cooperation and joint source-channel transmission in wireless...

COOPERATION AND JOINT SOURCE-CHANNEL

TRANSMISSION IN WIRELESS NETWORKS

by

Jing Wang

B.E., Zhejiang University, 2004

M.E., Zhejiang University, 2006

a Thesis submitted in partial fulfillment of

the requirements for the degree of

Doctor of Philosophy

in the

School of Engineering Science

c© Jing Wang 2010

SIMON FRASER UNIVERSITY

Summer 2010

All rights reserved. However, in accordance with the Copyright Act of Canada, this work

may be reproduced, without authorization, under the conditions for Fair

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with the law, particularly if cited appropriately.

APPROVAL

Name: Jing Wang

Degree: Doctor of Philosophy

Title of Thesis: Cooperation and Joint Source-Channel Transmission in Wire

less Networks

Examining Committee: Dr. Craig Scratchley

Chair

Dr. Jie Liang, Senior Supervisor

Dr. Sami (Hakam) Muhaidat, Supervisor

Dr. Daniel C. Lee, Supervisor

Dr. Ivan V. Bajic, Internal Examiner

Dr. Jun Chen, External Examiner

Assistant Professor of Electrical and Computer Engi

neering, McMaster University

Date Approved: May 4,2010

11

Last revision: Spring 09

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Abstract

In this thesis, we study the problem of cooperation and joint source-channel transmission

in wireless networks, with an emphasis on some fundamental information-theoretic aspects.

The majority of this thesis focuses on the analysis of fundamental performance lim-

itations of joint source-channel transmission in wireless cooperative networks. We made

three major contributions in this topic. The first contribution is a study on the end-to-

end distortion of joint source-channel transmission in multi-relay cooperative systems, in

terms of the distortion exponent at high signal-to-noise ratio (SNR). Building upon results

from the diversity-multiplexing tradeoff (DMT) analysis, the achievable distortion expo-

nents of multi-relay cooperative systems with layered coding and transmission strategies

are obtained. We next propose to improve the achievable distortion exponent by employing

limited channel state feedback in the multiple-relay system. We show that combining a

simple feedback scheme with single-rate coding outperforms the best known non-feedback

layered transmission strategies with only a few bits of feedback information. The third part

focuses on the recently proposed two-way relaying cooperative networks, where two users

communicate in both directions with the help of one relay. We introduce and analyze a

new concept - achievable distortion exponent region, which characterizes the end-to-end

distortions of both users and addresses the multiuser nature of the two-way communication

system. In addition, we extend the DMT analysis to two-way relaying cooperative networks

and obtain the DMT regions of various bidirectional cooperation protocols.

This thesis also investigates the cross-layer resource allocation in wireless systems. We

consider transmitting a layer-coded source over a slow fading channel using the broadcast

strategy, where the channel state information is not available at the transmitter. An efficient

iterative algorithm is proposed to minimize the end-to-end distortion by jointly solving the

power allocation problem and the channel discretization problem at an arbitrary SNR.

iii

Acknowledgments

I would like to express my sincere gratitude to my senior supervisor, Professor Jie Liang,

whose insightful vision and broad knowledge in signal processing and multimedia commu-

nications have guided me in working towards my doctoral degree at SFU. Dr. Liang always

inspires me to look deeper into fundamental problems by posing thoughtful questions. Being

an understanding and caring supervisor, Dr. Liang has provided me an incredible opportu-

nity and great flexibility to pursue my research. This thesis would not have been possible

without his invaluable guidance, generous support, and heart-warming encouragement.

I am grateful to my supervisor, Professor Sami Muhaidat, for introducing me to the

field of cooperative communications. Our discussion on relay selection and various other

topics in wireless communications was one of the initial impetuses that prompt me to study

the topics of this thesis. I would also like to thank Professor Daniel Lee for serving on

my supervisory committee and Professor Ivan Bajic for serving as the internal examiner of

my thesis. I have benefited greatly from their informative courses: Personal and Mobile

Communications taught by Dr. Lee, and Information Theory taught by Dr. Bajic. Their

inspiring lectures opened a window for me into these fascinating research fields and helped

me attack some interesting research problems.

It is also my honor to thank the external examiner of my thesis, Prof. Jun Chen, at

McMaster University and the defense chair, Dr. Craig Scratchley, for their time and efforts.

It is a pleasure to thank many of my friends for making my life of the past four years

a most wonderful and memorable one. Special thanks go to the current and past members

of our lab, Upul Samarawickrama, Quoqian Sun, Yuemeng Chen, and Xiaoyu Xiu to name

but a few, for their help and lively discussions on research, campus life, and beyond.

Finally, and above all, I owe my deepest gratitude to my parents for their enduring love,

limitless support, and unconditional sacrifice.

iv

Contents

Approval ii

Abstract iii

Acknowledgments iv

Contents v

List of Tables ix

List of Figures x

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline and Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Notations and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Review 8

2.1 Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Wireless Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Wireless channel model . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Channel capacity and outage . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.3 Diversity-multiplexing tradeoff . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Distortion Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Cooperative Communications and Relay Channels . . . . . . . . . . . . . . . 20

2.5 Two-way Communications and Bidirectional Relaying . . . . . . . . . . . . . 24

v

2.6 Distortion Minimization of Joint Source-Channel Transmission in Fading

Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Distortion Exponents of Multi-relay Cooperative Networks 31

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Distortion Exponents of Layered Source Coding with Progressive Transmission 33

3.4 Distortion Exponents of Broadcast Strategy . . . . . . . . . . . . . . . . . . . 36

3.4.1 Repetition-based cooperation . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.2 Relay-selection-based cooperation . . . . . . . . . . . . . . . . . . . . . 45

3.4.3 Space-time-coded cooperation . . . . . . . . . . . . . . . . . . . . . . 49

3.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.A Proof of Lemma 3.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.B Proof of Theorem 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.C Proof of Lemma 3.B.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.D Proof of Theorem 3.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.E Proof of Lemma 3.4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.F Proof of Theorem 3.4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Distortion Exponents of Multi-relay Cooperation with Limited Feedback 67

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3 Distortion Exponents of Amplify-and-forward Based Protocols . . . . . . . . 70

4.3.1 Orthogonal amplify-and-forward protocol . . . . . . . . . . . . . . . . 72

4.3.2 Nonorthogonal amplify-and-forward protocol . . . . . . . . . . . . . . 75

4.3.3 Sequential slotted amplify-and-forward protocol . . . . . . . . . . . . . 78

4.4 Distortion Exponents of Decode-and-forward Based Protocols . . . . . . . . . 80

4.4.1 Orthogonal selection decode-and-forward protocol . . . . . . . . . . . 81

4.4.2 Nonorthogonal selection decode-and-forward protocol . . . . . . . . . 83


4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.A Optimality of Equating Linear Terms in (4.11) . . . . . . . . . . . . . . . . . 90


vi

4.C Proof of Theorem 4.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5 Distortion Exponents of Two-way Relaying Cooperative Networks 97

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3 Distortion Exponent Outer Bound and One-way Relaying Strategies . . . . . 100

5.3.1 Outer bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3.2 One-way relaying strategies . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4 Distortion Exponents of MABC Protocols with Single-rate Source-Channel

Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.4.1 Decode-and-forward MABC protocol . . . . . . . . . . . . . . . . . . . 105

5.4.2 Amplify-and-forward MABC protocol . . . . . . . . . . . . . . . . . . 110

5.4.3 Compress-and-forward MABC protocol . . . . . . . . . . . . . . . . . 112

5.5 Distortion Exponents of TDBC Protocols with Single-rate Source-Channel

Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.5.1 Decode-and-forward TDBC protocol . . . . . . . . . . . . . . . . . . . 114

5.5.2 Amplify-and-forward TDBC protocol . . . . . . . . . . . . . . . . . . . 118


5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.A Proof of Theorem 5.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124


5.C Proof of Theorem 5.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.D Proof of Theorem 5.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6 Finite-SNR End-to-end Distortion Minimization 139

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.3 An Interative Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.3.1 Rate allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.3.2 Channel discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.3.3 Algorithm description . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.4 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 146

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

vii

7 Conclusions 152

7.1 Conlusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Bibliography 156

viii

List of Tables

1.1 List of notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 List of acronyms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

ix

List of Figures

2.1 Joint source and channel coding. . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Separate source and channel coding. . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Diversity-multiplexing tradeoffs of various cooperation protocols. . . . . . . . 22

2.4 Transmission phases of a two-way relaying system with (a) one-way relaying

strategy, (b) the MABC protocol, (c) the TDBC protocol. . . . . . . . . . . . 25

2.5 Layered source coding with broadcast strategy. . . . . . . . . . . . . . . . . . 28

3.1 System model of an m-relay cooperative system. . . . . . . . . . . . . . . . . 32

3.2 Layered source coding with progressive transmission using (a) repetition-

based cooperation, (b) relay-selection-based cooperation, (c) distributed space-

time-coded cooperation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Layered source coding with broadcast strategy using (a) repetition-based co-

operation, (b) relay-selection-based cooperation, (c) distributed space-time-

coded cooperation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Distortion exponent vs. channel allocation ratio t at various bandwidth ratios

b of layered coding with broadcast strategy using the distributed space-time-

coded protocol for a 2-relay cooperative system. . . . . . . . . . . . . . . . . . 52

3.5 Distortion exponent vs. bandwidth ratio of layered coding with broadcast

strategy using the distributed space-time-coded protocol for t = 1/2 and

t = t∗ (optimal). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.6 Distortion exponent vs. bandwidth ratio of layered source coding with pro-

gressive transmission for multi-relay cooperative systems. . . . . . . . . . . . 54

3.7 Distortion exponent vs. bandwidth ratio of layered source coding with broad-

cast strategy for multi-relay cooperative systems. . . . . . . . . . . . . . . . . 54

x

3.8 Comparison of various coding and transmission strategies for a 3-relay coop-

erative system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.9 Outage region of the distributed space-time-coded protocol with broadcast

strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1 System model of an m-relay cooperative system with limited feedback from

the destination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Distortion exponents of the NAF protocol and the OAF protocol with differ-

ent feedback resolution L for a 2-relay cooperative system. . . . . . . . . . . . 86

4.3 Distortion exponents of the SAF protocol with feedback resolution L = 8 and

∞ for different transmission slots M for a 2-relay cooperative system under

the relay isolation assumption. . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.4 Comparison of the distortion exponents of the NDF protocol (solid curves)

and the ODF protocol (dashed curves) with feedback resolution L = 1, 2, 4,∞for a 3-relay cooperative system. . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.5 Comparison of the distortion exponents of various multi-relay cooperation

protocols for a 2-relay cooperative system. M = 3 is used in the sequential

SAF protocol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.6 Comparison of the distortion exponents of the NAF protocols with feedback

resolution L = 2, 4, 8, 16 and number of relays m = 3, 4, 5, 6, 7. . . . . . . . . . 89

5.1 System model of a two-way relaying communication system. . . . . . . . . . . 98

5.2 Equivalent system model for the outer bound. . . . . . . . . . . . . . . . . . . 101

5.3 Comparison of various source-channel transmission schemes in a two-way re-

laying cooperative system for b = 1. . . . . . . . . . . . . . . . . . . . . . . . 121

5.4 Comparison of various source-channel transmission schemes in a two-way re-

laying cooperative system for b = 8. Note that the outer bound is achieved

by the one-way BS strategy. Also, the curves of the AF/CF-based MABC

protocol and the DF-based MABC protocol coincide. . . . . . . . . . . . . . . 121

5.5 Comparison of various source-channel transmission schemes in a symmetric-

rate two-way relaying cooperative system. . . . . . . . . . . . . . . . . . . . . 123

5.6 The region of 1− θ23 ≤ 1−tt (1− θ31) (light gray area) for (θ23, θ31) ∈ R2+ and

t > 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xi

5.7 The region of 1− θ23 ≤ 1−tt (1− θ31) (light gray area) for (θ23, θ31) ∈ R2+ and

t ≤ 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.8 The outage set O212 of the DF-based TDBC protocol. (a) r1 ≥ t3, (b) r1 < t3. 133

6.1 The optimized power allocation γi, rate allocation Ri, and discrete channel

fading gains si of Rician fading channels with different Rician K-factors:

K = 0, 4, 32, 64. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.2 Minimum expected end-to-end distortion achieved by different methods with

M = 10 and 1000 for a Rayleigh fading channel. . . . . . . . . . . . . . . . . 148

6.3 Minimum expected end-to-end distortion achieved by different methods with

M = 10 for the SISO Nakagami fading channels. . . . . . . . . . . . . . . . . 149

6.4 The convergence behavior of the proposed iterative algorithm. . . . . . . . . . 150

xii

Chapter 1

Introduction

1.1 Background and Motivation

The past decades have witnessed a rapid evolution of wireless communication systems thanks

to the dramatic progress in the very-large-scale integration (VLSI) circuit design and the

advances in communication theory, data compression, networking, and signal processing

techniques. Wireless communication has become a vibrant and fast evolving research area.

While wireless systems are advantageous over wired systems in terms of energy efficiency,

availability, and flexibility, the hostile nature of wireless environment also gives rise to many

challenging problems in both theory and practical system design.

A wireless channel is subject to a large degree of unreliability such that the communica-

tion signal may experience significant attenuation and delay as it is transmitted over wireless

mediums. This is known as the notorious fading effect. Fading occurs as the signal travels

over a long distance or is obstructed by large objects on the propagation path. It is also

caused by the multipath propagation effect as the signals undergone different attenuations

and different delays may add up destructively at the receiver. In general, the fading varies

with time, space and frequency, and is often modeled as a random process. Transmission

strategies that effectively combat the fading effect are therefore essential in improving the

reliability and efficiency of wireless systems.

Due to the broadcast nature of the wireless medium, when a node transmits in a wireless

network, all of its neighboring nodes will receive the transmitted signal. As multiple users

communicate through the same transmission medium, they may compete for network re-

sources and create interference to the unintended receivers. On the other hand, interaction

1

CHAPTER 1. INTRODUCTION 2

like this also creates opportunities for cooperation among users. In the emerging cooper-

ative communication systems [1, 2], users are allowed to jointly encode, decode, or relay

others’ messages via cooperation. The broadcast nature of omnidirectional mobile anten-

nas incurs no additional cost of transmit power for communicating with both the ultimate

receiver and the partner nodes. In this way, virtual antenna arrays are formed and each

message is passed through multiple independent links, which thus significantly increases the

transmission reliability and throughput.

The performance of a wireless communication system is also greatly influenced by the

knowledge of the channel state information (CSI) at the transmitter side (CSIT) and at the

receiver side (CSIR). Here, CSI refers to the knowledge about the actual status of the wireless

channel. CSI plays an important role in many sophisticated transmission techniques such as

linear precoding and beamforming. In both theory and practice, (relatively) accurate CSIR

can be measured by sending training sequences known a priori to the receiver whereas CSIT

is usually obtained via feedback. However, due to practical limitations such as link capacity

and delay constraint, obtaining arbitrarily accurate feedback information is often infeasible

in most wireless systems. Furthermore, it is known that Shannon’s source-channel separation

theorem does not hold when full CSIT is not available [3]. Therefore, to achieve an optimal

performance, joint source-channel transmission approaches are in general required. The

cross-layer wireless network design approach addresses this issue by jointly optimizing the

allocation of network resources such as power and bandwidth based on the wireless channel

conditions and the quality of service (QoS) requirements.

Driven by the increasing demands for reliable universal connectivity, higher through-

put, and wireless applications with stringent delay and energy constraints such as wireless

broadband multimedia services, future wireless systems must employ advanced algorithms

and techniques to offer better reliability and higher data rate. However, as we have already

seen, while rapid progresses have been made, wireless system performance is still limited

due to the many challenging issues mentioned above. This prompts us to study the funda-

mental performance limitations of wireless communication systems, which will provide us

a complete picture and some useful insights into this intricate problem, in particular, the

joint impact of key factors such as fading, user cooperation and CSI feedback on the sys-

tem performance. Furthermore, as we take into consideration both the source and channel

characteristics in the joint source-channel system design, it is also important to adopt an

appropriate measure that characterizes the end-to-end performance of the overall system.


The main purpose of this thesis is to gain a better understanding of the theoretical

performance limitations of wireless communication systems by jointly investigating user

cooperation, CSI feedback and source-channel transmission strategies. In particular, we

1. investigate the layered source coding and transmission in multi-relay cooperative net-

works;

2. investigate the impact of feedback information on the transmission in multi-relay co-

operative networks;

3. study the performance limits of a two-way relaying cooperative network where two

users communicate with the help of one relay;

4. develop an efficient algorithm for minimizing the distortion of joint source-channel

transmission over fading channels.

Our work reveals many challenging problems in both theory and practice in cooperative

communications and joint source-channel transmission. It also provides interesting implica-

tions in the design of transmission strategies and source-channel coding in wireless systems.

1.2 Outline and Main Contributions

The topic of this thesis is cooperation and joint source-channel transmission in wireless

networks. The problem is studied from two perspectives: First, we study the end-to-end

performance limitations of multi-relay cooperative communication systems and two-way

relaying cooperative systems in terms of the distortion exponent. Second, we study the

cross-layer resource allocation for joint source-channel transmission over fading channels.

The outline and main contributions of the thesis are listed below.

In Chapter 2, we provide a brief review of important background materials related to

the thesis. We first review the information-theoretic aspects of a wireless communication

system. We then introduce the important concepts that will be used extensively throughout

the thesis, including channel capacity and outage, diversity-multiplexing tradeoff (DMT),

distortion exponent, cooperative communications, two-way relaying communications, and

distortion minimization in joint source-channel transmission.

In Chapter 3, we consider the transmission of a Gaussian source over a cooperative

network with multiple relays, and analyze the end-to-end distortion at high signal-to-noise


ratio (SNR), in terms of the distortion exponent. Our contributions in this chapter are

threefold. First, we extend the existing distortion exponent analyses of relay networks

[4, 5, 6] to cooperative networks with an arbitrary number of relays. Secondly, we derive

the distortion exponents when the layered source coding with progressive transmission or

broadcast strategy is used in multi-relay networks under three cooperation protocols, based

on repetition, relay selection, and space-time coding, respectively. Our analyses reveal the

impacts of the number of relays, bandwidth ratio and cooperation protocol on the distortion

exponent. Thirdly, as an important addition to the DMT analysis, we prove the successive

refinability of the DMTs of the three multi-relay cooperation protocols. The material in

this chapter has appeared in [7, 8].

In Chapter 4, we investigate the impact of feedback information on the distortion expo-

nent of joint source-channel transmission over a multi-relay cooperative network. Limited

channel state feedback is combined with separate source and channel coding to help the

transmission. Various orthogonal and nonorthogonal multi-relay cooperation protocols are

considered, including the orthogonal amplify-and-forward (AF) or decode-and-forward (DF)

protocols, the nonorthogonal AF/DF protocols, and the slotted AF protocol. We derive the

optimal distortion exponents of all cases, and illustrate the effect of the feedback resolution,

bandwidth ratio as well as number of relays on the distortion exponent. It is shown that the

feedback scheme outperforms the best known non-feedback strategies for multi-relay coop-

erative systems with only a few bits of feedback information. The material in this chapter

has appeared in [9, 10].

Chapter 5 extends the distortion exponent analysis to a three-node half-duplex bidi-

rectional relaying network, where two users communicate in both directions with the help

of one relay. The relay employs AF, DF, or compress-and-forward (CF) based two-way

cooperation protocols. We analyze the distortion exponents for both users. The different

transmission rates of the two users necessitate the study of a new concept - the achievable

distortion exponent region of the system. We first derive an outer bound on the distortion

exponent region of two-way relaying communications, which is tight at large bandwidth

ratio. We then obtain the optimal distortion exponent pairs of conventional one-way relay-

ing strategies and AF/DF based two-way relaying protocols with single-rate coding. The

material in this chapter has appeared in [11, 12, 13].

In Chapter 6, we study the distortion minimization problem in transmitting a Gaussian

signal over a slow fading channel. The channel state information is assumed to be only known


at the receiver. The source is layer-coded and transmitted using the broadcast strategy.

We investigate the optimal power and rate allocation to minimize the expected end-to-

end distortion of the reconstructed signal at the receiver. An efficient iterative algorithm is

proposed to jointly solve the rate allocation problem and the channel discretization problem.

Numerical results show that the proposed algorithm outperforms the schemes using fixed

channel discretization by a large margin. Meanwhile, the computational cost of our method

is lower than those of the joint optimization approaches that involve partial exhaustive

search. The material in this chapter has appeared in [14].

Finally, we summarize the work in this thesis and present the conclusions in Chapter 7.

We also discuss several possible future research directions.

Contributions outside the scope of the thesis

In addition to the materials reported above, we briefly summarize our contributions on

multiple description coding, which are also related to the topic of the thesis, but are not

formally included. Research along this line will be discussed in Section 7.2 as possible

directions for future works.

Multiple description coding (MDC) [15] is an attractive technique of combating trans-

mission errors. In MDC, the source signal is encoded into several coded streams called de-

scriptions, which are sent to the receiver via different network paths. Judiciously designed

redundancies are introduced in all descriptions such that an arbitrary subset of descriptions

can be used to reconstruct the original signal, and the reconstruction quality improves with

the number of descriptions received.

We proposed a prediction-compensated multiple description coding (PC-MDC) frame-

work for two-band filter banks in [16, 17], in which the coefficients in each subband are split

into two descriptions. Each description also includes the prediction residuals of the data in

the other description. The designs of the optimal orthogonal and biorthogonal filter banks

are formulated in a unified framework, and both one-level and multiple-level decompositions

are analyzed. We also proposed a weighted reconstruction method in [18] to further improve

the reconstruction quality of the proposed PC-MDC scheme. In [16, 17, 19], we applied the

PC-MDC method to multiple description image coding using both H.264 intra frame coding

and JPEG 2000 image coding. Simulation results show that our method achieves better

or comparable performance than that of the latest MDC results while the complexity is

reduced.


1.3 Notations and Acronyms

In this section we define the notations and acronyms used throughout this thesis.

R The set of real numbers.R+ The set of positive real numbers.Rn+ The set of n-dimensional real vectors with positive coordinates.A A calligraphic uppercase letter denotes a set.Ac The complementary set of a set A.|A| The cardinality of a set A.An The n-ary Cartesian power of a set A.an A sequence of scalars a1, a2, · · · , an.a A boldface lowercase letter denotes a vector.A A boldface uppercase letter denotes a matrix.I The identity matrix.0 The null matrix.AT The transpose of a matrix A.AH The conjugate transpose of a matrix A.A−1 The inverse of a matrix A.det(A) The determinant of a matrix A..= The exponential equality [20]: f(a) .= ab denotes b = lima→∞

log f(a)log a .

a+ Denotes max(a, 0) for a real number a.dae The smallest integer no less than the real number a.Pr{A} The probability of event A occuring.E(x) The expected value of a random variable x.CN (0, 1) zero-mean unit-variance circularly symmetric complex Gaussian

Table 1.1: List of notations.


AF amplify-and-forwardAWGN additive white Gaussian noiseBC broadcast channelBS broadcast strategyc.d.f. cumulative density functionCF compress-and-forwardCSI channel-state informationCSIR channel-state information at the receiverCSIT channel-state information at the transmitterDDF dynamic decode-and-forwardDF decode-and-forwardDMT diversity-multiplexing tradeoffi.i.d. independent and identically distributedKKT Karush-Kuhn-TuckerLS layered source coding with progressive transmissionMABC multiple-access broadcastMAC multiple-access channelMIMO multiple-input multiple-outputMISO multiple-input single-outputMSE mean-squared errorNAF nonorthogonal amplify-and-forwardNDF nonorthogonal selection decode-and-forwardOAF orthogonal amplify-and forwardODF orthogonal selection decode-and-forwardp.d.f. probability density functionSAF slotted amplify-and-forwardSIMO single-input multiple-outputSISO single-input single-outputSNR signal-to-noise ratioTDBC time-division broadcast

Table 1.2: List of acronyms.

Chapter 2

Review

This chapter reviews some basic definitions and important concepts that are used extensively

throughout the thesis. We begin with a brief introduction to some important information-

theoretic concepts of communications over wireless channels. Discussions of related works

are also provided as motivational background materials.

2.1 Communication Systems

We first revisit a communication system in its information-theoretic aspect. To introduce the

basic concepts, we consider the simple but quite general scenario where a single transmitter

and receiver pair communicates over a discrete channel. Mathematically, a discrete channel

is characterized by an input alphabet X , an output alphabet Y, and a probability transition

matrix p(y|x) that expresses the probability of observing the output symbol y ∈ Y given

that a symbol x ∈ X is sent. The probability distribution of the channel output y may

depend not only on the channel input at that time but also the previous channel inputs or

outputs. The communication procedure can then be described as follows: The transmitter

maps (encodes) a sequence of source samples into some sequence of channel symbols to

be sent over the channel. The encoder may map the sequence of source samples directly

into the input of the channel (joint source and channel coding) as illustrated in Fig. 2.1,

or it may first compress the source samples into an efficient representation, then perform

the appropriate channel coding to send it over the channel (separate source and channel

coding) as illustrated in Fig. 2.2. The output sequence of the channel is random but has a

distribution that depends on the input sequence and the probability transition matrix. The

8

CHAPTER 2. REVIEW 9

Channel p(y|x) DecoderEncoderKsKs Nx Ny

Channel p(y|x) DecoderEncoderKsKs Nx Ny

Figure 2.1: Joint source and channel coding.

SourceEncoder

Ks w NxChannelEncoder

SourceDecoder

ChannelDecoder

NyChannelp(y|x)

w KsSourceEncoder

Ks w NxChannelEncoder

SourceDecoder

ChannelDecoder

NyChannelp(y|x)

w Ks

Figure 2.2: Separate source and channel coding.

receiver attempts to recover (decode) the source samples from the output sequence using

either joint source-channel decoding or separate source-channel decoding.

In Shannon’s 1948 seminal paper “A mathematical theory of communication” [21], the

two-stage separate source and channel coding is proved to be as good as any other joint

source-channel coding method, in terms of the maximum amount of information that can

be reliably communicated over the channel. The result was initially stated for reliable

communications with stationary memoryless sources and channels in [21], and was later

revisited in the context of transmission with a distortion measure in [22] and for more general

classes of sources and channels in [23]. The source-channel separation theorem implies

that source coding and channel coding can be designed independently in a communication

system. In particular, source encoder/decoder can be designed without knowing the channel.

Although the separation theorem, which is true for considerable classes of sources and

channels, does not always hold, the separation approach has been widely adopted in practical

communication systems due to the great reduction in complexity.

More precisely, in the source coding stage of the separate source and channel coding

system, a discrete-time source s1, s2, · · · , sK with K samples (e.g. image, video, or speech

signal) is mapped to an index (message) w drawn from a finite alphabetW = {1, 2, · · · ,M}.It is clear that on average each source sample is represented by dlog2 Me

K bits, and the rate

of the source code is thus defined to be Rs = dlog2MeK bits per source sample.

In the channel coding stage, the message w is mapped into a sequence of N channel

symbols x1, x2, · · · , xN to be transmitted over the channel, for which we say the communi-

cation consumes N channel uses. x1, x2, · · · , xN is referred to as a length-N codeword. The

set of all codewords is called a codebook, which is known to both transmitter and receiver.

Since the communication system attempts to convey dlog2Me bits of information through

CHAPTER 2. REVIEW 10

the channel after N channel uses, the rate of the channel code is thus R = dlog2MeN bits per

channel use.

We now introduce an important concept called the bandwidth ratio, which is the ratio

between the channel bandwidth and the source bandwidth, or the number of channel uses

per source sample. In the aforementioned separate source and channel coding system, K

source samples are transmitted in N channel uses, which corresponds to a bandwidth ratio

of b = NK .

The receiver observes an output sequence y1, y2, · · · , yN and attempts to estimate which

message has been sent. This stage is known as channel decoding. An error is declared if the

decoded massage w 6= w. The corresponding error probability is denoted by Pe , Pr{w 6=w}. In the source decoding stage, the receiver tries to recover the source based on the

decoded message w, and the result is the reconstructed source samples s1, s2, · · · , sK . In

this thesis, we measure the overall end-to-end performance of the system using the single-

letter squared-error distortion between the original and reconstructed source samples, which

is defined to be

D =1K

K∑i=1

(si − si)2. (2.1)

Consider a block of continuous-amplitude, independent and identically distributed (i.i.d.),

complex Gaussian source samples s1, s2, · · · , sK with zero mean and variance σ2

2 in each di-

mension. Assuming there is no decoding error (w = w), the squared-error distortion (2.1)

is then lower-bounded by the following well-known distortion rate function [3]

D(R) = σ22−Rs = σ22−bR. (2.2)

The distortion lower bound D(R) is achievable when the block length K is sufficiently large.

That is, there exists at least one source code with rate Rs such that for sufficiently large

K, the average distortion D is less than D(R) + δ, where δ > 0 is arbitrarily small. When

a decoding error occurs (w 6= w), the minimum mean squared error (MMSE) estimation is

employed, which results in an average distortion of σ2.

This average squared-error distortion measure and the Gaussian source model are widely

used in both theory and practice, which are particularly useful for multimedia sources such

as image and video signals.

The above model characterizes a point-to-point communication system, which is perhaps

the most classical but also the simplest one. In this thesis, we will deal with single-user


systems with multiple cooperative nodes as well as two-user bidirectional communication

systems. Furthermore, in addition to the single-rate source-channel code described above,

we will also consider layered codes where source samples are mapped into multiple messages

w1, · · · , wn, which are then coded into a single sequence to be transmitted and successively

decoded at the receiver. The information-theoretic properties of these systems and coding

techniques will be discussed in more details in Section 2.4 - 2.6 or when appropriate.

2.2 Wireless Systems

We now review some useful information-theoretic properties and performance measures of

communication over wireless systems.

2.2.1 Wireless channel model

The variation of wireless channels is usually divided into two types [24]: large-scale fad-

ing and small-scale fading. The large-scale fading comes from the path loss as the signal

travels over distance and the shadowing as the signals are obstructed by large objects on

the propagation path. The small-scale fading is caused by the many different paths that

the transmitted signals may propagate to the receiver. As the signals undergo different

attenuations and delays, they may add up constructively or destructively at the receiver.

The large-scale fading effect changes slowly and can usually be regarded as fixed within the

duration of a symbol or a codeword while the small-scale multipath fading is more relevant

to the reliability and efficiency of wireless systems.

Due to its time-varying nature, a wireless channel is often modeled as a linear time-

varying system [24]. Assume the waveform that carries the input channel symbol x is

bandlimited to B Hz. Using the approach of the sampling theorem, the discrete-time base-

band equivalent input/output model of the wireless channel can be written as [24]

y[k] =∑i

hi[k]x[k − i] + n[k], (2.3)

where y[k] is the (sampled) channel output, hi[k] is the ith time-varying complex channel

filter tap (or the channel coefficient), and n[k] is the low-pass filtered additive noise, all at

sampling instant kB . For notational simplicity, we refer to such sampling instant as time k

or the kth time slot.


Two important parameters regarding the wireless channel are the coherence time and

the coherence bandwidth [24]. The coherence time is defined to be the interval over which

the channel coefficient hi changes significantly as a function of time. In the wireless commu-

nication literature, fading channels are usually categorized into fast fading and slow fading.

However, the definitions of these two terms often vary. For example, the term “slow (or

slowly) fading” is often used to describe the case where the symbol duration is shorter than

the coherence time of the channel [25, 26]. It is also sometimes used as a synonym for “large-

scale fading” [27]. Throughout this thesis, we adopt the definition in [24]. We will refer to

a channel as slow fading if the coherence time is longer than the delay constraint (codeword

length) of the application, and fast fading otherwise. Therefore, whether a channel is slow

or fast fading is also application-dependent. For example, a slow fading channel for voice

or video communication, which usually has a strict delay constraint, may actually be fast

fading for file downloading applications.

While the coherence time reflects how fast the channel varies in time, the coherence

bandwidth Bc dictates how fast it varies in frequency. Bc is defined to be the reciprocal

of the multipath delay spread, which is the largest difference in propagation time between

any two separate paths [24]. If the bandwidth of the transmitted waveform is much smaller

than the coherence bandwidth of the channel, i.e., B � Bc, all the frequency components

of the transmitted signal experience almost the same attenuation and delay. In this case the

channel is said to be frequency-nonselective, or flat fading, which can be represented by a

single time-varying complex channel coefficient. Otherwise, the channel is called frequency-

selective, and is usually represented by a channel filter with multiple taps. The flat fading

channel model can be expressed as follows

y[k] = h[k]x[k] + n[k]. (2.4)

It has been discovered that the inherent frequency diversity in a frequency-selective channel

can be exploited by techniques such as orthogonal frequency division multiplexing (OFDM)

and code division multiple access (CDMA) to provide robust transmission. For example,

by employing the OFDM technique, the wideband signal is divided into many narrowband

subcarriers, each experiencing flat fading rather than frequency-selective fading. Therefore,

we focus exclusively on flat fading channels in this thesis.

The performance of a wireless communication system relies heavily on the knowledge of

the channel state information (CSI) at the transmitter side (CSIT) and at the receiver side


(CSIR). Here, CSI refers to the information about the realization of the channel coefficient h.

CSI plays an important role in many sophisticated transmission techniques such as adaptive

modulation, linear precoding, beamforming, etc.

In both theory and practice, CSIR can be estimated to a certain level of accuracy by

sending training sequences known a priori to the receiver, provided that the channel does

not change significantly until the next training sequence is sent. Therefore, obtaining CSIR

is usually considered relatively easy; and we will always assume perfect CSIR in this thesis.

On the other hand, CSIT is usually obtained via dedicated feedback links. Due to prac-

tical limitations such as link capacity and delay constraint, obtaining arbitrarily accurate

feedback information is very difficult in most wireless systems, if not impossible; and in

most cases transmitters can only acquire coarsely quantized CSI through low-rate feedback

links. For a blind transmitter (with no CSIT), the channel coefficient is often modeled as a

random variable based on the collected statistics. A widely used statistical channel model

is the Rayleigh fading model [26], where the channel coefficient is modeled as a zero-mean

circularly symmetric complex Gaussian random variable. Rayleigh fading models a rich

scattering environment with a lot of reflection paths and no direct line-of-sight component.

Other frequently used fading models include the Rician fading model and the Nakagami

fading model [26].

2.2.2 Channel capacity and outage

Channel capacity is perhaps the most important information-theoretic limitation of reliable

communications over noisy channels, which is introduced by Claude Shannon in [21]. In

[21], Shannon proved the following fundamental result, known as Shannon’s channel coding

theorem: reliable communication between a transmitter and a receiver is possible if and

only if the transmission rate R is below a certain quantity called channel capacity, denoted

by C. Specifically, for every rate R < C, there exists a code such that the probability

of decoding error Pr{w 6= w} can be made arbitrarily small, provided that the codeword

length N is sufficiently large. We will refer to such a code as a capacity-achieving code.

Channel capacity is essentially the maximum rate of reliable communication supported by

the channel, which is also referred to as the maximum achievable rate.

Finding the exact expression of the capacity of a channel in its most general form has

been a longstanding problem for many channels such as the broadcast channel [28] and relay

channel [29]. The simplest and most well-understood channel is perhaps the discrete-time


additive white Gaussian noise (AWGN) channel

y[k] = x[k] + n[k], (2.5)

where x[k] and y[k] are the channel input and output at time k, respectively. n[k] is the

additive white Gaussian noise with an average power N0. Assuming a transmit power of P

joules per symbol, the capacity of the discrete-time AWGN channel is well-known to be [3]

C = log(1 + γ) bits per channel use (or bits/s/Hz), (2.6)

where γ , PN0

is the received SNR.1

Using the AWGN channel as a building block, we now introduce the capacity of the

point-to-point fading channel, which will be extensively used in the thesis.

Consider the flat fading model in (2.4). In the case of slow fading, the channel coefficient

h[k] is random but remains constant at all time k during the transmission of a codeword,

which can be expressed as

y[k] = h x[k] + n[k]. (2.7)

Slow fading channels are often studied in the framework of compound channels [24, 30],

where the transition probability pθ(y|x) is parameterized by θ ∈ Θ. For example, the slow,

flat fading channel in (2.7) can be viewed as a compound channel parameterized by the

channel coefficient h. Conditioned on the channel realization h, the fading channel becomes

an AWGN channel with received SNR |h|2PN0

. Define γ , PN0

as the average received SNR

for a normalized channel, i.e., the channel coefficient is normalized to have a variance of 1.

The maximum rate of reliable communication supported by the channel is thus

C(h) = log(1 + |h|2γ) bits per channel use. (2.8)

The following comments are in order.

