cooperation and joint source-channel transmission in wireless...
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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
Dealing. Therefore, limited reproduction of this work for the purposes of private
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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
4.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.A Optimality of Equating Linear Terms in (4.11) . . . . . . . . . . . . . . . . . 90
4.B Proof of Theorem 4.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
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
5.6 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.A Proof of Theorem 5.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.B Proof of Theorem 5.4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
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.
CHAPTER 1. INTRODUCTION 3
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
CHAPTER 1. INTRODUCTION 4
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
CHAPTER 1. INTRODUCTION 5
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.
CHAPTER 1. INTRODUCTION 6
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.
CHAPTER 1. INTRODUCTION 7
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
CHAPTER 2. REVIEW 11
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.
CHAPTER 2. REVIEW 12
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
CHAPTER 2. REVIEW 13
(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
CHAPTER 2. REVIEW 14
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.
CHAPTER 2. REVIEW 15
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
CHAPTER 2. REVIEW 16
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
CHAPTER 2. REVIEW 17
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
CHAPTER 2. REVIEW 18
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
CHAPTER 2. REVIEW 19
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.
CHAPTER 2. REVIEW 20
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
CHAPTER 2. REVIEW 21
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
CHAPTER 2. REVIEW 22
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
CHAPTER 2. REVIEW 23
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
CHAPTER 2. REVIEW 24
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
CHAPTER 2. REVIEW 25
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.
CHAPTER 2. REVIEW 26
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
CHAPTER 2. REVIEW 27
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
CHAPTER 2. REVIEW 28
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
CHAPTER 2. REVIEW 29
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
CHAPTER 2. REVIEW 30
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.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 33
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
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 34
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
Source Relay 1 Relay m
1+m
Ntn
1+m
Ntn
1+m
Ntn
11
+m
Nt
11
+m
Nt
11
+m
Nt
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
Source Relay 1 Relay m
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
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
(b)
Layer 1 1R
channel uses:
Source All Relays All Relays
Layer 1 1R Layer 2 2R Layer 2 2R Layer n nR Layer n nR
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
Layer 1 1R Layer 2 2R Layer 2 2R Layer n nR Layer n nR
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.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 35
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].
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 36
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.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 37
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 ,
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 38
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
Source Relay 1 Relay m
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
Source Relay 1 Relay m
(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*
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*
(b)
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 Relay(s) in D1
Relay(s) in DnSource
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 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.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 39
∑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
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 40
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.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 41
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.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 42
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
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 43
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)
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 44
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
refinable in terms of the DMT curve. The successive decoding diversity gain is given by
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)
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 45
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)).
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 46
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.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 47
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)
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 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).
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 49
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
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 50
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)
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 51
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.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 52
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).
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 53
Corollary 3.4.10. The space-time-coded cooperation protocol with t = 12 is successively
refinable in terms of the DMT curve. The successive decoding diversity gain is given by
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.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 54
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.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 55
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,
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 56
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.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 57
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)
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 58
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.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 59
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)
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 60
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)
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 61
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)
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 62
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}.
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 63
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)
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 64
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).
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 65
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)
CHAPTER 3. MULTI-RELAY COOPERATIVE NETWORKS 66
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.
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 69
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
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 70
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.
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 71
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.
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 72
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)
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 73
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.
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 74
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.
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 75
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)
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 76
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.
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 77
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
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 78
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
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 79
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)
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 80
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,
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 81
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.
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 82
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-
back, m relays, and bandwidth ratio b is lower-bounded by
∆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.
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 83
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)
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 84
Proof. The proof is given in Appendix 4.C.
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-
back, m relays, and bandwidth ratio b is lower-bounded by
∆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)
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 85
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)
4.5 Results and Discussions
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.
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 86
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
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 87
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.
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 88
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
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 89
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
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 90
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).
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 91
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)
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 92
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).
4.B Proof of Theorem 4.4.1
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).
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 93
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)
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 94
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).
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 95
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)
CHAPTER 4. MULTI-RELAY COOPERATION WITH LIMITED FEEDBACK 96
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.
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 99
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 100
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
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 101
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
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 102
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 103
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.
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 104
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)
The achievable distortion exponents (∆1,∆2) satisfy 0 ≤ ∆1 ≤ ∆∗1 and 0 ≤ ∆2 ≤ ∆∗2.
Combining ∆∗1 and ∆∗2 in (5.13) by eliminating t leads to the claimed result.
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)
The achievable distortion exponents (∆1,∆2) satisfy 0 ≤ ∆1 ≤ ∆∗1 and 0 ≤ ∆2 ≤ ∆∗2.
Combining ∆∗1 and ∆∗2 in (5.13) by eliminating t leads to the claimed result.
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 105
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]
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 106
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 107
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.
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 108
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 109
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 110
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 111
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
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 112
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.
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 113
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 114
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 115
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 116
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)
Proof. The proof is given in Appendix 5.C.
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.
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 117
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).
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 118
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 119
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 120
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.
5.6 Results and Discussions
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 121
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
Outer boundOne−way SROne−way LSOne−way BSMABC, AF/CFMABC, DFTDBC, AFTDBC, DF
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.
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 122
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
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 123
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
Outer boundOne−way SROne−way LSOne−way BSMABC, AF/CFMABC, DFTDBC, AFTDBC, DF
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
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 124
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 125
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)+.
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 126
(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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 127
5.B Proof of Theorem 5.4.5
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 128
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 129
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 .
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 130
(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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 131
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 132
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 133
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.
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 134
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 135
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)
which leads to the following region of (r1, r2):
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 136
• Case 3: 0 ≤ r1 <1−t2
3 and 1−t13 ≤ r2 < 1− t1.
The optimal triplet (t1, t2, t3) is found to be
t1 =1− r2
2, t2 =
1 + 3r2
4, t3 =
1− r2
4, (5.149)
which leads to the following region of (r1, r2):
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 .
The optimal triplet (t1, t2, t3) is found to be
t1 =1 + 3r1
4, t2 =
1− r1
2, t3 =
1− r1
4, (5.152)
which leads to the following region of (r1, r2):
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.
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 137
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)
CHAPTER 5. TWO-WAY RELAYING COOPERATIVE NETWORKS 138
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
CHAPTER 6. FINITE-SNR END-TO-END DISTORTION MINIMIZATION 141
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)
CHAPTER 6. FINITE-SNR END-TO-END DISTORTION MINIMIZATION 142
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).
CHAPTER 6. FINITE-SNR END-TO-END DISTORTION MINIMIZATION 143
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.
CHAPTER 6. FINITE-SNR END-TO-END DISTORTION MINIMIZATION 144
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)
CHAPTER 6. FINITE-SNR END-TO-END DISTORTION MINIMIZATION 145
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
CHAPTER 6. FINITE-SNR END-TO-END DISTORTION MINIMIZATION 146
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
CHAPTER 6. FINITE-SNR END-TO-END DISTORTION MINIMIZATION 147
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
CHAPTER 6. FINITE-SNR END-TO-END DISTORTION MINIMIZATION 148
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
CHAPTER 6. FINITE-SNR END-TO-END DISTORTION MINIMIZATION 149
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
CHAPTER 6. FINITE-SNR END-TO-END DISTORTION MINIMIZATION 150
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)
CHAPTER 6. FINITE-SNR END-TO-END DISTORTION MINIMIZATION 151
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
CHAPTER 7. CONCLUSIONS 154
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
CHAPTER 7. CONCLUSIONS 155
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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