VEL NO OTOCOLS PR OR F THE O-HOP TW HALF-DUPLEX Y …...Co op e erativ orks w Net. 2 1.2 The o-Hop w...

NOVEL PROTOCOLS FOR THE TWO-HOP HALF-DUPLEX RELAY

NETWORK

by

Nikola Zlatanov

M. S i., Ss. Cyril and Methodius University, 2010

Dipl. Eng., Ss. Cyril and Methodius University, 2007

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

The Fa ulty of Graduate and Postdo toral Studies

(Ele tri al and Computer Engineering)

THE UNIVERSITY OF BRITISH COLUMBIA

(Van ouver)

July 2015

© Nikola Zlatanov, 2015

Abstra t

Wireless ommuni ation has enabled people to be onne ted from anywhere and at

any time. This has had a profound impa t on human so iety. Currently, wireless

ommuni ation is performed using the ommuni ation proto ols developed for ellu-

lar and wireless lo al area networks. Although these proto ols support a broad range

of mobile servi es, they do not fully exploit the apa ity of the underlying networks

and annot satisfy the exponential growth in demand for higher data rates and more

reliable onne tions. Therefore, new ommuni ation proto ols have to be developed

for general wireless networks in order to meet this demand. Ultimately, these pro-

to ols will have to be able to rea h the fundamental limits of information ow in

wireless networks, i.e., the network apa ity. However, due to the omplexity of the

problem, it is urrently not known how to design su h proto ols for general wireless

networks. Therefore, in order to get insight into this problem, as a rst step, ommu-

ni ation proto ols for very simple wireless networks have to be devised. Later, the

gained knowledge an be exploited to design proto ols for more omplex networks.

In this thesis, we propose new ommuni ation proto ols for the simplest half-

duplex relay network, whi h is also the most basi building blo k of any wireless

network, the two-hop half-duplex relay network. This network is omprised of a

sour e, a half-duplex relay, and a destination where a dire t sour e-destination link

is not available. For the onsidered relay network, we propose three novel ommuni-

ation proto ols. The rst proposed proto ol a hieves the apa ity of the onsidered

ii

Abstra t

network when fading on the sour e-relay and relay-destination links is not present.

The se ond and third proto ols signi antly improve the average data rate and the

outage probability, respe tively, of the onsidered network when both the sour e-relay

and the relay-destination links are ae ted by fading.

iii

Prefa e

Chapters 24 of this thesis are based on works performed under the supervision of

Prof. Robert S hober and the ollaboration with Prof. Petar Popovski, Aalborg

University, Denmark, and Vahid Jamali, Friedri h-Alexander-Universität Erlangen-

Nürnberg, Germany.

Unless otherwise stated, for all hapters and orresponding papers, I ondu ted

the literature survey on related topi s, identied the hallenges, and performed the

analyses and simulations. I wrote all paper drafts for whi h I am the rst author.

My supervisor guided the resear h, validated the analyses, and gave omments on

improving the manus ripts. The ollaborators' ontributions are listed below:

1. Vahid Jamali gave omments for improving the paper related to Chapter 2 and

validated the analyses.

2. Prof. Petar Popovski suggested the system model for the paper related to

Chapter 3.

Two papers related to Chapter 2 have been submitted for publi ation:

• N. Zlatanov, V. Jamali, and R. S hober, Capa ity of the Two-Hop Half-Duplex

Relay Channel, Submitted for publi ation.

• N. Zlatanov, V. Jamali, and R. S hober, On the Capa ity of the Two-Hop

Half-Duplex Relay Channel, Pro . of IEEE Globe om, San Diego, CA, USA,

De . 2015.

iv

Prefa e

Two papers related to Chapter 3 have been published:

• N. Zlatanov, R. S hober, and P. Popovski, Buer-Aided Relaying with Adap-

tive Link Sele tion, IEEE Journal on Sele ted Areas in Communi ations, vol.

31, no. 8, pp. 1530-1542, Aug. 2013.

• N. Zlatanov, R. S hober, and P. Popovski, Throughput and Diversity Gain

of Buer-Aided Relaying, Pro . of IEEE Globe om 2011, Houston, TX, De .

2011.

Two papers related to Chapter 4 have been published:

• N. Zlatanov and R. S hober, Buer-Aided Relaying with Adaptive Link Sele -

tion - Fixed and Mixed Rate Transmission, IEEE Transa tions on Information

Theory, vol. 59, no. 5, pp. 2816-2840, May 2013.

• N. Zlatanov and R. S hober, Buer-Aided Relaying with Mixed Rate Trans-

mission, Pro . of IEEE IWCMC 2012, Limassol, Cyprus, Aug. 2012 (Invited

Paper).

I have also o-authored other resear h works whi h have been published or sub-

mitted for publi ation during my time as a Ph.D. student at UBC. These works are

listed in Appendix D.

v

Table of Contents

Abstra t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Prefa e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

List of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

A knowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Dedi ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Cooperative Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The Two-Hop HD Relay Channel . . . . . . . . . . . . . . . . . . . . 4

1.3 Motivation of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 The Two-Hop HD Relay Channel Without Fading . . . . . . 5

1.3.2 The Two-Hop HD Relay Channel With Fading . . . . . . . . 7

1.4 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 15

vi

Table of Contents

2 Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fad-

ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2 Mathemati al Modelling of the HD Constraint . . . . . . . . 22

2.2.3 Mutual Information and Entropy . . . . . . . . . . . . . . . . 24

2.3 Capa ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1 The Capa ity . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.2 A hievability of the Capa ity . . . . . . . . . . . . . . . . . . 29

2.3.3 Simpli ation of Previous Converse Expressions . . . . . . . . 36

2.4 Capa ity Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4.1 Binary Symmetri Channels . . . . . . . . . . . . . . . . . . . 38

2.4.2 AWGN Channels . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.5 Numeri al Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.5.1 BSC Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.5.2 AWGN Links . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.6 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 Buer-Aided RelayingWith Adaptive Re eption-Transmission: Adap-

tive Rate Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Preliminaries and Ben hmark S hemes . . . . . . . . . . . . . . . . . 57

3.3.1 Adaptive Re eption-Transmission Proto ol and CSI Require-

ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

vii

Table of Contents

3.3.2 Transmission Rates and Queue Dynami s . . . . . . . . . . . 58

3.3.3 A hievable Average Rate . . . . . . . . . . . . . . . . . . . . 60

3.3.4 Conventional Relaying . . . . . . . . . . . . . . . . . . . . . . 60

3.4 Optimal Adaptive Re eption-Transmission Proto ol for Fixed Powers 63

3.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 63

3.4.2 Optimal Adaptive Re eption-Transmission Proto ol . . . . . 64

3.4.3 De ision Threshold . . . . . . . . . . . . . . . . . . . . . . . 69

3.4.4 Rayleigh Fading . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4.5 Real-Time Implementation . . . . . . . . . . . . . . . . . . . 71

3.5 Optimal Adaptive Re eption-Transmission and Optimal Power Allo-

ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5.1 Problem Formulation and Optimal Power Allo ation . . . . . 72

3.5.2 Finding λ and ρ . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.5.3 Rayleigh Fading . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.5.4 Real-Time Implementation . . . . . . . . . . . . . . . . . . . 76

3.6 Delay-Limited Transmission . . . . . . . . . . . . . . . . . . . . . . . 77

3.6.1 Average Delay . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.6.2 Buer-Aided Proto ol for Delay Limited Transmission . . . . 79

3.7 Numeri al and Simulation Results . . . . . . . . . . . . . . . . . . . 81

3.7.1 Delay-Un onstrained Transmission . . . . . . . . . . . . . . . 81

3.7.2 Delay-Constrained Transmission . . . . . . . . . . . . . . . . 84

3.8 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4 Buer-Aided RelayingWith Adaptive Re eption-Transmission: Fixed

and Mixed Rate Transmission . . . . . . . . . . . . . . . . . . . . . . 87

4.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

viii

Table of Contents

4.2 System Model and Channel Model . . . . . . . . . . . . . . . . . . . 89

4.3 Preliminaries and Ben hmark S hemes . . . . . . . . . . . . . . . . . 90

4.3.1 Adaptive Re eption-Transmission and CSI Requirements . . 91

4.3.2 Transmission Rates and Queue Dynami s . . . . . . . . . . . 93

4.3.3 Link Outages and Indi ator Variables . . . . . . . . . . . . . 95

4.3.4 Performan e Metri s . . . . . . . . . . . . . . . . . . . . . . . 96

4.3.5 Performan e Ben hmarks for Fixed Rate Transmission . . . . 97

4.3.6 Performan e Ben hmarks for Mixed Rate Transmission . . . . 99

4.4 Optimal Buer-Aided Relaying for Fixed Rate Transmission Without

Delay Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 102

4.4.2 Throughput Maximization . . . . . . . . . . . . . . . . . . . 103

4.4.3 Performan e in Rayleigh Fading . . . . . . . . . . . . . . . . 112

4.5 Buer-Aided Relaying for Fixed Rate Transmission With Delay Con-

straints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.5.1 Average Delay . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.5.2 Adaptive Re eption-Transmission Proto ol for Delay Limited

Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.5.3 Throughput and Delay . . . . . . . . . . . . . . . . . . . . . 117

4.5.4 Outage Probability . . . . . . . . . . . . . . . . . . . . . . . . 122

4.6 Mixed Rate Transmission . . . . . . . . . . . . . . . . . . . . . . . . 125

4.6.1 Optimal Adaptive Re eption-Transmission Proto ol Without

Power Allo ation . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.6.2 Optimal Adaptive Re eption-Transmission Poli y With Power

Allo ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

ix

Table of Contents

4.6.3 Mixed Rate Transmission With Delay Constraints . . . . . . 135

4.6.4 Conventional Relaying With Delay Constraints . . . . . . . . 136

4.7 Numeri al and Simulation Results . . . . . . . . . . . . . . . . . . . 137

4.7.1 Fixed Rate Transmission . . . . . . . . . . . . . . . . . . . . 138

4.7.2 Mixed Rate Transmission . . . . . . . . . . . . . . . . . . . . 142

4.8 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5 Summary of Thesis and Future Resear h Topi s . . . . . . . . . . . 147

5.1 Summary of the Results . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Appendi es

A Proofs for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

A.1 Proof That the Probability of Error at the Relay Goes to Zero When

(2.25) Holds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

A.2 Proof That the Probability of Error at the Destination Goes to Zero

When (2.26) Holds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A.3 Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

B Proofs for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

B.1 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 169




x

Table of Contents

B.5 Proof of Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

C Proofs for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

C.1 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 178


C.3 Proof of Lemma 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

C.4 Proof of Lemma 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 187


C.6 Proof of Lemma 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

C.7 Proof of Lemma 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 191






D Other Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

xi

List of Figures

1.1 The two-hop relay network omprised of a sour e (S), a relay (R), and

a destination (D). Sin e there is no dire t sour e-destination link, the

sour e transmits a message to the destination only via the relay. . . . 4

2.1 Two-hop relay hannel. . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 To a hieve the apa ity in (2.18), transmission is organized in N + 1

blo ks and ea h blo k omprises k hannel uses. . . . . . . . . . . . . 29

2.3 Example of generated swit hing ve tor along with input/output ode-

words at sour e, relay, and destination. . . . . . . . . . . . . . . . . 35

2.4 Blo k diagram of the proposed hannel oding proto ol for time slot

i. The following notations are used in the blo k diagram: C1|r and C2

are en oders, D1 and D2 are de oders, I is an inserter, S is a sele tor,

B is a buer, and w(i) denotes the message transmitted by the sour e

in blo k i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Comparison of rates for the BSC as a fun tion of the error probability

Pε1 = Pε2 = Pε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.6 Example of proposed input distributions at the relay pV (x2). . . . . . 49

2.7 Sour e-relay and relay destination links are AWGN hannels with P1/σ21 =

P2/σ22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

xii

List of Figures

2.8 Sour e-relay and relay destination links are AWGN hannels with P1/σ21/10 =

P2/σ22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1 The two-hop HD relay network with fading on the S-R and R-D links.

s(i) and r(i) are the instantaneous SNRs of the S-R and R-D links in

the ith time slot, respe tively. . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Average rates a hieved with buer-aided relaying (BAR) with adap-

tive re eption-transmission and with onventional relaying with and

without buer for ΩSR = 0.9 and ΩRD = 1.1. . . . . . . . . . . . . . . 82

3.3 Estimated ρe(i) as a fun tion of the time slot i. . . . . . . . . . . . . 83

3.4 Average rate with buer-aided relaying with adaptive re eption-transmission

with and without power allo ation for ΩS = 0.1 and ΩR = 1.9 . . . . 83

3.5 Average rate of BAR with adaptive re eption-transmission for dierent

average delay onstraints. . . . . . . . . . . . . . . . . . . . . . . . . 84

3.6 Average delay until time slot i for T0 = 5 and γ = 20 dB . . . . . . . . 85

4.1 Ratio of the throughputs of buer-aided relaying and Conventional

Relaying 1, τ/τfixedconv,1, vs. γ. Fixed rate transmission without delay

onstraints. γS = γR = γ, S0 = R0 = 2 bits/symb, and ΩR = 1. . . . . 138

4.2 Outage probability of buer-aided (BA) relaying and Conventional

Relaying 1 vs. γ. Fixed rate transmission without delay onstraints.

γS = γR = γ, S0 = R0 = 2 bits/symb, and ΩR = 1. . . . . . . . . . . . 139

4.3 Throughputs of buer-aided (BA) relaying and Conventional Relaying

2 vs. γ. Fixed rate transmission with delay onstraints. γS = γR = γ,

S0 = R0 = 2 bits/symb, ΩR = 1, and ΩS = 1. . . . . . . . . . . . . . . 140

xiii

List of Figures

4.4 Outage probability of buer-aided (BA) relaying and Conventional

Relaying 2 vs. γ. Fixed rate transmission with delay onstraints. γS =

γR = γ, S0 = R0 = 2 bits/symb, ΩR = 1, and ΩS = 1. . . . . . . . . . 141

4.5 Throughput of buer-aided relaying with adaptive re eption-transmission

and Conventional Relaying 1 vs. Γ. Mixed rate transmission without

delay onstraints. ΩS = 10, ΩR = 1, and S0 = 2 bits/symb. . . . . . . 143

4.6 Throughput of buer-aided relaying with adaptive re eption-transmission

and onventional relaying vs. Γ. Mixed rate and xed rate transmis-

sion with delay onstraint. ET = 5 time slots, γS = γR = Γ, S0 = 2

bits/symb, and ΩS = ΩR = 1. . . . . . . . . . . . . . . . . . . . . . . 144

C.1 Markov hain for the number of pa kets in the queue of the buer if

the link sele tion variable di is given by (4.59). . . . . . . . . . . . . . 188

C.2 Markov hain for the number of pa kets in the queue of the buer if

the link sele tion variable di is given by (4.61) or (4.63). . . . . . . . 190

xiv

List of Abbreviations

AWGN Additive White Gaussian Noise

BA Buer-Aided

BAR Buer-Aided Relaying

BSC Binary Symmetri Channel

CDF Cumulative Distribution Fun tion

CSI Channel State Information

CSIT Channel State Information at Transmitter

DF De odeandForward

i.i.d. Independent and Identi ally Distributed

FD Full-Duplex

HD Half-Duplex

LTE Long Term Evolution

PMF Probability Mass Fun tion

PDF Probability Density Fun tion

RV Random Variable

SNR SignaltoNoise Ratio

WiMAX Worldwide Interoperability for Mi rowave A ess

xv

List of Notation

E· Statisti al expe tation

Pr· Probability of an event

I(·; ·) Mutual information

H(·) Entropy

xvi

A knowledgments

First, I would like to express my deep and sin ere gratitude to my advisor, Prof.

Robert S hober, for his endless support and invaluable advi e during my Ph.D. study.

I am extraordinarily lu ky to have had Prof. S hober as my advisor. This thesis would

not have been possible without him. I am forever indebted to Prof. S hober.

I am grateful to my o-supervisor, Prof. Lutz Lampe, for his help and onstant

support. I would also like to spe ially thank Prof. Zoran Hadzi-Velkov for his overall

guidan e, onstant support, and invaluable advi e. I have also greatly beneted from

the ollaboration and support of Prof. George Karagiannidis, Prof. Petar Popovski,

Prof. Ljup o Ko arev, Vahid Jamali, and Dr. Derri k Wing Kwan Ng.

Endless gratitude and admiration goes to my mother, Gu a Zlatanova, my father,

Todor Zlatanov, and my sister, Zori a Zlatanova, for their relentless support and

in redible sa ri e. Without their love and wisdom, I would not be where I am

today.

Finally, I would like to thank my wife, Ljupka Zlatanova, who has always been

there for me. I am grateful for her un onditional support, her endless love, and

ex eptional sa ri e. Without her, none of this would be possible.

This work was supported nan ially by The University of British Columbia, The

Killam Trusts, the Natural S ien es and Engineering Resear h Coun il of Canada

(NSERC), and the German A ademi Ex hange Servi e (DAAD).

xvii

Dedi ation

To My Family.

xviii

Chapter 1

Introdu tion

Wireless ommuni ation has enabled people to be onne ted from anywhere and

at any time. This has had a profound impa t on human so iety. Currently, wireless

ommuni ation is performed using the ommuni ation proto ols developed for ellular

and wireless lo al area networks. Although these proto ols support a broad range

of mobile servi es, they do not fully exploit the apa ity of the underlying networks

and annot satisfy the exponential growth in demand for higher data rates and more

reliable onne tions. Therefore, new ommuni ation proto ols have to be developed

for general wireless networks in order to meet this demand. Ultimately, these proto ol

will have to be able to rea h the fundamental limits of information ow in wireless

networks, i.e., the network apa ity. Unfortunately, our urrent understanding of

the network apa ity is very poor even for very simple networks [1. Therefore, we

are urrently unable to design proto ols whi h rea h the apa ity of general wireless

networks. Hen e, in order to get insight into this problem, we rst have to devise

ommuni ation proto ols for very simple wireless networks, e.g., networks omprised

of one sour e, one relay, and one destination, and then use the gained knowledge

to design proto ols for more omplex networks, e.g., networks omprised of multiple

sour es, relays, and destinations. These proto ols should take into a ount pra ti al

limitations su h as half-duplex (HD) re eption and transmission in HD relaying and

self-interferen e in full-duplex (FD) relaying, and ultimately, should be able the rea h

the apa ity of the underlaying networks. In the following, we briey review the state

1

Chapter 1. Introdu tion

of the art in relay networks and motivate the work in this thesis.