1. C(h) is a function of the random channel coefficient h, which is thus also random.

Furthermore, since no rate is supported when h = 0 as C(h) = 0, strictly speaking,

slow fading channel has zero capacity as long as |h|2 is not bounded away from zero.

2. Assume CSIT is available, i.e., the transmitter knows h exactly. By Shannon’s chan-

nel coding theorem, the transmitter can encode data at any rate R < C(h) with

1Throughout this thesis, we always assume a logarithm of base 2 unless otherwise mentioned.


an arbitrarily small probability of error by choosing a proper code. Hence, reliable

communication is always possible.

3. Assume a blind transmitter with no CSIT, which encodes data at a rate R bits per

channel use. If the channel coefficient is such that C(h) < R, then the decoding error

probability cannot be made arbitrarily small no matter what channel codes are used.

In this case, the system is said to be in outage. The outage probability is defined to be

Pout = Pr {C(h) < R} = Pr{

log(1 + |h|2γ) < R}. (2.9)

To achieve this outage probability, the transmitter needs a code that can achieve

reliable communication over all channels whose channel coefficients h satisfy log(1 +

|h|2γ) > R. Such a code is said to be universal for the given class of channels. For the

slow fading channel in (2.7), the universal code design problem is known to be the same

as the code design problem for the weakest channel, that is, a capacity-achieving code

for the channel that is just strong enough to support the target rate R automatically

achieves reliable communication over all stronger channels [24].

In the case of fast fading, the decoding delay constraint is much longer than the coherence

time, i.e., the codeword may span multiple coherence periods. A commonly used model for

fast fading is the block fading model, where the channel coefficient h is assumed to be

constant during each coherence period (a block) and is i.i.d. across different coherence

periods. By coding over L such blocks, the maximum average supported rate is [24]

C(h1, · · · , hL) =1L

L∑l=1

log(1 + |hl|2γ), (2.10)

where hl is the channel realization in the lth block. As L → ∞, the capacity of the block

fading channel is therefore

C = E[log(1 + |h|2γ)

]. (2.11)

2.2.3 Diversity-multiplexing tradeoff

In Section 2.2.1 and Section 2.2.2, we reviewed some useful information-theoretic properties

of communication over a single-input single-output (SISO) wireless channel. By employing

multiple antennas at the transmitter and the receiver, we are able to build a multiple-input

multiple-output (MIMO) system, which is shown as a promising approach to improve the


performance of wireless communications in fading channels. Two special cases are the single-

input multiple-output (SIMO) system and the multiple-input single-output (MISO) system,

where only one antenna is used at the transmitter or the receiver.

Earlier works on multiple-antenna systems concentrated on using multiple antennas to

extract diversity to combat channel fading. It has been shown that, in a system with multiple

transmit and receive antennas, if the fading is independent across different antenna pairs,

more reliable transmission can be achieved by sending signals carrying the same information

through different paths [24]. Denote γ as the average received SNR. The diversity order

d defines the asymptotic decay rate of the error probability at high SNR (γ → ∞), i.e.,

the average error probability can be made to decay like γ−d, in contrast to the γ−1 for the

single-antenna fading channel [24]. In a system with m transmit and n receive antennas,

assuming the channels between individual antenna pairs are i.i.d. Rayleigh fading, the

maximum asymptotic decay rate, know as the diversity gain, is d∗ = mn [24]. Space-time

coding [31, 32, 33] is known as an effective technique to exploit such diversities in wireless

multiple-antenna systems.

The underlying idea in transmit or receive diversity is to average over multiple path

gains to combat channel fading. On the other hand, the multiple paths between individual

transmit-receive antenna pairs also increase the degrees of freedom available for communi-

cations [34, 35]. By transmitting independent information in parallel through the multiple

paths, the data rate of the system can be increased. This effect is referred to as spatial

multiplexing [24]. It has been shown in [35] that, for a system with m transmit antennas,

n receive antennas, and i.i.d. Rayleigh-fading links between each antenna pair, the capac-

ity of the system can be made to scale like min{m,n} log γ in the high-SNR regime, where

min{m,n} is the multiplexing gain. Many schemes have been proposed to exploit the spatial

multiplexing gain as well, e.g., the vertical Bell Labs space-time architecture (V-BLAST)

[36].

Having observed that the multiple-antenna system provides two types of gains and max-

imizing one type of gain may not necessarily maximize the other, a new perspective has

been put forth in [20] by Zheng and Tse, where it is shown that both diversity gain and

multiplexing gain can be simultaneously achieved, however, to achieve higher gain of one

type comes at the price of sacrificing the other. This essentially suggests that there is a fun-

damental tradeoff between the diversity gain and multiplexing gain, which is referred to as

the diversity-multiplexing gain tradeoff (DMT) [20]. The DMT concept not only brings new


insights into understanding the overall multiple-antenna system, but also provides an easy

way to compare the performance between diversity-based and multiplexing-based schemes.

The DMT analysis of multiple-antenna systems has since drawn much attention and has

become an active research area.

We now present the formal definitions of multiplexing gain and diversity gain from [20].

Definition 2.2.1. A scheme is said to achieve spatial multiplexing gain r and diversity gain

d if the data rate R satisfies

limγ→∞

R

log γ= r, (2.12)

and the average error probability Pe satisfies

− limγ→∞

logPelog γ

= d. (2.13)

For each r, define d∗(r) to be the supremum of the diversity advantage achieved over all

schemes. d∗(r) is referred to as the DMT curve, or simply DMT.

For flat, slow fading channels, let the data rate be R = r log γ. It has been shown that

the outage probability Pout satisfies [20]

d∗(r) = − limγ→∞

logPout

log γ, (2.14)

That is, the DMT d∗(r) is achieved by using any capacity-achieving code with rate R =

r log γ. Eq. (2.14) can also be written as Pout.= γ−d

∗(r), where “ .=” is the exponential

equality [20] (see also Table 1.1).

The successive refinablity of DMT

In [37], Diggavi and Tse considered the problem of transmitting multiple streams over a

wireless system. They studied the successive refinement of the DMT curves via diversity-

embedded codes such as the broadcast codes [38], with which all streams (layers) can si-

multaneously operate on the optimal DMT curve, i.e., all layers can achieve the optimal

diversity gains for any multiplexing gain allocation. The definition of the successive refine-

ment of DMT is given as follows:

Definition 2.2.2. Consider transmitting L streams simultaneously over a wireless system

at data rates R1, R2, · · · , RL. Let d∗(r) be the DMT of the wireless system. d∗(r) is said to


be successively refinable if there exists a scheme such that for the lth stream, the data rate

Rl satisfies

limγ→∞

Rllog γ

= rl, (2.15)

and the average error probability Pl satisfies

− limγ→∞

logPllog γ

= d∗(r1 + · · ·+ rl). (2.16)

That is, the optimal diversity gains (d∗(r1), d∗(r1 + r2), · · · , d∗(r1 + · · ·+ rL)) can be simul-

taneously achieved for any multiplexing gain allocation (r1, r2, · · · , rL).

The successive refinability is an important property when multiple levels of reliability

are desired in a single user channel. The successive refinability of the DMT curves for MISO

/ SIMO systems as well as certain single-relay cooperative systems has been established in

[37] and [4]. However, to the best of our knowledge, the successive refinements of the DMTs

for multi-relay cooperative systems is still not well understood, and will be investigated in

Chapter 3, Section 3.4.

2.3 Distortion Exponent

In Section 2.2, we introduced the fading models of wireless channels and information-

theoretic performance limitations of the wireless system measured by capacity, outage prob-

ability and the DMT. Among these performance measures, the DMT reveals that high data

rate and high reliability are two conflicting design parameters in wireless communication

systems. Hence, neither capacity nor outage probability alone gives a complete picture of

the performance of wireless systems. In this section, we introduce a concept called distortion

exponent, which reflects the combined effect of rate and reliability and is particularly useful

in characterizing the overall performance of wireless systems.

Inspired by the concept of diversity gains in [20], some researchers have recently applied

the ideas of the DMT to the problem of transmitting a discrete-time analog-amplitude source

over slow fading channels [39, 40, 41, 42]. As is discussed in Section 2.2.2, when the CSI

is not fully known to the transmitter, transmissions may suffer from the decoding outage

effect, i.e., the receiver may not be able to decode the received data if the channel is in deep

fading. This also results in larger distortions in the reconstructed sources. Furthermore, it

is known that Shannon’s source-channel separation theorem [3] in general requires full CSI


to be known at the transmitter. Therefore, designing source and channel codes separately

does not necessarily lead to optimal performance for such systems and a joint source-channel

approach is in general needed, for example, the sophisticated hybrid digital-analog (HDA)

coding scheme [43, 44, 45] and the broadcast strategy [28, 46, 38].

Note that an outage occurs when the data rate is greater than the maximum supported

rate for reliable communication over the channel. On the other hand, by the distortion-

rate function, a higher data rate allows the data compression method to achieve a smaller

distortion in the reconstructed source. These two observations confirm the conflict between

high data rate and high reliability (low outage probability) in a wireless system as suggested

by the DMT analysis. Accordingly, an end-to-end performance measure is needed for the

overall system.

It was first proposed in [39] to use the distortion exponent as a performance measure of a

communication system. The distortion exponent is defined to be the asymptotic exponential

decay rate of the expected end-to-end distortion in the high-SNR regime as follows [39]:

Definition 2.3.1 (Distortion exponent). Consider communicating a source signal over wire-

less channels. Let D be the mean-squared error distortion between the source signal and its

reconstruction at the destination. Let γ be the average received SNR. The distortion exponent

of the reconstructed source is defined to be

∆ = − limγ→∞

logDlog γ

. (2.17)

The distortion exponent ∆ is essentially the slope of the expected end-to-end distortion

on a log-log scale at high SNR. A larger distortion exponent reflects a fast decay rate of the

end-to-end distortion and accordingly a more efficient coding and transmission scheme.

Distortion exponent analysis for joint source-channel transmission over wireless channels

has since raised much interest in the research community. The distortion exponent is shown

to be closely related to the DMT analysis. Various results have been reported for the

distortion exponent of transmission over MIMO fading channels [47, 48, 49, 50] and fading

relay channels [4, 5]. In [6], Seddik et al. studied the distortion exponent of a multi-hop

and multi-relay system with main focus on the two-relay system and two-description source

coding. In [7], we studied the distortion exponents of multi-relay cooperative networks with

an arbitrary number of relays. Our recent works [11, 12, 13] extend the distortion exponent

analysis to two-way fading relay channels.


In a single-user system, the distortion exponent ∆ can be expressed as a function of the

bandwidth ratio b, which is defined as the ratio between the channel bandwidth and the

source bandwidth. Note that a smaller b suggests a more efficient coding scheme since it

consumes less channel bandwidth to transmit a given source. Whereas distortion exponent

analysis [47, 48, 4, 49] reveal that larger bandwidth ratio is in general necessary in achieving

larger distortion exponent. This hence reflects a tradeoff between the distortion exponent

and the bandwidth ratio in communication systems.

The distortion exponent analysis provides new insight into the problem of joint source-

channel transmission over fading channels. For example, for transmission over MIMO fading

channels, [48] shows that a simple HDA scheme is sufficient to achieve the optimal distor-

tion exponent at very small bandwidth ratios while [49] shows that the broadcast strategy

proposed in [46] is optimal in the distortion exponent sense for a range of sufficiently large

bandwidth ratios. It is further shown in [50] that, with quantized channel state informa-

tion at the transmitter the achievable distortion exponent can be significantly improved.

These schemes have also been extended to single-relay fading channels in [4, 5]. However,

to the best of our knowledge, the distortion exponent of multi-relay cooperative networks

remains uninvestigated in general (besides the special case of the two-relay system with

two-description source coding in [6]), which will be one of the main topics of this thesis.

2.4 Cooperative Communications and Relay Channels

Due to certain practical limitations such as the device size, deploying multiple antennas at

the same node of a communication link may sometimes be difficult or even infeasible. In

order to overcome this limitation, a promising technique called cooperative communication

is proposed in [1, 2] to facilitate robust transmission and provide higher throughput in

both mobile cellular systems and wireless ad hoc networks. Cooperative communications

exploit the spatial diversity inherent in multiuser systems by allowing users to cooperate and

relay others’ messages. In this way, virtual antenna arrays are formed and each message is

passed through multiple independent links and thus significantly increases the transmission

reliability.

Early works related to cooperative communications were studied in the context of relay

channels [51, 29], where a source node communicates with a destination node with the help

of a relay node. The relay node is fully devoted to helping the source node and has no its


own information to send. Despite its simplicity, the capacity of such a general relay channel

remains unknown. Recently, cooperative communication has gained increasing popularity

in the research community and has been extensively investigated in [1, 2, 52, 53, 54, 55, 56].

In the literature, cooperative systems or relaying systems can be categorized to be either

half-duplex, where the relay cannot transmit and receive at the same time, or full-duplex,

where the relay can transmit and receive simultaneously.

We first introduce three basic relaying schemes that can be employed at the relay:

1) Amplify-and-forward (AF) [53]: the relay amplifies its received signal on a symbol-

by-symbol basis, and forwards it to the destination. The AF scheme does not require

additional computations at the relay, hence has very low complexity. However, the

noise in the received signal at the relay is also forwarded in the relaying stage.

2) Decode-and-forward (DF) [53]: the relay attempts to fully decode its received signal.

If the decoding is successful, the signal is then re-encoded using a possibly different

codebook, and is sent to the destination. Otherwise, the relay remains silent. The DF

scheme requires full decoding and hence has high complexity. However, the noise at

the relay can be completely eliminated upon successful decoding.

3) Compress-and-forward (CF) [29]: different from the AF and DF schemes, the relay

does not perform full decoding, nor does it simply amplify the received signal. Instead,

it transmits compressed (quantized) versions of the received signals to the destination.

The destination decodes the message using the Wyner-Ziv coding mechanism [57].

The CF scheme is one of the fundamental coding strategies for relay channels, which

is shown to achieve the capacity region of the degraded relay channel by Cover and

El Gamal [29].

All cooperation protocols considered here involve two-phase communication, where the

source broadcasts to the relay(s) and the destination in the first phase (the broadcast phase),

and the relay(s) and/or the source transmit to the destination in the second phase (the

cooperation phase). A protocol is said to be orthogonal if the source node does not transmit

to the destination in the cooperation phase, otherwise the protocol is nonorthgonal.

Depending on the time duration for each node to participate in the communication,

cooperation protocols can also be categorized into static protocols and dynamic protocols.

In a static protocol, the transmission time of each node is independent of the channel


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Multiplexing gain, r

Div

ersi

ty g

ain,

d

MISO upper bound [14]repetition codes [33]space−time codes [33] /relay selection [43]NAF [36]slotted SAF [46]ODF [41]NDF [41]DDF [36]

Figure 2.3: Diversity-multiplexing tradeoffs of various cooperation protocols.

coefficients. Whereas in a dynamic protocol such as the dynamic DF (DDF) protocol [55],

the relay listens to the source until the received information is sufficient to ensure successful

decoding, which can be realized by using rateless codes, for example, LT codes [58] or Raptor

codes [59]. The duration of transmission in the DDF protocol hence is dynamic and depends

on the actual source-relay channel strength. In addition, a protocol is referred to as a fixed

protocol if the transmission time of each node is independent of its transmission rate, and

is said to be a variable protocol otherwise.

As multiple-antenna systems, the performance of a cooperative system can also be char-

acterized by its DMT curve. As an example from Table I and Table II in [60], we plot the

DMTs achieved by some representative cooperation protocols for a two-relay cooperative

system in Fig. 2.3. Their DMTs are also compared with the transmit diversity upper bound,

i.e., the DMT of a 3 × 1 MISO system. In the following, we give a brief introduction of

these cooperation protocols. More detailed discussions will be given along with the analysis

in Chapter 3 and Chapter 4.

Laneman and Wornell [52] proposed to achieve multi-user cooperative diversity through

two fixed, static, and orthogonal protocols based on repetition coding and distributed space-

time coding, respectively. In the repetition-based scheme, each relay takes turns to forward

the signal received from the source node while in the space-time-coding-based scheme, all


participating relays decode the received signal and reencode it using a distributed space-time

code and transmit simultaneously to the destination. The distributed space-time coding

based cooperation protocol has also been studied in [61] with AF relaying. A simple yet

effective cooperation scheme called relay selection is proposed in [62, 63, 64] for multi-relay

cooperative systems. Instead of letting all relays participate in the cooperation as in [52],

only the “best” relay is chosen to forward the information sent by the source, which hence

makes efficient use of network resources and simplifies the system design. However, it can

be seen from Fig. 2.3 that although the schemes proposed in [52] and [62] can achieve the

maximum possible diversity gain of the system, they are not optimal in the high multiplexing

gain regime. In fact, it is clear that these simple schemes are sub-optimal except at zero

multiplexing gain.

The nonorthogonal amplify-and-forward (NAF) protocol was proposed by Nabar et al. in

[54], which allows the source node to continue to transmit in the cooperation phase. In [55],

it is shown that the NAF protocol has improved DMT performance when compared with the

orthogonal protocols presented in [52]. Yang and Belfiore [65] studied the slotted amplify-

and-forward (SAF) protocol and proposed a sequential SAF scheme that further exploits

the diversity gain in the high multiplexing gain regime. The sequential SAF is shown to

outperform the NAF scheme for the case of two relays. It also approaches the transmit

diversity bound under the relay isolation assumption as the number of transmission slots

increases. In a recent work [60], Elia et al. studied both the variable and orthogonal selection

DF protocol (ODF) and the variable and nonorthogonal selection DF protocol (NDF) by

optimally choosing the time durations of the broadcast and cooperative phases according to

the transmission rates. As shown in Fig. 2.3, these variable and/or nonorthgonal protocols

achieve better DMTs than that of the distributed space-time coding based scheme or the

relay-selection based scheme, especially in the high multiplexing gain regime.

All the aforementioned cooperation protocols are static protocols. The DDF protocol

proposed in [55] employs rateless codes at the source node and requires one acknowledgement

bit from the relay node to inform the source when successful decoding is performed at the

relay. The DDF protocol is strictly optimal in terms of the DMT for all multiplexing gains

less than 1m+1 when m relays present, which is illustrated in Fig. 2.3 for the case m = 2.

While it is not clear whether DDF is still optimal or not for higher multiplexing gains,

there is no known scheme that outperforms DDF under the same CSI assumption [60]. An

enhanced DDF protocol (E-DDF) is proposed in [66], where the relay is required to decode


only a part of the source message. Under a much more relaxed assumption that the relay

knows full CSI of all links, it is shown in [67] that CF relaying is DMT-optimal at all

multiplexing gains.

2.5 Two-way Communications and Bidirectional Relaying

Two-way communication is a classical concept [68], where a pair of users on both sides of a

communication link have independent messages to transmit to each other. Recently, there

has been a growing interest in the field of two-way relaying communications, also known

as bidirectional relaying [69, 70, 71, 72, 73], where two users communicate simultaneously

in both directions with the help of one intermediate relay. Instead of simply letting the

relay take turns to forward each user’s information as in traditional one-way cooperative

relaying, intelligent processing is performed at both the source and the relay nodes to control

the interferences and improve the system throughput.

There are two major classes of two-way relaying protocols, namely, the two-phase pro-

tocol, also known as the multiple-access broadcast (MABC) protocol [74, 75, 76], and the

three-phase protocol, also known as the time-division broadcast (TDBC) protocol [75, 76].

We demonstrate by an example the transmission procedure of the two-way relaying com-

munication in a three-node half-duplex system in Fig. 2.4. Note that in this example we

assume there are no direct links between the two users so that they cannot communicate

directly and all messages must be forwarded by the relay. However, both MABC and TDBC

protocols apply to the case where there are direct links between the two users as well.

As illustrated in Fig. 2.4 (a), to complete one round of two-way relaying communi-

cation, the traditional one-way relaying strategy requires four transmission phases. The

transmissions in the one-way relaying strategy are all orthogonal, i.e., there is no signal

collision at any receivers, which, however, also limits the system throughput. To circumvent

this limitation, the MABC protocol and the TDBC protocol reduce the number of required

transmission phases to two and three, respectively, by introducing controlled interference

into the transmitted signals. More precisely, in the MABC protocol, both source nodes

transmit simultaneously to the relay in the first phase, and the relay processes its received

signal and broadcasts back to both source nodes in the second phase; in the TDBC protocol

each user uses one of the first two phases to transmit to the relay and/or the other user (de-

pending on whether direct links present or not), and the relay broadcasts back in the third


RelayUser 1 User 2

1T 3T2T

1T 3T2T

1T 3T2T

1T 3T2T

Phase 1

Phase 2

Phase 3

Phase 4

RelayUser 1 User 2

1T 3T2T

1T 3T2T

1T 3T2T

1T 3T2T

Phase 1

Phase 2

Phase 3

Phase 4

(a)RelayUser 1 User 2

1T3T 2T

1T 3T 2T

Phase 1

Phase 2

RelayUser 1 User 2

1T3T 2T

1T 3T 2T

Phase 1

Phase 2

(b) RelayUser 1 User 2

1T3T 2T

1T 3T 2T

1T3T 2T

Phase 1

Phase 2

Phase 3

RelayUser 1 User 2

1T3T 2T

1T 3T 2T

1T3T 2T

Phase 1

Phase 2

Phase 3

(c)

Figure 2.4: Transmission phases of a two-way relaying system with (a) one-way relayingstrategy, (b) the MABC protocol, (c) the TDBC protocol.


phase. The transmissions in each phase of the MABC protocol and the TDBC protocol are

illustrated in Fig. 2.4 (b) and Fig. 2.4 (c), respectively. It is not hard to see that due to

the half-duplex constraint, the MABC protocols cannot utilize the direct links between the

sources as in the TDBC protocols even such links exist. However, since the MABC protocol

only has two transmission phases, it may still be preferred when the bandwidth resource

is limited. Moreover, neither user 1 nor user 2 is able to transmit during the last phase

in the MABC protocol or in the TDBC protocol, which implies an inherent orthogonality

of the half-duplex two-way relaying communication. Details of the MABC and the TDBC

protocols with AF, DF or CF relaying will be studied in more details in Sec. 5.4 and Sec.

5.5, respectively.

Many efficient protocols have been proposed and investigated for two-way relaying coop-

erative networks. For example, Rankov and Wittneben studied the AF-based and DF-based

MABC protocols in [74], which are shown to effectively mitigate the loss in spectral effi-

ciency caused by the half-duplex constraint. The AF-based two-way relaying protocol is

studied from the analog network coding perspective in [77, 78], where network coding [79]

is performed to physical signals directly in the wireless channel. In [75], Kim et al. investi-

gated three DF-based two-way relaying protocols, where the physical layer network coding

[80, 81, 82] is performed to combine the decoded messages at the relay node to achieve

higher spectral efficiency. A feedback-based DF protocol is proposed in [83] and [84], where

quantized CSI is obtained at the transmitters through feedback links. In [85] and [86],

the CF-based two-way relaying protocols are studied in terms of the information-theoretic

achievable rates.

Although originated from a classical setup, the two-way relaying system is still not well

understood in general. For example, the capacity region of the general two-way relay chan-

nel remains unknown. In terms of the DMT analysis, while the DMTs of one-way relaying

cooperative systems have been extensively studied in the literature (see the comprehensive

summary in [60]), there have been only sporadic results reported on two-way relaying coop-

erative system due to its much more involved multiuser nature. In [76], the authors studied

the finite-SNR DMT of the AF-based two-way relaying protocols, which characterizes the

tradeoff between the system outage probability and the sum rate. The DMT of the DF-

based two-way relaying protocols have been studied in [83] and [84], where the two sources

are assumed to transmit at the same rate. In [87], the DMT of the CF-based two-way

relaying protocol is studied, where the DMT upper bound of the half-duplex two-way CF


relaying system is obtained. A dynamic CF protocol is also studied in [88], which is shown

to achieve the optimal DMT of a MIMO full-duplex two-way relay channel. As an important

addition to the DMT analysis, we will study in this thesis the DMT of half-duplex two-way

relaying networks with various AF/DF/CF based cooperation protocols, based on which we

will also derive the distortion exponent of the two-way relaying systems.

2.6 Distortion Minimization of Joint Source-Channel Trans-

mission in Fading Channels

The distortion exponent is a useful tool for analyzing the overall system performance and

comparing various transmission strategies. However, it only reflects the asymptotic behavior

of the source distortion and ignores any scaling in power and rate due to the limited scope

of the high-SNR regime. For a general system and an arbitrary SNR, one still has to solve

the associated distortion minimization problem to fully characterize the optimal design

parameters, e.g., the power and rate allocation. In this thesis, we will also investigate the

distortion minimization of layer-coded sources transmitted over SISO fading channels, which

is a simple yet representative example and also of practical relevance.

Successive refinement of information [89, 90] is a rate-distortion mechanism that approx-

imates the source data in a progressive manner, i.e., the data is first approximated by a

few bits of information, the accuracy of the approximation is improved as more information

becomes available. Therefore, the more information is provided, the less distortion is ob-

served in the reconstructed source. The source is successively refinable if a rate-distortion

optimal description of the data can be obtained at any approximation stage. That is, if a

source is first described at rate R1, and then refined at rate R2, the minimum achievable

distortion is the same as if the source were initially described at rate R1 +R2. It is shown in

[89] that Gaussian sources are successively refinable with squared-error distortion. Recently,

[91] shows that all i.i.d. sources are successively refinable under the squared-error distortion

measure with a limited constant rate loss.

An example of utilizing the successive refinement property in source coding is the lay-

ered source coding, where a coarse description is coded in the first layer, and each following

layer contains the refinment information of the preceding layers. The layered source cod-

ing mechanism has been widely applied in practical applications for image and video data

compressions, for example, the embedded zerotree wavelet algorithm (EZW) [92] and the


Decodable

layers

Virtual receiver

Virtual receiver

Virtual receiver

Transmitter

Layer MM PR ,

Source

Layer 11, PR

s

11,γR

jjR γ,

MMR γ,

j

M

M

1

1Decodable

layers

Virtual receiver

Virtual receiver

Virtual receiver

Transmitter

Layer MM PR ,

Source

Layer 11, PR

s

11,γR

jjR γ,

MMR γ,

j

M

M

1

1

Virtual receiver

Virtual receiver

Virtual receiver

Transmitter

Layer MM PR ,

Source

Layer 11, PR

s

11,γR

jjR γ,

MMR γ,

j

M

M

1

1

Figure 2.5: Layered source coding with broadcast strategy.

set partitioning in hierarchical trees (SPIHT) algorithm [93] for still image coding, the em-

bedded block coding with optimized truncation (EBCOT) [94] in JPEG2000 image coding

standard [95], the 3-D SPIHT video coding algorithm [96], the fine granularity scalability in

MPEG-4 video coding standard [97] and the scalable extension of H.264/AVC video coding

standard [98]. The layered source coding framework is a natural choice for source transmis-

sion over fading channels, as the amount of received information (coded layers) is dictated

by the channel realization, which allows a progressive quality improvement of the recon-

structed data. The successive refinement property of layered source coding also suggests

unequal error protection (UEP) [99, 100, 101] on the transmitted layers, i.e., the layers that

convey important information need to be protected the most and are decodable under the

most severe fading, while layers that carry refinement information are protected less and

decodable when the fading is less severe.

The broadcast strategy, also known as superposition coding [46, 38], is an effective

approach to transmit layer-coded sources over fading channels when the CSI is unknown

at the transmitter. In the broadcast strategy, the transmitter views the actual receiver

as a number of virtual receivers. The communication channel therefore can be modeled

as a broadcast channel (BC) as illustrated in Fig. 2.5. The virtual receivers are ordered

according to their channel strengths. For example, each virtual receiver in a single-antenna

SISO fading channel is associated with a fading state, and the channel strength is given by

the corresponding channel power gain [102].

In the broadcast strategy, the source signal is layer-coded into M layers where layer i

carries the refinement information of its preceding layer i − 1 and is intended for the ith


virtual receiver, i = 1, · · · ,M . Each source layer is then coded using an independent channel

codebook and assigned a transmit power. More precisely, the symbol of the ith layer, xi, is

coded by a channel code of rate Ri and is transmitted with a power of Pi. All layers are

superimposed and transmitted simultaneously to the virtual receivers. The superimposed

transmitted symbol at time k is

x[k] =M∑i=1

√Pixi[k]. (2.18)

The decoder performs successive decoding. That is, the layers are decoded in order

starting from the first layer (the base layer) by treating the rest layers as noise. The decoded

layer is subtracted from the received signal before decoding the next layer. The decoding

procedure continues until the decoder fails to decode one layer (layer j in Fig. 2.5). That

layer and all the subsequent layers will be declared in outage. The maximum number of

layers successfully decoded thus depends on the actual channel fading realization.

In order to minimize the expected distortion at the receiver, it is essential to find the

optimal power allocation {Pi} and the rate allocation {Ri}, which suggests a cross-layer

design approach. The problem of the end-to-end distortion minimization for transmitting a

layer-coded source using the broadcast strategy was initially studied by Sesia et al. in [103]

for SISO fading channels with continuous fading distributions. The optimization problem

involves quantizing the continuous fading distribution into discrete fading states, known

as channel discretization, and an algorithm was proposed for the case where each layer is

assumed to have equal rate. Etemadi and Jafarkhani [104] proposed an iterative algorithm

that allows unequal rate allocation by separating the optimization problem into the rate

allocation subproblem and the channel discretization subproblem. Their solution for the

rate allocation subproblem involves exhaustive search over the space of coding rates, which

thus provides near-optimal performance. However, this also results in high computational

complexity that grows with the size of the search space.

Recently, a recursive algorithm is proposed in [105] and [102] to solve this optimization

problem under the assumption that the fading distribution is discrete and known at the

transmitter. The worst case complexity of their algorithm is of O(2M ), where M is the

number of fading states. A more efficient algorithm is proposed in [106], which finds the

optimal power allocation in linear time O(M). The solution obtained by [102] or [106]

is globally optimal when the fading distribution is discrete. However, to the best of our


knowledge, efficient algorithms that optimize the distortion of broadcast strategy in fading

channels with continuous fading distributions are still unknown.

Chapter 3

Distortion Exponents of

Multi-relay Cooperative Networks

3.1 Introduction

In this chapter, we extend the distortion exponent analysis to multi-relay cooperative net-

works. We derive the distortion exponents when the layered source coding with progressive

transmission or the broadcast strategy is used in multi-relay networks under three orthogonal

cooperation protocols, based on repetition coding [52], relay selection [62], and space-time

coding [52], respectively. Our analyses reveal the impacts of the number of relays, band-

width ratio and cooperation protocol on the distortion exponent. As an important addition

to the DMT theory, we also prove the successive refinability of the diversity-multiplexing

tradeoffs of the three multi-relay cooperation protocols.

Although it is well known that nonorthogonal cooperation schemes such as the NAF

protocol [54] and the SAF protocol [65] are usually more efficient than the orthogonal

cooperation schemes, the derivation of the optimal distortion exponents for nonorthogonal

protocols remains an open problem, and it is still not clear how to effectively combine the

layered source coding based schemes such as the broadcast strategy with these cooperation

protocols even in the single-relay case [4]. Hence, our main focus here is still the orthogonal

protocols as in [52] and [62]. Later in Chapter 4, we will show that with only a few bits

of feedback information, the nonorthogonal protocols can be efficiently combined with the

single-rate source and channel coding to offer improved performance.

31

CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 32

1,sh

S D

1

dh ,1

msh , dmh ,

dsh ,

dh ,22,sh2

m

Relay

Source Destination

1,sh

S D

1

dh ,1

msh , dmh ,

dsh ,

dh ,22,sh2

m

Relay

Source Destination

Figure 3.1: System model of an m-relay cooperative system.

This chapter is organized as follows: In Section 3.2, we present the system model. After

that, we combine the three multi-relay cooperation protocols with layered source coding

using the progressive transmission and the broadcast strategy, respectively. The achievable

distortion exponents of these schemes are derived in Section 3.3 and Section 3.4, respectively.

The performance comparisons are given in Section 3.5, followed by the summary in Section

3.6.

3.2 System Model

We consider a wireless communication system where a source transmits information to a

destination with the help of m relays. All nodes are equipped with single antenna. The

system model is shown in Fig. 3.1. Each relay is half-duplex and employs the AF or DF

relaying protocol [53]. We assume no CSIT and perfect CSIR at all nodes.

The source {sk}∞k=1 is assumed to be zero-mean, unit-variance, i.i.d. complex Gaussian.

With layered source coding, a block of K source samples are encoded into n layers, which

are then transmitted in N channel uses. Let Rj bits per channel be the channel code rate

of the jth layer. The bandwidth ratio is b = N/K, which corresponds to a source code

rate of bRj bits per sample for the jth layer. We assume that K is large enough to design

source codes that can approach the rate-distortion bound of the source signal1, and N is

large enough to design a fixed-rate channel code that can be transmitted reliably if the

1In fact, since our main interest is the distortion exponent, it is not always necessary for K to be largeenough to approach the rate-distortion bound. A smaller K that achieves larger distortions with the sameexponential decay rate will also suffice.


instantaneous capacity is greater than the communication rate. We further assume that

the channel is in flat, slow fading so that the channel gain is random but remains constant

during all N channel uses, which is also known as the quasi-static scenario [24].

The channel is assumed to be Rayleigh fading and statistically symmetric, i.e., the

channel coefficients hs,d, hs,i and hi,d, i = 1, ...,m are i.i.d. complex Gaussian random

variables with zero mean and unit variance. The additive noise at each node is modeled

as zero-mean unit-variance circularly symmetric complex Gaussian (CN (0, 1)). We assume

all nodes have the same transmitting power, and denote the average received SNR at the

destination node to be γ. Although practical channels are usually asymmetric, it has been

pointed out in [4] and [5] that, due to the high SNR assumption, considering a symmetric

system does not affect the asymptotic exponent results.

We consider three multi-relay cooperation protocols, namely repetition-based coopera-

tion (RP), relay-selection-based cooperation (RS), and space-time-coded cooperation (ST).

The DF relaying is used in RS and ST, whereas RP uses either AF or DF relaying.

The DMTs of an m-relay cooperative system with these cooperation protocols are given

by [52], [62] and [54], respectively, as follows,

d∗RP (r) = (m+ 1)(1− (m+ 1)r)+, (3.1)

d∗RS(r) = (m+ 1)(1− 2r)+, (3.2)

d∗ST (r) = (m+ 1)(1− 2r)+, (3.3)

where x+ , max{x, 0}.

3.3 Distortion Exponents of Layered Source Coding with Pro-

gressive Transmission

We first study the distortion exponent of the layered source coding with progressive trans-

mission (LS) for an m-relay cooperative system. To transmit the layer-coded source over

an m-relay cooperative network, the LS strategy is combined with the RP, RS or ST co-

operation protocol. Each coded layer is transmitted sequentially, i.e., with progressive

transmission. It is not hard to see that in the LS strategy, each layer is still transmitted

in the same way as a conventional single-layer signal, with the only difference being that

instead of consuming all N channel uses, each layer is now allocated a fraction tj of total


Layer 1 1R

channel uses:

Source Relay 1 Relay m

Layer 1 1R Layer 1 1R Layer n nR Layer n nR Layer n nR


1+m

Ntn

1+m

Ntn

1+m

Ntn

11

+m

Nt

11

+m

Nt

11

+m

Nt

Layer 1 1R

channel uses:


Layer 1 1R Layer 1 1R Layer n nR Layer n nR Layer n nR


1+m

Ntn

1+m

Ntn

1+m

Ntn

11

+m

Nt

11

+m

Nt

11

+m

Nt

(a)

Layer 1 1R

channel uses:

Source Selected Relay

Layer 1 1R Layer 2 2R Layer 2 2R Layer n nR Layer n nR

Source Source

2/Ntn 2/Ntn2/1Nt 2/1Nt 2/2Nt 2/2Nt

Selected Relay Selected Relay

Layer 1 1R

channel uses:

Source Selected Relay


Source Source

2/Ntn 2/Ntn2/1Nt 2/1Nt 2/2Nt 2/2Nt

Selected Relay Selected Relay

(b)

Layer 1 1R

channel uses:

Source All Relays All Relays


Source Source All Relays

2/1Nt 2/1Nt 2/2Nt 2/2Nt 2/Ntn 2/Ntn

Layer 1 1R

channel uses:

Source All Relays All Relays


Source Source All Relays

2/1Nt 2/1Nt 2/2Nt 2/2Nt 2/Ntn 2/Ntn

(c)

Figure 3.2: Layered source coding with progressive transmission using (a) repetition-basedcooperation, (b) relay-selection-based cooperation, (c) distributed space-time-coded coop-eration.

channel uses, tj ≥ 0,∑n

j=1 tj = 1. Therefore, the LS strategy can be easily incorporated

into any wireless systems without changing the transmission protocols. Note that this is a

major difference and also an advantage over the broadcast strategy that will be studied in

Section 3.4. Meanwhile, the channel allocation fractions t1, · · · , tn offer additional degrees

of freedom, which can be optimized to improve the performance.

