

proves system throughput. This network coding is sometimes called digital network coding

or hard-decision network coding, since the coding is done on digital/hard bits [34]. The joint

network-channel coding can also be designed to exploit the broadcast nature of the wireless

channel [29]. As a message sent by one source may reach both the relay and the other source,

the relay forwards incrementally redundant information that provides additional error protec-

tion at the two sources. This approach is called joint network-channel coding [29]. Almost

all work in the area of two-way relaying assumes that the relay always decodes messages

from the two users correctly.

The same joint network-channel coding approach for a multiple-access relay channel sys-

tem is discussed in [30, 31, 35]. The multiple-access relay channel consists of two sources,

one dedicated relay and one destination. In [30, 36], the authors consider the design of a

joint network-channel code based on turbo codes and allocation of transmission time among

the two sources and relay. Three transmissions from the two sources and the relay are re-

quired (instead of four if no network coding is used). It is reported in [29, 31, 30] that joint

1. Introduction

network-channel coding exploits the redundancy from the relay more efficiently than sepa-

rate network-channel coding (where the relay forwards the incremental messages of the two

sources separately using two time slots). The results presented in [29, 31, 30] were based

on simulation, and the channels between the source(s) and relay were assumed to be ideal,

i.e., the relay always decodes messages from the two sources. Moreover, the relay has no

messages of its own and serves the sources only. For non-ideal source-relay channels, we

have performed the outage analysis of this scheme in [37].

Recall that in conventional cooperation systems, the available time slots are divided into two

subslots: one for the first-phase transmission and one for the second-phase transmission. The

use of such orthogonal and interference-free channels simplifies receiver implementation but

results in loss of spectral efficiency (as messages of high information rate should be trans-

mitted in each phase). To form low-rate codes that overcome the loss in spectral efficiency,

the idea of algebraic superposition of either channel codes [22], modulated bits [26], or su-

perposition in analog domain [38] (also called analog network coding) are introduced. The

basic idea is the source, for example, assuming knowledge of the partner’s previous message,

always sends the superposition of its own current and partner’s previous message; where the

superposition could be after channel coding, on modulated bits, or in the analog domain

(e.g., using dirty paper coding as in [38]). The partner extracts the source’s message from

the received superimposed message, does further superposition of its current message with

the source’s message, and further transmits to the source and destination. We see that, as

the source transmits for its own and simultaneously forwards for its partner, there is no need

for two-phased transmissions. The destination implements some kind of iterative decoding

on hard or soft information or use dirty paper decoding. Complexity of the receiver is one

drawback of this approach.

For the three-node cooperation, the work in [39] proposed a cooperation protocol in which

the partner concatenates its own and the source’s information bits to form longer information

bits (as opposed to modulo-2 summation of the two information bits). The longer informa-

tion bits form systematic bits of a systematic code. The partner channel codes the longer

information bits and transmits the codeword. In this way, in a single transmission a user

transmits its own message, forwards for its partner, and transmits parity bits that protect

messages of the two users. The notion of network coding is embodied on the parity bits as it

is the result of the information bits from the two users. It was shown, by analysis and simu-

lations, that this network coding scheme is more tolerant to poor inter-user channels than the

repetition-coding-based cooperation.

Katti et al. proposed, for the two-way relay channel system, to include wireless interference

through analog network coding in [34]. This scheme allows the two sources to transmit si-

multaneously so that they interfere at the relay node; the relay forwards the interfered signals.

1.2. Network-Coded Cooperation

Each source, given knowledge of its own message, can extract the other source’s message

from the message received from the relay. The scheme is called analog network coding be-

cause the relay mixes analog signals, not bits. We see that only two time slots are required to

complete the information exchange between nodes and this gives an improvement in spec-

tral efficiency as compared to the digital network coding discussed above, which requires

three slots. Note the analogy between analog network coding and its digital counterpart.

In digital network coding, the sources transmit using orthogonal channels; the relay super-

imposes (mixes) the content of the messages/packets and broadcasts the mixed version. In

analog network coding, the sources transmit simultaneously and the wireless channel mixes

these signals. Instead of forwarding mixed packets, the relay amplifies and forwards mixed

signals. Drawbacks of this approach are: the two sources should synchronize their transmis-

sions such that they interfere at the relay with no delay; this is difficult, if not impossible

to achieve. Moreover, like any analog scheme, there is no error checking at the relay which

makes error propagation inevitable.

Zhang et al. also proposed a similar approach to analog network coding and used the name

physical layer network coding [40]. The authors explained, through a proper design of mod-

ulation and demodulation techniques, how mapping/de-mapping of modulated/demodulated

symbols at the physical layer of relaying nodes can emulate the modulo-2 summation of

digital bit streams as in digital network coding. Historically, network coding is implemented

at the network layer of Open Systems Interconnection (OSI) architecture [41, 42]; however,

the use of analog, physical-layer, and joint network-channel network codings pushes the

implementation down to the physical layer.

The use of soft-bit information in network-coding-based cooperation is another recent de-

velopment [21, 43, 44]. Soft-bit information refers to the confidence interval of a received

bit and is computed from the LLR value of that bit. For the multiple-access relay chan-

nel system, the soft-bit-based protocols are proposed mainly to overcome one drawback of

the hard-decision network coding protocols in [30, 31, 35], where the relay must correctly

receive transmissions from the two sources. In this regard, for Additive White Gaussian

Noise (AWGN) channels, Koetter et al. investigate the network coding gain even when trans-

missions to the relay cannot be recovered correctly [44]. Instead of decoding the messages

received from the two sources, the relay computes the LLR values of the two messages and

performs an operation on the LLR values that emulates the modulo-2 summation in network

coding. As the resulting LLR takes on continuous values, it is modulated using one of the

known analog modulation techniques and the signal is forwarded to the destination. At the

destination, an iterative decoding algorithm based on the three observations (two from the

two sources and one from the relay) is run. Simulation is used to measure performance

and Bit Error Rate (BER) results for various sources-relay, relay-destination, and source-

1. Introduction

Network-coded

cooperation

Topology

Two-way relay channel

Multiple-access relay channel

Three-node cooperative transmission

Relayed information

Incremental redundant symbols

Independent messages

Analog vs. digital domain

Analog network coding (users transmit

simultaneously and no error detection mechanism)

Continuous network coding

(work on soft bits)

Digital network coding (based on decode-and-forward

and hard bits and may use CRC-based error checking)

Coding level

Packet level network coding

Symbol level

network coding

Other considerations

Analysis- and simulation-based results? YES

Non-ideal source-relay channel? YES

AWGN and fading channel models? YES

Figure 1.6.: Classification of network-coded cooperation schemes. Shaded areas indicate the

focus in this thesis.

destinations channels SNR values are presented (unfortunately, no comparison with other

protocols is presented). Pu et al. have also investigated the multiple-access relay channel

system with soft-bit information and used the name continuous network coding [43]. In their

approach, the soft-bit-based information forwarded by the relay is used as incremental re-

dundancy information at the destination. The authors reported that relaying a network-coded

version of the soft information gives significant gain over traditional network coding (i.e.,

based on hard-decision at the relay).

Finally, a network of one source communicating to one destination using multiple relays is

introduced in [32]. In this scheme, the relays forward selected groups of bits of a packet

instead of the entire bits of a packet, and it can be extended to a network of multiple sources

and multiple destinations. Katti et al. called the scheme MIXIT [32]. The basic motivation

is to minimize the packet discarded by relaying nodes in the event of incorrect reception, as

some of the bits in a packet are likely to be correct. MIXIT works as follows: at a source

node, a given packet is divided into sub-packets (which are groups of bits also called sym-

bols), network code the symbols, and send the coded symbols to the relays and destination.

1.3. Thesis Motivation and Contributions

A relaying node estimates the correct reception of each coded symbol using soft-bit infor-

mation, discards those that are more likely to be incorrect, and forwards the remaining coded

symbols. The destination also estimates and collects symbols that are more like to be correct,

and re-computes the original message using these symbols. The network coding approach is

also called symbol-level network coding. By forwarding symbols that are more likely to be

correct, it is reported that MIXIT achieves higher throughput than the traditional approach,

were only correctly received packets are forwarded. Figure 1.6 summarizes the various pro-

tocols revised above and the shaded areas indicate the focus of this work.

1.3. Thesis Motivation and Contributions

In the previous section, we have revised some of the network-coded cooperation protocols

related to this work. The advantages and limitations of each protocol were mentioned. In

the following, we point out some of the common assumptions in most of the network-coded

cooperation papers and limitations of the assumptions.

1.3.1. Thesis motivation

Network coding in general, and its application in cooperative transmission in particular, is

a recent field of study. So far, research is mainly focused on introducing the concept of

network coding in various application scenarios and designing new protocols. In the de-

sign of a new protocol, various parameters and assumptions come into play. At an initial

stage of research, keeping the overhead of the assumptions as small as possible, at the same

time elaborating on the concept is a procedure usually followed; and this is also the case

in network-coded cooperation. While reviewing existing work during the inception of this

thesis, we came across few assumptions in already published papers that, to our believe, are

unrealistic. Investigating network-coded cooperation by making more realistic assumptions

was the initial motivation of this thesis. In the following, we point out some of the limiting

assumptions common to most previous work.

∙The inter-user channels are assumed reliable such that relaying nodes2reliably decode

received messages and always forward network-coded message only [29, 35]. This

assumption simplifies the protocol design as no decision and adaptation strategies are

required in the case of reception failure. Moreover, the assumption simplifies analysis

(e.g., outage analysis). One notes that the decoding requirement is most likely fulfilled

2Partner in the case of three-node cooperation and dedicated relay in the case of multiple-access relay channel

and two-way relay channel systems.

1. Introduction

when the source and relaying node are close to each other; restricting the two nodes to

be closer to each other is a bottleneck as it inhibits further study in terms of node de-

ployment, partner selection, and coverage area extension (which requires the relaying

node to be located anywhere in the network where the chance of decoding all the time

may not be possible).

∙Recent results show that network-coded cooperation performs better than repetition-

coding-based cooperation when the inter-user channels are of poor quality [13, 39];

this contradicts the ideal inter-user channel assumption.

∙A thorough information theoretic analysis of outage probability, which leads to a fur-

ther study of outage behavior and diversity-multiplexing tradeoff, is missing. Simula-

tion is usually used to investigate protocols.

Based on the above three observations, our initial motivation, in a nut shell, was to theo-

retically analyze outage probability of the network-coded cooperation when the inter-user

channels are non-ideal. Using the developed outage probability expression, we have investi-

gated the performance of network-coded cooperation. A summary of the main contributions

of this thesis are presented next.

1.3.2. Thesis contributions

One problem when considering non-ideal inter-user channels was that the protocols have to

be redesigned to take decision in the case of decoding failure (i.e., should adapt to decoding

errors). This adaptiveness has to be reflected in the analysis, which makes the analysis at first

challenging and difficult to approach. For non-ideal inter-user channels, here are the main

achievements in this work:

1. The outage behavior of network-coded cooperation protocol is examined by deriving

its outage probability; this is reported in Chapter 4 and is already published in [45, 13].

Outage probability helps to study the protocol independent of any particular coding

scheme and is also shown to be a lower bound on block error rate for sufficiently large

block lengths [10]. To make the outage probability analysis more tractable and con-

venient for exposition, quasi-static (or block) Rayleigh fading channels, orthogonal

transmission, and half-duplex constraints are assumed. Approximating the outage re-

sult at high SNR values, we shown that this protocol achieves full diversity (order two

for two users) asymptotically in user transmit power [45, 46].

1.3. Thesis Motivation and Contributions

2. We investigate the outage behavior for various inter-user and uplink (between a trans-

mitting node and destination) channel qualities; we compare various cooperative pro-

tocols based on the inter-user channels. Based on the outage results, network-coded

cooperation protocols are found to be suitable when the inter-user channels are lower

quality; when the inter-user channels are good, protocols without network coding per-

form better [13].

3. The outage results are further extended to study the diversity-multiplexing tradeoff and

the coverage area extension and results are also published in [46].

Once outage results are available, the next step is to address energy efficiency of network-

coded cooperation in wireless sensor networks. The motivation is that cooperation helps

wireless nodes to achieve spatial diversity, which allows to reach equal error rate at lower

transmit power. However, relaying redundant messages consumes considerable energy at

both transmitting and receiving nodes. Striking a balance between the diversity gain and

additional consumed energy is the intention of this step. By defining appropriate energy

consumption metrics, a study is conducted if and when cooperation is energy efficient than ;

detailed discussion is given in Chapter 5.

The outage behavior of incremental redundancy network-coded cooperation is investigated

next; results are reported in Chapter 6 and also published in [47]. As explained above, in

incremental redundancy network-coding, the network coding is embedded into the channel

coding such that the redundancy in the network code is used to support the channel code for

better error protection [29, 30]. With this in mind, the steps followed and results obtained

are as follows.

1. Two decoding approaches, namely joint network-channel decoding and individual

network-channel decoding, are proposed first. In the former, one ‘big’ codeword is

formed from all symbols received in the two phases, and the information bits of both

source and partner are obtained from a single decoding of this big codeword. In the

latter approach, the source’s and partner’s information bits are recovered by two inde-

pendent decodings.

2. The outage behavior, for quasi-static Rayleigh fading channels, orthogonal transmis-

sion, and half-duplex constraints, is studied next. The outage results show that this

scheme also achieves full diversity order of two.

3. Then, using the outage result, ‘optimal’ rate and energy allocations that minimize

the outage probability are studied. The results show that outage performance is more

sensitive to the energy allocation than to the rate allocation.

1. Introduction

All in all, the outage probability analysis, based on realistic non-ideal channel assumption, of

both types of network-coded cooperations help to investigate if and under what condition(s)

cooperation benefits from network coding. Also, the outage probability approximations at

high SNR values paved the way for further study in diversity-multiplexing tradeoff, energy

efficiency computation, and rate and power allocation. Finally, our analysis approach has

been followed in recent publications, to mention but few: [48, 49, 50, 51].

1.4. Organization of the Thesis

The rest of the thesis is organized as follows. In the following chapter, we present an

overview of the salient work in network coding, encoding and decoding operations in a

linear network coding, and symbol- and packet-level network coding. Chapter 3 overviews

the system and channel models used in this work. It also discusses the layering issue in

wireless networks and briefly explains some of the performance measures used in the study.

In Chapter 4, we compute the outage probability for network-coded cooperation and show

the performance benefits, in comparison with the point-to-point transmission. The diversity-

multiplexing trade-off and coverage area are also discussed. By defining appropriate energy

efficiency metrics for sensor networks, the energy efficiency of network-coded cooperation

is studied in Chapter 5. The model takes into account all the transmission and processing

energy costs and shows when network-coded cooperation is better than point-to-point trans-

mission. The outage behavior of the incremental redundancy network-coded cooperation is

investigated in Chapter 6; moreover, optimal energy and rate allocations were investigated.

Chapter 7 summarizes the findings of this thesis and gives the main conclusions. In addition,

some recommendations for future research are also outlined.

2. Introduction to Network Coding

Network coding is a relatively new field of study which, initially, was proposed to increase

network throughput. It is concerned with coding at a node in a network with error-free

point-to-point links. This error-free-links requirement distinguishes network coding from

channel coding, which is designed for error protection in noisy links [52]. In this chapter, we

will give a brief overview of network coding. Section 2.1 introduces network coding using

a butterfly network and reviews some of the fundamental research in the area. A type of

network coding, called linear network coding, is discussed in Section 2.2 and the encoding

and decoding operations are explained. Section 2.3 highlights the interaction of network

coding and channel coding in error-prone networks.

2.1. Network Coding in Error-Free Networks

Consider a multicast network, where information is usually transmitted from a source node

to each sink node through a chain of intermediate (also called routing or forwarding) nodes

by a method known as store-and-forward. In this method, data received from an input link

of a routing node is stored and a copy is forwarded to the next node via an output link [53].

In this method, routing nodes only replicate incoming data without processing them.

In order to improve the throughput of a network, Ahlswede et al. recently introduced network

coding as an alternative approach to simple forwarding of data at routing nodes and demon-

strated its advantage over the store-and-forward approach [41]. In a network with error-free

links, one general definition of network coding is an arbitrary mapping of data at the inputs

of a node to output data [52]. The error-free-links assumption distinguishes the function of

network coding from that of channel coding, which is designed for error protection in noisy

links. In network coding, a node in the network is allowed to combine (usually linearly)

several messages it has received or created into one or several outgoing messages and for-

wards to nodes to which it is connected [41, 54, 42]. The sink node, after receiving sufficient

number of encoded messages, extracts the messages that were originally intended for it.

2. Introduction to Network Coding











 

























































































 





⊕

⊕⊕

⊕













(a ) (b )

Figure 2.1.: Butterfly topology where node 𝑆is a source with two messages, 𝑢1and 𝑢2. Both

sink nodes, denoted as 𝑡1and 𝑡2, need to receive both messages and all links are

assumed to have a capacity of one packet per unit of time. (a) Transmission using

the store-and-forward approach. (b) Using network coding, one can send 𝑢1⊕𝑢2

down the middle link from node Cto node Dand deliver the two messages.

The principle of network coding is best explained with the butterfly example depicted in

Figure 2.1 [41]. The butterfly network illustrates that, by sending fewer messages than the

store-and-forward approach, network coding can increase throughput in multicast networks.

Assume that all links are error free and have a capacity of one packet per unit of time. The

source, S, wants to deliver messages 𝑢1and 𝑢2to two sink nodes 𝑡1and 𝑡2. Figure 2.1 (left)

depicts the conventional store-and-forward approach. Router nodes Cand Dcan deliver

either 𝑢1to 𝑡2or 𝑢2to 𝑡1. The link between nodes Cand Dacts as a bottleneck as it has to

be used twice and only one packet can be sent at a time. The other links are used once.

Figure 2.1 (right) depicts the solution with network coding. Node Cperforms a modulo-2

addition of the two incoming messages to get 𝑢3=𝑢1⊕𝑢2and forwards 𝑢3to both sinks via

node D. Sink 𝑡1gets 𝑢1from node Aand can recover 𝑢2with another modulo-2 addition as

𝑢2=𝑢1⊕(𝑢1⊕𝑢2). Similarly, 𝑡2recovers 𝑢1as 𝑢1=𝑢2⊕(𝑢1⊕𝑢2). Contrary to the store-

and-forward solution, all links have to be used once and one channel use is saved through

network coding. Network coding can also be used to improve robustness, complexity, and

security of a network and a detailed discussion is available in [52].

Once the concept of network coding is clear, we briefly discuss the history of network coding

by revising some of the pioneer work that has greatly contributed to the emergence of net-

work coding.1The concept of network coding originates from the seminal paper of Ahlswede

et al. [41]. Their work focused on improving network multicast capacity for applications in

1An exhaustive list of literature on network coding is available at [55].

2.2. Linear Network Coding

computer networks (e.g., Internet backbone). In an error-free point-to-point network (as-

suming that the effect of the channel noise is removed by using powerful channel codes or

retransmission in the link layer), the authors in [41] proved that a source can multicast 𝑘mes-

sages to a set of sinks, provided the min-cut between the source and each sink has capacity

𝑘per unit time.

The work by Li et al. has laid the theoretical foundation for constructing network coding by

showing that the min-cut capacity for the multicast problem can always be achieved using

alinear network code [54]. They presented how to explicitly construct the linear network

codes. This focused attention on linear codes in particular raised the question of whether

they can be used to solve a wider array of network coding problems [56]. Their proof of the

existence of a linear solution can be viewed as the first deterministic algorithm for network

coding. However, its run time is exponential in the size of the network.

Koetter et al. established a simple and effective algebraic framework for network coding

[42]. This reduced the problem of finding a linear solution for a general network coding

problem to finding a non-zero point for a multivariate polynomial. Ho et al. proposed a

random linear network coding algorithm [57]. Each node in the network independently and

randomly encodes the received messages over some finite field. They gave a lower bound

on the probability that an independent, random linear code design at every node achieves the

multicast capacity. The probability approaches one as the size of the finite field approaches

infinity. The main benefits of random network codes are distributed implementation without

coordination between nodes in the network and robustness with respect to network change

or channel failure. In the next section, we will discuss the encoding and decoding operations

in linear network codes.

2.2. Linear Network Coding

Much of the work in network coding has concentrated around Linear Network Coding (LNC)

[52]. LNC requires messages, being communicated through the network, to be accompanied

by some degree of extra information, in this case, a vector of encoding coefficients. In

packet networks, data is divided into packets, network coding is applied to the content of the

packets (hereafter, we call this content the information vector), and the extra information can

be placed in the packet header [52].

In LNC, each node in the network forwards a new packet on its outgoing link. This packet

is formed by linearly combining earlier received information vectors (or locally generated

information vectors if it is a source node) and then appending the encoding vector [58].

2. Introduction to Network Coding

Assume that each information vector consists of 𝑙bits. When the information vectors to be

combined do not have the same size, the shorter ones are padded with trailing 0s. For coding,

the 𝑠consecutive bits of an information vector are treated as a symbol over the Galois field

𝔽𝑞with 𝑞= 2𝑠; hence, an information vector consists of 𝑚=𝑙

𝑠symbols. The coding

coefficients are from the finite field and addition and multiplication are performed over the

field 𝔽𝑞.

2.2.1. Encoding

Assume that 𝑛original information vectors, 𝑥1,...,𝑥𝑛, are generated by a source node 𝑘;

we call these vectors source vectors. In LNC, each encoded vector from the source, denoted

as 𝑦𝑘∈𝔽𝑚

𝑞, is associated with a sequence of coefficients 𝑔𝑘= (𝑔𝑘,1,...,𝑔𝑘,𝑛)∈𝔽𝑛

𝑞, also

called global encoding vector.2The encoding vectors are used to linearly combine the source

vectors 𝑥𝑖in order to produce the encoded vector 𝑦𝑘as

𝑦𝑘=

𝑛

∑

𝑖=1

𝑔𝑘,𝑖 ⋅𝑥𝑖.(2.1)

Note that encoded vectors have the same size as the source vectors and contain only a fraction

of the information contained in the source vectors. To recover the 𝑛source vectors at a sink

node, the source should perform the encoding at least 𝑛times, each time using a new set

of encoding vectors to insure that the encoded packets are linearly independent. Moreover,

knowledge of the global encoding vector is required at the sink node. For this reason, the

source forwards a packet containing both the global encoding vector, 𝑔𝑘, and the encoded

vector, 𝑦𝑘. Hereafter, we use a 2-tuple (𝑔𝑘, 𝑦𝑘)to represent a transmitted packet.

The encoding in Equation (2.1) is on a packet level as the summation is done on each packet.

In symbol-level network coding, the summation has to occur for every symbol of the source

vectors, and each symbol of the encoded vector is given by

𝑦𝑘,𝑚 =

𝑛

∑

𝑖=1

𝑔𝑘,𝑖 ⋅𝑥𝑖,𝑚 (2.2)

where 𝑥𝑖,𝑚 and 𝑦𝑘,𝑚 are the 𝑚𝑡ℎ symbols of the vectors 𝑥𝑖and 𝑦𝑘, respectively.

At forwarding nodes, it is not necessary to decode received packets (i.e., recover the source

vectors) in order to create new encoded packets. Instead, the same encoding operation as

in the source node can be applied recursively to already encoded (and received) packets.

Consider that a routing node 𝑟is connected to the source 𝑘and has received and stored a set

2If not stated otherwise, vectors are row vectors.

2.2. Linear Network Coding

of ℎencoded packets, namely (𝑔1

𝑘, 𝑦1

𝑘),...(𝑔ℎ

𝑘, 𝑦ℎ

𝑘), directly from the source. The superscripts

in (𝑔𝑗

𝑘, 𝑦𝑗

𝑘)are added to differentiate the various packets generated at the source 𝑘. The

forwarding node may generate a new packet (𝑔𝑟, 𝑦𝑟)by picking a local encoding vector

𝑙𝑟= (𝑙𝑟,1,...,𝑙𝑟,ℎ)∈𝔽ℎ

𝑞and performing the linear combination

𝑦𝑟=

ℎ

∑

𝑗=1

𝑙𝑟,𝑗 ⋅𝑦𝑗

𝑘.(2.3)

One can substitute for each 𝑦𝑗

𝑘from Equation (2.1) and show that the encoded vector 𝑦𝑟

is also a linear sum of the source vectors, 𝑥𝑖, and the corresponding encoding vectors are

formed from the global encoding vector 𝑔𝑘(at the source) and the local encoding vector 𝑙𝑟

(at the routing node). The new global encoding vector, 𝑔𝑟, to be transmitted together with 𝑦𝑟

is not simply equal to 𝑙𝑟. Instead, it is shown in [58] that it is also a linear combination of the

received global encoding vector and is given as

𝑔𝑟=

ℎ

∑

𝑗=1

𝑙𝑟,𝑗 ⋅𝑔𝑗

𝑘.(2.4)

To summarize, at an intermediated node the linear combination is always applied to both the

encoded vectors and global encoding vectors; this operation is repeated at several routing

nodes in the network. By doing so, we ensure that, even after several packet combinations

at different routing nodes, the global encoding vector always gives the linear combination of

the source vectors.

Example 2.2.1.1. Let us consider the butterfly example of Figure 2.1, where the field is

𝔽2={0,1}and a symbol is a bit. In this field, addition corresponds to bitwise modulo-2

summation and multiplication corresponds to bitwise and. The linear combination sent by

node C, after receiving 𝑥1=𝑢1and 𝑥2=𝑢2, is 𝑢1⊕𝑢2. Part (a) of Figure 2.2 shows the

global encoding vectors used at the source and routing nodes.

Example 2.2.1.2. Consider again LNC over 𝔽2, where a source node has three source vec-

tors, namely 𝑥1= (1111),𝑥2= (1100) and 𝑥3= (1010). Figure 2.3 shows a routing node in

the network that has received three network-coded packets (𝑔1, 𝑦1),(𝑔2, 𝑦2)and (𝑔2, 𝑦2). As

𝑔1has only a single one at position one, the payload (i.e., encoded vector) 𝑦1is equal to 𝑥1.

From the global encoding vector 𝑔2, we learn that the payload 𝑦2is equal to 𝑥2⊕𝑥3. Like-

wise, 𝑔3indicates that 𝑦3is computed from 𝑥1⊕𝑥2. The node generates its outgoing packet

(𝑔𝑘, 𝑦𝑘)by choosing a local encoding vector; in this case 𝑙𝑘= (110). From Equation 2.1, the

payload 𝑦𝑘is then given by 𝑦𝑘= 1111 ⊕0110 = 1001. The same linear combination is used

to produce the new global encoding vector,𝑔𝑘, which is, 𝑔𝑘=𝑔1⊕𝑔2= 100 ⊕011 = 111.

2. Introduction to Network Coding

 

 

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⊕

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⊕

(

)

(

)

(

)

(

)

(

)

(

)











Encoding coeffcients

(a ) (b )

Figure 2.2.: Linear network coding in the butterfly network example of Figure 2.1. The

global encoding vectors are shown next to the links.

Figure 2.3.: Local encoding over the Galois field 𝔽2.

2.2.2. Decoding

To recover the source vectors, the sink node should receive as many encoded packets as

possible. Consider that it has received 𝑚packets, denoted as (𝑔𝑘, 𝑦𝑘), where 1⩽𝑘⩽𝑚. As

discussed before, each encoded vector within a packet is a linear combination of the source

vectors. With the global encoding vectors included in the network-coded packet, the source

vectors 𝑥𝑖can be decoded by solving the system of linear equations {𝑦𝑘=∑𝑛

𝑖=1 𝑔𝑘,𝑖𝑥𝑖}for

received packets (𝑔𝑘, 𝑦𝑘),1⩽𝑘⩽𝑚. Writing this system in matrix form leads to

⎛

⎜

⎝

𝑦1

𝑦𝑚

⎞

⎟

⎠

|{z }

𝑌

=⎛

⎜

⎝

𝑔1

𝑔𝑚

⎞

⎟

⎠

|{z }

𝐺

⋅⎛

⎜

⎝

𝑥1

𝑥𝑛

⎞

⎟

⎠

|{z }

𝑋

(2.5)

2.3. Network Coding and Channel Coding

where 𝑌,𝐺, and 𝑋are matrices of dimension 𝑚×1,𝑚×𝑛, and 𝑛×1, respectively. With 𝑌

and 𝐺extracted from the received packets, the unknowns are the components of the source

vector, 𝑋. This is a linear system with 𝑚equations and 𝑛unknowns. As long as 𝑚⩾𝑛

and at least 𝑛of the vectors 𝑔𝑖are linearly independent (also called innovative), the decoding

matrix 𝐺formed by these vectors has rank 𝑛. Taking 𝑛of these linearly independent equa-

tions (accordingly the dimension of 𝐺will be 𝑛×𝑛), the source vectors 𝑥𝑖can be recovered

by inverting 𝐺, i.e., 𝑋=𝐺−1⋅𝑌. Hence, one task of the network is to ensure that it delivers

at least 𝑛linearly independent packets to the sink. One approach is to have each node in

the network select the local encoding vector over the field 𝐹2𝑠, uniformly at random in a

completely independent and decentralized manner. This is called random network coding

and a detailed discussion is available in [59, 60]. A well understood way to solve a system

of linear equations is via Gaussian elimination. For the Example 2.2.1.2, the decoding can

be done intuitively as follows.

Example 2.2.2.1. Suppose the node in Figure 2.3 is interested in obtaining the source vectors

𝑥1,𝑥2and 𝑥3. As all three global encoding vectors 𝑔𝑖are linearly independent, it is guar-

anteed that decoding is possible. We see from 𝑔1that 𝑦1=𝑥1= (1111) and have decoded

the first packet. Packet 𝑔3states that 𝑦3=𝑥1⊕𝑥2. We can then obtain 𝑥2by computing

𝑥2=𝑦3⊕𝑥1= (0011) ⊕(1111) = (1100). With the same argument we can then decode

the last packet by evaluating 𝑥3=𝑦2⊕𝑥2= (0110) ⊕(1100) = (1010).

2.3. Network Coding and Channel Coding

In the previous section, we have described network coding for multicast in error-free net-

works. However, in wireless networks links are not error-free and packet loss arises in the

network for various reasons. To mention but a few: link outage, buffer overflow, and collision

[52]. There are a number of ways to deal with such losses. The most straightforward is to set

up a system of acknowledgments, where packets received by the sink are acknowledged by a

message sent back to the source and, if the source does not receive the acknowledgment for

a particular packet, it retransmits the packet. Alternatively, channel coding is a method that

is sometimes used. Through careful design, the network coding can also be used to support

the purpose of the channel coding, i.e., error protection. The interaction of network coding

and channel coding will be the focus of this section. In the following, we briefly describe

the interaction of these two coding schemes. A significant part of this section is taken from

[30].

2. Introduction to Network Coding

Channel

Decoder

Channel

Decoder

Channel

Encoder Network

Encoder

Physical

Layer

Physical

Layer

Network

Layer

;

Figure 2.4.: Network encoding at a node for a system with separate network-channel coding.

2.3.1. Separate network-channel coding

According to the OSI model, channel coding is part of the physical layer whereas network

coding is located in the network layer [30]. The purpose of network coding is different from

channel coding. Channel coding is performed to protect the communication over point-

to-point links against transmission errors. Network coding is performed with the aim to

transfer information efficiently through the (error-free) network. The layered architecture

model separates the two encodings and simplifies the complexity of the system because

the system design is split into the following two tasks: channel coding for a point-to-point

communication and network coding for the error-free point-to-point network. We call this

approach separate network-channel coding.

In order to illustrate separate network-channel coding, consider a node with two incoming

links which performs network encoding (example node Cin Figure 2.1). Figure 2.4 depicts

the channel coding and network coding at this node. First, the two incoming packets are

channel decoded in the physical layer to obtain the information packets 𝑖1and 𝑖2. The phys-

ical layer delivers 𝑖1and 𝑖2to the network layer. Then, the network encoder in the network

layer outputs 𝑖3=𝑖1⊕𝑖2and then 𝑖3is delivered to the physical layer. Finally, the channel

encoder generates the packet 𝑢3. Assuming a channel encoder with generator matrix G, the

output of the network encoder can be expressed as

𝑢3= (𝑖1⊕𝑖2)⋅G.(2.6)

A sink node, such as node 𝑡1in Figure 2.1 which performs network decoding, works ac-

cording to the same principle. First, the incoming packets are channel decoded in the phys-

2.3. Network Coding and Channel Coding

ical layer to obtain 𝑖1and 𝑖3. Then, 𝑖2=𝑖1⊕𝑖3is obtained by the network decoder in

the network layer. Chapter 4 investigates the implementation and performance of separate

network-channel coding in cooperative transmission.

2.3.2. Joint network-channel coding

Joint network-channel coding is a more general approach than separate network-channel

coding. This approach tries to merge the network and channel codings in such a way that

the network coding also provides support for error protection. Instead of guaranteeing the

error-free transmission for each point-to-point link, the interest here is to guarantee error-free

decoding at the sink nodes. With this approach, a node has to decode the data using the input

from all incoming links, i.e., error-free decoding at the sink can be possible even if error-free

decoding of a single link is not possible. In joint network-channel coding, the network-coded

packets are designed to carry redundant information.

Analogous to joint source-channel coding where the remaining redundancy after the source

encoding helps the channel code to combat noise, joint network-channel coding allows to

exploit the redundancy in the network code to support the channel code for a better error

protection. Accordingly, the implementation of the network coding is pushed to a layer

below the network layer. Let us re-consider the channel coding given in Equation (2.6).

Assuming a linear channel encoder, the encoding can be written as

𝑢3= (𝑖1⊕𝑖2)⋅G=𝑖1⋅G⊕𝑖2⋅G.(2.7)

We learn from Equation (2.7) that the network coding can be done on channel-coded bits,

i.e., the network coding is performed after channel coding. This is a (joint) realization of the

two encodings on the same layer. As a second example, consider next how joint network-

channel coding can be used to provide additional redundancy. The encoding in Equation

(2.6) is written once again in the form

𝑢3= [𝑖1, 𝑖2][G1G2

0G2]= [𝑖1⋅G1,(𝑖2⊕𝑖2)⋅G2].(2.8)

where 𝐺1and 𝐺2are linear encoding matrices. We see that the first part of the encoded

packet, namely 𝑖1𝐺1, forms the systematic bits in a systematic code and the remaining

network-coded bits, namely (𝑖2⊕𝑖2)𝐺2, forms the parity bits. Chapter 6 presents a detailed

study on the implementation and performance of joint network-channel coding in coopera-

tive transmission.