This hapter is organized as follows. In Se tion 1.1, we briey introdu e the on-

ept of ooperative networks. In Se tion 1.2, we des ribe the simplest HD ooperative

network, the two-hop HD relay network. In Se tion 1.3, we motivate this thesis by

reviewing some of the best known proto ols for the two-hop HD relay network and

present their orresponding a hievable data rates and outage probabilities. In Se -

tion 1.4, we summarize the ontributions made in this thesis. The thesis organization

is provided in Se tion 1.5.

1.1 Cooperative Networks

For improving our understanding of the network apa ity, it has long been realized

that we have to move away from analyzing networks as a olle tion of dis onne ted

point-to-point ommuni ations, and fo us our attention on analyzing networks as

one system in whi h all nodes mutually ooperate in order for the information ow

in the network to rea h its fundamental limit, i.e., to rea h the network apa ity

[1. Thereby, the network nodes must assist ea h other by willingly a ting as relays

and use their own resour es to forward the information of other nodes [2, [3. In

parti ular, when a sour e (e.g., mobile phone) transmits a pa ket to a destination

(e.g., base station) wirelessly, all surrounding nodes (e.g., other mobile phones) whi h

overhear this pa ket, should pro ess it, and retransmit the pro essed pa ket to the

intended destination, thus helping in the transmission pro ess. It has been shown

that ooperation among the nodes of a network signi antly improves the data rate

and/or reliability of the network, and as a result, a host of ooperative te hniques

have been proposed [2-[51. Due to the benets of ooperation, it is expe ted that

future wireless ommuni ation systems will in lude some form of ooperation between

2


network nodes. In fa t, simple relaying s hemes have been/are being in luded in re-

ent/future wireless standards su h as the Worldwide Interoperability for Mi rowave

A ess (WiMAX) and Long Term Evolution (LTE) Advan ed standards [6, [49, [50.

In ooperative wireless networks, ea h relay node an perform re eption and trans-

mission either in FD or HDmode [7. In the FD mode, the relay transmits and re eives

at the same time and in the same frequen y band, whereas in the HD mode, trans-

mission and re eption o ur in the same frequen y band but not at the same time

or at the same time but in dierent frequen y bands. Although ideal FD relaying

a hieves a higher data rate than HD relaying, given the limitations of urrent radio

implementations, ideal FD relaying is not possible due to strong self-interferen e [7.

More pre isely, the transmit signal of an FD node onstitutes strong interferen e for

the re eived signal at the same node, thus potentially preventing the FD node from

su essfully de oding the re eived messages and thereby severely degrading its per-

forman e. Although re ently there has been a lot of eort in developing FD nodes

whi h an redu e self-interferen e [52, [53, it is still not possible to attenuate the

self-interferen e to a level whi h makes it negligible [54. Therefore, HD relaying is

still a preferred hoi e in pra ti e due to the mu h simpler hardware implementation

and the absen e of self-interferen e.

In urrent HD relaying proto ols, re eption and transmission at the HD relays

is organized in two su essive time slots. In the rst time slot, the relay re eives

data transmitted by a sour e, and in the se ond time slot the relay forwards the

re eived data to a destination. Su h xed s heduling of re eption and transmission

in HD relaying has be ome a ommonly a epted prin iple, i.e., it has almost be ome

an axiom. Resear hers have long thought that the apa ity limits of ooperative HD

relay networks an be obtained with xed s heduling of re eption and transmission at

3


S R D

Figure 1.1: The two-hop relay network omprised of a sour e (S), a relay (R), and a

destination (D). Sin e there is no dire t sour e-destination link, the sour e transmits

a message to the destination only via the relay.

the relays [18. However, as we will show in this thesis, xed s heduling of re eption

and transmission at HD relays is not optimal and results in signi ant performan e

losses.

1.2 The Two-Hop HD Relay Channel

Given our urrent knowledge, we are still not able to design ommuni ation pro-

to ols whi h rea h the apa ity of general HD relay networks [1. Hen e, in order

to in rease our knowledge, we have to rst investigate ommuni ation proto ols for

very simple HD relay networks. As a onsequen e, in this thesis, we will investigate

ommuni ation proto ols for the simplest HD relay network, shown in Fig. 1.1, whi h

we refer to as the two-hop HD relay hannel or as the two-hop HD relay network,

inter hangeably. The two-hop HD relay hannel onsists of a sour e, a HD relay, and

a destination, and there is no dire t link between the sour e and the destination.

Due to the HD onstraint, the relay annot transmit and re eive at the same time.

Moreover, sin e there is no dire t sour e-destination link, the sour e has to transmit

its information to the destination via the relay. The network shown in Fig. 1.1 is

not only the simplest relay network, but is also the most basi building blo k of any

ooperative network. Therefore, by understanding how to improve the performan e

of this network, we will get insight into how to improve the performan e of general

HD relay networks. In the following, we motivate this thesis by providing a brief

4


overview of previous results for the two-hop HD relay hannel.

1.3 Motivation of the Thesis

Although extensively investigated, the apa ity of the two-hop HD relay hannel is

not fully known nor understood. In parti ular, a apa ity expression whi h an be

evaluated is not available and an expli it oding s heme whi h a hieves the apa ity

is not known either. Hen e, only oding s hemes whi h a hieve rates stri tly lower

than the apa ity are known. To motivate this thesis, in the following, we briey

review previous results for the data rate and the outage probability of the two-hop

HD relay hannel in the absen e and presen e of fading.

1.3.1 The Two-Hop HD Relay Channel Without Fading

Consider the system model in Fig. 1.1. Assume that both the sour e-relay and relay-

destination links are general memoryless hannels whi h are not ae ted by fading,

i.e., the hannels' statisti s do not hange with time. For this relay hannel, the sour e

wants to transmit a message via the HD relay to the destination in n hannel uses

1

with the largest possible data rate for whi h the destination an reliably de ode the

transmitted message. Currently, a oding s heme whi h a hieves the largest known

data rate is des ribed in [18 and [48. In parti ular, the ommuni ation is performed

in n → ∞ hannel uses and is organized in two su essive time slots. In the rst

and se ond time slot the hannel is used nξ and n(1 − ξ) times, respe tively, where

0 < ξ < 1. In the rst time slot, the sour e transmits to the relay a odeword

omprised of nξ symbols and with a rate equal to the apa ity of the sour e-relay

hannel denoted by CSR. The relay de odes the re eived data, re-en odes it into a

1

A hannel use is equivalent to the duration of one symbol.

5


odeword omprised of n(1 − ξ) symbols and with a rate equal to the apa ity of

the relay-destination hannel, denoted by CRD, and transmits it to the destination.

Thereby, by optimizing ξ for rate maximization, the following data rate is a hieved

Rconv =CSRCRD

CSR + CRD. (1.1)

As an be seen from the dis ussion above, this ommuni ation proto ol has a

xed s heduling of re eption and transmission at the relay. However, su h xed

s heduling of re eption and transmission at the relay was shown to be suboptimal

in [8. In parti ular, in [8, it was shown that if the xed s heduling of re eption

and transmission at the HD relay is abandoned, then additional information an

be en oded in the relay's re eption and transmission swit hing pattern whi h would

yield a data rate larger than (1.1). Moreover, it was argued in [8 that the data rate

a hieved with the en oding of information in the relay's re eption and transmission

swit hing pattern would be the apa ity of the two-hop HD relay hannel in the

absen e of fading. However, the results for the apa ity of the two-hop HD relay

hannel in [8, as well in the literature, are in omplete. In parti ular, a apa ity

expression whi h an be evaluated still has not been provided and an expli it oding

s heme whi h a hieves the apa ity rate, or any rate larger than (1.1), is still not

known. Therefore, although expli it upper bounds on the apa ity exists [8, it is

still unknown exa tly how mu h larger the apa ity is ompared to the rate in (1.1).

Motivated by the above dis ussion, in Chapter 2, we derive a new easy-to-evaluate

expression for the apa ity of the two-hop HD relay hannel based on simplifying

previously derived onverse expressions. In ontrast to previous results, this apa ity

expression an be easily evaluated. Moreover, we propose a oding s heme whi h an

a hieve the apa ity. In parti ular, we show that a hieving the apa ity requires the

6


relay to swit h between re eption and transmission in a symbol-by-symbol manner.

Thereby, the relay does not only send information to the destination by transmitting

information- arrying symbols but also with the zero symbol resulting from the relay's

silen e during re eption. Furthermore, we show that the apa ity is signi antly

higher than the rate in (1.1).

1.3.2 The Two-Hop HD Relay Channel With Fading

For the onsidered HD relay network in Fig. 1.1, assume that both sour e-relay and

relay-destination links are additive white Gaussian noise (AWGN) hannels ae ted

by slow fading. Assume that the fading is a stationary and ergodi random pro ess.

Moreover, assume that time is divided into N → ∞ time slots su h that during

one time slot the fading on both sour e-relay and relay-destination links remains

onstant and hanges from one time slot to the next. Let CSR(i) and CRD(i) denote

the apa ities of the sour e-relay and relay-destination hannels in the i-th time slot,

respe tively. Furthermore, let CSR and CRD denote the average apa ities of the

sour e-relay and relay-destination hannels, respe tively, given by

CSR = ECSR(i)(a)= lim

N→∞

1

N

N∑

i=1

CSR(i) (1.2)

CRD = ECRD(i)(a)= lim

N→∞

1

N

N∑

i=1

CRD(i), (1.3)

where E· denotes expe tation and (a) follows from the assumed ergodi ity.

For this network, in the following, we briey review ommuni ation proto ols

whi h a hieve the best known average data rate and the best known outage proba-

bility, respe tively.

7


Best Known Data Rate

A ommuni ation proto ol whi h a hieves the highest known data rate for this net-

work was proposed in [18. In parti ular, the ommuni ation is performed in N → ∞

time slots. During one time slot, the hannel is used n → ∞ times. The proposed

proto ol in [18 is as follows. In the rst Nξ time slots, the sour e transmits to the

HD relay a odeword omprised of Nnξ symbols, where 0 < ξ < 1, and with rate

equal to the average apa ity of the sour e-relay hannel CSR. The relay de odes

the re eived data, re-en odes it into a odeword omprised on Nn(1 − ξ) symbols

and with rate equal to the average apa ity of the relay-destination hannel CRD and

transmits it to the destination. Thereby, by optimizing ξ for rate maximization, the

following data rate is a hieved

Rconv,1 =CSRCRD

CSR + CRD

. (1.4)

In order to a hieve the rate in (1.4), the destination has to wait for N → ∞ time

slots before it an de ode the re eived odeword. This may not be pra ti al for a

host of appli ations. To redu e the delay, and yet a hieve the same rate as (1.4), the

following proto ol an be used [51. Both sour e and relay transmit odewords whi h

span one time slot and are omprised of n → ∞ symbols. The relay is equipped

with an innite size buer. The ommuni ation is performed in N → ∞ time slots,

and is as follows. In ea h time slot i, where 1 ≤ i ≤ ξN , the sour e transmits to

the relay a odeword with rate CSR(i). The relay de odes the re eived odewords,

stores the information in its buer, and then sends the a umulated information to

the destination in the following (1 − ξ)N slots. In parti ular, in ea h time slot i,

where ξN + 1 ≤ i ≤ N , the relay transmits to the destination a odeword with rate

CRD(i). By optimizing ξ for rate maximization, the a hieved data rate is identi al

8


to the one in (1.4). In this proto ol, the destination has to wait for ξN time slots

before it an start de oding the rst re eived odeword. However, sin e N → ∞, this

proto ol may also be unpra ti al.

To redu e the delay even further, the following proto ol an be used [18. The

ommuni ation is performed in N → ∞ time slots. In time slot i, the sour e and relay

transmit odewords whi h span ξ(i) and 1− ξ(i) fra tions of time slot i, respe tively,

and are omprised of ξ(i)n and (1 − ξ(i))n symbols, respe tively, where n → ∞. In

the ξ(i) fra tion of time slot i, the sour e sends a odeword with rate CSR(i) to the

HD relay. Then, in the remaining 1− ξ(i) fra tion of time slot i, the relay re-en odes

the re eived information and sends it to the destination with rate CRD(i). As a

result, the overall rate transmitted from sour e to destination during time slot i is

R(i) = minξ(i)CSR(i), (1− ξ(i))CRD(i). By optimizing ξ(i) for rate maximization,

the following maximum rate is a hieved in time slot i

R(i) =CSR(i)CRD(i)

CSR(i) + CRD(i). (1.5)

Thereby, during N → ∞ time slots, the average rate a hieved with this ommuni a-

tion proto ol is given by

Rconv,2 = E

CSR(i)CRD(i)

CSR(i) + CRD(i)

. (1.6)

A hieving (1.6) requires the odeword lengths to be variable and adopted for ea h

fading state, whi h may not be desirable in pra ti e. In that ase, the above proto ol

an be modied by setting ξ(i) = 1/2, ∀i, and thereby the following average rate an

be a hieved

Rconv,3 =1

2E minCSR(i), CRD(i) . (1.7)

9


Comparing (1.4), (1.6), and (1.7) we observe that Rconv,1 ≥ Rconv,2 ≥ Rconv,3

holds. However, to realize Rconv,1 and Rconv,2, an innite delay and adaptive odeword

lengths must be introdu ed, respe tively.

As seen from the dis ussion above, all four proto ols have a xed s hedule of the

re eption and transmissions at the relay. We refer to these proto ols as onventional

relaying proto ols throughout this thesis. To in rease the average data rate, in

Chapter 3, we develop a HD relaying proto ol in whi h the HD relay adaptively

hooses whether to re eive or transmit a odeword in a given time slot based on

the instantaneous qualities of the sour e-relay and relay-destination links. This new

approa h requires the relay to have a buer, and therefore, the new proto ol is referred

to as buer-aided relaying with adaptive re eption-transmission. We will show that

buer-aided relaying with adaptive re eption-transmission a hieves rates whi h are

signi antly higher than the rates in (1.4), (1.6), and (1.7). In the following, we

briey des ribe buer-aided relaying with adaptive re eption-transmission and review

previous works on this subje t.

Buer-Aided Relaying With Adaptive Re eption-Transmission

Buer-aided relaying with adaptive re eption-transmission belongs to a lass of om-

muni ation proto ols for wireless HD relay networks where the HD relays use their

buers to adaptively hoose whether to re eive or transmit a pa ket in a given time

slot based on the instantaneous qualities of their respe tive re eiving and transmitting

hannels.

In onventional relaying proto ols, the relays employ a prexed s hedule of trans-

mission and re eption, independent of the quality of the transmitting and re eiving

hannels. This prexed s heduling may lead to a signi ant performan e degrada-

tion in wireless systems, where the quality of the transmitting and re eiving hannels

10


varies with time, sin e it may prevent the relays from exploiting the best transmitting

and the best re eiving hannels. Clearly, performan e ould be improved if the link

with the higher quality ould be used in ea h time slot. This an be a hieved via

a buer-aided relaying proto ol whi h does not have a prexed s hedule of re ep-

tion and transmission. In parti ular, buer-aided relaying with adaptive re eption-

transmission an exploit the stronger of the re eiving and transmitting hannels in

ea h time slot, and thereby improve the performan e.

We devised the on ept of buer-aided relaying with adaptive re eption-transmis-

sion in [55 and showed that signi ant improvement of the average data rate and

the outage probability are possible ompared to onventional relaying. Later, in [56

and [57 we investigated buer-aided relaying with adaptive re eption-transmission

for the two-hop HD relay network, and these two papers onstitute the basis of Chap-

ters 3 and 4, respe tively. The works in [55-[57 led to other extension. For example,

buer-aided relaying with adaptive re eption-transmission were also proposed for the

two-hop HD relay network with bit interleaved oded modulation and orthogonal

frequen y division multiplexing in [58 and with statisti al quality of servi e on-

straints in [59, for two-way relaying in [60-[63, for the HD relay hannel with a

dire t sour e-destination link in [64, [65, for the multihop relay network in [66,

[67, for two sour e and two destination pairs sharing a single HD relay in [68, for

se ure ommuni ation for two-hop HD relaying and HD relay sele tion in [69 and

[70, respe tively, for amplify-and-forward relaying in [71, for energy harvesting in

[72, for HD relay-sele tion in [73-[75, and for hybrid FD/HD in [76.

In the following, we review HD relay proto ols for xed rate transmission.

11


Best Known Outage Probability

For the onsidered relay network in Fig. 1.1, assume that both sour e and relay do

not have hannel state information at the transmitter (CSIT) and therefore have to

transmit odewords with a xed data rate R0. Moreover, assume that both sour e

and relay transmit odewords whi h span one time slot and are omprised of n→ ∞

symbols. In this ase, a hannel apa ity in the stri t Shannon sense does not exist

[77. In other words, not all transmitted odewords an be de oded at the respe tive

re eivers, and for the unde odable re eived odewords the system is said to be in

outage. An appropriate measure for su h systems is the outage probability whi h is

the fra tion of unde odable odewords at the re eiver [77.

For this s enario, a ommuni ation proto ol was proposed in [12 for the two-hop

HD relay hannel. Thereby, the sour e transmits odewords to the HD relay in odd

time slots, and the HD relay retransmits the re eived data to the destination in even

time slots. This proto ol, a hieves the following outage probability

Pout = PrCSR(i) < R0 OR CRD(i) < R0, (1.8)

where Pr· denotes probability. In this thesis, in Chapter 4, we will show that we

an improve the outage probability in (1.8) signi antly, using a novel buer-aided

relaying proto ol with adaptive re eption-transmission spe i ally designed for xed

rate transmission and improvement of the outage probability.

1.4 Contributions of the Thesis

This thesis presents novel HD relaying proto ols for the two-hop HD relay hannel.

In the following, we list the main ontributions of this thesis.

12


1. We derive a new easy-to-evaluate expression for the apa ity of the two-hop

HD relay hannel in the absen e of fading based on simplifying previously de-

rived onverse expressions. Compared to previous results, this apa ity expres-

sion an be easily evaluated. Moreover, we propose a oding s heme whi h

an a hieve the apa ity. In parti ular, we show that a hieving the apa ity

requires the relay to swit h between re eption and transmission in a symbol-

by-symbol manner. Thereby, the relay does not only send information to the

destination by transmitting information- arrying symbols but also with the

zero symbol resulting from the relay's silen e during re eption. As examples,

we derive simplied apa ity expressions for the following two spe ial ases: 1)

The sour e-relay and relay-destination links are both binary-symmetri han-

nels (BSCs); 2) The sour e-relay and relay-destination links are both AWGN

hannels. For these two ases, we numeri ally ompare the apa ity with the

rate a hieved by onventional relaying where the relay re eives and transmits

in a odeword-by- odeword fashion and swit hes between re eption and trans-

mission in a stri tly alternating manner. Our numeri al results show that the

apa ity is signi antly larger than the rate a hieved with onventional relaying

for both the BSC and the AWGN hannel.