The three transmission strategies, namely LS with RP, LS with RS, and LS with ST

are illustrated in Fig. 3.2, where the source transmits a layer to the destination and all

relays first; the participating relay(s) then forwards the received layer to the destination.

The procedure repeats until all n layers have been transmitted. More precisely, in the RP-

based scheme, each relay takes turns to forward the signal received from the source node;

in the RS-based scheme, only one selected relay forwards the received signal; in the ST-

based scheme, all participating relays decode the received signal and reencode it using a

distributed space-time code and transmit simultaneously to the destination.


Denote the outage probability of layer j as P jout. We can write the expected end-to-end

distortion as follows

D =n∑j=0

(P j+1

out − Pjout

)2−b

∑jk=1 tkRk , (3.4)

where we define P 0out = 0 and Pn+1

out = 1. Note that in the layered source coding, if layer j

is lost, layers j + 1, · · · , n all become useless. Hence, the transmission rates have to satisfy

Rj < Rj+1 so that P jout < P j+1out .

Let Rj = rj log γ, where rj is the multiplexing gain. By (2.14), we can write P jout.=

γ−d∗(rj). The expected end-to-end distortion is then

D.=

n∑j=0

[γ−d

∗(rj+1) − γ−d∗(rj)]· γ−b

∑jk=1 tkrk

.=n∑j=0

γ−d∗(rj+1)−b

∑jk=1 tkrk

.= γ−min0≤j≤n{d∗(rj+1)+b∑jk=1 tkrk},

(3.5)

where the second and third exponential equalities are due to the fact that the summation

is dominated by the term with the slowest decay (largest exponent) at high SNR. For the

second dot equality to hold, we also enforce the constraint rj < rj+1 such that d∗(rj+1) <

d∗(rj).

The maximum distortion exponent of the LS strategy with n-layer source coding and a

DMT d∗(r) can now be obtained by solving the following optimization problem

∆n = maxrj ,tj

min0≤j≤n

{d∗(rj+1) + b

j∑k=1

tkrk

}s.t. 0 ≤ r1 < r2 < · · · < rn ≤ 1,

t1 + t2 + · · ·+ tn = 1,

tj ≥ 0, j = 1, 2, · · · , n.

(3.6)

Note that given bandwidth ratio b, knowing the DMT curve d∗(r) is sufficient to determine

the optimal achievable distortion exponent for the LS strategy no matter the underlying

system has single relay or multiple relays. This also suggests that the distortion exponent of

the multi-relay system can be directly obtained from the results of the single-relay system

[4].


As shown in (3.1), (3.2) and (3.3), the DMT curves of all the three protocols share the

same linear form d∗(r) = a − cr, with a = m + 1, c = (m + 1)2 for the RP-based scheme

and a = m+ 1, c = 2(m+ 1) for the RS and ST based schemes. We can thus directly apply

the following lemma from [49], which gives the distortion exponent of the LS strategy with

a general linear DMT.

Lemma 3.3.1 ([49]). Consider the source transmission over a wireless system with DMT

d∗(r) = a− cr for r ∈ (0, a/c) and some a > 0 and c > 0. The optimal distortion exponent

achieved by the LS strategy with n-layer source coding and bandwidth ratio b is given by

∆n = a

(1−

(c

c+ bn

)n). (3.7)

and in the limit of infinite layers (n→∞)

∆∞ = a(1− e−b/c). (3.8)

Applying Lemma 3.3.1 with the DMTs in (3.1), (3.2) and (3.3), the distortion exponents

of the LS strategy with RP, RS and ST protocols for n-layer source coding are found to be

∆nLS−RP = (m+ 1)

(1−

((m+ 1)2

(m+ 1)2 + bn

)n), (3.9)

∆nLS−RS = (m+ 1)

(1−

(2(m+ 1)

2(m+ 1) + bn

)n), (3.10)

∆nLS−ST = (m+ 1)

(1−

(2(m+ 1)

2(m+ 1) + bn

)n), (3.11)

(3.12)

and in the limit of infinite layers (n→∞)

∆∞LS−RP = (m+ 1)(1− e−b

(m+1)2 ), (3.13)

∆∞LS−RS = (m+ 1)(1− e−b

2(m+1) ), (3.14)

∆∞LS−ST = (m+ 1)(1− e−b

2(m+1) ). (3.15)

3.4 Distortion Exponents of Broadcast Strategy

In this section, we consider the broadcast strategy [38] for an m-relay cooperative system.

In the broadcast strategy, n coded layers are superimposed and transmitted simultaneously.


The superimposed signal is denoted as x =∑n

j=1√γjxj , where xj is the coded channel

symbol of the jth layer, γj is the power allocated to layer j.

The decoder performs successive decoding [38]. That is, the layers are decoded in order

starting from the first layer. The decoded layer is subtracted from the received signal before

decoding the next layer. The decoding procedure continues until the decoder fails to decode

one layer. That layer and all the subsequent layers will be declared in outage.

To transmit the superimposed signal x over an m-relay cooperative network, the broad-

cast strategy is combined with the RP, RS, and ST multi-relay cooperation protocols. In

protocols with AF relaying, each relay amplifies and repeats its received signal, whereas in

protocols with DF relaying, each participating relay tries to decode and forward as many

layers as possible using successive decoding. We denote in the DF relaying the set of relays

at which layer j is successfully decoded to be Dj = {ijk}, ijk ∈ {1, 2, · · · ,m}, k = 1, . . . ,Nj ,

where Nj is the cardinality of Dj . We refer to Dj as the decoding set of the jth layer.

Note that a special case is Dj = ∅, for which no cooperation for the jth layer is available.

The three schemes, namely broadcast strategy with RP, broadcast strategy with RS, and

broadcast strategy with ST are illustrated in Fig. 3.3, where Rj and Dj are the coding rate

and the decoding set of the jth layer, respectively. i∗ is the “best” relay that is chosen to

participate in the RS-based cooperation. The selection criterion will be discussed in more

details in Section 3.4.2.

In the broadcast strategy, the successive decoding diversity gain is required to char-

acterize the achievable distortion exponent [4]. Define Ojd to be the set of channel states

h = (hs,d, {hs,i}mi=1, {hi,d}mi=1) for which layer j is the first layer in outage at the destination,

i.e., all layers before layer j can be decoded. We refer to Ojd as the set of conditional outage

events, or the conditional outage set of layer j. The corresponding conditional outage prob-

ability is P jd , Pr{h ∈ Ojd}. Due to successive decoding, the overall set of channel states

that result in an outage of layer j at the destination can be written as Ojd =⋃jk=1O

kd . We

refer to Ojd as the overall outage set of layer j. The overall outage probability for layer j at

the destination d is therefore P jd , Pr{h ∈ Ojd}.The successive decoding diversity gain for layer j with broadcast strategy is then [4]

dSD(rj) = − limγ→∞

log P jdlog γ

, (3.16)

where rj is the multiplexing gain of the jth layer. Let d∗(r) be the DMT of the underlying

cooperation protocol. d∗(r) is successively refinable if dSD(rj) = d∗(rj) [37], where rj ,


Layer 1 11, γR Layer 1 11, γR Layer 1 11, γR

Layer n nnR γ, Layer n nnR γ,Layer n nnR γ,

N/(m+1) channel uses N/(m+1) channel uses N/(m+1) channel uses


Layer 1 11, γR Layer 1 11, γR Layer 1 11, γR

Layer n nnR γ, Layer n nnR γ,Layer n nnR γ,

N/(m+1) channel uses N/(m+1) channel uses N/(m+1) channel uses


(a)

Layer 1 11, γR Layer 1 11, γR

Layer n nnR γ,Layer n nnR γ,

N/2 channel uses N/2 channel uses

Source Selected Relay i*




Source Selected Relay i*

(b)




Source Relay(s) in D1

Relay(s) in DnSource




Source Relay(s) in D1

Relay(s) in DnSource

(c)

Figure 3.3: Layered source coding with broadcast strategy using (a) repetition-based coop-eration, (b) relay-selection-based cooperation, (c) distributed space-time-coded cooperation.


∑jk=1 rk.

It is shown in [4] that at high SNR, the distortion exponent of the broadcast strategy

for a given multiplexing gain allocation is governed by

∆ = min0≤j≤n

{dSD(rj+1) + b

j∑k=1

rk

}. (3.17)

It turns out that ∆ can be characterized as a function of the bandwidth ratio b in the high-

SNR regime. It thus reflects a tradeoff between the spectral efficiency and the asymptotic

overall performance (via end-to-end distortion) of the system, and hence provides useful

guidance in the cooperative system design.

In the following, we study the distortion exponents of an m-relay cooperative system

with the three proposed schemes. The approach we take is as follows: We first derive the

conditional and the overall outage probabilities P jd and P jd of layer j, respectively. The

successive decoding diversity gain dSD(r) is then obtained using (3.16). We then show that

dSD(rj) = d∗(rj) for each of these cooperation protocols, thereby proving the successive

refinablity of the DMTs. Based on these results, we derive the corresponding achievable

distortion exponents according to (3.17).

3.4.1 Repetition-based cooperation

We now study the distortion exponents of the repetition-based (RP) cooperation protocols

with AF/DF relaying using broadcast strategy. We show that the optimal distortion expo-

nent can be achieved by employing a power allocation rule similar to the one proposed for

the single-relay system in [4]. Also, combined with this power allocation scheme, the broad-

cast strategy can be used to successively refine the DMT curves of the RP-based protocols.

As will be shown in the following sections, this is also true for the RS-based and ST-based

protocols for multiple relays.

Amplify-and-forward protocol

We first study the repetition-based cooperation protocol for multiple relays with AF relaying

using broadcast strategy. The transmission is done in two phases. In the first phase of

transmission, the source node broadcasts the superimposed layers to the destination and all

relay nodes. Each relay then amplifies the received signal under its power constraint and


retransmits to the destination. The repetition occurs on orthogonal channels, e.g., through

time-sharing as illustrated in Fig. 3.3 (a).

Define xj =∑n

k=j

√γkxk to be the superimposed signal of layer j, ..., n, where xk repre-

sents layer k, γk is the power allocated to layer k. Assuming layer 1 to layer j−1 have been

decoded and subtracted, the remaining received signal of layer j at the destination can then

be written as

y = h√γjxj + An, (3.18)

where

y = [ys,d, y1,d, · · · , ym,d]T ,

h = [hs,d, h1,dg1hs,1, · · · , hm,dgmhs,m]T ,

n = [xj+1, ns,1, · · · , ns,m, ns,d, n1,d, · · · , nm,d]T ,

(3.19)

and A = [h [0 G]T I], where 0 is an m×1 zero vector, G is an m×m diagonal matrix whose

ith diagonal entry is hi,dgi, I is an (m+ 1)× (m+ 1) identity matrix. ys,d and yi,d are the

signals received by the destination from the source and the ith relay node, respectively. ns,iis the additive noise at the ith relay. ns,d and ni,d are the additive noise at the destination.

All noises are assumed to be i.i.d. CN (0, 1). gi =√

γγ|hs,i|2+1

is the processing gain for the

ith relay to satisfy the power constraint.

The maximum rate that layer j can be reliably communicated to the destination given

that all previous layers have been successfully decoded and subtracted is then found to be

Cjd =1

m+ 1log det(I + γjhhH(AE[nnH ]AH)−1)

=1

m+ 1log(

1 + s γj1 + s γj+1

),

(3.20)

where γj =∑n

k=j γk, and

s , |hs,d|2 +m∑i=1

γ|hs,i|2|hi,d|2

γ|hs,i|2 + γ|hi,d|2 + 1. (3.21)

We first present in the following lemma the optimal distortion exponent achieved by

the broadcast strategy for a class of successive decoding diversity gains. The results of this

lemma have been partly reported in [4] and [49] for several special cases such as SIMO/MISO

systems and single-relay AF/DF protocols. Here, we state the general result for clarity and

completeness.


Lemma 3.4.1. Let dSD(rj) = (a − crj)+ for some a > 0 and c > 0 be the successive

decoding diversity gain achieved by an n-layer broadcast strategy for a cooperative system,

where x+ , max{x, 0}, rj ≥ 0, j = 1, · · · , n. The optimal distortion exponent at bandwidth

ratio b is given by 2

∆n =ab

c·

1−(bc

)n1−

(bc

)n+1 (3.22)

and in the limit of infinite layers (n→∞),

∆∞ = limn→∞

∆n =

ab/c, 0 ≤ b < c

a, b ≥ c(3.23)

Proof. The proof is given in Appendix 3.A.

Next, we prove the successive refinability of the repetition-based multi-relay cooperation

protocol with AF relaying, which has not been reported in literature. To do this, the

following power allocation scheme is introduced, which is a direct extension of the single-

relay cooperation power allocation rule proposed in [4],

γj =

γρj−1 − γρj , 1 ≤ j ≤ n− 1

γρn−1 j = n(3.24)

where ρ0 = 1, and ρj = 1− α∑j

k=1 rk, 1 ≤ j ≤ n− 1. For repetition-based cooperation, we

let α = m+ 1.

Theorem 3.4.2. The repetition-based cooperation protocol with AF relaying is successively

refinable in terms of the DMT curve. The successive decoding diversity gain is given by

dAFSD−RP (rj) = d∗RP (rj) = (m+ 1)(1− (m+ 1)rj)+. (3.25)

Proof. The proof is given in Appendix 3.B.

We then use the result in (3.25) to derive the distortion exponent of the repetition-based

cooperation protocol in the following theorem.

2Throughout the thesis, we define f(x0)g(x0)

= limx→x0f(x)g(x)

when g(x) = 0 at x = x0. The validity of the use

of this definition can be readily justified for all cases we study in this thesis.


Theorem 3.4.3. The optimal distortion exponent of the AF-and-repetition-based coopera-

tion protocol with the broadcast strategy, in terms of the number of relays m, the number of

layers n, and bandwidth ratio b, is

∆AF,nSD−RP =

b

m+ 1·

1−(

b(m+1)2

)n1−

(b

(m+1)2

)n+1

. (3.26)

In the limit of infinite number of layers (n→∞), we have

∆AFSD−RP =

b/(m+ 1), 0 ≤ b < (m+ 1)2

(m+ 1), b ≥ (m+ 1)2(3.27)

Proof. By using Lemma 3.4.1 with dSD(r) = dAFSD−RP (r) in (3.25), the maximum distortion

exponent of repetition-based cooperation with AF relaying is solved to be (3.26), and, in

the limit of infinite layers, (3.27).

A brief comparison with the corresponding single AF relay result in Thm. 4.5 in [5]

suggests that adding more relays effectively increases the largest achievable distortion expo-

nent, that is, from 2 to m+ 1 when b is large. However, the condition b ≥ (m+ 1)2 in (3.27)

also reveals that the improved distortion exponent comes at an extra bandwidth cost; hence

employing more relays may not always be beneficial in the repetition-based cooperation.

The detailed comparisons and discussions will be presented in Section 3.5.

Decode-and-forward protocol

We next study the distortion exponent of the repetition-based cooperation with DF re-

laying. The transmission is done in two phases. In phase I, the source node broadcasts

the information to the destination as well as all relays. In phase II, all relays try to fully

decode and repeat the message they receive. It has been illustrated in [55] that the DF-and-

repetition-based protocol achieves the same DMT curve as that of AF. However, we cannot

directly claim that these two schemes have the same achievable distortion exponent when

the broadcast strategy is used, as we need to show that they achieve the same successive

decoding diversity gain in this case.

Recall that the set of relays at which layer j is successfully decoded is defined to be

Dj = {ijk}, ijk ∈ {1, 2, · · · ,m}, k = 1, . . . ,Nj . Assuming the first j − 1 layers have been


decoded and subtracted from the received signal at the destination, the remaining received

signal of layer j at the destination can be written as

y = h√γjxj + [h I]n, (3.28)

where

y = [ys,d, yij1,d, · · · , y

ijNj,d

]T ,

h = [hs,d, hij1,d, · · · , h

ijNj,d

]T ,

n = [xj+1, ns,d, nij1,d, · · · , n

ijNj,d

]T .

(3.29)

Here, ys,d and yijk,d

are the signals received by the destination from source s and Relay ijk,

respectively. xj+1 is the interference term, ns,d and nijk,d

are the additive noise at destination,

which are assumed to be i.i.d. CN (0, 1). I is an (Nj + 1)× (Nj + 1) identity matrix.

Note that each relay node can choose not to forward those layers that it cannot decode,

i.e., the interference term could be smaller than xj+1. However, as in [4], we assume all

layers j + 1, · · · , n always act as the interference when decoding the jth layer, which gives

a lower bound to the maximum achievable rate and only degrades the performance.

Similar to the way that Ojd and Ojd are defined in Section 3.2, we define Oji and Oji to

be the sets of conditional and overall outage events for layer j at Relay i, respectively. Also,

we define Ojd|Dj and Ojd|Dj to be the conditional and overall sets of outage events for layer j

at the destination d conditioned on Dj , respectively. The relationships among these outage

sets are given by

Oji =j⋃

k=1

Oki , Ojd|Dj =j⋃

k=1

Okd|Dj . (3.30)

The corresponding outage probabilities P ji , P ji , P jd|Dj , and P jd|Dj are defined accordingly.

The overall successive decoding outage probability for layer j at the destination d can

now be written as

P jd =∑Dj

P jd|Dj · Pr{Dj}

=∑Dj

P jd|Dj ·∏i∈Dj

(1− P ji ) ·∏i 6∈Dj

P ji .(3.31)


We define P jd.= γ−d

DFSD−RP (rj), where dDFSD−RP (r) is the successive decoding diversity gain of

the repetition-based cooperation with DF relaying.

To find P jd and dDFSD−RP (r), we start with investigating the outage event sets Oji and

Ojd|Dj , which can be written as

Oji ={hs,i : Cji < Rj

},

Ojd|Dj ={(hs,d, {hi,d}i∈Dj

): Cjd|Dj < Rj

},

(3.32)

where Cji and Cjd|Dj are the maximum achievable rates of communicating jth layer to Relay

i and destination d, respectively, given that the first j − 1 layers can be decoded.

It is straightforward to show that

Cji =1α

log(

1 + |hs,i|2γj1 + |hs,i|2γj+1

),

Cjd|Dj =1α

log

(1 + (|hs,d|2 +

∑i∈Dj |hi,d|

2)γj1 + (|hs,d|2 +

∑i∈Dj |hi,d|

2)γj+1

),

(3.33)

where α = m+ 1.

After obtaining the expressions of the outage sets and the maximum achievable rates,

we are now ready to show the following theorem.

Theorem 3.4.4. The repetition-based cooperation protocol with DF relaying is successively


dDFSD−RP (rj) = d∗RP (rj) = (m+ 1)(1− (m+ 1)rj)+. (3.34)

Proof. The proof is given in Appendix 3.D.

Comparing (3.34) to (3.25), we see that dDFSD−RP (r) = dAFSD−RP (r). Since the distortion

exponent is determined by the successive decoding diversity gain, it is immediately clear that

the DF and AF have the same maximum distortion exponent with the broadcast strategy,

which leads to the following theorem.

Theorem 3.4.5. Under the broadcast strategy, the repetition-based cooperation protocol with

DF relaying achieves the same distortion exponent as that of the repetition-based cooperation

with AF relaying, i.e.,

∆DF,nSD−RP = ∆AF,n

SD−RP ,

∆DFSD−RP = ∆AF

SD−RP ,(3.35)


where ∆AF,nSD−RP and ∆AF

SD−RP are given in (3.26) and (3.27), respectively.

In the repetition-based cooperation, each relay employs the same codebook as that at

the source node. More generally, independently generated codebooks can be employed at

the relays so that, Ijd, the maximum achievable rate of communicating the jth layer to

destination d given that the first j − 1 layers can be decoded, becomes the following sum of

logarithmic terms

Cjd =1

m+ 1

∑i∈{s}

⋃Dj

log(

1 + |hi,d|2γj1 + |hi,d|2γj+1

). (3.36)

By Jensen’s inequality [3], the maximum achievable rate in (3.36) is larger than that of

the repetition-based cooperation in (3.33). This is related to utilizing the parallel channel

coding, and hence is more efficient than the repetition-based schemes [52]. Although the

distortion exponent analysis can be readily extended to the parallel coding case, we will focus

on several more bandwidth efficient schemes such as relay selection and space-time-coded

cooperation, which will be studied in the rest of this chapter.

3.4.2 Relay-selection-based cooperation

In this section, we analyze the distortion exponent of relay-selection-based multi-relay co-

operation protocols with DF relaying. The transmission is done in two phases in the relay-

selection-based scheme. In phase I, the source node broadcasts the information to all relays

and the destination. In phase II, only the “best” relay is chosen to participate in the co-

operation. This requires 1 bit of control information to be sent from a node with full CSI

to each relay. In this work, we assume full CSI at the destination and adopt Policy I of

[62] to be the relay selection criterion. Under this policy, the “best” relay i∗ is the relay

that maximizes the function hi = min{|hs,i|2, |hi,d|2} among all Relay i, i = 1, ...,m. This

selection policy is shown to achieve the relay selection DMT bound.

In relay selection, the decoding set Dj = ∅ or {i∗}. Thus, the analysis proceeds in the

same fashion as that of the repetition-based cooperation. As in the repetition-based coop-

eration, we define the outage sets Oji , Oji , O

jd|Dj , and Ojd|Dj according to (3.30) and (3.32).

The corresponding outage probabilities P ji , P ji , P jd|Dj , and P jd|Dj are defined accordingly.

Note that now α = 2 in (3.33) since both the source and the selected relay can use half of

the total channel uses to send information (see Fig. 3.3 (b)).


However, the results of repetition-based cooperation with DF relaying cannot be directly

reused because the related channel coefficients |hs,i∗ |2 and |hi∗,d|2 are no longer i.i.d. expo-

nential random variables. In fact, (|hs,i∗ |2, |hi∗,d|2) is themth (or the largest) conditionally N-

ordered statistics [107] of i.i.d. unit-variance exponential random vectors {(|hs,i|2, |hi,d|2)}mi=1

with measurable function N(x1, x2) = min{x1, x2}, whose joint probability density function

is given by [107]

f|hs,i∗ |2,|hi∗,d|2(x, y) = me−(x+y)(1− e−2 min{x,y})m−1. (3.37)

We first introduce the following lemma, which will be used to derive the successive

decoding diversity gain of the relay-selection-based protocol.

Lemma 3.4.6. Suppose {xi}mi=1, {yi}mi=1 and z are 2m+ 1 i.i.d. exponential random vari-

ables with mean 1/λ. Let

i∗ = arg max1≤i≤m

min{xi, yi}. (3.38)

Let ξ, θ and ζ be the exponential orders of x = xi∗, y = yi∗ and z, respectively, i.e., x = γ−ξ,

y = γ−θ, z = γ−ζ . Then

a) The probability POξ that ξ belongs to some set Oξ is characterized by

POξ , Pr{ξ ∈ Oξ}.= γ−ξ

∗(3.39)

where ξ∗ = infξ∈Oξ∩R+

mξ.

b) The probability POβ that (ζ, θ) belongs to some set Oβ is characterized by

POβ , Pr{(ζ, θ) ∈ Oβ}.= γ−β

∗(3.40)

where β∗ = inf(ζ,θ)∈Oβ∩R2+

ζ +mθ.

Proof. The proof is given in Appendix 3.E.

Remark: Lemma 3.4.6 can be easily generalized to the i.n.i.d. case, i.e., {xi}mi=1, {yi}mi=1

and z are independent but not identically distributed exponential random variables, while

(3.39) and (3.40) still hold.

The successive decoding diversity gain of the relay-selection-based protocol can now be

derived as follows.


Theorem 3.4.7. The relay-selection-based cooperation protocol with DF relaying is succes-

sively refinable in terms of the DMT curve. The successive decoding diversity gain is given

by

dSD−RS(rj) = d∗RS(rj) = (m+ 1)(1− 2rj)+. (3.41)

Proof. Since we only consider i = i∗ and Dj = ∅ or {i∗}, the overall successive decoding

outage probability for layer j at the destination d in (3.31) can now be simplified as follows

P jd = P jd|Dj={i∗} · (1− Pji∗) + P jd|Dj=∅ · P

ji∗ . (3.42)

In order to derive the outage probability P jd , we only need to find P ji∗ , Pjd|Dj=∅, and

P jd|Dj={i∗}.

Let |hs,d|2 = γ−ζ , |hs,i∗ |2 = γ−ξ, and |hi∗,d|2 = γ−θ. For the power allocation scheme in

(3.24) with α = 2, by (3.95) in Appendix 3.D, we immediately have

P jd|Dj=∅ , Pr{ζ ∈ Ojd|Dj=∅

}.= γ−(1−2rj)

+. (3.43)

By using Lemma 3.4.6, we obtain

P ji , Pr{ξ ∈ Oji∗

}.= γ−ξ

∗, (3.44)

where ξ∗ = infξ∈Oj

i∗∩R+mξ, and

P jd|Dj={i∗} , Pr{

(ζ, θ) ∈ Ojd|Dj={i∗}}.= γ−β

∗, (3.45)

where β∗ = inf(ζ,θ)∈Oj

d|Dj={i∗}∩R2+ζ +mθ.

By the same arguments as in the proof of the repetition-based cooperation with DF

relaying (Appendix 3.D), it can be shown that

infξ∈Oj

i∗∩R+mξ = m(1− αrj)+, (3.46)

inf(ζ,θ)∈Ojd∩R2+

ζ +mθ = (m+ 1)(1− αrj)+. (3.47)

As a result, the conditional outage probabilities are characterized by

P ji∗.= γ−(1−2rj)

+, (3.48)


P jd|Dj={i∗}.= γ−(m+1)(1−2rj)

+. (3.49)

Due to successive decoding, we have

P jd|Dj.= max{P j−1

d|Dj , Pjd|Dj}

.= P jd|Dj ,

P ji∗.= max{P j−1

i∗ , P ji∗}.= P ji∗ .

(3.50)

where Dj = ∅ or {i∗}.Plugging the outage probabilities results in (3.48), (3.49) and (3.50) into (3.42), we then

have

P jd.= γ−(m+1)(1−2rj)

+(1− γ−m(1−2rj)

+)

+ γ−(1−2rj)+γ−m(1−2rj)

+

.= γ−(m+1)(1−2rj)+

, γ−dDFSD−RS(rj)

(3.51)

Compared with (3.2), this suggests dDFSD−RS(rj) = d∗RS(rj). Hence the DMT of relay-

selection cooperation is successively refinable.

We now present the distortion exponent results of the relay-selection-based cooperation

protocol.

Theorem 3.4.8. The optimal distortion exponent of the relay-selection-based cooperation

protocol with the broadcast strategy, in terms of the number of relays m, the number of layers

n, and bandwidth ratio b, is

∆DF,nSD−RS =

b

2·

1−(

b2(m+1)

)n1−

(b

2(m+1)

)n+1

. (3.52)

In the limit of infinite number of layers (n→∞), we have

∆DFSD−RS =

b/2, 0 < b < 2(m+ 1)

(m+ 1), b ≥ 2(m+ 1)(3.53)

Proof. By using Lemma 3.4.1 with dSD(r) = dDFSD−RS(r), the distortion exponent of relay-

selection-based cooperation can be found to be (3.52), and, in the limit of infinite layers,

(3.53).


3.4.3 Space-time-coded cooperation

Space-time coding is useful to exploit the diversity in cooperative networks. The distributed

space-time coding based cooperation has been investigated in [52] and [54] for both AF

and DF protocols. In this section, we propose to extend the system framework in [52] by

combining the DF-based distributed space-time-coded protocol with the broadcast strategy.

We prove the successive refinability of the DMT, and obtain the distortion exponent of the

corresponding multi-relay cooperative system.

The transmission is done in two phases. In phase I (the broadcast phase), the source

node broadcasts the information to the destination and all relays, which is the same as all

previously studied protocols. In phase II (the cooperation phase), all relays try to fully

decode and forward the message they receive simultaneously on the same channel. To

facilitate the broadcast strategy, each layer is coded separately at all relays using a suitable

distributed space-time code. All space-time-coded layers are transmitted simultaneously to

the destination using the broadcast strategy.

It has been shown in [60] that, by optimally allocating the channel uses between the

broadcast phase and the cooperation phase, the DMT curves of the orthogonal protocols

can be improved. In [4], the unequal division of channel uses is studied for the single-relay

DF protocol. However, no distortion exponent results were reported since the corresponding

DMT is not successively refinable [4]. It is also suggested that further investigation is

required for this kind of extension even in the single-relay case.

In this section, we combine the distributed space-time-coded protocol with the optimal

channel use allocation. The source now transmits for the first tN channel uses, and the relay

nodes transmit for the rest (1 − t)N channel uses, 0 ≤ t ≤ 1. If t = 1/2, the total channel

uses are evenly divided between the source and the relay transmissions. The relays re-encode

the message using distributed space-time codes that are independent of the codebook used

at the source.

As in the repetition case, we define the outage sets Oji , Oji , O

jd|Dj , and Ojd|Dj as well

as the corresponding outage probabilities P ji , P ji , P jd|Dj , and P jd|Dj according to (3.30) and

(3.32). The overall outage probability P jd is again given by (3.31).

Assume the first j − 1 layers have been successfully decoded at the destination. The

maximum achievable rate of communicating the jth layer to Relay i and destination d are


now

Cji = t log

(1 + |hs,i|2γsj

1 + |hs,i|2γsj+1

),

Cjd|Dj = t log

(1 + |hs,d|2γsj

1 + |hs,d|2γsj+1

)

+ (1− t) log

(1 +

∑i∈Dj |hi,d|

2γrj

1 +∑

i∈Dj |hi,d|2γrj+1

),

(3.54)

where γsj and γrj are the power allocated to the jth layer at the source node and all relay

nodes, respectively. γsj =∑n

k=j γsk and γrj =

∑nk=j γ

rk. Again, we assume all layers j +

1, · · · , n always act as the interference when decoding the jth layer, which gives a lower

bound to the maximum achievable rate.

The following power allocation schemes are applied to the source and the relay nodes,

respectively.

For the source node, we have

γsj =

γρsj−1 − γρ

sj , 1 ≤ j ≤ n− 1

γρsn−1 , j = n

(3.55)

where ρs0 = 1, and ρsj−1 − ρsj ≥ rj/t, 1 ≤ j ≤ n− 1.

For all relay nodes, we have

γrj =

γρrj−1 − γρ

rj , 1 ≤ j ≤ n− 1

γρrn−1 , j = n

(3.56)

where ρr0 = 1, and ρrj−1 − ρrj ≥ rj/(1− t), 1 ≤ j ≤ n− 1.

We first derive the successive decoding diversity gain of the space-time-coded cooperation

protocol with broadcast strategy under the proposed power allocation scheme.

Theorem 3.4.9. With the proposed source and relay power allocation (3.55) and (3.56),

the successive decoding diversity gain of space-time-coded cooperation protocol with broadcast

strategy is

dSD−ST (rj) = min0≤k≤m

min {fk(rj), gk(rj)} , (3.57)

where

fk(rj) , (m− k + 1)(ρsj−1 −

rjt

)+ kρrj−1,

gk(rj) , (m− k + 1)ρsj−1 + kρrj−1 −(

k

1− t+m− kt

)rj .

(3.58)


Proof. The proof is given in Appendix 3.F.

The distortion exponent can then be expressed as follows

∆nSD−ST , min

0≤j≤n

{dSD−ST (rj+1) + b

j∑i=1

ri

}. (3.59)

The maximum distortion exponent in (3.59) does not appear to be analytically tractable

in general. Instead, we formulate it as the following optimization problem that maximizes

the distortion exponent for a finite number of layers.

max . ∆nSD−ST

s.t. ρsj−1 − ρsj ≥ rj/t, 1 ≤ j ≤ n− 1,

ρrj−1 − ρrj ≥ rj/(1− t), 1 ≤ j ≤ n− 1,

0 ≤ rj ≤ 1, 1 ≤ j ≤ n

(3.60)

Note that both fk and gk are linear in ρsj , ρrj and rj . Given the channel allocation

parameter t, the problem in (3.60) can be recast and efficiently solved as the following

linear programming problem

max . ∆nSD−ST

s.t. fk(rj+1) + brj ≥ ∆nSD−ST , 0 ≤ j ≤ n, 0 ≤ k ≤ m

gk(rj+1) + brj ≥ ∆nSD−ST , 0 ≤ j ≤ n, 0 ≤ k ≤ m

ρsj−1 − ρsj ≥ rj/t, 1 ≤ j ≤ n− 1,

ρrj−1 − ρrj ≥ rj/(1− t), 1 ≤ j ≤ n− 1,

0 ≤ rj ≤ 1, 1 ≤ j ≤ n

(3.61)

Fig. 3.4 shows an example of the numerically computed distortion exponent with respect

to different channel allocation ratio t at various bandwidth ratios b for a 2-relay cooperative

system with distributed space-time-coded cooperation. It is observed that the optimal t∗

is usually achieved near t = 1/2 and increases with b. However, the improvement in terms

of distortion exponents is only marginal compared to the distortion exponent achieved at

t = 1/2. This observation is also confirmed in Fig. 3.5, where the distortion exponents of

distributed space-time-coded cooperation with n = 5 layers and different numbers of relays

are plotted for t = t∗ and t = 1/2, respectively.

We next focus on the special case where t = 12 , i.e., the same amount of channel uses are

allocated to both transmission phases. The successive decoding diversity gain in this case

can be obtained analytically and is given in the following corollary.


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−0.5

0

0.5

1

1.5

2

2.5

3

Channel allocation ratio, t

Dis

tort

ion

expo

nent

, Δ

b = 8

b = 10

b = 7

b = 9b = 6

b = 0

b = 1

b = 3

b = 2

b = 4

b = 5

Figure 3.4: Distortion exponent vs. channel allocation ratio t at various bandwidth ratios bof layered coding with broadcast strategy using the distributed space-time-coded protocolfor a 2-relay cooperative system.

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Bandwidth ratio, b

Dis

tort

ion

expo

nent

, Δ

optimalt = 1/2

4 relays

2 relays

3 relays

Figure 3.5: Distortion exponent vs. bandwidth ratio of layered coding with broadcaststrategy using the distributed space-time-coded protocol for t = 1/2 and t = t∗ (optimal).


Corollary 3.4.10. The space-time-coded cooperation protocol with t = 12 is successively


dSD−ST (rj) = d∗ST (rj) = (m+ 1)(1− 2rj)+. (3.62)

Proof. Let ρsj = ρrj = 1 − 2rj in (3.57). After some simple manipulations, we obtain

dSD−ST (rj) = (m+ 1)(1− 2rj)+ = d∗ST (rj). This proves that the space-time-coded cooper-

ation protocol with t = 12 is successively refinable in the DMT sense.

Note that d∗ST (r) = d∗RS(r). The maximum distortion exponent of the system achieved

by the broadcast strategy can be directly followed from Corollary 3.4.10.

Corollary 3.4.11. Under the broadcast strategy, the space-time-coded cooperation protocol

with t = 12 achieves the same maximum distortion exponent as that of the relay-selection-

based cooperation, i.e.,

∆nSD−ST = ∆DF,n

SD−RS ,

∆SD−ST = ∆DFSD−RS ,

(3.63)

where ∆DF,nSD−RS and ∆DF

SD−RS are given in (3.52) and (3.53), respectively.

The above analysis relies on the random coding argument. In practice, since only a

randomly selected subset of n relays actually transmit among all m relays, it is required

that the space-time code designed for a maximum of m transmit antennas is also able to

offer diversity n when there are only n < m arbitrary antennas transmitting. It turns out

that the orthogonal space-time block codes [33] and the random space-time codes [108] have

this property as shown in [109, 108]. Thus, the space-time-coded cooperation protocols can

be readily deployed in practice using these codes.

3.5 Results and Discussions

The distortion exponents of the LS strategy and the broadcast strategy (BS) with the

repetition-based cooperation (RP), the relay-selection (RS) and the space-time coding (ST)

for infinite layers and different numbers of relays are shown in Fig. 3.6 and Fig. 3.7,

respectively. The channel use allocation ratio t is 12 in the space-time-coded cooperation.