2. Introduction to Network Coding

2.4. Summary

Network coding generalizes routing and increases throughput and robustness in communi-

cation networks. We have discussed this with the help of the butterfly network. Moreover,

discussion on one type of network coding, namely linear network coding, and its encoding

and decoding operations are presented. With wireless networks in mind, the interaction of

the network and channel coding is also briefly discussed. We also pointed out that, through

joint realization of the network and channel codes, network coding can be used to generate

redundant information that can support the channel codes. In the next chapters, we will show

how separate and joint network-channel codings can be used in cooperative transmission net-

works for the purpose of diversity gain; we will also study the outage behavior of the two

realizations.

3. Wireless Channels and Networks

A transmission in a wireless channel experiences significant attenuation, called path loss, as

well as self-interference, resulting in fading, induced by multipath propagation of a transmit-

ted signal. Moreover, nodes in a wireless network share a common transmission medium,

which leads to interference from users operating in the same spectrum. The presence of atten-

uation, interference, and fading makes the design of wireless networks particularly complex

and challenging. To reliably communicate over longer distance, these channel impairments

require increasing transmission power, bandwidth, and/or receiver complexity.

The objective of this chapter is to give some background on wireless channels and networks.

Considering the point-to-point transmission, we explain a general system model, channel

coding, and modulation operations in Section 3.1. Section 3.2 discusses interference, atten-

uation, and fading in the wireless channel and presents how these impairments are modeled.

The information-theoretic capacity of a fading channel, assuming various channel state infor-

mation, is presented in Section 3.3. The conventional and incremental-redundancy network-

coded cooperations, which were introduced in Chapter 1, are revisited in Section 3.4. Finally,

a brief discussion of existing network architectures and cross-layer considerations in coop-

erative wireless networks is given in Section 3.5.

3.1. System Model of the Point-to-Point

Transmission

In today’s wireless network implementations, channel coding and modulation are two of the

widely used techniques. Channel coding is used to detect and possibly correct transmission

errors whereas modulation is used to transmit messages into the wireless channel. For a

general point-to-point communication system shown in Figure 3.1, we next explain channel

coding and also introduce notations relevant for upcoming discussions.

3. Wireless Channels and Networks

Channel

encoder Modulator Channel

Demodulator

and

detector

Channel

decoder

u c x y

Source Destination

Channel

)

xp(y

Figure 3.1.: System model for the point-to-point transmission.

3.1.1. Forward error correction with channel coding

Many wireless communication systems require a reliable communication, in the sense that a

destination node obtains messages with minimal error. The use of Forward Error Correction

(FEC) with channel coding is one way to protect messages against channel impairments.

Consider the system diagram in Figure 3.1 and suppose the source generates a message u

that consists of 𝑘independent and uniformly distributed bits. The channel encoder encodes

the message

u= (u1,u2,...,u𝑘)

and outputs a message

c= (c1,c2,...,c𝑛)

of 𝑛⩾𝑘coded bits (or symbols). The block cis referred to as codeword, whereas the

ensemble from which cis chosen is called codebook. The channel encoder includes 𝑛−𝑘

redundant bits in order to allow the channel decoder at the destination to detect and correct

(within some bound) erroneous bits without the need to ask the source for retransmission.

Hence, FEC is usually applied in situations where retransmissions are relatively costly, too

late, or impossible. The code rate of the channel code is defined as 𝑅𝑐=𝑘/𝑛. Channel

coding should provide reliable communication with as little redundancy as possible. A small

amount of redundancy means to use a high code rate and vice versa.

A practically important special cases are so called linear codes. In linear channel codes, the

channel encoder is defined by the generator matrix, G, and the output of this encoder is given

c=u⋅G.(3.1)

At the destination, the task of the channel decoder is to generate the estimate ˆ

ufrom the

erroneous reception ˆ

c, which is the output of the demodulator and detector. For the linear

block code given in Equation (3.1), the decoding matrix 𝐻, also called parity check matrix,

can be designed to fulfill G⋅HT= 0, where the Tin the superscript indicates the transpose

of the matrix. Accordingly, codeword cgenerated by Equation (3.1) should fulfill c⋅HT=

3.1. System Model of the Point-to-Point Transmission

u⋅G⋅HT= 0. The destination multiplies a received codeword with the transpose of the

parity check matrix. If the product is zero, then the codeword is a valid codeword (i.e., either

received correctly or undetectable error(s) occurred). If the product is different from zero,

then the codeword is received in error.

3.1.2. Modulation and demodulation

Modulation is used in order to transmit coded bits over the wireless channel, as it is not

possible to directly send bits into this channel. A modulator takes 𝐿coded bits from the

output of a channel encoder and maps them, depending on the 𝐿bits, into one of 2𝐿wave-

forms that will be transmitted over the channel. The waveforms are selected such that their

characteristics, e.g., bandwidth, matches that of the channel. Due to practical constraints,

the 2𝐿waveforms usually have a similar shape but differ in amplitude (e.g., in Amplitude

Shift Keying (ASK)), phase (e.g., in Phase Shift Keying (PSK)), and/or frequency (e.g., in

Frequency Shift Keying (FSK)). Although the modulated signals are often continuous-time

and passband (i.e., centered at carrier frequencies ranging from kHz to GHz), it is often

conceptually convenient to model them as discrete-time and baseband (i.e., centered at 0

Hz) signals. Baseband-equivalent models are convenient because they suppress the issues

of frequency up- and down-conversion and discrete-time models are appealing because ar-

chitectures designed for them can be efficiently implemented in digital signal processing

hardware.

Assuming discrete-time and basedband-equivalent representation of a signal, the modulator

can be described as an alphabet 𝒳of 2𝐿complex numbers and it is not necessary to consider

the shape of the waveform. In reference to Figure 3.1, the modulator maps 𝐿coded bits

of the channel encoder into modulated symbol x𝑖, where x𝑖is from alphabet 𝒳. Each x𝑖

is used to scale the amplitude of the waveform in ASK, shift the phase in PSK, or shift

the frequency in FSK. To mention two examples: Binary Phase-Shift Keying (BPSK) with

alphabets 𝒳2={−1,+1}, where each symbol carries 𝐿= 1 bit, and Quadrature Phase

Shift Keying (QPSK) with 𝒳4={−𝑗, −1, 𝑗, 1}, where each symbol carries 𝐿= 2 bit. In

a block-based transmission of Figure 3.1, the modulator maps the codeword cof 𝑛coded

symbols to a block

x= (x1,x2,...,x𝑚)

of 𝑚=𝑛/𝐿 symbols where each x𝑖is from the 𝒳.

The rate of both the channel encoder and modulator is defined as 𝑅=𝑘/𝑚 = (𝑘/𝑛)⋅

(𝑛/𝑚) = 𝑅𝑐⋅𝐿. We note that by increasing 𝐿, the data rate can be increased as a single

modulated symbol contains more number of coded bits. However, when the energy of the

3. Wireless Channels and Networks

waveform is fixed, the Euclidean distance between the constellation points decreases and the

probability of wrong detection increases as well. The Euclidean distance can be increased by

increasing the energy of the waveform; however, this may not be desired for energy efficiency

reasons or to decrease interference to other nodes. This energy vs. rate relationship becomes

even more interesting in the context of cooperative transmission and will be addressed in

Chapter 5.

The modulator sends the block xthrough the channel. The channel outputs a block

y= (y1,y2,...,y𝑚),

which is a distorted version of the block x. Based on the block y, the detector and demod-

ulator generates the estimate ˆ

c, which is further processed by the channel decoder. The aim

of channel coding and modulation is to minimize the bit errors between uand ˆ

ugiven the

allowed rate 𝑅and other constraints, for example the transmission power or energy.

3.2. Wireless Channel Models

A wireless channel generally suffers from large-scale fading and small-scale fading. Also,

interference from nodes transmitting in the same spectrum and noise generated at a destina-

tion node cause significant signal distortions. Large-scale fading is attributed to path loss,

which is the loss because of the separation of the source and destination, and shadowing,

which is observed when moving over several tens of wavelengths. Small-scale fading is a

random effect observed in the temporal and spatial dimensions, which can be categorized

as slow fading vs. fast fading and flat fading vs. frequency selective fading. In the follow-

ing, we will describe the significant channel distortions affecting wireless transmissions and

provide their fairly general mathematical description.

3.2.1. Noise and interference

Thermal noise is the main type of noise at the destination generated by thermal agitation of

electrons in the receiver circuit [30]. The received signal at the destination is passed through

a bandpass filter with a bandwidth large enough not to distort the transmitted waveform. On

the other hand, interference results when nodes in the network use the same radio frequency

band. In cellular mobile radio communication systems, for example, frequency is reused

so that users in geographically separated cells can use the same frequency. This introduces

co-channel interference coming from the cells using the same carrier frequency. There is

3.2. Wireless Channel Models

also adjacent channel interference due to partial spectral overlap between neighboring radio

channels.

Noise is usually modeled as additive (superimposed on the signal), white (has a flat power

spectral density within the bandwidth), and Gaussian distributed (due to the central limit

theorem and the fact that noise is the cumulative result of contributions from a large number

of independent sources). Noise according to this model is called Additive White Gaussian

Noise (AWGN). The AWGN channel model, in baseband-equivalent form, is given as

y[𝑚] = x[𝑚] + z[𝑚](3.2)

where x[𝑚]is the transmitted signal, y[𝑚]is the received signal at the destination, and −∞ <

𝑚 < ∞is the time index. The term z[𝑚]is a zero-mean AWGN process with variance 𝑁

and captures the effects of thermal noise and interferences.

3.2.2. Fading channels

In free space propagation, the signal power decay is proportional to the square of the prop-

agation distance. In more general settings, a signal can travel from the source to destination

over multiple paths. This phenomenon is referred to as multipath propagation and arises

because a propagating signal reflects off, refracts through, and diffuses around objects in

the channel environment. The multiple copies of the transmitted signal might add construc-

tively, thereby increasing the SNR, or destructively, thereby decreasing the SNR and this

phenomena is generally called fading. The fading in wireless channels can be categorized as

large-scale fading and small-scale fading [2, 61].

Large-scale fading channels

Large-scale fading represents the average signal power attenuation due to separation of the

source and destination, called propagation path loss, or scattering from prominent terrain

contours (e.g. hills, forests, billboards, clumps of buildings, etc.) between the transmitter

and receiver, called shadowing [62]. Shadowing is observed when moving over large areas

and the receiver is often represented as being “shadowed” by such prominences. Large-scale

fading is considered constant in time for a specific non-changing environment, distance, and

frequency.1

1Strictly speaking, path loss is a deterministic effect whereas shadowing is a random effect observed in the

spatial dimension. However, for low-mobility communication scenario assumption, the effect of shadowing

is observed when moved over large geographic area and it is usually incorporated into the path loss.

3. Wireless Channels and Networks

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized received power 10 ·log10 ³Pr(d)

Pr(d0)´in (dB)

Relative distance d

n=2

n=6

n=4

n=3

Figure 3.2.: Normalized received power, in dB, vs. distance.

In practical indoor and outdoor radio channels, the average signal power decays with dis-

tance, with some path-loss exponent, 𝑛, greater than two. The free-space transmission for-

mula of Friis allows to calculate the received power, 𝑃𝑟(𝑑), at the destination when the source

transmits with power, 𝑃𝑡, as [30, 62]

𝑃𝑟(𝑑) = 𝑃𝑡⋅(𝜆

4𝜋𝑑)𝑛

⋅𝐺𝑡⋅𝐺𝑟=𝛼𝑃𝐿𝑃𝑡(3.3)

where 𝐺𝑡and 𝐺𝑟are the gains of the transmitter and receiver antennas, 𝜆is the wavelength

of the signal, and 𝑑is the separation distance of the source and destination. The path-loss

exponent, typically 2⩽𝑛⩽6, depends on frequency, antenna height, and propagation en-

vironment and is equal to 2 for free-pace propagation. The term 𝛼PL =𝐺𝑡⋅𝐺𝑟⋅(𝜆

4𝜋𝑑 )𝑛is

called the path-loss coefficient and represents the effect of the loss because of the separation,

frequency, and the antenna gains. Equation (3.3) is valid only in the far-field of the transmis-

sion antenna, where the distance 𝑑is larger than the Fraunhofer distance, 𝑑𝐹=2𝐷2

𝜆, where

𝐷is the size of the transmission antenna. At distance 𝑑, the received power is related to a

power received at a reference distance, 𝑑0, by

𝑃𝑟(𝑑) = 𝑃𝑟(𝑑0)⋅(𝑑0

𝑑)𝑛

(3.4)

where 𝑃𝑟(𝑑0)is the received power at the reference distance. Sometimes, it is convenient to

assume the reference distance 𝑑0= 1 unit and work on the normalized power (𝑃𝑟(𝑑)

𝑃𝑟(𝑑0))=

𝑑)𝑛. This representation will be used in later chapters when we address node deployment

3.2. Wireless Channel Models

and coverage area extension issues. The normalized power, in dB, is plotted in Figure 3.2

for various values of the path-loss coefficient, 𝑛.

The effect of large-scale fading on the transmitted signal is modeled to be multiplicative and

the channel model in Equation (3.2) can be modified as

y[𝑚] = √𝑃𝑟(𝑑)

𝑃𝑡⋅x[𝑚] + z[𝑚] = √𝛼PL ⋅x[𝑚] + z[𝑚].(3.5)

One can define the instantaneous SNR of the received signal, x[𝑚], as the ratio of the received

signal power and the noise power and is given by

𝛾=𝑃𝑟(𝑑)

𝑁=𝛼PL ⋅𝑃𝑡

𝑁.(3.6)

where 𝑁is the noise variance. The term 𝑃𝑡

𝑁is sometimes called transmit SNR.

Example 3.2.2.1. Consider the uplink transmission in the first phase of the cooperative net-

work shown in Figure 3.3, where the source and partner are located at a distance of 450

meters and 100 meters from the destination, respectively. Assume that each node is trans-

mitting at a carrier frequency 𝑓𝑐= 5.3GHz, has access to 𝑊= 100 MHz bandwidth, both

nodes transmit at a power 𝑃𝑡= 50 mW (17 dBm), and the noise level is -94 dBm.

Using the free space path-loss model given in Equation (3.3), the received signal, in dB, at

the destination is computed as

𝑃𝑟(𝑑) = 𝑃𝑡+𝛼PL,(3.7)

where the path-loss term 𝛼PL = 20 log10 (𝜆

4𝜋𝑑 )+𝐺𝑎, where 𝐺𝑎=𝐺𝑡+𝐺𝑟= 7 dB represents

the combined antennas gain, all in dB, and 𝜆=𝑐

𝑓𝑐, where 𝑐is the speed of the light. One can

compute that the partner and source, respectively located at distance of 100 and 450 meters

from the destination, experience path loss 𝛼PL = 87 and 𝛼PL = 100 dB, respectively. The

received power level of the partner is -77 dBm while that of the source is -90 dBm. Note that

the received power at the partner, because of its location, is greater than at the source. We

will use this example in the next sections to motivate relaying in cooperative networks.

Small-scale fading channels

For most practical channels, the above propagation model is inadequate to describe the chan-

nel and predict system performance. Small-scale fading occurs due to multipath components

in a channel, and refers to changes in signal amplitude and phase that can be experienced

3. Wireless Channels and Networks

First phase Second phase

S P D

SPD

Figure 3.3.: Cooperative transmission in the uplink transmission of the mobile communica-

tion system based on repetition coding; 𝑖𝑠is the source’s message.

as a result of small changes in the spatial separation between the source and destination. In

addition to the multipath propagation, other factors that affect small-scale fading are speed

of the destination, speed of surrounding objects, and signal bandwidth. Small-scale fading

manifests itself in two mechanisms, namely:

∙Time spreading of the signal and

∙Time-variant behavior of the channel.

Based on either the time-spreading or time-variant nature of the channel, fading is further

classified as follows.

Flat fading vs. frequency-selective fading Assume that a narrow pulse (e.g., im-

pulse signal) is transmitted from the source. Because of the multipath nature of the wireless

channel, the destination receives a sequence of delayed pulses of various magnitude and

delay, i.e., spreading of the impulse in time. For a single transmitted impulse, the excess de-

lay 𝑇𝑚between the first and last received component represents the maximum delay during

which the multipath signal power falls to some threshold level below that of the strongest

component.

Viewed in the frequency domain, for an impulse signal of infinite bandwidth, the bandwidth

of the received series of impulses is finite. This means that the channel has filtered out

some frequency content of the signal. The coherence bandwidth,𝑓0, is a statistical mea-

sure of the range of frequencies over which the channel passes all spectral components with

approximately equal gain and linear phase. Excess delay and coherence bandwidth are ap-

proximately related by

𝑓0≈1

𝑇𝑚

.(3.8)

In a multipath environment, the time spreading causes a signal to undergo either flat fading or

frequency-selective fading. In flat fading, the coherence bandwidth of the channel is greater

than the bandwidth of the transmitted signal, 𝑓, such that all frequency components of the

3.2. Wireless Channel Models

signal will experience the same magnitude of fading. This means 𝑓0> 𝑓 ≈1

𝑇𝑠, where 1

𝑇𝑠is

the symbol rate and is nominally taken to be equal to 𝑓. Flat fading could also be viewed,

in the time domain, to be the result of a multipath propagation whose excess delay, 𝑇𝑚, is so

small compared to the symbol duration, 𝑇𝑠, that they add up to one undistorted signal (or the

received signal is not distorted by Inter-symbol Interference (ISI)).

In frequency-selective fading, the coherence bandwidth of the channel is smaller than the

bandwidth of the signal, i.e., 𝑓0<1

𝑇𝑠, such that different frequency components of the

signal will be affected differently by the channel (or experience decorrelated fading). In the

time domain, the multipath components of the signal will have significant time dispersion

compared to the symbol period and this results in ISI.

Fast fading vs. slow fading In wireless channels, the time-varying nature of the chan-

nel is caused by changes in the propagation path (because of a relative motion between the

source and destination and/or by movement of objects within the channel). Thus, for a trans-

mitted signal the destination sees variations in the signal’s amplitude and phase. The time-

variant mechanism will be characterized in the time domain by the channel coherence time

𝑇𝑐, which is a measure of the expected time duration over which the channel is essentially

invariant.

The coherence time determines whether the channel can be described as slow fading or

fast fading. In slow fading channel, the coherence time of the channel is greater than the

symbol duration of the transmitted signal, i.e., 𝑇𝑐>> 𝑇𝑠. In this regime, the amplitude and

phase change imposed by the channel can be considered roughly constant over the period of

channel use. In contrast, fast fading occurs when the coherence time of the channel is small

relative to the symbol duration of the transmitted signal, i.e., 𝑇𝑐< 𝑇𝑠. In this regime, the

amplitude and phase change imposed by the channel varies considerably over the period of

channel use.

In the fast-fading channel, the source, using time diversity, may take advantage of channel

variations. Although a deep fade may temporarily erase some of a transmitted codeword,

use of channel coding coupled with successfully transmitted bits during other time instances

can allow the erased bits to be recovered. In a slow-fading channel, it is not possible to use

time diversity because the transmitter sees only a single realization of the channel within its

delay constraints (i.e., coherence time). A deep fade therefore lasts the entire duration of

transmission and cannot be mitigated using coding.

The coherence time of the channel is related to a quantity known as the Doppler spread of the

channel. When a user (or reflectors in its environment) is moving, the user’s velocity causes

a shift in the frequency of the signal transmitted along each signal path. This phenomenon

3. Wireless Channels and Networks

is known as the Doppler shift. Signals traveling along different paths can have different

Doppler shifts, such that when they add-up at the destination, the resulting signal will have

a broader (and possibly shifted) bandwidth than the transmitted signal. This is known as the

Doppler spread, represented as 𝑓𝑑, and measures this spectral broadening of the signal. In

general, coherence time is inversely related to Doppler spread and typically expressed as

𝑓𝑑≈𝑉

𝜆≈𝑘

𝑇𝑐

,(3.9)

where 𝑉is the relative velocity, 𝜆is the signal wavelength, and 𝑘is a constant taking values

in the range of 0.25 to 0.5.

Model of small-scale fading Small-scale fading is sometimes called Rayleigh fading

because if the multiple reflective paths are large in number and there is no line-of-sight

signal component, the envelope of the received signal is statistically described by a Rayleigh

probability density function given as:

𝑝(𝑥) = {𝑥

𝜎2⋅exp (−𝑥2

2𝜎2)𝑥⩾0;

0otherwise

(3.10)

where 𝑥is the amplitude of the received signal, and 2𝜎2is the mean power of the multi-

path signal envelope. When there is a dominant non-fading signal component, such as a

line-of-sight propagation path, the small-scale fading envelope is described by a Rician dis-

tribution.

Flat and slow fading (which is one case of small-scale fading where there is no ISI and

the channel remains the same during the period of channel use), like the large-scale fading,

results in multiplicative distortion of the transmitted signal, x[𝑚][62]. For this channel, let

𝛼represents the fading attenuation term. The channel model given in Equation (3.5) can be

modified to include the flat fading as

y[𝑚] = √𝑃𝑟(𝑑)

𝑃𝑡⋅𝛼⋅x[𝑚] + z[𝑚] = ℎ⋅x[𝑚] + z[𝑚](3.11)

where ℎis a new fading coefficient that combines both the large-scale fading and flat fading.

Figure 3.4 depicts this channel model. Note that Equation (3.11) assumes an ideal coherent

detection (i.e., the channel fading is sufficiently slow that the phase shift can be estimated

from the received signal without error). In a case that the separation distance of the source

and receiver is of interest (e.g. in a coverage area study), the channel model in Equation

3.3. Information Theory – Fading Channel Capacity

x y

⋅

P)(

Figure 3.4.: Channel model with small-scale fading coefficient 𝛼and noise samples z.

(3.11) can be represented as

y[𝑚] = ℎ

𝑑𝑛/2⋅x[𝑚] + z[𝑚].(3.12)

The instantaneous SNR of the received signal, x[𝑚], which was defined in Equation (3.6)

can now be written as

𝛾=∣𝛼∣2⋅𝛼𝑃𝐿 ⋅𝑃𝑡

𝑁=∣ℎ2∣

𝑑𝑛⋅𝑃𝑡

𝑁(3.13)

and the average SNR, Γ, is given by

Γ = 𝐸[𝛾] = 𝐸[∣ℎ2∣]

𝑑𝑛⋅𝑃𝑡

𝑁(3.14)

where 𝐸[⋅]is the expectation operation. For ∣𝛼∣Rayleigh-distributed random variable repre-

senting the magnitude of the small-scale fading term, the random variable ∣𝛼∣2, correspond-

ing to the signal’s power, is exponentially distributed.

3.3. Information Theory – Fading Channel Capacity

The aim of a communication system is to provide a reliable communication with as little

overhead as possible. Reliable communication means that an arbitrarily small probability

of decoding error can be achieved. Information theory defines capacity of a channel and

provides a limit for it. Re-considering the system model in Figure 3.1, let xand ybe random

variables representing the channel input and output with alphabets 𝒳and 𝒴, respectively.

The alphabets 𝒳and 𝒴can assume either discrete values or continuous values.

3. Wireless Channels and Networks

3.3.1. Capacity of Additive White Gaussian Noise Channel

Consider a discrete-input, continuous-output channel. A block x= (x1,x2,...,x𝑚)is trans-

mitted by the modulator, where each input symbol, x𝑖, for 1⩽𝑖⩽𝑚, is drawn from the

alphabet 𝒳. The AWGN channel is modeled by

y𝑖=x𝑖+z𝑖,

where z𝑖is the zero-mean Gaussian random variable with variance 𝑁. This channel is a

typical example of a discrete-input, continuous-output channel. The output y𝑖of the channel

is continuous (or unquantized) and can assume any value on the real line, i.e., 𝒴= (−∞,∞).

Assume that the power constraint on the input block is given as

𝑚

∑

𝑖=1

𝐸[x2

𝑖]/𝑚 ⩽𝑃𝑡.(3.15)

This channel is described by the conditional probability density function2𝑝(y𝑖∣x𝑖). The

capacity of this channel is the maximum rate 𝑅=𝑘/𝑚 bits per channel use3for which,

for sufficiently large 𝑚, there exists a u-to-xmapping (encoder and modulator) and a y-to-ˆ

mapping (detector, demodulator, and decoder) so that the error probability 𝑃𝑟[ˆ

u∕=u]can

be made as close to 0 as desired (but not necessarily exactly 0). The capacity, in bits per

channel use, is the maximum average mutual information between the discrete input x𝑖∈ 𝒳

and output y𝑖∈(−∞,∞)and is given as

𝐶= max

𝑃(x𝑖)𝐼(x𝑖;y𝑖)

= max

𝑃(x𝑖)∑

x𝑖∈𝒳 ∫∞

−∞

𝑝(y𝑖∣x𝑖)𝑃(x𝑖) log2

𝑝(y𝑖∣x𝑖)

𝑝(y𝑖)𝑑y𝑖[bits/channel use],(3.16)

where 𝑃(x𝑖)is the probability that x𝑖is sent, and 𝐼(x𝑖;y𝑖)is referred to as mutual information

and physically represents the amount of information that can be deduced about x𝑖, based on

observing y𝑖.

Capacity is essentially the maximum, over the input distribution of x𝑖, of the amount of

information about x𝑖that can be inferred from y𝑖. Note that if 𝒴is from a discrete alphabet,

then 𝑝(y𝑖∣x𝑖)and 𝑝(y𝑖)in Equation (3.16) are replaced by the probability functions 𝑃(y𝑖∣x𝑖)

and 𝑃(y𝑖), respectively, and also the integral is replaced by a sum. From Equation (3.16), an

2Also called transition probability density function.

3Bits per channel use means bits per input symbol into the channel. If a symbol enters the channel every 𝑇𝑠

seconds (for every symbol period a symbol is transmitted), the channel capacity, in bits per second, is 𝐶

𝑇𝑠

where 𝐶is the capacity in bits per channel use.

3.3. Information Theory – Fading Channel Capacity

arbitrarily reliable communication can be achieved for all rates 𝑅 < 𝐶. For rates 𝑅 > 𝐶, the

probability of error is strictly bounded above zero. It is shown that the best x𝑖that maximizes

the capacity in Equation (3.16) is Gaussian with zero mean and variance 𝑃𝑡(power constraint

in Equation (3.15)) and the capacity is given as [63]

𝐶=1

2log2(1 + 𝑃𝑡

𝑁) = 1

2log2(1 + 𝛾)[bits/channel use] (3.17)

where 𝛾=𝑃𝑡

𝑁is the instantaneous SNR. The capacity in Equation (3.19) is sometimes

called called Shannon capacity or the instantaneous capacity. In a case that both x𝑖and

z𝑖=z𝑖,𝑅 +𝑗z𝑖,𝐼 are complex with z𝑖,𝑅 and z𝑖,𝐼 are independent, real, Gaussian random

variables with variance 𝑁/2and 𝑗=√−1, the Shannon capacity becomes

𝐶= log2(1 + 𝛾)[bits/channel use].(3.18)

The Nyquist-Shannon sampling theorem states that a complex signal, x𝑖, of bandwidth 𝑊

can be represented by approximately 𝑊complex samples per second, and the channel ca-

pacity in Equation (3.18), in [bits/second], is then

𝐶=𝑊⋅log2(1 + 𝛾)[bits/second].(3.19)

3.3.2. Capacity of flat and slow fading channels

Consider the case that the channel is flat fading, slow fading, and AWGN, whose channel

model is given by Equation (3.12) as

y𝑖=ℎ

𝑑𝑛/2x𝑖+z𝑖

where 1⩽𝑖⩽𝑚, the same power constraint as given by Equation (3.15). The only dif-

ference between a fading channel and the AWGN channel is the random channel gain, ℎ,

and the Shannon capacity formula for the AWGN channel works for fading channel as well,

i.e.,

𝐶= log2(1 + ∣ℎ∣2

𝑑𝑛⋅𝑃𝑡

𝑁)= log2(1 + 𝛾)[bits/channel use].(3.20)

The capacity in Equation (3.20) is plotted in Figure 3.5 as the curve labeled “Gaussian”.

Equation (3.20) is based on a general complex Gaussian input x𝑖with zero-mean and vari-

ance 𝑃𝑡assumption. In a digitally modulated system, however, x𝑖assumes only a limited

3. Wireless Channels and Networks

−10 −5 0 5 10 15 20

0.5

1.5

2.5

3.5

SNR γ in dB

Channel capacity [bits/use]

BPSK

QPSK

8−PSK

Gaussian

Figure 3.5.: Channel capacity in [bits/channel use] for input symbols x𝑖from a Gaussian

distribution, BPSK, QPSK, or 8-PSK modulation.

number of values. For example, in M-ary phase-shift keying (M-PSK) x𝑖is uniform over the

alphabet

𝒳={√𝐸𝑠,√𝐸𝑠exp𝑗2𝜋/𝑀 ,...,√𝐸𝑠exp𝑗2𝜋(𝑀−1)/𝑀 },

where 𝐸𝑠=𝑃𝑡is the (average) per-symbol-energy. The capacity of the complex AWGN

channel with M-PSK is derived in [63, 64]. Figure 3.5 depicts the capacity of BPSK, QPSK,

and 8 Phase Shift Keying (8-PSK). From this plot, we note the following points

1. Sending at full rate through the channel is possible only at high SNR values (called

high-SNR region). This is the region of interest in the study of cooperative protocols

and will be used in the next chapters, when we compute outage probability of various

protocols.

2. The capacity of the Gaussian input is the upper bound. Moreover, at high SNRs, one

must use large symbol sets, while at low SNRs it seems that BPSK suffices [63].

3.3.3. Channel state information and channel capacity

Channel State Information (CSI) is an important issue affecting the design and analysis of

communication systems. CSI refers to knowledge of channel realization (or the channel co-

efficient, ℎ) at a node in a network. Depending on which node in the network has knowledge

of the channel, here are the possible cases to consider.

3.3. Information Theory – Fading Channel Capacity

CSI available at the destination This is a standard assumption in many wireless sys-

tems and is reasonable when the channel is slow fading. In slow fading channels, the des-

tination may use, for example, training signals such as pilot tones to “learn” the channel.

The capacity of the fading channel with destination side information is given by Equation

(3.20).

CSI available at the source This assumption is reasonable when the channel is fading

slowly and when the destination can share the CSI with the source by means of a separate

feedback channel. In this case, the source can adjust its power level and rate according to

the channel state. One adaptation rule would be to conserve power by not transmitting when

the CSI falls below a certain threshold, and transmit if the CSI lies above the threshold. Any

adaptation must be performed subject to an appropriate power constraint. In this case, the

Shannon capacity expression can be written in the form

𝐶= max

𝑃𝑡(⋅)log2(1 + ∣ℎ∣2

𝑑𝑛⋅𝑃𝑡(ℎ)

𝑁)(3.21)

where 𝑃𝑡(⋅)represents the power allocation function subject to an average power constraint

𝐸[𝑃𝑡(ℎ)] ⩽𝑃𝑡.

No CSI at the source and destination In reality, the CSI is never exactly known to

the source or destination. Here, the capacity depends not only on the marginal density of

the channel (Rayleigh, Rician, etc.), but also on the temporal correlation of the channel and

computing the channel capacity is an open problem.

3.3.4. Ergodic fading channels

In stationarity channel, the statistical properties of ℎ(such as its mean and variance) will not

change over time. When ℎis also ergodic, then the time average of the channel coefficient ℎ

is equal to its ensemble average and these prosperities can be deduced from a single and suf-

ficiently long sample of the process ℎ. In other words, the randomness of ℎcan be averaged

out (removed) over time so that long-term constant bit rates can be supported (like AWGN

channels). In general, an ergodic channel is defined as a channel whose ℎvaries (or fades

fast enough) over a (possibly finite but very long) codeword but all its moments are the same

from codeword to codeword. In that case, the capacity is given as

𝐶=𝐸[log2(1 + ∣ℎ∣2

𝑑𝑛⋅𝑃

𝑁)] (3.22)

3. Wireless Channels and Networks

where 𝐸[⋅]denotes the expectation with respect to ℎand this can be computed using the

distribution of ℎ. Equation (3.22) means that as the mean of the fading statistics can be

observed with high reliability, such a channel can support any rate not exceeding the capacity

𝐶. In terms of diversity, these temporal variations allows the coding strategy to fully exploit

temporal diversity.

3.3.5. Non-ergodic fading channels and outage probability

Consider a slow fading channel where the channel state is random but held constant for the

duration of a codeword. This channel is called a non-ergodic channel or, sometimes block-

fading channel. In non-ergodic channel, the randomness of the channel gain can not be

averaged out (removed) over time so that long-term constant bit rates can not be supported.

Shannon capacity is not a useful performance measure because, for example, if a channel is

in a deep fade, then the channel state, ℎ, as well as the capacity will be zero. However, since

the channel realization is random and kept constant over the codeword transmission, there is

a non-zero probability that a given transmission rate can be supported by the channel, and

this probability is called outage probability. The largest rate of reliable communication at a

certain outage probability is called the outage capacity.

Outage probability helps to examine the tradeoff between a fixed rate and the probability

that this rate is achievable over the channel. For example, for a fixed desired rate, 𝑅, those

channel states with

𝐶= log2(1 + ∣ℎ∣2

𝑑𝑛⋅𝑃𝑡

𝑁)< 𝑅 (3.23)

will not support the rate, i.e., there is an outage. The event {log2(1 + ∣ℎ∣2

𝑑𝑛⋅𝑃𝑡

𝑁)< 𝑅}is

called an outage event, and the outage probability is the probability that this event occurs. In

block-fading channels, lack of temporal variation in the channel state over a codeword dura-

tion prevents a coding strategy from exploiting temporal diversity. In the fast fading channel,

in contrast, outage can be avoided due to the ability to average over the time variation of

the channel, and one can define a capacity at which arbitrarily reliable communication is

possible.