2. For the two-hop HD relay hannel when both sour e-relay and relay-destination

links are ae ted by fading, we propose a new relaying proto ol employing adap-

tive re eption-transmission, i.e., in any given time slot, based on the hannel

state information of the sour e-relay and the relay-destination links a de ision

is made whether the relay should re eive or transmit. In order to avoid data loss

at the relay, adaptive re eption-transmission requires the relay to be equipped

with a buer su h that data an be queued until the relay-destination link

13


is sele ted for transmission. We study both delay-un onstrained and delay-

onstrained transmission. For the delay-un onstrained ase, we hara terize

the optimal adaptive re eption-transmission s hedule, derive the orresponding

a hievable rate, and develop an optimal power allo ation s heme. For the delay-

onstrained ase, we propose a modied buer-aided proto ol whi h satises a

predened average delay onstraint at the expense of a lower data rate. Our

analyti al and numeri al results show that buer-aided relaying with adaptive

re eption-transmission with and without a delay onstraint a hieve signi ant

rate gains ompared to onventional relaying proto ols with and without buers

where the relay employs a xed s hedule for re eption and transmission.

3. For the two-hop HD relay hannel when both the sour e-relay and relay-destinat-

ion links are ae ted by fading, we propose two new buer-aided relaying

s hemes with dierent requirements regarding the availability of CSIT. In the

rst s heme, neither the sour e nor the relay have full CSIT, and onsequently,

both nodes are for ed to transmit with xed rates. In ontrast, in the se -

ond s heme, the sour e does not have full CSIT and transmits with xed rate

but the relay has full CSIT and adapts its transmission rate a ordingly. In

the absen e of delay onstraints, for both xed rate and mixed rate transmis-

sion, we derive the throughput-optimal buer-aided relaying proto ols whi h

sele t the relay to either re eive or transmit based on the instantaneous qual-

ities of the sour e-relay and relay-destination links. In addition, for the de-

lay onstrained ase, we develop buer-aided relaying proto ols with adaptive

re eption-transmission that a hieve a predened average delay. Compared to

onventional relaying proto ols, the proposed buer-aided proto ols with adap-

tive re eption-transmission a hieve large performan e gains. In parti ular, for

14


xed rate transmission, we show that the proposed proto ol a hieves a diver-

sity gain of two as long as an average delay of more than three time slots an

be aorded. In ontrast, onventional relaying proto ols a hieve a diversity

gain of one. Furthermore, for mixed rate transmission with an average delay

of ET time slots, a multiplexing gain of r = 1 − 1/(2ET) is a hieved.

As a by-produ t of the onsidered adaptive re eption-transmission proto ols,

we also develop a novel onventional relaying proto ol for mixed rate trans-

mission whi h yields the same multiplexing gain as the proto ol with adaptive

re eption-transmission. Hen e, for mixed rate transmission, for su iently large

average delays, buer-aided HD relaying with and without adaptive re eption-

transmission does not suer from a multiplexing gain loss ompared to FD

relaying.

1.5 Organization of the Thesis

In the following, we provide a brief overview of the remainder of this thesis.

In Chapter 2, we derive the apa ity of the two-hop HD relay hannel when

the sour e-relay and relay-destination links are not ae ted by fading. Thereby, we

rst formally dene the hannel model. Then, we introdu e a new expression for

the apa ity of the onsidered relay hannel, prove that it satises the onverse,

and introdu e an expli it hannel oding s heme whi h a hieves this apa ity. We

also investigate the apa ity for the spe ial ases when the sour e-relay and relay-

destination links are both BSCs and AWGN hannels, respe tively, and numeri ally

ompare the derived apa ity expressions with the rate a hieved by onventional

relaying.

In Chapter 3, we introdu e a novel relaying proto ol, whi h we refer to as buer-

15


aided relaying with adaptive re eption-transmission, for improving the average data

rate of the two-hop HD relay hannel when the sour e-relay and relay-destination

links are AWGN hannels ae ted by fading. Thereby, we rst formally dene the

onsidered system and hannel models. Then, we formulate optimization problems

for maximization of the a hievable average rate of buer-aided relaying with and

without power allo ation. From these optimization problems we derive the optimal

buer-aided relaying proto ols whi h maximize the data rate. Sin e these proto ols

require unlimited delay, we also propose a heuristi buer-aided relaying proto ol

whi h limits the average delay.

In Chapter 4, we introdu e novel buer-aided relaying proto ols for improving the

outage probability of the two-hop HD relay hannel when the sour e-relay and relay-

destination links are AWGN hannels ae ted by fading. Thereby, we investigate two

system models. In the rst system, neither the sour e nor the relay have full CSIT,

and onsequently, both nodes are for ed to transmit with xed rates. In ontrast,

in the se ond system model, the sour e does not have full CSIT and transmits with

xed rate but the relay has full CSIT and adapts its transmission rate a ordingly.

For both system models, we introdu e buer-aided relaying proto ols with adaptive

re eption-transmission for delay un onstrained and delay onstrained transmission,

respe tively. The proto ols for delay un onstrained and delay onstrained transmis-

sions are analyzed and on lusions are drawn.

Chapter 5 summarizes the ontributions of this thesis and outlines areas of future

resear h.

Appendi es A - C ontain proofs of theorems and lemmas used in this thesis.

16

Chapter 2

Capa ity of the Two-Hop

Half-Duplex Relay Channel in the

Absen e of Fading

2.1 Introdu tion

Throughout this hapter, we assume that the two-hop HD relay hannel is not ae ted

by fading, i.e., the statisti s of the sour e-relay and relay-destination hannels do not

hange with time.

The apa ity of the two-hop FD relay hannel without self-interferen e has been

derived in [5 (see the apa ity of the degraded relay hannel). On the other hand,

although extensively investigated, the apa ity of the two-hop HD relay hannel is

not fully known nor understood. The reason for this is that a apa ity expression

whi h an be evaluated is not available and an expli it oding s heme whi h a hieves

the apa ity is not known either. Currently, for HD relaying, detailed oding s hemes

exist only for rates whi h are stri tly smaller than the apa ity, see [18 and [48. To

a hieve the rates given in [18 and [48, the HD relay re eives a odeword in one time

slot, de odes the re eived odeword, and re-en odes and re-transmits the de oded

information in the following time slot, see Se tion 1.3.1 for more details. However,

17

Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading

su h xed swit hing between re eption and transmission at the HD relay was shown to

be suboptimal in [8. In parti ular, in [8, it was shown that if the xed s heduling of

re eption and transmission at the HD relay is abandoned, then additional information

an be en oded in the relay's re eption and transmission swit hing pattern yielding

an in rease in data rate. In addition, it was shown in [8 that the HD relay hannel

an be analyzed using the framework developed for the FD relay hannel in [5. In

parti ular, results derived for the FD relay hannel in [5 an be dire tly applied to

the HD relay hannel. Thereby, using the onverse for the degraded relay hannel in

[5, the apa ity of the dis rete memoryless two-hop HD relay hannel is obtained as

[8, [9, [78

C = maxp(x1,x2)

min

I(X1; Y1|X2) , I(X2; Y2), (2.1)

where I(·; ·) denotes the mutual information, X1 and X2 are the inputs at sour e and

relay, respe tively, Y1 and Y2 are the outputs at relay and destination, respe tively,

and p(x1, x2) is the joint probability mass fun tion (PMF) of X1 and X2. Moreover, it

was shown in [8, [9, [78 that X2 an be represented as X2 = [X ′2, U ], where U is an

auxiliary random variable with two out omes t and r orresponding to the HD relay

transmitting and re eiving, respe tively. Thereby, (2.1) an be written equivalently

as

C = maxp(x1,x′

2,u)min

I(X1; Y1|X′2, U) , I(X

′2, U ; Y2), (2.2)

where p(x1, x′2, u) is the joint PMF of X1, X

′2, and U . However, the apa ity expres-

sions in (2.1) and (2.2), respe tively, annot be evaluated sin e it is not known how

X1 and X2 nor X1, X′2, and U are mutually dependent, i.e., p(x1, x2) and p(x1, x

′2, u)

are not known. In fa t, the authors of [78, page 2552 state that: Despite knowing

18


the apa ity expression (i.e., expression (2.2)), its a tual evaluation is elusive as it

is not lear what the optimal input distribution p(x1, x′2, u) is. On the other hand,

for the oding s heme that would a hieve (2.1) and (2.2) if p(x1, x2) and p(x1, x′2, u)

were known, it an be argued that it has to be a de ode-and-forward strategy sin e

the two-hop HD relay hannel belongs to the lass of the degraded relay hannels

dened in [5. Thereby, the HD relay should de ode any re eived odewords, map

the de oded information to new odewords, and transmit them to the destination.

Moreover, it is known from [8 that su h a oding s heme requires the HD relay to

swit h between re eption and transmission in a symbol-by-symbol manner, and not

in a odeword-by- odeword manner as in [18 and [48. However, sin e p(x1, x2) and

p(x1, x′2, u) are not known and sin e an expli it oding s heme does not exist, it is

urrently not known how to evaluate (2.1) and (2.2) nor how to en ode additional

information in the relay's re eption and transmission swit hing pattern and thereby

a hieve (2.1) and (2.2).

Motivated by the above dis ussion, in this hapter, we derive a new expression for

the apa ity of the two-hop HD relay hannel based on simplifying previously derived

onverse expressions. In ontrast to previous results, this apa ity expression an be

easily evaluated. Moreover, we propose an expli it oding s heme whi h a hieves the

apa ity. In parti ular, we show that a hieving the apa ity requires the relay indeed

to swit h between re eption and transmission in a symbol-by-symbol manner as pre-

di ted in [8. Thereby, the relay does not only send information to the destination by

transmitting information- arrying symbols but also with the zero symbols resulting

from the relay's silen e during re eption. In addition, we propose a modied od-

ing s heme for pra ti al implementation where the HD relay re eives and transmits

at the same time (i.e., as in FD relaying), however, the simultaneous re eption and

19


transmission is performed su h that the self-interferen e is ompletely avoided. As

examples, we ompute the apa ities of the two-hop HD relay hannel for the ases

when the sour e-relay and relay-destination links are both binary-symmetri han-

nels (BSCs) and additive white Gaussian noise (AWGN) hannels, respe tively, and

we numeri ally ompare the apa ities with the rates a hieved by onventional relay-

ing where the relay re eives and transmits in a odeword-by- odeword fashion and

swit hes between re eption and transmission in a stri tly alternating manner. Our

numeri al results show that the apa ities of the two-hop HD relay hannel for BSC

and AWGN links are signi antly larger than the rates a hieved with onventional

relaying.

We note that the apa ity of the two-hop HD relay hannel was also investigated

in [79 as a spe ial ase of the multi-hop HD relay hannel, but only for the ase when

all involved links are error-free BSCs.

The rest of this hapter is organized as follows. In Se tion 2.2, we present the

hannel model. In Se tion 2.3, we introdu e a new expression for the apa ity of the

onsidered hannel, expli itly show the a hievability of the derived apa ity, and prove

that the new apa ity expression satises the onverse. In Se tion 2.4, we investigate

the apa ity for the ases when the sour e-relay and relay-destination links are both

BSCs and AWGN hannels, respe tively. In Se tion 2.5, we numeri ally evaluate the

derived apa ity expressions and ompare them to the rates a hieved by onventional

relaying. Finally, Se tion 2.6 on ludes the hapter.

2.2 System Model

The two-hop HD relay hannel onsists of a sour e, a HD relay, and a destination,

and the dire t link between sour e and destination is not available, see Fig. 2.1. Due

20


Source Relay Destinat.X1 X2Y1 Y2

Figure 2.1: Two-hop relay hannel.

to the HD onstraint, the relay annot transmit and re eive at the same time. In the

following, we formally dene the hannel model.

2.2.1 Channel Model

The dis rete memoryless two-hop HD relay hannel is dened by X1, X2, Y1, Y2,

and p(y1, y2|x1, x2), where X1 and X2 are the nite input alphabets at the en oders

of the sour e and the relay, respe tively, Y1 and Y2 are the nite output alphabets

at the de oders of the relay and the destination, respe tively, and p(y1, y2|x1, x2) is

the PMF on Y1 × Y2 for given x1 ∈ X1 and x2 ∈ X2. The hannel is memoryless

in the sense that given the input symbols for the i-th hannel use, the i-th output

symbols are independent from all previous input symbols. As a result, the onditional

PMF p(yn1 , yn2 |x

n1 , x

n2 ), where the notation a

nis used to denote the ordered sequen e

an = (a1, a2, ..., an), an be fa torized as p(yn1 , yn2 |x

n1 , x

n2 ) =

∏ni=1 p(y1i, y2i|x1i, x2i).

For the onsidered hannel and the i-th hannel use, let X1i and X2i denote the

random variables (RVs) whi h model the input at sour e and relay, respe tively, and

let Y1i and Y2i denote the RVs whi h model the output at relay and destination,

respe tively.

In the following, we model the HD onstraint of the relay and dis uss its ee t on

some important PMFs that will be used throughout this hapter.

21


2.2.2 Mathemati al Modelling of the HD Constraint

Due to the HD onstraint of the relay, the input and output symbols of the relay

annot assume non-zero values at the same time. More pre isely, for ea h hannel

use, if the input symbol of the relay is non-zero then the output symbol has to be

zero, and vi e versa, if the output symbol of the relay is non-zero then the input

symbol has to be zero. Hen e, the following holds

Y1i =

Y ′1i if X2i = 0

0 if X2i 6= 0,(2.3)

where Y ′1i is an RV that take values from the set Y1.

In order to model the HD onstraint of the relay more onveniently, we represent

the input set of the relay X2 as the union of two sets X2 = X2R ∪ X2T , where X2R

ontains only one element, the zero symbol, and X2T ontains all symbols in X2

ex ept the zero symbol. Note that, be ause of the HD onstraint, X2 has to ontain

the zero symbol. Furthermore, we introdu e an auxiliary random variable, denoted

by Ui, whi h takes values from the set t, r, where t and r orrespond to the relay

transmitting a non-zero symbol and a zero symbol, respe tively. Hen e, Ui is dened

as

Ui =

r if X2i = 0

t if X2i 6= 0.(2.4)

Let us denote the probabilities of the relay transmitting a non-zero and a zero symbol

for the i-th hannel use as PrUi = t = PrX2i 6= 0 = PUiand PrUi = r =

PrX2i = 0 = 1 − PUi, respe tively. We now use (2.4) and represent X2i as a

22


fun tion of the out ome of Ui. Hen e, we have

X2i =

0 if Ui = r

Vi if Ui = t,(2.5)

where Vi is an RV with distribution pVi(x2i) that takes values from the set X2T , or

equivalently, an RV whi h takes values from the set X2, but with pVi(x2i = 0) = 0.

From (2.5), we obtain

p(x2i|Ui = r) = δ(x2i), (2.6)

p(x2i|Ui = t) = pVi(x2i), (2.7)

where δ(x) = 1 if x = 0 and δ(x) = 0 if x 6= 0. Furthermore, for the derivation of

the apa ity, we will also need the onditional PMF p(x1i|x2i = 0) whi h is the input

distribution at the sour e when the relay transmits a zero (i.e., when Ui = r). As

we will see in Theorem 2.1, the distributions p(x1i|x2i = 0) and pVi(x2i) have to be

optimized in order to a hieve the apa ity. Using p(x2i|Ui = r) and p(x2i|Ui = t),

and the law of total probability, the PMF of X2i, p(x2i), is obtained as

p(x2i) = p(x2i|Ui = t)PUi+ p(x2i|Ui = r)(1− PUi

)

(a)= pVi

(x2i)PUi+ δ(x2i)(1− PUi

), (2.8)

where (a) follows from (2.6) and (2.7). In addition, we will also need the distribution

of Y2i, p(y2i), whi h, using the law of total probability, an be written as

p(y2i) = p(y2i|Ui = t)PUi+ p(y2i|Ui = r)(1− PUi

). (2.9)

23


On the other hand, using X2i and the law of total probability, p(y2i|Ui = r) an be

written as

p(y2i|Ui = r) =∑

x2i∈X2

p(y2i, x2i|Ui = r)

=∑

x2i∈X2

p(y2i|x2i, Ui = r)p(x2i|Ui = r)

(a)=∑

x2i∈X2

p(y2i|x2i, Ui = r)δ(x2i) = p(y2i|x2i = 0, Ui = r)

(b)= p(y2i|x2i = 0), (2.10)

where (a) is due to (2.6) and (b) is the result of onditioning on the same variable

twi e sin e if X2i = 0 then Ui = r, and vi e versa. On the other hand, using X2i and

the law of total probability, p(y2i|Ui = t) an be written as

p(y2i|Ui = t) =∑

x2i∈X2

p(y2i, x2i|Ui = t) =∑

x2i∈X2

p(y2i|x2i, Ui = t)p(x2i|Ui = t)

(a)=

∑

x2i∈X2T

p(y2i|x2i, Ui = t)pVi(x2i)

(b)=

∑

x2i∈X2T

p(y2i|x2i)pVi(x2i), (2.11)

where (a) follows from (2.7) and sin e Vi takes values from set X2T , and (b) follows

sin e onditioned on X2i, Y2i is independent of Ui. In (2.11), p(y2i|x2i) is the distri-

bution at the output of the relay-destination hannel onditioned on the relay's input

X2i.

2.2.3 Mutual Information and Entropy

For the apa ity expression given later in Theorem 2.1, we need I(X1; Y1|X2 = 0),

whi h is the mutual information between the sour e's input X1 and the relay's output

Y1 onditioned on the relay having its input set to X2 = 0, and I(X2; Y2), whi h is

24


the mutual information between the relay's input X2 and the destination's output

Y2.