0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

Bandwidth ratio, b

Dis

tort

ion

expo

nent

, Δ

3−relay RS/ST3−relay RP2−relay RS/ST2−relay RPsingle−relay AF/DFDT

Figure 3.6: Distortion exponent vs. bandwidth ratio of layered source coding with progres-sive transmission for multi-relay cooperative systems.

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

Bandwidth ratio, b

Dis

tort

ion

expo

nent

, Δ

3−relay RS / ST3−relay RP2−relay RS / ST2−relay RPsingle−relay AF/DFDT

Figure 3.7: Distortion exponent vs. bandwidth ratio of layered source coding with broadcaststrategy for multi-relay cooperative systems.


0 2 4 6 8 10 12 14 16 180

0.5

1

1.5

2

2.5

3

3.5

4

Bandwidth ratio, b

Dis

tort

ion

expo

nent

, Δ

Upper boundBS−RS / BS−STBS−RPLS−RS / LS−STLS−RPBS−DTLS−DT

Figure 3.8: Comparison of various coding and transmission strategies for a 3-relay cooper-ative system.

The direct transmission (DT) (no cooperation) and single-relay results from [4] are also

plotted as reference.

By (3.13), the maximum distortion exponents of the LS strategy with all three multi-

relay protocols approach the same upper bound m + 1 as the bandwidth ratio b increases.

Note that the upper bound increases with the number of relays, which demonstrates the

advantage of multi-relay systems. However, as shown in Fig. 3.6, with the same number of

relays, the distortion exponent of the RP increases much slower than that of the RS and

ST. Furthermore, as the number of relay increases, the performance improvement of the RP

can only be observed at very large b. This is due to the inefficient bandwidth utilization of

the repetition-based scheme.

Similarly, in the BS case, the maximal distortion exponent m + 1 increases with the

number of relays, which again confirms the benefit of multiple relays. When m > 1, with

the same number of relays, the three multi-relay protocols can achieve the same maximum.

However, the RS and ST methods reach the maximum when b ≥ 2(m+ 1), whereas the RP

method attains the maximum at b ≥ (m+1)2. Before that, the distortion exponent is b/2 in

RS and ST, whose slope is independent of m. However, the distortion exponent of the RP is

b/(m+ 1), whose slope reduces as the increase of m. Therefore when more relays are used,


the RP only gives better performance at very large b, which is similar to the observation in

the LS case. Note also that at small b, DT outperforms all cooperation-based schemes.

By assuming the source signal to be available at all relay nodes, the cooperative system

becomes a (m + 1) × 1 MISO system, whose distortion exponent upper bound is given by

[49]

∆MISO = min{b,m+ 1}. (3.64)

This is also a distortion exponent upper bound of the m-relay cooperative system.

In Fig. 3.8, we compare the maximal achievable distortion exponents of all the studied

schemes and the upper bound in (3.64) for a 3-relay cooperative system. Both LS and the

broadcast strategy (BS) have infinite coding layers.

It can be seen that with the same protocol, the BS always outperforms LS, and achieves

the upper bound in (3.64) much faster than the LS. This demonstrates the advantage of

the broadcast strategy over the progressive scheme. However, at low bandwidth ratio, the

improvement is limited, and the LS scheme might be preferred for its simplicity.

When b ≤ 1, only the DT achieves the upper bound. At medium bandwidth ratio, all the

studied schemes fail to approach the upper bound. This is partly because the cooperation

protocols considered are in general not optimal for multi-relay systems.

Further improvement can be expected by using sophisticated schemes with better DMTs

such as the dynamic decode-and-forward (DDF) protocol [55]. However, it is still not clear

how to effectively combine the BS with these protocols even in the single-relay setup [4].

Another possible extension is to combine the BS with more efficient static protocols, e.g., the

NAF in [54] and the sequential SAF in [65]. This however requires further investigations on

the successive refinability of the DMTs of these protocols, which remains an open problem.

3.6 Summary

In this chapter, we study the end-to-end distortion of wireless cooperative systems. Different

from most of the current work, we focus on the multi-relay scenario and related coopera-

tion strategies including repetition-based cooperation, relay-selection-based cooperation and

space-time-coded cooperation. We derive the asymptotic distortion exponent of these co-

operation schemes with broadcast strategy and AF or DF relaying. We also establish the

successive refinability of the DMT curves of the multi-relay cooperative system with these

protocols.


3.A Proof of Lemma 3.4.1

In [4], it is shown that for a cooperative system with successive decoding diversity gain

dSD(r), the optimal multiplexing gain allocation that leads to the maximum achievable

distortion exponent is the solution of the following set of equations, provided that {rj} are

feasible,

brn = dSD(r1 + · · ·+ rn)

dSD(r1 + · · ·+ rn) + brn−1 = dSD(r1 + · · ·+ rn−1)

· · ·

dSD(r1 + r2) + br1 = dSD(r1)

(3.65)

The corresponding optimal distortion exponent is given by ∆ = dSD(r1).

Assuming rj ≤ a/c, we then have dSD(rj) = a − crj . The solution of (3.65) is then

solved to be

r1 =a

c· 1− (b/c)

1− (b/c)n+1 ,

rj = (b/c)j−1 r1, j = 2, · · · , n.(3.66)

It can be verified that rj = ac ·

1−(b/c)j

1−(b/c)n+1 ≤ ac and rj ≥ 0 for all b. Hence, r1, · · · , rn are

feasible and form the optimal multiplexing gain allocation.

The optimal distortion exponent is then given by ∆n = dSD(r1) = a − cr1, which can

be verified to be (3.22) and, in the limit of infinite layers, (3.23). This completes the proof.

3.B Proof of Theorem 3.4.2

We first introduce the following lemmas:

Lemma 3.B.1. Suppose x0, {xi}mi=1 and {yi}mi=1 are 2m + 1 i.i.d. exponential random

variables with mean 1/λ. Define

x = x0 +m∑i=1

γxiyiγxi + γyi + 1

, x0 +12

m∑i=1

zi. (3.67)

Let ζ and θi be the exponential orders of x0 and zi, respectively, i.e., x0 = γ−ζ , zi = γ−θi.

The probability POx that (ζ, θ1, · · · θm) belongs to some set Ox is characterized by

POx , Pr{(ζ, θ1, · · · , θm) ∈ Ox}.= γ−β

∗, (3.68)


where β∗ = inf(ζ,θ1,··· ,θm)∈Ox∩R(m+1)+

ζ +∑m

i=1 θi.

Proof. The proof is given in Appendix 3.C.

Lemma 3.B.2. Denote x = {x1, x2, · · · , xn}. Let

Ox ={x : [a−min{x}]+ − [b−min{x}]+ < a− b

},

where x+ , max{x, 0}, and a > b. Then for any mi ≥ 0, i = 1, 2, · · · , n,

infx∈Ox∩Rn+

m1x1 +m2x2 + · · ·+mnxn =n∑i=1

mib+. (3.69)

Proof. It is easy to verify that infx∈Ox∩Rn+

min{x} = b+. Therefore,

infx∈Ox∩Rn+

n∑i=1

mixi =n∑i=1

mi infx∈Ox∩Rn+

min{x} =n∑i=1

mib+.

Let θi = − log(

γ|hs,i|2|hi,d|2γ|hs,i|2+γ|hi,d|2+1

)/log γ, ζ = −log |hs,d|2/log γ, and assign Rj = rj log γ.

Note that this kind of change of variables is often used in the DMT-related analysis [20].

With the proposed power allocation scheme (3.24), the conditional outage set of layer j can

then be written as

Ojd ={

(ζ, {θi}) : Cjd < Rj

}={

(ζ, {θi}) :1

m+ 1log(

1 + s γρj−1

1 + s γρj

)< rj log γ

},

(3.70)

where Cjd is the maximum achievable rate defined in (3.20), and s , γ−ζ +∑m

i=1 γ−θi .

At high SNR (large γ), we have log(1+∑i γxi )

log γ ' [max{xi}]+ [20]. Ojd can then be written

as

Ojd ={

(ζ, {θi}mi=1) : (ρj−1 − ν)+ − (ρj − ν)+ < αrj}

(3.71)

for j = 1, · · · , n− 1, and

Ond ={

(ζ, {θi}mi=1) : (ρn−1 − ν)+ < αrn}, (3.72)

where ν , min (ζ, {θi}mi=1), α = m+ 1.


Since ρj = 1− αrj , by Lemma 3.B.2, we have

inf(ζ,{θi}mi=1)∈Ojd∩R(m+1)+

ζ +m∑i=1

θi = (m+ 1)(1− αrj)+. (3.73)

By using Lemma 3.B.1 and the result of (3.73), the conditional outage probability of

layer j is then

P jd , Pr{

(ζ, {θi}mi=1) ∈ Ojd}.= γ−(m+1)(1−αrj)+

. (3.74)

Define the overall outage probability of layer j to be P jd.= γ−d

AFSD−RP (rj). Due to succes-

sive decoding, we have

P jd.= max{P j−1

d , P jd}.= P jd . (3.75)

Therefore, the successive decoding diversity gain of repetition-based cooperation with

AF relaying is dAFSD−RP (rj) = (m + 1)(1 − (m + 1)rj)+. Compared with (3.1), we have

dAFSD−RP (rj) = d∗RP (rj). This also suggests that the DMT of repetition-based cooperation

with AF relaying is successively refinable.

3.C Proof of Lemma 3.B.1

Note that we are interested in the limiting case of γ →∞. As γ →∞, we can approximate

x by

x = x0 +m∑i=1

xiyixi + yi

, x0 +12

m∑i=1

zi, (3.76)

where zi is the harmonic mean of xi and yi.

The probability density function of the harmonic mean of two exponential random vari-

ables with mean 1/λ is given by [110]

fZ(z) =1λ2ze−z/4λ [K1(λz) +K0(λz)] , (3.77)

where Ki is the ith-order modified Bessel function of the second kind defined in [111].

Since zi = γ−θi , the p.d.f. of θi can be shown to be

fΘ(θ) = log(γ)γ−θfZ(γ−θ). (3.78)


A sum formula for Kn(z) is given by [111]

Kn(z) =12

(12z

)−n n−1∑k=0

(n− k − 1)!k!

(−z

2

4

)k+ (−1)n+1 ln

(z2

)In(z)

+ (−1)n12

(12z

)n·∞∑k=0

[ψ(k + 1) + ψ(n+ k + 1)]( z

2

4 )k

k!(n+ k)!,

(3.79)

where ψ is the digamma function, and

In(z) = e−z(z

2

)n 1F1

(n+ 1

2 ; 1 + 2n; 2z)

Γ(n+ 1)(3.80)

is the nth-order modified Bessel function of the first kind, where F is a hypergeometric

function, Γ is the gamma function.

Combine (3.77), (3.79) and (3.80), and plug the result into (3.78). Careful examination

reveals that fΘ(θ) is dominated by the γ−θ term. Hence,

fΘ(θ) .=

γ−θ, θ ≥ 0

0, θ < 0(3.81)

The following lemma characterizes the p.d.f. of the exponentially distributed random

variable x0.

Lemma 3.C.1 ([55]). Suppose x1, x2, · · · , xn are n i.i.d. exponential random variables with

probability density function f(x) = e−x. Let xi = γ−αi. The probability density function

(p.d.f.) of αi is

f(α) .=

γ−α, α ≥ 0

0, α < 0(3.82)

The probability POα that (α1, α2, · · · , αn) belongs to some set Oα is characterized by

POα , Pr{(α1, α2, · · · , αn) ∈ Oα}.= γ−α

∗, (3.83)

where α∗ = inf(α1,α2,··· ,αn)∈Oα∩Rn+

α1 + α2 + · · ·+ αn.

Since x0 = γ−ζ , by Lemma 3.C.1, we have f(ζ) .= γ−ζ . Therefore,

f(ζ, θ1, · · · , θm) = f(ζ)f(θ1) · · · f(θm) .= γ−(ζ+∑mi=1 θi). (3.84)


The probability POx that (ζ, θ1, · · · , θm) belongs to some set Ox is given by

POx =∫Oxf(ζ, θ1, · · · , θm)dζdθ1 · · · dθm

.=∫Oxγ−(ζ+

∑mi=1 θi)dζdθ1 · · · dθm.

(3.85)

By Laplace’s method [20], as γ → ∞, the integral∫Ox γ

−(ζ+∑mi=1 θi)dζdθ1 · · · dθm is

dominated by the term with the maximum exponent. Hence, we have

POx.= γ−β

∗. (3.86)

where β∗ = inf(ζ,θ1,··· ,θm)∈Ox∩R(m+1)+

ζ +∑m

i=1 θi. This completes the proof.

3.D Proof of Theorem 3.4.4

Let |hs,i|2 = γ−ξi . Assign Rj = rj log γ. We apply the same power allocation scheme (3.24)

as in the AF case at both the source node and the relay nodes. The conditional outage set

Oji in (3.32) can then be expanded as follows

Oji = {hs,i : Cji < Rj}

={ξi :

1α

log(

1 + γ−ξiγρj−1

1 + γ−ξiγρj

)< rj log γ

}.

(3.87)

At high SNR (large γ), we have log(1+γx)log γ ' x+ [20]. The preceding expression of Oji can

then be written as

Oji ={ξi : (ρj−1 − ξi)+ − (ρj − ξi)+ < αrj

}(3.88)

for j = 1, · · · , n− 1, and

Oni ={ξi : (ρn−1 − ξi)+ < αrn

}. (3.89)

Similarly, let |hs,d|2 = γ−ζ , and |hi,d|2 = γ−θi . Denote ν = min(ζ, {θi}i∈Dj

). We are

able to expand the conditional outage set Ojd|Dj in (3.32) as follows,

Ojd|Dj ={(ζ, {θi}i∈Dj

): (ρj−1 − ν)+ − (ρj − ν)+ < αrj

}(3.90)

for j = 1, · · · , n− 1, and

Ond|Dn ={

(ζ, {θi}i∈Dn) : (ρn−1 − ν)+ < αrn}, (3.91)


where we have used the fact that, at high SNR (large γ), log(1+∑i γxi )

log γ ' [max{xi}]+ [20].

Since ρj = 1− αrj , by Lemma 3.B.2, we have

infξi∈Oji∩R+

ξi = (1− αrj)+ , (3.92)

inf(ζ,{θi}i∈Dj )∈Oj

d|Dj∩R(Nj+1)+

ζ +∑i∈Dj

θi = (Nj + 1)(1− αrj)+. (3.93)

By our assumption, γ−ζ = |hs,d|2, γ−ξi = |hs,i|2, and γ−θi = |hi,d|2 are i.i.d. exponential

random variables with unit variance. Hence, by using Lemma 3.C.1 and the results from

(3.92) and (3.93), we have

P ji , Pr{ξi ∈ Oji

}.= γ−(1−αrj)+

, (3.94)

P jd|Dj , Pr{(ζ, {θi}i∈Dj

)∈ Ojd|Dj

}.= γ−(Nj+1)(1−αrj)+

. (3.95)

Due to successive decoding, we have

P jd|Dj.= max{P j−1

d|Dj , Pjd|Dj}

.= P jd|Dj ,

P ji.= max{P j−1

i , P ji }.= P ji .

(3.96)

Plugging (3.94), (3.95) and (3.96) into (3.31), we obtain

P jd.=∑Nj

(m

Nj

)·(γ−(1−(m+1)rj)

+)m−Nj

·(

1− γ−(1−(m+1)rj)+)Nj

· γ−(Nj+1)(1−(m+1)rj)+

.= γ−(m+1)(1−(m+1)rj)+

, γ−dDFSD−RP (rj).

(3.97)

Compared with (3.1), we have dDFSD−RP (rj) = d∗RP (rj). Hence, the DMT of repetition-

based cooperation with DF relaying is successively refinable, which is also the same as that

of the AF relaying.

3.E Proof of Lemma 3.4.6

By assumption, (x, y) = (xi∗ , yi∗) is the mth conditionally N-ordered statistics of i.i.d. unit-

variance exponential random vectors {(xi, yi)}mi=1 with measurable function N(xi, yi) =

min{xi, yi}.


The joint probability density function of (x, y) is then given by (3.37)

fX,Y (x, y) = mλ2e−λ(x+y)(1− e−2λmin{x,y})m−1. (3.98)

The marginal p.d.f. of X is therefore

fX(x) =∫ ∞

0fX,Y (x, y)dy

=∫ ∞

0mλ2e−λ(x+y)(1− e−2λmin{x,y})m−1dy

(3.99)

Instead of finding the exact expression of fX(x), we propose to derive the following upper

bound and lower bound, which are sufficient to characterize the asymptotic behavior of the

exponent of x.

Notice that the integrand in (3.99) is always non-negative. Replacing min{x, y} by x

thus gives an upper bound of fX(x)

fX(x) ≤∫ ∞

0mλ2e−λ(x+y)(1− e−2λx)m−1dy

= mλe−λx(1− e−2λx)m−1.

(3.100)

To obtain a lower bound of fX(x), we change the integration range of y from [0,∞) to

[x,∞). Hence,

fX(x) ≥∫ ∞x

mλ2e−λ(x+y)(1− e−2λmin{x,y})m−1dy

=∫ ∞x

mλ2e−λ(x+y)(1− e−2λx)m−1dy

= mλe−2λx(1− e−2λx)m−1.

(3.101)

Since x = γ−ξ, the p.d.f. of ξ can be shown to be

fΞ(ξ) = log(γ)γ−ξfX(γ−ξ). (3.102)

By using the upper bound and lower bound of fX(x) in (3.100) and (3.101), we can

bound fΞ(ξ) as follows

fΞ(ξ) ≤ mλ log(γ)γ−ξe−λγ−ξ

(1− e−2λγ−ξ)m−1,

fΞ(ξ) ≥ mλ log(γ)γ−ξe−2λγ−ξ(1− e−2λγ−ξ)m−1.(3.103)


By Taylor Expansion, we have

(1− e−2λγ−ξ)m−1 =

( ∞∑n=1

(−1)n+1 (2λ)n

n!γ−nξ

)m−1

(3.104)

Combining (3.103) and (3.104), it is then clear that both the upper bound and the lower

bound of fΞ(ξ) are dominated by the γ−mξ term. Hence,

fΞ(ξ) .={ γ−mξ, ξ ≥ 0

0, ξ < 0(3.105)

The probability POξ that ξ belongs to some set Oξ is given by

POξ =∫OξfΞ(ξ)dv .=

∫Oξγ−mξdξ. (3.106)

By Laplace’s method [20], as γ →∞, the integral∫Oξ γ

−mξdξ is dominated by the term

with the maximum exponent γ−mξ∗. Hence, we have

POξ.= γ−mξ

∗. (3.107)

where ξ∗ = infξ∈Oξ∩R+

mξ. This completes the proof of Lemma 3.4.6 (a).

Note that y and x have the same marginal p.d.f. due to the symmetry. Since y = γ−θ,

by (3.105), the p.d.f. of θ is then

p(θ) .= γ−mθ. (3.108)

Recall that z = γ−ζ is an exponential random variable, by Lemma 3.C.1, the p.d.f. of ζ

is given by

q(ζ) .= γ−ζ . (3.109)

Since y and z are independent, the joint p.d.f. of ζ and θ is therefore

g(ζ, θ) = p(θ)q(ζ) .= γ−(ζ+mθ). (3.110)

The probability POβ that (ζ, θ) belongs to some set Oβ is given by

POβ =∫Oβ

g(ζ, θ)dζdθ .=∫Oβ

γ−(ζ+mθ)dζdθ. (3.111)

By Laplace’s method, as γ → ∞, the integral∫Oβγ−(ζ+mθ)dζdθ is dominated by the

term with the maximum exponent. Hence, we have

POβ.= γ−β

∗, (3.112)

where β∗ = inf(ζ,θ)∈Oβ∩R2+

ζ +mθ. This completes the proof of Lemma 3.4.6 (b).


3.F Proof of Theorem 3.4.9

Let |hs,d|2 = γ−ζ , |hs,i|2 = γ−ξi , and |hi,d|2 = γ−θi . With the maximum achievable rate de-

fined in (3.54) and the power allocation scheme in (3.55) and (3.56), by the same arguments

in the repetition case (Appendix 3.D), we obtain the conditional outage set Oji as follows

Oji ={ξi : t

[(ρsj−1 − ξi)+ − (ρsj − ξi)+

]< rj

}(3.113)

for j = 1, · · · , n− 1, and

Oni ={ξi : t(ρsn−1 − ξi)+ < rn

}. (3.114)

Denote θ = min{θi}. The conditional outage event Ojd|Dj is then found to be

Ojd|Dj ={(ζ, {θi}i∈Dj

): t[(ρsj−1 − ζ)+ − (ρsj − ζ)+

]+ (1− t)

[(ρrj−1 − θ)+ − (ρrj − θ)+

]< rj

} (3.115)

for j = 1, · · · , n− 1, and

Ond|Dj ={(ζ, {θi}i∈Dj

): t(ρsn−1 − ζ)+ + (1− t)(ρrn−1 − θ)+ < rn

}. (3.116)

By Lemma 3.C.1, we have

P ji , Pr{ξi ∈ Oji

}.= γ−ξ

∗, (3.117)

where ξ∗ = infξi∈Oji∩R+

ξi = ρsj−1 −rjt . And,

P jd|Dj , Pr{(ζ, {θi}i∈Dj

)∈ Ojd|Dj

}.= γ−β

∗, (3.118)

where β∗ = inf(ζ,{θi}i∈Dj

)∈Oj

d|Dj∩R(Nj+1)+

ζ +∑

i∈Dj θi.

It can be verified that, under the given power allocation scheme, the outage set Ojd|Djis given by the shaded region in Fig. 3.9. Therefore, the infimum β∗ is achieved either at

point A or point B, i.e.,

β∗ = min{ρsj−1 −

rjt

+Nj · ρrj−1, ρsj−1 +Nj ·

(ρrj−1 −

rj1− t

)}(3.119)


Br

jρ

r

j 1−ρ

t

rjsj −−1ρs

jρ

θ

0 sj 1−ρ ζ

A

Br

jρ

r

j 1−ρ

t

rjsj −−1ρs

jρ

θ

0 sj 1−ρ ζ

A

Figure 3.9: Outage region of the distributed space-time-coded protocol with broadcast strat-egy.

Plugging (3.117), (3.118) and (3.96) into (3.31), we obtain

P jd.=∑Nj

(m

Nj

)·(γ−(ρsj−1−

rjt

))m−Nj·(

1− γ−(ρsj−1−

rjt

))Nj· γ−min

{ρsj−1−

rjt

+Nj ·ρrj−1, ρsj−1+Nj ·

(ρrj−1−

rj1−t

)}.= γ−minNj min

{fNj (rj), gNj (rj)

}, γ−dSD−ST (rj),

(3.120)

where fNj (rj) and gNj (rj) are given by (3.58) with k = Nj .By changing Nj to k, we then have

dSD−ST (rj) = min0≤k≤m

min {fk(rj), gk(rj)} . (3.121)

This completes the proof.

Chapter 4

Distortion Exponents of

Multi-relay Cooperation with

Limited Feedback

4.1 Introduction

In Chapter 3, we studied the distortion exponents of source transmission over multi-relay

cooperative networks with no CSIT. In [50], Kim et al. proposed a feedback-based scheme

for source transmission over MIMO fading channels where the receiver quantizes the channel

coefficients jointly into an integer index, which is then sent to the transmitter via a noise-

less zero-delay feedback link. The feedback scheme in [50] is later applied to single-relay

cooperative systems with DF relaying by the same authors in [5]. It is shown in [50] and [5]

that combining the limited channel state feedback with simple separate source and chan-

nel coding achieves better distortion exponent performance than that of the best layering

schemes in [49] and [4]. The simple structure of the feedback scheme is also more attractive

in practical systems.

In this chapter, we study the distortion exponent of the separate source and channel

coding with limited channel state feedback followed by various multi-relay cooperation pro-

tocols, including the OAF protocol [52], the NAF protocol [54], the sequential SAF protocol

[65], and the ODF/NDF protocols [60]. We derive the optimal distortion exponents for

all these schemes. The results illustrate the effect of the feedback resolution, bandwidth

67

CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 68

ratio, number of relays and cooperation protocols on the overall performance of multi-relay

cooperative systems in terms of the distortion exponent. It is also shown that the feedback

scheme outperforms the best known non-feedback strategies for multi-relay cooperative sys-

tems with only a few bits of feedback information.

Our work extends the distortion exponent study in [5] in the following two main aspects:

First, we investigate the feedback scheme with various cooperation protocols using both AF

and DF relaying, whereas only DF-based single-relay protocols have been considered in

[5]. Second, our work focuses primarily on multi-relay cooperative systems, which is more

general than the single-relay system studied in [5]. The corresponding distortion exponent

analysis is also more involved.

This chapter is organized as follows: In Section 4.2, we introduce the system model and

the limited feedback scheme. In Section 4.3, we combine the limited feedback with various

AF-based multi-relay cooperation protocols, and derive the corresponding achievable dis-

tortion exponents. The distortion exponents of DF-based multi-relay cooperation protocols

are investigated in Section 4.4. The performance comparisons of all schemes in terms of

their distortion exponents are given in Section 4.5. The work in this chapter is summarized

in Section 4.6.

4.2 System Model

We consider a wireless communication system where a source transmits information to a

destination with the help of m relays. All nodes are equipped with single antenna. The

system model is shown in Fig. 4.1. Each relay is half-duplex and employs the AF or DF

relaying protocol [53]. Perfect CSI of all communication links is assumed to be available at

the destination node.

The source {sk}∞k=1 is assumed to be zero-mean, unit-variance, independent and identi-

cally distributed (i.i.d.) complex Gaussian. As in Chapter 3, we assume K source samples

are transmitted in N channel uses. Hence the bandwidth ratio is b = N/K. As in Sec.

3.2, we assume that K is large enough to design source codes that can approach the rate-

distortion bound of the source signal and N is large enough to design a fixed-rate channel

code that can be transmitted reliably if the instantaneous capacity is greater than the com-

munication rate. We also assume the quasi-static fading scenario so that the channel gain

is random but remains constant during all N channel uses.


dh ,2

1,sh dh ,1

2,sh

dsh ,

msh , dmh ,

S D

1

*l

2

m

Relay

Source Destination

Feedback index

dh ,2

1,sh dh ,1

2,sh

dsh ,

msh , dmh ,

S D

1

*l

2

m

Relay

Source Destination

Feedback index

Figure 4.1: System model of an m-relay cooperative system with limited feedback from thedestination.

The channel is assumed to be Rayleigh fading and statistically symmetric, i.e., the

channel coefficients hs,d, hs,i and hi,d, i = 1, ...,m are all i.i.d. complex Gaussian random

variables with zero mean and unit variance. The additive noise at each node is modeled as

CN (0, 1). We assume all nodes have the same transmitting power, and denote the average

received SNR at the destination node to be γ. As stated in Chapter 3, since our interest

lies in the high SNR regime, it is sufficient to consider the symmetric system.

The feedback scheme we develop for the multi-relay system is an extension of that in

[5] for single-relay systems. In the feedback scheme, all nodes are equipped with a library

of L pairs of source-channel encoder and decoder, each has a coding rate of Rj = rj log γ

bits per channel use, where 0 < r1 < · · · < rL < 1 are the corresponding multiplexing

gains. The destination feeds back an index l ∈ {1, 2, · · · , L} based on the channel states

h = (hs,d, {hs,i}mi=1, {hi,d}mi=1) to the source node. In the case of DF relaying, the feedback

index is also broadcasted to all relay nodes. K is referred to as the feedback resolution, or

the feedback level. The feedback is assumed to be noiseless and zero-delay. Upon receiving

index l, the source node encodes the source symbols at a rate of R = Rl, and then transmits

the coded symbols to the destination node through cooperation.

Let C be the maximum rate that the coded channel symbols can be reliably commu-

nicated to the destination under a given cooperation scheme, which is governed by the

channel state vector h and is known to the receiver. The index mapping rule employed by


the destination node is then given as follows

l∗ = max1≤l≤L

l, s.t. Rl ≤ C, (4.1)

where l∗ is the feedback index sent by the destination node. That is, the destination informs

the source node to use the largest code rate possible for which the transmission will not be in

outage. If C < R1, i.e., the transmission will be in outage even the lowest code rate is used,

an arbitrary index will be sent since no reliable communication is possible. To distinguish it

from the other “true” indices, we always represent the arbitrary index by l∗ = 0 for clarity

purposes.

In the following sections, we study the achievable distortion exponents of the multi-relay

cooperative system with limited channel state feedback and different cooperation protocols.

The distortion exponent ∆ can be characterized as a function of the bandwidth ratio b, the

feedback resolution L, and the number of relays m in the high SNR regime. It reflects the

tradeoff between these system parameters in achieving the optimal asymptotic overall per-

formance (via end-to-end distortion), and hence provides useful guidance in the cooperative

system design.

4.3 Distortion Exponents of Amplify-and-forward Based Pro-

tocols

In this section, we derive the optimal distortion exponent for an m-relay cooperative system

with limited feedback. We study three multi-relay cooperation protocols with AF relaying,

namely, the OAF protocol [52], the NAF protocol [54], and the SAF protocol [65]. The

DMTs of these AF-based protocols were summarized in [60]. We list them as follows,

d∗OAF (r) = (m+ 1)(1− (m+ 1)r)+, (4.2)

d∗NAF (r) = (1− r)+ +m(1− 2r)+, (4.3)

d∗SAF (r) ≤ (1− r)+ +m(

1− M

M − 1r)+, (4.4)

where M is the number of transmission slots in the SAF protocol. Note that (4.4) in general

only characterizes a DMT upper bound of the SAF protocol, which however can be made

tight under certain constraints as will be discussed later.


The AF-based cooperation protocols we consider in this chapter are all fixed and static

protocols, i.e., under these protocols, each node transmits for a constant fraction of time

that does not depend on the code rate or the channel coefficients. Given the transmitting

power γ, the maximum achievable rate is then solely determined by the channel state vector

h, which we denote as C(h).

We first present in the following theorem a general result of the optimal distortion

exponent for the fixed and static AF relaying protocols with limited channel state feedback,

which has not been reported in the literature.

Theorem 4.3.1. The optimal distortion exponent of the fixed and static AF-based cooper-

ation protocol with L-level feedback is given by

∆ = sup0<r1<···<rL<1

min0≤l≤L

{d∗AF (rl+1) + brl} , (4.5)

where d∗AF (r) is the DMT of the corresponding AF-based cooperation protocol.

Proof. Define Oj , {h : C(h) < Rj} to be the jth outage set, that is, Oj is the set of all

channel states h for which the transmission is in outage at the destination given a coding

rate of Rj = rj log γ. The corresponding outage probability is then P jout , Pr{h ∈ Oj}.=

γ−d∗AF (rj).

Since R1 < · · · < RL, we have Oj ⊆ Oj+1, j = 0, · · · , L, where we define O0 = ∅ and

define OL+1 to be the set of all possible channel states h. By the construction of the feedback

index mapping rule, the probability that an feedback index l is sent by the destination is

defined to be

Pl , Pr{h ∈ Ol ∩ Ol+1}, l = 0, · · · , L, (4.6)

where l = 0 if an arbitrary index is sent, Ol is the complementary set of Ol.It can be shown that

Pl = Pr{h ∈ Ol ∩ Ol+1

}= Pr {h ∈ Ol+1} − Pr {h ∈ Ol}.= γ−d

∗AF (rl+1) − γ−d∗AF (rl)

.= γ−d∗AF (rl+1),

(4.7)

where we define r0 = 0 and rL+1 = 1.


As a result, the expected end-to-end distortion can be written as

D =L∑l=0

Pl2−bRl =L∑l=0

Pl2−brl log γ

.=L∑l=0

γ−(d∗AF (rl+1)+brl)

.= γ−min0≤l≤L{d∗AF (rl+1)+brl}.

(4.8)

The optimal distortion exponent is then

∆ = sup0<r1<···<rL<1

min0≤l≤L

{d∗AF (rl+1) + brl} . (4.9)

In the following, we derive the achievable distortion exponents of different multi-relay

cooperation protocols using the result in (4.5).

4.3.1 Orthogonal amplify-and-forward protocol

We first study the OAF protocol for multiple relays. The source encodes the signal at

rate Rl according to the feedback index l. In the first phase of transmission, the source

node broadcasts the signal to the destination as well as all relay nodes. Each relay then

amplifies its received signal under its power constraint and retransmits to the destination on

orthogonal channels. The processing gain for the ith relay to satisfy the power constraint

is given by

gi =√

γ

γ|hs,i|2 + 1. (4.10)

Plugging d∗AF (r) = d∗OAF (r) into (4.5), the optimal distortion exponent of the OAF

protocol with L-level feedback is then

∆KOAF = sup

0<r1<···<rL<1min

0≤l≤L{d∗OAF (rl+1) + brl} . (4.11)

In order to solve (4.11), the following lemma is introduced from [5], which we restate

using our notations.

Lemma 4.3.2 (Lemma 2 [5]). Let (r∗1, · · · , r∗L) be the solution to the system of linear

equations

a− cr1 = wr1 + a− cr2 = · · · = wrL−1 + a− crL = wrL, (4.12)


for some a, c, w > 0. Then

r∗L =a

c· 1− (w/c)L

1− (w/c)L+1, (4.13)

and

r∗l =a− wr∗L

c· 1− (w/c)l

1− (w/c), 1 ≤ l ≤ L− 1. (4.14)

It can be readily verified that the solution (r∗1, · · · , r∗L) always satisfies 0 < r1 < · · · < rL <ac

for any given a, c, w > 0.

The maximin optimization is a typical problem in the distortion exponent analysis.

Optimization problems that have a similar form as (4.11) have previously appeared in [47,

4, 49, 5]. Notice that d∗OAF (r) is a linear function of r ∈ [0, 1m+1 ]. We show in Appendix

4.A that the optimization problem in (4.11) is equivalent to the following linear program

max . ∆LOAF

s.t. d∗OAF (rl+1) + brl ≥ ∆LOAF , 0 ≤ l ≤ L,

0 ≤ r1 ≤ · · · ≤ rL ≤1

m+ 1.

(4.15)

We show in Appendix 4.A that, by the KKT conditions [112], the solution of (4.15) is

attained at (r∗1, · · · , r∗L), for which the set of functions {d∗OAF (r∗l+1) + br∗l }Ll=0 have the equal

value ∆LOAF , i.e.,

∆LOAF = d∗OAF (r∗1),

∆LOAF = d∗OAF (r∗2) + br∗1,

· · ·

∆LOAF = d∗OAF (r∗L) + br∗L−1,

∆LOAF = br∗L,

(4.16)

The maximum distortion exponent is thus given by ∆LOAF = br∗L.

Earlier statements that are partly related to this result have been made in [5, 4, 49].

However, no proofs or detailed explanations are given in those papers. In [47], a similar

result is obtained for the broadcast scheme in MIMO systems, but its conclusion can not

be directly used here.

We now present the maximum distortion exponent achieved by the OAF protocol in the

following theorem.


Theorem 4.3.3. The maximum distortion exponent achieved by the OAF protocol with

limited feedback, in terms of the number of relays m, the feedback resolution L, and bandwidth

ratio b, is

∆LOAF =

b

m+ 1·

1−(

b(m+1)2

)L1−

(b

(m+1)2

)L+1. (4.17)

In the limit of infinite feedback resolution (L→∞), we have

∆∞OAF =

b

m+1 , 0 ≤ b < (m+ 1)2,

m+ 1, b ≥ (m+ 1)2.(4.18)

Proof. Equating all terms {d∗OAF (rl+1) + brl}Ll=0 in (4.11) yields the system of linear equa-

tions in (4.12) with a = m+1, c = (m+1)2 and w = b. Applying Lemma 4.3.2, the solution

(r∗1, · · · , r∗L) is obtained as follows

r∗L =1

m+ 1·

1−(

b(m+1)2

)L1−

(b

(m+1)2

)L+1, (4.19)

and

r∗l =(m+ 1)− br∗L

(m+ 1)2·

1−(

b(m+1)2

)l1− b

(m+1)2

, 1 ≤ l ≤ L− 1, (4.20)

where we have 0 < r∗1 < · · · < r∗L <1

m+1 . Hence, the optimal distortion exponent is give by

∆LOAF = br∗L, which can be found to be (4.17), and in the limit of L→∞, (4.18).