Example 3.3.5.1. In this example, we will compute the outage probability in a block-fading

channel. We observe that the capacity is a random variable whose distribution depends on

the distribution of the fading coefficient. From the outage event {log2(1 + ∣ℎ∣2

𝑑𝑛⋅𝑃𝑡

𝑁)< 𝑅},

we deduce that ∣ℎ∣2<2𝑅−1

Γ𝑇, where ΓT=𝑃𝑡

𝑑𝑛𝑁. The outage probability, denoted as 𝑃𝑜𝑢𝑡,𝑝𝑝𝑡,

3.3. Information Theory – Fading Channel Capacity

is then computed as

𝑃𝑜𝑢𝑡,𝑝𝑝𝑡 = Pr [∣ℎ∣2<2𝑅−1

ΓT].(3.24)

If the channel state ∣ℎ∣is Rayleigh distributed, ∣ℎ∣2will have an exponential distribution with

parameter 1

Γℎ, where Γℎ=𝐸[∣ℎ∣2]. The outage probability in Equation (3.24) evaluates to

𝑃𝑜𝑢𝑡,𝑝𝑝𝑡 =∫2𝑅−1

Γ𝑇

Γℎ⋅exp (−ℎ

Γℎ)𝑑ℎ = 1 −exp (−2𝑅−1

ΓℎΓT)≈2𝑅−1

ΓℎΓ𝑇

.(3.25)

The approximation in Equation (3.25) holds for high ΓℎΓTvalues (or in the high SNR re-

gion). In block-fading channels, outage probability is used to compute diversity, an impor-

tant figure of merit that measures reliability of communication. Diversity measures the rate

at which the outage probability decays with respect to SNR. In Equation (3.25), this decay

is proportional to (ΓℎΓT)−1, and hence the diversity gain is 1. In cooperative transmission,

higher diversity gain is obtained by the use of relaying nodes in the networks.

Example 3.3.5.2. Re-consider the link budget calculation Example 3.2.2.1 whose network

was shown in Figure 3.3. Let us additionally assume that 𝐸[∣ℎ∣2]= 1 and a capacity 𝐶of

400 Mb/s is desired. Because of 𝐸[∣ℎ∣2]= 1 assumption, the average received power will

remain the same as calculated, i.e., -77 dBm for the partner and -90 dBm for the source.

From the Shannon formula 𝐶=𝑊log2(1+𝛾), we compute that a SNR of 12 dB is required

to support the desired rate. For a noise level of -94 dBm, either the source or partner will

not be in outage if the received signal power level is at least -82 (= -94 + 12) dBm. As

the received power level of the partner is -77 dBm while that of the source is -90 dBm, the

partner is not in outage; however, the source is in outage since the received power is 8 dB

below the required power level. However, one way for the source to avoid outage is to use

the partner as a relaying node such that the overall path loss is reduced.

3.3.6. Capacity vs. combining schemes

Consider the case of repetition coding where the destination node receives multiple copies

of the same codeword (either through retransmission from the source or other nodes in the

network, which correctly receive from the source and forward to the destination). Maximum-

Ratio Combining is used to combine the multiply received codewords. As a result of this

combining, the SNR of the codeword accumulates. If the destination receives 𝑚copies of

the codeword and if all channels are block fading, the Shannon capacity is given as

𝐶=1

𝑚⋅log2(1 +

𝑚

∑

𝑖=1

∣ℎ𝑖∣2

𝑑𝑛

𝑖⋅𝑃𝑡

𝑁)(3.26)

3. Wireless Channels and Networks

where ℎ𝑖and 𝑑𝑛

𝑖are the channel coefficient and separation distance from node 𝑖to the desti-

nation, respectively. The 1

𝑚is included if each transmission takes 1

𝑚of the available time.

If the source and relaying nodes in the network re-transmit incrementally redundant informa-

tion, the destination can employ code combining to combine parts of the received codeword.

By using redundant transmission, more information bits are delivered to the destination and

this is reflected as addition in the channel capacity. The amount of incremental information

need not necessarily be the same as the information sent at the first time. If each transmission

is finished in 𝛼𝑖of the available time and ∑𝑚

𝑖=1 𝛼𝑖= 1, then the capacity of this channel for

𝑚re-transmissions is given as

𝐶=

𝑚

∑

𝑖=1 [𝛼𝑖⋅log2(1 + ∣ℎ𝑖∣2

𝑑𝑛

𝑖⋅𝑃𝑡

𝑁)].(3.27)

3.4. Network-Coded Cooperation

As explained in the above sections, the presence of time-varying fading channels makes

the design of wireless networks complex. One way to combat the effect of fading is by

using spatial diversity. As explained in Section 1.1, network-coded cooperation is a cooper-

ative transmission approach that realizes spatial diversity. In this section, we will formulate

a generalized channel model and define general assumptions that will be used in the next

chapters.

3.4.1. System model for network-coded cooperation

Consider the system model of the network-coded cooperation shown in Figure 3.6. In

network-coded cooperation, in the first phase of cooperation, both the source and partner

send a codeword of 𝑁1=𝛼𝑁 symbols, where 𝛼is the cooperation level. In the conven-

tional network-coded cooperation of Chapter 4, 𝛼= 1/2and in the incremental network-

coded cooperation of Chapter 6, 0⩽𝛼⩽1.𝑁is the number of symbols per codeword

if the point-to-point transmission were used. Then both nodes, using orthogonal channels,

transmit their respective codewords to the destination and simultaneously try to decode each

other’s codeword.

If, for example, the source decodes the partner’s message correctly, then it forms a network-

coded message by linearly combining its own and the received message (details of the con-

tent of this message will be explained in the upcoming chapters). This network-coded mes-

3.4. Network-Coded Cooperation

First phase

Second phase

⊕

⊕ ⊕

⊕ i

⊕

⊕ ⊕

⊕ i

Figure 3.6.: System diagram of network-coded cooperation in the uplink transmission. Solid,

dashed, and dotted lines show the transmission of the source’s message, partner’s

message, and network-coded message, respectively.

sage of 𝑁2= (1 −𝛼)𝑁symbols is transmitted to the destination in the second phase. The

partner will also perform the same operation in the next cycle.

3.4.2. Equivalent channel model

In our model of the wireless channels, narrowband transmissions suffer from the effects of

frequency non-selective slow fading and additive noise. The coherence time of the fading is

long enough such that the fading does not change for the transmission of a large number of

(block of) transmitted symbols, i.e., block fading channel. The medium-access control in our

model imposes the practical system constraints of orthogonal transmission and half-duplex

relaying. We also assume that CSI is available at the destination, i.e., channel coefficients

are perfectly estimated at the destination, and perfect synchronization exists between nodes

and each receiving node is capable of coherent detection.

Consider a baseband-equivalent, discrete-time channel model for the continuous-time chan-

nel, and also consider 2𝑁consecutive uses of the channel, where 𝑁is a large integer is

utilized. For point-to-point transmission, our baseline for comparison, the channel model for

the transmission from the source to destination, is

𝑦𝑠,𝑑[𝑚] = ℎ𝑠,𝑑𝑖𝑠[𝑚] + 𝑧𝑑[𝑚](3.28)

for, say, 𝑚= 1,...,𝑁, where 𝑖𝑠[𝑚]is the source-transmitted signal, and 𝑦𝑠,𝑑[𝑚]is the

destination-received signal. The partner transmits for 𝑚=𝑁+ 1,...,2𝑁. The fading ℎ𝑠,𝑑

captures the effects of path loss, shadowing, and frequency non-selective fading, and 𝑧𝑑[𝑚]

captures the effects of receiver noise and other forms of interference in the system.

For network-coded cooperation, the source uses the channel for 𝑚= 1,...,𝛼𝑁 in the first

3. Wireless Channels and Networks

phase. The source-destination and source-partner channels are modeled as

𝑦𝑠,𝑑[𝑚] = ℎ𝑠,𝑑𝑥𝑠[𝑚] + 𝑧𝑑[𝑚]

𝑦𝑠,𝑝[𝑚] = ℎ𝑠,𝑝𝑥𝑠[𝑚] + 𝑧𝑝[𝑚].(3.29)

Likewise, for 𝛼𝑁 + 1,...,2𝛼𝑁 the partner sends its message to the destination and the

source. The partner-destination and partner-source channel models are given by

𝑦𝑝,𝑑[𝑚] = ℎ𝑝,𝑑𝑥𝑝[𝑚−𝛼𝑁] + 𝑧𝑑[𝑚]

𝑦𝑝,𝑠[𝑚] = ℎ𝑝,𝑠𝑥𝑝[𝑚−𝛼𝑁] + 𝑧𝑠[𝑚].(3.30)

Similarly, in the second phase of transmission, the destination receives from the source for

channel use 2𝛼𝑁 + 1,...,(1 + 𝛼)𝑁and from the partner for channel use (1 + 𝛼)𝑁+

1,...,2𝑁. In the case of conventional network-coded cooperation, the model of the source-

destination and partner-destination channels are given by

𝑦𝑠,𝑑[𝑚] = ℎ𝑠,𝑑 (𝑥𝑠[𝑚−2𝛼𝑁]⊕𝑥𝑝[𝑚−2𝛼𝑁]) + 𝑧𝑑[𝑚]

𝑦𝑝,𝑑[𝑚] = ℎ𝑝,𝑑 (𝑥𝑠[𝑚−(1 + 𝛼)𝑁]⊕𝑥𝑝[𝑚−(1 + 𝛼)𝑁]) + 𝑧𝑑[𝑚].(3.31)

In the case of incremental redundancy network-coded cooperation, the channel model in

Equation (3.31) should be modified a little as only a part of the message of the source and

partner are forwarded.

3.5. Cross-Layer Design in Cooperative Wireless

Network

In wireless networks, to reliably transmit messages among nodes, the network should mit-

igate channel impairments, e.g. fading and interference, and efficiently allocate and utilize

network resources, e.g. power and bandwidth. One approach to realize reliable communica-

tion is to partition the network architecture into a set of protocol layers. Figure 3.7 illustrates

layers in an existing wireless network architecture and indicates the functions they usually

serve. As examples, the Medium-Access Control Layer (MAC) manages interference in

the network and the Physical Layer (PHY) combats fading with channel coding, spread-

spectrum, or multiple antennas. In the next sections, we will discuss the existing network

architecture as well as considerations when cooperative transmission is included in the ex-

isting networks.

3.5. Cross-Layer Design in Cooperative Wireless Network

3.5.1. Existing wireless network architecture

In the point-to-point transmission, consider the source runs an application layer process that

wishes to transmit messages to an application layer process at the destination node. Messages

are encoded as data packets with appropriate headers that identify the application process,

the source node, and the destination node.4Roughly, the tasks allocated to each layer are

as follows. The application layer generates user messages, and conveys them through an

interface to the transport layer. These packets are buffered and sequenced by the transport

layer, typically Transmission Control Protocol (TCP), that implements both reliable end-to-

end connection as well as end-to-end congestion and flow control [63]. The release of packets

to the network layer is controlled by a reverse stream of TCP Acknowledgment (ACK)s from

the destination TCP process.

Finding routes (via a sequence of point-to-point links) to the destination, maintaining these

routes, and forwarding packets along these routes is the task of the network layer. The data

link layer ensures reliable packet transmission on a single point-to-point link. This layer

may include a MAC sublayer that regulates channel access [63]. At the data link layer of the

source, it is a common practice to append error-detecting codes, such as parity-bit or CRC, to

each packet. The CRC allows the data link layer at the destination to detect packet reception

errors. Sequence numbers may also be added to facilitate Automatic Repeat Request (ARQ)

retransmission protocols at the link layer. The retransmission is triggered by the result of

the error detecting code. If the destination determines that the packet is in error, it sends a

Negative Acknowledgment (NACK) to the source, otherwise it sends an ACK. In the former

case, the packet is retransmitted.

The physical layer PHY, which incorporates a majority of the analog circuitry and signal

processing, transmits signals at the source and receives and processes signals at the desti-

nation. Channel coding and modulation are two of the basic functionalities of the physical

layer. Other schemes that are used in the physical layer are bit-rate adaptation, channel se-

lection, and recently techniques such as MIMO smart antennas, cooperative transmission,

interference cancellation, and ultra-wide band transmission (UWB).

The throughput of ARQ protocols can be improved by combining them with channel coding

in the form of Hybrid ARQ. Hybrid ARQ lets erroneously received packets be collected

and combined in various ways before decoding. Packet combining can be based on hard

decisions or soft channel outputs. In soft channel outputs, noisy versions of the same packet

4In the context of network layering, a message is information/data that the application layer of the source

wishes to communicate and a packet generated and transmitted by a source and addressed to a particular

destination. Along the way, several intermediate nodes may contribute to the communication of the message

to the destination, but each node may transmit its own unique data packets [63].

3. Wireless Channels and Networks

Wireless Channel

Physical

Interface

Data Link

Medium Access Control

Network

Transport

Session

Application

Presentation

TCP

PHY

Application

Physical

Interface

Data Link

Medium Access Control

Network

Transport

Session

Application

Presentation

Figure 3.7.: Existing wireless network protocol stack.

are combined by MRC, EGC, or SC techniques. The transmitted packets can thus be viewed

as symbols of a repetition code. Incremental redundancy ARQ can also be realized, with both

hard decisions and soft channel outputs, e.g. using rate-compatible punctured convolutional

codes (RCPC), by first sending the highest rate code from the RCPC code family and then

sending additional bits as needed.

When Hybrid ARQ is employed, the line between the physical layer, link layer, and the MAC

sub layer is blurred (see the gray boxes of Figure 3.7) [63]. These three layers are lumped

together as a PHY layer and this combined PHY layer is just an interface queue that accepts

IP packets. This becomes even more complicated in the case of network-coded cooperation

as it involves interactions among the physical, link and network layers.

3.5.2. Cooperative wireless network architecture

Like the Hybrid ARQ which involves interaction of the physical and link layers, cooperative

networks need a more complex set of interactions between the physical, link, and network

layers. In cooperative networks, transmissions are not point-to-point and routing is not store-

and-forward. In a cooperative network, a packet received in error at the PHY layer is not nec-

essarily discarded; instead, such a packet may be saved by the link layer and subsequently

combined with other received packets or perhaps pushed up to the network layer as a packet

with errors (e.g. at a relaying node of the static decode-and-forward cooperation) [65]. Thus

in cooperative transmission, we distinguish between receiving a packet reliably or unreliably

3.5. Cross-Layer Design in Cooperative Wireless Network

Combiner Decoder Demodulator

Cooperative Decoder

store samples

CRC

fail

CRC

OK Send packets to

network layer,

discard samples

Buffer

Figure 3.8.: Cooperative transmission physical layer.

as a synonym for error-free or erroneous reception, respectively [63]. The reliability/unre-

liability decision could be based on a CRC check. The next section discusses the physical

layer at the destination and partner nodes of cooperative networks.

3.5.3. Destination node receiver

At the destination node of cooperative networks, possibly unreliable reception of multiple

transmitted packets are first combined and then decoded. As mentioned above, packet com-

bining can be based on either hard decision or soft decision channel outputs [63]. A coop-

erative transmission PHY layer receiver architecture is shown in Figure 3.8. As in a con-

ventional receiver, demodulation and sampling occurs in the demodulator module, yielding

a soft symbol output stream. The soft symbols of previously received packets are stored

in a buffer and a combiner merges the stored packets with the newly received packet. The

precise action of the combiner and the buffer depends on the type of cooperative protocol

implemented.

When diversity is achieved through repetition coding, the new packet is simply a copy of

a previously received packet and the combiner performs MRC on the soft symbols of the

packet copies. In this case, the buffer can store the soft symbols corresponding to a linear

combination of past received packets, regardless of how many packet copies are received for

a particular message. This applies to the conventional network-coded cooperation as well.

In this cooperation, in the case that the inter-user channel is good, two copies of a network-

coded packet are received at the destination from the source and partner. When the inter-user

channel has bad quality, the source may repeat its packet in the second phase. In either case,

the combiner of Figure 3.8 is used to combine twice received packets. The network layer

performs network decoding on correctly received packets sent from the decoder.

In incremental-redundancy-based relaying strategies (e.g. coded cooperation or incremental

redundancy network-coded cooperation), the new packet contains new coded symbols. These

approaches require the destination to store soft samples for all received packets for decoding.

3. Wireless Channels and Networks

The storage requirements of the buffer increase linearly with the number of received packets

for that message. Furthermore, the decoder component becomes considerably more complex

because decoding is based on multiple transmissions from multiple transmitters employing

multiple codebooks.

3.5.4. Partner node receiver

In decode-and-forward cooperation, the partner passes (reliably or unreliably) received pack-

ets to the network layer. The network layer reads the packet header, possibly modifies this

header, and sends the packet back to the PHY layer, from where it will be forwarded to

the destination. In network-coded cooperation, the network layer combines locally gener-

ated packets (i.e., from application layer) and received packets from the PHY layer. A key

observation is that all data packet transmissions are generated by the network layer. The

network-coded packet can be sent to the PHY and forwarded to the destination. In incre-

mentally redundant network-coded cooperation, the PHY can puncture the network-coded

packet received from the network layer and forwards incremental information to the destina-

tion.

3.6. Summary

In this chapter, considering the point-to-point transmission, we have in general explained

channel coding and modulation operations, discussed common causes of distortion in a wire-

less channel and how they are modeled, and gave background knowledge on the information-

theoretic capacity of fading channels. Section 3.1, Section 3.2, and Section 3.3 were written

to explain these points. The conventional and incremental-redundancy network-coded coop-

erations were revisited in Section 3.4, and the appropriate channel model was presented. The

last section of the chapter was dedicated to briefly discuss existing network architecture and

cross-layer considerations in cooperative wireless networks. In this section, we have seen

that cooperative networks require a more complex set of interactions between the physical,

link, and network layers.

4. Outage Behavior of

Network-Coded Cooperation

In Chapter 3, we have introduced network-coded cooperation as a cooperative transmission

protocol with additional network coding, as well as its extension which we call incremental

network-coded cooperation. In this chapter, we first examine the outage behavior of the

network-coded cooperation protocol by deriving its outage probability. Outage probability

helps to study the protocol independent of any particular coding scheme and is also shown

to be a lower bound on block error rate for sufficiently large block lengths [10]. To make

the outage probability analysis more tractable and convenient for exposition, quasi-static

(or block) Rayleigh fading channels, orthogonal transmission, and half-duplex constraints

are assumed. Approximating the outage result at high SNR values, we first show that this

protocol achieves full diversity (order two for two users) asymptotically in transmit power.

Second, we investigate the outage behavior for various inter-user (i.e., between transmitting

nodes) and uplink (between a transmitting node and destination) channel qualities; and we

compare various cooperative protocols based on the inter-user channels. Based on the outage

results, network-coded cooperation protocols are found to be suitable when the inter-user

channels are lower quality; when the inter-user channels are good, protocols without network

coding perform better. Third, the outage results are further extended to study the diversity-

multiplexing tradeoff and the coverage area extension.

Section 4.1 describes the system and channel models under investigation. The outage prob-

ability of the point-to-point transmission and network-coded cooperation (considering both

SC and MRC at the destination) are derived in Section 4.2. In Section 4.3, numerical results

are presented, results are discussed, and conclusions on the general outage behavior and

comparison results are drawn. Finally, the diversity-multiplexing tradeoff of the network-

coded cooperation and its coverage area analysis are investigated in Section 4.4 and Section

4.5, respectively.

4. Outage Behavior of Network-Coded Cooperation

First phase

Second phase

⊕i

Figure 4.1.: System diagram of network-coded cooperation. Solid, dashed, and dotted lines

show the transmission of S, P, and the network-coded codewords, respectively.

4.1. System model

We focus on the case of two cooperating terminals, called source (S) and partner (P), com-

municating to the same destination terminal (D) as depicted in Figure 4.1. In our model

of the wireless channel, transmissions suffer from the effect of quasi-static Rayleigh fading

channels, in which the fading remains constant over the two phases (also called block-fading

Rayleigh channels), and also additive noise. This model is appropriate for many types of

ad hoc and sensor networks in which the nodes move slowly, or are fixed but with the exact

geometry unknown at the time of design [10]. In addition, practical system constraints such

as orthogonal transmission and half-duplex constraints are considered (refer Figure 4.2). The

orthogonal transmission constraint allows for the system to be readily integrated into exist-

ing networks and makes the analysis of outage probability more tractable and convenient for

exposition. The orthogonality constraint is fulfilled by dividing the available bandwidth into

orthogonal channels and allocate these channels to the transmitting terminals. The medium-

access control (MAC) sublayer typically performs this function.

The system model considered in this work is shown Figure 4.1 whose timing diagram shown

in Figure 4.2. In network-coded cooperation, in the first phase of cooperation, the source and

partner first encode 𝑘of their information bits into a codeword of 𝑁1=𝛼𝑁 symbols, where

𝛼is the cooperation level1and 𝑁is the number of symbols per codeword if the point-to-point

transmission was used. Then both nodes, using orthogonal channels, transmit their respective

codewords to the destination and simultaneously try decoding each other’s codeword. If, for

example, the source node decodes the partner’s 𝑘bits correctly, then it forms network-coded

bits by linearly combining its own and the decoded bits. These network-coded bits are further

channel encoded (using the same or a different codebook) to form a network-coded codeword

of length 𝑁2= (1 −𝛼)𝑁, and the resulting codeword is transmitted to the destination in

the second phase. The partner will also perform the same operation. If decoding fails, then

the source (respectively the partner) transmits additional 𝑁2symbols for its own or remains

1In this chapter, we only consider the case 𝛼= 1/2.

4.1. System model

Source TX

own codeword

time, t

Partner TX

own codeword

Partner forwards

source’s codeword

Source forwards

partner’s codeword

First phase Second phase

αTαT (1-α)T (1-α)T

Figure 4.2.: Timing diagram of network-coded cooperation. 𝑇is the total time allocated to

transmit 𝑁symbols if the point-to-point transmission was used, and 𝛼is the

fraction of the time 𝑇in which users allocate to phase one. In network-coded

cooperation protocol 𝛼= 1/2and in incremental network-coded cooperation

0< 𝛼 ⩽1.

silent. Cyclic redundancy check (CRC) is assumed to detect any decoding error. Moreover,

incorporating one additional bit in the second-phase transmission would help the destination

to know the success of decoding at the source and partner. As the 𝑁1and 𝑁2symbols contain

the same 𝑛information bits as the 𝑁symbols of point-to-point transmission, for 𝛼=1

2the

information rates in both phases are 𝑛/𝑁1=𝑛/𝑁2= 2𝑛/𝑁, which are double the rate of

point-to-point transmission 𝑛/𝑁.

Once the destination receives the four codewords, it first combines the two network-coded

codewords using either SC, MRC, or EGC. Then the source’s, partner’s, and network-coded

codewords are decoded to get the respective information bits. Note that the network coding

at transmitting nodes is performed before channel coding and the network decoding at the

destination is performed after channel decoding. The network-coded cooperation can be seen

as a separate network-channel coding scheme in that the two codings are done separately.

If the information bits of the source and partner pass the CRC check, then the network-

coded bits are discarded. However, if for example the information bits from the source fail

the CRC check, then they can still be received by network decoding of the partner and the

network-coded bits, provided both are correctly recovered. A simple modulo-2 summation

can implement the network encoding and decoding at the transmitting nodes and destination

node, respectively. The same operation is performed to recover the partner’s information

bits. Intuitively, information bits are recovered using either of the two options, which means

that a maximum diversity order of two can be obtained. The actual value of the diversity

order depends on received SNR and transmission rate.

The last point concerns energy consumption as in [9], we confine the total energy2spent

to transmit 𝑁symbols is the same in both network-coded cooperation and point-to-point

2This energy, also called radiated energy, refers to the energy available at the transmitter antenna. In Chapter

5, we also include the processing energy at both transmitting and receiving nodes as well as the energy

spent at the power amplifier.

4. Outage Behavior of Network-Coded Cooperation

transmission. This is referred to total energy constraint, and this total energy is shared in the

two phases of cooperation. Let 𝛽be the fraction of the total energy allocated to the first phase.

If 𝐸𝑠is the radiated energy per symbol in point-to-point transmission, then 𝛽𝑁𝐸𝑠

𝛼𝑁 = 2𝛽𝐸𝑠is

the energy per symbol in phase one for 𝛼=1

2. Similarly, the energy per symbol in phase

two is 2(1 −𝛽)𝐸𝑠. We designate 2𝛽𝑘𝐸𝑠to represent the energy in phase 𝑘∈ {1,2}, and the

energy allocation term, 𝛽𝑘, takes on the value

𝛽𝑘={𝛽if k= 1;

(1 −𝛽)if k= 2.(4.1)

The SNR relationship in the two phases of the network-coded cooperation and point-to-

point transmission follow the same relations as the energy per symbol discussed above, and

are used to compute outage probability in the coming sections. Note that the SNR, which

takes the channel impairments and additive noise into account, is measured at the destination;

whereas the energy per symbol refers to the radiated energy at the transmitting nodes.

4.2. Outage Probability Computation

This section presents the outage probability of network-coded cooperation. First, as a base-

line for comparison, we consider point-to-point transmission between source and destina-

tion. An outage probability computation requires knowledge of the outage event, which

occurs when the channel capacity between the source and destination falls below a target

information rate, 𝑅. Since the channel capacity is a function of the fading coefficient of the

channel, it too is a random variable [5]. The outage event is converted into an equivalent

event defined in terms of the fading coefficients of the channel and the probability that this

outage event occurs is referred to as the outage probability. In terms of the Shannon formula,

𝐶(𝑠,𝑑)

𝑠= log(1 + 𝛾𝑠,𝑑)b/s/Hz is the capacity of the source-destination channel (represented

by the superscript (𝑠, 𝑑)) when the information of the source is transmitted (represented by

the subscript 𝑠) and 𝛾𝑠,𝑑 is the instantaneous SNR of the channel. The corresponding out-

age event is {𝐶(𝑠,𝑑)

𝑠< 𝑅}, where 𝑅is the information rate at which the source transmits, or

equivalently {𝛾𝑠,𝑑 <2𝑅−1}. The outage event probability or simply the outage probability

is thus defined as

𝑃(𝛾𝑠,𝑑 <2𝑅−1)=∫2𝑅−1

𝑝(𝛾𝑠,𝑑)𝑑𝛾𝑠,𝑑 (4.2)

where 𝑝(𝑥)denotes the probability density function (pdf) of random variable 𝑥. For the case

of Rayleigh fading, 𝛾has an exponential pdf with parameter 1

Γ𝑠,𝑑 , where Γ𝑠,𝑑 denotes the

4.2. Outage Probability Computation

mean value of SNR over the fading and accounts for the combination of transmit power and

large-scale path loss and shadowing effects [10]. The outage probability for Rayleigh fading

can thus be evaluated as

𝑃(𝛾𝑠,𝑑 <2𝑅−1)=∫2𝑅−1

Γ𝑠,𝑑

exp (−𝛾𝑠,𝑑

Γ𝑠,𝑑 )𝑑𝛾 = 1 −exp (−2𝑅−1

Γ𝑠,𝑑 ).(4.3)

Equation (4.3) implies that increasing Γ𝑠,𝑑 by 10 dB reduces the outage probability by only

a factor of 10. As will be discussed in the next section, the network-coded cooperation

protocol decrease the outage probability by roughly a factor of 100 when Γ𝑠,𝑑 is increased

by 10 dB, for SNR large, i.e., a diversity order of two is achievable.

4.2.1. Network-coded cooperation

In the outage analysis of network-coded cooperation, we assume symmetrical inter-user

channels (i.e., source-partner and partner-source channels have different instantaneous SNR

but may have the same average SNR). If both the source and partner act independently in

the second phase with no knowledge of whether each other’s codeword was correctly de-

coded, there are four possible cases of cooperation. In Case 1, both the source and partner

successfully decode each other, so that each user transmits the network-coded codeword in

the second phase, resulting in the full cooperation scenario. In Case 2, neither user suc-

cessfully decodes its partner’s first-phase transmission correctly and the system reverts to

the non-cooperative scenario for that pair of codewords. In Case 3, the partner successfully

decodes the source codeword, but not vice versa. Consequently, the partner transmits the

network-coded codeword which helps both nodes, but the source repeats its own codeword.

The two independent copies of the source are combined at the destination prior to decoding.

Case 4 is identical to Case 3 with the roles of the source and partner reversed. Clearly the

destination must know which of these four cases has occurred and this is achieved by in-

corporating one additional bit in the second-phase transmission and assuming that this bit is

received correctly.

Selection combining and maximum-ratio combing are the two types of combing considered

at the destination. In the former, from two received codewords the one with stronger SNR is

selected and decoded, and in the latter a codeword whose SNR is the weighted sum of the

received codewords is formed and decoded. We focus on the outage probability computation

of the source only; the same analysis holds for the partner.

4. Outage Behavior of Network-Coded Cooperation

Selection Combing at the Destination

The general approach to compute the outage probability is as follow: we first compute the

outage event and probability conditioned on the occurrence of each case, and then compute

the total probability by taking the sum of the probabilities in each case, assuming that each

case occurs independently.

For compactness of representation, let 𝑋(𝑖,𝑗)

𝑙be a codeword of node 𝑙(could be its own

codeword or the network-coded codeword) transmitted from node 𝑖and received by node 𝑗.

The subscript 𝑙takes on either 𝑠,𝑝, or 𝑠⊕𝑝to represent the source, partner, and network-

coded codewords, respectively. Instead of introducing additional notation, let us abuse 𝑋(𝑖,𝑗)

𝑙

such that it also represents the event that the transmission is working correctly; hence 𝑋(𝑗,𝑘)

𝑖

denotes the outage event. The outage event of a codeword in terms of Shannon channel

capacity as

𝑋(𝑗,𝑘)

𝑖∼

={𝐶(𝑖,𝑗)

𝑙(𝛾𝑖,𝑗) = log2(1 + 2𝛽𝑘𝛾𝑖,𝑗)<2𝑅}(4.4)

where 𝑖∈ {𝑠, 𝑝},𝑗∈ {𝑠, 𝑝, 𝑑},𝑖∕=𝑗,𝑘∈ {1,2}represents the two phases, and 𝛾𝑖,𝑗 is the

instantaneous SNR. Note that actually the capacity is also a function of 𝑅and 𝛽, were R is

the rate used in point-to-point transmission. Here 2𝑅has to be used since only half the time

is available per phase. Rearranging Equation (4.4), the outage event can also be written in

terms of the instantaneous, 𝛾𝑖,𝑗, as

𝑋(𝑗,𝑘)

𝑖∼

={𝛾𝑖,𝑗 <22𝑅−1

2𝛽𝑘}.(4.5)

For the Rayleigh fading channel assumption with 𝛾𝑖,𝑗 and Γ𝑖,𝑗 as the instantaneous SNR and

the average SNR between nodes 𝑖and 𝑗, respectively, the outage probability of the event in

Equation (4.5) is computed using Equation (4.2) and is written as follow

𝑃(𝑋(𝑗,𝑘)

𝑖)=𝑃(𝛾𝑖,𝑗 <22𝑅−1

2𝛽𝑘)= 1 −exp (−𝑔(𝑅)

𝛽𝑘Γ𝑖,𝑗 ).(4.6)

where 𝑔(𝑅) = 22𝑅−1

2. Equation (4.6) will be used to determine the outage of the inter-user

and first-phase source/partner-destination transmissions. Now let us look the four cases one-

by-one.

∙Case 1: Both the source and partner succeed in correctly decoding each other’s code-

word. In this case 𝑋𝑠and 𝑋𝑝in the first phase and 𝑋𝑠⊕𝑋𝑝∼

=𝑋𝑠⊕𝑝in the second

phase are transmitted by the source and partner (see Figure 4.3). The success event

4.2. Outage Probability Computation

First phase

Second phase

(s,d)

(s,p)

(p,s)

Xs⊕

⊕⊕

⊕p

(s,d)

Xs⊕

⊕⊕

⊕p

(p,d)

Figure 4.3.: Both users decode each others’ codeword correctly. In the first phase, the source

and partner transmit their own codewords 𝑋𝑠and 𝑋𝑝, respectively; in the second

phase both the source and partner transmit the network-coded codeword 𝑋𝑠⊕𝑝.

probability of the inter-user channel transmission is obtained by subtracting Equation

(4.6) from 1.

Under this condition, a successful recovery of the source’s information bits at the des-

tination depends on the success of the transmissions of the source and partner informa-

tion bits in the two phases. The destination recovers the source’s information bits either

from the direct transmission in the first phase or by network decoding3of the partner’s

information bits, received in the first phase, and the network-coded bits, provided both

are correctly recovered.

Using the above notation, the outage event of the source occurs if 𝑋𝑠and at least one

of 𝑋𝑝and 𝑋𝑠⊕𝑝(after selection combining) are incorrectly decoded, i.e.,

𝑋(𝑠,1) ∼

=𝑋(𝑠,𝑝)

𝑠∧𝑋(𝑝,𝑠)

𝑝∧𝑋(𝑠,𝑑)

𝑠∧(𝑋(𝑝,𝑑)

𝑝∨𝑋𝑆𝐶

𝑠⊕𝑝).(4.7)

where the subscript ‘1’ refers to the case one, ‘∧’and ‘∨’ are the logical ‘𝑎𝑛𝑑’ and ‘𝑜𝑟’

operators. 𝑋𝑆𝐶

𝑠⊕𝑝is the outage event of the network-coded codeword after selection

combining.

The events 𝑋(𝑠,𝑝)

𝑠and 𝑋(𝑝,𝑠)

𝑝in Equation (4.7) show the success of reception of the

source and partner codewords at the destination, respectively, and the corresponding

probabilities are computed using Equation (4.6). We see that even if 𝑋𝑠from the

first-phase transmission fails, the destination is still able to recover 𝑋𝑠by combining

𝑋𝑝and 𝑋𝑠⊕𝑝, provided both are correctly recovered.4Expressed equivalently, for the

source to be in outage 𝑋(𝑠,𝑑)

𝑠and at least one of 𝑋(𝑝,𝑑)

𝑝and 𝑋𝑆𝐶

𝑠⊕𝑝must be in outage.