The mutual information I(X1; Y1|X2 = 0) is obtained by denition as

I(

X1; Y1|X2 = 0)

=∑

x1∈X1

∑

y1∈Y1

p(y1|x1, x2 = 0)p(x1|x2 = 0) log2

(

p(y1|x1, x2 = 0)

p(y1|x2 = 0)

)

,

(2.12)

where

p(y1|x2 = 0) =∑

x1∈X1

p(y1|x1, x2 = 0)p(x1|x2 = 0). (2.13)

In (2.12) and (2.13), p(y1|x1, x2 = 0) is the distribution at the output of the sour e-

relay hannel onditioned on the relay having its input set to X2 = 0, and onditioned

on the input symbols at the sour e X1.

On the other hand, I(X2; Y2) is given by

I(X2; Y2) = H(Y2)−H(Y2|X2), (2.14)

where H(Y2) is the entropy of RV Y2, and H(Y2|X2) is the entropy of Y2 onditioned

on X2. The entropy H(Y2) an be found by denition as

H(Y2) = −∑

y2∈Y2

p(y2) log2(p(y2))

(a)= −

∑

y2∈Y2

[

p(y2|U = t)PU + p(y2|U = r)(1− PU)]

× log2[

p(y2|U = t)PU + p(y2|U = r)(1− PU)]

, (2.15)

where (a) follows from (2.9). Now, inserting p(y2|U = r) and p(y2|U = t) given in

25


(2.10) and (2.11), respe tively, into (2.15), we obtain the nal expression for H(Y2),

as

H(Y2) = −∑

y2∈Y2

[

PU

∑

x2∈X2T

p(y2|x2)pV (x2) + p(y2|x2 = 0)(1− PU)

]

× log2

[

PU

∑

x2∈X2T

p(y2|x2)pV (x2) + p(y2|x2 = 0)(1− PU)

]

. (2.16)

On the other hand, the onditional entropy H(Y2|X2) an be found based on its

denition as

H(Y2|X2) = −∑

x2∈X2

p(x2)∑

y2∈Y2

p(y2|x2) log2(p(y2|x2))

(a)= −PU

∑

x2∈X2T

pV (x2)∑

y2∈Y2

p(y2|x2) log2(p(y2|x2))

− (1− PU)∑

y2∈Y2

p(y2|x2 = 0) log2(p(y2|x2 = 0)), (2.17)

where (a) follows by inserting p(x2) given in (2.8). Inserting H(Y2) and H(Y2|X2)

given in (2.16) and (2.17), respe tively, into (2.14), we obtain the nal expression

for I(X2; Y2), whi h is dependent on p(x2), i.e., on pV (x2) and PU . To highlight the

dependen e of I(X2; Y2) with respe t to PU , in the following, we write I(X2; Y2) as

I(X2; Y2)∣

∣

PU.

We are now ready to present the apa ity of the onsidered hannel.

2.3 Capa ity

In this se tion, we provide an easy-to-evaluate expression for the apa ity of the

two-hop HD relay hannel, an expli it oding s heme that a hieves the apa ity, and

prove that the new apa ity expression satises the onverse.

26


2.3.1 The Capa ity

A new expression for the apa ity of the two-hop HD relay hannel is given in the

following theorem.

Theorem 2.1. The apa ity of the two-hop HD relay hannel is given by

C = maxPU

min

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

(1− PU), maxpV (x2)

I(X2; Y2)∣

∣

PU

, (2.18)

where I(

X1; Y1|X2 = 0)

is given in (2.12) and I(X2; Y2) is given in (2.14)-(2.17).

The optimal PU that maximizes the apa ity in (2.18) is given by P ∗U = minP ′

U , P′′U,

where P ′U is the solution of

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

(1− PU) = maxpV (x2)

I(X2; Y2)∣

∣

PU, (2.19)

where, if (2.19) has two solutions, then P ′U is the smaller of the two, and P ′′

U is the

solution of

∂(

maxpV (x2)

I(X2; Y2)∣

∣

PU

)

∂PU= 0. (2.20)

If P ∗U = P ′

U , the apa ity in (2.18) simplies to

C = maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

(1− P ′U) = max

pV (x2)I(X2; Y2)

∣

∣

PU=P ′U

, (2.21)

whereas, if P ∗U = P ′′

U , the apa ity in (2.18) simplies to

C = maxpV (x2)

I(X2; Y2)∣

∣

PU=P ′′U

= maxp(x2)

I(X2; Y2), (2.22)

whi h is the apa ity of the relay-destination hannel.

27


Proof. The proof of the apa ity given in (2.18) is provided in two parts. In the rst

part, given in Se tion 2.3.2, we show that there exists a oding s heme that a hieves

a rate R whi h is smaller, but arbitrarily lose to apa ity C. In the se ond part,

given in Se tion 2.3.3, we prove that any rate R for whi h the probability of error

an be made arbitrarily small, must be smaller than apa ity C given in (2.18). The

rest of the theorem follows from solving (2.18) with respe t to PU , and simplifying

the result. In parti ular, note that the rst term inside the min· fun tion in (2.18)

is a de reasing fun tion with respe t to PU . This fun tion a hieves its maximum

for PU = 0 and its minimum, whi h is zero, for PU = 1. On the other hand, the

se ond term inside the min· fun tion in (2.18) is a on ave fun tion with respe t to

PU . To see this, note that I(X2; Y2) is a on ave fun tion with respe t to p(x2), i.e.,

with respe t to the ve tor omprised of the probabilities p(x2), for x2 ∈ X2, see [80.

Now, sin e 1 − PU is just the probability p(x2 = 0) and sin e pV (x2) ontains the

rest of the probability onstrained parameters in p(x2), I(X2; Y2) is a jointly on ave

fun tion with respe t to pV (x2) and PU . In [81, pp. 87-88, it is proven that if f(x, y)

is a jointly on ave fun tion in both (x, y) and C is a onvex nonempty set, then

the fun tion g(x) = maxy∈C

f(x, y) is on ave in x. Using this result, and noting that

the domain of pV (x2) is spe ied by the probability onstraints, i.e., by a onvex

nonempty set, we an on lude that maxpV (x2)

I(X2, Y2)∣

∣

PUis on ave with respe t to PU .

Now, the maximization of the minimum of the de reasing and on ave fun tions

with respe t to PU , given in (2.18), has a solution PU = P ′′U , when the on ave

fun tion rea hes its maximum, found from (2.20), and when for this point, i.e., for

PU = P ′′U , the de reasing fun tion is larger than the on ave fun tion. Otherwise,

the solution is PU = P ′U whi h is found from (2.19) and in whi h ase P ′

U < P ′′U

holds. If (2.19) has two solutions, then P ′U has to be the smaller of the two sin e

28


...

PSfrag repla ements

1 2 3 4 N N + 1

kn = Nk

Figure 2.2: To a hieve the apa ity in (2.18), transmission is organized in N + 1blo ks and ea h blo k omprises k hannel uses.

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

(1− PU) is a de reasing fun tion with respe t to PU . Now,

when P ∗U = P ′

U , (2.19) holds and (2.18) simplies to (2.21). Whereas, when P ∗U = P ′′

U ,

then maxpV (x2)

I(X2; Y2)∣

∣

PU=P ′′U

= maxpV (x2)

maxPU

I(X2; Y2) = maxp(x2)

I(X2; Y2), thereby leading

to (2.22).

2.3.2 A hievability of the Capa ity

In the following, we des ribe a method for transferring nR bits of information in

n + k hannel uses, where n, k → ∞ and n/(n + k) → 1 as n, k → ∞. As a result,

the information is transferred at rate R. To this end, the transmission is arried out

in N + 1 blo ks, where N → ∞. In ea h blo k, we use the hannel k times. The

numbers N and k are hosen su h that n = Nk holds. The transmission in N + 1

blo ks is illustrated in Fig. 2.2.

The sour e transmits messageW , drawn uniformly frommessage set 1, 2, ..., 2nR,

from the sour e via the HD relay to the destination. To this end, before the start of

transmission, message W is spilt into N messages, denoted by w(1), ..., w(N), where

ea h w(i), ∀i, ontains kR bits of information. The transmission is arried out in the

following manner. In blo k one, the sour e sends message w(1) in k hannel uses to

the relay and the relay is silent. In blo k i, for i = 2, ..., N , sour e and relay send

messages w(i) and w(i− 1) to relay and destination, respe tively, in k hannel uses.

In blo k N + 1, the relay sends message w(N) in k hannel uses to the destination

29


and the sour e is silent. Hen e, in the rst blo k and in the (N + 1)-th blo k the

relay and the sour e are silent, respe tively, sin e in the rst blo k the relay does not

have information to transmit, and in blo k N+1, the sour e has no more information

to transmit. In blo ks 2 to N , both sour e and relay transmit, while meeting the

HD onstraint in every hannel use. Hen e, during the N + 1 blo ks, the hannel is

used k(N + 1) times to send nR = NkR bits of information, leading to an overall

information rate given by

limN→∞

limk→∞

NkR

k(N + 1)= R bits/use. (2.23)

A detailed des ription of the proposed oding s heme is given in the following,

where we explain the rates, odebooks, en oding, and de oding used for transmission.

Rates: The transmission rate of both sour e and relay is denoted by R and given

by

R = C − ǫ, (2.24)

where C is given in Theorem 2.1 and ǫ > 0 is an arbitrarily small number. Note that

R is a fun tion of P ∗U , see Theorem 2.1.

Codebooks: We have two odebooks: The sour e's transmission odebook and

the relay's transmission odebook.

The sour e's transmission odebook is generated by mapping ea h possible binary

sequen e omprised of kR bits, where R is given by (2.24), to a odeword

2 x1|r

omprised of k(1 − P ∗U) symbols. The symbols in ea h odeword x1|r are generated

independently a ording to distribution p(x1|x2 = 0). Sin e in total there are 2kR

possible binary sequen es omprised of kR bits, with this mapping we generate 2kR

2

The subs ript 1|r in x1|r is used to indi ate that odeword x1|r is omprised of symbols whi h

are transmitted by the sour e only when Ui = r, i.e., when X2i = 0.

30


odewords x1|r ea h ontaining k(1 − P ∗U) symbols. These 2kR odewords form the

sour e's transmission odebook, whi h we denote by C1|r.

The relay's transmission odebook is generated by mapping ea h possible binary

sequen e omprised of kR bits, where R is given by (2.24), to a transmission odeword

x2 omprised of k symbols. The i-th symbol, i = 1, ..., k, in odeword x2 is generated

in the following manner. For ea h symbol a oin is tossed. The oin is su h that it

produ es symbol r with probability 1− P ∗U and symbol t with probability P ∗

U . If the

out ome of the oin ip is r, then the i-th symbol of the relay's transmission odeword

x2 is set to zero. Otherwise, if the out ome of the oin ip is t, then the i-th symbol

of odeword x2 is generated independently a ording to distribution pV (x2). The 2kR

odewords x2 form the relay's transmission odebook denoted by C2.

The two odebooks are known at all three nodes.

En oding, Transmission, and De oding: In the rst blo k, the sour e maps

w(1) to the appropriate odeword x1|r(1) from its odebook C1|r. Then, odeword

x1|r(1) is transmitted to the relay, whi h is s heduled to always re eive and be silent

(i.e., to set its input to zero) during the rst blo k. However, knowing that the

transmitted odeword from the sour e x1|r(1) is omprised of k(1 − P ∗U) symbols,

the relay onstru ts the re eived odeword, denoted by y1|r(1), only from the rst

k(1− P ∗U) re eived symbols. In Appendix A.1 , we prove that odeword x1|r(1) sent

in the rst blo k an be de oded su essfully from the re eived odeword at the relay

y1|r(1) using a typi al de oder [80 sin e R satises

R < maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

(1− P ∗U). (2.25)

In blo ks i = 2, ..., N , the en oding, transmission, and de oding are performed as

follows. In blo ks i = 2, ..., N , the sour e and the relay map w(i) and w(i − 1) to

31


the appropriate odewords x1|r(i) and x2(i) from odebooks C1|r and C2, respe tively.

Note that the sour e also knows x2(i) sin e x2(i) was generated from w(i− 1) whi h

the sour e transmitted in the previous (i.e., (i − 1)-th) blo k. The transmission of

x1|r(i) and x2(i) an be performed in two ways: 1) by the relay swit hing between

re eption and transmission, and 2) by the relay always re eiving and transmitting as

in FD relaying. We rst explain the rst option.

Note that both sour e and relay know the position of the zero symbols in x2(i).

Hen e, if the rst symbol in odeword x2(i) is zero, then in the rst symbol interval

of blo k i, the sour e transmits its rst symbol from odeword x1|r(i) and the relay

re eives. By re eiving, the relay a tually also sends the rst symbol of odeword

x2(i), whi h is the symbol zero, i.e., x21 = 0. On the other hand, if the rst sym-

bol in odeword x2(i) is non-zero, then in the rst symbol interval of blo k i, the

relay transmits its rst symbol from odeword x2(i) and the sour e is silent. The

same pro edure is performed for the j-th hannel use in blo k i, for j = 1, ..., k. In

parti ular, if the j-th symbol in odeword x2(i) is zero, then in the j-th hannel use

of blo k i the sour e transmits its next untransmitted symbol from odeword x1|r(i)

and the relay re eives. With this re eption, the relay a tually also sends the j-th

symbol of odeword x2(i), whi h is the symbol zero, i.e., x2j = 0. On the other

hand, if the j-th symbol in odeword x2(i) is non-zero, then for the j-th hannel

use of blo k i, the relay transmits the j-th symbol of odeword x2(i) and the sour e

is silent. Note that odeword x2(i) ontains k(1 − P ∗U) ± ε(i) symbols zeros, where

ε(i) > 0. Due to the strong law of large numbers [80, limk→∞

ε(i)/k = 0 holds, whi h

means that for large enough k, the fra tion of symbols zeros in odeword x2(i) is

1 − P ∗U . Hen e, for k → ∞, the sour e an transmit pra ti ally all

3

of its k(1 − P ∗U)

3

When we say pra ti ally all, we mean either all or all ex ept for a negligible fra tion

limk→∞ ε(i)/k = 0 of them.

32


symbols from odeword x1|r(i) during a single blo k to the relay. Let y1|r(i) denote

the orresponding re eived odeword at the relay. In Appendix A.1, we prove that

the odewords x1|r(i) sent in blo ks i = 2, . . . , N an be de oded su essfully at the

relay from the orresponding re eived odewords y1|r(i) using a typi al de oder [80

sin e R satises (2.25). Moreover, in Appendix A.1, we also prove that, for k → ∞,

the odewords x1|r(i) an be su essfully de oded at the relay even though, for some

blo ks i = 2, ..., N , only k(1− P ∗U)− ε(i) symbols out of k(1− P ∗

U) symbols in ode-

words x1|r(i) are transmitted to the relay. On the other hand, the relay sends the

entire odeword x2(i), omprised of k symbols of whi h a fra tion 1 − P ∗U are zeros,

to the destination. In parti ular, the relay sends the zero symbols of odeword x2(i)

to the destination by being silent during re eption, and sends the non-zero symbols

of odeword x2(i) to the destination by a tually transmitting them. On the other

hand, the destination listens during the entire blo k and re eives a odeword y2(i).

In Appendix A.2 , we prove that the destination an su essfully de ode x2(i) from

the re eived odeword y2(i), and thereby obtain w(i− 1), sin e rate R satises

R < maxpV (x2)

I(X2; Y2)∣

∣

∣

PU=P ∗U

. (2.26)

In a pra ti al implementation, the relay may not be able to swit h between re ep-

tion and transmission in a symbol-by-symbol manner, due to pra ti al onstraints

regarding the speed of swit hing. Instead, we may allow the relay to re eive and

transmit at the same time and in the same frequen y band similar to FD relaying.

However, this simultaneous re eption and transmission is performed while avoid-

ing self-interferen e sin e, in ea h symbol interval, either the input or the output

information- arrying symbol of the relay is zero. This is a omplished in the follow-

ing manner. The sour e performs the same operations as for the ase when the relay

33


swit hes between re eption and transmission. On the other hand, the relay transmits

all symbols from x2(i) while ontinuously listening. Then, the relay dis ards from

the re eived odeword, denoted by y1(i), those symbols for whi h the orresponding

symbols in x2(i) are non-zero, and only olle ts the symbols in y1(i) for whi h the

orresponding symbols in x2(i) are equal to zero. The olle ted symbols from y1(i)

onstitute the relay's information- arrying re eived odeword y1|r(i) whi h is used for

de oding. Codeword y1|r(i) is ompletely free of self-interferen e sin e the symbols in

y1|r(i) were re eived in symbol intervals for whi h the orresponding transmit symbol

at the relay was zero.

In the last (i.e., the (N +1)-th) blo k, the sour e is silent and the relay transmits

w(N) by mapping it to the orresponding odeword x2(i) from set C2. The relay

transmits all symbols in odeword x2(i) to the destination. The destination an

de ode the re eived odeword in blo k N + 1 su essfully, sin e (2.26) holds.

Finally, sin e both relay and destination an de ode their respe tive odewords

in ea h blo k, the entire message W an be de oded su essfully at the destination

at the end of the (N + 1)-th blo k.

Coding Example

In Fig. 2.3, we show an example for ve tors x1|r, x1, y1, y1|r, x2, and y2, for k = 8

and P ∗U = 1/2, where x1 ontains all k input symbols at the sour e in luding the

silen es. From this example, it an be seen that x1 ontains zeros due to silen es for

hannel uses for whi h the orresponding symbol in x2 is non-zero. By omparing x1

and x2 it an be seen that the HD onstraint is satised for ea h symbol duration.

The blo k diagram of the proposed oding s heme is shown in Fig. 2.4. In par-

ti ular, in Fig 2.4, we show s hemati ally the en oding, transmission, and de oding

at sour e, relay, and destination. The ow of en oding/de oding in Fig. 2.4 is as

34


PSfrag repla ements

x22 x23 x25 0 x28

x11|r

x11|r

x12|r

x12|r

x13|r

x13|r

x14|r

y11

y11

y14

y14

y16

y16

y17

y12 y13 y15 y17 y18

0000

000

x14|r

x2:

:

x1:

y1:

x1|r:

y1|r:

y2:y21 y22 y23 y24 y25 y26 y27 y28

Figure 2.3: Example of generated swit hing ve tor along with input/output ode-

words at sour e, relay, and destination.