It is worth noting that the expression of the achievable distortion exponents of the OAF

protocol is of exactly the same form as the distortion exponent of the broadcast strategy

with the repetition-based cooperation for m-relay cooperative systems (see Section 3.4.1,

Chapter 3), where instead of single-layer coding, an L-layer superposition coding scheme

(the broadcast strategy) is employed without using any feedback. Hence, combining limited

feedback with the OAF protocol may not seem quite appealing since it does not improve

the known achievable distortion exponents for multi-relay cooperative systems. However,

the feedback scheme still enjoys a simplified system design. Moreover, in the following,

we will study sophisticated cooperation protocols such as the NAF protocol and the SAF

protocol. It is still not clear how to effectively combine the broadcast strategy with these

protocols, whereas an improved performance can be easily obtained by using only a few bits

of feedback information.


4.3.2 Nonorthogonal amplify-and-forward protocol

The NAF protocol proposed by Nabar et al. [54] has been shown to achieve the best

DMT among all AF-based schemes for the half-duplex single-relay channel by Azarian et

al. [55]. In the following, we apply the NAF protocol to the multi-relay system and derive

the corresponding distortion exponent.

In the NAF scheme, the source transmits during all N symbol intervals. Each relay

participate in the transmission for Nm symbol intervals in a round-robin fashion. Specifically,

each participating relay listens to the source node for the first N2m symbol intervals, and

then forwards its received signal to the destination node using AF relaying in the secondN2m symbol intervals.

As in the OAF case, by letting d∗AF (r) = d∗NAF (r) in (4.5), the optimal distortion

exponent of the NAF protocol is then

∆LNAF = sup

0<r1<···<rL<1min

0≤l≤L{d∗NAF (rl+1) + brl} . (4.21)

Unlike the OAF protocol, as will be shown later, an explicit form of the distortion

exponent for the NAF protocol cannot be obtained at all bandwidth ratios in general.

Instead, we propose an algorithm that solves (4.21) efficiently.

Although the problem in (4.21) is not a convex optimization problem since d∗NAF (r) is

not concave, it is indeed true that d∗NAF (r) is a linear function of r for r ∈ [0, 12 ] or [1

2 , 1].

Assuming 0 < r1 < · · · < rl∗ <12 ≤ rl∗+1 < · · · < rL < 1 for a finite number of resolution

L and 0 ≤ l∗ ≤ L, the problem in (4.21) can be formulated as the following optimization

problem

∆LNAF = max

0≤l∗≤L∆l∗ (4.22)


where

∆l∗ = maxr1,··· ,rL

min{

(1− r1) +m(1− 2r1),

(1− r2) +m(1− 2r2) + br1,

· · · ,

(1− rl∗) +m(1− 2rl∗) + brl∗−1,

(1− rl∗+1) + brl∗ ,

· · · ,

(1− rL) + brL−1,

brL

}s.t. 0 < r1 < · · · < rl∗ <

12≤ rl∗+1 < · · · < rL < 1.

(4.23)

The problem in (4.23) can be recast and efficiently solved as a linear program for each l∗.

The distortion exponent ∆LNAF in (4.22) can then be obtained by solving L+1 subproblems

defined by (4.23) with l∗ = 0, · · · , L.

In the limit of L → ∞, the distortion exponent ∆KNAF can be found explicitly. In this

case, we have a continuum of feedback levels, which are indexed by r ∈ [0, 1]. The distortion

exponent associated with feedback level r is then d∗NAF (r) + br. By the proof of Corollary 3

in [5], the dominant distortion exponent is the minimum of d∗NAF (r) + br over [0, 1], which

is found to be

∆∞NAF = minr∈[0,1]

d∗NAF (r) + br

= min

{minr∈[0, 1

2](1− r) +m(1− 2r) + br, min

r∈[ 12,1]

(1− r) + br

}

=

b, 0 ≤ b < 1

(b+ 1)/2, 1 ≤ b < 2m+ 1

m+ 1, b ≥ 2m+ 1

(4.24)

Since there is no closed-form solution for (4.21) in general, in the following, we provide a

lower bound on the optimal distortion exponent for finite level of feedbacks, which is exact

when b is sufficiently large.


Proposition 4.3.4. The optimal distortion exponent of the NAF protocol with L-level feed-

back, m relays, and bandwidth ratio b is lower-bounded by

∆LNAF =

b(m+ 1)2m+ 1

·1−

(b

2m+1

)L1−

(b

2m+1

)L+1. (4.25)

The lower bound ∆LNAF is exact, i.e., ∆L

NAF = ∆LNAF , when b ≥ b∗, where b∗ = 2m + 1 if

L = 2m+ 1, otherwise, b∗ is the root of the equation

bL+1 − 2(m+ 1)bL + (2m+ 1)L = 0 (4.26)

that satisfies b∗ > 0 and b∗ 6= 2m+ 1.

Proof. The proof of the existence and uniqueness of b∗ is straightforward and hence omitted.

We impose the constraint 0 < r1 < · · · < rL ≤ 12 under which d∗NAF (ri) = (1 − ri) +

m(1− 2ri) is a linear function of ri. Note that we can also constrain d∗NAF (ri) to be linear

by assuming 12 ≤ r1 < · · · < rL < 1. However, in this case we have d∗NAF (ri) = 1 − ri,

which corresponds to the direct transmission, i.e., no cooperation is utilized. Therefore,

this constraint significantly limits the performance especially in the large bandwidth ratio

regime, for which we will obtain a much looser lower bound.

Following the same approach in deriving the distortion exponent for the OAF protocol,

it can be shown that under the additional constraint, the solution of the problem in (4.21)

is attained by equating the functions {d∗NAF (rl+1) + brl}Ll=0. Applying Lemma 4.3.2, the

solution (r∗1, · · · , r∗L) is then obtained as follows

r∗L =m+ 12m+ 1

·1−

(b

2m+1

)L1−

(b

2m+1

)L+1(4.27)

and

r∗l =(m+ 1)− br∗L

2m+ 1·

1−(

b2m+1

)l1− b

2m+1

, 1 ≤ l ≤ L− 1. (4.28)

The corresponding distortion exponent is then ∆LNAF = br∗L, which is given by (4.25).

For the solution to be feasible and optimal, the rate constraint 0 < r∗1 < · · · < r∗L ≤12

has to be satisfied. Since 0 < r∗1 < · · · < r∗L is guaranteed by Lemma 4.3.2, we only require


r∗L ≤12 . Define f(b) , bL+1 − 2(m + 1)bL + (2m + 1)L. By simple algebraic manipulation,

the constraint r∗L ≤12 can be rewritten in terms of the bandwidth ratio b as follows

f(b) ≤ 0 if b < 2m+ 1,

f(b) ≥ 0 if b > 2m+ 1.(4.29)

When b = 2m+ 1, we have

limb→2m+1

r∗L =m+ 12m+ 1

· L

L+ 1≤ 1

2⇒ L ≤ 2m+ 1. (4.30)

Using standard calculus arguments, it can be shown that the constraints (4.29) and (4.30)

are satisfied for any b ≥ b∗. As a result, ∆LNAF is optimal in the range [b∗,∞). In the range

[0, b∗], ∆LNAF serves as an achievable lower bound to the optimal distortion exponent.

For L < 3, b∗ can be computed explicitly. We present these results in the following

corollary

Corollary 4.3.5. For L = 1 and b ≥ b∗ = 1, the optimal distortion exponent of the NAF

protocol with limited feedback is given by

∆1NAF =

b(m+ 1)b+ 2m+ 1

. (4.31)

For L = 2 and b ≥ b∗ = 1+√

8m+52 , the optimal distortion exponent of the NAF protocol with

limited feedback is given by

∆2NAF =

b(m+ 1)b+ 2m+ 1

·1 + b

2m+1

1 + b2m+1 +

(b

2m+1

)2 . (4.32)

4.3.3 Sequential slotted amplify-and-forward protocol

The SAF protocol proposed in [65] is shown to outperform the NAF protocol in terms of the

DMT for a two-relay system. It is shown in [65] that the upper bound to the DMT of the

SAF protocol approaches the transmit diversity upper bound as the number of transmission

slots M increases. A sequential SAF scheme is proposed in [65], which is shown to achieve

the SAF DMT upper bound in (4.4) for the two-relay case with transmission slot M = 3. For

an arbitrary number of relays and transmission slot M , the DMT upper bound is achieved

under the assumption of relay isolation, i.e., each relay does not overhear the signals sent


by other relays. In the following, we propose to combine the sequential SAF protocol with

the L-level feedback scheme to improve the distortion exponent performance of an m-relay

cooperative system.

In the sequential SAF scheme, one transmission frame is composed of M slots. The

source transmits during all M slots. Starting from the second slot, there is one and only

one relay scheduled to forward its received signal in the previous slot to the destination

node using AF relaying. It is shown in [65] that a round-robin scheduling is sufficient for

the sequential SAF protocol to achieve the DMT upper bound under the relay isolation

assumption.

As before, the optimal distortion exponent is given by (4.5) with d∗AF (r) = d∗SAF (r) as

follows

∆LSAF = sup

0<r1<···<rL<1min

0≤l≤L{d∗SAF (rl+1) + brl} . (4.33)

Note that the expression of the upper bound of d∗SAF (r) in (4.4) is of the same form as that

of d∗NAF (r). Hence, an upper bound of ∆LSAF can be efficiently solved by formulating an

optimization problem as that in (4.22). Furthermore, all distortion exponent results of the

SAF protocol can be derived in the same fashion as that of the NAF protocol. Therefore,

we directly state the following results and omit the proofs.

Proposition 4.3.6. The distortion exponent of the sequential SAF protocol, in the limit of

infinite feedback resolutions (L→∞), is upper bounded by

∆∞SAF =

b, 0 ≤ b < 1

((M − 1)b+ 1)/M, 1 ≤ b < Mm/(M − 1) + 1

m+ 1, b ≥Mm/(M − 1) + 1

(4.34)

The following proposition characterizes a distortion exponent lower bound of the sequen-

tial SAF protocol with isolated relays.

Proposition 4.3.7. The optimal distortion exponent of the sequential SAF protocol with

L-level feedback, m isolated relays, and bandwidth ratio b is lower-bounded by

∆LSAF =

b(m+ 1)αm+ 1

·1−

(b

αm+1

)L1−

(b

αm+1

)L+1, (4.35)


where α , M/(M − 1). The lower bound ∆LSAF is exact, i.e., ∆L

SAF = ∆LSAF , when b ≥ b∗,

where b∗ = αm+ 1 if L = αm+ 1, otherwise, b∗ is the root of the equation

bL+1 − α(m+ 1)bL + (α− 1)(αm+ 1)L = 0 (4.36)

that satisfies b∗ > 0 and b∗ 6= αm+ 1.

Again, we would like to emphasize that the distortion exponent results obtained here

are in general upper bounds for the sequential SAF protocol. However, as will be shown in

Section 4.5, the obtained upper bound is much tighter than the general distortion exponent

upper bound for m-relay systems. More importantly, all upper bounds can be achieved by

the sequential SAF protocol with two-relay and three-slot transmission.

4.4 Distortion Exponents of Decode-and-forward Based Pro-

tocols

We now study the decode-and-forward relaying protocols. We investigate both the orthog-

onal selection DF protocol (ODF) and the nonorthogonal selection DF protocol (NDF)

proposed in [60] with limited feedback for an m-relay cooperative system. The DMTs of

the two DF-based protocols are found in [60]

d∗ODF (r) =

(m+ 1)(1− 2m+1m+1 r), 0 ≤ r ≤ m

2m+1

(m+1)(1−r)mr+1 , m

2m+1 ≤ r ≤ 1(4.37)

d∗NDF (r) =

(m+ 1)− 1+2m+√

1+4m2

2 r, 0 ≤ r < βm

(m+1−r)(1−r)(m−1)r+1 , βm ≤ r ≤ 1

(4.38)

where βm , 1+2m−√

1+4m2

2 .

Since decoding and encoding are also performed at the relay node, the destination feeds

back an index l based on the channel states h = (hs,d, {hs,i}mi=1, {hi,d}mi=1) to both the source

node as well as all relay nodes. Upon receiving index l, the source node encodes the source

symbols at a rate of R = Rl using the lth encoder from the library.

The transmission is then done in two phases. In Phase 1, the source node broadcasts

the signal to the destination node and all relays nodes using a fraction t of a total number

of N channel uses, 0 ≤ t ≤ 1. Different from the previously studied AF-based protocols,


here the fraction t is optimized with respect to the multiplexing gain r. We will use the

notation t(r) to indicate an optimized fraction. Each participating relay tries to decode the

message based on its received signal using the lth decoder. If the decoding was successfully,

the relay re-encodes the message using the lth encoder and transmits to the destination in

Phase 2 using (1 − t)N channel uses. Otherwise, the relay remains silent. In orthogonal

schemes, there is no transmission between the source node and the destination node in the

second phase, whereas in the non-orthogonal case, the source node may transmit additional

symbols using the remaining (1− t)N channel uses. The destination then decodes based on

the received signals in both phases.

Note that the maximum achievable rate of the DF-based protocols also depends on the

multiplexing gain r since the optimized t is a function of r. Let C(h, t(r)) be the maximum

achievable rate of the DF-based cooperation scheme given the channel state vector h and

the fraction t. As in the AF case, we define Oj = {h : C(h, t(rj)) < Rj} to be the jth outage

set. The corresponding outage probability is P jout , Pr{h ∈ Oj}.= γ−d

∗DF (rj), where d∗DF (r)

is the DMT of the DF-based cooperation protocol.

In the fixed relaying schemes, i.e., when t is a fixed number that does not depend on r,

it can be easily shown that Oj ⊆ Oj+1 as in the AF case. By using the same arguments,

the distortion exponent of the DF-based protocols can be readily shown to be (4.5) with

the DMT d∗AF (r) replaced by d∗DF (r). In the following, we will show that this conclusion

also holds for the ODF protocol and the NDF protocol with an optimized fraction t(r). We

then use this result to derive the optimal distortion exponents of the DF-based protocols.

4.4.1 Orthogonal selection decode-and-forward protocol

We first consider the ODF protocol for multiple relays. Given a coding rate of Rj = rj log γj ,

denote the set of relays at which the received signal is successfully decoded to be Dj = {ijk},ijk ∈ {1, 2, · · · ,m}, k = 1, . . . ,Nj , where Nj is the cardinality of Dj . We refer to Dj as the

decoding set. Note that a special case is Dj = ∅, for which no cooperation is available.

The maximum achievable rate of the ODF scheme conditioned on Dj can then be written

as follows [60]

C(h, t(rj),Dj) = t(rj) · log(

1 + γ|hs,d|2)

+ (1− t(rj)) · log(

1 + γ∑i∈Dj

|hi,d|2). (4.39)

We now present the optimal distortion exponent achieved by the ODF protocol with

limited feedback in the following theorem.


Theorem 4.4.1. The optimal distortion exponent of the ODF protocol in the multi-relay

cooperative systems with L-level feedback is given by

∆LODF = sup

0<r1<···<rL<1min

0≤l≤L{d∗ODF (rl+1) + brl} . (4.40)

Proof. The proof is given in Appendix 4.B, which utilizes the same technique in the proof of

Proposition 2 in [5]. It is however more involved due to the presence of multiple relays.

It is worth noting that the distortion exponent of the single-relay DF relaying protocol

reported in [5] has the same expression as that in (4.40), which is in fact a special case of

the general multi-relay ODF protocol we investigate here.

We present the following lower bound of ∆LODF , which is exact when b is sufficiently

large.

Proposition 4.4.2. The optimal distortion exponent of the ODF protocol with L-level feed-


∆LODF =

b(m+ 1)2m+ 1

·1−

(b

2m+1

)L1−

(b

2m+1

)L+1. (4.41)

The lower bound ∆LODF is exact, i.e., ∆L

ODF = ∆LODF , when b ≥ b∗, where b∗ = 2m + 1 if

L = 2m+ 1, otherwise, b∗ is the root of the equation

mbL+1 − (m+ 1)(2m+ 1)bL + (2m+ 1)L+1 = 0 (4.42)

that satisfies b∗ > 0 and b∗ 6= 2m+ 1.

Proof. The proof follows along the lines of the proof of Proposition 4.3.4 by imposing the

constraint 0 < r1 < · · · < rL ≤ m2m+1 , which is a straightforward extension, and hence is

omitted.

Although ∆LODF has the same expression as that of ∆L

NAF in (4.25), since the corre-

sponding b∗’s are different, the distortion exponents of the ODF protocol and the NAF

protocol are different in general. However, it does suggest that the ODF protocol and the

NAF protocol have the same distortion exponent performance when the bandwidth ratio is

large.


The distortion exponent of the ODF protocol in the limit of infinite feedback resolution

(L→∞) is given by

∆∞ODF = minr∈[0,1]

d∗ODF (r) + br. (4.43)

By considering the two cases r ∈ [0, m2m+1 ] and r ∈ [ m

2m+1 , 1] separately, we obtain

∆r∈[0, m2m+1

] =

1 + m2m+1b, 0 ≤ b < 2m+ 1

m+ 1, b ≥ 2m+ 1

∆r∈[ m2m+1

,1] =

b, 0 ≤ b < 12(m+1)m

√b− m+1

m − bm , 1 ≤ b < (2m+1)2

(m+1)2

1 + m2m+1b, b ≥ (2m+1)2

(m+1)2

(4.44)

The distortion exponent of the ODF protocol with L→∞ is then given by

∆∞ODF = minr∈[0,1]

{∆r∈[0, m

2m+1],∆r∈[ m

2m+1,1]

}

=

b, 0 ≤ b < 12(m+1)m

√b− m+1

m − bm , 1 ≤ b < (2m+1)2

(m+1)2

1 + m2m+1b,

(2m+1)2

(m+1)2 ≤ b < 2m+ 1

m+ 1, b ≥ 2m+ 1

(4.45)

4.4.2 Nonorthogonal selection decode-and-forward protocol

We now consider the NDF protocol for multiple relays. The maximum achievable rate

C(h, t) of the NDF scheme conditioned on the decoding set Dj is given by [60]

C(h, t(rj),Dj) = t(rj) · log(

1 + γ|hs,d|2)

+ (1− t(rj)) · log(

1 + γ|hs,d|2 + γ∑i∈Dj

|hi,d|2).

(4.46)

The optimal distortion exponent achieved by the NDF protocol with limited feedback is

presented in the following theorem.

Theorem 4.4.3. The optimal distortion exponent of the NDF protocol in the multi-relay

cooperative systems is given by

∆LNDF = sup

0<r1<···<rL<1min

0≤l≤L{d∗NDF (rl+1) + brl} . (4.47)



We present the following lower bound of ∆LNDF , which is exact when b is sufficiently

large.

Proposition 4.4.4. The optimal distortion exponent of the NDF protocol with L-level feed-


∆LNDF =

b(m+ 1)c

· 1− (b/c)L

1− (b/c)L+1, (4.48)

where c , 1+2m+√

1+4m2

2 . The lower bound ∆LNDF is exact, i.e., ∆L

NDF = ∆LNDF , when

b ≥ b∗, where b∗ = c if L = c, otherwise, b∗ is the root of the equation

mbL+1 − c(m+ 1)bL + cL+1 = 0 (4.49)

that satisfies b∗ > 0 and b∗ 6= c.

Proof. The proof follows along the lines of the proof of Proposition 4.3.4 by imposing the

constraint 0 < r1 < · · · < rL ≤ c = 1+2m+√

1+4m2

2 , which is a straightforward extension, and

hence is omitted.

The distortion exponent of the NDF protocol in the limit of infinite feedback resolution

(L→∞) is given by

∆∞NDF = minr∈[0,1]

d∗NDF (r) + br. (4.50)

We obtain, for m ≥ 2,

∆r∈[0,βm] =

1 + 1+2m−√

1+4m2

2 b, 0 ≤ b < βm

m+ 1, b ≥ βm

∆r∈[βm,1] =

b, 0 ≤ b < 1(m+1−r∗)(1−r∗)

(m−1)r∗+1 + br∗, 1 ≤ b < θm

1 + 1+2m−√

1+4m2

2 b, b ≥ θm

(4.51)


where

βm ,1 + 2m+

√1 + 4m2

2,

θm ,m3[

(m− 1)1+2m−√

1+4m2

2 + 1]2

(m− 1)− 1m− 1

,

r∗ ,m

m− 1

√m

1 + b(m− 1)− 1m− 1

.

(4.52)

It can be verified that 1 < θm < βm. Hence, the distortion exponent with L→∞ is

∆∞NDF = minr∈[0,1]

{∆r∈[0,βm],∆r∈[βm,1]

}

=

b, 0 ≤ b < 1(m+1−r∗)(1−r∗)

(m−1)r∗+1 + br∗, 1 ≤ b < θm

1 + 1+2m−√

1+4m2

2 b, θm ≤ b < βm

m+ 1, b ≥ βm

(4.53)


Recall that the distortion exponent upper bound ∆MISO = min{b,m+1} given in Eq. (3.64)

for the m-relay cooperative system assumes perfect CSIT. It is thus also an upper bound of

the feedback-based multi-relay cooperative system.

The distortion exponents for the NAF protocol with feedback resolution L = 2, 4, 8,∞are plotted in Fig. 4.2 for a 2-relay cooperative system. They are compared with that of the

OAF protocol with infinite feedback resolution L = ∞ and the upper bound ∆UB. It can

be seen that with as few as 3 bits of feedback (L = 8), the performance of the NAF protocol

gets very close to the limiting case (L → ∞). It also outperforms the OAF protocol with

infinite feedback resolution at almost all bandwidth ratio b. The corresponding distortion

exponent lower bounds of the NAF protocols are plotted in Fig. 4.2 as well. As expected,

the lower bound becomes exact when the bandwidth ratio b is large.

We show in Fig. 4.3 the distortion exponents for the SAF protocol with feedback reso-

lution L = 8 and ∞ for different number of transmission slots M for a 2-relay cooperative

system under the relay isolation assumption. It can be seen that as the number of trans-

mission slots M increases, the achievable distortion exponents gradually converges to the

upper bound.


0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

Bandwidth ratio, b

Dis

tort

ion

expo

nent

, ∆

Upper boundNAF, L−level feedbackNAF, lower boundOAF, infinite feedback level

L = ∞L = 8

L = 4

L = 2L = 1 (no feedback)

Figure 4.2: Distortion exponents of the NAF protocol and the OAF protocol with differentfeedback resolution L for a 2-relay cooperative system.

It is worth noting that, the distortion exponent of the NAF protocol does not converge

to the upper bound in (3.64) even with the perfect CSI at the source node, i.e., when the

feedback resolution L → ∞, whereas in the sequential SAF protocol, as the number of

transmission slots M → ∞, the distortion exponent can approach the upper bound even

with finite L. However, the optimality of the sequential SAF protocol is achieved under the

relay isolation assumption. Hence, further investigation of the general SAF protocol is still

needed.

In Fig. 4.4, we compare the distortion exponents of the NDF protocol and the ODF

protocol with feedback resolution L = 1, 2, 4,∞ for a 3-relay cooperative system. It can

be seen that the NDF protocol in general outperforms the ODF protocol. However, the

improvement is not significant.

We next compare the distortion exponent performance of the investigated multi-relay

cooperation protocols. The achievable distortion exponents of various cooperation protocols

for a 2-relay cooperative system with infinite feedback resolution are plotted in Fig. 4.5. The

broadcast strategy with relay-selection-based cooperation (BS-RS) in the infinite number of

coding layers studied in Chapter 3 is also included for comparison purpose. The number of

transmission slots is M = 3 in the sequential SAF protocol so that the DMT upper bound in


0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

Bandwidth ratio, b

Dis

tort

ion

expo

nent

, Δ

Upper bound

sequential SAF, L = ∞sequential SAF, L = 8

M = 2

M = 5

M = 3

Figure 4.3: Distortion exponents of the SAF protocol with feedback resolution L = 8 and∞for different transmission slots M for a 2-relay cooperative system under the relay isolationassumption.

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

Bandwidth ratio, b

Dis

tort

ion

expo

nent

, ∆

Upper bound L = ∞L = 4

L = 2

L = 1 (no feedback)

Figure 4.4: Comparison of the distortion exponents of the NDF protocol (solid curves) andthe ODF protocol (dashed curves) with feedback resolution L = 1, 2, 4,∞ for a 3-relaycooperative system.


0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

Bandwidth ratio, b

Dis

tort

ion

expo

nent

, Δ

Upper boundNDFODFSAFNAFOAFBS−RS

Figure 4.5: Comparison of the distortion exponents of various multi-relay cooperation pro-tocols for a 2-relay cooperative system. M = 3 is used in the sequential SAF protocol.

(4.4) is achieved with a simple round-robin scheduling. It is observed that in the limiting case

(L → ∞), the SAF protocol outperforms all the other protocols for almost all bandwidth

ratio b and achieves the upper bound when b ≥ 4, whereas at small bandwidth ratio (b / 3),

the NDF protocol is slightly better. Furthermore, the feedback scheme in general shows

an improvement distortion exponent performance over the broadcast-strategy-based scheme

when combined with different protocols except for the simple OAF protocol.

As shown in Fig. 4.5, even with perfect CSI at the source node (L→∞), all the stud-

ied schemes fail to approach the distortion exponent upper bound at medium bandwidth

ratio (1 ≤ b ≤ 4). One reason is that the distortion exponent upper bound (3.64) assumes

arbitrary cooperation between the source node and all relay nodes. It is still not clear

whether such an upper bound is tight at all bandwidth ratios or not. Hence, further inves-

tigation on tighter bounds is required. Another reason is that the protocols considered are

in general not optimal for multiple-relay systems. Further improvement can be expected

by using advanced schemes with better DMTs such as the dynamic decode-and-forward

(DDF) protocol [55]. Another possible extension is to combine the limited feedback with

sophisticated joint source-channel coding schemes such as the layered source coding with

progressive transmission, the broadcast strategy, or the hybrid digital-analog (HDA) scheme


0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Bandwidth ratio, b

Dis

tort

ion

expo

nent

, Δ

m = 3, L = 2

m = 3, L ↑m ↑, L = 2

m = 4,5,6,7

L = 4,8,16

Figure 4.6: Comparison of the distortion exponents of the NAF protocols with feedbackresolution L = 2, 4, 8, 16 and number of relays m = 3, 4, 5, 6, 7.

[4, 47, 48, 49]. Improvement can also be expected by employing the power control technique

as in [5]. However, efficient combinations of these schemes with multi-relay protocols are

much more involved and remain to be topics for future study.

Finally, we demonstrate by an example the effect of the feedback resolution and the

number of relays on the achievable distortion exponents. Fig. 4.6 shows the distortion

exponents achieved by the NAF protocol with feedback resolution L = 2, 4, 8, 16 and number

of relays m = 3, 4, 5, 6, 7. It is observed that at small bandwidth ratios, increasing the

feedback resolution is more effective in improving the distortion exponent than employing

more relays. The benefit of additional relays is marginal due to the bandwidth limitation.

At large bandwidth ratios, the distortion exponent is in general dominated by the number

of relays m. Hence increasing the feedback resolution can only help approach the upper

bound m+ 1 whereas adding more relays offers much greater performance gain.

4.6 Summary

In this chapter, we study the end-to-end distortion of wireless cooperative systems with

limited feedback. Different from most of the current work, we focus on the multi-relay


scenario and related cooperation strategies such as the orthogonal/nonorthogonal amplify-

and-forward protocols, the sequential slotted amplify-and-forward protocol, and the or-

thogonal/nonorthogonal decode-and-forward protocols. We derive the achievable distortion

exponents in all cases. The results show that a few bits feedback allows the simple sepa-

rate source and channel coding scheme to outperform the non-feedback schemes such as the

broadcast strategy. The feedback-based scheme also enjoys a simplified system design.

4.A Optimality of Equating Linear Terms in (4.11)

We consider here a general form of the optimization problem in (4.11) with relaxed constraint

0 ≤ r1 ≤ · · · ≤ rL ≤ 1 as follows

g = max0≤r1≤···≤rL≤1

min0≤l≤L

{f(rl+1) + wrl} , (4.54)

where w > 0, f(r) = (a − cr)+ for some c > a > 0, r0 , 0 and rL+1 , 1. If the solution

(r∗1, · · · , r∗L) of (4.54) satisfies the strict inequality constraint 0 < r1 < · · · < rL < 1, then

it is also the solution of the original problem in (4.11) with a = m + 1, c = (m + 1)2, and

w = b.

We first show that the solution (r∗1, · · · , r∗L) of the optimization problem in (4.54) is the

same as that of the following problem

g = sup0≤r1≤···≤rL≤ac

min0≤l≤L

{f(rl+1) + wrl} . (4.55)

To see this, assume (r∗1, · · · , r∗L) satisfies 0 ≤ r∗1 ≤ · · · ≤ r∗l−1 ≤ac ≤ r∗l ≤ · · · ≤ r∗L ≤ 1

for some 1 ≤ l ≤ L. Since f(r∗l ) = f(r∗l+1) = · · · = f(r∗L) = 0, we then have

f(r∗l ) + wr∗l−1 ≤wa

c≤ f(r∗l+1) + wr∗l ≤ · · · ≤ wr∗L. (4.56)

This suggests that

minl−1≤l≤L

{f(r∗l+1) + wr∗l

}= wr∗l−1 ≤

wa

c. (4.57)

It is then clear that the choice of (r∗l , r∗l+1, · · · , r∗L) does not change the optimal objective g∗

as long as ac ≤ r∗l ≤ · · · ≤ r∗L ≤ 1. Therefore, we can always let r∗l = r∗l+1 = · · · = r∗L = a

c .

That is to say, the optimization can be constrained over [0, ac ] without losing any optimality.

As a result, the optimization problem in (4.54) is then equivalent to that in (4.55).


We now derive a closed-form solution of the equivalent problem in (4.55). Note that

under the new constraint 0 ≤ r1 ≤ · · · ≤ rL ≤ ac , f(r) = a− cr becomes a linear function of

r. The optimization problem in (4.55) can then be recast as the following linear program

max . g

s.t. f(rl+1) + wrl ≥ g, 0 ≤ l ≤ L,

0 ≤ r1 ≤ · · · ≤ rL ≤a

c.

(4.58)

The Lagrangian of (4.58) is then

J = −g + λ0(g + cr1 − a)

+ λ1(g + cr2 − wr1 − a)

· · ·

+ λL−1(g + crL − wrL−1 − a)

+ λL(g − wrL)

+L−1∑l=0

ξl(rl − rl+1) + ξL(rL −a

c),

(4.59)

where (λ0, · · · , λL) and (ξ0, · · · , ξL) are the associated Lagrange multipliers.

Letting ∂J /∂rl = 0 and ∂J /∂g = 0, we obtain

∂J∂rl

= λl−1c− λlw + ξl − ξl−1 = 0, l = 1, · · · , L, (4.60)

and∂J∂g

= −1 + λ0 + · · ·+ λL = 0. (4.61)

Assume the solution (r∗1, · · · , r∗L) satisfies

0 < r1 < · · · < rL ≤a

c. (4.62)

By Karush-Kuhn-Tucker (KKT) conditions, we require that ξ∗0 = · · · = ξ∗L = 0. The

Langrange multiplier λ∗l is then solved from (4.60) and (4.61) as follows

λ∗l =(w/c)l∑Li=1(w/c)i

, l = 1, · · · , L. (4.63)


Since all λ∗l > 0, by the KKT conditions, the optimal (g∗, r∗1, · · · , r∗L) has to satisfy

g = a− cr1

g = wr1 + a− cr2

· · ·

g = wrL−1 + a− crL

g = wrL

(4.64)

The solution of (4.64) is given by Lemma 4.3.2 as follows

r∗l =

ac ·

1−(w/c)L

1−(w/c)L+1 , l = L,

a−wr∗Lc · 1−(w/c)l

1−(w/c) , 1 ≤ l ≤ L− 1,(4.65)

and

g∗ = wr∗L =aw

c· 1− (w/c)L

1− (w/c)L+1. (4.66)

It can be readily verified that (r∗1, · · · , r∗L) given by (4.65) always satisfies 0 < r1 <

· · · < rL < ac for any a, c, w > 0, which justifies our assumption in (4.62). Hence, the

KKT conditions are satisfied, and (r∗1, · · · , r∗L) is the solution of the optimization problem

in (4.55). Furthermore, since (r∗1, · · · , r∗L) satisfies the strict inequality constraint 0 < r1 <

· · · < rL < 1, it is also the solution of the original problem in (4.11).


To simplify the notation, we use tj to denote the optimized t(rj). We first define the overall

outage set of channel states h for a given coding rate Rj as follows

Oj =⋃Dj

ODj , (4.67)

where

ODj ,{

h : C(h, tj ,Dj) < rj log γ,

tj log(1 + γ|hs,i|2

)< rj log γ, i /∈ Dj ,

tj log(1 + γ|hs,i|2

)≥ rj log γ, i ∈ Dj

} (4.68)

is the outage set of h given that the decoding set is Dj . C(h, tj ,Dj) is the maximum

achievable rate defined in (4.39).


The probability Pl that a feedback index l is sent by the destination node is then

Pl = Pr{h ∈ Ol ∩ Ol+1 ∩ · · · ∩ OL

}, (4.69)

where Ol is the complementary set of Ol.We now show that Pl

.= γ−d∗ODF (rl+1). This result will be used in deriving the expected

end-to-end distortion, and consequently the distortion exponent.

It is clear that

Pl = Pr{h ∈ Ol ∩ Ol+1 ∩ · · · ∩ OL

}≤ Pr {h ∈ Ol+1}.= γ−d

∗ODF (rl+1).

(4.70)

Hence, we have Pl ≤ γ−d∗ODF (rl+1).

To find a lower bound of Pl, we extend the approach that was used in the proof of

Proposition 2 of [5] to multiple relays. Let hs,d = γ−ζ , hs,i = γ−ξi , and hi,d = γ−θi . By

standard large deviation arguments [20], we have, in the limit of γ →∞,

ODj ={

(ζ, θ1, · · · , θm, ξ1, · · · , ξm) :

tj(1− ξi)+ ≥ rj , i ∈ Dj ,

tj(1− ξi)+ < rj , i /∈ Dj ,

tj(1− ζ)+ + (1− tj)(1− mini∈Dj{θi})+ < rj

} (4.71)

and

Pl.= γ− infOl∩Ol+1∩···∩OL∩R(2m+1)+

ζ+∑mi=1(θi+ξi)

≥γ−(ζ∗+∑mi=1 θ

∗i +∑mi=1 ξ

∗i ),

(4.72)

for any (ζ∗, θ∗i , ξ∗i ) ∈ Ol ∩ Ol+1 ∩ · · · ∩ OL ∩ R(2m+1)+.

We now choose (ζ∗, θ∗i , ξ∗i ) as follows

ζ∗ = 1− rl+1

tl+1+ ε, ξ∗i = 1− rl+1

tl+1+ ε, θ∗i = 0, (4.73)

for all i and some ε > 0.

Define

t(r) =

m+1

(2m+1) , 0 ≤ r ≤ m2m+1

1+mr(m+1) ,

m2m+1 ≤ r ≤ 1

(4.74)


such that d∗ODF (r) = (m + 1)(

1− rt(r)

). It can be verified that r

t(r) is a monotonically

increasing function of r, which is in the range of [0, 1].

Therefore, for sufficiently small ε, we have

tl(1− ζ∗) = tl(1− ξ∗i ) = tl

(rl+1

tl+1− ε)≥ rl, (4.75)

and

tj(1− ζ∗) = tj(1− ξ∗i ) = tj

(rl+1

tl+1− ε)< rj , (4.76)

for j = l + 1, · · · , L.

Note that

Ol =⋃Dl

ODl =⋂Dl

ODl (4.77)

and

Oj =⋃Dj

ODj (4.78)

for j = l + 1, · · · , L.

We then have the following results:

• Eq. (4.75) suggests that (ζ∗, θ∗i , ξ∗i ) /∈ ODl , ∀Dl. Hence, by (4.77), (ζ∗, θ∗i , ξ

∗i ) ∈ Ol.

• Eq. (4.76) suggests that (ζ∗, θ∗i , ξ∗i ) ∈ ODj=∅. Hence, by (4.78), (ζ∗, θ∗i , ξ

∗i ) ∈ Oj ,

j = l + 1, · · · , L.

• Since rt(r) ∈ [0, 1], it then follows that ζ∗, ξ∗i ≥ 0 for sufficiently small ε. Hence,

(ζ∗, θ∗i , ξ∗i ) ∈ R(2m+1)+.

To summarize, we have shown that (ζ∗, θ∗i , ξ∗i ) ∈ Ol ∩Ol+1∩ · · · ∩OL∩R(2m+1)+. Hence

Pl ≥ γ−(ζ∗+∑mi=1 θ

∗i +∑mi=1 ξ

∗i )

= γ−(m+1)(1−

rl+1tl+1

)

= γ−d∗ODF (rl+1).