The selection combining is implemented by picking one of the two copies of the

network-coded bits with higher instantaneous SNR value. The outage event of the

3The network decoding at the destination is identical to the network coding at transmitting nodes.

4Note that (𝑥⊕𝑦)⊕𝑥=𝑦and (𝑥⊕𝑦)⊕𝑦=𝑥.

4. Outage Behavior of Network-Coded Cooperation

selection combining, 𝑋𝑆𝐶

𝑠⊕𝑝, for the given values of 𝑅and 𝛽, is then written as

𝑋𝑆𝐶

𝑠⊕𝑝∼

={max (𝛾𝑠,𝑑 , 𝛾𝑝,𝑑)<𝑔(𝑅)

𝛽2}.(4.8)

Equation (4.8) states that, given the instantaneous SNR of the two received codewords

as 𝛾𝑠,𝑑 and 𝛾𝑝,𝑑, the larger of the two SNRs is selected first, and then compared to 𝑔(𝑅)

to determine the outage of the source after the selection combining. From Equation

(4.8), the outage condition “if the maximum of the two received SNR is less then the

threshold 𝑔(𝑅)” is equivalent to “if both SNRs are less than the threshold”. Accord-

ingly, Equation (4.8) can be re-written in the form

𝑋𝑆𝐶

𝑠⊕𝑝∼

={𝛾𝑠,𝑑 <𝑔(𝑅)

𝛽2}∧{𝛾𝑝,𝑑 <𝑔(𝑅)

𝛽2}.(4.9)

Assuming that the source-destination and partner-destination channels are uncorre-

lated, the probability that the event in (4.9) occurs can be written in terms of the indi-

vidual probabilities5of the two transmissions as

𝑃(𝑋𝑆𝐶

𝑠⊕𝑝)=(1−exp (−𝑔(𝑅)

𝛽2Γ𝑠,𝑑 ))(1−exp (−𝑔(𝑅)

𝛽2Γ𝑝,𝑑 )).(4.10)

With the selection combining outage probability quantified in Equation (4.10), the

probability that the event in Equation (4.7) occurs is then6

𝑃(𝑋(𝑠,1))=𝑃(𝑋(𝑠,𝑝)

𝑠)𝑃(𝑋(𝑝,𝑠)

𝑝)𝑃(𝑋(𝑠,𝑑)

𝑠)[𝑃(𝑋(𝑝,𝑑)

𝑝) + 𝑃(𝑋𝑆𝐶

𝑠⊕𝑝)(1 −𝑃(𝑋(𝑝,𝑑)

𝑝))].

(4.11)

The probabilities 𝑃(𝑋(𝑖,𝑗)

𝑙)and 𝑃(𝑋𝑆𝐶

𝑠⊕𝑝)in Equation (4.11) can be computed from

Equations (4.6) and (4.10), respectively. The outage probability for the partner node

can be computed similarly.

Asymptotic Analysis and Diversity Order

To examine the behavior of the outage probability at high-SNR values, where the

promised full diversity order of two is achieved for two-users cooperative schemes,

we use the approximation 1−exp−1

𝑥≈1

𝑥for large values of 𝑥. Consequently, the

5For two independent events 𝐴and 𝐵, the probability 𝑃(𝐴∩𝐵) = 𝑃(𝐴/𝐵)𝑃(𝐵) = 𝑃(𝐴)𝑃(𝐵).

6In Equation (4.11), for two events 𝐴and 𝐵, the probability property 𝑃(𝐴∪𝐵) = 𝑃(𝐴)+𝑃(𝐵)−𝑃(𝐴∩𝐵)

is used for the term inside [..].

4.2. Outage Probability Computation

outage probability in Equation (4.6) is approximated at high Γ𝑖,𝑗 values as

𝑃(𝑋(𝑗,𝑘)

𝑖) = 1 −exp (−𝑔(𝑅)

𝛽𝑗Γ𝑗,𝑘 )≈𝑔(𝑅)

𝛽𝑘Γ𝑖,𝑗

.(4.12)

Similarly, for the network-coded codeword given in Equation (4.10)

𝑃(𝑋𝑆𝐶

𝑠⊕𝑝)≈(𝑔(𝑅)

𝛽2)21

Γ𝑠,𝑑Γ𝑝,𝑑

.(4.13)

Substituting Equations (4.12) and (4.13) into Equation (4.11), we get

𝑃(𝑋(𝑠,1))≈(1−𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )(1−𝑔(𝑅)

𝛽1Γ𝑝,𝑠 )

|{z }

inter-user txs.

(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )

|{z }

direct tx.

[(𝑔(𝑅)

𝛽1Γ𝑝,𝑑 )+𝑔2(𝑅)

𝛽2

2Γ𝑠,𝑑Γ𝑝,𝑑 (1−𝑔(𝑅)

𝛽1Γ𝑝,𝑑 )]

|{z }

network decoding

=(1−𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )(1−𝑔(𝑅)

𝛽1Γ𝑝,𝑠 )(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )(𝑔(𝑅)

𝛽1Γ𝑝,𝑑 )

[1 + 𝛽1

𝛽2

𝑔(𝑅)

Γ𝑠,𝑑 (1−𝑔(𝑅)

𝛽1Γ𝑝,𝑑 )].(4.14)

The [1−𝑔(𝑅)

𝛽𝑘Γ𝑖,𝑗 ]terms in (4.14) approach 1 for high Γ𝑖,𝑗 values. For symmetrical

uplink channels, i.e., Γ𝑠,𝑑 = Γ𝑝,𝑑, the outage probability reduces to

𝑃(𝑋(𝑠,1))≈(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )2[1 + 𝛽1

𝛽2

𝑔(𝑅)

Γ𝑠,𝑑 ]≈(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )2

.(4.15)

From the outage probability approximation given in Equation (4.15), we observe the

following points:

1. When the inter-user channels are reliable, the outage-probability decay is pro-

portional to the square of Γ𝑠,𝑑; hence a diversity order of two is achievable using

network-coded cooperation.

2. The outage probability is less sensitive to the inter-user channels variation.

3. At high-SNR values, the contribution of the direct transmission from the source

is dominant compared to the contribution of the network coding. As will be

discussed in the later sections when comparing the network-coding-based coop-

eration approach with other cooperative approaches, the former performs better

when the inter-user channels are reciprocal and the uplink channels are of poor

4. Outage Behavior of Network-Coded Cooperation

First phase Second phase

(p,d)

(s,d)

(p,d)

Figure 4.4.: Neither user decods its partner’s codeword correctly. The source transmits and

partner transmit codewords 𝑋𝑠and 𝑋𝑝, respectively, to the destination in phase

one and repest the same codeword in the second phase.

quality, i.e., at lower SNR values.

4. Controlling the energy allocation term 𝛽is one way to optimize the performance

of the network-coded cooperation.

∙Case 2: Neither user decodes its partner’s codeword correctly and hence the source

and partner repeat their own codewords in the second phase (Figure 4.4). For a block-

fading channel assumption where the fading is the same over the two phases, the

strength of the received SNR is dictated by the energy allocation term 𝛽. Under this

condition, selection combining is performed on the source’s codewords received over

the two phases (unlike on the network-coded codeword in the above case) and an

outage occurs to the source if 𝑋𝑠is incorrectly decoded from either of the two trans-

missions. In terms of the instantaneous SNR, the outage event occurs if both 2𝛽1𝛾𝑠,𝑑

and 2𝛽2𝛾𝑠,𝑑 are less than the threshold 𝑔(𝑅), and for a given value of 𝛽, this event is

written as

𝑋𝑆𝐶

𝑠∼

={𝛾𝑠,𝑑 <min (𝑔(𝑅)

𝛽1

,𝑔(𝑅)

𝛽2)}.(4.16)

Note that, unlike the event in Equation (4.10), in Equation (4.16) a single random

variable, 𝛾𝑠,𝑑, is compared to the minimum of two fixed threshold values, namely 𝑔(𝑅)

𝛽1

and 𝑔(𝑅)

𝛽2. The probability that the event in Equation (4.16) occurs is then

𝑃(𝑋𝑆𝐶

𝑠)= 1 −exp (−𝑔(𝑅)

Γ𝑠,𝑑

min (1

𝛽1

𝛽2))

≈𝑔(𝑅)

Γ𝑠,𝑑

min (1

𝛽1

𝛽2).(4.17)

The second line in Equation (4.17) is the result of a high-SNR approximation. The

overall outage event is then

𝑋(𝑠,2) ∼

=𝑋(𝑠,𝑝)

𝑠⋅𝑋(𝑝,𝑠)

𝑝⋅𝑋𝑆𝐶

𝑠.(4.18)

4.2. Outage Probability Computation

The outage probability 𝑃(𝑋(𝑠,2))that the event in (4.18) occurs is then given as

𝑃(𝑋(𝑠,2)) = exp (−𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )exp (−𝑔(𝑅)

𝛽1Γ𝑝,𝑠 )

|{z }

inter-user txs.

[1−exp (−𝑔(𝑅)

Γ𝑠,𝑑

min (1

𝛽1

𝛽2))]

|{z }

direct txs.

≈(𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )(𝑔(𝑅)

𝛽1Γ𝑝,𝑠 )[𝑔(𝑅)

Γ𝑠,𝑑

min (1

𝛽1

𝛽2)].(4.19)

The second line in Equation (4.19) is the result of a high SNR approximation. From the

outage probability approximation given in Equation (4.19), we observe the following

points:

1. When the inter-user channels are unreliable, the outage probability decays pro-

portionally to Γ𝑠,𝑑 (not to the square of Γ𝑠,𝑑 as in case one above); hence, even at

high SNR values, no diversity gain is obtained in such a case. The result would

have been different if the channels were time varying per phase, i.e., the scheme

would have benefited from time diversity as there is no diversity gain by repeat-

edly sending on a correlated channel (i.e., block-fading channels).

2. As a consequence of the above observation, if we allow nodes to remain silent

when they fail to decode their partner’s codeword, the outage probability in Equa-

tion (4.19) reduces to

𝑃(𝑋(𝑠,2))≈(𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )(𝑔(𝑅)

𝛽1Γ𝑝,𝑠 )(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 ).(4.20)

where the outage probability is more or less the same. In correlated channels,

it is better to remain silent instead of repeating own message, and the former is

more advantageous if energy saving is a concern as in sensor networks.7

3. Minimizing the outage probability given in Equation (4.19), among other factors,

requires minimizing the minimum of 1

𝛽1=1

𝛽and 1

𝛽2=1

(1−𝛽). As the former and

the latter are decreasing and increasing functions of 𝛽, respectively, allocating the

same power in the two phases, i.e., 𝛽=1

2is an optimal choice when the inter-

user channels are poor quality. Of course, optimal allocation of energy should

consider all four cases.

∙Case 3: The partner correctly decodes the source’s codeword, but the source cannot

correctly decode the partner’s codeword (see Figure 4.5). In the second phase, the

source transmits its own codeword, 𝑋𝑠, and the partner transmits 𝑋𝑠⊕𝑝. Selection

combining done on the source’s codeword and its outage event and probability are

7The energy efficiency in sensor network discussion in Chapter 5 is based on the idea of remaining silent.

4. Outage Behavior of Network-Coded Cooperation

First phase Second phase

(p,d)

(s,d)

s⊕

⊕⊕

⊕p

(p,d)

Figure 4.5.: The partner correctly decodes the source’s codeword, but the source cannot cor-

rectly decode the partner’s codeword. In the second phase, the source transmits

its own codeword 𝑋𝑠whereas the partner transmits the network-coded codeword

𝑋𝑠⊕𝑝.

given as in Case 2 of Equations (4.16) and (4.17), respectively. The overall outage

event of this case is given as

𝑋(𝑠,3) ∼

=𝑋(𝑠,𝑝)

𝑠∧𝑋(𝑝,𝑠)

𝑝∧𝑋𝑆𝐶

𝑠∧[𝑋(𝑝,𝑑)

𝑝∨𝑋(𝑝,𝑑)

𝑠⊕𝑝].(4.21)

As in Equations (4.8) and(4.16), one can write the event [𝑋(𝑝,𝑑)

𝑝∨𝑋(𝑝,𝑑)

𝑠⊕𝑝]in Equation

(4.21) in the form

𝑋(𝑝,𝑑)

𝑝∨𝑋(𝑝,𝑑)

𝑠⊕𝑝∼

=𝑋(𝑝,𝑑)

𝑝,𝑠⊕𝑝∼

={𝛾𝑝,𝑑 <max (𝑔(𝑅)

𝛽1

,𝑔(𝑅)

𝛽2)}.(4.22)

and the probability that this event occurs is written similar to the probability in Equa-

tion (4.17). Therefore, the overall outage probability in this case is then

𝑃(𝑋(𝑠,3))=𝑃(𝑋(𝑠,𝑝)

𝑠)𝑃(𝑋(𝑝,𝑠)

𝑝)𝑃(𝑋𝑆𝐶

𝑠)𝑃(𝑋(𝑝,𝑑)

𝑝,𝑠⊕𝑝).(4.23)

Substituting the individual outage probability results at high-SNR regime, Equation

(4.23) becomes

𝑃(𝑋(𝑠,3))≈(1−𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )(𝑔(𝑅)

𝛽1Γ𝑝,𝑠 )

|{z }

inter-user txs.

(𝑔(𝑅)

Γ𝑠,𝑑 )min (1

𝛽1

𝛽2)

|{z }

direct tx.

(𝑔(𝑅)

Γ𝑝,𝑑 )max (1

𝛽1

𝛽2)

|{z }

network coding

.(4.24)

Note that in Equation (4.24), max (1

𝛽1,1

𝛽2)min (1

𝛽1,1

𝛽2)=(1

𝛽1𝛽2). For the case that

the source-destination and partner-destination uplink channels have the same average

SNR, i.e., Γ𝑠,𝑑 = Γ𝑝,𝑑, and the inter-user channels are symmetrical, Equation (4.24)

4.2. Outage Probability Computation

s⊕

⊕⊕

⊕p

(

)

(p,d)

First phase Second phase

(p,d)

(s,d)

Figure 4.6.: The source correctly decodes the partner’s codeword, but not vice versa. In the

second phase, the partner transmits its own codeword 𝑋𝑝whereas the source

transmits the network-coded codeword 𝑋𝑠⊕𝑝.

can be further approximated as

𝑃(𝑋(𝑠,3))≈(𝑔(𝑅)

𝛽1Γ𝑝,𝑠 )(𝛽1

𝛽2)(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )2

.(4.25)

From the outage probability result in Equation (4.25), we conclude the following

points:

1. Like Case 1, diversity order of two is achievable as the outage probability decay

is proportional to the square of Γ𝑠,𝑑.

2. For asymmetric cooperation (i.e., one node cooperates but not the other), the out-

age probability depends on the inter-user channels quality, and this is unlike Case

1above, where both the source and partner send the network-coded codeword

in the second phase. This means that network coding is more beneficial when

asymmetric cooperation exists; this is because the network-coded codeword sent

by either the source or partner benefits both the source and partner.

3. Like the above two cases, the energy allocation term plays an important role in

controlling the outage probability.

∙Case 4: The source correctly decodes the partner’s codeword, but the partner cannot

decode the source’s codeword correctly (see Figure 4.6). Here, 𝑋𝑝is received twice

and its outage event, considering selection combining, is then

𝑋𝑆𝐶

𝑝∼

={𝛾𝑝,𝑑 <min (𝑔(𝑅)

𝛽1

,𝑔(𝑅)

𝛽2)} (4.26)

and the probability that this event occurs is then

𝑃{𝑋𝑆𝐶

𝑝} ≈ 𝑔(𝑅)

Γ𝑝,𝑑

min (1

𝛽1

𝛽2).(4.27)

4. Outage Behavior of Network-Coded Cooperation

The overall outage event under this case is given as

𝑋(𝑠,4) ∼

=𝑋(𝑠,𝑝)

𝑠∧𝑋(𝑝,𝑠)

𝑝∧𝑋(𝑠,𝑑)

𝑠∧[𝑋𝑆𝐶

𝑝∨𝑋(𝑠,𝑑)

𝑠⊕𝑝]

∼

=𝑋(𝑠,𝑝)

𝑠∧𝑋(𝑝,𝑠)

𝑝∧[(𝑋(𝑠,𝑑)

𝑠∧𝑋(𝑠,𝑑)

𝑠⊕𝑝)∨(𝑋(𝑠,𝑑)

𝑠∧𝑋𝑆𝐶

𝑝)].(4.28)

As in Case 3 above, the even (𝑋(𝑠,𝑑)

𝑠∧𝑋(𝑠,𝑑)

𝑠⊕𝑝)in Equation (4.28) is written in the

form

𝑋(𝑠,𝑑)

𝑠∨𝑋(𝑠,𝑑)

𝑠⊕𝑝∼

=𝑋(𝑠,𝑑)

𝑠,𝑠⊕𝑝∼

={𝛾𝑠,𝑑 <max (𝑔(𝑅)

𝛽1

,𝑔(𝑅)

𝛽2)}.(4.29)

The overall outage probability to user S is computed from Equation (4.28) as

𝑃(𝑋(𝑠,4))≈(𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )[𝑔(𝑅)

Γ𝑠,𝑑

max (1

𝛽1

𝛽2)+(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )2

min (1,𝛽1

𝛽2)].(4.30)

Performing the same manipulations as above, the outage probability in Equation (4.30)

reduces to the form

𝑃(𝑋(𝑠,4))≈(𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )min (1,𝛽1

𝛽2)(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )(4.31)

From the outage probability result in Equation (4.31), we conclude the following

points:

1. In this case, only diversity order of one is achievable as the outage probability

decay is proportional to Γ𝑠,𝑑.

2. As in Case 3 above, for asymmetric cooperation the outage probability depends

on the inter-user channels’ quality, which indicates that network coding is more

beneficial in asymmetric cooperation cases. In this case the network-coded code-

word sent by the source benefits both the source and partner.

3. The energy allocation term plays an important role in controlling the outage prob-

ability.

Total Outage Expression

In the four cases seen so far, we computed the outage probability conditioned on the oc-

currence of each case. Assuming that the occurrence events of the four cases are mutually

exclusive (for independent inter-user channels), the total outage probability, 𝑃(𝑋𝑠), is the

4.2. Outage Probability Computation

sum of the outage probabilities in the four cases, and its high-SNR approximation is given

𝑃(𝑋𝑠)=𝑃(𝑋(𝑠,1))+𝑃(𝑋(𝑠,2))+𝑃(𝑋(𝑠,3))+𝑃(𝑋(𝑠,4))

≈(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )2

|{z }

Case 1

+(𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )(𝑔(𝑅)

𝛽1Γ𝑝,𝑠 )[𝑔(𝑅)

Γ𝑠,𝑑

min (1

𝛽1

𝛽2)]

|{z }

Case 2

(𝑔(𝑅)

𝛽1Γ𝑝,𝑠 )( 1

𝛽1𝛽2)(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )2

|{z }

Case 3

+(𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )min (1,𝛽1

𝛽2)(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )

|{z }

Case 4

(4.32)

Plotted in Figure 4.7 are the exact and approximation (given in Equation (4.32)) outage prob-

ability results when the average SNR of all the uplink and inter-user channels are the same,

i.e., Γ𝑠,𝑝 = Γ𝑝,𝑠 = Γ𝑠,𝑑 = Γ𝑝,𝑑. One can see that at higher Γ𝑠,𝑑 values, the approximation

captures the exact result, and the approximation error is on the order of 1 dB.

Asymptotic Analysis and Diversity Order

To determine the diversity order achieved by network-coded cooperation, the mean SNR Γ𝑖,𝑗

is re-parameterize by decoupling the user transmit power from the physical impairments of

the channel itself as in [66]:

Γ𝑖,𝑗 ⇒Γ𝑇Γ𝑖,𝑗 (4.33)

where Γ𝑇is the ratio of transmit power to the received noise, and Γ𝑖,𝑗 is a finite constant

accounting for large-scale, path-loss, and shadowing effects. Further, we assume that Γ𝑇is

the same for both the source and partner, and the relative differences in quality between the

various channels are captured by the Γ𝑖,𝑗 values. Thus, by expressing outage probability as a

function of 1/Γ𝑇, and then letting Γ𝑇→ ∞(e.g., the high-SNR regime), the diversity order

is given by the smallest exponent of 1/Γ𝑇.

To obtain the outage probability as a function of 1

Γ𝑇, we re-write Equation (4.32) by collect-

ing like-order terms of 1

Γ𝑇, and this results in the following expression:

𝑃(𝑋𝑠)≈1

Γ2

𝑇⋅[𝑔(𝑅)

𝛽1Γ𝑠,𝑑 ]2

+O(1

Γ3

𝑇)(4.34)

where O(1

Γ3

𝑇)denotes the higher-order terms. It is interesting to note that, in the high-SNR

regime, the dependence of outage probability of the source on the inter-user channels, i.e.,

Γ𝑠,𝑝 and Γ𝑝,𝑠, appears only in the third-order term. So, we once again see from Equation

4. Outage Behavior of Network-Coded Cooperation

−10 −5 0 5 10 15 20 25 30

10−4

10−3

10−2

10−1

100

Mean source−destination SNR, Γs,d (dB)

Outage Probability

Point−to−point

Network coding, exact outage

Network coding, approximate outage

Figure 4.7.: Outage probability curves of the point-to-point transmission and the network-

coded cooperation (both approximate and exact).

(4.34) that the outage probability decays proportional to the square of the source-destination

SNR, hence the diversity order of two can be achieved.

Reciprocal inter-user channels

In reciprocal inter-user channels, the instantaneous SNRs 𝛾𝑠,𝑝 =𝛾𝑝,𝑠, such that Case 3 and

Case 4 vanish. The overall outage probability becomes

𝑃(𝑋𝑠)≈(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )2

+(𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )(𝑔(𝑅)

𝛽1Γ𝑝,𝑠 )[𝑔(𝑅)

Γ𝑠,𝑑

min (1

𝛽1

𝛽2)].(4.35)

We see from Equation (4.35) that outage probability is dependent on the partner-destination

channel only in Case 1. Hence, in the case of reciprocal inter-user channels, the cooperative

protocol does not benefit much from network coding.

Maximum-Ratio Combing at the Destination

So far, the outage behavior of the network-coded cooperation was investigated assuming

selection combining at the destination. Next, we study the outage behavior when MRC

is used to combine identical codewords at the destination. In its simplest form, MRC is

realized by adding the instantaneous SNR of each codeword received from the two channels.8

This accumulation of the instantaneous SNR increases the rate at which the destination can

8A more complex implementation of MRC requires estimation of the channel coefficients and then taking the

weighted sum of the received codewords.

4.2. Outage Probability Computation

reliably decode the codeword (in other words, increases the channel capacity). Following

similar four cases as in selection combining, the outage results are presented next.

∙Case 1: Both the source and partner decode each others’ codeword correctly. As

the network-coded codeword 𝑋𝑠⊕𝑝is received twice from the source and partner, its

instantaneous SNR accumulates at the destination. We can define an outage event as

𝑋𝑀𝑅𝐶

𝑠⊕𝑝∼

={(𝛾𝑠,𝑑 +𝛾𝑝,𝑑)<𝑔(𝑅)

𝛽2}.(4.36)

This event depends on two random variables 𝛾𝑠,𝑑 and 𝛾𝑏,𝑑, which are exponentially dis-

tributed under a Rayleigh fading channel assumption. Let uand vbe two exponentially

distributed random variables with parameters 𝜆𝑢and 𝜆𝑣, and w = u + v. Referring to

[5, appendix I], the probability 𝑃{w⩽𝑤}is given as

𝑃(w⩽𝑤) = {1−[( 𝜆𝑢

𝜆𝑢−𝜆𝑣)exp (−𝜆𝑣𝑤) + (𝜆𝑣

𝜆𝑣−𝜆𝑢)exp (−𝜆𝑢𝑤)]

1−(1 + 𝜆𝑤)exp (−𝜆𝑤)(4.37)

where the first equation is when 𝜆𝑢∕=𝜆𝑣and the second for 𝜆𝑢=𝜆𝑤=𝜆. In our

case, setting 𝑤=𝑔(𝑅)

𝛽2,𝜆𝑢=1

Γ𝑠,𝑑 ,𝜆𝑣=1

Γ𝑝,𝑑 , and w=𝛾𝑠,𝑑 +𝛾𝑝,𝑑, the probability

𝑃(𝑋𝑀𝑅𝐶

𝑠⊕𝑝)is readily computed. For identical source/partner-destination channels,

i.e., Γ𝑠,𝑑 = Γ𝑝,𝑑, the outage probability at high SNR is approximated as

𝑃(𝑋𝑀𝑅𝐶

𝑠⊕𝑝)=𝑃((𝛾𝑠,𝑑 +𝛾𝑝,𝑑)<𝑔(𝑅)

𝛽2)

= 1 −(1 + 1

Γ𝑠,𝑑

𝑔(𝑅)

𝛽2)exp (−1

Γ𝑠,𝑑

𝑔(𝑅)

𝛽2)≈1

2(𝑔(𝑅)

𝛽2Γ𝑠,𝑑 )2

.(4.38)

We see the high-SNR approximation in Equation (4.38) (which is for MRC at the

destination) differs from the approximation in Equation (4.13) (which is for selection

combing at the destination) by a factor of 1

2. The outage event probability of the

source is the same as the probability given in Equation (4.11) except that the term

𝑃(𝑋𝑆𝐶

𝑠⊕𝑝)is replaced by 𝑃(𝑋𝑀𝑅𝐶

𝑠⊕𝑝). For symmetrical inter-user channels, the outage

probability in this case is approximated as follows:

𝑃(𝑋(𝑠,1))≈(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )2[1 + 1

𝛽1

𝛽2

𝑔(𝑅)

Γ𝑠,𝑑 ]≈(𝑔(𝑅)

𝛽1Γ𝑠,𝑑 )2

.(4.39)

This is the same as for selection combining at the destination, and it is not surprising

because at high-SNR regime, the direct transmission from the the source dictates the

outage behavior. The conclusions for the Case 1 of the selection combining holds true

for MRC as well.

4. Outage Behavior of Network-Coded Cooperation

∙Case 2: Neither the source nor the partner decodes each other’s codeword correctly.

Each node repeats its own codeword and MRC is performed at the destination. The

resulting instantaneous SNR of the source’s codeword at the destination is 𝛽1𝛾𝑠,𝑑 +

𝛽2𝛾𝑠,𝑑 =𝛾𝑠,𝑑, and the outage probability is computed as

𝑃(𝑋𝑀𝑅𝐶

𝑠)=𝑃(𝛾𝑠,𝑑 < 𝑔(𝑅)) ≈𝑔(𝑅)

Γ𝑠,𝑑

.(4.40)

Note that the difference between the above equation and Equation (4.17) (which is for

selection combining) is that the term min (1

𝛽1,1

𝛽2), which accounts for the energy

allocation, is missing in the former. The overall outage probability, for symmetrical

inter-user channels, is given as

𝑃(𝑋(𝑠,2))≈(𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )2(𝑔(𝑅)

Γ𝑠,𝑑 ).(4.41)

From the outage probabilities given in Equations (4.40) and (4.41), we note the fol-

lowing points:

1. As a result of the MRC operation at the destination, the outage probability of the

uplink transmission is independent of the energy allocation.

2. No diversity gain is obtained in the case of unreliable inter-user channels, as

the outage is a function of the source-destination instead of the source/partner-

destination channels.

∙Case 3: The partner correctly decodes the source’s codeword, but the source cannot

decode the partner’s codeword correctly. In this case, 𝑋𝑠is received twice from the

source and its outage probability is given by Equation (4.40), and the codeword 𝑋𝑠⊕𝑝

is received from the partner only. The overall outage probability is approximated as

𝑃(𝑋(𝑠,3))≈(𝑔(𝑅)

𝛽1Γ𝑝,𝑠 )

|{z }

inter-user txs.

max (1

𝛽1

𝛽2)

|{z }

power allo.

(𝑔(𝑅)

Γ𝑠,𝑑 )2

|{z }

uplink tx.

.(4.42)

∙Case 4: The source correctly decodes the partner’s codeword, but the partner cannot

decode the source’s codeword correctly. In this case, 𝑋𝑝is received twice from the

partner and its outage probability is given by Equation (4.40) with Γ𝑠,𝑑 replaced by

Γ𝑝,𝑑. The outage probability is similar to the probability of Case 4 given in Equation

(4.31) above, and at high SNR, symmetrical inter-user, and identical uplink channels,

4.3. Numerical Results and Discussion

Static Adaptive

Repetition

coding

Network

coding

Repetition

coding

Network

coding

Coded

cooperation

Remain

silent

Repeat

own

codeword

Decode-and-forward

Network

coding

Remain

silent

Repeat

own

codeword

Repeat

partner’s

codeword

Repeat

partner’s

symbols

Repeat

own

symbols

YN N Y N N

Figure 4.8.: A block diagram showing the various protocols considered for comparison, ‘𝑌’

and ‘𝑁’ stands for ‘𝑌 𝐸𝑆’ and ‘𝑁𝑂’ to mean that decoding of partner’s code-

word is successful or not successful.

it is approximated as

𝑃(𝑋(𝑠,4))≈(𝑔(𝑅)

𝛽1Γ𝑠,𝑝 )

|{z }

inter-user txs.

max (1

𝛽1

𝛽2)

|{z }

energy allo.

(𝑔(𝑅)

Γ𝑠,𝑑 )

|{z }

uplink tx.

.(4.43)

The total outage probability is the sum of the outage probabilities under Cases 1-4. The

asymptotic behavior of the total outage probability can be written as in (4.34) and after

summing the probabilities in each case and simplifying, we get

𝑃(𝑋𝑠)≈1

Γ2

𝑇⋅[𝑔(𝑅)

𝛽1Γ𝑠,𝑑 ]2

+O(1

Γ3

𝑇).(4.44)

Comparing on the asymptotic outage probabilities of (4.34) and (4.44), we conclude that at

high-SNR regime, both selection combining and maximum-ratio combining behave the same

and also that both achieve diversity order of two.

4.3. Numerical Results and Discussion

4.3.1. Basic assumptions and parameters

In this section, the outage probability results for the network-coded cooperation and point-to-

point transmission are presented. For ease of exposition, we set Γ𝑠,𝑝 = Γ𝑝,𝑠, i.e., symmetrical

4. Outage Behavior of Network-Coded Cooperation

−10 −5 0 5 10 15 20 25 30

10−4

10−3

10−2

10−1

100

Mean source−destination SNR, Γs,d (dB)

Outage Probability

Direct transmission

RC−based, Γs,p= 0 dB

RC−based, Γs,p= 10 dB

RC−based, Γs,p= 30 dB

NC−based, Γs,p= 0 dB

NC−based, Γs,p= 10 dB

NC−based, Γs,p= 30 dB

Figure 4.9.: Outage probability vs. Γ𝑠,𝑑 results of static protocols without (solid line) and

with (dashed lines) network coding. Results are based on Γ𝑠,𝑝 = 0, 10, and 30

dB; NC stands for network coding.

inter-user channels (this is a true assumption for reciprocal inter-user channels, and reason-

able for independent inter-user channels since path loss is a reciprocal phenomenon, and

large-scale shadowing, i.e., from buildings or other large obstructions, is also in many cases

[10]). As a result of this, we note that the outage probabilities for both the source and partner

are equal if the uplink channels have equal mean SNR, i.e., Γ𝑠,𝑑 = Γ𝑝,𝑑. Unless stated other-

wise, we assume an information rate of 𝑅= 1/4, energy allocations of 𝛽= 1/2,Γ𝑠,𝑑 = Γ𝑝,𝑑

channels, and all plots are results of the exact outage probability (i.e., not the high SNR

approximations).

4.3.2. List of investigated protocols

The list of protocols considered for comparison are shown in the block diagram of Figure

4.8, and a detailed description of each protocol can be referred from [13]. Here is a brief

description of how the protocols are categorized. Based on their level of adaptiveness when

decoding of each other’s codeword fails, decode-and-forward-based protocols can be cate-

gorized as static or adaptive. In static protocols, the partner always forwards the source’s

codeword using either repetition or network coding. In the group of adaptive protocols, the

partner decides whether to forward or not, depending on its success of decoding the source’s

codeword. If successful, then it may forward using network coding, repetition coding, or

4.3. Numerical Results and Discussion

−10 −5 0 5 10 15 20 25 30

10−6

10−5

10−4

10−3

10−2

10−1

100

Mean source−destination SNR, Γs,d (dB)

Outage Probability

Direct transmission

RC−based, Γs,p= 0 dB

RC−based, Γs,p= 10 dB

RC−based, Γs,p= 30 dB

NC−based, Γs,p= 0 dB

NC−based, Γs,p= 10 dB

NC−based, Γs,p= 30 dB

Figure 4.10.: Outage probability vs. Γ𝑠,𝑑 results of adaptive protocols based on repetition

coding (solid lines) and network coding (dashed lines), both with the option to

remain silent if decoding fails. Γ𝑠,𝑝 =0, 10, and 30 dB are used; RC stands for

repetition coding.

coded cooperation; if decoding fails, then the partner has the options to transmit its own

codeword (symbols for coded cooperation) or remain silent [13].

4.3.3. Numerical results

Below, the numerical results for the protocols listed in Figure 4.8 are presented. Figure 4.9

depicts the outage probability vs. Γ𝑠,𝑑 (dB) of static protocols based on repetition and net-

work coding, and for various inter-user channel qualities. We see that at lower Γ𝑠,𝑝 values

(example 0 dB), the performance of these protocols is closer to the point-to-point transmis-

sion; and at higher Γ𝑠,𝑝 values (example 30 dB), the performance improves substantially and

a diversity order of two, taking point-to-point transmission as reference, can be achieved.