PSfrag repla ements

w(i− 1)

w(i− 1)

w(i− 1) w(i)

w(i) x1|r(i)

x1(i) y1(i)

y1|r(i)x2(i)

x2(i)

x2(i) y2(i)

C1|r

C2

C2

I

S D1

D2

BChannel 1 Channel 2

Source Relay Destination

Figure 2.4: Blo k diagram of the proposed hannel oding proto ol for time slot i.The following notations are used in the blo k diagram: C1|r and C2 are en oders, D1

and D2 are de oders, I is an inserter, S is a sele tor, B is a buer, and w(i) denotesthe message transmitted by the sour e in blo k i.

35


follows. Messages w(i− 1) and w(i) are en oded into x2(i) and x1|r(i), respe tively,

at the sour e using the en oders C2 and C1|r, respe tively. Then, an inserter I is

used to reate the ve tor x1(i) by inserting the symbols of x1|r(i) into the positions

of x1(i) for whi h the orresponding elements of x2(i) are zeros and setting all other

symbols in x1(i) to zero. The sour e then transmits x1(i). On the other hand, the

relay, en odes w(i− 1) into x2(i) using en oder C2. Then, the relay transmits x2(i)

while re eiving y1(i). Next, using x2(i), the relay onstru ts y1|r(i) from y1(i) by

sele ting only those symbols for whi h the orresponding symbol in x2(i) is zero. The

relay then de odes y1|r(i), using de oder D1, into w(i) and stores the de oded bits

in its buer B. The destination re eives y2(i), and de odes it using de oder D2, into

w(i− 1).

2.3.3 Simpli ation of Previous Converse Expressions

As shown in [8, the HD relay hannel an be analyzed with the framework developed

for the FD relay hannel in [5. Sin e the onsidered two-hop HD relay hannel

belongs to the lass of degraded relay hannels dened in [5, the rate of this hannel,

for some p(x1, x2), is upper bounded by [5, [8

R ≤ min

I(

X1; Y1|X2

)

, I(

X2; Y2)

. (2.27)

On the other hand, I(

X1; Y1|X2

)

an be simplied as

I(

X1; Y1|X2

)

= I(

X1; Y1|X2 = 0)

(1− PU) + I(

X1; Y1|X2 6= 0)

PU

(a)= I

(

X1; Y1|X2 = 0)

(1− PU), (2.28)

36


where (a) follows from (2.3) sin e when X2 6= 0, Y1 is deterministi ally zero thereby

leading to I(

X1; Y1|X2 6= 0)

= 0. Inserting (2.28) into (2.27), we obtain that for

some p(x1, x2), the following holds

R ≤ min

I(

X1; Y1|X2 = 0)

(1− PU) , I(

X2; Y2)

. (2.29)

Now, sin e I(

X1; Y1|X2 = 0)

(1−PU) is a fun tion of p(x1|x2 = 0), and no other fun -

tion inside themin· fun tion in (2.29) is dependent on the distribution p(x1|x2 = 0),

the right hand side of (2.29) and thereby the rate R an be upper bounded as

R ≤ min

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

(1− PU) , I(

X2; Y2)

, (2.30)

where maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

exists sin e the mutual information I(

X1; Y1|X2 =

0)

is a on ave fun tion with respe t to p(x1|x2 = 0). On the other hand, I(

X2; Y2)

is a fun tion of p(x2) whi h is given in (2.8) as a fun tion of pV (x2) and PU . Hen e,

I(

X2; Y2)

is also a fun tion of pV (x2) and PU . Now, sin e in the right hand side of

(2.30) only I(

X2; Y2)

is a fun tion of pV (x2), and sin e I(

X2; Y2)

is a on ave fun tion

of pV (x2) (see proof of Theorem 2.1 for the proof of on avity), we an upper bound

the right hand side of (2.30) and obtain a new upper bound for the rate R as

R ≤ min

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

(1− PU) , maxpV (x2)

I(

X2; Y2)

∣

∣

∣

PU

. (2.31)

Now, both the rst and the se ond term inside the min· fun tion in (2.31) are

dependent on PU . If we maximize (2.31) with respe t to PU , we obtain a new upper

37


bound for rate R as

R ≤ maxPU

min

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

(1− PU) , maxpV (x2)

I(

X2; Y2)

∣

∣

∣

PU

, (2.32)

where the maximum with respe t to PU exists sin e the rst and the se ond terms in-

side the min· fun tion in (2.32) are monotoni ally de reasing and on ave fun tions

with respe t to PU , respe tively (see proof of Theorem 2.1 for proof of on avity).

This on ludes the proof that the new apa ity expression in Theorem 2.1 satises

the onverse.

2.4 Capa ity Examples

In the following, we evaluate the apa ity of the onsidered relay hannel when the

sour e-relay and relay-destination links are both BSCs and AWGN hannels, respe -

tively.

2.4.1 Binary Symmetri Channels

Assume that the sour e-relay and relay-destination links are both BSCs, where X1 =

X2 = Y1 = Y2 = 0, 1, with probability of error Pε1 and Pε2, respe tively. Now,

in order to obtain the apa ity for this relay hannel, a ording to Theorem 2.1, we

rst have to nd maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

and maxpV (x2)

I(X2; Y2). For the BSC, the

expression for maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

is well known and given by [5

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

= 1−H(Pε1), (2.33)

38


where H(Pε1) is the binary entropy fun tion, whi h for probability P is dened as

H(P ) = −P log2(P )− (1− P ) log2(1− P ). (2.34)

The distribution that maximizes I(

X1; Y1|X2 = 0)

is also well known and given by

[5

p(x1 = 0|x2 = 0) = p(x1 = 1|x2 = 0) =1

2. (2.35)

On the other hand, for the BSC, the only symbol in the set X2T is symbol 1, whi h

RV V takes with probability one. In other words, pV (x2) is a degenerate distribution,

given by pV (x2) = δ(x2 − 1). Hen e,

maxpV (x2)

I(X2; Y2) = I(X2; Y2)∣

∣

∣

pV (x2)=δ(x2−1)(2.36)

= H(Y2)∣

∣

∣

pV (x2)=δ(x2−1)−H(Y2|X2)

∣

∣

∣

pV (x2)=δ(x2−1). (2.37)

For the BSC, the expression for H(Y2|X2) is independent of X2, and is given by [5

H(Y2|X2) = H(Pε2). (2.38)

On the other hand, in order to nd H(Y2)∣

∣

∣

pV (x2)=δ(x2−1)from (2.16), we need the

distributions of p(y2|x2 = 0) and p(y2|x2 = 1). For the BSC with probability of error

Pε2, these distributions are obtained as

p(y2|x2 = 0) =

1− Pε2 if y2 = 0

Pε2 if y2 = 1,(2.39)

39


and

p(y2|x2 = 1) =

Pε2 if y2 = 0

1− Pε2 if y2 = 1.(2.40)

Inserting (2.39), (2.40), and pV (x2) = δ(x2−1) into (2.16), we obtainH(Y2)∣

∣

pV (x2)=δ(x2−1)

as

H(Y2)∣

∣

pV (x2)=δ(x2−1)= −A log2(A)− (1− A) log2(1−A), (2.41)

where

A = Pε2(1− 2PU) + PU . (2.42)

Inserting (2.38) and (2.41) into (2.36), we obtain maxpV (x2)

I(X2; Y2) as

maxpV (x2)

I(X2; Y2) = −A log2(A)− (1−A) log2(1− A)−H(Pε2). (2.43)

We now have the two ne essary omponents required for obtaining P ∗U from (2.18),

and thereby obtaining the apa ity. This is summarized in the following orollary.

Corollary 2.1. The apa ity of the onsidered relay hannel with BSCs links is given

by

C = maxPU

min

(1−H(Pε1))(1− PU),−A log2(A)− (1− A) log2(1−A)−H(Pε2)

(2.44)

40


and is a hieved with

pV (x2) = δ(x2 − 1) (2.45)

p(x1 = 0|x2 = 0) = p(x1 = 1|x2 = 0) = 1/2. (2.46)

There are two ases for the optimal P ∗U whi h maximizes (2.44). If PU found from

4

(1−H(Pε1))(1− PU) = −A log2(A)− (1− A) log2(1− A)−H(Pε2) (2.47)

is smaller than 1/2, then the optimal P ∗U whi h maximizes (2.44) is found as the

solution to (2.47), and the apa ity simplies to

C = (1−H(Pε1))(1− P ∗U) = −A∗ log2(A

∗)− (1−A∗) log2(1−A∗)−H(Pε2),

(2.48)

where A∗ = A|PU=P ∗U. Otherwise, if PU found from (2.47) is PU ≥ 1/2, then the

optimal P ∗U whi h maximizes (2.44) is P ∗

U = 1/2, and the apa ity simplies to

C = 1−H(Pε2). (2.49)

Proof. The apa ity in (2.44) is obtained by inserting (2.33) and (2.43) into (2.18).

On the other hand, for the BSC, the solution of (2.20) is P ′′U = 1/2, whereas (2.19)

simplies to (2.47). Hen e, using Theorem 2.1, we obtain that if P ′U ≤ P ′′

U = 1/2,

then P ∗U = P ′

U , where P′U is found from (2.47), in whi h ase the apa ity is given by

(2.21), whi h simplies to (2.48) for the BSC. On the other hand, if P ′U > P ′′

U = 1/2,

then P ∗U = P ′′

U = 1/2, in whi h ase the apa ity is given by (2.22), whi h simplies

4

Solving (2.47) with respe t to PU leads to a nonlinear equation, whi h an be easily solved

using e.g. Newton's method [82.

41


to (2.49) for the BSC.

2.4.2 AWGN Channels

In this subse tion, we assume that the sour e-relay and relay-destination links are

AWGN hannels, i.e., hannels whi h are impaired by independent, real-valued, zero-

mean AWGN with varian es σ21 and σ2

2, respe tively. More pre isely, the outputs at

the relay and the destination are given by

Yk = Xk +Nk, k ∈ 1, 2, (2.50)

where Nk is a zero-mean Gaussian RV with varian e σ2k, k ∈ 1, 2, with distribution

pNk(z), k ∈ 1, 2, −∞ ≤ z ≤ ∞. Moreover, assume that the symbols transmitted

by the sour e and the relay must satisfy the following average power onstraints

5

∑

x1∈X1

x21 p(x1|x2 = 0) ≤ P1 and

∑

x2∈X2T

x22 pV (x2) ≤ P2. (2.51)

Obtaining the apa ity for this relay hannel using Theorem 2.1, requires expressions

for the fun tions maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

and maxpV (x2)

I(X2; Y2) = maxpV (x2)

[

H(Y2) −

H(Y2|X2)]

. For the AWGN hannel, the expressions for the mutual information

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

and the entropy H(Y2|X2) are well known and given by

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

=1

2log2

(

1 +P1

σ21

)

(2.52)

H(Y2|X2) =1

2log2

(

2πeσ22

)

, (2.53)

5

If the optimal distributions p(x1|x2 = 0) and pV (x2) turn out to be ontinuous, the sums in

(2.51) should be repla ed by integrals.

42


where, as is well known, for AWGN I(X1; Y1|X2 = 0) is maximized when p(x1|x2 =

0) is the zero mean Gaussian distribution with varian e P1. On the other hand,

H(Y2|X2) is just the dierential entropy of Gaussian RV N2, whi h is independent of

p(x2), i.e., of pV (x2). Hen e,

maxpV (x2)

I(X2; Y2) = maxpV (x2)

H(Y2)−1

2log2

(

2πeσ22

)

(2.54)

holds and in order to nd maxpV (x2)

I(X2; Y2) we only need to derive maxpV (x2)

H(Y2). Now,

in order to nd maxpV (x2)

H(Y2), we rst obtain H(Y2) using (2.16) and then obtain the

distribution pV (x2) whi h maximizesH(Y2). Finding an expression forH(Y2) requires

the distribution of p(y2|x2). This distribution is found using (2.50) as

p(y2|x2) = pN2(y2 − x2). (2.55)

Inserting (2.55) into (2.16), we obtain H(Y2) as

H(Y2) = −

∫ ∞

−∞

[

PU

∑

x2∈X2T

pN2(y2 − x2)pV (x2) + pN2(y2)(1− PU)

]

× log2

[

PU

∑

x2∈X2T

pN2(y2 − x2)pV (x2) + pN2(y2)(1− PU)

]

dy2, (2.56)

where, sin e p(y2|x2) is now a ontinuos probability density fun tion, the summation

in (2.16) with respe t to y2 onverges to an integral as

∑

y2

→

∞∫

−∞

dy2. (2.57)

We are now ready to maximizeH(Y2) in (2.56) with respe t to pV (x2). Unfortunately,

obtaining the optimal pV (x2) whi h maximizes H(Y2) in losed form is di ult, if

43


not impossible. However, as will be shown in the following lemma, we still an

hara terize the optimal pV (x2), whi h is helpful for numeri al al ulation of pV (x2).

Lemma 2.1. For the onsidered relay hannel where the relay-destination link is an

AWGN hannel and where the input symbols of the relay must satisfy the average

power onstraint given in (2.51), the distribution pV (x2) whi h maximizes H(Y2) in

(2.56) for a xed PU < 1 is dis rete, i.e., it has the following form

pV (x2) =K∑

k=1

pkδ(x2 − x2k), (2.58)

where pk is the probability that symbol x2 will take the value x2k, for k = 1, ..., K,

where K ≤ ∞. Furthermore, pk and x2k given in (2.58), must satisfy

K∑

k=1

pk = 1 and

K∑

k=1

pkx22k = P2. (2.59)

In the limiting ase when PU → 1, distribution pV (x2) onverges to the zero-mean

Gaussian distribution with varian e P2.

Proof. Please see Appendix A.3.

Remark 2.1. Unfortunately, there is no losed-form expression for distribution pV (x2)

given in the form of (2.58), and therefore, a brute-for e sear h has to be used in order

to nd x2k and pk, ∀k.

44


Now, inserting (2.58) into (2.56) we obtain maxpV (x2)

H(Y2) as

maxpV (x2)

H(Y2) = −

∞∫

−∞

(

PU

K∑

k=1

p∗kpN2(y2 − x∗2k) + (1− PU)pN2(y2)

)

× log2

(

PU

K∑

k=1

p∗kpN2(y2 − x∗2k) + (1− PU)pN2(y2)

)

dy2,

(2.60)

where p∗V (x2) =∑K

k=1 p∗kδ(x2 − x∗2k) is the distribution that maximizes H(Y2) in

(2.56). Inserting (2.60) into (2.54), we obtain maxpV (x2)

I(X2; Y2). Using (2.52) and

maxpV (x2)

I(X2; Y2) in Theorem 2.1, we obtain the apa ity of the onsidered relay hannel

with AWGN links. This is onveyed in the following orollary.

Corollary 2.2. The apa ity of the onsidered relay hannel where the sour e-relay

and relay-destination links are both AWGN hannels with noise varian es σ21 and σ2

2,

respe tively, and where the average power onstraints of the inputs of sour e and relay

are given by (2.51), is given by

C =1

2log2

(

1 +P1

σ21

)

(1− P ∗U)

(a)= −

∞∫

−∞

(

P ∗U

K∑

k=1

p∗kpN2(y2 − x∗2k) + (1− P ∗U)pN2(y2)

)

× log2

(

P ∗U

K∑

k=1

p∗kpN2(y2 − x∗2k) + (1− P ∗U)pN2(y2)

)

dy2 −1

2log2(2πeσ

22),

(2.61)

where the optimal P ∗U is found su h that equality (a) in (2.61) holds. The apa ity

in (2.61) is a hieved when p(x1|x2 = 0) is the zero-mean Gaussian distribution with

varian e P1 and p∗V (x2) =

∑Kk=1 p

∗kδ(x2−x

∗2k) is a dis rete distribution whi h satises

45


(2.59) and maximizes H(Y2) given in (2.60).

Proof. The apa ity in (2.61) is obtained by inserting (2.60) into (2.54), then inserting

(2.54) and (2.52) into (2.18), and nally maximizing with respe t to PU . For the

maximization of the orresponding apa ity with respe t to PU , we note that P′U <

P ′′U = 1 always holds. Hen e, the apa ity is given by (2.21), whi h for the Gaussian

ase simplies to (2.61). To see that P ′′U = 1, note the relay-destination hannel is

an AWGN hannel for whi h the mutual information is maximized when p(x2) is a

Gaussian distribution. From (2.8), we see that p(x2) be omes a Gaussian distribution

if and only if PU = 1 and pV (x2) also assumes a Gaussian distribution.

2.5 Numeri al Examples

In this se tion, we numeri ally evaluate the apa ities of the onsidered HD relay

hannel when the sour e-relay and relay-destination links are both BSCs and AWGN

hannels, respe tively. As a performan e ben hmark, we use the maximal a hievable

rate of onventional relaying [48. Thereby, the sour e transmits to the relay one

odeword with rate maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

in 1 − PU fra tion of the time, where

0 < PU < 1, and in the remaining fra tion of time, PU , the relay retransmits the

re eived information to the destination with rate maxp(x2)

I(X2; Y2), see [18 and [48.

The optimal PU , is found su h that the following holds

Rconv = maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

(1− PU) = maxp(x2)

I(X2; Y2)PU . (2.62)

46


Employing the optimal PU obtained from (2.62), the maximal a hievable rate of

onventional relaying an be written as

Rconv =

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

×maxp(x2)

I(X2; Y2)

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

+maxp(x2)

I(X2; Y2). (2.63)

2.5.1 BSC Links

For simpli ity, we assume symmetri links with Pε1 = Pε2 = Pε. As a result, P∗U < 1/2

in Corollary 2.1 and the apa ity is given by (2.48). This apa ity is plotted in

Fig. 2.5, where P ∗U is found from (2.47) using a mathemati al software pa kage, e.g.

Mathemati a. As a ben hmark, in Fig. 2.5, we also show the maximal a hievable

rate using onventional relaying, obtained by inserting

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0)

= maxp(x2)

I(X2; Y2) = 1−H(Pε) (2.64)

into (2.63), where H(Pε) is given in (2.34) with P = Pε. Thereby, the following rate

is obtained

Rconv =1

2

(

1−H(Pε))

. (2.65)

As an be seen from Fig. 2.5, when both links are error-free, i.e., Pε = 0, onventional

relaying a hieves 0.5 bits/ hannel use, whereas the apa ity is 0.77291, whi h is 54%

larger than the rate a hieved with onventional relaying. This value for the apa ity

an be obtained by inserting Pε1 = Pε2 = 0 in (2.47), and thereby obtain

C = 1− P ∗U

(a)= H(P ∗

U). (2.66)

47


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Pε

Rate

(bits/use)

CapacityConventional relaying

Figure 2.5: Comparison of rates for the BSC as a fun tion of the error probability

Pε1 = Pε2 = Pε.