(4.79)

Since the upper bound and lower bound of Pl are both dominated by the term γ−d∗ODF (rl+1),

we can conclude that Pl.= γ−d

∗ODF (rl+1).


As in Eq. (4.7) in the proof of Theorem 4.3.1, the expected end-to-end distortion can

be written as

D =L∑l=0

Pl2−bRl =L∑l=0

Pl2−brl log γ

.=L∑l=0

γ−(d∗ODF (rl+1)+brl)

.= γ−min0≤l≤L{d∗ODF (rl+1)+brl}.

(4.80)

The optimal distortion exponent for the ODF protocol is then found to be

∆LODF = sup

0<r1<···<rL<1min

0≤l≤L{d∗ODF (rl+1) + brl} . (4.81)

This completes the proof.

4.C Proof of Theorem 4.4.3

The proof follows along the same lines as that of the proof of Theorem 4.4.1. Therefore, we

only briefly state the proof sketch by pointing out the main differences.

As in the ODF case, we define the outage sets Oj and ODj according to (4.67) and

(4.68), repsectively. The maximum achievable rate C(h, tj ,Dj) in the expression of ODj is

defined in (4.46). The probability that an index l is sent is again

Pl = Pr{h ∈ Ol ∩ Ol+1 ∩ · · · ∩ OL

}. (4.82)

As in (4.70), an upper bound of Pl can be found to be Pl ≤ γ−d∗NDF (rl+1).

To find a lower bound of Pl, we let hs,d = γ−ζ , hs,i = γ−ξi , and hi,d = γ−θi . By standard

large deviation arguments, we have, in the limit of γ →∞,

ODj ={

(ζ, θ1, · · · , θm, ξ1, · · · , ξm) :

tj(1− ξi)+ ≥ rj , i ∈ Dj ,

tj(1− ξi)+ < rj , i /∈ Dj ,

tj(1− ζ)+ + (1− tj) · (max{1− ζ, 1−mini∈D{θi}})+ < rj

} (4.83)

Define

t(r) =

m√

1+4m2, 0 ≤ r ≤ 1+2m−

√1+4m2

2

1+(m−1)rm+1−r , 1+2m−

√1+4m2

2 ≤ r ≤ 1(4.84)


It can be readily shown that (m + 1)(1 − rt(r)) ≤ d∗NDF (r) for r ∈ [0, 1], and r

t(r) is a

monotonically increasing function of r, which is in the range of [0, 1].

Let ζ∗ = 1 − rl+1

tl+1+ ε, ξ∗i = 1 − rl+1

tl+1+ ε, θ∗i = 0 for all i and some ε > 0. Following

along the lines of the proof of the ODF case, we are able to show that (ζ∗, θ∗i , ξ∗i ) ∈ Ol ∩

Ol+1 ∩ · · · ∩ OL ∩ R(2m+1)+. Hence

Pl ≥ γ−(ζ∗+∑mi=1 θ

∗i +∑mi=1 ξ

∗i )

= γ−(m+1)(1−

rl+1tl+1

)

≥ γ−d∗NDF (rl+1).

(4.85)

By using the upper bound and lower bound of Pl, we can conclude that Pl.= γ−d

∗NDF (rl+1).

The expected end-to-end distortion can then be written as

D =L∑l=0

Pl2−bRl =L∑l=0

Pl2−brl log γ

.=L∑l=0

γ−(d∗NDF (rl+1)+brl)(4.86)

The optimal distortion exponent for the NDF protocol is therefore

∆LNDF = sup

0<r1···<rL<1min

0≤l≤L{d∗NDF (rl+1) + brl} . (4.87)

Chapter 5

Distortion Exponents of Two-way

Relaying Cooperative Networks

5.1 Introduction

In the previous chapters, we investigated two classes of single-user, one-way cooperative

systems with no CSIT or limited channel state feedback, and characterize the end-to-end

performance of the systems by the distortion exponent.

In this chapter, we consider a half-duplex two-way relaying cooperative system, where

two users communicate simultaneously in both directions with the help of one relay at

possibly different rates. Again, we focus on the study of the high-SNR system performance

using distortion exponent, which has not been reported in the literature for two-way relaying

cooperative networks. Different from that of the single-user system, the distortion exponent

pairs achieved by the two users in a two-way relaying system form a distortion exponent

region. To the best of our knowledge, our work is also the first to study such distortion

exponent regions for wireless communication systems, which is another major contribution

of this work. We first derive an outer bound on the distortion exponent region of the two-

way relaying cooperative system. Second, we obtain the optimal distortion exponent pairs

of various transmission schemes, including conventional one-way relaying strategies and two-

way AF/DF/CF relaying protocols with single-rate coding. The distortion exponent results

illustrate the effect of the bandwidth ratio and relaying strategies on the overall performance

97

CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 98

1T

Relay

User 1 User 2

31h23h

32h13h

3T

2T21h

12h1T

Relay

User 1 User 2

31h23h

32h13h

3T

2T21h

12h

Figure 5.1: System model of a two-way relaying communication system.

of the two-way relaying cooperative system. It is shown that, even with the simple single-

rate coding, the two-way relaying protocols can still achieve an improved performance over

sophisticated one-way relaying strategies such as the layered source coding with progressive

transmission or the broadcast strategy [4, 7] at small bandwidth ratio. We also derive

the achievable DMTs of AF, DF, and CF based two-way relaying protocols, which is an

important extension of the DMT theory to the two-way relay channels.

This chapter is organized as follows: In Section 5.2, we present the system model and

the preliminaries on the distortion exponent and two-way relaying cooperative systems.

In Section 5.4, we study the achievable distortion exponent regions of various two-phase

two-way relaying protocols. The achievable distortion exponent regions of three-phase two-

way relaying protocols are studied in Section 5.5. The performance comparison of different

transmission strategies and cooperation protocols are given in Section 5.6. The work in this

chapter is summarized in Section 5.7.

5.2 System Model

We consider a three-node two-way relaying wireless communication system, where two source

nodes T1 and T2 communicate in both directions with the help of a relay node T3. The

system model is shown in Fig. 5.1. All nodes are equipped with single antenna and operate

in half-duplex mode. We assume no CSIT at the sources.

We consider both the multiple-access broadcast (MABC) protocol and the time-division

broadcast (TDBC) protocol for the two-way relaying system (see Section 2.5 for the introduc-

tion of the MABC and TDBC protocols). The relay node employs AF, DF or CF relaying.


The MABC and the TDBC protocols with these relaying strategies will be discussed in more

details in Sec. 5.4 and Sec. 5.5, respectively.

As in Chapter 3 and 4, we consider symmetric, flat, slow fading systems. The channels

are assumed to be Rayleigh fading and statistically symmetric, i.e., the channel coefficients

hij , i, j = 1, 2, 3, i 6= j, are i.i.d. complex Gaussian random variables with zero mean and

unit variance. The additive noises at each receiver are all modeled as CN (0, 1). We assume

all nodes have the same transmitting power. The average received SNRs at both sources

are then equal and are denoted by γ.

Denote s1 and s2 as the source signals transmitted by T1 and T2, respectively, both

are assumed to be memoryless, zero-mean, unit-variance complex Gaussian. Denote R1

and R2 as the transmitted data rates of s1 and s2, respectively. We refer to the system

as a symmetric-rate system if R1 = R2. Assume each user transmits K source samples

in N channel uses. The bandwidth ratio of each user is then b = N/K. Without losing

much generality, this equal bandwidth ratio assumption makes the analysis tractable while

still captures the conceptual nature of the original problem. As before, we assume that

both K and N are large enough so that the source codes can approach the rate-distortion

bound of the source signal and the fixed-rate channel code can be transmitted reliably if

the instantaneous capacity is greater than the communication rate. We also consider the

quasi-static scenario where the channel gain is random but remains constant during the

transmission.

Denote s1 and s2 as the reconstructed source samples at T2 and T1, respectively. The

expected end-to-end distortions are then

D1 = E[(s1 − s1)2], D2 = E[(s2 − s2)2], (5.1)

which are the mean-squared errors between the source signals and their reconstructions at

the destinations. The corresponding distortion exponents are defined by (2.17)

∆1 = − limγ→∞

logD1

log γ, ∆2 = − lim

γ→∞

logD2

log γ, (5.2)

which form the achievable distortion exponent pair (∆1,∆2).

For a pair of codes whose rates (R1, R2) grow as (r1 log γ, r2 log γ) with multiplexing

gains r1 and r2, 0 ≤ r1, r2 ≤ 1, we define the diversity gain pair (d1(r1, r2), d2(r1, r2)) of the

two-way relay channel as follows

d1(r1, r2) = − limγ→∞

logP 1out

log γ, d2(r1, r2) = − lim

γ→∞

logP 2out

log γ, (5.3)


where P 1out and P 2

out are the outage probabilities of the transmissions from T1 to T2 and from

T2 to T1, respectively. The DMT pair (d∗1(r1, r2), d∗2(r1, r2)) is defined as the Pareto optimal

set of (d1(r1, r2), d2(r1, r2)) over all possible families of codes. More precisely, consider

two pairs of codes (f1, f2) and (g1, g2) whose rates both grow as (r1 log γ, r2 log γ) with

multiplexing gains r1 and r2. Denote (df11 (r1, r2), df2

2 (r1, r2)) and (dg11 (r1, r2), dg2

2 (r1, r2)) as

the diversity gain pairs achieved by using codes (f1, f2) and (g1, g2), respectively. We say

that (f1, f2) is better than (g1, g2) if df11 (r1, r2) ≥ dg1

1 (r1, r2) and df22 (r1, r2) ≥ dg2

2 (r1, r2),

where at least one inequality holds strictly. A pair of codes is then said to be Pareto optimal

if there are no better codes, and we refer to the corresponding achieved diversity gain pair

(d∗1(r1, r2), d∗2(r1, r2)) as a Pareto optimal diversity gain pair. The set of all Pareto optimal

diversity gain pairs forms the DMT region.

In the following sections, we study the optimal distortion exponent pair (∆1,∆2) of a two-

way relaying cooperative system with various coding and transmission strategies. Different

from previous works in one-way cooperative communications, we propose and study in this

chapter the new concept of achievable distortion exponent region, which reveals not only

the relationship between the spectral efficiency and the asymptotic overall performance (via

end-to-end distortion) as in [4, 5, 7], but also the tradeoff between different users due to the

impact of possible interferences in a two-way relay channel.

5.3 Distortion Exponent Outer Bound and One-way Relay-

ing Strategies

5.3.1 Outer bound

We first derive in the following theorem a distortion exponent outer bound for the half-

duplex two-way relay channel.

Theorem 5.3.1. The distortion exponent pair (∆1,∆2) of a half-duplex two-way relay chan-

nel is outer-bounded by

∆1 + ∆2 ≤ b, 0 ≤ ∆1 ≤ 2, 0 ≤ ∆2 ≤ 2. (5.4)

Proof. To obtain an outer bound, we assume T1 and T3 fully cooperate, i.e., T1 and T3

always share the same information. The original system then reduces to a two-node half-

duplex two-way MIMO system without any relay, as shown in Fig. 5.2, which consists of a


12h

23h

32h

21h

Relay

User 1

CCooperation node

1T

3TUser 2

2T

12h

23h

32h

21h

Relay

User 1

CCooperation node

1T1T

3TUser 2

2T

Figure 5.2: Equivalent system model for the outer bound.

2× 1 MISO channel for the transmission from T1 to T2 and a 1× 2 SIMO channel for the

transmission from T2 to T1. The same technique is used to derive a DMT upper bound of

the half-duplex two-way relaying system in [84]. We further assume that perfect CSIT is

available at all nodes.

The distortion exponent upper bound of an M × 1 MISO system or a 1 ×M SIMO

system is given in Theorem 3.1 [49] as follows

∆MISO/SIMO = min{b,M}. (5.5)

Assume the cooperation node C transmits for a fraction t of time and T2 transmits for

the remaining fraction 1− t of time, t ∈ [0, 1]. It can be found that the distortion exponent

upper bounds of the transmission in each direction of the equivalent two-way MIMO system

are

∆∗1 = min {2, bt} , ∆∗2 = min {2, b(1− t)} . (5.6)

Eq. (5.6) is a two-user extension of the distortion exponent upper bound for single-user

one-way MIMO channels in (5.5), where t = 1. The proof is similar to that in [49], and thus

is omitted.

The achievable distortion exponent pair satisfies 0 ≤ ∆1 ≤ ∆∗1 and 0 ≤ ∆2 ≤ ∆∗2.

Combining ∆∗1 and ∆∗2 in (5.6) by eliminating t leads to the claimed result.

In the symmetric-rate case, T1 and T2 transmit at the same rate with multiplexing gains

r1 = r2. Due to the symmetry, the maximum achievable distortion exponent pair lies on


the 45◦ line ∆1 = ∆2, whose intersection with the outer bound (5.4) is the corresponding

distortion exponent upper bound, that is, (min{b2 , 2},min

{b2 , 2}

). This corresponds to an

optimal channel use allocation t = 12 . The result is summarized in the following corollary.

Corollary 5.3.2. The distortion exponent of a symmetric-rate half-duplex two-way relay

channel is upper-bounded by

∆∗ = min{b/2, 2}. (5.7)

Comparing (5.7) with the distortion exponent upper bound ∆∗ = min {b, 2} for a single-

relay one-way relay channel in Theorem 3.2 [4], we notice that the following: First, the

maximum of the upper bounds in both cases is 2, which, as will later be shown, is achiev-

able when the bandwidth ratio b is large. This suggests that the maximum achievable

performance of each user in the two-way relaying system is the same as that in a single-

user one-way relaying system, even though the resources are now shared between two users.

Second, under stringent bandwidth limitation, i.e., the bandwidth ratio is small, due to the

half-duplex constraint, in the optimal case, each user uses only half of the total channel uses

for transmission in two-way relaying, which leads to the factor 12 in b

2 , and hence limits the

single-user performance. It is worth noting that the half-duplex constraint on the source

node does not affect the overall performance in one-way relaying. However, its impact can

not be neglected in two-way relaying since each node now acts as both a sender and a

receiver.

5.3.2 One-way relaying strategies

We now try to approach the outer bound in (5.4) by using traditional one-way relaying based

strategies, where the relay takes turns to forward each user’s information. The communica-

tion consists of two independent one-way relay-assisted communications in each direction,

i.e., from T1 to T2 and from T2 to T1. The relay employs the one-way AF or DF cooper-

ation protocol [53]. Overall four phases of transmissions are needed to complete one round

of two-way communication.

We assume the transmission from T1 to T2 uses a fraction t of time, and the transmission

from T2 to T1 uses a fraction 1− t of time. The two corresponding DMTs when AF or DF

relaying protocol is employed at the relay node can be found to be

d∗1(r1) = 2(

1− 2r1

t

)+

, d∗2(r2) = 2(

1− 2r2

1− t

)+

. (5.8)


The DMTs above are direct extensions of those of the one-way AF and DF protocols in [53],

where t = 1. The proof is similar to that in [53], and hence is omitted.

We combine the one-way AF/DF relaying protocol with three transmission techniques,

namely the single-rate coding (SR), the layered source coding with progressive transmission

(LS), and the broadcast strategy (BS) [4]. The LS and BS schemes have been shown to

effectively improve the distortion exponent of one-way relaying communications [4, 7].

The following theorem gives the distortion exponents achieved by one-way AF/DF re-

laying protocol with the SR based transmission.

Theorem 5.3.3. The achievable distortion exponent region of the single-rate coding with

one-way AF/DF relaying protocol for a half-duplex two-way relay channel at bandwidth ratio

b is given by

∆1

2−∆1+

∆2

2−∆2≤ b

4,

0 ≤ ∆1 ≤2bb+ 4

,

0 ≤ ∆2 ≤2bb+ 4

.

(5.9)

Proof. Since the two one-way transmissions are independent of each other, it can be shown

that, the maximum distortion exponent for the transmission from T1 to T2 with the SR

scheme is

∆∗1 =2btbt+ 4

. (5.10)

Similarly, the maximum distortion exponent for the transmission from T2 to T1 is found to

be

∆∗2 =2b(1− t)b(1− t) + 4

. (5.11)

∆∗1 and ∆∗2 are obtained from the DMTs in Eq. (5.8) by generalizing the distortion exponents

of the single-rate coding for one-way relaying in Thm. 3.1 [4], where t = 1. The proof is

similar and hence is omitted.

The achievable distortion exponents (∆1,∆2) satisfy 0 ≤ ∆1 ≤ ∆∗1 and 0 ≤ ∆2 ≤ ∆∗2.

Combining Eq. (5.10) and Eq. (5.11) leads to the claimed results.

Similarly, we derive the achievable distortion exponent region of the LS strategy with

one-way AF/DF relaying protocol.


Theorem 5.3.4. The achievable distortion exponent region of the LS strategy with one-way

AF/DF relaying protocol for a half-duplex two-way relay channel at bandwidth ratio b is

given by

log(

22−∆1

)+ log

(2

2−∆2

)≤ b

4,

0 ≤ ∆1 ≤ 2(1− e−b/4),

0 ≤ ∆2 ≤ 2(1− e−b/4),

(5.12)

where the LS strategy has infinite coding layers.

Proof. Similar to the SR case, by generalizing the distortion exponents of the LS strat-

egy for one-way relaying in Corollary 4.1 [4], the maximum distortion exponents for the

transmissions from T1 to T2 and from T2 to T1 with the LS strategy are found to be

∆∗1 = 2(1− e−bt/4), ∆∗2 = 2(1− e−b(1−t)/4). (5.13)



We present in the following theorem the achievable distortion exponent of the BS strat-

egy.

Theorem 5.3.5. The achievable distortion exponent region of the BS strategy with one-

way AF/DF relaying protocol for a half-duplex two-way relay channel at bandwidth ratio b

is given by

∆1 + ∆2 ≤ b/2, 0 ≤ ∆1 ≤ 2 ≤ ∆2 ≤ 2, (5.14)

where the BS strategy has infinite coding layers.

Proof. By generalizing the distortion exponents of the BS strategy for one-way relaying in

Thm. 4.5 [4], the maximum distortion exponents for the transmissions from T1 to T2 and

from T2 to T1 with the BS strategy are found to be

∆∗1 = min{

2,bt

2

}, ∆∗2 = min

{2,b(1− t)

2

}. (5.15)




A quick examination reveals that with the conventional one-way AF/DF relaying, the

BS strategy achieves the distortion exponent outer bound (5.4) when the bandwidth ratio

b ≥ 8. This also shows that the outer bound is tight at large bandwidth ratio. However,

these one-way relaying schemes are in general not optimal at small bandwidth ratio as will

be shown later.

5.4 Distortion Exponents of MABC Protocols with Single-

rate Source-Channel Coding

In this section, we study the achievable distortion exponents of various MABC protocols

with single-rate source-channel coding for a half-duplex two-way relaying system. We first

derive the DMT of the studied two-way relaying protocol. Based on the obtained DMT,

we derive the corresponding achievable distortion exponent region using single-rate source-

channel coding.

Note that the direct links between the sources are not utilized in the MABC protocols

due to the half-duplex constraint. Therefore, the same distortion exponent results also apply

to the half-duplex two-way relaying system with no direct links between the source nodes.

5.4.1 Decode-and-forward MABC protocol

We first study the DF-based MABC protocol in [75] with single-rate coding. Let user Ti

encode the source si using a channel code of rate Ri = ri log γ bits per channel use, where

ri is the multiplexing gain, and denote the coded symbol as xi, i = 1, 2.

In Phase 1 of the transmission, both source nodes transmit simultaneously to the relay

node for a fraction t of time, t ∈ (0, 1), which resembles a multiple-access channel (MAC).

The received signal at the relay node is then y3 =√γh13x1 +

√γh23x2 + n3, where n3 is

the additive noise. The relay decodes the signals x1 and x2 using joint maximum-likelihood

(ML) decoding, and re-encodes them using a joint codebook. The coded symbol x3 is then

sent to both users in the second phase using the remaining fraction 1 − t of time, which

resembles a broadcast channel (BC). The received signal at source node Ti, i = 1, 2, is then

yi =√γh3,ix3 + ni, where ni is the additive noise at Ti.

The achievable rate region of the two-way relay channel with DF relaying is given in [75]


as follows

R1 < min{t log(1 + γ|h13|2), (1− t) log(1 + γ|h32|2)

},

R2 < min{t log(1 + γ|h23|2), (1− t) log(1 + γ|h31|2)

},

R1 +R2 < t log(1 + γ|h13|2 + γ|h23|2).

(5.16)

Lemma 5.4.1. Given the fraction of channel use t, the achievable DMTs of the DF-based

MABC protocol for a half-duplex two-way relay channel are given by

d1(r1, r2) = min{

1− r1

t, 1− r2

t, 2− 2(r1 + r2)

t, 1− r1

1− t

}+

, (5.17)

d2(r1, r2) = min{

1− r1

t, 1− r2

t, 2− 2(r1 + r2)

t, 1− r2

1− t

}+

. (5.18)

Proof. We first consider the transmission from T1 to T2. Recall that the MABC proto-

col consists of the MAC phase and the BC phase. The probability that the MAC phase

transmission is in outage is characterized by [113]

PMAC−out.= γ−d

∗MAC(r1,r2), (5.19)

where

d∗MAC(r1, r2) = min{

1− r1

t, 1− r2

t, 2− 2(r1 + r2)

t

}+

. (5.20)

Assume the MAC phase transmission is not in outage, i.e., the signals transmitted from

T1 and T2 are both successfully decoded at the relay. After the BC phase transmission, the

signal is in outage at node T2 if

(1− t) log(1 + γ|h32|2

)< R1. (5.21)

Let |hij |2 = γ−θij , i, j = 1, 2, 3 and i 6= j. Since R1 = r1 log γ and R2 = r2 log γ, by

standard large deviation arguments [20], the outage probability of the BC phase transmission

can then be characterized by

P 1BC−out = Pr{(1− t) log

(1 + γ|h32|2

)< r1 log γ}

.= γ−(1− r11−t)

+

.(5.22)


The overall probability that the transmission from T1 to T2 is in outage is then

P 1out = PMAC−out + (1− PMAC−out) · P 1

BC−out

.= PMAC−out + P 1BC−out

.= γ−min

{1− r1

t,1− r2

t,2− 2(r1+r2)

t

}+

+ γ−(1− r11−t)

+

.= γ−min

{1− r1

t,1− r2

t,2− 2(r1+r2)

t,1− r1

1−t

}+

, γ−d1(r1,r2),

(5.23)

where d1(r1, r2) is corresponding achievable DMT.

Using the same arguments, the probability that the transmission from T2 to T1 is in

outage can be found to be

P 2out

.= γ−min

{1− r1

t,1− r2

t,2− 2(r1+r2)

t,1− r2

1−t

}+

, γ−d2(r1,r2),(5.24)

where d2(r1, r2) is corresponding achievable DMT.

For complex Gaussian signals, the distortion-rate function is given by D(Rs) = 2−Rs

[3], where Rs is the source coding rate. With single-rate coding, the expected end-to-end

distortion of the reconstructed signal at T2 is found to be

D1 = (1− P 1out) · 2−bR1 + P 1

out.= γ−br1 + γ−d1(r1,r2)

.= γ−min{br1,d1(r1,r2)}

, γ−∆1 ,

(5.25)

where ∆1 is the corresponding distortion exponent.

Plugging the DMT d1(r1, r2) in (5.17) into (5.25), we then have

∆1 = min{br1, 1−

r1

t, 1− r2

t, 2− 2(r1 + r2)

t, 1− r1

1− t

}+

. (5.26)

Similarly, the distortion exponent of the reconstructed signal at T1 is found to be

∆2 = min{br2, 1−

r1

t, 1− r2

t, 2− 2(r1 + r2)

t, 1− r2

1− t

}+

. (5.27)

It is not immediately clear whether closed-form solutions or efficient algorithms exist

for obtaining all Pareto optimal distortion exponent pairs (∆1,∆2) over (r1, r2, t). Instead,

we derive an outer bound of (∆1,∆2) in the following theorem, which is tight when the

bandwidth ratio b is large.


Theorem 5.4.2. The distortion exponent pair (∆1,∆2) of the DF-based MABC protocol

with single-rate source-channel coding at bandwidth ratio b is outer-bounded as follows

0 ≤ ∆1 ≤b

b+ 2, 0 ≤ ∆2 ≤

b

b+ 2. (5.28)

This outer bound is tight when b ≥ 4.

Proof. We first bound ∆1 in (5.26) by its global maximum. Note that ∆1 is a non-increasing

function of r2, to obtain the global maximum of ∆1, we can always let r2 = 0. The global

maximum of ∆1 can then be obtained by solving the following linear-fractional program,

which can be recast and solved as a linear program [112]

maxr1,t

min{br1, 1−

r1

t, 1− r1

1− t

}s.t. r1 ≤ t ≤ 1− r1,

0 ≤ t ≤ r1,

0 ≤ r1 ≤ 1.

(5.29)

The closed-form solution of (5.29) can then be obtained as follows

r∗1 =1

b+ 2, t∗ =

12, (5.30)

for which the maximum of ∆1 is given by ∆∗1 = bb+2 .

It turns out that for ∆∗1 to be achievable, it is not always necessary for r2 to be zero.

In fact, it can be verified that ∆∗1 can be achieved with r∗1 and t∗ in (5.30) as long as the

following conditions all hold:

max{br∗1, 1−

r∗1t∗, 1− r∗1

1− t∗

}≤ 1− r2

t∗,

max{br∗1, 1−

r∗1t∗, 1− r∗1

1− t∗

}≤ 2− 2(r∗1 + r2)

t∗,

r2 ≤ t∗ ≤ 1− r2,

r∗1 + r2 ≤ t∗,

0 ≤ r2 ≤ 1.

(5.31)

Solving the above inequalities leads to the following constraints on the optimal r2:

r∗2 ∈

[0, b4(b+2) ], 0 ≤ b < 4,

[0, 1b+2 ], b ≥ 4.

(5.32)


Therefore, ∆∗1 is achievable as long as the triplet (r∗1, r∗2, t∗) satisfies both (5.30) and (5.32).

Similarly, it can be shown that the maximum of ∆2 is given by ∆∗2 = bb+2 , where the

optimal triplet (r∗1, r∗2, t∗) satisfies

r∗1 ∈

[0, b4(b+2) ], 0 ≤ b < 4,

[0, 1b+2 ], b ≥ 4,

r∗2 =1

b+ 2, t∗ =

12.

(5.33)

An outer bound of the distortion exponent region is thus

0 ≤ ∆1 ≤b

b+ 2, 0 ≤ ∆2 ≤

b

b+ 2. (5.34)

For the outer bound to be tight, the optimal distortion exponent pair (∆∗1,∆∗2) has to

be achievable. From (5.30), (5.32) and (5.33), we see that ∆∗1 and ∆∗2 can be achieved

simultaneously by letting r1 = r2 = 1b+2 and t = 1

2 when b ≥ 4.

We now consider the symmetric-rate case (r1 = r2 = r), where the maximum distortion

exponents can be found explicitly.

It has been shown that the DMTs of the two-way DF relaying in the symmetric-rate

case satisfy [84]

d∗1(r, r) = d∗2(r, r) , d∗(r) =

1− 2r, 0 ≤ r < 16 ,

2− 4rβ∗ ,

16 ≤ r ≤

13 ,

(5.35)

where β∗ ,5r+1−

√(5r+1)2−16r

2 .

With single-rate source-channel coding, the expected end-to-end distortion of the recon-

structed signal at T2 is again given by (5.25). By letting d1(r1, r2) = d∗(r), the achiev-

able distortion exponent can be written as ∆1 = min{br, d∗(r)}, which also suggests that

∆1 = ∆2.

The maximum distortion exponent is thus

∆∗ = maxr∈[0, 1

3]min{br, d∗(r)}

= max

{minr∈[0, 1

6]

{br, 1− 2r

}, minr∈[ 1

6, 13

]

{br, 2− 4r

β∗

}}

=

3(b+2)−

√(b−2)2+32

2(b+5) , 0 ≤ b < 4,

bb+2 , b ≥ 4.

(5.36)


5.4.2 Amplify-and-forward MABC protocol

We now study the achievable distortion exponent region of the AF-based MABC protocol

[74]. In the first time slot, T1 and T2 transmit their coded symbols x1 and x2 to the relay T3

simultaneously. The received signal at the relay node is again y3 =√γh13x1 +

√γh23x2 +n3,

where n3 is the additive noise.

In the second time slot, the relay scales the received signal y3 by a factor of g and

broadcasts it back to T1 and T2. We assume short-term power constraint and the amplifying

factor g is chosen to be [74]

g =

√1

γ|h13|2 + γ|h23|2 + 1. (5.37)

The received signals at T1 and T2 are then given by

y1 =√γh31gh13x1 +

√γh31gh23x2 + h31gn3 + n1, (5.38)

y2 =√γh32gh13x1 +

√γh32gh23x2 + h32gn3 + n2, (5.39)

respectively, where ni is the additive noise at Ti.

Assume full CSI of all links is available at T1 and T2 at the time of decoding. This

strong assumption about CSI is required by most AF-based two-way relaying protocols,

and is often made in the literature, for example, in [74] and [76]. Since T1 already knows

x1, it can then subtract x1 (the back-propagating self-interference) from the received signal

y1, which is known as the self-interference cancellation [74]. x2 is then decoded from the

residual signal y1 =√γh31gh23x2 + h31gn3 + n1.

Similarly, after self-interference cancellation is performed at T2, and x1 is then decoded

from the residual signal y2 =√γh32gh13x1 + h32gn3 + n2.

The achievable rates of the two-way AF relaying are then given by [74] as follows

R1 <12

log(

1 +γ2|h32|2|h13|2

1 + γ(|h32|2 + |h23|2 + |h13|2)

), (5.40)

R2 <12

log(

1 +γ2|h31|2|h23|2

1 + γ(|h31|2 + |h13|2 + |h23|2)

). (5.41)

Theorem 5.4.3. The DMTs of the AF-based MABC protocol for a half-duplex two-way

relay channel are given by

d∗1(r1, r2) = (1− 2r1)+, d∗2(r1, r2) = (1− 2r2)+. (5.42)


Proof. The proof is given in Appendix 5.A.

It is worth pointing out that an achievable DMT of AF-based MABC protocol has

previously been reported in [76]. However, their result assumes equal rate for the two sources

and characterizes the tradeoff between the sum rate and the system outage probability, which

is hence a special case of (5.42) by restricting r1 = r2. Moreover, we provide an alternative

approach to derive the DMTs by using the large-deviation arguments, which is different

from the finite-SNR analysis in [76].

Theorem 5.4.4. The distortion exponent region (∆1,∆2) of the AF-based MABC protocol

with single-rate source-channel coding at bandwidth ratio b is given by

0 ≤ ∆1 ≤b

b+ 2, 0 ≤ ∆2 ≤

b

b+ 2. (5.43)

Proof. As in the DF case (Eq. (5.25)), with single-rate source-channel coding, the distortion

exponent of the reconstructed signal at T2 can be found to be

∆1 = min{br1, d∗1(r1, r2)}, (5.44)

where d∗1(r1, r2) is the corresponding DMT.

By using the DMT results in Eq. (5.42) Theorem 5.4.3, the maximum of ∆1 is thus

found to be

∆∗1 = maxr1∈[0,1]

min{br1, (1− 2r1)+} =b

b+ 2. (5.45)

Similarly, the maximum distortion exponent of the signal received at T1 is found to be

∆∗2 = maxr2∈[0,1]

min{br1, (1− 2r2)+} =b

b+ 2. (5.46)

Since ∆∗1 and ∆∗2 are optimized independently over r1 and r2, the optimal distortion

exponent pair (∆∗1,∆∗2) is then achievable. The achievable distortion exponent pair satisfies

0 ≤ ∆1 ≤ ∆∗1 and 0 ≤ ∆2 ≤ ∆∗2. Combining Eq. (5.45) and Eq. (5.46) leads to the claimed

result.

The following comments are in order. 1) Unlike in the DF case, the maximum distortion

exponents (∆∗1,∆∗2) of the AF-based MABC protocol can always be achieved simultaneously.

This is due to the complete cancellation of the self-interference, which however requires

perfect CSI at the sources at the time of decoding. 2) The AF-based MABC protocol does


not require fully decoding both signals at the relay node, which contributes to an improved

maximum multiplexing gain of 12 compared to 1

3 as in the DF-based MABC protocol [84].

The improved bandwidth efficiency also allows the AF-based protocol to achieve a better

distortion exponent than that of the DF-based protocol, especially at low bandwidth ratio.

Whether an improved performance is still possible for the two-way AF relaying with limited

CSI remains a topic that needs further investigation, and is beyond the scope of this work.

5.4.3 Compress-and-forward MABC protocol

In [67], it has been shown that the CF scheme achieves the DMT of the one-way relay

channel for both the full-duplex and the half-duplex case. We now consider the CF-based

MABC protocol, whose achievable rate region and DMT have been previously studied in

[85, 87, 86].

In the first phase of the CF-based MABC protocol, T1 and T2 transmit simultaneously

to the relay T3 for a fraction t of time, t ∈ (0, 1). The received signal at the relay node is

hence y3 =√γh13x1 +

√γh23x2 + n3, where n3 is the additive noise.

Different from the AF and DF based protocols, the relay compresses (quantizes) the

received signal y3 into y3. Using the optimal Gaussian quantizers (in the rate-distortion

sense), the relationship between y3 and y3 can be modeled by the following equivalent

channel [87, 88]

y3 = y3 + nq, (5.47)

where nq is the compression noise, which is assumed to be circularly symmetric complex

Gaussian with variance σ2q .

The quantization index w3 is then mapped to a codeword x3 and broadcasted back to

T1 and T2 in the second phase of transmission using a fraction 1− t of time. The received

signals at T1 and T2 are then given by

y1 =√γh31x3 + n1, y2 =

√γh32x3 + n2, (5.48)

where ni is the additive noise at Ti.

After phase 2 transmission, the decoder at T1 knows the signals x1 and y1. By treat-

ing x1 as the correlated side information, x2 can be decoded using the Wyner-Ziv coding

mechanisms [85, 86]. Similarly, x1 is decoded at at T2 by using x2 as the correlated side

information.


The achievable rate pair (R1, R2) of the CF-based MABC protocol is shown to be [85, 86]

R1 < t log(

1 +γ|h13|2

1 + σ2q

), (5.49)

R2 < t log(

1 +γ|h23|2

1 + σ2q

), (5.50)

where the fraction t and the compression noise nq satisfy

t log(

1 +γ|h13|2 + 1

σ2q

)< (1− t) log

(1 + γ|h32|2

), (5.51)

t log(

1 +γ|h23|2 + 1

σ2q

)< (1− t) log

(1 + γ|h31|2

). (5.52)

We can rewrite (5.51) and (5.52) as follows

σ2q >

γ|h13|2 + 1

(1 + γ|h32|2)1−tt − 1

, σ2q1 , (5.53)

σ2q >

γ|h23|2 + 1

(1 + γ|h31|2)1−tt − 1

, σ2q2 . (5.54)

The general case of CF protocol is difficult to analyze due to the multiple variables

that are involved. Instead, we consider a suboptimal fixed and static CF protocol for its

tractability, where the fraction t of the transmission time is a constant that does not depend

on either the code rate or the channel coefficients. This assumption allows a better insight

into the general problem. Also, we assume that the quantizer is designed to have a quanti-

zation error with variance σ2q = min{σ2

q1, σ2q2} such that the achievable rates are maximized.

The achievable rates of the CF scheme can then be written as

R1 < t log(

1 +γ|h13|2

1 + σ2q

)= max

{t log

(1 +

γ|h13|2

1 + σ2q1

), t log

(1 +

γ|h13|2

1 + σ2q2

)},

R2 < t log(

1 +γ|h23|2

1 + σ2q

)= max

{t log

(1 +

γ|h23|2

1 + σ2q1

), t log

(1 +

γ|h23|2

1 + σ2q2

)}.

(5.55)


Theorem 5.4.5. The DMTs of the fixed and static CF-based MABC protocol for a half-

duplex two-way relay channel are given by

d∗1(r1, r2) = (1− 2r1)+, d∗2(r1, r2) = (1− 2r2)+. (5.56)

Proof. The proof is given in Appendix 5.B.

Comparing (5.56) to (5.42), we see that the AF and CF protocols achieve the same DMT.

Since the distortion exponent with single-rate source-channel coding is determined by the

diversity gains d∗1(r1, r2) and d∗2(r1, r2), it is immediately clear that the CF and AF protocols

achieve the same maximum distortion exponent, which leads to the following theorem.