Static protocols are relatively simple to implement as the destination does not need to know

decoding results at the source and partner, and the partner (respectively the source) also

does not need to worry about errors contained in the codeword it has received from the

source (respectively partner). This simplicity of implementation, if supported by the pres-

ence of reliable inter-user channels, would make static protocols attractive. Within static

protocols, the performance of the protocol with network coding is poor as compared to the

4. Outage Behavior of Network-Coded Cooperation

−10 −5 0 5 10 15 20

10−4

10−3

10−2

10−1

100

Mean source−destination SNR, Γs,d (dB)

Outage Probability

Direct transmission

RC−based AdDF, β = 1/2

RC−based AdDF, β = 2/3

NC−based AdDF, β = 1/2

NC−based AdDF, β = 2/3

Coded cooperation, α = 1/2

Coded cooperation, α = 3/4

Figure 4.11.: Outage probability vs. Γ𝑠,𝑑 results of adaptive protocols based on repetition

coding (solid lines) and network coding (dashed lines), both with the option

to repeat their own codewords if decoding fails, and coded cooperation (dotted

lines). Γ𝑠,𝑝 = 0 dB; 𝛽= 1/2and 2/3 for repetition and network-coding-based

protocols; 𝛽= 1/2, and 𝛼= 1/2and 3/4 for coded cooperation. AdDF stands

for adaptive DF.

repetition-coding-based protocol; hence there is no incentive to use network coding in static

protocols.

Next, let us investigate the option of remaining silent during decoding failure in adaptive

protocols, and for that we consider the two groups of protocols: repetition and network

coding. The outage probability results of the two protocols are shown in Figure 4.10. At

lower Γ𝑠,𝑝 values of 0 and 10 dB, the protocol with network coding performs better than

the repetition-coding-based protocol; this improvement is appreciable when Γ𝑠,𝑑 is greater

than 10 dB. On the other hand, the repetition-coding-based protocol performs better than

its network-coded counterpart when the inter-user channels are of high quality (example 30

dB); and at this Γ𝑠,𝑝 value both protocols achieve a diversity order of two.

Now, let us include coded cooperation protocols on top of the repetition-coding and network-

coding-based adaptive protocols, both with the option of repeating own codeword, whose

comparison is shown in Fig. 4.10. For clarity of presentation, three separate plots are made

for three inter-user channel qualities. Figure 4.11 is when Γ𝑠,𝑝 = 0 dB, Figure 4.12 is when

Γ𝑠,𝑝 =10 dB, and Figure 4.13 is when Γ𝑠,𝑝 = 30 dB. From Figure 4.11 and Figure 4.12,

we see that at lower Γ𝑠,𝑝 values, namely 0 and 10 dB, the network-coding-based protocol

outperforms the other two protocols. Also, for the first two protocols, we see the advantage

of allocating more energy in the first phase (i.e., higher 𝛽value). This is logical in that at

4.3. Numerical Results and Discussion

−10 −5 0 5 10 15 20

10−4

10−3

10−2

10−1

100

Mean source−destination SNR, Γs,d (dB)

Outage Probability

Direct transmission

RC−based AdDF, β = 1/2

RC−based AdDF, β = 2/3

NC−based AdDF, β = 1/2

NC−based AdDF, β = 2/3

Coded cooperation, α = 1/2

Coded cooperation, α = 3/4

Figure 4.12.: Outage probability vs. Γ𝑠,𝑑 results for the same three adaptive protocols con-

sidered in Fig. 4.11; the only exception is here Γ𝑠,𝑝 = 10 dB used. In both

figures, the curves of coded cooperation with 𝛼= 1/2and repetition coding

with 𝛽= 1/2appear overlapping.

lower inter-user SNR values, the source and partner should allocate more power in the first

phase than in the second phase. In the coded cooperation protocol, a larger 𝛼shows a better

performance. Here, the increased redundancy in the first phase (i.e., large 𝛼) compensates for

the low Γ𝑠,𝑝. With 𝛼= 3/4it achieves higher performance than the repetition-coding-based

protocol.

In Figure 4.13, the comparison is done for Γ𝑠,𝑝 = 30 dB (a high-quality link). The network-

coded cooperation with 𝛽= 1/2performs worse; and coded cooperation with 𝛼= 1/2

performs relatively better than the other protocols. It is interesting that compared to the

cases with lower Γ𝑠,𝑝 (Figures 4.11 and 4.12), here the effect of 𝛼has reversed. Due to the

good inter-user links, here it is beneficial to allocate more redundancy to the second phase,

i.e., choosing a smaller 𝛼than with low Γ𝑠,𝑝.

Finally, let us investigate if there is an advantage by remaining silent instead of repeating

one’s own codeword in repetition and network-coding-based adaptive protocols. The results

in Figure 4.14 show that, in repetition-coding-based protocol, at lower Γ𝑠,𝑝 values (example

0 dB) there is a slight advantage by repeating one’s own codeword during decoding failure;

and at Γ𝑠,𝑝 values higher than 0 dB, there is only negligible advantage by repeating one’s

own codeword, and the curves for Γ𝑠,𝑝 10 dB and 30 dB appear overlapping. In the network-

coding-based protocols, as illustrated in Figure 4.15, there is a clear advantage of repeating

one’s own codeword at lower Γ𝑠,𝑝 values (example 0 dB); and at a moderate value of 10

4. Outage Behavior of Network-Coded Cooperation

−10 −5 0 5 10 15 20

10−4

10−3

10−2

10−1

100

Mean source−destination SNR, Γs,d (dB)

Outage Probability

Direct transmission

RC−based AdDF, β = 1/2

RC−based AdDF, β = 2/3

NC−based AdDF, β = 1/2

NC−based AdDF, β = 2/3

Coded cooperation, α = 1/2

Coded cooperation, α = 3/4

Figure 4.13.: Outage probability vs. Γ𝑠,𝑑 results for the same three adaptive protocols consid-

ered in Fig. 4.11 and Fig. 4.12 above. The only difference is, here Γ𝑠,𝑝 =30 dB

is used.

dB, the performance improvement is evident at higher values of Γ𝑠,𝑑. In the presence of a

reliable source-relay link (example Γ𝑠,𝑝 =30 dB), like the results in Figure 4.14, there is

negligible performance gain by repeating one’s own codeword during decoding failure; this

is not surprising as decoding failure at the relay, at high Γ𝑠,𝑝 values, rarely happens.

4.3.4. Conclusion and remarks

In this section, we have presented the outage probability results of the static and adaptive

cooperative protocols without and with network coding, and inter-user-channels-based per-

formance comparisons are done. Based on the results, we draw the following conclusions.

∙Static protocols are found to achieve full diversity, provided the destination decodes

the source’s first-phase transmission before attempting to decode from the combined

codeword.

∙Static protocols do not benefit from network coding.

∙In adaptive protocols, contrary to static protocols, incorporating network coding de-

livers performance improvement, and this improvement is noticeable when the inter-

source channels are poor quality.

4.4. Diversity-Multiplexing Tradeoff

−10 −5 0 5 10 15 20 25 30

10−6

10−5

10−4

10−3

10−2

10−1

100

Mean source−destination SNR, Γs,d (dB)

Outage Probability

Direct transmission

Remaining silent, Γs,r= 0 dB

Remaining silent, Γs,r= 10 dB

Remaining silent, Γs,r= 30 dB

Repeating own message, Γs,r= 0 dB

Repeating own message, Γs,r= 10 dB

Repeating own message, Γs,r= 30 dB

Figure 4.14.: Outage probability vs. Γ𝑠,𝑑 results of repetition-coding-based adaptive proto-

cols with the options of either remaining silent (solid lines) or repeating one’s

own codeword (dashed lines) when decoding failure occurs. The curves with

identical Γ𝑠,𝑝 values of 10 dB and 30 dB appear overlapping with each other.

∙With reliable inter-user channels, the coded cooperation protocol, with low coopera-

tion level 𝛼, performs relatively better. In unreliable channels, higher values of 𝛼give

better performance.

∙In repetition-coding-based protocols, when the inter-user channels are unreliable, re-

peating one’s own codeword during decoding failure offers slight advantage over re-

maining silent, but this advantage vanishes even at moderate inter-user channels (ex-

ample 10 dB).

∙But in network-coded cooperation protocols, there is a clear advantage of repeating

one’s own codeword; and this advantage is significant at poor inter-user link qualities.

∙We have demonstrated the effect of the energy allocation on the performance of adap-

tive protocols. Based on the results, allocating more energy in the first phase gives

better performance.

4.4. Diversity-Multiplexing Tradeoff

Generally, MIMO and cooperative transmission schemes provide both diversity gain and

multiplexing gain. But there is a fundamental tradeoff between how much of each type of

gain a cooperative scheme can get, and diversity-multiplexing tradeoff is one performance

4. Outage Behavior of Network-Coded Cooperation

−10 −5 0 5 10 15 20 25 30

10−6

10−5

10−4

10−3

10−2

10−1

100

Mean source−destination SNR, Γs,d (dB)

Outage Probability

Direct transmission

Remaining silent, Γs,r= 0 dB

Remaining silent, Γs,r= 10 dB

Remaining silent, Γs,r= 30 dB

Repeating own message, Γs,r= 0 dB

Repeating own message, Γs,r= 10 dB

Repeating own message, Γs,r= 30 dB

Figure 4.15.: Outage probability versus Γ𝑠,𝑑 results of network-coding-based adaptive proto-

cols with the option of either remaining silent (solid lines) or repeating one’s

own codeword (dashed lines) during decoding failure.

measure of this tradeoff. This tradeoff illustrates the relationship between reliability of data

transmission in terms of diversity gain, and spectral efficiency in terms of multiplexing gain

[5, 67].

In point-to-point transmission with the source’s information rate 𝑅(bits/second/Hz), mul-

tiplexing gain helps to examine the high spectral-efficiency regime as SNR becomes large.

This is done by allowing 𝑅to grow with increasing SNR [5]. For slower growth, the outage

results essentially behave like fixed 𝑅with sufficiently large SNR, while for faster growth,

the outage probability tend to 1. In other words, the multiplexing gain 𝑚is defined as [67]

𝑚:= lim

Γ𝑠,𝑑→∞

𝑅(Γ𝑠,𝑑)

log2(Γ𝑠,𝑑).(4.45)

The diversity gain 𝑑is defined as

𝑑:= −lim

Γ𝑠,𝑑→∞

log2𝑃𝑋𝑠(Γ𝑠,𝑑, 𝑅)

log2(Γ𝑠,𝑑)(4.46)

where 𝑃𝑋𝑠is the outage probability of the source. Larger 𝑑implies more robustness to fading

(faster decay in the outage probability with increasing SNR). The relationship between di-

versity gain and multiplexing gain can be characterized by mapping 𝑑as a function of 𝑚. At

large Γ𝑠,𝑑 values, if we approximate the outage probability of the point-to-point transmission

4.5. Coverage Area Analysis

by 𝑃𝑋𝑠≈2𝑅

Γ𝑠,𝑑 , then we get

𝑑:= −lim

Γ𝑠,𝑑→∞

log2𝑃𝑋𝑠(Γ𝑠,𝑑, 𝑅)

log2(Γ𝑠,𝑑)≈1−𝑚 . (4.47)

In network-coded cooperation with selection combing at the destination, substituting (4.32)

into (4.46) and using (4.45), the diversity-multiplexing tradeoff of this scheme is given as

𝑑: = −lim

Γ𝑠,𝑑→∞

log2𝑃𝑋𝑠(Γ𝑠,𝑑, 𝑅)

log2(Γ𝑠,𝑑)=−lim

Γ𝑠,𝑑→∞ 2[log2(𝑔(𝑅)) −log2(𝛽)−log2(Γ𝑠,𝑑)

log2(Γ𝑠,𝑑)]

≈2(1 −2𝑚).(4.48)

Figure 4.16 shows the tradeoff curves based on Equations (4.47) and (4.48). We see that

the maximum multiplexing gain of the network-coded cooperation is 1

2, and this loss in rate

is expected as total available time per codeword in the point-to-point transmission is split

into two for the two phases of cooperation. As a result, to send the same information bits

in each phase of cooperation and point-to-point transmission, the codewords in the former

should have double the information rate of the latter. To conclude, we see that comparing the

network-coded cooperation and point-to-point transmission, the latter is good for multiplex-

ing gain and bad in terms of reliability of transmission, and vice versa.

4.5. Coverage Area Analysis

So far, we have seen the diversity gain of using network coding in cooperative transmission.

In large networks, e.g., ad hoc networks, cooperative transmission can be implemented to

expand coverage area (i.e., increase the range of signal radiation); however, this increase in

coverage also increases the interference range. To implement cooperative transmission in

a network of multiple nodes, partner node selection mechanisms are required and a study

on coverage area gives some clues. Moreover, relative location of nodes and coverage area

extension are other performance measures that can be used to compare various protocols

[17, 68, 69, 70].

As pointed out in the previous sections, the level of cooperation between the source and

partner node depends on the quality of inter-user channels, which in turn depend, among

other factors, on relative location of the two nodes. Network-coded cooperation performs

better than the repetition coding counterpart when the quality of the inter-user channels is

poor (i.e., when, for example, the partner is closer to the destination than the source). In [17],

it was found that the latter scheme works better when the partner is closer to the source than

the destination. In the following, we will show that the former scheme preforms better when

4. Outage Behavior of Network-Coded Cooperation

0 0.2 0.4 0.6 0.8 1

0.4

0.8

1.2

1.6

Multiplexing gain, m

Diversity gain, d

Point−to−point transmission

Network−coded cooperation

Figure 4.16.: Diversity-multiplexing tradeoff plots of the network-coding-based cooperative

protocol (solid line) and point-to-point transmission scheme (dashed line).

the partner is located closer to the destination than the source. Specifically, the coverage area

and location of the partner node, where outage is minimized, is studied in this section; the

exact outage probability results are used (see [13, 46] for further details). In the course of

discussion, the following questions will be addressed:

∙For the given network topology (i.e., location of the source, partner, and destination

nodes), which cooperative protocol to use?

∙In which geographic region does the network-coded cooperation perform better than

the repetition-coding-based protocol, and vice versa?

∙Within the network-coding-based (respectively the repetition coding ) protocol, how

do the static and adaptive protocols perform?

To address these points, the following approach is followed: the channel coefficient is split

into path-loss and fading coefficients. Then in all channels, the fading coefficient is assumed

to have unity mean power, but the pathloss varies as it depends on node locations. The

source-destination separation and the transmit power at both the source and partner are fixed.

The partner’s location is varied, such that the inter-user and partner-destination link quality

vary because of the pathloss. To aid the discussion, the following three terms are defined as

follows.

Definition Coverage area is defined as the region or area in which the partner can be placed

4.5. Coverage Area Analysis

0.001

0.003

0.005

0.001

0.003

0.005

Partner location in X coordinate

Partner location in Y coordinate

Source

Destination

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

−0.8

−0.6

−0.4

−0.2

0.2

0.4

0.6

0.8

Static and repetition coding

Static and network−coded cooperation

Figure 4.17.: Outage probability contours of static protocols: repetition coding (dashed lines)

and with network coding (solid lines). The source and the destination are

placed at the coordinates (0,0) and (0,1), respectively.

such that for a given resource allocation (i.e transmission power and bandwidth) and end-to-

end spectral efficiency, the outage probability (or ratio of outage probabilities) is less than or

equal to some threshold value.

Definition Intra-cooperation gain is the ratio of outage probabilities of two protocols; a

gain of unity demarcates the region into two parts, where one protocol performs better than

the other.

Definition Transmitter cluster is formed when the partner is deployed closer to the source

than the destination. Receiver cluster is formed when the partner is deployed closer to the

destination than the source.

To plot the coverage area, the transmit SNR at both the source and partner is set to 20 dB

and the source-destination distance is taken as a reference, i.e., 𝑑𝑠,𝑑 = 1 such that the other

channels’ normalized pathloss become 𝑞2

𝑖,𝑗 =(1

𝑑𝑖,𝑗 )𝛼

, where 𝑖,𝑗∈ {𝑠, 𝑝, 𝑑}and 𝛼is the

pathloss coefficient.

Figure 4.17 depicts the outage probability contours of the repetition coding (dashed lines)

and network-coding-based (solid lines) static protocols. For a given outage probability value,

4. Outage Behavior of Network-Coded Cooperation

0.05

0.5

1.2

0.05

0.5

1.2

Partner location in X coordinate

Partner location in Y coordinate

Source Destination

−3 −2 −1 0 1 2 3 4 5

−5

−4

−3

−2

−1

Network−coded / Repetition coding, both in static

Network−coded / Repetition coding, both in adaptive

Figure 4.18.: Intra-cooperation gain contours, where the gain is computed by dividing

the outage probability results of the network-coding-based static protocol by

the repetition coding static protocol (dashed lines) and network-coding-based

adaptive protocol by the repetition coding adaptive protocol (solid lines).

e.g. 0.001, the area span by the network coding protocol is larger than the repetition-coding-

based protocol. As long as the partner is confined to these areas, we are guaranteed that the

outage probability does not exceed 0.001. The probability contours of the network-coding-

based and repetition-coding-based static protocol are approximately concentric to the coor-

dinates (0, 0.7) and (0, 0.45), respectively. The coordinates (0, 0.7) and (0, 0.45) can also be

seen as the outage contours where the outage probability approaches 0. Hence, the network-

coding-based static protocol is more appropriate when a node closer to the destination is

selected as the partner; at such locations the quality of the inter-user channels is poor and

the uplink channels are more asymmetrical. In the repetition coding protocol, a node closer

to the source (or in the center) should be selected as the partner. This gives more choice of

selecting partner.

Shown in Figure 4.18 is the intra-cooperation gain contours of the static protocols (solid

lines) and adaptive protocols (dashed lines). The gain is computed by dividing the outage

probability of the network-coding-based and repetition coding protocols when both work

either in static or adaptive manner. These contours help to answer the question, given the

location of the relay, is it better to use the repetition coding or network-coding-based proto-

col. When the gain is greater than 1 (outside the unity-gain contour), the repetition coding

protocol performs better; when it is less than 1 (inside the unity-gain contour), the protocol

4.5. Coverage Area Analysis

0.75

0.95

0.75

0.95

Partner location in X coordinate

Partner location in Y coordinate

Source Destination

−3 −2 −1 0 1 2 3

−3

−2

−1

3Adaptive / static, both repetition coding

Adaptive / static, both with network−coded coop.

Figure 4.19.: Intra-cooperative gain contours, where the gain is computed by dividing the

outage probability of the repetition coding adaptive protocol by the repetition

coding static protocol (solid lines) and the network-coding-based adaptive pro-

tocol by the network-coding-based static protocol (dashed lines).

with network coding performs better. From the figure, we note that the region in which the

gain is greater than 1 (or the repetition coding protocols perform better) is located closer to

the source, and the region in which the gain is less than 1 is located closer to the destination.

Moreover, the gain is less than one for sufficiently large geographic area.

Finally shown in Figure 4.19 is the intra-cooperative gain contours, where the comparison is

performed within the repetition-coding-based and network-coding-based protocols, i.e., by

dividing the outage probability of the repetition coding (respectively with network coding)

adaptive protocol by the the repetition coding (respectively with network coding) static pro-

tocol. These contours help to illustrate the advantage, within either the repetition coding or

network-coding-based protocol, when we switch from static to adaptive. We see that as we

go from the bigger to the smaller contours, the intra-cooperation gain approaches unity. This

means that over such large area, adaptive protocols are more suited than the static protocols.

We conclude that, in general, adaptive protocols outperform their static counterparts over

wider geographic area.

4. Outage Behavior of Network-Coded Cooperation

4.6. Summary

In this chapter, we have derived the outage probability of the network-coded cooperation

protocol and examined its outage behavior. All channels are assumed to be block-fading

Rayleigh distributed; moreover, orthogonal transmission and half-duplex constraints are con-

sidered. The system model was given in Section 4.1. The outage derivation was presented

in Section 4.2; approximating this outage result at high SNR values, we have shown that the

protocol achieves full diversity order 2. The outage behavior of various cooperative protocols

is compared in Section 4.3 and we showed that network-coded cooperation is suitable when

the inter-user channels are lower quality. When the inter-user channels are good, protocols

without network coding perform better. Based on the outage probability result, the diversity-

multiplexing tradeoff and the coverage area extension of the protocol were studied in Section

4.4 and Section 4.5. Based the coverage area results, network-coded cooperation static pro-

tocol is more appropriate when a node closer to the destination is selected as the partner; and

at such locations the quality of the inter-user channels is poor and the uplink channels are

more asymmetrical. In the repetition coding protocol, a node closer to the source (or in the

center) should be selected as the partner.

5. Energy Efficiency in Wireless

Sensor Networks with

Network-Coded Cooperation:

Diversity-Energy Saving tradeoff

In energy-constrained WSNs, designing energy efficient transmission protocols is a key re-

quirement. As discussed in Chapter 4, network-coded cooperation allows wireless nodes

to exploit spatial diversity, which reduces transmission energy or increases communication

reliability. However, relaying redundant messages consumes considerable energy at both

transmitting and receiving nodes. Hence, one has to strike a balance between the diversity

gain and additional energy expenditure when designing cooperative protocols in WSN. This

chapter investigates the energy consumption of network-coded cooperation in WSN. First,

for a given error rate requirement at the destination node, an energy consumption model is

formulated. The model describes the average energy per information bit and takes into ac-

count transmission, reception, and processing energy spent at all cooperating nodes. Second,

the impact of various parameters of the model, for example node separation distance, on the

energy consumption will be evaluated.

Section 5.1 presents a brief introduction to energy considerations in a WSN that uses network-

coded cooperation. A typical WSN transceiver circuit is presented in Section 5.2. The energy

consumption modeling of point-to-point transmission is described in Sections 5.3. For the

network-coded cooperation, general assumptions are presented in Section 5.4 and the aver-

age energy consumption model is derived in Section 5.5. In this section, we formulate the

energy consumption as an optimization problem and present numerical results. Finally, an

alternative formulation of energy consumption, called energy efficiency model is given in

Section 5.6 and results for various payload sizes and source-partner separation distance are

presented.

5. Energy Efficiency in Wireless Sensor Networks

−10 −5 0 5 10 15 20 25 30

10−4

10−3

10−2

10−1

100

Mean source−destination SNR, Γs,d (dB)

Outage Probability

Point−to−point transmission

Network−coded cooperation

Figure 5.1.: Outage probability vs. average source-destination SNR Γ𝑠,𝑑 of network-coded

cooperation.

5.1. Introduction

This section presents the motivation to study energy consumption in network-coded cooper-

ation as well as a brief literature survey on related studies.

5.1.1. Motivation

A WSN is generally composed of a collection of sensor nodes that are capable of data collec-

tion, signal processing, and wireless communication; this network is expected to be widely

applicable in various fields [71]. The sensor nodes in WSNs are small and low cost de-

vices that typically operate with small batteries for which replacement, if not impossible,

is very difficult and expensive [72]. Consequently, minimizing energy consumption is one

of the primary objectives in WSN design [73]. Energy-efficient communication techniques

typically focus on minimizing transmission energy only, which is reasonable in long-range

applications where this energy is dominant in total energy consumption. However, in short-

range applications such as sensor networks, energy consumed in transmitting, receiving, and

processing circuitry can constitute a significant portion of the total consumed energy. The

circuit energy consumption includes Analog-to-Digital Converter (ADC), Digital-to-Analog

Converter (DAC), frequency synthesizer, mixer, Low Noise Amplifier (LNA), power am-

plifier, and baseband Digital Signal Processor (DSP) (See Figure 5.3). Moreover, startup

5.1. Introduction

⊕

⊕⊕

⊕X

)

⊕

⊕⊕

⊕X

)

XXX

⊕

⊕⊕

⊕X

)(X

⊕

⊕⊕

⊕X

)

⊕

⊕⊕

⊕X

)

XXX

Figure 5.2.: Network-coded cooperation in WSN.

and idle mode (i.e. when a sensor is not transmitting or receiving) energy consumptions are

sometimes significant. Hence, a comprehensive study of energy efficiency in WSNs requires

a consideration of all these sources of energy consumption.

Network-coded cooperation can be used in WSNs to combat channel fading by diversity

gain, which can be tradedoff to provide savings in transmit energy or an increase in commu-

nication range. Figure 5.1 shows the diversity gain of network-coded cooperation in com-

parison to the point-to-point transmission. However, when nodes transmit cooperatively, we

need to consider the extra transmission and processing energy consumed in all cooperating

nodes as a result of the redundant transmissions at relaying nodes.

The extra energy consumption in network-coded cooperation is explained using the network

shown in Figure 5.2, where all transmissions from the source and partner in the two phases

are shown. Assume the following: Inter-user channels are reciprocal, transmitting nodes use

forward error correcting code, a relaying node remains silent if it fails to decode its partner’s

message, and all nodes are identical. Moreover, nodes once started up will be awake during

the two phases. Idle mode occurs when either a node remains silent while its partner is

transmitting, to avoid collision due to the orthogonality assumption, or when a node is not

able to decode its partner’s message and will remain silent in the second phase. Table 5.1

compares the network-coded cooperation and point-to-point transmission in terms of total

number of transmissions, receptions, decodings, startup and idle times (including the source,

partner, and destination).

We see that the total number of transmissions, processing, and receptions (accordingly the

energy spent for processing these transmissions) are different in the two systems. Within the

network-coded cooperation, the numbers differ based on the inter-user channel’s quality. In

fading wireless channels, quality of a channel is a random process that follows the behavior

of the channel. Hence, energy consumption in cooperative transmission is also a random

process that follows the quality of the inter-user channels. A stochastic approach is required

5. Energy Efficiency in Wireless Sensor Networks

Table 5.1.: Network-coded cooperation vs. point-to-point transmissions.

Number of . . . Cooperative transmission Point-to-point transmission

Good S-P link Bad S-P link

Transmissions 4 2 2

Receptions 6 4 2

Encodings 4 2 2

Decodings 5 4 2

Idle times 2 6 2

Start-ups 3 3 3

when modeling energy consumption in these networks. From the curves in Figure 5.1 and

the number of transmissions in Table 5.1, there is obviously a tradeoff between the diversity

gain and the loss due to additional transmitting, receiving, and processing energy. This needs

to be taken into consideration in a network design [74]. We will address such tradeoff and

characterize the gain of cooperation under such extra overhead in energy consumption.

In this chapter, we first formulate an energy consumption model for a given error rate (i.e,

Packet Error Rate or Bit Error Rate) at the destination. Based on the model, we compute the

optimal radiated power at the source and partner that fulfills the error rate requirement and

minimizes the total energy consumption. Second, we define an energy efficiency metric as

the product of reliability of reception at the destination and the ratio of the useful energy to

the total energy. The useful energy is the energy spent on information (or payload) bits and

the total energy is the energy spent on the payload bits, overhead bits, encoding, decoding,

idle mode, and start-up mode. Using this model, for a given radiated power at the trans-

mitting nodes, the impact of parameters (such as payload length) on the energy efficiency is

investigated. As will be shown, the energy consumption and efficiency models are, among

other parameters, functions of node locations in a network. Finding an optimal location of

the partner that maximizes these metrics will be one of the points to be investigated and gives

additional insight on partner node selection for cooperation.

5.1.2. Literature survey

Liu et al. analyzed the energy efficiency of MIMO transmissions in WSNs considering the

tradeoff between diversity and multiplexing gains [73]. The results show that, with proper

design, the energy efficiency of MIMO can be higher than that of the traditional Single-Input

Single-Output (SISO) transmissions and the optimal energy efficiency is usually achieved

when both the diversity gain and the multiplexing gain are jointly considered. An energy

consumption model for a MIMO system with Alamouti diversity codes is developed in [72]

5.2. Wireless Sensor Networks

and the best modulation and transmission strategy to minimize the total energy consump-

tion is analyzed. The results show that BPSK-modulated SISO is more energy-efficient than

MIMO systems when transmission distance is short. However, by allowing the constellation

size to be optimally chosen, the energy efficiency of MIMO systems can be drastically in-

creased. Sadek et al. developed an analytic framework to study the energy consumption for

a class of relaying schemes based on ARQ [74]. The results show that for small separation

between the source and destination, point-to-point transmission is more energy-efficient than

relaying; moreover, an equal power allocation performs as good as optimal power allocation

for some scenarios. The effects of the relay location and the number of employed relays on

energy efficiency are also investigated. Simic et al. developed a heuristic for optimal partner

choice and power allocation in a selection decode-and-forward relaying [75, 76] protocol.

The authors presented that the partner-destination and the inter-user channels have roughly

equal influence on optimal energy consumption.

5.2. Wireless Sensor Networks

In this section, we briefly present a block diagram of a WSN transceiver circuit and link layer

data packet structure.

5.2.1. WSN Transceiver circuits

The block diagram of a typical WSN transceiver circuit is shown in Figure 5.3. The trans-

mitter circuit includes DAC, channel encoder, filters, mixers, frequency synthesizer, local

oscillator, and power amplifier. Similarly, the receiver circuit is composed of filters, LNA,

local oscillator, mixers, ADC, and decoder. 𝑃𝑡𝑒,𝑃𝑎𝑚𝑝,𝑃𝑟𝑎𝑑,𝑠, and 𝑃𝑟𝑒𝑐 represent the power

spent in the transmitter circuit, power amplifier, radiated power, and the receiver circuit, re-

spectively. The radiated power 𝑃𝑟𝑎𝑑,𝑠 is usually varied to fulfill the Quality of Service (QoS)

requirement (e.g., BER, Packet Error Rate (PER)) at a receiving node.

Assume that the transceiver circuitry works on a multi-mode basis, i.e, when there is a signal

to transmit all circuits work in active mode, when there is no signal to transmit they work

in idle mode, when the major circuit components are turned off in sleep mode, and when

switching from sleep mode to active mode there is a start-up mode. This multi-mode oper-

ation provides a significant saving of energy when sleep mode is employed [77]. Note that

the transition from active mode to sleep mode is short enough to be negligible, however,

the start-up process may be slow due to the finite Phase-Lock Loop (PLL) settling time in a

frequency synthesizer.

5. Energy Efficiency in Wireless Sensor Networks

Pte Pamp Prec

Pout

Transmitter ckt:

Encoder, DAC,

filters, mixer, LO

Noise figure

K bits

Power amplifier Receiver ckt: filters, LNA,

LO, mixers, IFD, ADC,

decoder

out,s

rec,d

rad,s

Figure 5.3.: Block diagram of a typical WSN transceiver circuit.

5.2.2. Packet structure in point-to-point transmission

Consider the link layer data packet structure shown in Figure 5.4. It consists of 𝜑header

bits, 𝑙payload bits, and 𝜏trailer bits. The header field generally includes the current segment

number, higher layer packet identifier, and the source and destination identifiers.1However,

for typical WSN applications 𝜑is expected to be only few bytes. The payload contains

information and CRC bits, and the trailer is composed of parity bits and is used for error

control of both the header and payload bits.

Figure 5.4.: The link-layer packet format.

5.3. Energy Consumption of Point-to-Point

Transmission

In the following, we develop a relationship between the energy dissipated in the amplifier

and received SNR at the receiving node. The strength of the received SNR determines the

error rate (i.e, outage probability or PER) at the receiving node. The bit rate, symbol rate

(i.e., after channel coding), and information rate (also called code rate) are represented as 𝑅𝑏

1In network-coded cooperation, the header field may contain bits required to set up cooperation (e.g., partner

selection) and to implement network coding.

5.3. Energy Consumption of Point-to-Point Transmission

[bits/s], 𝑅𝑠[symbol/s], and 𝑅𝑐[bit/symbol], respectively. The three rates are related to each

other as 𝑅𝑏=𝑅𝑠𝑅𝑐.

5.3.1. Power amplifier calibration

Consider the transceiver circuit shown in Figure 5.3, where the source transmits to the des-

tination. For the power amplifier at the source, a typical power consumption model is given

as [72]

𝑃𝑎𝑚𝑝 =𝜁

𝜂𝑃𝑟𝑎𝑑,𝑠 (5.1)

where 𝑃𝑎𝑚𝑝 is the consumed power, 𝑃𝑟𝑎𝑑,𝑠 is the radiated power, 𝜁is the drain efficiency of

the RF power amplifier, and 𝜂is the peak-to-average ratio, which depends on the modulation

scheme used.2The radiated power determines the error rate at the receiving node.

The wireless channel between the source and destination is assumed to be slow fading and

Rayleigh distributed with AWGN. Following the link budget analysis approach of [3] and

the model developed in Chapter 3, the average received power 𝑃𝑟𝑒𝑐,𝑑 at the output of the

destination is related to the transmit power 𝑃𝑟𝑎𝑑,𝑠 according to the path-loss model

𝑃𝑟𝑒𝑐,𝑑 =𝐺𝑡𝐺𝑟(𝜆

4𝜋)2

𝐸{∣ℎ𝑠,𝑑∣2}(1

𝑁𝑓)(1

𝑑𝛼

𝑠,𝑑 )𝑃𝑟𝑎𝑑,𝑠

=𝜉(1

𝑑𝛼

𝑠,𝑑 )𝑃𝑟𝑎𝑑,𝑠 (5.2)

where 𝜆is the carrier wavelength, 𝑑𝑠,𝑑 is the distance between the source and destination,

2⩽𝛼 < 5is the channel path-loss exponent, 𝑁𝑓is the receiver noise figure that accounts

for attenuation in the receiver, 𝐺𝑡and 𝐺𝑟are the transmitter and receiver antennas’ gains, and

𝐸{∣ℎ𝑠,𝑑∣2}is the expected value of the fading coefficient. The term 𝜉in (5.2) represents

𝜉=𝐺𝑡𝐺𝑟(𝜆

4𝜋)2(1

𝑁𝑓)𝐸{∣ℎ𝑠,𝑑∣2}.(5.3)

The average received SNR at the destination, represented by Γ𝑠,𝑑, is written in terms of the

radiated power as

(𝑃𝑟𝑒𝑐,𝑑

𝑁)=𝜉(1

𝑘𝑇𝑜𝐵)(1

𝑑𝛼

𝑠,𝑑 )𝑃𝑟𝑎𝑑,𝑠 = Γ𝑠,𝑑 (5.4)

2An alternative power consumption model is 𝑃𝑎𝑚𝑝 =𝛼𝑎𝑚𝑝 +𝛽𝑎𝑚𝑝𝑃𝑟𝑎𝑑,𝑠, where 𝛼𝑎𝑚𝑝 and 𝛽𝑎𝑚𝑝 are con-

stants depending on process technology and amplifier architecture [77].