Solving (a) in (2.66) with respe t to P ∗U and inserting the solution for P ∗

U ba k into

(2.66), yields C = 0.77291. We note that this value was rst reported in [9, page

327.

2.5.2 AWGN Links

For the AWGN ase, the apa ity is evaluated based on Corollary 2.2. However, sin e

for this ase the optimal input distribution at the relay p∗V (x2) is unknown, i.e., the

values of p∗k and x∗2k in (2.61) are unknown, we have performed a brute for e sear h for

the values of p∗k and x∗2k whi h maximize (2.61). Two examples of su h distributions

6

are shown in Fig. 2.6 for two dierent values of the SNR P1/σ21 = P2/σ

22. Sin e

we do not have a proof that the distributions obtained via brute-for e sear h are

a tually the exa t optimal input distributions at the relay that a hieve the apa ity,

the rates that we obtain, denoted by CL, are lower than or equal to the a tual

6

Note that these distributions resemble a dis rete, Gaussian shaped distribution with a gap

around zero.

48


−20 −10 0 10 200

0.1

0.2

0.3

0.4

x2k

pk

p(x2|t) for 10 dBp(x2|t) for 15 dB

Figure 2.6: Example of proposed input distributions at the relay pV (x2).

apa ity. These rates are shown in Figs. 2.7 and 2.8, for symmetri links and non-

symmetri links, respe tively, where we set P1/σ21 = P2/σ

22 and P1/σ

21/10 = P2/σ

22,

respe tively. We note that for the results in Fig. 2.7, for P1/σ21 = P2/σ

22 = 10 dB and

P1/σ21 = P2/σ

22 = 15 dB, we have used the input distributions at the relay shown in

Fig. 2.6. In parti ular, for P1/σ21 = P2/σ

22 = 10 dB we have used the following values

for p∗k and x∗2k

p∗k = [0.35996, 0.11408, 2.2832× 10−2, 2.88578× 10−3, 2.30336× 10−4,

1.16103× 10−5, 3.69578× 10−7],

x∗2k = [2.62031, 3.93046, 5.24061, 6.55077, 7.86092, 9.17107, 10.4812],

and for P1/σ21 = P2/σ

22 = 15 dB we have used

p∗k = [0.212303, 0.142311, 8.12894×10−2, 3.95678×10−2, 1.64121×10−2, 5.80092×10−3

1.7472×10−3, 4.48438×10−4, 9.80788×10−5, 1.82793×10−5, 2.90308×10−6, 3.92889×

10−7],

x∗2k = [3.40482, 5.10724, 6.80965, 8.51206, 10.2145, 11.9169, 13.6193, 15.3217,

17.0241, 18.7265, 20.4289, 22.1314].

The above values of p∗k and x∗2k are only given for x∗2k > 0, sin e the values of p∗k and

x∗2k when x∗2k < 0 an be found from symmetry, see Fig. 2.6.

49


−10 −5 0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

P1/σ21 = P2/σ

22 (in dB)

Rate

(bits/use)

CLRGaussRconv

Unachievable bound from [8] and [78]

Figure 2.7: Sour e-relay and relay destination links are AWGN hannels with P1/σ21 =

P2/σ22.

−10 −5 0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

P1/σ21/10 = P2/σ

22 (in dB)

Rate

(bits/use)

CLRGaussRconv

Unachievable bound from [8] and [78]

Figure 2.8: Sour e-relay and relay destination links are AWGN hannels with

P1/σ21/10 = P2/σ

22 .

50


In Figs. 2.7 and 2.8, we also show the rate a hieved when instead of an optimal

dis rete input distribution at the relay p∗V (x2), f. Lemma 2.1, we use a ontinuous,

zero-mean Gaussian distribution with varian e P2. Thereby, we obtain the following

rate

RGauss =1

2log2

(

1 +P1

σ21

)

(1− PU)

(a)= −

∞∫

−∞

(

PU pG(y2) + (1− PU)pN2(y2))

× log2(

PU pG(y2) + (1− PU)pN2(y2))

dy2 −1

2log2(2πeσ

22),

(2.67)

where PU is found su h that equality (a) holds and pG(y2) is a ontinuous, zero-mean

Gaussian distribution with varian e P2 +σ22 . From Figs. 2.7 and 2.8, we an see that

RGauss ≤ CL, whi h was expe ted from Lemma 2.1. However, the loss in performan e

aused by the Gaussian inputs is moderate, whi h suggests that the performan e

gains obtained by the proposed proto ol are mainly due to the exploitation of the

silent (zero) symbols for onveying information from the HD relay to the destination

rather than the optimization of pV (x2).

As ben hmark, in Figs. 2.7 and 2.8, we have also shown the maximal a hievable

rate using onventional relaying, obtained by inserting

maxp(x1|x2=0)

I(

X1; Y1|X2 = 0, U = r)

=1

2log2

(

1 +P1

σ21

)

(2.68)

and

maxpV (x2)

I(X2; Y2|U = t) =1

2log2

(

1 +P2

σ22

)

(2.69)

51


into (2.63), whi h yields

Rconv =1

2

log2

(

1 + P1

σ21

)

log2

(

1 + P2

σ22

)

log2

(

1 + P1

σ21

)

+ log2

(

1 + P2

σ22

) . (2.70)

Comparing the rates CL and Rconv in Figs. 2.7 and 2.8, we see that for 10 dB ≤

P2/σ22 ≤ 30 dB, CL a hieves 3 to 6 dB gain ompared to Rconv. Hen e, large per-

forman e gains are a hieved using the proposed apa ity proto ol even if suboptimal

input distributions at the relay are employed.

Finally, as additional ben hmark in Figs. 2.7 and 2.8, we show the una hievable

upper bounds reported in [8 and [78, given by

CUpper = maxPU

min

1

2log2

(

1 +P1

σ21

)

(1− PU) ,1

2log2

(

1 +P2

σ22

)

PU +H(PU)

.

(2.71)

As an be seen from Figs. 2.7 and 2.8, this bound is loose for low SNRs but be omes

tight for high SNRs.

2.6 Con lusion

We have derived an easy-to-evaluate expression for the apa ity of the two-hop HD

relay hannel without fading based on simplifying previously derived onverse ex-

pressions. Moreover, we have proposed an expli it oding s heme whi h a hieves

the apa ity. In parti ular, we showed that the apa ity is a hieved when the re-

lay swit hes between re eption and transmission in a symbol-by-symbol manner and

when additional information is sent by the relay to the destination using the zero

symbol impli itly sent by the relay's silen e during re eption. Furthermore, we have

evaluated the apa ity for the ases when both links are BSCs and AWGN hannels,

52


respe tively. From the numeri al examples, we have observed that the apa ity of

the two-hop HD relay hannel is signi antly higher than the rates a hieved with

onventional relaying proto ols.

53

Chapter 3

Buer-Aided Relaying With Adaptive

Re eption-Transmission: Adaptive

Rate Transmission

3.1 Introdu tion

The apa ity of the two-hop HD relay network when the sour e-relay and relay-

destination links are AWGN hannels ae ted by fading is not known, and only

a hievable rates have been reported in the literature so far, see Se tion 1.3.2. In this

hapter, we present new a hievable average rates for this network whi h are larger

than the best known average rates. These new average rates are a hieved with a

buer-aided relaying proto ol with adaptive re eption-transmission.

In this hapter, we onsider buer-aided relaying with adaptive re eption-transmis-

sion for the two-hop HD relay network when the sour e-relay and relay-destination

links are AWGN hannels ae ted by fading. In parti ular, in any given time slot,

based on the hannel state information (CSI) of the sour e-relay and the relay-

destination link a de ision is made on whether the relay transmits or re eives. For

the two-hop HD relay network, this is equivalent to sele ting either the sour e-relay

or relay-destination link for transmission in a given time slot, i.e., equivalent to

54

Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission

link sele tion. We onsider the ases of delay-un onstrained and delay- onstrained

transmission. For the delay-un onstrained ase, we optimize the adaptive re eption-

transmission proto ol and the power allo ated to the sour e and the relay for data

rate maximization. Interestingly, the optimal adaptive re eption-transmission poli y

requires only knowledge of the instantaneous CSI of the onsidered time slot and

the statisti al CSI of the involved links. However, the instantaneous CSI of past

and future time slots and the state of the relay's buer are not required for opti-

mal re eption-transmission. For the delay- onstrained ase, we propose a heuristi

buer-aided proto ol with adaptive re eption-transmission whi h limits the average

delay and a hieves a rate lose to the rate a hieved without a delay onstraint. This

proto ol only requires the instantaneous CSI of both links, and an be easily imple-

mented in real-time. Our analyti al and simulation results show, in good agreement,

that buer-aided relaying with adaptive re eption-transmission an a hieve signif-

i ant performan e gains ompared to onventional relaying with or without buer

[51, [18, as long as a ertain delay an be tolerated.

The remainder of the hapter is organized as follows. In Se tion 3.2, the onsid-

ered system and hannel models are presented. The proposed adaptive re eption-

transmission proto ol for buer-aided relaying is introdu ed in Se tion 3.3, and opti-

mized for rate maximization and power allo ation in Se tions 3.4 and 3.5, respe tively.

In Se tion 3.6, we propose an adaptive re eption-transmission proto ol that limits the

delay. Numeri al results are presented in Se tion 3.7, and some on lusions are drawn

in Se tion 3.8.

55


s(i) r(i)

S R D

Figure 3.1: The two-hop HD relay network with fading on the S-R and R-D links.

s(i) and r(i) are the instantaneous SNRs of the S-R and R-D links in the ith time

slot, respe tively.

3.2 System Model

We onsider the two-hop HD relay network shown in Fig. 3.1. For simpli ity of presen-

tation, in this and the following hapter, we denote the sour e, relay, and destination

by S, R, and D, respe tively, and the sour e-relay and relay-destination links by S-R

and R-D links, respe tively. We assume that the HD relay is equipped with an unlim-

ited buer. The sour e sends odewords to the relay, whi h de odes these odewords,

possibly stores the information in its buer, and eventually sends the information to

the destination. Throughout this hapter, we assume that the S-R and R-D links

are AWGN hannels ae ted by fading and that the sour e has always data to trans-

mit. We assume that time is divided into N → ∞ slots of equal lengths. In the ith

time slot, the transmit powers of sour e and relay are denoted by PS(i) and PR(i),

respe tively, and the instantaneous (squared) hannel gains of the S-R and R-D links

are denoted by hS(i) and hR(i), respe tively. The hannel gains hS(i) and hR(i) are

modeled as mutually independent, non-negative, stationary, ergodi , and ontinuous

random pro esses with expe ted values EhS(i) , ΩSR and EhR(i) , ΩRD. We

assume slow fading su h that the hannel gains are onstant during one time slot

but hange from one time slot to the next due to e.g. the mobility of the involved

nodes and/or frequen y hopping. The instantaneous link SNRs of the S-R and R-D

hannels in the ith time slot are given by s(i) , γS(i)hS(i) and r(i) , γR(i)hR(i),

respe tively. Here, γS(i) = PS(i)/σ2nR

and γR(i) = PR(i)/σ2nD

denote the transmit

56


SNRs without fading of the sour e and the relay, respe tively, and σ2nR

and σ2nD

are

the varian es of the omplex AWGN at the relay and the destination, respe tively.

The average SNRs re eived at relay and destination are denoted by ΩSR , Es(i)

and ΩRD , Er(i), respe tively. Throughout this hapter, we assume transmission

with apa ity a hieving odes. Hen e, the transmitted odewords by sour e and relay

span one time slot, are omprised of n → ∞ omplex symbols whi h are generated

independently a ording to the zero-mean omplex ir ular-invariant Gaussian dis-

tribution. In time slot i, the varian e of sour e's and relay's odewords are PS(i) and

PR(i), and their data rate will be determined in the following se tion.

In the following, we outline the general buer-aided adaptive re eption-transmis-

sion proto ol.

3.3 Preliminaries and Ben hmark S hemes

In this se tion, we des ribe the general buer-aided adaptive re eption-transmission

proto ol for the two-hop HD rely network. Later, we optimize the general buer-aided

adaptive re eption-transmission proto ol for average rate maximization and thereby

obtain the proposed buer-aided proto ol.

3.3.1 Adaptive Re eption-Transmission Proto ol and CSI

Requirements

The general buer-aided adaptive re eption-transmission proto ol is as follows. At

the beginning of ea h time slot, the relay de ides to either re eive a odeword from

the sour e or to transmit a odeword to the destination, i.e., whether to sele t the

S-R or R-D link for transmission in a given time slot i. On e the relay makes the

57


de ision, it broad asts its de ision

7

to the other nodes before transmission in time

slot i begins. If they are sele ted for transmission, the sour e and the relay transmit

odewords spanning one time slot with rates whi h are adapted to the apa ity of

their respe tive links. For sele tion of the re eption-transmission and of the rate

adaptation, the nodes require CSI knowledge as will be detailed in the following.

CSI requirements: The relay node requires knowledge of the instantaneous hannel

gains hS(i) and hR(i) in order to make the de ision of whether it should re eive or

transmit. In addition, if the S-R link is sele ted for transmission, the sour e requires

knowledge of hS(i) in order to adapt the rate of its odeword. On the other hand, if

the R-D link is sele ted for transmission, the relay the destination requires knowledge

of hR(i) for de oding. In a given time slot i, this CSI an be obtained by three pilot

symbol transmissions, one from sour e and destination, respe tively, and one from

the relay. Furthermore, we assume that the noise varian e σ2nR

is known at sour e

and relay, and that the noise varian e σ2nD

is known at relay and destination.

3.3.2 Transmission Rates and Queue Dynami s

In the following, we dene the rates of the odewords transmitted by the sour e and

relay in a given time slot i, and determine the state of the queue at the buer of the

relay.

Sour e transmits relay re eives: If the sour e is sele ted for transmission in

time slot i, it transmits one odeword with rate

SSR(i) = log2(1 + s(i)). (3.1)

7

The de ision ontains an information of one bit whi h is: should relay re eive or transmit in

time slot i.

58


Hen e, the relay re eives SSR(i) bits/symb from the sour e and appends them to the

queue in its buer. The number of bits/symb in the buer of the relay at the end of

time slot i is denoted by Q(i) and given by

Q(i) = Q(i− 1) + SSR(i). (3.2)

Relay transmits destination re eives: If the relay transmits in time slot i,

the number of bits/symb transmitted by the relay is given by

RRD(i) = minlog2(1 + r(i)), Q(i− 1), (3.3)

where we take into a ount that the maximal number of bits/symb that an be send

by the relay is limited by the number of bits/symb in its buer and the instantaneous

apa ity of the R-D link. The number of bits/symb remaining in the buer at the

end of time slot i is given by

Q(i) = Q(i− 1)− RRD(i), (3.4)

whi h is always non-negative be ause of (3.3).

Be ause of the HD onstraint, we have RRD(i) = 0 when the sour e transmits

and the relay re eives, and we have SSR(i) = 0 when the relay transmits.

In the following, we determine the average data rate re eived at the destination

during N → ∞ time slots.

59


3.3.3 A hievable Average Rate

Sin e we assume the sour e has always data to transmit, the average number of

bits/symb that arrive at the destination per time slot is given by

RSD = limN→∞

1

N

N∑

i=1

RRD(i), (3.5)

i.e., RSD is the a hievable average rate of the onsidered ommuni ation system.

The main goal in this hapter is the maximization of RSD by optimizing the relay's

re eption and transmission in ea h time slot and the transmit power allo ated to

sour e and relay.

3.3.4 Conventional Relaying

For omparison purpose, we provide the a hievable average rate of two baseline

s hemes and provide the CSI requirements. Thereby, we assume that the transmit

powers at the sour e and the relay are xed, i.e., PS(i) = PS, PR(i) = PR, ∀i.

Conventional Relaying With Buer [18, [51

In onventional relaying with buer as proposed in [18, [51, the relay re eives data

from the sour e in the rst ξN time slots, where 0 < ξ < 1, and sends this umulative

information to the destination in the next (1 − ξ)N slots, where N → ∞. The

orresponding a hievable average rate is given in (1.4), where after setting Elog2(1+

s(i)) = CSR and Elog2(1 + r(i)) = CRD, we obtain the following rate

Rconv,1 =Elog2(1 + s(i))Elog2(1 + r(i))

Elog2(1 + s(i))+ Elog2(1 + r(i)). (3.6)

CSI Requirements: In order to a hieve (3.6) using onventional relaying with

60


buer as proposed [51, the sour e and relay have to a quire the CSI of the sour e-

relay link in the rst ξN time slots, and the relay and destination have to a quire

the CSI of the relay-destination link in the following (1 − ξ)N time slot. Thereby,

two pilot symbol transmissions are required per time slot. Compared to buer-aided

relaying with adaptive re eption-transmission, this proto ol requires one pilot symbol

transmission less.

Conventional Relaying Without Buer

The a hievable average rate of onventional relaying without buer, where the relay

re eives a odeword in ξ(i) fra tion of time slot i and transmits the re eived infor-

mation in the remaining 1 − ξ(i) fra tion of time slot i, where 0 < ξ(i) < 1, is given

in (1.6). After setting log2(1+ s(i)) = CSR(i) and log2(1+ r(i)) = CRD(i), we obtain

from (1.6) the following average data rate

Rconv,2 = E

log2(1 + s(i))× log2(1 + r(i))

log2(1 + s(i)) + log2(1 + r(i))

. (3.7)

However, to a hieve (3.7), the lengths of odewords have to vary and to be adapted

to the fading state of the hannels in ea h time slot, whi h may not be desirable in

pra ti e. In that ase, by setting ξ(i) = 1/2, ∀i, the odeword lengths ould be xed,

and thereby, the following average rate is a hieved

Rconv,3 =1

2E minlog2(1 + s(i), log2(1 + r(i)) . (3.8)

CSI Requirements: In order to a hieve (3.7) and (3.8) using onventional relaying

without buer, the sour e, relay, and destination have to a quire the CSI of both the

S-R and R-D links in ea h time slot. Thereby, three pilot transmissions are required

61


per time slot. In addition, the CSI of S-R and R-D links have to be feedba k

to destination and sour e, respe tively, in ea h time slot. In omparison, buer-

aided relaying with adaptive re eption-transmission also requires three pilot symbol

transmissions per time slot, however, it requires only one bit of feedba k information

per time slot. On the other hand, sin e the CSI of the S-R and R-D links are real

numbers, the two feedba ks required for onventional relaying without buer must

ontain large (ideally innite) number of information bits.