Theorem 5.4.6. With single-rate source-channel coding, the achievable distortion exponent

region (∆1,∆2) of the CF-based MABC protocol is the same as that of the AF-based MABC

protocol, that is, at bandwidth ratio b,

0 ≤ ∆1 ≤b

b+ 2, 0 ≤ ∆2 ≤

b

b+ 2. (5.57)

5.5 Distortion Exponents of TDBC Protocols with Single-

rate Source-Channel Coding

In this section, we study the achievable distortion exponents of various TDBC protocols with

single-rate source-channel coding for a half-duplex two-way relaying system. Unlike in the

MABC protocols, the direct links between the source nodes can be utilized in the TDBC

protocols. As will be shown later, this additional degree of freedom in general offers an

improvement in the distortion exponent, which makes the TDBC protocols more preferable

under our system model.

5.5.1 Decode-and-forward TDBC protocol

We first study the DF-based TDBC protocol in [75] with single-rate coding. Again, let user

Ti encode the source si using a channel code of rate Ri = ri log γ bits per channel use, where

ri is the multiplexing gain, i = 1, 2. In the first phase, T1 broadcasts the coded symbol x1

to T2 and T3 using a fraction t1 of time. The received signals at T2 and T3 are then

y12 =√γh12x1 + n1

2, y13 =√γh13x1 + n1

3, (5.58)


respectively, where n12 and n1

3 are the additive noise.

In the second phase, T2 broadcasts the coded symbol x2 to T1 and the relay T3 using a

fraction t2 of time. The received signals at T1 and T3 are then

y11 =√γh21x2 + n1

1, y23 =√γh23x2 + n2

3, (5.59)

respectively, where n11 and n2

3 are the additive noise.

The relay decodes x1 and x2 from the received signals y13 and y2

3, and re-encodes them

using a joint codebook. The coded symbol x3 is then sent to both users in the third phase

using the remaining t3 = 1− t1− t2 fraction of time. The received signal at source node Ti,

i = 1, 2, is then y2i =√γh3,ix3 + n2

i , where n2i is the additive noise at Ti.

We first derive the DMT of the DF-based TDBC protocol, which has been studied

recently in [83] for a symmetric-rate system. However, their result does not fully characterize

the performance of each user, and can be treated as a special case of our result.

Theorem 5.5.1. The DMTs of the DF-based TDBC protocol are given by, when d∗1(r1, r2) 6=0 and d∗2(r1, r2) 6= 0,

d∗1(r1, r2) =

2− 5r1, (r1, r2) ∈ A1

2(1−r1−r2)1−r2+2r1

, (r1, r2) ∈ A2

2(1− 2r11−r2 ), (r1, r2) ∈ A3

2(1−r1)1+3r1

, (r1, r2) ∈ A4

(5.60)

d∗2(r1, r2) =

2− 5r2, (r1, r2) ∈ A1

2(1−r1−r2)1−r1+2r2

, (r1, r2) ∈ A2

2(1−r2)1+3r2

, (r1, r2) ∈ A3

2(1− 2r21−r1 ), (r1, r2) ∈ A4

(5.61)

where

A1 = {(r1, r2) : 0 ≤ r1 < 1/5, 0 ≤ r2 < 1/5},

A2 = {(r1, r2) : 4r1 + r2 ≥ 1, r1 + 4r2 ≥ 1, r1 + r2 < 1},

A3 = {(r1, r2) : 4r1 + r2 < 1, 1/5 ≤ r2 < 1},

A4 = {(r1, r2) : r1 + 4r2 < 1, 1/5 ≤ r1 < 1}.

(5.62)


The corresponding optimal fractions of channel uses are

(t1, t2, t3) =

(25 ,

25 ,

15), (r1, r2) ∈ A1

(1+2r1−r23 , 1+2r2−r1

3 , 1−r1−r23 ), (r1, r2) ∈ A2

(1−r22 , 1+3r2

4 , 1−r24 ), (r1, r2) ∈ A3

(1+3r14 , 1−r1

2 , 1−r14 ), (r1, r2) ∈ A4

(5.63)


In Theorem 5.5.1, we restrict that d∗1(r1, r2) 6= 0 and d∗2(r1, r2) 6= 0. It can be easily

shown that, when d∗1(r1, r2) = 0, by letting t1 → 0, we have

d∗2(r1, r2) =

2− 3r2, r1 ≥ 13 , 0 ≤ r2 <

13

2(1−r2)1+r2

, 2r1 + r2 ≥ 1, 13 ≤ r2 < 1

(5.64)

Similarly, if d∗2(r1, r2) = 0, the optimal d∗1(r1, r2) is obtained by letting t2 → 0, which is

given by

d∗1(r1, r2) =

2− 3r1, 0 ≤ r1 <13 , r2 ≥ 1

3

2(1−r1)1+r1

, r1 + 2r2 ≥ 1, 13 ≤ r1 < 1

(5.65)

It can be verified that (5.64) and (5.65) reduce to the optimal DMTs of the one-way

variable orthogonal selection decode-and-forward relaying protocol in [60]. In fact, since we

allow the DMT of the transmission of one direction to be zero, it is natural to allocate almost

all available transmission slots to the transmission of the other direction. The optimized

system then resembles the conventional one-way relaying system. The orthogonality comes

from the nature of the three-phase protocol.

As in the DF-based MABC protocol, we derive an outer bound of (∆1,∆2) in the

following theorem, which is tight when the bandwidth ratio is large.

Theorem 5.5.2. The distortion exponent pair (∆1,∆2) of the DF-based TDBC protocol

with single-rate source-channel coding at bandwidth ratio b is outer-bounded as follows

0 ≤ ∆1 ≤2bb+ 5

, 0 ≤ ∆2 ≤2bb+ 5

. (5.66)

This outer bound is tight when b ≥ 5.


Proof. We first consider the transmission from T1 to T2. As in Eq. (5.44), the distortion

exponent of the reconstructed signal at T2 is given by

∆1 = min{br1, d

∗1(r1, r2)

}. (5.67)

We now bound ∆1 by its global maximum ∆∗1, which is given by

∆∗1 = max1≤i≤4

∆∗1,i, (5.68)

where

∆∗1,i , max(r1,r2)∈Ai

min{br1, d

∗1(r1, r2)

}(5.69)

is the local maximum of ∆1 when (r1, r2) ∈ Ai.Plugging in the expression of d∗1(r1, r2) in Eq. (5.60), we are able to solve each ∆∗1,i

analytically. As a result, the global maximum of ∆1 is obtained as

∆∗1 =

−(b+2)+

√(b+2)2+24b

6 , 0 ≤ b < 5,

2bb+5 , b ≥ 5,

(5.70)

where the optimal multiplexing gain pair (r∗1, r∗2) satisfiesr∗1 = −(b+2)+

√(b+2)2+24b

6b , r∗2 ≤1−r∗1

4 , 0 ≤ b < 5,

r∗1 = 2b+5 , r

∗2 ≤ 1

5 , b ≥ 5.(5.71)

Similarly, the global maximum of ∆2 is obtained as

∆∗2 =

−(b+2)+

√(b+2)2+24b

6 , 0 ≤ b < 5,

2bb+5 , b ≥ 5,

(5.72)

where the optimal multiplexing gain pair (r∗1, r∗2) satisfiesr∗1 ≤

1−r∗24 , r∗2 = −(b+2)+

√(b+2)2+24b

6b , 0 ≤ b < 5,

r∗1 ≤ 15 , r

∗2 = 2

b+5 , b ≥ 5.(5.73)

Noticing that −(b+2)+√

(b+2)2+24b

6 ≤ 2bb+5 when 0 ≤ b < 5, an outer bound of the achiev-

able distortion exponent pair is thus

0 ≤ ∆1 ≤2bb+ 5

, 0 ≤ ∆2 ≤2bb+ 5

. (5.74)

To show that this outer bound is exact when b ≥ 5, we only need to show (∆∗1,∆∗2) is

achievable. This can be realized by letting r∗1 = r∗2 = 2b+5 . In this case, we have (r∗1, r

∗2) ∈ A1,

and the corresponding optimal (t1, t2, t3) is then (25 ,

25 ,

15).


In the symmetric-rate case (R1 = R2), by letting r1 = r2 = r in (5.60) and (5.61), the

corresponding DMT is found to be, which agrees with that in [84]

d∗(r) =

2− 5r, 0 ≤ r < 15 ,

2−4r1+r , r ≥ 1

5 .(5.75)

As in the MABC case, the optimal distortion exponent pair is then given by (∆∗,∆∗),

which can be found explicitly as follows

∆∗ = max0≤r≤1

{br, d∗(r)} =

−(b+4)+

√(b+4)2+8b

2 , 0 ≤ b < 5,

2bb+5 , b ≥ 5.

(5.76)

5.5.2 Amplify-and-forward TDBC protocol

We next study the AF-based TDBC protocol in [76]. In the first time slot, T1 transmits to

both T2 and T3, whose received signals are y12 and y1

3, respectively, as in Eq. (5.58). In the

second time slot, T2 transmits to both T1 and T3, whose received signals are y11 and y2

3,

respectively, as in Eq. (5.59).

The relay linearly combines the received signals y13 and y2

3 as follows

x3 = g1y13 + g2y

23 =√γg1h13x1 +

√γg2h23x2 + g1n

13 + g2n

23, (5.77)

where the amplifying factors g1 and g2 are chosen as follows to satisfy the relaying power

constraint with 0 < η < 1 [76]

g1 =√

η

γ|h13|2 + 1, g2 =

√(1− η)

γ|h23|2 + 1. (5.78)

The combined signal y3 is then broadcasted to T1 and T2. The received signals at T1

and T2 in the third time slot are then given by

y21 =√γh31g1h13x1 +

√γh31g2h23x2 + h31g1n

13 + h31g2n

23 + n2

1, (5.79)

y22 =√γh32g1h13x1 +

√γh32g2h23x2 + h32g1n

13 + h32g2n

23 + n2

2, (5.80)

where n2i is the additive noise at Ti.

After self-interference cancellation, the residual signals obtained at T2 and T1 are given

by

y21 =√γh31g2h23x2 + h31g1n

13 + h31g2n

23 + n2

1,

y22 =√γh32g1h13x1 + h32g1n

13 + h32g2n

23 + n2

2.(5.81)


The achievable rates of the three-phase two-way AF relaying are then given by [76] as

follows

R1 <13

log(

1 + γ|h12|2 +γ2|h32|2|g1|2|h13|2

γ|h32|2(|g1|2 + |g2|2) + 1

), (5.82)

R2 <13

log(

1 + γ|h21|2 +γ2|h31|2|g2|2|h23|2

γ|h31|2(|g1|2 + |g2|2) + 1

). (5.83)

In [76], a DMT of the AF-based TDBC protocol is reported, which characterizes the

relationship between the sum rate and the system outage probability, i.e., the probability

that either user is in outage. The corresponding DMT is given by [76]

d(r) = (2− 3r)+. (5.84)

In the following theorem, we also derive the DMTs of the AF-based TDBC protocol.

Different from the system perspective in [76], our results address the problem on a finer

scale and characterize the achievable performance of each user.

Theorem 5.5.3. The DMTs of the AF-based TDBC protocol for a half-duplex two-way

relay channel are given by

d∗1(r1, r2) = (2− 6r1)+, d∗2(r1, r2) = (2− 6r2)+. (5.85)

Proof. The proof is given in Appendix 5.D.

Although the DMT presented in [76] (Eq. (5.84)) characterizes the two-way relaying

system from a different perspective, it is not hard to obtain their result from ours. To see

this, let r = r1 + r2 be the multiplexing gain corresponding to the sum rate and denote

the outage probabilities of each user to be P 1out and P 2

out, respectively. The system outage

probability is therefore

Pout = 1− (1− P 1out) · (1− P 2

out).= 1− (1− γ−(2−6r1)+

) · (1− γ−(2−6r2)+)

.= γ−min{(2−6r1)+,(2−6r2)+}.

(5.86)

The corresponding DMT is hence

d(r) = maxr1+r2=r,r1≥0,r2≥0

min{(2− 6r1)+, (2− 6r2)+} = (2− 3r)+, (5.87)


which is the same as (5.84). This also suggests that our result is more fundamental than

that in [76].

We now use the DMTs in (5.85) to derive the achievable distortion exponent region of

the AF-baed TDBC protocol with single-rate coding.

Theorem 5.5.4. The distortion exponent region (∆1,∆2) of the AF-based TDBC protocol

with single-rate source-channel coding at bandwidth ratio b is given by

0 ≤ ∆1 ≤2bb+ 6

, 0 ≤ ∆2 ≤2bb+ 6

. (5.88)

Proof. As in (5.44), the distortion exponent of the reconstructed signal at T2 is given by

∆1 = min{br1, d∗1(r1, r2)}, (5.89)

where d∗1(r1, r2) is the corresponding DMT.

By Theorem 5.5.3, the maximum of ∆1 is thus given by

∆∗1 = maxr1∈[0,1]

min{br1, (2− 6r1)+} =2bb+ 6

. (5.90)

Similarly, the maximum distortion exponent of the signal received at T1 is found to be

∆∗2 = maxr2∈[0,1]

min{br2, (2− 6r2)+} =2bb+ 6

. (5.91)

The achievable distortion exponent pair satisfies 0 ≤ ∆1 ≤ ∆∗1 and 0 ≤ ∆2 ≤ ∆∗2.

Combining Eq. (5.90) and Eq. (5.91) leads to the claimed result.

As in the AF-based MABC protocol, the maximum distortion exponents ∆∗1 and ∆∗2 of

the AF-based TDBC protocol can be simultaneously achieved at any bandwidth ratio. This

is due to the complete cancellation of the self-interference, which however requires perfect

CSI at the sources at the time of decoding.


The distortion exponents of different cooperation protocols for a two-way relaying coop-

erative network are plotted in Fig. 5.3 and Fig. 5.4 for bandwidth ratio b = 1 and 8,

respectively. The maximum achievable distortion exponents of the DF-based MABC and

TDBC protocols are numerically computed when b = 1, and are given explicitly by (5.28)


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distortion exponent, Δ1

Dis

tort

ion

expo

nent

, Δ2

Outer boundOne−way SROne−way LSOne−way BSMABC, AF/CFMABC, DFTDBC, AFTDBC, DF

Figure 5.3: Comparison of various source-channel transmission schemes in a two-way relay-ing cooperative system for b = 1.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Distortion exponent, Δ1

Dis

tort

ion

expo

nent

, Δ2


Figure 5.4: Comparison of various source-channel transmission schemes in a two-way relay-ing cooperative system for b = 8. Note that the outer bound is achieved by the one-way BSstrategy. Also, the curves of the AF/CF-based MABC protocol and the DF-based MABCprotocol coincide.


and (5.66), respectively, when b = 8. Note that in Fig. 5.4, the outer bound is achieved

by the one-way BS strategy, and the curves of the AF/CF-based MABC protocol and the

DF-based MABC protocol coincide.

It can be seen from Fig. 5.3 and Fig. 5.4 the distortion exponent region of the DF-

based MABC protocol is outer-bounded by that the AF-based MABC protocol. This is

because no full decoding is required at the relay node in the AF-based protocol and complete

cancellation of self-interference is always performed at the source node, both contribute to

the improved performance of the AF-based MABC protocol. This however imposes the much

stronger CSI requirement, and the performance of the AF or CF protocol under the limited

CSI assumption remains a topic for future study. On the contrary, it can be seen that the

AF-based TDBC protocol is always outperformed by the DF-based TDBC protocol, even

though the stronger CSI assumption is still made for the AF-based protocol. This is because

unlike in the MABC protocol, there is no interference in decoding the received signals at the

relay in the DF-based TDBC protocol. Furthermore, the DF-based TDBC protocol is able

to allocate channel uses for different phases more flexibly, whereas the number of channel

uses is restricted to be the same for all three phases in the AF-based TDBC protocol.

Comparing the two-way relaying protocols with one-way relaying strategies, we see in

Fig. 5.3 that, at small bandwidth ratio, even with the simple single-rate coding (SR), an

improved performance of the two-way relaying protocol can still be observed when compared

with the sophisticated one-way relaying schemes such as the LS and BS strategies due to

the improved spectral efficiency. However, this improvement rapidly diminishes at large

bandwidth ratio, as in Fig. 5.4, where the performance loss of the single-rate coding to

the more sophisticated layered source coding offsets the gain from the two-way relaying

protocol. This observation is made clearer in Fig. 5.5 by looking at the symmetric-rate

system, where the maximum distortion exponent at a given bandwidth ratio b is equal for

both users and can be found at the intersection between the 45◦ line ∆1 = ∆2 and the

boundary of the corresponding distortion exponent region. It can also be seen that, with

single-rate coding, the two-way relaying protocol is strictly better than the one-way relaying

protocol in a symmetric-rate system.

We are also interested in the performance comparison between the MABC and the TDBC

protocols in the studied two-way relaying cooperative system. From Fig. 5.4 and Fig. 5.5,

it can be seen that the TDBC protocols are in general better than the MABC protocols,

especially when the bandwidth ratio is large. This is not unexpected since the MABC


0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Bandwidth ratio, b

Dis

tort

ion

expo

nent

, Δ


Figure 5.5: Comparison of various source-channel transmission schemes in a symmetric-ratetwo-way relaying cooperative system.

protocols cannot utilize the direct links between the sources as in the TDBC protocols, which

in turn limits the maximum achievable distortion exponent to be one. Nevertheless, it is still

possible for the MABC protocols to outperform the TDBC protocols in the low bandwidth

ratio regime by taking advantage of the spectrally efficient multiple-access channel in phase

1 transmission, as shown in Fig. 5.3. However, the improvement is only marginal, and can

only be observed when compared with the less efficient AF-based TDBC protocol. Although

the MABC protocol appears to be not efficient under our system model, it has been shown

that it in general outperforms the TDBC protocol for the case where there is no direct

link between the two sources [75, 76, 86]. The distortion exponent analysis performed in

this chapter can be easily extended to this no-direct-link case, which however is beyond the

scope of this thesis.

From Fig. 5.4 and Fig. 5.5, it can be seen that the outer bound is achieved by the

one-way BS strategy when the bandwidth ratio b is large. However, all studied schemes

fail to approach the outer bound at small bandwidth ratio. This is mainly because: First,

the cooperation protocols studied in this work are not DMT-optimal over all multiplexing

gains, which hence limits the distortion exponent performance. Second, since the upper

bound assumes full CSI at all nodes as well as perfect cooperation between one source node


and the relay, it is still not clear whether such optimality can be achieved or not. Therefore,

further investigation on sophisticated transmission strategies that improve the distortion

exponent of two-way relay channel in the low bandwidth ratio regime remains a topic for

future study. One possible improvement is to combine layered source coding based schemes

such as the BS strategy with the two-way relaying protocols. Another possible direction

will be to utilize the limited feedback as in [83] to allow partial CSIT at the sources.

5.7 Summary

In this chapter, we propose and study the new concept of the distortion exponent region

of source transmission in half-duplex two-way relaying cooperative networks. We derive

an outer bound on the distortion exponent region of two-way relaying communications,

which is tight at large bandwidth ratio. We obtain the optimal distortion exponent pairs

of conventional one-way relaying strategies and single-rate coding with various two-way

relaying protocols, including the MABC protocols and the TDBC protocols. We also obtain

the DMTs of the studied two-way relaying protocols.

5.A Proof of Theorem 5.4.3

According to the achievable rate region of the two-way AF relaying given in (5.40) and (5.41),

we define the sets of channel states h = {h13, h23, h32, h31} for which the transmissions are

in outage at T2 and T1, respectively, as follows

O1 ={

h :12

log(

1 +γ2|h32|2|h13|2

1 + γ(|h32|2 + |h23|2 + |h13|2)

)< R1

},

O2 ={

h :12

log(

1 +γ2|h31|2|h23|2

1 + γ(|h31|2 + |h13|2 + |h23|2)

)< R2

}.

(5.92)

We first consider the transmission from T1 to T2. Let the code rate R1 = r1 log γ, where

r1 is the multiplexing gain. Let |hij |2 = γ−θij . By standard large deviation argument [20]

and algebraic manipulation, we obtain

O1 ={

(θ32, θ23, θ13) :12

log(

1 +γ2−θ32−θ13

1 + γ1−θ32 + γ1−θ23 + γ1−θ13

)< r1 log γ

}.={

(θ32, θ23, θ13) : max{1− θ32, 1− θ23, 1− θ13, 2− θ32 − θ13}+

−max{1− θ32, 1− θ23, 1− θ13}+ < 2r1

},

(5.93)


where we have used following high-SNR approximation: log(1+∑i γxi )

log γ ' [max{xi}]+ at large

γ [20].

By Laplace’s method [20], the corresponding outage probability is then characterized by

P 1out , Pr{(θ32, θ23, θ13) ∈ O1}

.= γ−d∗1(r1,r2), (5.94)

where

d∗1(r1, r2) = inf(θ32,θ23,θ13)∈O1∩R3+

θ32 + θ23 + θ13 (5.95)

is the achievable DMT.

Without loss of generality, we assume θ32 > θ13, then

O1.={

(θ32, θ23, θ13) : max{1− θ23, 1− θ13, 2− θ32 − θ13}+

−max{1− θ23, 1− θ13}+ < 2r1

}.

(5.96)

It can be readily verified that if any one of θ32, θ23 or θ13 is greater than one, we can

always let the other two θij to be zero so that the infimum in (5.95) is simply given by

inf(θ32,θ23,θ13)∈O1∩R3+

θ32 + θ23 + θ13 = 1. (5.97)

Otherwise, we have θ32 ≤ 1, θ23 ≤ 1, and θ13 ≤ 1. Hence, 1 − θ13 ≤ 2 − θ32 − θ13, and

the outage set can be written as

O1.={

(θ32, θ23, θ13) : max{1− θ23, 2− θ32 − θ13}+

−max{1− θ23, 1− θ13}+ < 2r1

}.

(5.98)

We consider the following two cases separately

(1) θ23 > θ13

In this case, we have 0 ≤ 1− θ23 < 1− θ13 ≤ 2− θ23 − θ13.

It can then be found that

O1.={

(θ32, θ23, θ13) : (2− θ32 − θ13)− (1− θ13) < 2r1

}={

(θ32, θ23, θ13) : 1− θ32 < 2r1

}.

(5.99)

Therefore,

inf(θ32,θ23,θ13)∈O1∩R3+

θ32 + θ23 + θ13 = (1− 2r1)+.


(2) θ23 ≤ θ13

In this case, we have 0 ≤ 1− θ13 ≤ 1− θ23.

It can then be shown that

O1.={

(θ32, θ23, θ13) : max{1− θ23, 2− θ32 − θ13}

− (1− θ23) < 2r1

}={

(θ32, θ23, θ13) : (1− θ32 + θ23 − θ13)+ < 2r1

}.

(5.100)

Therefore,

inf(θ32,θ23,θ13)∈O1∩R3+

θ32 + θ23 + θ13 = θ∗32 + θ∗23 + θ∗13

= (1− 2r1)+,

where θ∗23 = 0, and θ∗32 + θ∗13 = (1− 2r1)+.

To summarize, the achievable DMT is given by

d∗1(r1, r2) = inf(θ32,θ23,θ13)∈O1∩R3+

θ32 + θ23 + θ13

= (1− 2r1)+,

(5.101)

where the infimum is achieved when θ32 = (1− 2r1)+ and θ13 = θ23 = 0.

Similarly, we can show that the achievable DMT of the transmission from T2 to T1 is

given by

d∗2(r1, r2) = inf(θ31,θ23,θ13)∈O2∩R3+

θ31 + θ23 + θ13

= (1− 2r2)+,

(5.102)

where the infimum is achieved when θ31 = (1− 2r2)+ and θ13 = θ23 = 0.

Note that the dominate outage events occur at θ31 = (1 − 2r2)+, θ32 = (1 − 2r1)+,

and θ13 = θ23 = 0 for both d∗1(r1, r2) and d∗2(r1, r2). Hence, d∗1(r1, r2) and d∗2(r1, r2) can be

simultaneously achieved.

The achievable DMTs of the two-phase two-way DF relaying protocol are therefore given

by

d∗1(r1, r2) = (1− 2r1)+, d∗2(r1, r2) = (1− 2r2)+. (5.103)



We first consider the transmission from T1 to T2. According to the achievable rate region

defined in (5.55), the corresponding outage probability is then

P 1out = Pr

{max

{I1

1 , I21

}≤ R1

}= Pr

{σ2q1 < σ2

q2

}· Pr

{I1

1 ≤ R1

}+ Pr

{σ2q1 ≥ σ

2q2

}· Pr

{I1

2 ≤ R1

},

(5.104)

where

I11 , t log

(1 +

γ|h13|2

1 + σ2q1

), I2

1 , t log(

1 +γ|h13|2

1 + σ2q2

), (5.105)

and σ2q1 and σ2

q2 are defined in (5.53) and (5.54), respectively.

Note that |h13|2, |h23|2, |h31|2, and |h32|2 are i.i.d. random variables. Also, the vaule

of t does not depend on the channel coefficients |hij |. By symmetry, it is easy to see that

Pr{σ2q1 < σ2

q2

}= Pr

{σ2q1 ≥ σ

2q2

}= 1

2 . Therefore, we have

P 1out =

12

Pr{I1

1 ≤ R1

}+

12

Pr{I2

1 ≤ R1

}. (5.106)

By (5.55), we define the following outage sets of channel states h = {h13, h23, h32, h31}:

O11 , {h : I1

1 < R1} ={

h : t log(

1 +γ|h13|2

1 + σ2q1

)< R1

},

O21 , {h : I2

1 < R2} ={

h : t log(

1 +γ|h13|2

1 + σ2q2

)< R1

}.

(5.107)

Let |hij |2 = γ−θij . Let R1 = r1 log γ. By standard large deviation arguments and

algebraic manipulation, the outage sets O11 and O2

1 can be rewritten as follows

O11 =

{(θ13, θ32) : (1− θ13)+ +

1− tt

(1− θ32)+

−max{

1− θ13,1− tt

(1− θ32)}+

<r1

t

},

(5.108)

O21 =

{(θ13, θ31, θ23) :

max{

(1− θ13)+ +1− tt

(1− θ31)+, (1− θ23)+}

−max{

1− θ23,1− tt

(1− θ31)}+

<r1

t

}.

(5.109)


By Laplace’s method, we have

P 1out =

12(Pr{I1

1 ≤ R1

}+ Pr

{I2

1 ≤ R1

}).= γ−d

11(r1,r2) + γ−d

21(r1,r2)

.= γ−min{d11(r1,r2),d2

1(r1,r2)}

, γ−d∗1(r1,r2),

(5.110)

where d∗1(r1, r2) is the corresponding DMT, and

d11(r1, r2) = inf

(θ13,θ32)∈O11∩R2+

θ13 + θ32,

d21(r1, r2) = inf

(θ13,θ31,θ23)∈O21∩R3+

θ13 + θ31 + θ23.(5.111)

We now derive d11(r1, r2) and d2

1(r1, r2). It can be easily shown that

d11(r1, r2) = inf

(θ13,θ32)∈O11∩R2+

θ13 + θ32

= min{

(1− r1

t)+, (1− r1

1− t)+}.

(5.112)

It is clear that for any triplet (θ13, θ31, θ23) ∈ O21 ∩ R3+ where θ13 > 1 or θ31 > 1 or

θ23 > 1, we always have d21(r1, r2) ≥ 1. Since d1

1(r1, r2) ≤ 1, it is then sufficient to consider

only the case where θ13 ≤ 1, θ31 ≤ 1, and θ23 ≤ 1 when obtaining d21(r1, r2).

We now consider the following two cases separately

(1) 1− θ23 ≤ 1−tt (1− θ31)

In this case, we have

O21 =

{(θ13, θ31, θ23) : 1− θ13 <

r1

t

}(5.113)

Hence, θ13 > 1− r1t .

From Fig. 5.6 and Fig. 5.7, it can be seen that, when 1 − θ23 ≤ 1−tt (1 − θ31),

min{θ31 + θ23} is achieved at either point A or (0, 0) depending on the value of t.

Hence, min{θ31 + θ23} = (2t−1t )+, θ31 ≥ 0, θ23 ≥ 0.

Therefore,

d21(r1, r2) = inf

(θ13,θ31,θ23)∈O11∩R3+

θ13 + θ31 + θ23

= (1− r1

t)+ + (

2t− 1t

)+.

(5.114)


t

t

−−

1

21

t

t 12 −

ξ

ζ

0

A

t

t

−−

1

21

t

t 12 −

ξ

ζ

0

A

Figure 5.6: The region of 1 − θ23 ≤ 1−tt (1 − θ31) (light gray area) for (θ23, θ31) ∈ R2+ and

t > 12 .

t

t

−−

1

21

t

t 12 −

ξ

ζ

0

B

t

t

−−

1

21

t

t 12 −

ξ

ζ

0

B

Figure 5.7: The region of 1 − θ23 ≤ 1−tt (1 − θ31) (light gray area) for (θ23, θ31) ∈ R2+ and

t ≤ 12 .


(2) 1− θ23 >1−tt (1− θ31)

In this case, we obtain

O21 =

{(θ13, θ31, θ23) : θ23 − θ13 +

1− tt

(1− θ31) <r1

t

}(5.115)

It can be shown that

d21(r1, r2) = inf

(θ13,θ31,θ23)∈O11∩R3+

θ13 + θ31 + θ23

= θ∗13 + θ∗31 + θ∗23,

(5.116)

where θ∗23 = 0, and

θ∗13 + θ∗31 = min{(1− t− r1)+

t, (1− r1

1− t)+}.

To summarize, the achievable DMT is given by

d1(r1, r2) = min{d11(r1, r2), d2

1(r1, r2)}

= min{

(1− r1

t)+, (1− r1

1− t)+,

(1− t− r1)+

t,

(1− r1

t)+ + (

2t− 1t

)+}.

(5.117)

It can be shown that the maximum of d1(r1, r2) is obtained when t = 12 . Therefore,

d∗1(r1, r2) = (1−2r1)+, where the dominant outage event occurs when θ13 = θ32 = (1−2r1)+.

Similarly, we can show that the maximum achievable DMT of the transmission from T2

to T1 is given by d∗2(r1, r2) = (1 − 2r2)+, where the dominant outage event occurs when

θ23 = θ31 = (1− 2r2)+.

Note that d∗1(r1, r2) and d∗2(r1, r2) can be achieved simultaneously. The DMTs of the

two-phase two-way CF relaying protocol is therefore given by

d∗1(r1, r2) = (1− 2r1)+, d∗2(r1, r2) = (1− 2r2)+. (5.118)

5.C Proof of Theorem 5.5.1

We first consider the transmission from user T1 to user T2 with the help of the relay T3.

Define the set of channel coefficients h for which the transmission from T1 to T3 is in

outage to be

O13 ={

h : t1 log(1 + |h13|2) < R1

}(5.119)


Define the set of channel coefficients h for which the transmission from T1 to T2 is in

outage given that the signal is not successfully decoded at the relay to be

O112 =

{h : t1 log(1 + |h12|2) < R1

}(5.120)

The set of channel coefficients for which the transmission from T1 to T2 is in outage

given that the signal is successfully decoded at the relay is given by

O212 =

{h : t1 log(1 + |h12|2) + t3 log(1 + |h32|2) < R1

}(5.121)

As a result, the overall outage probability of the transmission from T1 to T2 is given by

Pout = Pr{

h ∈ O13 ∪ O112

}+ Pr

{h ∈ O13 ∪ O2

12

}, (5.122)

where O13 is the complementary set of O13.

Let |hij |2 = γ−θij , i, j = 1, 2, 3 and i 6= j. Let R1 = r1 log γ and R2 = r2 log γ. By

standard large deviation arguments, we obtain

O13 = {θ13 : t1(1− θ13)+ < r1}

O112 = {θ12 : t1(1− θ12)+ < r1}

O212 = {(θ12, θ32) : t1(1− θ12)+ + t3(1− θ32)+ < r1}

(5.123)

Using Laplace’s method [20], we have

Pr{

h ∈ O13 ∪ O112

}.= γ− inf

(θ13,θ12)∈O13∪O112∩R2+

θ13+θ12

, γ−d11(r1,r2),

(5.124)

Pr{

h ∈ O13 ∪ O212

}.= γ− inf

(θ13,θ12,θ32)∈O13∪O212∩R3+

θ13+θ12+θ32

, γ−d21(r1,r2).

(5.125)

The overall outage probability is then

Pout.= γ−d

11(r1,r2) + γ−d

21(r1,r2) .= γ−min{d1

1(r1,r2),d21(r1,r2)}. (5.126)

It can be readily shown that

inf(θ13,θ12)∈O13∪O1

12∩R2+θ13 + θ12 = (1− r1

t1)+ + (1− r1

t1)+

= 2(1− r1

t1)+.

(5.127)


Hence, d11(r1, r2) = 2(1− r1

t1)+.

Note that if t1 ≤ r1, we always have d11(r1, r2) = 0, and hence min{d1

1(r1, r2), d21(r1, r2)} =

0. This leads to an overall outage probability of Pout.= γ0, which does not decay with the

SNR. To avoid this, we always choose t1 > r1.

Since the channels are statistically independent, we have

d21(r1, r2) = inf

(θ13,θ12,θ32)∈O13∪O212∩R3+

θ13 + θ12 + θ32

= infθ13∈O13∩R+

θ13 + inf(θ12,θ32)∈O2

12∩R2+θ12 + θ32

(5.128)

Since t1 > r1, it can be easily shown that

infθ13∈O13∩R+

θ13 = 0. (5.129)

If θ12 ≥ 1, we have

inf(θ12,θ32)∈O2

12∩R2+θ12 + θ32 = 1 + (1− r1

t3)+. (5.130)

If θ32 ≥ 1, we have

inf(θ12,θ32)∈O2

12∩R2+θ12 + θ32 = (1− r1

t1)+ + 1. (5.131)

If θ12 < 1 and θ32 < 1, the set O212 is illustrated in Fig. 5.8 for r1 ≥ t3 and r1 < t3,

respectively. It can be readily shown that inf θ12 + θ32 is achieved at either point A or point

B, i.e., if r1 ≥ t3,

inf(θ12,θ32)∈O2

12∩R2+θ12 + θ32 = min

{2− r1

t1,t1 + t3 − r1

t1

}, (5.132)

and if r1 < t3,

inf(θ12,θ32)∈O2

12∩R2+θ12 + θ32 = min

{2− r1

t1, 2− r1

t3

}. (5.133)

To summarize, combining the results from (5.129) to (5.132) with (5.128), we obtain

d21(r1, r2) =

t1+t3−r1

t1, t3 ≤ r1 < t1

2− r1t3, r1 < t3 < t1

2− r1t1, r1 < t1 < t3

=

max{2− r1t3, t1+t3−r1

t1}, t3 < t1, r1 < t1

2− r1t1, t3 ≥ t1, r1 < t1

(5.134)


B

3

131

t

rtt −+

1

131

t

rtt −+

32θ

0 12θ

A1

1

1

11t

r−

B

3

131

t

rtt −+

1

131

t

rtt −+

32θ

0 12θ

A1

1

1

11t

r−

(a)

B

3

131

t

rtt −+

1

131

t

rtt −+

32θ

0 12θ

A1

1

3

11t

r−

1

11t

r−

B

3

131

t

rtt −+

1

131

t

rtt −+

32θ

0 12θ

A1

1

3

11t

r−

1

11t

r−

(b)

Figure 5.8: The outage set O212 of the DF-based TDBC protocol. (a) r1 ≥ t3, (b) r1 < t3.


As a result, we have, if t1 ≤ 2t3,

d∗1(r1, r2) = min{d11(r1, r2), d2

1(r1, r2)}

= 2(1− r1

t1)+

(5.135)

and if 2t3 < t1,

d∗1(r1, r2) = min{d11(r1, r2), d2

1(r1, r2)}

=

2− r1t3, 0 ≤ r1 < t3

t1+t3−r1t1

, t3 ≤ r1 < t1 − t3

2(1− r1t1

), t1 − t3 ≤ r1 < t1

0, r1 ≥ t1

(5.136)

We now find the optimal t1 and t3 that maximizes d∗1(r1, r2) for a given t2.