5. Energy Efficiency in Wireless Sensor Networks

where 𝑁=𝑘𝑇𝑜𝐵is the noise power in Watt, 𝑘= 1.38𝑋10−23Joule/Kelvin is Boltzman’s

constant, 𝑇𝑜the noise temperature in Kelvin, and 𝐵the bandwidth in Hz. The power spectral

density of the receiver noise is given as 𝑁𝑜=𝑘𝑇𝑜Watt/Hz. Substituting Equation (5.1) into

Equation (5.4) and rearranging terms, the dissipated power at the amplifier circuit is written

𝑃𝑎𝑚𝑝 =𝜁

𝜂

𝜉𝑘𝑇𝑜𝐵 𝑑𝛼

𝑠,𝑑 (𝑃𝑟𝑒𝑐,𝑑

𝑁)

=𝐿′

𝑡𝑜𝑡𝑎𝑙 Γ𝑠,𝑑 𝑑𝛼

𝑠,𝑑 (5.5)

where 𝐿′

𝑡𝑜𝑡𝑎𝑙 =𝜁

𝜂

𝜉𝑘𝑇𝑜𝐵is an attenuation factor. Equation (5.5) relates the dissipated power

at the power amplifier to the received SNR and distance between the source and destination.

In Equation (3.25), it was shown that the received SNR given in (5.4) is approximately related

to the outage probability, 𝑃𝑜𝑢𝑡,𝑝𝑝𝑡, as

Γ𝑠,𝑑 ≈𝑔(𝑅)

𝑃𝑜𝑢𝑡,𝑝𝑝𝑡

(5.6)

where 𝑔(𝑅) = 2𝑅𝑐−1and 𝑅𝑐is the code rate. This equation helps us to compute the required

SNR at the destination for a given outage probability which represents the required QoS. If

a closed form expression relating PER and Γ𝑠,𝑑 exists, then the PER Equation (5.6) can be

used instead of outage probability. Substituting (5.6) into (5.5), we get

𝑃𝑎𝑚𝑝 ≈𝑔(𝑅)

𝑃𝑜𝑢𝑡,𝑝𝑝𝑡

𝐿′

𝑡𝑜𝑡𝑎𝑙 𝑑𝛼

𝑠,𝑑 (5.7)

Equation (5.6) and (5.7) show that, for a given location of nodes and information rate 𝑅𝑐,

the dissipated power at the source can be varied to meet the desired outage probability. This

expression will be used when energy efficiency is computed in the next section.

5.3.2. Energy consumption formulation

Network-coded cooperation involves the transmission of both the source’s and partner’s

packet to the destination. To make a fair comparison, in point-to-point transmission we

consider that both the source and partner directly transmit to the destination, instead of a sin-

gle transmission from either the source or partner. The consumed energy per payload bit can

be formulated following the approach in [78]. In Table 5.1, the energy spent per information

5.3. Energy Consumption of Point-to-Point Transmission

bit in the transmitter circuitry of the source and partner, given as 𝐸𝑡, is modeled as

𝐸𝑡=1

2𝑙[(2𝑃𝑡𝑒 +𝑃𝑑

𝑎𝑚𝑝,𝑠 +𝑃𝑑

𝑎𝑚𝑝,𝑝)(𝜑+𝑙+𝜏

𝑅𝑠

) + 2𝑃𝑖𝑑𝑙𝑒𝑇𝑖𝑑𝑙𝑒 + 2𝑃𝑠𝑡𝑎𝑟𝑡𝑇𝑠𝑡𝑎𝑟𝑡 + 2𝐸𝑒𝑛𝑐]

𝑙[(𝑃𝑡𝑒 +1

2(𝑃𝑑

𝑎𝑚𝑝,𝑠 +𝑃𝑑

𝑎𝑚𝑝,𝑝))(𝜑+𝑙+𝜏

𝑅𝑠

) + 𝑃𝑖𝑑𝑙𝑒𝑇𝑖𝑑𝑙𝑒 +𝑃𝑠𝑡𝑎𝑟𝑡𝑇𝑠𝑡𝑎𝑟𝑡 +𝐸𝑒𝑛𝑐](5.8)

where 𝑃𝑑

𝑎𝑚𝑝,𝑠 and 𝑃𝑑

𝑎𝑚𝑝,𝑝 are the power dissipated in the amplifier circuitry of the source and

partner node. 𝑃𝑠𝑡𝑎𝑟𝑡 and 𝑇𝑠𝑡𝑎𝑟𝑡 are the startup power and time in the transmitter circuit; 𝐸𝑒𝑛𝑐 is

the energy to encode the packet. Moreover, 𝑃𝑖𝑑𝑙𝑒 and 𝑇𝑖𝑑𝑙𝑒 are the power and time spent in idle

mode. 𝐸𝑒𝑛𝑐 depends on the type of code and the implementation of the encoding algorithm

(i.e, hardware or software) and is zero if no encoding is used. In Equation (5.8), the term

inside the square brackets represents the total energy to send two packets of 𝜑+𝑙+𝜏bits.

This energy is divided by 2𝑙, the total number of information bits of the source and partner,

to get the energy spent per information bit. Similarly, the total receiving energy per payload

bits is given as

𝐸𝑟=1

2𝑙[2𝑃𝑟𝑒𝑐(𝜑+𝑙+𝜏

𝑅𝑠

) + 𝑃𝑠𝑡𝑎𝑟𝑡𝑇𝑠𝑡𝑎𝑟𝑡 + 2𝐸𝑑𝑒𝑐]

𝑙[𝑃𝑟𝑒𝑐(𝜑+𝑙+𝜏

𝑅𝑠

) + 1

2𝑃𝑠𝑡𝑎𝑟𝑡𝑇𝑠𝑡𝑎𝑟𝑡 +𝐸𝑑𝑒𝑐](5.9)

where 𝐸𝑑𝑒𝑐 is the energy to decode a packet. The energy to communicate (i.e, transmit and

receive) one bit of information, 𝐸𝑝𝑝𝑡, is then

𝐸𝑝𝑝𝑡 =𝐸𝑡+𝐸𝑟

=𝑘′

|{z}

useful energy

+𝑘′

(𝜑+𝜏)

𝑙+𝑘′

2+𝐸𝑒𝑛𝑐 +𝐸𝑑𝑒𝑐

𝑙

|{z }

additional energy

(5.10)

where 𝑘′

1and 𝑘′

2are constants for a given transceiver and symbol rate 𝑅𝑠. For the RFM-

TR1000 transceiver circuit, 𝑘1′and 𝑘2′were calculated to be 1.85𝜇J/bit and 24.86𝜇J, respec-

tively. The contribution of 𝑘2′is high at high data rate and short payload [78].

The “useful energy”, 𝑘′

1, refers to energy spent to communicate an information bit, while

“additional energy” refers to the energy spent because of circuit inefficiency (i.e, start-up

energy), link-layer protocol design (i.e, idle mode), and error control (i.e, encoding and de-

5. Energy Efficiency in Wireless Sensor Networks

coding). If we further assume that 𝑃𝑑

𝑎𝑚𝑝,𝑠 =𝑃𝑑

𝑎𝑚𝑝,𝑝 =𝑃𝑑

𝑎𝑚𝑝, then

𝑘′

1=1

𝑅𝑠[𝑃𝑡𝑒 +1

2(𝑃𝑑

𝑎𝑚𝑝,𝑠 +𝑃𝑑

𝑎𝑚𝑝,𝑝) + 𝑃𝑟𝑒𝑐]=1

𝑅𝑠[𝑃𝑡𝑒 +𝑃𝑑

𝑎𝑚𝑝 +𝑃𝑟𝑒𝑐]

𝑘2′=𝑃𝑠𝑡𝑎𝑟𝑡𝑇𝑠𝑡𝑎𝑟𝑡 +1

2𝑃𝑠𝑡𝑎𝑟𝑡𝑇𝑠𝑡𝑎𝑟𝑡 +𝑃𝑖𝑑𝑙𝑒𝑇𝑖𝑑𝑙𝑒 =3

2𝑃𝑠𝑡𝑎𝑟𝑡𝑇𝑠𝑡𝑎𝑟𝑡 +𝑃𝑖𝑑𝑙𝑒𝑇𝑖𝑑𝑙𝑒.(5.11)

If the outage probability is fixed, then 𝑃𝑑

𝑎𝑚𝑝 can be computed from (5.7) (provided separation

distance and 𝐿′

𝑡𝑜𝑡𝑎𝑙 are known) and substituted into (5.10) to get the total energy consumption.

Using this approach, the total consumed energy that fulfills the outage probability require-

ment (or QoS specified by the BER or PER at the destination ) can easily be computed.

5.3.3. Energy efficiency formulation

The energy consumption formulated above specifies the total energy per information bit

for a given QoS. Sometimes, instead of fixing the error rate and computing the energy

consumption, we may be interested to know the joint “optimal” energy consumption and

error rate of the system. The energy efficiency metric that takes percentage of energy spent

to communicate the payload and reliability of reception at the destination is defined as

𝜂=[𝑘′

1𝑙

𝑘′

1(𝜑+𝑙+𝜏) + 𝑘2′+𝐸𝑒𝑛𝑐 +𝐸𝑑𝑒𝑐 ](1 −𝑃𝑜𝑢𝑡,𝑝𝑝𝑡)(5.12)

where 𝑘′

1𝑙

𝑘′

1(𝜑+𝑙+𝜏)+𝑘2′+𝐸𝑒𝑛𝑐+𝐸𝑑𝑒𝑐 accounts for the energy throughput and (1 −𝑃𝑜𝑢𝑡,𝑝𝑝𝑡)is the

packet success rate which accounts for the transmission reliability. Energy efficiency ex-

presses the ratio of the energy to communicate the payload bits to the total energy spent and

conditions this energy ratio on the probability of successfully receiving a packet. Using this

formula, for various values of radiated power, the percentage of energy spent to communi-

cate the information bit is computed. Considering a black box where the total energy is the

input and the energy to communicate the payload as the output, the term in the square bracket

of Equation (5.12) accounts for the ratio of the output energy to the input energy, hence the

name energy efficiency.

5.4. General Assumptions in Network-Coded

Cooperation

In this section, we present general assumptions that will be used for energy consumption

formulation in network-coded cooperation. Let the radiated powers in cooperative transmis-

5.4. General Assumptions in Network-Coded Cooperation

sion be 𝑃𝑐,𝑖

𝑎𝑚𝑝,𝑗, with phase 𝑖∈ {1,2}and node 𝑗∈ {𝑠, 𝑝}. The total power spent in the two

phases is given as 𝑃𝑐

𝑎𝑚𝑝,𝑗 =𝑃𝑐,1

𝑎𝑚𝑝,𝑗 +𝑃𝑐,2

𝑎𝑚𝑝,𝑗. Similarly, in the point-to-point transmission

𝑃𝑑

𝑎𝑚𝑝,𝑗 represents the radiated power from node 𝑗. For the analysis in the following section,

let us make the following general assumptions.

∙System or user level: In system level, we compute the energy consumption/efficiency

considering the successful reception of both users’ packets. On the other hand, in user

level we will be concerned with one user only and compute the consumption/efficiency

of that user.

∙Same bit rate and code rate in both the point-to-point transmission and the two phases

of cooperation: The total time spent to send one packet in cooperation is twice that of

the point-to-point transmission (see Figure 5.5). This assumption benefits cooperation

as there is no loss in spectral efficiency (i.e, same coding rate like in point-to-point)

because of cooperation. In Chapter 4, the cooperative scheme was designed to operate

at twice the code rate so that the total time in the two phases was the same as the

point-to-point (so that same radiated energy in both cooperation and point-to-point).

We note in Figure 5.5 that same time T is required to send one packet in each phase

of cooperation or the point-to-point transmission. The radiated energy in point to

point transmission is given as 𝐸𝑑

𝑎𝑚𝑝,𝑠 =T𝑃𝑑

𝑎𝑚𝑝,𝑠. Similarly, the radiated energy in

cooperation is 𝐸𝑐

𝑎𝑚𝑝,𝑠 =𝐸𝑐,1

𝑎𝑚𝑝,𝑠 +𝐸𝑐,2

𝑎𝑚𝑝,𝑠 =T(𝑃𝑐,1

𝑎𝑚𝑝,𝑠 +𝑃𝑐,2

𝑎𝑚𝑝,𝑠) = T𝑃𝑐

𝑎𝑚𝑝,𝑠, where

𝐸𝑐,1

𝑎𝑚𝑝,𝑠 and 𝐸𝑐,2

𝑎𝑚𝑝,𝑠 are the radiated energy in phase one and two, respectively. We see

that the sum of radiated energy in the two phases is given by the sum of the radiated

power in the two phases times the packet duration.

∙Reciprocal inter-user channel: In Chapter 4, it was defined that in reciprocal inter-

user channels the received instantaneous SNR at the source and partner are the same.

When the channel attenuations are the same, the reciprocal channels case happens if

the radiated power of the two users, in each phase, are the same, i.e, 𝑃𝑐,𝑖

𝑎𝑚𝑝,𝑠 =𝑃𝑐,𝑖

𝑎𝑚𝑝,𝑝 =

time

Cooperation

Phase IIPhase I

Point-to-point

dsamp,

(n,

c,1 samp,

2c, samp,

Figure 5.5.: Timing diagram of cooperative and point-to-point transmissions. (𝑛, 𝑘)repre-

sent the number of symbols, 𝑛, and the informations bits, 𝑘, in one packet.

5. Energy Efficiency in Wireless Sensor Networks

samp

pamp

samp

pamp

amp

1,c

amp

2,c

amp

Point-to-point

Source Partner

Coop.

Phase I

Phase II

∑

Selected

parameters

amp

PPP

+= 2,1,

Figure 5.6.: Summary of parameters used in energy allocation.

𝑃𝑐,𝑖

𝑎𝑚𝑝. Because of this assumption, we will have only two cases for analysis, i.e, either

both nodes cooperate or both will not cooperate.

In general, we can allocate the same or different radiated energy in cooperation and point-to-

point transmission. Moreover, within the cooperation scheme, the same or different radiated

energy can be used in the two phases. The following two points explain the energy allocation

in cooperation vs. point-to-point and phase one vs. phase two.

∙Proportion of the total radiated energy in cooperation and point-to-point: We consider

that 𝐸𝑐

𝑎𝑚𝑝 =𝜙𝐸𝑑

𝑎𝑚𝑝. Substituting for 𝐸𝑐

𝑎𝑚𝑝 and 𝐸𝑑

𝑎𝑚𝑝, we get the relationship 𝑃𝑐

𝑎𝑚𝑝 =

𝑃𝑐,1

𝑎𝑚𝑝+𝑃𝑐,2

𝑎𝑚𝑝 =𝜙𝑃𝑑

𝑎𝑚𝑝, where 0⩽𝜙=𝐸𝑐

𝑎𝑚𝑝

𝐸𝑑

𝑎𝑚𝑝 =𝑃𝑐

𝑎𝑚𝑝

𝑃𝑑

𝑎𝑚𝑝 <∞is the proportion of the total

energy (or power) in cooperation and point-to-point transmissions. When 𝜙= 1, i.e,

the radiated energy in both cooperation and point-to-point transmissions is the same,

the PER at the destination will be different for the two schemes. Alternatively, one can

set 𝜙∕= 1, i.e., different energy allocation in the two schemes, and the transmit powers

in the two schemes are varied to deliver the same PER at the destination.

∙Energy allocation per phase: Assume 𝛽of the total available energy is used in phase

one and (1 −𝛽)in the second phase. This implies that

𝑃𝑐,𝑖

𝑎𝑚𝑝 ={𝛽𝜙𝑃𝑑

𝑎𝑚𝑝 phase one, i.e. i =1;

(1 −𝛽)𝜙𝑃𝑑

𝑎𝑚𝑝 phase two, i.e. i =2.(5.13)

A value of 𝛽= 0.5means equal allocation in the two phases, which is usually assumed

in the conventional cooperative approaches.

To conclude, the three parameters of interest are the radiated powers in the point-to-point

transmission, 𝑃𝑑

𝑎𝑚𝑝, phase one of cooperation, 𝑃𝑐,1

𝑎𝑚𝑝, and phase two of cooperation, 𝑃𝑐,2

𝑎𝑚𝑝.

5.5. Energy Consumption in Network-Coded Cooperation

Figure 5.6 summarizes these parameters. These parameters are related to each other as

𝑃𝑐,1

𝑎𝑚𝑝 =𝛽𝜙𝑃𝑑

𝑎𝑚𝑝

𝑃𝑐,2

𝑎𝑚𝑝 = (1 −𝛽)𝜙𝑃𝑑

𝑎𝑚𝑝.(5.14)

5.5. Energy Consumption in Network-Coded

Cooperation

In this section, the energy consumption of the network-coded cooperation protocol will be

formulated. Based on the above assumptions, the reciprocal inter-user channels suggest that

there are two cases to consider, i.e, when the inter-user channels are good or bad so that

transmissions are either correctly or incorrectly decoded.

5.5.1. Good inter-user channel

In the case of good inter-user channel, there is full cooperation between the source and

partner. Figure 5.7 shows states of the source, partner, and destination in the two phases.

Referring to Table 5.1, the total energy per information bit, taking transmissions of both the

source and partner into account, is formulated as

𝐸𝑐𝑜𝑜𝑝,1=1

2𝑙[(4𝑃𝑡𝑒 + 2𝜙𝑃𝑑

𝑎𝑚𝑝 + 6𝑃𝑟𝑒𝑐)(𝜑+𝑙+𝜏

𝑅𝑠)+ 3𝑃𝑠𝑡𝑇𝑠𝑡]+

2𝑙[2𝑃𝑖𝑑𝑇𝑖𝑑 + 4𝐸𝑒𝑛𝑐 + 5𝐸𝑑𝑒𝑐]

=𝑘1+𝑘1+ 2𝑘1(𝜑+𝜏

𝑙)+𝑃𝑟𝑒𝑐 −𝜙𝑃𝑑

𝑎𝑚𝑝

𝑅𝑠(𝜑+𝑙+𝜏

𝑙)+𝑘2

|{z }

Energy due to add. tx, rx, and processing

(5.15)

where 𝑘1=𝑃𝑡𝑒+𝜙𝑃 𝑑

𝑎𝑚𝑝+𝑃𝑟𝑒𝑐

𝑅𝑠and 𝑘2=𝑘′

2+2𝐸𝑒𝑛𝑐+5

2𝐸𝑑𝑒𝑐

𝑙.𝑃𝑡𝑒,𝑃𝑎𝑚𝑝,𝑃𝑟𝑒𝑐, and 𝑅𝑠are the power,

in Watts, spent at the transmitter circuit, power amplifier, receiving circuit, and symbol rate

in symbols/s, respectively. Also 𝑃𝑠𝑡 and 𝑇𝑠𝑡 are the power and time at the startup mode.

For 𝜙= 1, the term 𝑘′

1=𝑘1where 𝑘′

1is defined in Equation (5.11). The term 2𝜙𝑃𝑑

𝑎𝑚𝑝 in

Equation (5.15) is the sum of the radiated power from the source and partner over the two

phases, i.e., 2𝜙𝑃𝑑

𝑎𝑚𝑝 =𝑃𝑐,1

𝑎𝑚𝑝,𝑠 +𝑃𝑐,2

𝑎𝑚𝑝,𝑠 +𝑃𝑐,1

𝑎𝑚𝑝,𝑝 +𝑃𝑐,2

𝑎𝑚𝑝,𝑝.

From Equation (5.15), we note that extra power is spent on cooperation because of redundant

transmissions, overhead and error correction bits, encoding and decoding, circuit inefficiency

5. Energy Efficiency in Wireless Sensor Networks

Transmits

Receives

Receives Idle

Idle

Receives Receives Receives Receives

Figure 5.7.: Diagram showing the states of the source, partner, and destination when good-

quality inter-user channel exist.

(start up), and idle mode (to avoid collision). Equation (5.15) can also be written as

𝐸𝑐𝑜𝑜𝑝,1=𝐸𝑝𝑝𝑡 +(𝑘′

1+𝑃𝑟𝑒𝑐 −𝑃𝑑

𝑎𝑚𝑝(2 −𝜙)

𝑅𝑠)(𝜑+𝑙+𝜏

𝑙)+𝐸𝑒𝑛𝑐 +3

2𝐸𝑑𝑒𝑐

𝑙

|{z }

Loss due to cooperation

(5.16)

where 𝐸𝑝𝑝𝑡 is the energy needed for point-to-point transmission defined in Equation (5.10).

From this equation, we see the loss of energy by using cooperation instead of point-to-point

transmission. On the other hand, we exploit diversity by cooperation which can be used to

reduce PER and BER.

System-level outage probability

The energy consumption formulated in Equation (5.15) is valid when the inter-user channels

are of good quality. The probability that good quality inter-user channels occur (or the prob-

ability that the consumed energy is given by Equation (5.15)) is obtained from the outage

probability of the inter-user transmissions following an identical approach as in Chapter 4.

The system-level outage event occurs when the destination is unable to decode the transmis-

sions both from the source and partner. At the destination, let 𝑋𝑠,𝑋𝑝, and 𝑋𝑠⊕𝑋𝑝represent

the outage events of the source’s packet, partner’s packet, and network-coded packet, respec-

tively. The overall outage event is written as

𝑋𝑠∧𝑋𝑝∼

=[𝑋𝑠∧[𝑋𝑝∨𝑋𝑠⊕𝑋𝑝]]∨[𝑋𝑝∧𝑋𝑠⊕𝑋𝑝](5.17)

where bars indicate the outage of an event; ∧and ∨are logical ‘AND’ and ‘OR’ oper-

ations, respectively. The probability that the event in Equation (5.17) occurs, given by

𝑃(𝑋𝑠∧𝑋𝑝) = 𝑃𝑠𝑦𝑠,1, is then

𝑃𝑠𝑦𝑠,1=𝑃(𝑋𝑠)[𝑃(𝑋𝑝) + 𝑃(𝑋𝑠⊕𝑋𝑝)] + 𝑃(𝑋𝑝)𝑃(𝑋𝑠⊕𝑋𝑝)(1 −2𝑃(𝑋𝑠)).(5.18)

100

5.5. Energy Consumption in Network-Coded Cooperation

Transmits

Idle

Transmits

Idle

Receives

Receives Idle

Idle

IdleIdleReceives Receives

Figure 5.8.: Diagram showing the states of the source, partner, and destination when bad-

quality inter-user channel exist.

At high SNR, this outage probability is approximated as

𝑃𝑠𝑦𝑠,1≈[𝑘3

𝑃𝑑

𝑎𝑚𝑝 ]2

𝑑𝛼

𝑠,𝑑𝑑𝛼

𝑝,𝑑.(5.19)

The term 𝑘3=[𝑔(𝑅)

𝛽𝜙 𝐿′

𝑡𝑜𝑡𝑎𝑙]where 𝐿′

𝑡𝑜𝑡𝑎𝑙 and 𝑔(𝑅)are given in Equations (5.5) and (5.6).

User-level outage probability

For the case of good inter-user channels, the approximation in Equation (5.19) also applies

to user-level outage probability, denoted as 𝑃𝑢𝑠𝑒,1.

5.5.2. Bad inter-user channel

Following an identical approach as in the case of good-quality inter-user channel and assum-

ing that nodes remain silent during the second phase when decoding fails, only two packets

will be transmitted, i.e., one by the source and one by the partner (see Figure 5.8 and also

Table 5.1). This case of no cooperation is similar to the point-to-point transmission except

that there are two additional receptions and decodings as well as 4 idle times. The destina-

tion tries to recover the source’s and partner’s packets from the first phase transmissions and

remains idle in the second phase. The total energy in this case is then given as

𝐸𝑐𝑜𝑜𝑝,2=𝑘′

1+𝑘′

1(𝜑+𝜏

𝑙)+(𝑃𝑟𝑒𝑐 −𝑃𝑑

𝑎𝑚𝑝(1 −𝛽𝜙)

𝑅𝑠)(𝜑+𝑙+𝜏

𝑙)+𝑘′′

|{z }

Energy due to overhead, add. rx., and processing

(5.20)

101

5. Energy Efficiency in Wireless Sensor Networks

with new constant 𝑘′′

2=𝑘2

′+2𝑃𝑖𝑑𝑙𝑒𝑇𝑖𝑑𝑙𝑒+𝐸𝑒𝑛𝑐+2𝐸𝑑𝑒𝑐

𝑙. To see the energy loss by using cooperation

instead of point-to-point transmission, we can re-write Equation (5.20) as

𝐸𝑐𝑜𝑜𝑝,2=𝐸𝑝𝑝𝑡 +(𝑃𝑟𝑒𝑐 −𝑃𝑑

𝑎𝑚𝑝(1 −𝛽𝜙)

𝑅𝑠)(𝜑+𝑙+𝜏

𝑙)+2𝑃𝑖𝑑𝑙𝑒𝑇𝑖𝑑𝑙𝑒 +𝐸𝑑𝑒𝑐

𝑙

|{z }

Energy loss b/c of cooperation

(5.21)

System-level outage probability

In the case of bad inter-user channel, the system level outage probability, 𝑃𝑠𝑦𝑠,2, can be

calculated as follows. The outage event 𝑋𝑠∧𝑋𝑝in this case is written as

𝑋𝑠∧𝑋𝑝∼

=𝑋𝑠∧𝑋𝑝(5.22)

and the corresponding probability is then

𝑃𝑠𝑦𝑠,2≈[𝑘3

𝑃𝑑

𝑎𝑚𝑝 ]2

𝑑𝛼

𝑠,𝑑𝑑𝛼

𝑝,𝑑.(5.23)

where 𝑘3is defined in Equation (5.19). Note that this approximation is identical to that of

good inter-source channels case because, at high SNR values, the first-phase transmissions

dominates the outage behavior.

User-level outage probability

The user-level outage probability, as computed in the previous chapters, is given by

𝑃𝑢𝑠𝑒,2≈[𝑘3

𝑃𝑑

𝑎𝑚𝑝 ]𝑑𝛼

𝑠,𝑑.(5.24)

5.5.3. Average energy consumption

The average energy consumption is the weighted sum of the energy consumptions calculated

for the two cases above, where the weighing coefficients are the probabilities that the cases

occur (which in turn depend on inter-user channel condition). The average consumed energy,

102

5.5. Energy Consumption in Network-Coded Cooperation

𝐸𝑐,𝑡, is given as

𝐸𝑐,𝑡 =𝐸𝑐𝑜𝑜𝑝,1(1 −𝑃1) + 𝐸𝑐𝑜𝑜𝑝,2𝑃1

=𝐸𝑐𝑜𝑜𝑝,1+𝑃1(𝐸𝑐𝑜𝑜𝑝,2−𝐸𝑐𝑜𝑜𝑝,1)

=𝐸𝑐𝑜𝑜𝑝,1−𝑃1[(𝑘1−𝑃𝑑

𝑎𝑚𝑝(1 −𝜙(1 −𝛽))

𝑅𝑠)(𝜑+𝑙+𝜏

𝑙)+𝑘2−𝑘′′

𝑙]

(5.25)

where the outage probability 𝑃1=[𝑘3

𝑃𝑑

𝑎𝑚𝑝 ]𝑑𝛼

𝑠,𝑝 shows failure of the inter-user transmission,

𝐸𝑐𝑜𝑜𝑝,1and 𝐸𝑐𝑜𝑜𝑝,2are given by Equations (5.15) and (5.20), and 𝑘2−𝑘′′

2=𝐸𝑒𝑛𝑐 +1

2𝐸𝑑𝑒𝑐 −

2𝑃𝑖𝑑𝑙𝑒𝑇𝑖𝑑𝑙𝑒. Replacing for 𝐸𝑐𝑜𝑜𝑝,1and 𝐸𝑐𝑜𝑜𝑝,2, we get

𝐸𝑐,𝑡 =𝐸𝑝𝑝𝑡 +(𝜑+𝑙+𝜏

𝑙)(𝑘1(1 −𝑃1) + 𝑃𝑟𝑒𝑐 −𝑃𝑑

𝑎𝑚𝑝 [(2 −𝑃1) + 𝜙𝑃1(1 −𝛽)−𝜙]

𝑅𝑠)

|{z }

Because of cooperation

+𝐸𝑒𝑛𝑐(1 −𝑃1) + 1

2𝐸𝑑𝑒𝑐(3 −𝑃1) + 2𝑃𝑖𝑑𝑙𝑒𝑇𝑖𝑑𝑙𝑒𝑃1

𝑙

|{z }

Because of add. processing

(5.26)

To simplify Equation (5.26) further, we assume that the energy ratio 𝜙= 1 and 𝛽𝑃1<< 1.

Replacing the point-to-point transmission from Equation (5.10) into Equation (5.26), we

get

𝐸𝑐,𝑡 =(𝑃𝑑

𝑎𝑚𝑝(1 −𝑃1) + 𝑃𝑡𝑒(2 −𝑃1) + 𝑃𝑟𝑒𝑐(3 −𝑃1)

𝑅𝑠)(𝜑+𝑙+𝜏

𝑙)+𝑘4(5.27)

where the constant 𝑘4=

2𝑃𝑠𝑡𝑇𝑠𝑡+𝑃𝑖𝑑𝑙𝑒𝑇𝑖𝑑𝑙𝑒(1+2𝑃1)+𝐸𝑒𝑛𝑐(2−𝑃1)+ 1

2𝐸𝑑𝑒𝑐(5−𝑃1)

𝑙. Equation (5.27) ap-

proximates the average total energy consumed in network-coded cooperation and it is, among

other variables, a function of the amplifier power, 𝑃𝑑

𝑎𝑚𝑝. In the next section, 𝑃𝑑

𝑎𝑚𝑝 will be

expressed as a function of the average outage probability at the destination. The resulting

expression is substituted into Equation (5.26) (or Equation (5.27)) so that the total energy

consumption will become a function of the outage probability.

5.5.4. Average outage probability

The average outage probability (like the average energy consumption discussed above) is

the weighted sum of the outage probabilities calculated for the two cases, where the weigh-

103

5. Energy Efficiency in Wireless Sensor Networks

ing coefficients are the probabilities that the cases occur. The system level average outage

probability, denoted as 𝑃𝑜𝑢𝑡,𝑠𝑦𝑠, is given by

𝑃𝑜𝑢𝑡,𝑠𝑦𝑠 = (1 −𝑃1)𝑃𝑠𝑦𝑠,1+𝑃1𝑃𝑠𝑦𝑠,2

≈[𝑘3

𝑃𝑑

𝑎𝑚𝑝 ]2

𝑑𝛼

𝑠,𝑑𝑑𝛼

𝑝,𝑑 (5.28)

where 𝑘3is defined in Equation (5.19). We see that the case for full cooperation dominates

at high SNR values. If this outage probability is required to be less than a threshold value,

given by 𝑃∗

𝑜𝑢𝑡, then we get the condition

𝑃𝑑

𝑎𝑚𝑝 ⩾𝑘3√(𝑑𝑠,𝑑 𝑑𝑝,𝑑)𝛼

𝑃∗

𝑜𝑢𝑡

.(5.29)

Once the 𝑃∗

𝑜𝑢𝑡 is fixed, 𝑃𝑑

𝑎𝑚𝑝 is readily computed from Equation (5.29). Then using knowl-

edge of 𝑃𝑑

𝑎𝑚𝑝, the inter-user outage probability 𝑃1is calculated next and then finally all

substituted into Equation (5.26) to get the average consumed energy per information bit.

The same approach can be followed for the user level.

Similarly, the user-level average outage probability is approximated as

𝑃𝑜𝑢𝑡,𝑢𝑠𝑒 = (1 −𝑃1)𝑃𝑢𝑠𝑒,1+𝑃1𝑃𝑢𝑠𝑒,2

≈[𝑘3

𝑃𝑑

𝑎𝑚𝑝 ]2

𝑑𝛼

𝑠,𝑑𝑑𝛼

𝑝,𝑑 [(1−𝑘3

𝑃𝑑

𝑎𝑚𝑝

𝑑𝛼

𝑠,𝑝)+(𝑑𝑠,𝑝

𝑑𝑝,𝑑 )𝛼].(5.30)

If the desired outage probability, 𝑃∗

𝑜𝑢𝑡, is specified, then identical threshold expression as in

Equation (5.29) can be computed. Finally, we note that the total consumed energy formulated

in Equation (5.29) as well as the amplifier power 𝑃𝑑

𝑎𝑚𝑝 computed from either Equation (5.29)

or Equation (5.30) are functions of many variables. If we specifically consider their relation

with node separations, there is a non-linear relationship between 𝑑𝑠,𝑑 and 𝑑𝑝,𝑑 and the average

energy consumption and outage probability. In the following, we will pick 𝑑𝑠,𝑑 and 𝑑𝑝,𝑑 as

variables and study their impact on energy consumption.

5.5.5. Numerical results

In this section, numerical results of the energy consumption model are presented. The param-

eter values used for numerical results are: symbol rate 𝑅𝑠= 250 Kbit/s, code rate 𝑅𝑐= 0.8,

𝜂= 0.2,𝜉= 1,𝛼= 4,𝐺𝑡.𝐺𝑟=-10 dB, 𝛽= 0.65,𝑃𝑡𝑒 = 5.75 mW, 𝑃𝑟𝑒𝑐 = 11.13mW, 𝑃𝑠𝑡𝑎𝑟𝑡 =

60mW, 𝑇𝑠𝑡𝑎𝑟𝑡 = 466 𝜇s, and 𝑃𝑖𝑑𝑙𝑒 =100 mW. The source and destination are separated by 30

104

5.5. Energy Consumption in Network-Coded Cooperation

50 50

Partner location in X coordinate

Partner location in Y coordinate

Source

Destination

-20 -10 0 10 20 30 40 50 60 70 80

-60

-40

-20

Outage probability 0.0001

Outage probability 0.001

Figure 5.9.: Transmit power 𝑃𝑑

𝑎𝑚𝑝, in milliwatt, contours for various locations of the partner

node.

meters; however, the partner is located anywhere in the network and its position is specified

by its coordinate on the two axes. The output power from the amplifiers is variable.