Comparing (3.7) and (3.6), we observe that Rconv,2 ≤ Rconv,1 holds. However, to

realize this performan e gain, the relay has to be equipped with a buer of innite

size and and innite delay is introdu ed.

Rayleigh Fading

For the numeri al results shown in Se tion 3.7, we onsider the ase where the S-

R and R-D links are both Rayleigh faded, i.e., the probability density fun tions

(PDFs) of s(i) and r(i) are given by fs(s) = e−s/ΩSR/ΩSR and fr(r) = e−r/ΩRD/ΩRD,

respe tively. In this ase, Rconv,1, Rconv,2, and Rconv,3 given in (3.6), (3.7), and (3.8),

respe tively, an be obtained as

Rconv,1 =1

ln(2)

exp(

1ΩSR

)

E1

(

1ΩSR

)

exp(

1ΩRD

)

E1

(

1ΩRD

)

exp(

1ΩSR

)

E1

(

1ΩSR

)

+ exp(

1ΩRD

)

E1

(

1ΩRD

) , (3.9)

Rconv,2 =

∫ ∞

0

∫ ∞

0

log2(1 + s)× log2(1 + r)

log2(1 + s) + log2(1 + r)

e−s/ΩSR−r/ΩRD

ΩSRΩRDdsdr, (3.10)

62


and

Rconv,3 =1

2 ln(2)exp

(

ΩR + ΩS

ΩSΩR

)

E1

(

ΩR + ΩS

ΩSΩR

)

, (3.11)

respe tively, where E1(x) =∫∞

xe−t/t dt, x > 0, denotes the exponential integral

fun tion.

In the following, we optimize the general buer-aided proto ol with adaptive

re eption-transmission for rate maximization.

3.4 Optimal Adaptive Re eption-Transmission

Proto ol for Fixed Powers

To gain insight, we rst derive the optimal adaptive re eption-transmission poli y

and the orresponding a hievable average rate for the ase when the sour e and

relay transmit with xed powers, i.e., PS(i) = PS, PR(i) = PR, ∀i. Optimal power

allo ation will be dis ussed in Se tions 3.5.

3.4.1 Problem Formulation

In order to formulate the re eption and transmission at the relay in time slot i, we

introdu e a binary de ision variable di ∈ 0, 1. We set di = 1 if the R-D link is

sele ted for transmission in time slot i, i.e., the relay transmits and the destination

re eives. Similarly, we set di = 0 if the S-R link is sele ted for transmission in time

slot i, i.e., the sour e transmits and the relay re eives. Exploiting di, the number of

bits/symb send from the sour e to the relay and from the relay to the destination in

63


time slot i an be written in ompa t form as

SSR(i) = (1− di) log2(1 + s(i)) (3.12)

and

RRD(i) = di minlog2(1 + r(i)), Q(i− 1), (3.13)

respe tively. Consequently, the average rate in (3.5) an be rewritten as

RSD = limN→∞

1

N

N∑

i=1

di minlog2(1 + r(i)), Q(i− 1). (3.14)

The onsidered rate maximization problem an now be stated as follows: Find the op-

timal adaptive re eption-transmission poli y, i.e., the optimal sequen e di, ∀i, whi h

maximizes the a hievable average rate RSD given in (3.14).

3.4.2 Optimal Adaptive Re eption-Transmission Proto ol

Using notation from queueing theory [83, we dene the average arrival rate of

bits/symb per time slot arriving into the queue of the buer, denoted by A, and

the average departure rate of bits/symb time per slot departing out of the queue of

the buer, denoted by D, as

A , limN→∞

1

N

N∑

i=1

(1− di) log2(1 + s(i)) (3.15)

and

D , limN→∞

1

N

N∑

i=1

diminlog2(1 + r(i)), Q(i− 1), (3.16)

64


respe tively. We note that the average departure rate D is equal to the a hievable

average rate given in (3.14). In order to derive the optimal proto ol, we give the

following denition for an absorbing and non-absorbing queue.

Denition 3.1. An absorbing queue is a queue for whi h A > D holds. A non-

absorbing queue is a queue for whi h A = D holds.

In an absorbing queue a part of the arrival rate is absorbed (trapped) inside the

buer of unlimited size and therefore the departure rate D is smaller than the arrival

rate.

The following theorem hara terizes the optimal adaptive re eption-transmission

poli y in terms of the state of the queue in the buer of the relay.

Theorem 3.1. A ne essary ondition for the optimal adaptive re eption-transmission

poli y whi h maximizes the a hievable average rate is that the queue in the buer of

the relay is at the edge of non-absorbtion, i.e., the queue is non-absorbing but is at

the boundary of a non-absorbing and an absorbing queue.

Proof. Please refer to Appendix B.1.

Exploiting Theorem 3.1, we an establish a useful ondition that the optimal

adaptive re eption-transmission poli y has to fulll and a simplied expression for

the a hievable average rate. This is the subje t of the following theorem.

Theorem 3.2. The a hievable average rate in (3.14) is maximized when the following

identity holds

limN→∞

1

N

N∑

i=1

(1− di) log2(1 + s(i)) = limN→∞

1

N

N∑

i=1

di log2(1 + r(i)). (3.17)

65


Moreover, when (3.17) holds, the a hievable average rate in (3.14) is then given by

RSD = limN→∞

1

N

N∑

i=1

di log2(1 + r(i)) = Edi log2(1 + r(i)). (3.18)


Remark 3.1. A queue that meets ondition (3.17) is rate-stable sin e there is no

loss of information, i.e., the information that goes in the buer eventually leaves the

buer without any loss. Hen e, all of the information sent by the sour e is eventually

re eived at the destination without any loss.

Remark 3.2. By omparing (3.14) and (3.18) it an be seen that when (3.17) holds,

the average data rate be omes independent of the queue states Q(i), ∀i. The rea-

son for this is the following. When (3.17) holds and N → ∞, the number of time

slots in whi h the buer does not have enough data for transmission, and thereby

minQ(i − 1), log2(1 + r(i) = Q(i − 1) o urs, are negligeable ompared to the

number of time slots in whi h the buer does have enough data for transmission,

and thereby minQ(i − 1), log2(1 + r(i) = log2(1 + r(i)) o urs. In parti ular, as

shown in Appendix B.2, ondition (3.17) automati ally ensures that for N → ∞,

1N

∑Ni=1 di log2(1 + r(i)) = 1

N

∑Ni=1 diminlog2(1 + r(i)), Q(i − 1) is valid, i.e., the

impa t of event log2(1 + r(i)) > Q(i − 1), i = 1, . . . , N , is negligible. Hen e, when

(3.17) holds and N → ∞, we an pra ti ally onsider the relay is fully ba klogged.

We are now ready to derive the optimal adaptive re eption-transmission poli y

for buer-aided relaying without power allo ation. A ording to Theorem 3.2, the

poli y that maximizes the a hievable average rate RSD in (3.18) an be found inside

the set of poli ies that produ e a queue whi h satises (3.17). Thus, for N → ∞, we

66


formulate the following optimization problem:

Maximize :di

1N

∑Ni=1 di log2(1 + r(i))

Subject to : C1 : 1N

∑Ni=1(1− di) log2(1 + s(i)) = 1

N

∑Ni=1 di log2(1 + r(i))

C2 : di ∈ 0, 1, ∀i,

(3.19)

where onstraint C1 ensures that the sear h for the optimal poli y is ondu ted only

among those poli ies that satisfy (3.17) and C2 ensures that di ∈ 0, 1. We note

that C1 and C2 do not ex lude the ase that the relay is hosen for transmission if

log2(1 + r(i)) > Q(i − 1). However, a ording to Remark 3.2, C1 ensures that the

inuen e of event log2(1 + r(i)) > Q(i − 1) is negligible. Therefore, an additional

onstraint dealing with this event is not required. The solution of problem (3.19)

leads to the following theorem.

Theorem 3.3. The optimal poli y maximizing the a hievable average rate of buer-

aided relaying with adaptive re eption-transmission is given by

di =

1 if log2(1 + r(i)) ≥ ρ log2(1 + s(i))

0 otherwise

(3.20)

where ρ is a onstant, referred to as the de ision threshold, found su h that onstraint

C1 in (3.19) holds. The orresponding maximum rate, denoted by RSD,max, is found

by inserting (3.20) into (3.18).


Remark 3.3. Interestingly, we observe from Theorem 3.3 that the optimal de ision,

di, at time slot i, depends only on the instantaneous SNRs, s(i) and r(i), of that

time slot. Hen e, di does not depend on the state of the queue, Q(i), in any time

67


slot nor on the instantaneous SNRs in previous or future time slots. This makes

the proposed optimal sele tion poli y easy to implement. We note that the de ision

threshold, ρ, depends on the statisti al CSI of both involved links as will be established

in the next se tion. The independen e of the optimal adaptive re eption-transmission

poli y from non- ausal instantaneous CSI is aused by the relay being operated at the

edge of non-absorption, i.e., the relay node is pra ti ally fully ba klogged. Non- ausal

knowledge would only help buer management (i.e., ensuring that there is a su ient

number of bits/symb in the buer for up oming time slots), whi h is not required in

the onsidered regime.

Remark 3.4. In this hapter, we assume that the transmitting nodes have perfe t

CSI and apply adaptive rate transmission. However, we note that this is not ne -

essary for a hieving the maximum a hievable average rate in (3.18). In fa t, the

proposed adaptive re eption-transmission proto ol (3.20) also a hieves the maximum

a hievable average rate in (3.18) if sour e and relay transmit long odewords that

span (ideally innitely) many time slots (and onsequently innitely many fading

states). In this ase, both the sour e and the relay an transmit with onstant rate

RSD,max = E(1 − di) log2(1 + s(i)) = Edi log2(1 + r(i)), where di is given in

(3.20), and rate adaptation is not ne essary. The rst odeword is transmitted by

the sour e without adaptive re eption-transmission and de oded by the relay. For all

subsequent odewords, adaptive re eption-transmission is performed based on (3.20)

and sour e and relay transmit parts of a long odeword whenever they are sele ted

for transmission. The disadvantage of this approa h is that the long odewords inher-

ently introdu e (ideally innitely) long delays and the generalization of this approa h

to the delay- onstrained ase is di ult. Therefore, in this hapter, we onsider adap-

tive rate transmission and assume that one odeword spans only one time slot (and

68


onsequently one fading state).

3.4.3 De ision Threshold

The de ision threshold ρ an be omputed based on the following lemma.

Lemma 3.1. The de ision threshold ρ is found as the solution of

∫ ∞

0

[∫ ∞

G(r)

log2(1 + s)fs(s)ds

]

fr(r)dr =

∫ ∞

0

[∫ ∞

H(s)

log2(1 + r)fr(r)dr

]

fs(s)ds,

(3.21)

where fs(s) and fr(r) are the PDFs of s(i) and r(i), respe tively, and G(r) = (1 +

r)1ρ − 1 and H(s) = (1 + s)ρ − 1.

Proof. Due to the ergodi ity, the left hand side of (3.17) is the expe tation of variable

(1 − di) log2(1 + s(i)). This variable is nonzero only when di = 0. From (3.20)

we observe that di = 0 if ρ log2(1 + s(i)) > log2(1 + r(i)), whi h is equivalent to

s(i) > G(r). Therefore, the domain of integration for al ulating the expe tation of

(1− di) log2(1+ s(i)) is s(i) > G(r) and r(i) > 0, whi h leads to the left hand side of

(3.21). Using a similar approa h, the right hand side of (3.21) is obtained from the

right hand side of (3.17). This on ludes the proof.

Remark 3.5. Eq. (3.21) reveals that the de ision threshold ρ depends indeed on the

statisti al properties of both involved links as was already alluded to in Remark 3.3.

3.4.4 Rayleigh Fading

For on reteness, we provide in this subse tion expressions for ρ and the orrespond-

ing maximum a hievable average rate RSD,max for Rayleigh fading links. Thus, by

69


inserting fs(s) = e−s/ΩSR/ΩSR and fr(r) = e−r/ΩRD/ΩRD into (3.21), we obtain

1

ln(2)

∫ ∞

0

[

exp

(

−(r + 1)

1ρ − 1

ΩSR

)

ln(

(r + 1)1ρ

)

+ e1

ΩSRE1

(

(r + 1)1ρ

ΩSR

)

]

e−r/ΩRD

ΩRD

dr

=1

ln(2)

∫ ∞

0

[

exp

(

−(s + 1)ρ − 1

ΩRD

)

ln ((s+ 1)ρ) + e1

ΩRDE1

(

(s+ 1)ρ

ΩRD

)]

×1

ΩSRexp

(

−s

ΩSR

)

ds. (3.22)

The optimal de ision threshold ρ an be found numeri ally from (3.22). The orre-

sponding maximum a hievable average rate is obtained as

RSD,max =1

ln(2)

∫ ∞

0

[

exp

(

−(s+ 1)ρ − 1

ΩRD

)

× ln ((s+ 1)ρ)

+e1

ΩRDE1

(

(s+ 1)ρ

ΩRD

)]

1

ΩSRexp

(

−s

ΩSR

)

ds, (3.23)

where ρ is found from (3.22).

Spe ial ase (ΩSR = ΩRD)

For the spe ial ase ΩSR = ΩRD = Ω, we obtain from (3.22) ρ = 1, and the orre-

sponding maximal a hievable average rate is

RSD,max =1

ln(2)exp

(

1

Ω

)

E1

(

1

Ω

)

−1

2 ln(2)exp

(

2

Ω

)

E1

(

2

Ω

)

. (3.24)

Comparing this average rate with the average rate a hieved with onventional re-

laying with a buer, f. (3.9), the gain of adaptive re eption-transmission an be

hara terized by

RSD,max

Rconv,1

= 2−exp

(

2Ω

)

E1

(

2Ω

)

exp(

1Ω

)

E1

(

1Ω

) ≥ 1, (3.25)

70


where the ratio RSD,max/Rconv,1 monotoni ally in reases from 1 to 1.5 as Ω de reases

from ∞ to zero.

3.4.5 Real-Time Implementation

In order to sele t whether the relay should re eive or transmit a ording to the

proto ol in Theorem 3.3, the relay has to ompute the onstant ρ. This onstant an

be omputed using Lemma 3.1, but this requires knowledge of the PDFs of the fading

gains of the two links before the start of transmission. Su h a priori knowledge may

not be available in pra ti e. In this ase, the relay has to estimate ρ in real-time

using only the CSI knowledge until time slot i. Sin e ρ is a tually the Lagrange

multiplier obtained by solving the optimization problem in (3.19), see Appendix B.3,

an a urate estimate of ρ an be obtained using the gradient des ent method [81. In

parti ular, using log2(1+s(i)) and log2(1+r(i)), the destination re ursively omputes

an estimate of ρ, denoted by ρe(i), as

ρe(i) =[

ρe(i− 1) + ψ(i)(De(i− 1)−Ae(i− 1))]∞

0, (3.26)

where [x]ba = minmaxx, a, b, Ae(i − 1) and De(i − 1) are real-time estimates of

the average arrival rate A and the average departure rate D, respe tively, omputed

as

Ae(i− 1) =i− 2

i− 1Ae(i− 2) +

1− di−1

i− 1log2(1 + s(i− 1)), i ≥ 2, (3.27)

De(i− 1) =i− 2

i− 1De(i− 2) +

di−1

i− 1log2(1 + r(i− 1)), i ≥ 2, (3.28)

where Ae(0) and De(0) are set to zero. In (3.26), ψ(i) is an adaptive step size whi h

ontrols the speed of onvergen e of ρe(i) to ρ. In parti ular, the step size ψ(i) is

71


some properly hosen monotoni ally de aying fun tion of i with ψ(1) < 1, see [81

for more details.

On e the relay has estimated s(i) and r(i), and omputed ρe(i), it sele ts the

a tive link, i.e., the value of di, a ording to Theorem 3.3.

3.5 Optimal Adaptive Re eption-Transmission and

Optimal Power Allo ation

So far, we have assumed that the sour e and relay transmit powers are xed. In this

se tion, we jointly optimize the power allo ation and adaptive re eption-transmission

poli ies for buer-aided relaying.

3.5.1 Problem Formulation and Optimal Power Allo ation

Our goal is to jointly optimize the link sele tion variable di and the powers PS(i)

and PR(i) in ea h time slot i su h that the a hievable average rate is maximized.

For onvenien e, we optimize in the following the transmit SNRs without fading

γS(i) and γR(i), whi h may be viewed as normalized powers, instead of the powers

PS(i) = γS(i)σ2nR

and PR(i) = γR(i)σ2nD

themselves. For a fair omparison, we limit

the average power onsumed by the sour e and the relay to Γ. This leads for N → ∞

72


to the following optimization problem:

Maximize :γS(i)≥0,γR(i)≥0,di

1N

∑Ni=1 di log2(1 + γR(i)hR(i))


∑Ni=1(1− di) log2(1 + γS(i)hS(i))

= 1N


C2 : di ∈ 0, 1

C3 : 1N

∑Ni=1(1− di)γS(i) +

1N

∑Ni=1 diγR(i) ≤ Γ

(3.29)

where onstraints C1 and C2 are identi al to the onstraints in (3.19) and C3 is the

joint sour e-relay power onstraint. The solution of Problem (3.29) is summarized in

the following theorem.

Theorem 3.4. The optimal (normalized) powers γS(i) and γR(i) and de ision vari-

able di maximizing the a hievable average rate of buer-aided relaying with adaptive

re eption-transmission while satisfying an average sour e-relay power onstraint are

given by

γS(i) =

ρ/λ− 1/hS(i) if hS(i) > λ/ρ

0 otherwise

(3.30)

γR(i) =

1/λ− 1/hR(i) if hR(i) > λ

0 otherwise

(3.31)

di =

1 if[

ln(

hR(i)λ

)

+ λhR(i)

− 1 > ρ ln(

ρλhS(i)

)

+ λhS(i)

− ρ

AND hR(i) > λ AND hS(i) >λρ

]

OR

[

hR(i) > λ AND hS(i) ≤λρ

]

0 otherwise

(3.32)

73


where ρ and λ are found su h that C1 and C3 in (3.29) hold with equality for N → ∞.