Note that d∗1(r1, r2) is an increasing function of t1 when t1 ≤ 2t3. When t1 > 2t3,

d∗1(r1, r2) is an increasing function of both t1 and t3 when 0 ≤ r1 < t3 and t1− t3 ≤ r1 < t1,

and is a decreasing function of t1 when t3 ≤ r1 < t1 − t3. Using these properties, the

maximum of d∗1(r1, r2) can be found as follows

d∗1(r1, r2) =

2(1− 3r1

2(1−t2)), 0 ≤ r1 <1−t2

3

2(1−t2−r1)1−t2+r1

, 1−t23 ≤ r1 < 1− t2

0, r1 ≥ 1− t2

(5.137)

where

t1 =

2(1−t2)

3 , 0 ≤ r1 <1−t2

3

1−t2+r12 , r1 ≥ 1−t2

3

(5.138)

t3 =

1−t2

3 , 0 ≤ r1 <1−t2

3

1−t2−r12 , r1 ≥ 1−t2

3

(5.139)

By using the same arguments, the optimal DMT for the transmission from T2 to T1 can

be found similarly as follows

d∗2(r1, r2) =

2(1− 3r2

2(1−t1)), 0 ≤ r2 <1−t1

3

2(1−t1−r2)1−t1+r2

, 1−t13 ≤ r2 < 1− t1

0, r2 ≥ 1− t1

(5.140)


where

t2 =

2(1−t1)

3 , 0 ≤ r2 <1−t1

3

1−t1+r22 , r2 ≥ 1−t1

3

(5.141)

t3 =

1−t1

3 , 0 ≤ r2 <1−t1

3

1−t1−r22 , r2 ≥ 1−t1

3

(5.142)

For a given pair of (r1, r2), (d∗1(r1, r2), d∗2(r1, r2)) is Pareto optimal if and only if there

exists (t1, t2, t3) such that Eqs. (5.138), (5.139), (5.141), and (5.142) are satisfied simulta-

neously. We consider the following cases separately.

• Case 1: 0 ≤ r1 <1−t2

3 and 0 ≤ r2 <1−t1

3 .

In this case, the only triplet (t1, t2, t3) that satisfies all constraints is solved to be

t1 =25, t2 =

25, t3 =

15, (5.143)

which leads to the following region of (r1, r2):

A1 ={

(r1, r2) : 0 ≤ r1 <15, 0 ≤ r2 <

15

}. (5.144)

Therefore, the optimal (d∗1, d∗2) when (r1, r2) ∈ A1 is

d∗1(r1, r2) = 2− 5r1, d∗2(r1, r2) = 2− 5r2. (5.145)

• Case 2: 1−t23 ≤ r1 < 1− t2 and 1−t1

3 ≤ r2 < 1− t1.

The optimal triplet (t1, t2, t3) is found to be

t1 =1 + 2r1 − r2

3, t2 =

1 + 2r2 − r1

3, t3 =

1− r1 − r2

3, (5.146)


A2 ={

(r1, r2) : 4r1 + r2 ≥ 1, r1 + 4r2 ≥ 1, r1 + r2 < 1}. (5.147)

The optimal (d∗1, d∗2) when (r1, r2) ∈ A2 is therefore

d∗1(r1, r2) =2(1− r1 − r2)1− r2 + 2r1

, d∗2(r1, r2) =2(1− r1 − r2)1− r1 + 2r2

. (5.148)


• Case 3: 0 ≤ r1 <1−t2

3 and 1−t13 ≤ r2 < 1− t1.


t1 =1− r2

2, t2 =

1 + 3r2

4, t3 =

1− r2

4, (5.149)


A3 ={

(r1, r2) : 4r1 + r2 < 1,15≤ r2 < 1

}. (5.150)

The optimal (d∗1, d∗2) when (r1, r2) ∈ A3 is then found to be

d∗1(r1, r2) = 2(

1− 2r1

1− r2

), d∗2(r1, r2) =

2(1− r2)1 + 3r2

. (5.151)

• Case 4: 1−t23 ≤ r1 < 1− t2 and 0 ≤ r2 <

1−t13 .


t1 =1 + 3r1

4, t2 =

1− r1

2, t3 =

1− r1

4, (5.152)


A4 ={

(r1, r2) : r1 + 4r2 < 1,15≤ r1 < 1

}. (5.153)

The optimal (d∗1, d∗2) when (r1, r2) ∈ A4 is therefore

d∗1(r1, r2) =2(1− r1)1 + 3r1

, d∗2(r1, r2) = 2(

1− 2r2

1− r1

). (5.154)

Combining (5.145)-(5.154) leads to the claimed result.

5.D Proof of Theorem 5.5.3

In this proof, we first derive the maximum achievable DMTs d∗1(r1, r2) and d∗2(r1, r2) for the

transmission from T1 to T2 and from T2 to T1, respectively. We then show that the two

maximum DMTs can be achieved simultaneously.

To find an upper bound on d1(r1, r2), we let η → 1 in g1 and g2 in Eq. (5.78) so that

g1 →√

1γ|h13|2+1

and g2 → 0. This corresponds to allocating almost all relaying power

for transmitting x1, and hence leads to an upper bound of d1(r1, r2). The upper bound of

d2(r1, r2) can be obtained similarly by letting η → 0.


With η = 1, the achievable rate of the transmission from T1 to T2 in (5.82) can now be

written as follows

R1 <13

log(

1 + γ|h12|2 +γ2|h32|2|h13|2

γ|h32|2 + γ|h13|2 + 1

), I1 (5.155)

Note that I1 is of the same form as the maximum achievable rate in the one-way AF relaying

[53]. The only difference is that the pre-log factor is 13 instead of 1

2 due to the three-phase

transmission. Following along the lines of the proof of the DMT for the one-way AF relaying

in [53], we immediately have the maximum achievable DMT d∗1(r1, r2) = (2 − 6r1)+. The

proof is straightforward, and hence is omitted.

Similarly, we can also show that d∗2(r1, r2) = (2− 6r2)+.

In the following, we show that the two DMT upper bounds can be simultaneously

achieved with any fixed value of η.

According to the achievable rate region in (5.82) and (5.83), we define the sets of channel

states h = {h13, h23, h32, h31, h12, h21} for which the transmissions are in outage at T2 and

T1, respectively, as follows

O1 ={

h :13

log(

1 + γ|h12|2 +γ2|h32|2|g1|2|h13|2

γ|h32|2(|g1|2 + |g2|2) + 1

)< R1

}, (5.156)

O2 ={

h :13

log(

1 + γ|h21|2 +γ2|h31|2|g2|2|h23|2

γ|h31|2(|g1|2 + |g2|2) + 1

)< R2

}. (5.157)

We first consider the transmission from T1 to T2. Let R1 = r1 log γ. Let |hij |2 = γ−θij ,

i, j = 1, 2, 3 and i 6= j. By standard large deviation arguments and algebraic manipulation,

the outage set O1 can then be written as follows

O1 ={{θij} :

13

log(

1 + γ1−θ12 +α

β

)< r1 log γ

}, (5.158)

where

α , η(γ3−θ32−θ13−θ23 + γ2−θ32−θ13),

β , (1− η)(γ2−θ32−θ13 + γ1−θ32) + η(γ2−θ32−θ23 + γ1−θ32)

+ γ2−θ13−θ23 + γ1−θ13 + γ1−θ23 + 1.

(5.159)

By Laplace’s method, the corresponding outage probability is then

P 1out , Pr{(θ13, θ23, θ32, θ12) ∈ O1}

.= γ− inf

(θ13,θ23,θ32,θ12)∈O1∩R4+θ13+θ23+θ32+θ12

, γ−d1(r1,r2).

(5.160)


It can be easily verified that the dominant exponent of α is smaller than that of β if any

one of θ13, θ23, or θ32 is greater than one. Therefore, in this case, we obtain

O1.={

(θ13, θ23, θ32, θ12) : (1− θ12)+ < 3r1

}. (5.161)

Hence,

inf(θ13,θ23,θ32,θ12)∈O1∩R4+

θ13 + θ23 + θ32 + θ12 = 1 + (1− 3r1)+, (5.162)

where the infimum is obtained when only one of θ13, θ23, θ32 is greater than one.

If θ13 < 1, θ23 < 1, and θ32 < 1, the dominant exponent of α/β is thus found to

be min{1 − θ31, 1 − θ23, 1 − θ13}. By standard large deviation argument and algebraic

manipulation, we have

O1 ={θij : max

{(1− θ12)+ min{1− θ31, 1− θ23, 1− θ13}

}< 3r1

}. (5.163)

Hence,

inf(θ13,θ23,θ32,θ12)∈O1∩R4+

θ13 + θ23 + θ32 + θ12 = 2(1− 3r1)+, (5.164)

where the infimum is obtained when θ12 = θ32 = (1− 3r1)+ and θ13 = θ23 = 0.

Since 2(1 − 3r1)+ ≤ 1 + (1 − 3r1)+, we have d1(r1, r2) = 2(1 − 3r1)+. Note that the

upper bound d∗1(r1, r2) is achieved.

Similarly, we show that for the transmission from T2 to T1,

d2(r1, r2) = inf(θ13,θ23,θ31,θ21)∈O2∩R4+

θ13 + θ23 + θ31 + θ21

= 2(1− 3r2)+,

(5.165)

where the infimum is achieved when θ21 = θ31 = (1 − 3r2)+ and θ13 = θ23 = 0. This also

achieves the upper bound d∗2(r1, r2).

Note that the dominant events occur at θ13 = θ23 = 0 for both d∗1(r1, r2) and d∗2(r1, r2).

Therefore d∗1(r1, r2) and d∗2(r1, r2) can be achieved simultaneously. The DMTs of the AF-

based TDBC protocol are therefore given by

d∗1(r1, r2) = (2− 6r1)+, d∗2(r1, r2) = (2− 6r2)+. (5.166)

Chapter 6

Finite-SNR End-to-end Distortion

Minimization

In this chapter, we consider the transmission of a Gaussian signal over a slow fading channel.

The channel state information is assumed to be only known at the receiver. The source is

layer-coded and transmitted using the broadcast strategy. Instead of considering the asymp-

totic distortion exponent, we study the optimization problem of minimizing the expected

end-to-end distortion of the reconstructed signal at the receiver at an arbitrary SNR. An

efficient iterative algorithm is proposed to jointly solve the rate allocation problem and the

channel discretization problem. Numerical results show that the proposed algorithm out-

performs the schemes using fixed channel discretization by a large margin. Meanwhile, the

computational cost of our method is lower than those of the joint optimization approaches

that involve partial exhaustive search.

6.1 Introduction

The broadcast strategy [38] is an effective approach to facilitate robust transmission in fading

channels when CSI is available at the transmitter. In the broadcast strategy, the information

is decoded adaptively according to the channel realization, i.e., more information can be

decoded when the fading is less severe, and vice versa.

In this chapter, we consider the problem of transmitting a discrete-time analog-amplitude

source over slow fading channels using the broadcast strategy. The performance measure in

139

CHAPTER 6. FINITE-SNR END-TO-END DISTORTION MINIMIZATION 140

which we are interested is the end-to-end distortion between the source signal and its re-

construction at the destination. We propose an efficient iterative algorithm to minimize the

expected end-to-end distortion of transmitting a Gaussian source over fading channels using

layered source coding and the broadcast strategy, also known as layered broadcast trans-

mission. The proposed algorithm jointly solves the rate allocation problem and the channel

discretization problem, whereas neither [102] nor [106] considers the optimal channel dis-

cretization explicitly. Numerical results show that by jointly optimizing the rate allocation

and channel discretization, the expected distortion can be significantly reduced compared

to those in [102] and [106] with uniform channel discretization. Furthermore, to achieve a

near-optimal performance, the proposed algorithm only needs a very few number of coding

layers, while a much larger number of coding layers is required in [102] and [106], which

makes our method attractive in practice. In terms of the complexity, each iteration step of

our algorithm has a complexity of O(M), where M is the number of fading states in the

current iteration, which is non-increasing. In addition, our method converges to within 1 dB

of the theoretical bound in less than 10 iterations. Therefore, its computational complexity

is much lower than that in [104].

6.2 System Model

We consider the problem of transmitting a memoryless, zero-mean, unit-variance complex

Gaussian source over a SISO fading channel, where the full CSI is available at the receiver,

and the transmitter only knows the fading distribution. The performance metric is the

mean-squared error between the source signal and its reconstruction at the destination.

The bandwidth ratio is denoted as b channel uses per source sample.

The channel is assumed to be in flat, slow fading with continuous fading state h. Denote

s = |h|2 as the channel power gain. Denote F (s) and f(s) as the cumulative distribution

function (c.d.f.) and the probability density function (p.d.f.) of s, respectively. F (s) and

f(s) are assumed to be continuous over [0,∞). The additive noise at the receiver is modeled

as zero-mean unit-variance circularly symmetric complex Gaussian. The transmitter has an

average power constraint γ.

The source is coded into M layers by layered source coding and transmitted using the

broadcast strategy [38]. In order to perform M -layer broadcast transmission over a channel


with continuous fading distribution, the encoder needs to pre-determine M discrete (quan-

tized) channel power gains s1 < s2 < · · · < sM , such that each layer is associated with

a channel power gain. The coding rate (nats per channel use) of the jth layer, which is

associated with sj , is then [38]

Rj = log(

1 +γj

1/sj + γj+1

), (6.1)

where the logarithm is taken to the base e (natural logarithm), γj+1 =∑M

i=j+1 γi is the

self-interference power, and γi is the power allocated to the ith layer. All M source layers

are then superimposed and transmitted simultaneously to the receiver. The superimposed

signal is x =∑M

j=1√γjxj , where xj is the coded symbol of the jth layer.

The decoder performs successive decoding. That is, the layers are decoded in order

starting from the first layer. The decoded layer is subtracted from the received signal before

decoding the next layer. The decoding procedure continues until the decoder fails to decode

one layer. As a result, to have exactly the first k layers successfully decoded requires that

the realized channel power gain s to be within [sk, sk+1) [38], where we define s0 = 0 and

sM+1 =∞.

For a layer-coded Gaussian signal, the distortion achieved when the first k layers are

successfully decoded is thus [106]

Dk = exp(− b

k∑j=1

Rj

)=

k∏j=1

(1 +

γj1/sj + γj+1

)−b. (6.2)

Our objective is then to minimize the distortion over the power allocation (γ1, γ2, · · · , γM )

and the discrete channel power gain parameters (s1, s2, · · · , sM ):

minM∑k=0

[F (sk+1)− F (sk)] ·Dk

s.t.M∑i=1

γi ≤ γ,

γi ≥ 0, i = 1, 2, · · · ,M,

si ≥ si−1, i = 1, 2, · · · ,M.

(6.3)


6.3 An Interative Algorithm

In this section, we consider the optimization problem in (6.3). Instead of optimizing the

objective function over the power allocation (γ1, γ2, · · · , γM ) directly, we reformulate the

problem as follows by utilizing an alternative characterization of the rates (R1, R2, · · · , RM )

given in [106]:

minM∑k=0

(F (sk+1)− F (sk)) exp(− b

k∑j=1

Rj

)

s.t.M∑k=1

( 1sk− 1sk+1

)exp

( k∑j=1

Rj

)− 1s1≤ γ,

si ≥ si−1, i = 1, 2, · · · ,M,

Ri ≥ 0, i = 1, 2, · · · ,M.

(6.4)

The optimal power allocation {γ∗i } can be solved by Eq. (6.1) after the optimal rates {R∗i }and channel power gains {s∗i } are obtained.

The Lagrangian of (6.4) is given by

L =M∑k=0

(F (sk+1)− F (sk)

)exp

(− b

k∑j=1

Rj

)

+ λ( M∑k=1

( 1sk− 1sk+1

)exp

( k∑j=1

Rj

)− 1s1− γ)

−M∑k=1

νkRk −M∑k=1

ξk(sk − sk−1),

(6.5)

where λ, νk and ξk are the associated Lagrange multipliers.

The Karush-Kuhn-Tucker (KKT) conditions require that ∂L∂Ri

= 0 and ∂L∂si

= 0 for the

optimal solution [114]. It is also worth noting that the the optimal {R∗i } and {s∗i } are

associated with the same optimal Lagrange multipliers.

The optimization problem (6.4) is in general nonlinear and nonconvex. In the follow-

ing, we consider the rate allocation and channel discretization subproblems separately. An

efficient iterative algorithm is then proposed for solving (6.4).


6.3.1 Rate allocation

Assume the quantized channel power gain parameters {si}Mi=1 are given. The objective of the

rate allocation subproblem is then to find the optimal {Ri}Mi=1 that minimizes the objective

function in (6.4). An efficient algorithm that solves this problem has been proposed in [106]

by directly solving the KKT conditions. In the proposed method, we use the algorithm in

[106] to solve the rate allocation subproblem. We briefly restate some of its useful results

in the following. Please refer to [106] for details of the algorithm.

Denote the set of layers that are assigned non-zero rates (effective layers) under optimal

rate allocation to be {ik}Lk=1. The following results are from [106]:

1. The sequence {κik}Lk=1 is monotonically increasing, i.e., κi1 < κi2 < · · · < κiL , where

κik ,b(F (sik+1

)− F (sik))1/sik − 1/sik+1

. (6.6)

2. The optimal rate allocation {Rik}Lk=1 is obtained by solving the following set of equa-

tions:

exp( k∑j=1

Rij

)=(κikλ0

)1/(b+1)

, k = 1, · · · , L (6.7)

where λ0 is the associated optimal Lagrange multiplier, which is related to the total

power γ by the first inequality constraint in (6.4).

3. The minimum expected distortion in (6.4) is achieved by continuous broadcasting,

i.e., when the number of layers goes to infinity (M →∞). The corresponding optimal

power allocation γ(s) is continuous and positive over the intervals where s2f(s) has

strictly positive derivatives. The upper bounds of these intervals satisfy the boundary

condition 1− F (s) = sf(s).

6.3.2 Channel discretization

The objective of the channel discretization subproblem is to find the optimal channel power

gains that minimize the objective function in (6.4) given the optimal rate allocation {Ri}Mi=1.

Note that it suffices to consider only the set of effective layers with non-zero rates {Rik}Lk=1,

and find the corresponding optimal {sik}Lk=1. Hence, the problem size can be reduced.


From (6.5), the KKT condition of ∂L∂sik

= 0 can be derived as follows

∂L∂sik

= f(sik)− λ αiks2ikwik− (ξik − ξik+1

) = 0 (6.8)

where we define ξiL+1 = 0 and

wik , exp(− b

k−1∑j=1

Rij

)− exp

(− b

k∑j=1

Rij

),

αik , exp( k∑j=1

Rij

)− exp

( k−1∑j=1

Rij

).

(6.9)

Let λ = λ0, the optimal Lagrange multiplier for the rate allocation subproblem. Clearly,

if sik is a positive solution of the equation

s2ikf(sik) = λ0

αikwik

, (6.10)

which satisfies sik ≥ sik−1, we can then set all ξik = 0, and the optimal channel power

gains are then given by {sik}Lk=1, provided that the power constraint is also satisfied. We

now present the following lemma, which states that such a solution always exists if certain

constraint is imposed on the initial channel power gains {si}Mi=1.

Lemma 6.3.1. Let sb be the largest channel power gain that satisfies the boundary condition

1 − F (sb) = sbf(sb). Suppose the discretized channel power gains satisfy s1 < s2 < · · · <sM ≤ sb. Let {Rik}Lk=1 be the solution of the rate allocation subproblem for the effective

layers, and λ0 be the corresponding Lagrange multiplier. A positive solution sik ≤ sb always

exists for (6.10) for all k, provided that the bandwidth ratio b ≥ 1.

Proof. For brevity, we only prove the case where there is only one channel power gain s∗

(s∗ < ∞) that satisfies the boundary condition so that sb = s∗. The generalization to the

case where sb = ∞ or the case where the equation 1 − F (s) = sf(s) has more than one

solutions is straightforward.

Note that s2f(s) is continuous over [0, sb] and is equal to 0 at s = 0. Since λ0αikwik

> 0,

for (6.10) to have a positive solution sik , we only need to show

λ0αikwik≤ s2

bf(sb). (6.11)


Plugging (6.7) and (6.9) into (6.10), we obtain

λ0αikwik

=κ

1/(b+1)ik

− κ1/(b+1)ik−1

κ−b/(b+1)ik−1

− κ−b/(b+1)ik

=b(κik/κik−1

)1/(b+1) − b(κik/κik−1

)−(κik/κik−1

)1/(b+1)· κikb

,bβ − bβb+1 − β

· κikb

(6.12)

where β , (κik/κik−1)1/(b+1) > 1, and κi0 , λ0.

It is easy to verify that 0 < bβ − b < βb+1 − β for any β > 1 and b ≥ 1. As a result, we

have

λ0αikwik

<κikb≤ κiL

b= siL(1− F (siL)). (6.13)

Consider g(s) = s(1− F (s)), whose derivative is given by

h(s) =dg

ds= 1− F (s)− sf(s). (6.14)

By the boundary condition 1− F (sb) = f(sb)sb, we have h(sb) = 0. Since h(0) = 1, by

our assumption, we have h(s) 6= 0 over [0, sb). Hence, h(s) ≥ 0 over [0, sb] with equality

achieved at sb, which suggests that s(1− F (s)) is monotonically increasing over [0, sb].

As a result, we have

siL(1− F (siL)) ≤ sb(1− F (sb)) = s2bf(sb) (6.15)

Combine (6.13) and (6.15), we have λ0αikwik≤ s2

bf(sb) for all k. Therefore, there exists a

positive sik ≤ sb such that λ0αikwik

= s2ikf(sik).

Note that Lemma 6.3.1 guarantees the existence of the optimal channel power gains

{sik}Lk=1, which is associated with the optimal rate allocation {Rik}Lk=1 through the Lagrange

multiplier λ0, provided that the initial channel power gains {si}Mi=1 are chosen properly.

It has to be pointed out that although the iterative algorithm in [104] also involves solving

an equation that is similar to Eq. (6.10) when optimizing the channel discretization, there

are two major differences between their approach and ours. First, to guarantee the existence

of a feasible solution {si}, an additional constraint of non-decreasing Ri is imposed in [104],

whose optimality, as mentioned in [104], is not justified. Whereas in our method, by Lemma

6.3.1, we only require that the initial channel power gains satisfy s1 < · · · < sM ≤ sb. Since


sb is the largest upper bound among those of the optimal power allocation intervals, there

is in fact no loss of optimality in our case by imposing this condition. Second, to optimize

the channel discretization, a bisection search is employed in [104], which further increases

the complexity. Therefore, our approach is also more computationally efficient.

We now propose an iterative approach to jointly solve the rate allocation and channel

discretization subproblems.

6.3.3 Algorithm description

1. Initialize l = 0. Choose an initial set of channel power gains {s0i } that satisfies the

constraint in Lemma 6.3.1.

2. Solve the rate allocation problem using the method in [106] for the optimal (effective)

rate allocation {Rlik} and the associated Lagrange multiplier λl.

3. Stop if the optimized distortion is satisfied with a desired accuracy or a pre-specified

number of iteration is achieved. Otherwise go to step 4.

4. Solve the channel discretization problem using {Rlik} and λl obtained from Step

2. The optimized channel fading states {sl+1ik} are the solution of the equations

(sl+1ik

)2f(sl+1ik

) = λlαlik/wlik . l = l + 1. Go to step 2.

The following comments are in order. 1) The power constraint is satisfied after step 2.

Hence the result is feasible. 2) Lemma 6.3.1 ensures that a feasible solution {slik} is obtained

in step 4, which is also a valid starting point for the next iteration.

6.4 Numerical Results and Discussions

In this section, we apply the iterative algorithm developed in Section 6.3.3 to channels with

different fading distributions. The bandwidth ratio is b = 2. The total number of iterations

is 20 in the proposed method. We first consider a SISO Rician fading channel, whose channel

power gain s has a p.d.f. of [26]

f(s) =(K + 1)e−K

sexp

(− (K + 1)s

s

)I0

(2

√K(K + 1)s

s

), (6.16)

where s is the mean of the channel power gain, I0(·) is the modified Bessel function of

the first kind with order zero. The Rician K-factor is defined as the ratio of powers of


0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

(a) si

γ i

0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

(b) si

Ri

K = 0K = 4K = 32K = 64

K = 0K = 4K = 32K = 64

Figure 6.1: The optimized power allocation γi, rate allocation Ri, and discrete channelfading gains si of Rician fading channels with different Rician K-factors: K = 0, 4, 32, 64.

the dominant component and the Rayleigh component. The Rician distribution reduces to

Rayleigh for K = 0, and approaches Gaussian when K →∞ [26]. Note that the Rician K-

factor is different from the length of source samples, which is also denoted by K in previous

Chapters. However, they should be clearly distinguishable from the context.

In order to perform M -layer broadcast transmission, an M -level uniform quantizer is

used for the initial channel discretization. The channel power gain s is truncated to be in

the range [sl, su] and quantized into M evenly spaced levels as follows

si , sl + (i− 1)su − slM − 1

, i = 1, · · · ,M, (6.17)

The M quantized channel power gains then serve as the intial values {s0i } in the proposed

algorithm.

Fig. 6.1 shows an example of the optimal power allocation γi and rate allocation Ri

obtained by the proposed algorithm for Rician fading channels with s = 1 and different

Rician K-factors. The total transmit power is γ = 0 dB. The layers are indexed by the

corresponding optimal discretized channel fading gains si. The truncation interval is [0, 2s],

and the number of initial quantization levels is M = 25. It can be seen that, depending on

the parameter K, the algorithm produces L = 2 ∼ 9 effective layers, which is much less than

the number of initial fading states. Furthermore, the larger K is, the less is the number of


31 32 33 34 35 36 37 38 39 40

−31

−30

−29

−28

−27

−26

−25

−24

Dis

tort

ion

(dB

)

SNR (dB)

Continuous broadcastingIterative approach: M = 10Iterative approach: M = 1000Uniform−bnd : M = 1000Uniform: M = 1000

Figure 6.2: Minimum expected end-to-end distortion achieved by different methods withM = 10 and 1000 for a Rayleigh fading channel.

effective layers. This is because the channel is more deterministic when K is large [26], and

hence only a few layers are needed to adapt to the fading in the broadcast strategy.

We next compare the proposed algorithm with two fixed-uniform-discretization-based

schemes, namely Uniform-bnd and Uniform, for which the algorithm in [106] is applied

after the uniform channel discretization. Although we only compare with the algorithm in

[106], the conclusions also apply to the algorithm in [102] since they essentially produce

the same results due to their global optimality for channels with discrete fading states.

The truncation intervals in Uniform-bnd and Uniform are [s∗l , s∗u] and [0, s∗u], respectively,

where s∗l and s∗u are the lower and upper bounds of the positive power allocation interval

for the given fading distribution, which can be calculated by Eq.(46) and Eq.(47) in [106].

The truncation interval of the proposed algorithm is [0, s∗u]. Fig. 6.2 shows the minimum

distortions achieved by the proposed algorithm (iterative approach), Uniform and Uniform-

bnd, with 10-level and 1000-level channel discretization, for a Rayleigh fading channel. The

continuous broadcasting distortion lower bound is also included as reference. We observe

that the performance of the proposed algorithm with 10-level channel discretization can

approach that of the optimal continuous broadcasting, which indicates that our method

is able to achieve a near-optimal performance with only a few coding layers. Whereas to


0 5 10 15 20 25 30 35 40−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

dist

ortio

n (d

B)

SNR (dB)

UniformUniform−bndContinuous broadcastingIterative approach

m = 1

m = 2

Figure 6.3: Minimum expected end-to-end distortion achieved by different methods withM = 10 for the SISO Nakagami fading channels.

achieve a similar distortion, around 1000-level channel discretization is required if the fixed-

uniform-discretization-based schemes are used, which corresponds to 1000-layer broadcast

transmission that is usually infeasible in practice. (The distortions achieved by Uniform

and Uniform-bnd with 10-level channel discretization are both larger than -20 dB, and

hence lie outside the figure. Still, they are worse than our method with 10-level channel

discretization.).

Another example we consider is the Nakagami fading channel, for which the p.d.f. of

the channel power gain s is [26]

f(s) =(ms

)m sm−1

Γ(m)exp

(−ms

s

), (6.18)

where s is the mean of the channel power gain, Γ(·) is the gamma function, and m is the

shape factor. Fig. 6.3 shows the expected distortions achieved by various optimization

methods, all with 10-level channel discretization, for the Nakagami fading channel with

s = 1 and different parameter m. It can be seen that, with the same number of quantized

channel power gains, the proposed algorithm outperforms the fixed-uniform-discretization-

based schemes by a large margin, especially at high signal-to-noise ratio (SNR). For example,

when m = 2, at SNR = 40 dB, the distortion achieved by the proposed algorithm is 29.5


0 10 20 30 40 50−24

−22

−20

−18

−16

−14

−12

−10

−8

iterations

Dis

tort

ion

(dB

)

Iterative approach: M = 10Iterative approach: M = 100Continuous broadcasting

Figure 6.4: The convergence behavior of the proposed iterative algorithm.

dB and 14.4 dB lower than that of Uniform and Uniform-bnd, respectively.

We next demonstrate by an example of the Rayleigh fading channel the fast convergence

behavior of the proposed iterative algorithm. The optimized expected distortion after each

step of iteration is plotted against the continuous broadcasting lower bound in Fig. 6.4 for

the 10-level and the 100-level channel discretization at SNR = 30 dB. It can be seen that

the expected distortion reduces rapidly in the first 2 ∼ 4 iterations, and then gradually

approaches the continuous broadcasting lower bound. In both cases, the gap between the

optimized distortion and the continuous broadcasting lower bound reduces below 1 dB after

the first 10 iterations. The same behavior can be observed for other channels as well.

Finally, we briefly comment on the algorithm complexity. Note that we do not compare

our algorithm with the iterative algorithm in [104] directly since both methods are able

to achieve a near-optimal performance that is very close to the distortion lower bound.

However, our approach is more preferable in terms of the computational complexity. It is

clear that each step of the proposed algorithm is of O(M) complexity, which is much lower

than that in [104], whose complexity of each iteration step is of O(M |R| log(λmax/ε)), where

|R| is the size of the search space for coding rate Ri and ε is the desired accuracy in searching

the optimal λ. More precisely, the major complexity reduction in our method comes from

the rate allocation part, where our method only requires to solve a set of equations (6.7)


whereas the method in [104] needs to perform an exhaustive search over the rate space R,

which is normally undesirable. Although it is possible for the non-iterative algorithm in

[106] to achieve a similar distortion as that of the proposed method with possibly lower

complexity by using a uniform quantizer with a large M ( ∼ 1000 in the Rayleigh fading

example in Fig. 6.2), as we have pointed out before, this approach eventually results in a

large number of coding layers, which is prohibited in most practical scenarios. Hence, our

algorithm is more feasible in practice.

6.5 Summary

In this chapter, we study the end-to-end distortion of transmitting a Gaussian source over

fading channels using layered broadcast transmission. An efficient iterative algorithm is

proposed to minimize the expected distortion by jointly optimizing the power allocation and

the channel discretization. Numerical results show that the proposed algorithm outperforms

the schemes using fixed channel discretization by a large margin. The proposed algorithm

is also shown to have low complexity in terms of both computational cost and practical

implementation.

Chapter 7

Conclusions

7.1 Conlusions

In this thesis, we investigate the joint source-channel transmission in wireless systems, in

particular the cooperative relaying networks. We study the end-to-end distortion of wireless

cooperative systems and extend the distortion exponent analysis to the multi-relay scenario

and two-way relay channels. Various coding and transmission strategies are investigated

along with different cooperation protocols.

We first derive the distortion exponent of the layered source coding with progressive

transmission and the broadcast strategy combined with repetition-based cooperation, relay-

selection-based cooperation and space-time-coded cooperation for a multi-relay cooperative

network. The layered source coding is shown to be an effective technique to improve the

distortion exponent of multi-relay cooperative systems. In terms of the cooperation pro-

tocols, the relay-selection-based cooperation and space-time-coded cooperation have both

demonstrated significant performance gains over the repetition-based cooperation. As an

important addition to the DMT theory, we also establish the successive refinability of the

DMT curves of these multi-relay cooperation protocols.

We next study the distortion exponent of multi-relay cooperative systems with limited

feedback. Single-rate separate source-channel coding is combined with various coopera-

tion strategies such as the orthogonal/nonorthogonal amplify-and-forward protocols, the

sequential slotted amplify-and-forward protocol, and the orthogonal/nonorthogonal decode-

and-forward protocols. It is shown that the feedback scheme outperforms the best known

non-feedback strategies for multi-relay cooperative systems with only a few bits of feedback

152

CHAPTER 7. CONCLUSIONS 153

information.

We also propose and study the new concept of the distortion exponent region of joint

source-channel transmission in half-duplex two-way relaying cooperative networks. We de-

rive an outer bound on the distortion exponent region of two-way relaying communications,

which is tight at large bandwidth ratio. We obtain the optimal distortion exponent pairs

of conventional one-way relaying strategies and single-rate coding with various two-way

relaying protocols, including the MABC protocols and the TDBC protocols. Our results

show that at small bandwidth ratio, even with the simple single-rate coding, an improved

performance of the two-way relaying protocol can be observed when compared with the

sophisticated layered-coding-based one-way relaying schemes due to the improved spectral

efficiency. However, at large bandwidth ratio, one-way relaying schemes with the layered

source coding in general offers better performance gain. These results reveal some impor-

tant tradeoffs in the two-way relaying communications and provide useful guidance in the

system design. We also obtain the DMTs of the studied two-way relaying protocols.

Finally, we consider the finite-SNR end-to-end distortion minimization of source trans-

mission over SISO fading channels using the broadcast strategy. An efficient iterative al-

gorithm is proposed to minimize the expected distortion by jointly optimizing the power

allocation and the channel discretization. Numerical results show that the proposed algo-

rithm outperforms the schemes using fixed channel discretization by a large margin. The

proposed algorithm is also shown to have low complexity in terms of both computational

cost and practical implementation.

7.2 Future Work

The works in this thesis also reveal some interesting topics for future research. One question

that has not been fully answered is: whether the obtained distortion exponent upper bounds

(outer bounds) are tight or not? Although most of these bounds have been shown to be

tight in certain regimes, for example, when the bandwidth ratio is large, it is still not clear

whether this is true in general or not. Therefore, the investigation on tighter distortion

exponent upper bounds remains an important topic for future study. Current derivations

of the upper bounds assume arbitrary cooperation between cooperative nodes without any

restrictions, which is clearly overly ideal. By adding back system constraints such as the

transmission delay between cooperative nodes, we would expect to obtain better distortion


exponent bounds for the cooperative system.

Another particularly challenging problem is to find theoretical as well as practical coding

and transmission schemes that are distortion exponent optimal or near optimal in the general

cooperative system. This requires not only a relaying protocol that is DMT-optimal, but also

a sophisticated coding and transmission scheme that is able to fully exploit such optimality

in order to be distortion exponent optimal, e.g., through the successive refinability property.

Generalizing the results in this thesis to multiple-relay systems with multiple-antenna nodes

would be another important extension.

As mentioned early in Section 1.2, multiple description coding is an effective coding

technique to combating transmission errors. However, in the literature (see [15] and ref-

erences therein), multiple description systems have been studied (almost) exclusively for

“on-off” channels, i.e., the channel supports either a given transmission rate or no rate at

all. Among the sporadic reports on the distortion exponent of multiple description coding in

joint source-channel transmission over fading channels, Laneman et al. [39] show that, in the

case of separate source and channel encoding, combining multiple description encoding with

joint source-channel decoding achieves the best distortion exponent performance among all

source-channel coding schemes for both continuous-state and on-off parallel fading chan-

nels. The distortion exponent of multiple description systems has also been investigated in

[6] for fading relay channel. However, the distortion exponent of multiple description sys-

tems remains largely uninvestigated and would be an interesting research direction. Another

possible direction along this line would be to study the multiple description system design

for cooperative communications based on our previous work in prediction-compensated mul-

tiple description coding and multiple description filter banks [16, 17, 18, 19], which is also

of more practical relevance.

The study of two-way relay channels in this thesis is the first step to extend the distortion

exponent analysis to multiuser systems. It aims to provide some useful insight in analyzing

other multiuser systems such as the multiple-access channel, the broadcast channel, and

the interference channel. The distortion exponent region of general multi-user networks

remains an open problem. A better understanding of the effects of user cooperation and

user interference is crucial in studying this problem. Designing intelligent schemes that

can effectively control the interference and improve the system throughput would be a

worthwhile and useful topic.

In this thesis, we have mostly focused on the high-SNR regime. While the asymptotic


analysis provides a neat solution, thus giving useful insight into the complicated general

problems, it does not necessarily give a complete picture about the overall system. In

particular, the DMT as well as the distortion exponent analysis ignore any constant scaling

in power and rate, which are important for any practical communication systems. We

addressed the distortion minimization problem at a more practical range of SNR for various

types of SISO fading channels. Further investigation along this direction would include the

extension of our result to multi-user, multiple-relay, multiple-antenna systems.

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