For a propagation environment specified by 𝑘3and the network geometry specified by a

range of potential partner locations, Figure 5.9 depicts the contour plot of the optimal trans-

mit power 𝑃𝑑

𝑎𝑚𝑝, in milliwatt, for system-level outage probability of either 0.0001 or 0.001.

A given contour specifies that, to operate at a fixed outage probability, the required transmit

power is the same for a range of locations defined by the contour. We see that the optimal

transmit power generally increases the further away the partner is from the source and des-

tination nodes and the contours are concentric around the location of the destination. Thus,

higher energy savings are expected from network-coded cooperation when the partner is in

the vicinity of the destination.

For each location of the partner node whose optimal transmit power is shown Figure 5.9,

we next compute and plot the average energy consumption per information bit at each point.

Figure 5.10 is a three-dimensional plot of this average consumed energy, given by Equation

(5.27), when the system-level outage probability is fixed to 0.0001. Examining this energy

plot, we can draw the following conclusions.

Firstly, Figure 5.10 confirms that the closer a potential partner is to both the source and

destination, the smaller the consumed energy from network-coded cooperation. Moreover,

the energy consumption contours on the horizontal axis approximately form circles centered

about the destination. This implies that from energy consumption perspective, the source

105

5. Energy Efficiency in Wireless Sensor Networks

−100

−50

100

150

200

−200

−100

100

200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Partner location in X coordinate

Partner location in Y coordinate

Consumed energy [mJ]

Figure 5.10.: Average consumed energy, in mill Joule, for system-level outage probability of

0.0001.

should pick a partner located in the vicinity of the destination. Secondly, we observe that the

lower the consumed energy, the smaller the diameter of the corresponding contour in Figure

5.10. This indicates that the partner choice region is smaller for a lower energy budget.

Finally, in a special case of linear topology where the partner is located in a straight line from

the source to the destination, Figure 5.11 depicts the energy consumption for two values of

the outage probability. We note that the consumption is minimal when the partner is located

in the vicinity of the destination. And if the reliability of transmission is low, e.g., outage

probability of 0.001, the consumption is also lower.

5.5.6. Formulation as an optimization problem

The energy consumption in Equation (5.26) can be formulated as an optimization problem.

As an example, the optimal amplifier power 𝑃𝑎𝑚𝑝 that fulfills the outage probability require-

ment at the destination and also minimizes the total energy consumption, can be formulated

min

𝑃𝑑

𝑎𝑚𝑝,𝛽,𝜙 𝐸𝑐,𝑡(𝑃𝑑

𝑎𝑚𝑝, 𝛽, 𝜙)subject to 𝑃𝑜𝑢𝑡,𝑠𝑦𝑠 ⩽𝑃∗

𝑜𝑢𝑡.(5.31)

For presentation convenience, let the amplifier power 𝑥=𝑃𝑑

𝑎𝑚𝑝, the inter-user outage prob-

ability 𝑃1=𝑘3𝑑𝛼

𝑠,𝑝

𝑃𝑑

𝑎𝑚𝑝 =ℎ1

𝑥, the source-destination outage probability 𝑘3𝑑𝛼

𝑠,𝑑

𝑃𝑑

𝑎𝑚𝑝 =ℎ2

𝑥, and also

106

5.5. Energy Consumption in Network-Coded Cooperation

−50 0 50 100 150

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Soure−partner separation in meter

Average energy per bit [mJ/bit]

Outage probability 0.0001

Outage probability 0.001

Figure 5.11.: Average consumed energy in mill Joules for system-level outage probability of

0.0001.

the partner-destination transmission 𝑘3𝑑𝛼

𝑝,𝑑

𝑃𝑑

𝑎𝑚𝑝 =ℎ3

𝑥. Substituting for 𝑃1, the objective function

defined by Equation (5.27) can be re-arrange into

𝐸𝑐,𝑡(𝑥)≈1

𝑅𝑠[(2𝑃𝑡𝑒 + 3𝑃𝑟𝑒𝑐 −ℎ1) + 𝑥(2 −𝜙′)−ℎ1

𝑥(𝑃𝑡𝑒 +𝑃𝑟𝑒𝑐)](𝜑+𝑙+𝜏

𝑙)+𝑘3.

(5.32)

Similarly, the outage probability constraint of the user level can be written in the form

𝑃𝑜𝑢𝑡,𝑢𝑠𝑒 ≈ℎ2ℎ3

(𝑃𝑑

𝑎𝑚𝑝)2(1 −ℎ1

𝑃𝑑

𝑎𝑚𝑝

) + ℎ1ℎ2

(𝑃𝑑

𝑎𝑚𝑝)2=1

𝑥2(ℎ1ℎ2+ℎ2ℎ3)−ℎ1ℎ2ℎ3

𝑥3⩽𝑃∗

𝑜𝑢𝑡.(5.33)

This is a constrained optimization problem in variable 𝑥. Introducing a Lagrangian function

Λ(𝑥, 𝜆) = 𝐸𝑐,𝑡(𝑥) + 𝜆(𝑃𝑜𝑢𝑡,𝑢𝑠𝑒 −𝑃∗

𝑜𝑢𝑡), where 𝜆is a constant and setting the derivative

𝑑Λ(𝑥, 𝜆) = 0 yields two system of equations. Introducing a change of variables 1

𝑥=𝑦, the

first system of equation is given by

∂Λ(𝑥, 𝜆)

∂𝜆 = (ℎ1ℎ2+ℎ2ℎ3)𝑦2−(ℎ1ℎ2ℎ3)𝑦3−𝑃∗

𝑜𝑢𝑡 = 0.(5.34)

Likewise, the second system of equation is computed as

∂Λ(𝑥, 𝜆)

∂𝑥 =∂𝐸𝑐,𝑡

∂𝑥 +𝜆∂(𝑃𝑜𝑢𝑡,𝑢𝑠𝑒𝑟 −𝑃∗

𝑜𝑢𝑡)

∂𝑥 = 0 (5.35)

107

5. Energy Efficiency in Wireless Sensor Networks

where the derivatives of the total power consumption 𝐸𝑐,𝑡 and the outage probability 𝑃𝑜𝑢𝑡,𝑢𝑠𝑒

with respect to the transmit power 𝑥are given by

∂𝐸𝑐,𝑡

∂𝑥 =1

𝑅𝑠[(2 −𝜙′) + ℎ1

𝑥2(𝑃𝑡𝑒 +𝑃𝑟𝑒𝑐)](𝜑+𝑙+𝜏

𝑙),(5.36)

and

∂(𝑃𝑜𝑢𝑡,𝑢𝑠𝑒 −𝑃∗

𝑜𝑢𝑡)

∂𝑥 =−2(ℎ1ℎ2+ℎ2ℎ3)

𝑥3+3ℎ1ℎ2ℎ3

𝑥4.(5.37)

Substituting Equations (5.36) and (5.37) into Equation (5.35), we get the following equation

in polynomial form

2−𝜙′

𝑅𝑠(𝜑+𝑙+𝜏

𝑙)−ℎ1

𝑅𝑠

(𝑃𝑡𝑒 +𝑃𝑟𝑒𝑐)(𝜑+𝑙+𝜏

𝑙)𝑦2−2𝜆(ℎ1ℎ2+ℎ2ℎ3)𝑦3+

3𝜆(ℎ1ℎ2ℎ3)𝑦4= 0 (5.38)

which can be written in a compact form as

𝑔0−𝑔2𝑦2−𝜆𝑔3𝑦3+𝜆𝑔4𝑦4= 0 (5.39)

under the outage constraint from Equation (5.34)

(ℎ1ℎ2+ℎ2ℎ3)𝑦2−(ℎ1ℎ2ℎ3)𝑦3⩽𝑃∗

𝑜𝑢𝑡.(5.40)

This constraint equation is a polynomial of order three, so it may be solvable depending on

the constants ℎ1,ℎ2, and ℎ3. In general, this optimization problem is nonlinear with respect

to the decision variable 𝑦and numerical optimization techniques are required.

5.6. Energy Efficiency Formulation

The energy formulation in the above section focuses on the consumed energy for a given

error rate. If we want to study the effect of certain system parameters, for example size

of redundancy bits for error protection, on energy consumption, we need to define a metric

that takes into account both energy and robustness of the system. In this section, we will

formulate energy efficiency of network-coded cooperation and study the consumed energy

over a range of error rates instead of a single error rate. We focus on system-level energy

efficiency.

108

5.6. Energy Efficiency Formulation

5.6.1. Good inter-user channel

In the case of good quality inter-user channel, the energy efficiency, based on information

bits only, is computed from Equation (5.15) as

𝜂1=⎡

⎣𝑘1𝑙

2𝑘1(𝜑+𝑙+𝜏) + 𝑃𝑟𝑒𝑐−𝜙𝑃 𝑑

𝑎𝑚𝑝

𝑅𝑠(𝜑+𝑙+𝜏) + 𝑘2𝑙⎤

⎦(1 −𝑃𝑠𝑦𝑠,1)(5.41)

where 𝑃𝑠𝑦𝑠,1is the system-level outage probability computed from Equation (5.17). The term

within the square brackets is the ratio of the energy spent for the payload bits only versus

the total energy, whereas the term (1 −𝑃𝑠𝑦𝑠,1)on the right accounts for the reliability of

reception. If we stick to the size of payload example above, the more redundancy bits we

send for error protection, the smaller the energy ratio becomes, but the larger the reliability

of the system gets. Therefore, we need to know the optimal payload size that maximizes the

energy efficiency. The same can be said about using cooperation instead of point-to-point

transmission. The additional transmission, processing, and reception decrease the energy

ratio but increases the reliability. To illustrate this point, the energy efficiency can also be

defined according to Equation (5.16) as

𝜂1=⎡

⎣𝐸𝑝𝑝𝑡

𝐸𝑝𝑝𝑡 +(𝜑+𝑙+𝜏

𝑙)(𝑘1+𝑃𝑟𝑒𝑐−𝑃𝑑

𝑎𝑚𝑝(2−𝜙)

𝑅𝑠)+(𝐸𝑒𝑛𝑐+3

2𝐸𝑑𝑒𝑐

𝑙)⎤

⎦(1 −𝑃𝑠𝑦𝑠,1).(5.42)

5.6.2. Bad inter-user channel

In the case of bad inter-user channels, the energy efficiency is computed from Equation (5.20)

and we get

𝜂2=⎡

⎣𝑘1′

(𝑘1′+𝑃𝑟𝑒𝑐−𝑃𝑑

𝑎𝑚𝑝(1−𝛽𝜙)

𝑅𝑠)(𝛼+𝑙+𝜏

𝑙)+𝑘′′

2⎤

⎦(1 −𝑃𝑠𝑦𝑠,2)(5.43)

5.6.3. Average energy efficiency

Following the same approach as in Section 5.5.3 above, the average energy efficiency is the

weighted sum of the energy consumptions calculated for the two cases above, where the

109

5. Energy Efficiency in Wireless Sensor Networks

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Code rate, Rc

Energy effciency, in percent

Network−coded cooperation

Point−to−point transmisison

Figure 5.12.: Energy efficiency versus code rate.

weighing coefficient is the probability with which each case occurs. Defining 𝜂𝑎𝑣𝑔 as the

average efficiency, we can write it in the form

𝜂𝑎𝑣𝑔 = (1 −𝑃1)𝜂1+𝑃1𝜂2

=𝜂1+𝑃1(𝜂2−𝜂1)(5.44)

where the probability 𝑃1shows the outage of the inter-user channel transmission. A closed-

form expression for 𝜂𝑎𝑣𝑔 can be found by substituting Equations (5.42) and (5.43) into Equa-

tion (5.44). Note that the average energy efficiency is a function of transmission power, the

energy consumption at the transmitter, receiver, power amplifier, encoding/decoding circuits,

node location/separation, and the packet size including overhead bits.

5.6.4. Numerical Results

In this section, we present numerical energy efficiency results for network-coded coopera-

tion. Results are based on (𝑛, 𝑘, 𝑡)linear block codes with 𝑡error correcting capability. Pa-

rameter values used for computation are the same as in Section 5.5.5. The effect of payload

size on energy efficiency is investigated first. The code rate is varied from 0< 𝑅𝑐=𝑘

𝑛⩽1.

Figure 5.12 shows energy efficiency, in percent, vs. the code rate. We note that as the rate

increases, the energy efficiency increases because more energy is spent on the payload. On

the other hand, as the rate approaches 1, the efficiency falls down to 0 as the error rate of the

system increases. For the given choice of parameters, the maximum efficiency is around 9

percent for a code rate in the range of 0.7.

110

5.7. Summary

−50 0 50 100 150

Source−partner separation in meter

Energy effciency, η in [%]

Network−coded cooperation

Point−to−point transmisison

Figure 5.13.: Energy efficiency versus source-partner separation.

Next, for linear network geometry where the partner is located between the source and des-

tination that are separated by 100 meters, we investigate the effect of partner location on

energy consumption in Figure 5.13. We see that the energy efficiency becomes optimal

when the partner is located halfway between the source and partner and it decreases as the

partner approaches either the source or destination, unlike for energy consumption where the

partner should be placed in the vicinity of the destination.

5.7. Summary

In this chapter, we investigated the energy consumption of network-coded cooperation in

WSN. We defined the energy consumption model as a performance measure that computes

the average energy per information bit and for a given error rate at the destination. Based

on this model, the energy consumption was studied and the impact of network geometry on

the consumption was investigated. From the numerical results, the energy consumption is

optimal when the partner is located in the vicinity of the destination. Section 5.5 presents

the details of the assumptions, model formulation, and numerical results. Next, we studied

the energy efficiency taking both the energy consumption and reliability of reception into

account; this is presented in Section 5.6. From the numerical results, we have found out that

for an optimal energy efficiency, the partner should be located halfway in between the source

and destination. To sum it up, optimal energy consumption requires the partner to be located

near the destination; however, optimal energy efficiency requires the partner to be located

halfway between the source and destination.

111

5. Energy Efficiency in Wireless Sensor Networks

112

6. Incremental Redundancy

Network-Coded Cooperation

This chapter studies the outage behavior of incremental redundancy network-coded coop-

eration, which is an extension of the network-coded cooperation discussed in Chapter 4.

To differentiate the two network-coding-based cooperations, in the following, we call the

network-coded cooperation conventional network-coded cooperation. In the incremental

network-coded cooperation, the source, for example, after receiving the partner’s codeword

in the first phase, combines its own and its partner’s parity symbols using network coding

and transmits these bits to the destination in the second phase. These network-coded parity

symbols are code combined with codeword(s) received in the first phase to form a stronger

codeword (i.e., are used as incremental information at the destination and this is unlike the

conventional network-coded cooperation where these symbols are decoded on their own).1

The principle of incremental redundancy network-coding is that the network coding is em-

bedded into the channel coding such that the redundancy in the network code is used to

support the channel code for better error protection [29, 30].

Two decoding approaches are proposed in this chapter: joint network-channel decoding and

individual network-channel decoding. In the former approach, one ‘big’ codeword is formed

from all symbols received in the two phases and the information bits of both source and part-

ner are obtained from a single decoding of this big codeword; however, in the latter approach

the source’s information bits are recovered by decoding a codeword which is formed from

code combining of its codeword received in the first phase and the network-coded parity

symbols; the same approach is used to decode the partner’s information bits [13, 31].

We first study the outage behavior of the incremental network-coded cooperation assuming

quasi-static Rayleigh fading channels, orthogonal transmission, and half-duplex constraints.

The outage results show that this scheme also achieves full diversity order of two. Then,

using the outage result, we investigate ‘optimal’ rate and energy allocations that minimize

1This is similar to coded cooperation where the source transmits incrementally redundant symbols for the

partner, and vice versa.

113

6. Incremental Redundancy Network-Coded Cooperation

the outage probability. The results show that outage performance is more sensitive to the

energy allocation than to the rate allocation.

Section 6.1 describes the system model and the joint network-channel encoding/decoding

operations. The outage probability of the incremental network-coded cooperation is derived

in Section 6.2; here, the same four cases approach as in Chapter 4 is followed. In Section 6.3,

numerical results are presented, discussions based on the results are made, and conclusions

on the general outage behavior and rate and power allocation are drawn. Finally, summary

of the chapter is presented in Section 6.4.

6.1. System Model, Joint Network-Channel Coding

6.1.1. System Model

The system diagram of the incremental network-coded cooperation scheme is shown in Fig-

ure 6.1. Like other cooperative schemes, each source encodes 𝑘of its information bits2into

a codeword of 𝑁1=𝛼𝑁 symbols, where 0< 𝛼 ⩽1is the cooperation level and 𝑁is the

number of symbols per codeword if the point-to-point transmission were used.3The source

and partner, using two time slots, transmit their own codeword to the destination and simul-

taneously decode each other’s codeword. If the source, for example, decodes the partner’s 𝑘

bits, then it independently generates parity symbols for its own bits as well as for the part-

ner’s bits, each of length 𝑁2= (1 −𝛼)𝑁. Then it linearly combines (i.e network codes)

these two groups of parity symbols to form network-coded parity symbols which are sent to

the destination in the second phase. The network-coded parity symbols can also be generated

by network coding the information bits of the source and partner first, and then channel cod-

ing the resulting network-coded bits by using a different codebook, i.e., performing network

coding followed by channel coding. If decoding fails, then the source transmits additional

𝑁2symbols for its own data or remains silent. The cooperation level can be adjusted to

control the number of parity symbols, and the control can be based on various requirements,

e.g., inter-user channel condition.

One basic difference between the conventional and incremental network-coded cooperation

is the following: in the former, the network-coded codeword is independently channel de-

coded at the destination and then, depending on the successful recovery of one of the two

users’ information bits, network decoding follows, i.e., there are two levels of decoding:

2In this chapter, information bits and symbols refer to bits before and after channel coding, respectively.

3A cooperation levels of 𝛼= 0.5and 𝛼= 1 corresponds to the conventional cooperation and point-to-point

transmission, respectively.

114

6.1. System Model, Joint Network-Channel Coding

First phase

Second phase

⊕i

Figure 6.1.: System diagram of incremental network-coded cooperation. Solid, dashed,

and dotted lines show the transmission of the source (S), partner (P), and the

network-coded parity symbols, respectively.

channel decoding followed by network decoding. However, in the latter the network-coded

parity symbols are jointly decoded with codewords received in the first phase, i.e., there

is one level of decoding as the network decoding is embedded into the channel decoding.

Another difference is if we compare them from a layering perspective: the joint network-

channel decoding behavior forces the implementation of incremental network-coded coop-

eration down to the physical layer as the network decoding is part of the channel decoding.

On the other hand, in the conventional network-coded cooperation, the channel decoding is

done at the physical layer and the network decoding is done at a layer above the physical

(usually network layer) making its implementation a cross layer one.

Finally, the energy per symbol relationship in the incremental network-coded cooperation is

as follows. Let 𝐸𝑇=𝑁𝐸𝑠be the energy budget to transmit 𝑁symbols in the point-to-point

transmission, where 𝐸𝑠is the radiated energy per symbol in point-to-point transmission; also

let 𝛽be the fraction of 𝐸𝑇allocated in phase one, where 0< 𝛽 ⩽1. If we designate 𝛽𝑘𝐸𝑠

to represent the energy per symbol in phase 𝑘∈ {1,2}, and the energy allocation term, 𝛽𝑘,

takes on value

𝛽𝑘={𝛽

𝛼if k= 1;

1−𝛽

1−𝛼if k= 2.(6.1)

Note that the energy allocation term in Equation (6.1) is different from the allocation term of

the conventional network-coded cooperation given in Equation (4.1) which is not a function

of 𝛼. Moreover, identical allocation by both the source and partner is assumed in this work.

The SNR relationship in each phase of the incremental network-coded cooperation and the

point-to-point transmission follow the same relation as the energy per symbol relationships

given in Equation (6.1). The SNR in the two phases are function of the energy allocation

and cooperation level, where the latter controls the information rate in the two phases. One

way to optimize performance of this cooperative scheme, e.g., outage probability, is by joint

optimization of the energy and cooperation level (rate).

115

6. Incremental Redundancy Network-Coded Cooperation

6.1.2. Joint network-channel encoding

In this subsection, the implementation of joint network-channel encoding is explained. In

general, the focus of this chapter is to study the outage behavior of the incremental network-

coded cooperation, and the following discussion is only to mention the existence of codes

that perform the joint network-channel encoding and decoding. Designing efficient encoding

and decoding schemes is beyond the scope of this work. Moreover, for ease of exposition,

discussions are based on binary linear block codes.

In a linear block coding scheme at the transmitting node 𝑙and phase 𝑘, where 𝑙∈ {𝑠, 𝑝},

let 𝑖𝑙represent 𝑛information bits, G𝑙,𝑘 be the generator matrix of node 𝑙, and 𝑋𝑙,𝑘 be the

resulting codeword/symbols. Assume that the source and partner have obtained each other’s

information bits from first-phase transmissions. The joint network-channel encoding at the

source is given as

[𝑋𝑠,1, 𝑋𝑠,2] = [𝑖𝑠, 𝑖𝑝][𝐺𝑠,1𝐺𝑠,2

0𝐺𝑝,2]

= [𝑖𝑠𝐺𝑠,1,(𝑖𝑠𝐺𝑠,2⊕𝑖𝑝𝐺𝑝,2)] .(6.2)

In a case that identical encoding matrices are used at the source and partner, 𝐺𝑠,1=𝐺𝑝,1=

𝐺1and 𝐺𝑠,2=𝐺𝑝,2=𝐺2and Equation (6.2) takes the following form

[𝑋𝑠,1, 𝑋𝑠,2] = [𝑖𝑠, 𝑖𝑝][𝐺1𝐺2

0𝐺2]= [𝑖𝑠, 𝑖𝑝]𝐺𝑠

= [𝑖𝑠𝐺1,(𝑖𝑠⊕𝑖𝑝)𝐺2].(6.3)

where 𝐺𝑠is a matrix made of 𝐺1and 𝐺2. In Equation (6.3), [𝑖𝑠𝐺1]represents the systematic

and channel-coded parity symbols (achieved by designing the encoding matrix 𝐺1such that

[𝑖𝑠𝐺1]contains both systematic and parity symbols for the source), whereas [(𝑖𝑠⊕𝑖𝑝)𝐺2]

represents the network-coded parity symbols (as these parity symbols are made of the in-

formation bits of both the source and partner). The generator matrices 𝐺1and 𝐺2can be

designed4to have dimensions 𝑛×𝛼𝑁 and 𝑛×(1 −𝛼)𝑁, respectively. This means that the

cooperation level 𝛼can be controlled by changing the size of the generator matrices. If the

source fails to decode the partner’s codeword, 𝑖𝑝in Equation (6.3) will be a zero vector and

[𝑖𝑠𝐺1, 𝑖𝑠𝐺2]results from the encoding and the source transmits incremental information for

its own. Another option is for the source to remain silent during decoding failure such that

4Though it may not be realistic to design these matrices for an arbitrary value of the cooperation level, 𝛼.

116

6.1. System Model, Joint Network-Channel Coding

[𝑖𝑠𝐺1]will be transmitted. Similarly, the encoding operation at the partner is

[𝑋𝑝,1, 𝑋𝑝,2] = [𝑖𝑠, 𝑖𝑝][0𝐺2

𝐺1𝐺2]= [𝑖𝑠, 𝑖𝑝]𝐺𝑝

= [𝑖𝑝𝐺1,(𝑖𝑠⊕𝑖𝑝)𝐺2].(6.4)

Note that the codewords in Equations (6.3) and (6.4) contain information bits of both the

source and partner, and if the size of 𝐺1and 𝐺2are fixed, then the joint encoding results

in a higher rate codeword compared to [𝑖𝑠𝐺1, 𝑖𝑠𝐺2]and [𝑖𝑝𝐺1, 𝑖𝑝𝐺2]. The property that we

have parity bits from both the channel coding and network coding will be exploited in the

decoding procedure. For the design of the joint decoder, let us assume that there exists a

‘virtual’ joint encoder that generates the codewords in Equations (6.2) and (6.4), and then

design the corresponding parity check matrix. The joint encoding can be represented as:

[𝑖𝑠𝐺1,(𝑖𝑠⊕𝑖𝑝)′𝐺2, 𝑖𝑝𝐺1]= [𝑖𝑠, 𝑖𝑝][𝐺1𝐺20

0𝐺2𝐺1]

=𝐼𝐺𝑠𝑝 (6.5)

where 𝐼= [𝑖𝑠, 𝑖𝑝]and 𝐺𝑠𝑝 is the generalized generator matrix. A parity check matrix, 𝐻𝑠𝑝,

can be designed to fulfill 𝐺𝑠𝑝𝐻𝑇

𝑠𝑝 = 0 and using 𝐻𝑠𝑝, a single decoding will give both 𝑖𝑠

and 𝑖𝑝. The prime notation in Equation (6.5) shows the possible MRC on the network-

coded symbols and this will be apparent when the decoding operation in the next section is

discussed. If 𝐺1is a systematic generator matrix, i.e., 𝐺1= [𝐼𝑛, 𝑃1]where 𝐼𝑛is 𝑛×𝑛identity

matrix and 𝑃1is 𝑛×(𝛼𝑁 −𝑛)matrix. If we also let the parity matrix 𝐺2=𝑃1, substituting

𝐺1and 𝐺2into Equation (6.5) and re-arranging columns, the joint encoder matrix 𝐺𝑠𝑝 will

take the form

𝐺𝑠𝑝 =⎡

⎣𝐼𝑛0.

.𝑃10𝑃1

0𝐼𝑛

.0𝑃1𝑃1⎤

⎦= [𝐼2𝑛, 𝑃](6.6)

where 𝐼2𝑛is an identity matrix of dimension 2𝑛×2𝑛and 𝑃is the new parity matrix of

dimension 3(𝛼𝑁 −𝑛).

Example 6.1.2.1. To illustrate the above points, let us consider the (2,5) systematic code

with generator matrix 𝐺1given by

𝐺1=[10101

01110]= [𝐼2, 𝑃1](6.7)

where 𝐼2is a 2×2identity matrix. The encoding matrix at the source, 𝐺𝑠, is constructed

117

6. Incremental Redundancy Network-Coded Cooperation

from 𝐺1and 𝐺2as follows

𝐺𝑠=⎡

⎢

⎣

10101101

01110110

00000101

00000110

⎤

⎥

⎦

.(6.8)

We see that 5 and 3 symbols will be sent by the source in phases one and two, respectively,

such that 𝑁= 8 and the condition 𝛼𝑁 = 5 and (1 −𝛼)𝑁= 3 should be fulfilled. Solving

for 𝛼, we get that the cooperation level should be 5

8. The joint encoding matrix 𝐺𝑠𝑝 is then

formed as

𝐺𝑠𝑝 =⎡

⎢

⎣

1 0 0 0 101000101

0 1 0 0 110000110

0010000101101

0 0 0 1 000110110

⎤

⎥

⎦

= [𝐼4, 𝑃].(6.9)

The encoding matrix 𝐺𝑠𝑝 is in systematic form and it also models the encoding (distributed)

operation at the two nodes as a single encoding operation. Suppose the information bits of

the source and partner are 𝑖𝑠= [𝑢11, 𝑢12]and 𝑖𝑝= [𝑢21, 𝑢22], respectively, and if these bits

are jointly encoded using the generator matrix given in Equation (6.9), we get the codeword

[𝑖𝑠, 𝑖𝑝]𝐺as

⎡

⎢

⎣𝑢11, 𝑢12, 𝑢21, 𝑢22

|{z }

systematic bits

, 𝑢11 ⊕𝑢12, 𝑢12, 𝑢11, 𝑢21 ⊕𝑢22, 𝑢22, 𝑢21,

| {z }

channel-coded parity symbols

𝑢11 ⊕𝑢12 ⊕𝑢21 ⊕𝑢22, 𝑢12 ⊕𝑢22, 𝑢11 ⊕𝑢21

|{z }

network-coded parity symbols

⎤

⎥

⎦.(6.10)

The above codeword consists of systematic bits, channel-coded bits (i.e., coding bits from

the same sources), and network-coded bits (i.e coding bits from the two users). In the case

that the source and partner fail to decode each others’ codewords and if nodes remain silent

during decoding error, then part of the channel-coded parity symbols and the entire network-

coded parity symbols will vanish from the codeword. In actual implementation, there are

more efficient designs of the coding and parity check matrices e.g., using Low Density parity

check (LDPC) codes or modifying Turbo codes [29].

118

6.1. System Model, Joint Network-Channel Coding

6.1.3. Joint network-channel decoding

The destination collects codeword/symbols received from the source and partner and per-

forms joint network-channel decoding. First, using MRC the destination combines the

network-coded parity symbols, which are received over the two phases from the source and

partner. Second, a ‘big’ codeword [𝑖𝑠𝐺1,(𝑖𝑠⊕𝑖𝑝)′𝐺2, 𝑖𝑠𝐺1]is formed by code combining

of all the received symbols, where the prime notation shows the MRC on the network-coded

parity symbols [13, 31]. For the generator matrix 𝐺𝑠𝑝 = [𝐼2𝑛∣𝑃], the parity check matrix

can be designed as 𝐻𝑠𝑝 = [𝑃𝑇∣𝐼𝑁−𝑛]such that 𝐺𝑠𝑝𝐻𝑇

𝑠𝑝 = 0, where 𝑇stands for the trans-

pose of a matrix. Returning to the encoding matrix example given in Equation (6.9), the

corresponding parity check matrix is given as

𝐻𝑠𝑝 =

⎡

⎢

⎣

1 1 0 0 100000000

0 1 0 0 010000000

1 0 0 0 001000000

0 0 1 1 000100000

0 0 0 1 000010000

0 0 1 0 000001000

1 1 1 1 000000100

0 1 0 1 000000010

1 0 1 0 000000001

⎤

⎥

⎦

=[𝑃𝑇, 𝐼9].(6.11)

Error checking is done by multiplying the codeword in Equation (6.10) with the transpose of

the parity check matrix in Equation (6.11) and we see that the product yields a zero vector

if no error exists in each bit. A non zero vector means an error exists in one or more bit

positions.

Alternatively, the source and partner information bits can be recovered through individual

network-channel decoding. In individual decoding, the information bits of the source and

partner are decoded separately (i.e., two decodings are required) and two approaches can be

followed

1. In approach one, the source’s information bits are recovered by decoding a codeword

which is formed by code combining of the source’s codeword received in the first

phase and the network-coded parity symbols. This approach considers the partner’s

information bits, which are contained in the network-coded parity symbols, as if they

were transmission errors on these parity bits, and tries to decode the source’s bits by

exploiting the redundancy in the channel-coded and network-coded parity symbols. To

explain the above point, consider a codeword given by Equation (6.3) is transmitted

119

6. Incremental Redundancy Network-Coded Cooperation

and received at the destination as

[𝑖𝑠𝐺1⊕𝑒1,(𝑖𝑠⊕𝑖𝑝)𝐺2⊕𝑒2] = [𝑖𝑠𝐺1⊕𝑒1, 𝑖𝑠𝐺2⊕𝑒′

2](6.12)

where 𝑒1,𝑒2, and 𝑒′

2=𝑖𝑝𝐺2⊕𝑒2are errors because of the channel fading and noise.

This codeword can be assumed as if it were transmitted from the source where the

codeword at the time of transmission was 𝑖𝑠[𝐺1, 𝐺2] = 𝑖𝑠𝐺𝑠, where 𝐺𝑠is the encoding

matrix. The parity check matrix can be designed accordingly.

This decoding, from the outset, appears to perform inferior as the decoder is attempting

to decode information bits of one user without knowledge of the information of the

other user (or tries to decode the information from higher rate codeword), but as will

be seen from the numerical results, it can still achieve full diversity gain. Moreover,

the outage probability computation of this approach is relatively simpler than the joint

decoding and it also helps to investigate the behavior of incremental network-coded

cooperation. Hence, the outage analysis in the next section is based on this approach.

2. The second approach is to decode the source’s information bits by removing the part-

ner’s information bits from the network-coded parity symbols first, and then utilize the

redundant symbols from the first and second phase transmissions. This assumes that

the partner information bits are decoded a priori from its first-phase transmission. In

the case that knowledge of the partner’s information bits is not available, the source’s

information bits are decoded from the first-phase transmission only. The decoding re-

quirement that “the destination decodes the partner’s information bits” can be fulfilled

if, for example, the partner is located closer to the destination such that the destination

can decode the partner’s codeword.

6.2. Outage Behavior of Incremental Redundancy

Network-Coded Cooperation

In this section, we compute the outage probability of incremental network-coded cooperation

assuming individual decoding at the destination. The analysis of joint decoding can be done

in the same manner except that it is less tractable. The same four-cases analysis, based on

success of inter-user channels transmissions, will be followed.

120

6.2. Outage Behavior of Incremental Redundancy Network-Coded Cooperation

6.2.1. Inter-user transmission

Consider the transmission from the source to the partner in phase one. The source transmits

its codeword of 𝛼𝑁 symbols carrying 𝑛information bits, such that the codeword rate is

𝑛

𝛼𝑁 =𝑅

𝛼, where 𝑅=𝑛

𝑁is the code rate if the point-to-point transmission were used. The

Shannon capacity of this channel is given by 𝐶(𝑠,𝑝)

𝑠= log2(1+𝛽1𝛾𝑠,𝑝), where 𝛾𝑠,𝑝 =∣ℎ𝑠,𝑝∣2𝑃𝑡

𝑁

is the instantaneous SNR of the channel with the transmit SNR 𝑃𝑡

𝑁. Using the notation of

Chapter 4, the outage event of this source-partner transmission is

𝑋(𝑠,𝑝)

𝑠∼

={𝐶(𝑠,𝑝)

𝑠<𝑅

𝛼}(6.13)

and, for ∣ℎ𝑠,𝑝∣2exponentially distributed, the probability that this event occurs is known to

𝑃(𝑋(𝑠,𝑝)

𝑠)= 1 −exp (−2𝑅/𝛼 −1

𝛽1Γ𝑠,𝑝 )≈2𝑅/𝛼 −1

𝛽1Γ𝑠,𝑝

(6.14)

where Γ𝑠,𝑝 is the average of 𝛾𝑠,𝑝 and the last term in (6.14) is the result of a high-SNR-

regime approximation. Equation (6.14) is also valid for the transmission from the partner

to the source with Γ𝑠,𝑝 replaced by Γ𝑝,𝑠. In the case of symmetrical inter-user channels, i.e.,

when 𝛾𝑠,𝑝 ∕=𝛾𝑝,𝑠 but Γ𝑠,𝑝 = Γ𝑝,𝑠, the source-partner and partner-source transmissions will

have the same outage probability.

6.2.2. Uplink transmission

The success of decoding at the destination depends on the four transmissions in the two

phases, which in turn depend on the quality of the inter-user channels. To determine the

outage, we need to identify parts of the received codewords/symbols that will be used for

decoding. In the following, we perform outage analysis for the source only; by symmetry

the same approach holds true for the partner. Depending on inter-user transmission success,

here are the four cases to consider.

∙Case 1: Both the source and partner decode each others’ codewords correctly. The

destination will receive each user’s codeword from the first phase and the network-

coded parity symbols, (𝑖𝑠⊕𝑖𝑝)𝐺2, in the second phase (see Figure 6.2). For the

individual decoding, a codeword [𝑖𝑠𝐺1,(𝑖𝑠⊕𝑖𝑝)′𝐺2]of length 𝑁is formed, where

the prime denotes MRC. As this codeword contains 2𝑛information bits, its code rate

is 2𝑛

𝑁= 2𝑅. The two parts of this codewords can be viewed as if they were received

121

6. Incremental Redundancy Network-Coded Cooperation

(

)

⊕i

(

)

Figure 6.2.: Codewords sent by the source (shaded) and partner in the two phases of Case 1.

from parallel channels whose capacities add up [10]. The outage event for the source,

in terms of the Shannon capacity, is written as

𝑋𝑠∼

={𝛼log2(1 + 𝛽1𝛾𝑠,𝑑) + (1 −𝛼) log2(1 + 𝛽2(𝛾𝑠,𝑑 +𝛾𝑝,𝑑)) <2𝑅}

∼

={log2[(1 + 𝛽1𝛾𝑠,𝑑)𝛼(1 + 𝛽2(𝛾𝑠,𝑑 +𝛾𝑝,𝑑))(1−𝛼)]<2𝑅}.(6.15)

The sum of capacities 𝛼log2(1+𝛽1𝛾𝑠,𝑑)and (1−𝛼) log2(1+𝛽2(𝛾𝑠,𝑑+𝛾𝑝,𝑑)) in Equation

(6.15) reflects the code combining (i.e., incremental redundancy symbols) and the term

𝛽2(𝛾𝑠,𝑑+𝛾𝑝,𝑑)that adds SNR values accounts for the MRC on the network-coded parity

symbols [12, 10]. The first phase uses a fraction 𝛼of the total 𝑁allocated symbols,

while the second frame uses 1−𝛼, in other words the cooperation level 𝛼denotes the

fraction of time that the network is in the first phase; it is hence used in Equation (6.15)

to reflect that [12, 10]. The outage probability, corresponding to the event in Equation

(6.15), is then

𝑃(𝑋𝑠)=𝑃{(1 + 𝛽1𝛾𝑠,𝑑)𝛼(1 + 𝛽2(𝛾𝑠,𝑑 +𝛾𝑝,𝑑))(1−𝛼)<22𝑅}

=∫ ∫𝐴

Γ𝑠,𝑑Γ𝑝,𝑑

exp (−𝛾𝑠,𝑑

Γ𝑠,𝑑 −𝛾𝑝,𝑑

Γ𝑝,𝑑 )𝑑𝛾𝑠,𝑑𝑑𝛾𝑝,𝑑 (6.16)

where 𝑃(𝑋𝑠)is the outage probability of the source, which actually is a function of

𝛾𝑠,𝑑, 𝛾𝑝,𝑑, 𝛼, and 𝛽. The region 𝐴is defined by

𝐴≡{(1 + 𝛽1𝛾𝑠,𝑑)𝛼(1 + 𝛽2(𝛾𝑠,𝑑 +𝛾𝑝,𝑑))(1−𝛼)<22𝑅}.(6.17)

In Equation (6.16), the exponential term represents the joint probability density func-

tion (PDF) of the random variables 𝛾𝑠,𝑑 and 𝛾𝑝,𝑑 and this joint PDF is integrated over

𝐴, which defines all possible values of 𝛾𝑠,𝑑 and 𝛾𝑝,𝑑. Following similar approaches to

[10] and using a combination of Taylor’s series and the high-SNR approximation of

the exponential function, it can be shown that the outage probability in (6.16) can be

approximated as

𝑃(𝑋𝑠)≈(22𝑅/(1−𝛼)−1)2

Γ𝑠,𝑑Γ𝑝,𝑑

(1 −𝛼)2

(1 −𝛽).(6.18)

Derivation of this probability is shown in Appendix A.1. The overall outage proba-

bility under this case is the product of the probability given in (6.18) and the proba-

122

6.2. Outage Behavior of Incremental Redundancy Network-Coded Cooperation

(1-α)N

Figure 6.3.: Codewords sent by the source (shaded) and partner in two phases of Case 2.

bility of occurrence of case 1, which in turn is the product of the success probabili-

ties of the inter-user channels from Equation (6.14) which are given as 𝑃(𝑋(𝑠,𝑝)

𝑠)=

1−𝑃(𝑋(𝑠,𝑝)

𝑠)and 𝑃(𝑋(𝑝,𝑠)

𝑝)= 1 −𝑃(𝑋(𝑝,𝑠)

𝑝). The overall outage probability

at the destination, defined as 𝑃(𝑋(𝑠,1)), where ‘1’ indicates the case number, is then

given as

𝑃(𝑋(𝑠,1))≈[1−2𝑅/𝛼 −1

𝛽1Γ𝑠,𝑝 ][1−2𝑅/𝛼 −1

𝛽1Γ𝑝,𝑠 ]

|{z }

inter-user txs.

[(22𝑅/(1−𝛼)−1)2

Γ𝑠,𝑑Γ𝑝,𝑑

(1 −𝛼)2

(1 −𝛽)]

|{z }

up-link txs.

.(6.19)

For symmetrical inter-user channels, i.e., Γ𝑠,𝑝 = Γ𝑝,𝑠 so that 𝑃(𝑋(𝑠,𝑝)

𝑠)=𝑃(𝑋(𝑝,𝑠)

𝑝),

Equation (6.19) reduces to the form

𝑃(𝑋(𝑠,1))≈[1−2𝑅/𝛼 −1

𝛽1Γ𝑠,𝑝 ]2[(22𝑅/(1−𝛼)−1)2

Γ𝑠,𝑑Γ𝑝,𝑑

(1 −𝛼)2

(1 −𝛽)].(6.20)

If we re-parametrize Γ𝑖,𝑗 = Γ𝑇Γ𝑖,𝑗 as in Equation (4.33) and for 𝛽1Γ𝑠,𝑝 >> 1, the

outage in Equation (6.20) will become

𝑃(𝑋(𝑠,1))≈1

Γ2

𝑇[(22𝑅/(1−𝛼)−1)2

Γ𝑠,𝑑Γ𝑝,𝑑

(1 −𝛼)2

(1 −𝛽)].(6.21)

From this outage probability approximation, we observe the following points:

1. When the inter-user channels are good quality, the outage-probability decay is

proportional to the square of Γ𝑇, the transmit SNR; hence full diversity order of

two is achievable using the incremental network-coded cooperation.

2. Controlling both the cooperation level 𝛼and the energy allocation term 𝛽is one

way to optimize the performance of this cooperative scheme.

∙Case 2: Neither the source nor partner decodes each other’s codeword correctly. Each

user transmits additional symbols for its own (see Figure 6.3). At the destination, a

codeword [𝑖𝑠𝐺1, 𝑖𝑠𝐺2]is formed, and the rate of this codeword is 𝑅as it contains the

123

6. Incremental Redundancy Network-Coded Cooperation

information bits of the source only. The outage event in this case is then

𝑋𝑠∼

={𝛼log2(1 + 𝛽1𝛾𝑠,𝑑) + (1 −𝛼) log2(1 + 𝛽2𝛾𝑠,𝑑)< 𝑅}

∼

={log2[(1 + 𝛽1𝛾𝑠,𝑑)𝛼(1 + 𝛽2𝛾𝑠,𝑑)(1−𝛼)]< 𝑅}.(6.22)

The corresponding outage probability is then

𝑃(𝑋𝑠)=𝑃{(1 + 𝛽1𝛾𝑠,𝑑)𝛼(1 + 𝛽2𝛾𝑠,𝑑)(1−𝛼)<2𝑅}

=∫𝐵

Γ𝑠,𝑑

exp (−𝛾𝑠,𝑑

Γ𝑠,𝑑 )𝑑𝛾𝑠,𝑑 (6.23)

where 𝐵is the contour defined by

𝐵≡{(1 + 𝛽1𝛾𝑠,𝑑)𝛼(1 + 𝛽2𝛾𝑠,𝑑)(1−𝛼)<2𝑅}.(6.24)

The term (1 + 𝛽1𝛾𝑠,𝑑)𝛼(1 + 𝛽2𝛾𝑠,𝑑)(1−𝛼)that defines the contour can be approximated

using Taylor’s series in one variable. Taking the first order terms, it reduces to the

form 1 + 𝛾𝑠,𝑑 such that the contour becomes 𝐵≡{𝛾𝑠,𝑑 <2𝑅−1}. The probability in

Equation (6.24) can be easily computed. The high SNR approximation of the overall

outage probability in this case is then approximated as

𝑃(𝑋(𝑠,2))≈[2𝑅/𝛼 −1

𝛽1Γ𝑠,𝑝 ][2𝑅/𝛼 −1

𝛽1Γ𝑝,𝑠 ][2𝑅−1

Γ𝑠,𝑑 ]

=[2𝑅/𝛼 −1

𝛽1Γ𝑠,𝑝 ]2[2𝑅−1

Γ𝑠,𝑑 ].(6.25)

In Equation (6.25), the first two terms in the box bracket show the outage for the

inter-user transmissions, whereas the the third term is the probability of the uplink

transmission obtained from Equation (6.23). The equation in the second line in (6.25)

results when Γ𝑠,𝑝 = Γ𝑝,𝑠. From this outage probability approximation, we observe the

following points:

1. When the inter-user channels are unreliable, the outage-probability decay is pro-

portional to Γ𝑠,𝑑 (not to the square of Γ𝑠,𝑑 as in case 1 above); hence, even at high

SNR values, no diversity gain is obtained in such a case.

2. The outage probability is sensitive to variation of the inter-user channels’ quality.

3. The outage of the uplink transmission is the same as that of the point-to-point

transmission and is independent of 𝛼and 𝛽. If we allow nodes to remain silent

when they fail to decode their partner’s codeword, the outage probability in Equa-

tion (6.23) would have reduced to 2𝑅/𝛼−1

𝛽1Γ𝑠,𝑑 . If we further set 𝛼=𝛽, then this

124

6.2. Outage Behavior of Incremental Redundancy Network-Coded Cooperation

G1ipG1

(1-α)N

⊕ip)G2

Figure 6.4.: Codewords sent by the source (shaded) and partner in two phases of Case 3.

probability becomes 2𝑅/𝛼−1

Γ𝑠,𝑑 , which is greater than 2𝑅−1

Γ𝑠,𝑑 and still a function of

𝛼. So, unlike the conventional network-coded cooperation, sending extra par-

ity symbols in the case that nodes fail to decode partner’s transmission helps to

reduce outage probability (actually this is an inherent behavior of incremental

redundancy coding systems).

∙Case 3: The partner can correctly decode the source’s codeword, but not vice versa. In

this case, the partner helps the source but the source transmits for its own (see Figure

6.4), and the parity symbols received from the source and partner differ in content and

will be used as different incremental information. The source’s information bits are

decoded from a codeword [𝑖𝑠𝐺1,(𝑖𝑠⊕𝑖𝑝)𝐺2, 𝑖𝑠𝐺2]which is of length (2 −𝛼)𝑁and

code rate 2𝑅

(2−𝛼). The outage event, in terms of the Shannon capacity, is defined as

𝑋𝑠∼

={𝛼log2(1 + 𝛽1𝛾𝑠,𝑑) + (1 −𝛼) log2[(1 + 𝛽2𝛾𝑠,𝑑)(1 + 𝛽2𝛾𝑝,𝑑)] <2𝑅

(2 −𝛼)}

∼

={log2[(1 + 𝛽1𝛾𝑠,𝑑)𝛼[(1 + 𝛽2𝛾𝑠,𝑑)(1 + 𝛽2𝛾𝑝,𝑑)](1−𝛼)]<2𝑅

(2 −𝛼)}(6.26)

where the capacities 𝛼log2(1 + 𝛽1𝛾𝑠,𝑑),(1 −𝛼) log2(1 + 𝛽2𝛾𝑠,𝑑), and (1 −𝛼) log2(1 +

𝛽2𝛾𝑝,𝑑)account for receptions from the three channels. The outage probability is

𝑃(𝑋𝑠)=∫ ∫𝐶

Γ𝑠,𝑑Γ𝑝,𝑑

exp (−𝛾𝑠,𝑑

Γ𝑠,𝑑 −𝛾𝑝,𝑑

Γ𝑝,𝑑 )𝑑𝛾𝑠,𝑑𝑑𝛾𝑝,𝑑 (6.27)

where 𝐶is the region defined by

𝐶≡{(1 + 𝛽1𝛾𝑠,𝑑)𝛼[(1 + 𝛽2𝛾𝑠,𝑑)(1 + 𝛽2𝛾𝑝,𝑑)](1−𝛼)<22𝑅/(2−𝛼)}.(6.28)

Following identical steps and approximations as shown in the Appendix A.2, it can be

shown that

𝑃(𝑋𝑠)≈(1 −𝛼)2

(1 −𝛽)Γ𝑠,𝑑Γ𝑝,𝑑 [22𝑅/[(2−𝛼)(1−𝛼)] −1]2.(6.29)

125

6. Incremental Redundancy Network-Coded Cooperation

(1-α)N

⊕i

(1-α)N

Figure 6.5.: Codewords sent by the source (shaded) and partner in two phases of Case 4.

and the total probability in this case is gven as

𝑃(𝑋(𝑠,3))≈[1−2𝑅/𝛼 −1

𝛽1Γ𝑠,𝑝 ][2𝑅/𝛼 −1

𝛽1Γ𝑝,𝑠 ](1 −𝛼)2

(1 −𝛽)Γ𝑠,𝑑Γ𝑝,𝑑 [22𝑅/[(2−𝛼)(1−𝛼)] −1]2

.(6.30)

For the case that 𝛽1Γ𝑠,𝑝 >> 1and Γ𝑠,𝑑 = Γ𝑝,𝑑, Equation (6.30) becomes

𝑃(𝑋(𝑠,3))≈[2𝑅/𝛼 −1

𝛽1Γ𝑝,𝑠 ][(1 −𝛼)2

(1 −𝛽)][22𝑅/[(2−𝛼)(1−𝛼)] −1]2[1

Γ𝑠,𝑑 ]2

.(6.31)

From the outage probability result in Equation (6.31), we conclude the following

points:

1. Like Case 1, diversity order of two is achievable as the outage probability decay

is proportional to the square of Γ𝑠,𝑑.

2. For asymmetric cooperation (i.e., one node cooperates but not the other), the

outage probability depends on the two inter-user channels quality.

3. Like the above two cases, the rate and energy allocation terms play an important

role in controlling the outage probability.

∙Case 4: The source correctly decodes the partner’s codeword, but not vice versa. In

this case, the source helps the partner but the partner transmits additional symbols for

its own (See Figure 6.5) and a codeword [𝑖𝑠𝐺1,(𝑖𝑠⊕𝑖𝑝)𝐺2]of code rate 2𝑅is formed

at the destination, where the additional network-coded parity symbols are received

from the source. It can be shown that the total outage even under this case is given as

𝑋𝑠∼

={𝛼log2(1 + 𝛽1𝛾𝑠,𝑑) + (1 −𝛼) log2(1 + 𝛽2𝛾𝑠,𝑑)<2𝑅}

∼

={(1 + 𝛽1𝛾𝑠,𝑑)𝛼(1 + 𝛽2𝛾𝑠,𝑑)(1−𝛼)<22𝑅}(6.32)

and the corresponding probability is

𝑃(𝑋(𝑠,4))≈[1−2𝑅/𝛼 −1

𝛽1Γ𝑝,𝑠 ][2𝑅/𝛼 −1

𝛽1Γ𝑠,𝑝 ][22𝑅−1

Γ𝑠,𝑑 ].(6.33)

126

6.2. Outage Behavior of Incremental Redundancy Network-Coded Cooperation

0 5 10 15 20 25 30

10−6

10−5

10−4

10−3

10−2

10−1

Mean source−destination SNR, Γs,d (dB)

Outage Probability

Point−to−point (ant. & sim.)

Coop. Γs,p= 0 dB (ant.)

Coop. Γs,p= 0 dB (sim.)

Coop. Γs,p= 10 dB (ant.)

Coop. Γs,p= 10 dB (sim.)

Coop. Γs,p= 30 dB (ant.)

Coop. Γs,p= 30 dB (sim.)

Figure 6.6.: Outage probability vs. uplink SNR Γ𝑠,𝑑 results when the inter-user SNR Γ𝑠,𝑝 =

0, 10, and 30 dB. Solid lines are for uncorrelated (analysis) and dashed lines

for autocorrelated channels (simulation). For the point-to-point, both the ana-

lytic and simulated results match; coop, ant. and sim. stands for cooperation,

analytical, and simulation, respectively.

We note from Equation (6.33) that, unlike the case of conventional network-coded co-

operation, there is no diversity gain in this case as both receptions are from the source

over a time-correlated channel (because of the block fading channel assumption).

The total outage probability, 𝑃(𝑋(𝑠,𝑇)), is the sum of the overall probabilities given in the

four cases above. Following the parametrization stated above, i.e., Γ𝑖,𝑗 = Γ𝑖,𝑗 ⋅Γ𝑇, the total

outage probability is shown to be

𝑃(𝑋(𝑠,𝑇))≈1

Γ2

𝑇[[2𝑅/𝛼 −1

𝛽1Γ𝑠,𝑝 ][22𝑅−1

Γ𝑠,𝑑 ]+(22𝑅/(1−𝛼)−1)2(1 −𝛼)2

Γ𝑠,𝑑Γ𝑝,𝑑(1 −𝛽)]+𝑂(1

Γ3)(6.34)

where 𝑂(1

Γ3)denotes the higher-order terms. We can infer from (6.34) that the outage prob-

ability decays proportional to the inverse square of Γ𝑇; hence at higher SNR, full diversity

order of two can be obtained. Also note that, in the high-SNR regime, the dependence of

outage probability on the inter-user channels Γ𝑠,𝑝 and Γ𝑝,𝑠 also appears in the second-order

terms, and this is in contrast to the conventional network-coded cooperation.

127

6. Incremental Redundancy Network-Coded Cooperation

0.2

0.4

0.6

0.8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Rate allocation, α

Energy allocation, β

Outage probability

Figure 6.7.: Three dimensional plot of the outage probability taking 𝛼and 𝛽as the other

variables. Results are based on 𝑅= 0.25 (bits/s/Hz) and all channels having

identical average SNR, i.e Γ𝑠,𝑑 = Γ𝑠,𝑝 = Γ𝑝,𝑑 =10 dB.

6.3. Result and Discussion

In this section, numerical and simulation results based on 𝑅= 0.25 (bits/s/Hz), symmetrical

inter-user, and symmetrical uplink (i.e., Γ𝑠,𝑑 = Γ𝑝,𝑑) channels are presented. The point-to-

point transmission is plotted to serve as a baseline for comparison.

Figure 6.6 depicts the outage probability vs. Γ𝑠,𝑑 (dB) for 𝛼=𝛽= 0.5and three inter-user

channel values, namely Γ𝑠,𝑝 =0, 10, and 30 dB. We see that at lower Γ𝑠,𝑝 values (example 0

dB), which represent a relatively poor channel, the performance of the incremental network-

coded cooperation approaches that of the point-to-point; and at higher Γ𝑠,𝑝 values (example

30 dB), the performance improves substantially and a diversity order of two, taking the point-

to-point transmission as a baseline, is achieved. Hence, like other cooperative schemes,

this incremental network-coded cooperation also requires good quality inter-user channels

to achieve full diversity.

Shown in Figure 6.7 is the three-dimensional plot of the outage probability, with 𝛼and 𝛽

taking the other-two axises and Γ𝑠,𝑑 = Γ𝑝,𝑑 = Γ𝑠,𝑝 = 10 dB. We see that the outage results

are more sensitive to 𝛽variation than to 𝛼variation as the outage curve changes significantly

for a slight change in the energy allocation term 𝛽. Moreover, 𝛽= 0.5is a sufficient choice

over a range of 𝛼values. The worst performance happens as both parameters approach zero

and unity, where the latter substantiates the benefit of cooperation. The lesser sensitivity to

𝛼variation could also mean that this parameter employed to study the effect of other system

parameters such as location of nodes in the networks, where 𝛼could be varied in accordance

with nodes locations. Choosing 0.4< 𝛼 < 0.7practically minimizes outage. As a simple

128

6.4. Summary

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.005

0.01

0.015

0.02

0.025

Rate allocation, α

Energy allocation, β

Outage probability

Figure 6.8.: Three dimensional plot of the outage probability taking 𝛼and 𝛽as the other

variables. Results are based on 𝑅= 0.25 (bits/s/Hz), Γ𝑠,𝑑 = Γ𝑝,𝑑 =10 dB, and

Γ𝑠,𝑝 =0 dB, which represents a relatively poor link.

rule of thumb, when all the channels are on average identical, 𝛼=𝛽= 0.5(similar to the

conventional schemes) could be employing.

Finally shown in Fig. 6.8 is the outage plot when Γ𝑠,𝑝 = 0 db (i.e relatively poor inter-user

channels). The result shows that larger 𝛽values (more energy allocated in the first phase)

benefits the source, as at these inter-user channel conditions, the benefit of cooperation is

less significant (or 𝛽 > 0.5strengthens the inter-user channels).

6.4. Summary

In this chapter, we have introduced the incremental redundancy network-coded cooperation

protocol. This protocol embeds network coding into channel coding such that the redundancy

in the network code is used to support the channel code for better error protection. Descrip-

tion of the system model and the joint network-channel coding was presented in Section 6.1.

We derived the closed-form outage probability expression of this protocol in Section 6.2 and

presented numerical results in Section 6.3. The numerical results show that the performance

of the protocol mostly depends on the quality of the two inter-user channels. Based on the

outage result, energy and rate allocation are also investigated. In the case that the average

quality of all channels are identical, an allocation strategy that equally distributes these re-

sources in the two phases is sufficient. Moreover, the sensitivity of the system to energy

allocation varies significantly in comparison to the rate allocation.

129

6. Incremental Redundancy Network-Coded Cooperation

130

7. Conclusion

This thesis has presented two types of network-coded cooperation and analyzed their outage

probabilities. The main contributions that have been achieved in the course of this work as

well as direction for future research are described in the next sections.

7.1. Contribution of the Thesis

Most previous work in the area of network-coded cooperation is based on error-free inter-

user channels assumption; however, this is not a realistic assumption in wireless networks.

Moreover, a thorough performance analysis was missing and simulation was usually used

to investigate protocols. The motivation of this thesis was to perform outage probability

analysis of network-coded cooperation protocols based on more realistic assumptions and

investigate the outage behavior of the protocols. Using the analysis, the next objective of the

thesis was to study other features of network-coded cooperation, e.g., diversity-multiplexing

tradeoff. The contributions of the thesis are summarized and presented below according to

their order of appearance in the thesis.

7.1.1. Conventional network-coded cooperation

The outage behavior of the conventional network-coded cooperation was studied in Chapter

4. The main observations are presented as follows.

1. We proposed an implementation of an adaptive network-coded cooperation protocol.

The adaptiveness is required in a case that inter-user channels are erroneous and a

cooperating node has to take decisions (adapt) on what to do next. Then, the outage

behavior of this protocol is examined by deriving its outage probability. To make the

outage probability analysis more tractable and convenient for exposition, block fading

Rayleigh distributed channels, orthogonal transmission, and half-duplex constraints

are assumed. Approximating the outage result at high SNR values, we showed that

this protocol achieves full diversity order of two.

131

7. Conclusion

2. The outage behavior was investigated for various inter-user and uplink channel quali-

ties; we compared various cooperative protocols based on the inter-user channels. In

terms of the outage results, network-coded cooperation protocols are found to be su-

perior when the inter-user channels are lower quality; when the inter-user channels are

good, protocols without network coding perform better.

3. The outage results are further used to study the diversity-multiplexing tradeoff and the

coverage area in cooperative networks. Based on the coverage area results, a static

network-coded cooperation protocol is more appropriate when the partner is closer to

the destination. With repetition coding, a node closer to the source (or in the center)

should be selected as the partner.

7.1.2. Energy consumption of network-coded cooperation

In large networks where a source has potential partners in its surrounding to choose from,

metrics that provide insight on how to select a partner are required. One option would be

to select a partner node that minimizes the energy spent in the network. With energy min-

imization in mind, energy consumption of network-coded cooperation was investigated in

Chapter 5. The used energy consumption metric computes the average consumed energy per

information bit for a fixed error rate at the destination. Using the energy metric, we investi-

gated the energy consumption behavior and impact of network geometry in WSNs that use

network-coded cooperation. The main observations are presented as follow.

1. The average energy consumption is optimal when the partner node is located in the

vicinity of the destination rather than the source node.

2. A second metric, called energy efficiency, that takes into account both the energy con-

sumption and reliability of reception, was investigated next. From the numerical re-

sults, we found out that for an optimal energy efficiency, the partner should be located

halfway between the source and destination.

7.1.3. Incremental redundancy network-coded cooperation

The outage behavior of another variant of incremental redundancy network-coded coopera-

tion was investigated in Chapter 6. The results are summarized as follows.

1. We proposed two decoding approaches, namely joint network-channel decoding and

individual network-channel decoding, and demonstrated their implementation using

linear block coding.

132

7.2. Recommendations for Future Research

2. Considering individual network-channel decoding at the destination, we provided a

framework for computing the outage probability for quasi-static Rayleigh fading chan-

nels, orthogonal transmission, and half-duplex constraints. Then the outage behavior

was studied and the results show that this protocol also achieves full diversity order.

3. Using the outage result, the rate and energy allocations that minimize the outage prob-

ability were studied. The numerical results show that outage performance is more

sensitive to the energy allocation than to the rate allocation.

7.2. Recommendations for Future Research

There are plenty of open questions that need to be addressed before any potential deployment

of network coding in real networks. In the following, we propose a few promising research

areas.

1. The cooperative scenario considered in this thesis was based on three-nodes coopera-

tion. As a future work, protocol design and performance study in large scale networks

is one potential research direction. In this regard, [79] can be taken as a good start.

In our study, the network coding was on a Galois field of 𝔽2={0,1}. However, in

large networks one may require larger field sizes and studying the impact of field size

in network performance needs to be addressed.

2. In this thesis, we considered the decode-and-forward relaying strategy where decisions

are based on hard information bits. The use of soft-bit information in network-coded

cooperation is an interesting area to explore, e.g., [21]. Soft-information facilitates the

design of an amplify-and-forward-like protocol with network coding, where the cod-

ing is done by superposition of soft-bit information of locally generated and received

messages.

3. In our analysis, all channels were assumed to be block-fading and Rayleigh distributed.

In block fading channels cooperation does not benefit from time diversity. In a case of

fast fading channels, channel fading coefficients vary within a packet and the block-

fading assumption does not hold. Therefore, performance of network-coded coopera-

tion in fast fading channel environments could be one area of future study [80].

4. A cross-layer study of network-coded cooperation that includes PHY, data link, MAC,

and network layers is worth investigating.

133

7. Conclusion

5. In Chapter 6, we used linear block codes to demonstrate the joint network-channel

coding. Designing more efficient network-channel codes, e.g., using convolutional

codes, that improve performance is another future research direction.

6. In our approach, the partner discards a packet that is received in error even if parts

of this packet are received correctly. Soft-bit information and symbol-level network

coding may be used to filter out correctly received parts of a packet [32, 81]. A detailed

study in this area is our final recommendation.

134

A. Outage Probability Approximation

The outage probability approximations of the incremental redundancy network-coded coop-

eration are presented as follows.

A.1. Case 1

The outage probability expression given in Equation (6.18) on page 122 is derived next.

Re-writing Equation (6.16) once again

𝑃(𝑋𝑠)=𝑃{(1 + 𝛽1𝛾𝑠,𝑑)𝛼(1 + 𝛽2(𝛾𝑠,𝑑 +𝛾𝑝,𝑑))(1−𝛼)<22𝑅}

=∫ ∫𝐴

Γ𝑠,𝑑Γ𝑝,𝑑

exp (−𝛾𝑠,𝑑

Γ𝑠,𝑑 −𝛾𝑝,𝑑

Γ𝑝,𝑑 )𝑑𝛾𝑠,𝑑𝑑𝛾𝑝,𝑑.(A.1)

The term inside the braces is a function of 𝛾𝑠,𝑑 and 𝛾𝑝,𝑑 and this function is compared to the

threshold value 22𝑅. This probability can be computed by integrating the joint probability

density function (PDF) of the two random variable on a region 𝐴defined by

𝐴≡{(1 + 𝛽1𝛾𝑠,𝑑)𝛼(1 + 𝛽2(𝛾𝑠,𝑑 +𝛾𝑝,𝑑))(1−𝛼)<22𝑅}.(A.2)

To determine the range of the two variables, we extract 𝛾𝑝,𝑑 from 𝐴and get the range

0< 𝛾𝑝,𝑑 <1

𝛽2[22𝑅/(1−𝛼)

(1 + 𝛽1𝛾𝑠,𝑑)𝛼/(1−𝛼)−(1 + 𝛽2𝛾𝑠,𝑑)].(A.3)

As the term in bracket in Equation (A.3) should be greater than zero, 𝛾𝑠,𝑑 must satisfy

(1 + 𝛽2𝛾𝑠,𝑑)(1 + 𝛽1𝛾𝑠,𝑑)𝛼/(1−𝛼)<22𝑅/(1−𝛼)(A.4)

It is difficult to solve for 𝛾𝑠,𝑑 and 𝛾𝑝,𝑑 from (A.3) and (A.4) directly. Using Taylor’s series in

two variables and taking up to the first order terms in Equations (A.3), we get the ranges

0< 𝛾𝑝,𝑑 <1

𝛽2[22𝑅/(1−𝛼)−1]−[22𝑅/(1−𝛼)𝛽

1−𝛽+ 1]𝛾𝑠,𝑑 .(A.5)

135

A. Outage Probability Approximation

The right side of Equation (A.5) is a linear function of the variable 𝛾𝑠,𝑑. Similarly, Equation

(A.4) can be approximated as

0< 𝛾𝑠,𝑑 <[22𝑅/(1−𝛼)−1](1 −𝛼).(A.6)

Once these ranges are known, the integral in (A.1) can be evaluated and

𝑃(𝑋𝑠)≈(22𝑅/(1−𝛼)−1)2

Γ𝑠,𝑑Γ𝑝,𝑑

(1 −𝛼)2

(1 −𝛽),(A.7)

which is the outage probability approximation given in Equation (6.18), is obtained.

A.2. Case 2

Next the outage approximation given in Equation (6.29) is derived. Given the region 𝐶where

the variables 𝛾𝑠,𝑑 and 𝛾𝑝,𝑑 are defined as

𝐶≡{(1 + 𝛽1𝛾𝑠,𝑑)𝛼[(1 + 𝛽2𝛾𝑠,𝑑)(1 + 𝛽2𝛾𝑝,𝑑)](1−𝛼)<22𝑅/(2−𝛼)}(A.8)

and extracting 𝛾𝑝,𝑑 from 𝐶, we get the range

0< 𝛾𝑝,𝑑 <1

𝛽2[22𝑅/[(2−𝛼)(1−𝛼)]

(1 + 𝛽1𝛾𝑠,𝑑)𝛼/(1−𝛼)(1 + 𝛽2𝛾𝑠,𝑑)−1].(A.9)

As the right side of Equation (A.9) should be greater than zero, 𝛾𝑠,𝑑 must fulfill the following

condition.

(1 + 𝛽1𝛾𝑠,𝑑)𝛼/(1−𝛼)(1 + 𝛽2𝛾𝑠,𝑑)<22𝑅/[(2−𝛼)(1−𝛼)].(A.10)

Using again Taylor’s series in two variables and taking up to the first order terms, Equation

(A.9) can be approximated as

0< 𝛾𝑝,𝑑 <(1 −𝛼)

(1 −𝛽)[22𝑅/[(2−𝛼)(1−𝛼)] −1]−[22𝑅/[(2−𝛼)(1−𝛼)] 1

1−𝛽]𝛾𝑠,𝑑

=𝑎−𝑏𝛾𝑠,𝑑 (A.11)

where 𝑎=(1−𝛼)

(1−𝛽)[22𝑅/[(2−𝛼)(1−𝛼)] −1]and 𝑏=[22𝑅/[(2−𝛼)(1−𝛼)] 1

1−𝛽]. We see in Equation

(A.11) that, for given values of 𝛼and 𝛽, the term 𝑎−𝑏𝛾𝑠,𝑑 is a linear function of 𝛾𝑠,𝑑.

Similarly, Equation (A.10) can be approximated as

𝛾𝑠,𝑑 <[22𝑅/(2−𝛼)−1].(A.12)

136

A.2. Case 2

Once the ranges of 𝛾𝑠,𝑑 and 𝛾𝑠,𝑑 are known, the integral in Equation (6.27) can easily be

evaluated and the following result is obtained.

𝑃(𝑋𝑠)≈(1 −𝛼)2

(1 −𝛽)Γ𝑠,𝑑Γ𝑝,𝑑 [22𝑅/[(2−𝛼)(1−𝛼)] −1]2.(A.13)

137

A. Outage Probability Approximation

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