The orresponding maximum average rate is found by inserting (3.31) and (3.32) into

RSD,max = limN→∞

1

N

N∑

i=1

di log2(1 + γR(i)hR(i)). (3.33)


3.5.2 Finding λ and ρ

The following lemma establishes two equations from whi h the optimal λ and ρ an

be found.

Lemma 3.2. Denote the PDFs of hS(i) and hR(i) by fhS(hS) and fhR

(hR), respe -

tively. Let the transmit powers of the sour e and the relay in time slot i be given by

(3.30) and (3.31), respe tively, and the link sele tion variable di by (3.32). Then,

ρ and λ maximizing the a hievable average rate of buer-aided relaying with adap-

tive re eption-transmission and power allo ation are found from the following two

equations

∫ λ

0

[∫ ∞

λ/ρ

log2

(

ρhSλ

)

fhS(hS)dhS

]

fhR(hR)dhR

+

∫ ∞

λ

[∫ ∞

L1

(

ρhSλ

)

fhS(hS)dhS

]

fhR(hR)dhR

=

∫ λ/ρ

0

[∫ ∞

λ

log2

(

hRλ

)

fhR(hR)dhR

]

fhS(hS)dhS

+

∫ ∞

λ/ρ

[∫ ∞

L2

log2

(

hRλ

)

fhR(hR)dhR

]

fhS(hS)dhS , (3.34)

74


∫ λ

0

[∫ ∞

λ/ρ

(

ρ

λ−

1

hS

)

fhS(hS)dhS

]

fhR(hR)dhR

+

∫ ∞

λ

[∫ ∞

L1

(

ρ

λ−

1

hS

)

fhS(hS)dhS

]

fhR(hR)dhR

+

∫ λ/ρ

0

[∫ ∞

λ

(

1

λ−

1

hR

)

fhR(hR)dhR

]

fhS(hS)dhS

+

∫ ∞

λ/ρ

[∫ ∞

L2

(

1

λ−

1

hR

)

fhR(hR)dhR

]

fhS(hS)dhS = Γ (3.35)

where

L1 = −λ

ρW (−e(hR−λ)/(ρhR)−1(λ/hR)1/ρ),

L2 = −λ

W (−eρ−1−λ/hS (λ/(ρhS))ρ). (3.36)

Here, W (·) is the Lambert W -fun tion [84, whi h is available as built-in fun tion

in software pa kages su h as Mathemati a. The maximum a hievable average rate is

given by the left (and right) hand side of (3.34).


The onstants λ and ρ an be found oine sin e (3.34) and (3.35) only depend

on the statisti al properties of the S-R and the R-D links. Sin e these statisti al

properties hange on a mu h slower time s ale than the instantaneous hannel gains,

λ and ρ an be updated with a low rate.

75


3.5.3 Rayleigh Fading

For the spe ial ase of Rayleigh fading with fhS(hS) = e−hS/ΩSR/ΩSR and fhR

(hR) =

e−hR/ΩRD/ΩRD, (3.34) and (3.35) an be simplied to an be simplied to

1

ln(2)

[

(

1− e−λ/ΩRD

)

E1

(

λ

ρΩSR

)

+

∫ ∞

λ

e−L1/ΩSR ln

(

ρL1

λ

)

+ E1

(

L1

ΩSR

)

e−hR/ΩRD

ΩRD

dhR

]

=1

ln(2)

[

(

1− e−λ/(ρΩSR))

E1

(

λ

ΩRD

)

+

∫ ∞

λ/ρ

e−L2/ΩRD ln

(

L2

λ

)

+ E1

(

L2

ΩRD

)

e−hS/ΩSR

ΩSR

dhS

]

(3.37)

and

(

1− e−λ/ΩRD

)

ρ

λe−λ/(ρΩSR) −

E1

(

λρΩSR

)

ΩSR

+

∫ ∞

λ

ρ

λe−L1/ΩSR −

E1

(

L1

ΩSR

)

ΩSR

e−hR/ΩRD

ΩRD

dhR

+(

1− e−λ/(ρΩSR))

1

λe−λ/ΩRD −

E1

(

λΩRD

)

ΩRD

+

∫ ∞

λ/ρ

1

λe−L2/ΩRD −

E1

(

L2

ΩRD

)

ΩRD

e−hS/ΩSR

ΩSR

dhS = Γ, (3.38)

respe tively, where L1 and L2 are given in (3.36) and the maximum a hievable average

rate is given by the left (and right) hand side of equation (3.37).

3.5.4 Real-Time Implementation

Similar to the real-time implementation for nding ρ for the ase without power

allo ation des ribed in Se . 3.5.1, we an also onstru t a real-time implementation

76


for nding ρ and λ for the ase with power allo ation. In this ase, an estimate for

ρ is found in the same way as in Se . 3.5.1. On the other hand, an estimate for λ,

denoted by λe(i) is found as

λe(i) =[

λe(i− 1) + φ(i)(ΓeS(i− 1) + Γe

R(i− 1)− Γ]∞

0, (3.39)

where

ΓeS(i− 1) =

i− 2

i− 1ΓeS(i− 2) +

1− di−1

i− 1γS(i), i ≥ 2, (3.40)

ΓeR(i− 1) =

i− 2

i− 1ΓeR(i− 2) +

di−1

i− 1γR(i), i ≥ 2, (3.41)

where ΓeS(0) and Γe

R(0) are set to zero. In (3.39), φ(i) is an adaptive step size whi h

ontrols the speed of onvergen e of λe(i) to λ. In parti ular, the step size φ(i) is

some properly hosen monotoni ally de aying fun tion of i with φ(1) < 1, see [81 for

more details.

3.6 Delay-Limited Transmission

So far, we have assumed that there is no delay onstraint. In pra ti e, there is usually

some onstraint on the delay. In this se tion, we propose a buer-aided adaptive

re eption-transmission proto ol for delay onstrained transmission. For simpli ity,

we assume xed transmit powers, i.e., PS(i) = PS, PR(i) = PR, ∀i.

3.6.1 Average Delay

Sin e we assume that the sour e is ba klogged and has always information to transmit,

for the onsidered network, the transmission delay is aused only by the buer at the

77


relay. Let T (i) denote the delay of a bit of information that is transmitted by the

sour e in time slot i and re eived at the destination in time slot i + T (i), i.e., the

onsidered bit is stored for T (i) time slots in the buer. Then, a ording to Little's

law [85 the average delay ET (i) in number of time slots is given by

ET (i) =EQ(i)

A, (3.42)

where EQ(i) is the average queue length at the buer and A is the average arrival

rate into the queue.

The queue size at time slot i an be obtained as

Q(i) = Q(i− 1) + (1− di) log2(1 + s(i))− diminQ(i− 1), log2(1 + s(i)). (3.43)

Due to the re ursiveness of the expression in (3.43), it is di ult, if not impossible,

to obtain an analyti al expression for the average queue size EQ(i) for a general

buer-aided relaying poli y. Hen e, in ontrast to the ase without delay onstraint,

for the delay limited ase, it is very di ult to formulate an optimization problem

for maximization of the average rate subje t to some average delay onstraint. As

a result, in the following, we develop a simple heuristi proto ol for delay limited

transmission. In the proposed proto ol, the relay itself de ides whether it should

re eive or transmit in ea h time slot su h that the average delay onstraint is satised,

and informs the sour e and destination about the de ision. We note that the proposed

proto ol does not need any knowledge of the statisti s of the hannels. The proto ol

needs only the instantaneous CSI of the S-R and R-D links at the relay, and the

desired average delay T0. This allows for relatively easy real-time implementation of

the proposed proto ol for delay-limited transmission.

78


3.6.2 Buer-Aided Proto ol for Delay Limited Transmission

Before presenting the proposed heuristi proto ol for delay limited transmission, we

rst explain the intuition behind the proto ol.

Intuition Behind the Proto ol

Assume that we have a buer-aided proto ol whi h, when implemented in the on-

sidered network, enfor es the following relation

EQ(i)

A= T0, (3.44)

where T0 is the desired average delay. There are many ways to enfor e (3.44) at

the relay. Our preferred method for enfor ing (3.44) is to have the relay re eive and

transmit when Q(i)/A < T0 and Q(i)/A > T0, respe tively. In this way, Q(i)/A

be omes a random pro ess whi h exhibits u tuation around its mean value T0, and

thereby a hieves (3.44) in the long run. We are now ready to present the proposed

proto ol.

The Proposed Proto ol

Let T0 be the desired average delay onstraint of the system. At the beginning

of time slot i, sour e and destination transmit pilots in su essive pilot time slots.

This enables the relay to a quire the CSI of their respe tive S-R and R-D links,

respe tively. Using the a quired CSI, the relay omputes log2(1 + s(i)) and log2(1 +

r(i)). Next, using log2(1 + s(i)) and the amount of normalized information in its

buer, Q(i− 1), the relay omputes a variable ω(i) as follows

ω(i) = ω(i− 1) + ζ(i)

(

T0 −Q(i− 1)

Ae(i− 1)

)

, (3.45)

79


where Ae(i−1) is a real-time estimate of A, omputed using (3.27). In (3.45), ζ(i) is

the step size fun tion, whi h is some properly hosen monotoni ally de aying fun tion

of i with ζ(1) < 1. Now, using log2(1 + s(i)), log2(1 + r(i)), Q(i− 1), and ω(i), the

relay omputes di as

di =

1 if

1ω(i)

minQ(i− 1), log2(1 + r(i)) ≥ ω(i) log2(1 + s(i))

0 otherwise

(3.46)

The relay then broad asts a ontrol pa ket ontaining pilot symbols and information

about whether the relay re eives or transmits to the sour e and destination. From the

pa ket broad asted by the relay, both sour e and destination learn the S-R and R-D

links, respe tively, and learn whether the relay is s heduled to re eive or transmit.

If the relay is s heduled to transmit, then it extra ts information from its buer and

transmits a odeword to the destination with rate

RRD(i) = minQ(i− 1), log2(1 + r(i)).

However, if the relay is s heduled to re eive, then the sour e transmits a odeword

to the relay with rate SSR(i) = log2(1 + s(i)).

Remark 3.6. The required overhead of the proposed delay-limited proto ol is identi al

to the overhead of the proposed proto ol without delay onstraint.

Remark 3.7. Although on eptually simple, a theoreti al analysis of the a hievable

average rate of the proposed delay-limited proto ol is di ult. Thus, we will resort to

simulations to evaluate the performan e of the delay limiting proto ol in Se tion 3.7.

Remark 3.8. We note that the proposed proto ol for the delay- onstrained ase is

heuristi in nature. The sear h for other proto ols with possibly superior perfor-

80


man e is an interesting topi for future work. The proposed proto ols for the delay-

onstrained and the delay-un onstrained ase an serve as ben hmark and perfor-

man e upper bound for these new proto ols, respe tively.

3.7 Numeri al and Simulation Results

In this se tion, we evaluate the performan e of buer-aided relaying (BAR) with

adaptive re eption-transmission and ompare it with that of onventional relaying.

Throughout this se tion, we assume Rayleigh fading. All results shown in this se tion

have been onrmed by omputer simulations. However, the simulations are not

shown in all instan es for larity of presentation. We note that the simulation results

are independent of whether the fading is slot-by-slot orrelated or un orrelated.

3.7.1 Delay-Un onstrained Transmission

First, we assume that there are no delay onstraints and investigate the a hievable

average rates with and without power allo ation.

In Fig. 3.2, we show the a hievable average rates of buer-aided relaying with

adaptive re eption-transmission, RSD,max, without power allo ation, given in (3.23),

and the a hievable average rate of onventional relaying with a buer, Rconv,1, given

in (3.9), and without a buer with adaptive and xed odeword lengths Rconv,2 and

Rconv,3, respe tively, given in (3.10) and (3.11), respe tively, for ΩSR = 0.9, ΩRD =

1.1, and γs = γr = γ. Moreover, we have also shown the rate of buer-aided relaying

obtained via simulations. As an be seen from Fig. 3.2, the simulated and theoreti al

results mat h perfe tly. The gure shows that buer-aided relaying with adaptive

re eption-transmission leads to substantial gains ompared to onventional relaying.

In parti ular, for γ = 10 dB, the gain of buer-aided relaying with adaptive re eption-

81


−10 −5 0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

γ (in dB)

Averagerate

(inbits/symb)

BAR with adaptive reception-transmission - TheoryBAR with adaptive reception-transmission - SimulationConventional relaying with bufferConventional relaying without buffer, adaptive codeword lengthsConventional relaying without buffer, fixed codeword lengths

Figure 3.2: Average rates a hieved with buer-aided relaying (BAR) with adaptive

re eption-transmission and with onventional relaying with and without buer for

ΩSR = 0.9 and ΩRD = 1.1.

transmission over onventional relaying with a buer is 3 dB, and without a buer

with adaptive and xed odeword lengths is 4 dB and 6 dB, respe tively.

For the parameters adopted in Fig. 3.2, we show in Fig. 3.3 the orresponding

onstant ρ obtained using Lemma 3.1, and the orresponding estimated parameter

ρe(i) obtained using the re ursive method in (3.26) as fun tions of time for γ = 0

dB. As an be seen from Fig. 3.3, the estimated parameter ρe(i) onverges relatively

qui kly to ρ.

In Fig. 3.4, we investigate the gains a hieved with power allo ation for a system

with ΩS = 0.1 and ΩR = 1.9. Thereby, we ompare the performan es of buer-aided

relaying with adaptive re eption-transmission with and without power allo ation.

For buer-aided relaying with adaptive re eption-transmission and power allo ation

the average rate, power allo ation, and adaptive re eption-transmission poli y were

obtained as des ribed in Theorem 3.4 and Lemma 3.2 in Se tion 3.5. As an be seen

from Fig. 3.4, and as expe ted, power allo ation in reases the average rate.

82


50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

Time slot i

ρ

ρe(i)ρ

Figure 3.3: Estimated ρe(i) as a fun tion of the time slot i.

−5 0 5 10 15 200

0.5

1

1.5

2

2.5

Γ (in dB)

Average

rate

(inbits/symb)

With power allocationWithout power allocation

Figure 3.4: Average rate with buer-aided relaying with adaptive re eption-

transmission with and without power allo ation for ΩS = 0.1 and ΩR = 1.9

83


−5 0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

γ (in dB)

Averagerate

(inbits/symb)

BAR with adaptive reception-transmission, T0 → ∞BAR with adaptive reception-transmission, T0 = 5BAR with adaptive reception-transmission, T0 = 3BAR with adaptive reception-transmission, T0 = 2Conventional relaying, adaptive codeword lengths T0 = 1Conventional relaying, fixed codeword lengths T0 = 1

Figure 3.5: Average rate of BAR with adaptive re eption-transmission for dierent

average delay onstraints.

3.7.2 Delay-Constrained Transmission

We now turn our attention to delay-limited transmission and investigate the perfor-

man e of the proposed buer-aided proto ol for this ase. Furthermore, we assume

xed transmit powers for the sour e and the relay.

In Fig. 3.5, we plot the a hievable average rate for buer-aided relaying with

adaptive re eption-transmission without and with a delay onstraint, as a fun tion of

γ, for ΩSR = ΩRD = 1. This numeri al example shows that for an average delay of 5,

3, and 2 time slots, the rate of the delay onstrained proto ol is within 0.75, 1.5, and

2.5 dB from the rate of the proto ol without a delay onstraint (i.e., T0 → ∞). For

omparison, we have also plotted the average rate of onventional relaying without

buer with adaptive and xed odeword lengths, respe tively, whi h require a delay

of one time slot. Fig. 3.5 shows that for an average delay of 5, 3, and 2 time slots,

the rate of the delay onstrained proto ol is within 3, 2.5, and 1.5 dB from the rate

a hieved with onventional relaying with adaptive odeword lengths, and within 5,

84


0 100 200 300 400 5001

2

3

4

5

6

Time slot i

Average

delay

untiltimeslot

i

T0=5

Figure 3.6: Average delay until time slot i for T0 = 5 and γ = 20 dB .

4.5, and 3.5 dB from the rate a hieved with onventional relaying with xed odeword

lengths. Considering the more stringent feedba k requirements for CSI a quisition of

onventional relaying without buer, see Se tion 3.3.4, this example learly shows the

potential of buer-aided relaying for pra ti al delay-limited transmission s enarios.

Furthermore, for the parameters adopted in Fig. 3.5, we have plotted the average

delay of the proposed delay-limited proto ol until time slot i in Fig. 3.6, for the ase

when T0 = 5 time slots, and γ = 15 dB. The average delay until time slot i, denoted

by T (i) is omputed as

T (i) =

∑ij=1Q(i)

∑ij=1(1− di) log2

(

1 + s(i)) ,

i.e., the queue size and the arrival rates are both averaged from the rst to the

i-th time slot. Fig. 3.6 shows that with the proto ol proposed for delay limited

transmission, the average delay until time slot i onverges to the desired delay T0

relatively fast. Moreover, after the average delay has rea hed T0, it exhibits relatively

small u tuations around T0.

85


3.8 Con lusions

In this hapter, we proposed a novel adaptive re eption-transmission proto ol for

relays with buers. In ontrast to onventional relaying, where the sour e and the

relay transmit a ording to a pre-dened s hedule regardless of the fading state, in

the proposed s heme, always the node with the relatively stronger link is sele ted for

transmission. For delay-un onstrained transmission, we derived the optimal adap-

tive re eption-transmission poli y for the ases of xed and variable sour e and relay

transmit powers. In both ases, the optimal poli y for a given time slot only depends

on the instantaneous CSI of that time slot and the statisti al CSI of the involved

links. For delay- onstrained transmission, we proposed a buer-aided proto ol whi h

ontrols the delay introdu ed by the buer at the relay. This proto ol needs only

instantaneous CSI and does not need statisti al CSI of the involved links, and an

be implemented in real-time. Our analyti al and simulation results showed that

buer-aided relaying with adaptive re eption-transmission with and without delay

onstraints is a promising approa h to in rease the a hievable average data rate om-

pared to onventional relaying.

86

Chapter 4

Buer-Aided Relaying With Adaptive

Re eption-Transmission: Fixed and

Mixed Rate Transmission

4.1 Introdu tion

In this hapter, we onsider the two-hop HD relay network where the sour e-relay

and the relay-destination links are AWGN hannels ae ted by fading, and assume

that that the sour e and/or the relay do not have CSIT and therefore have to trans-

mit odewords with a xed data rate. Moreover, we assume that the transmitted

odewords span one fading state. In this ase, a hannel apa ity in the stri t Shan-

non sense does not exist and an appropriate measure for su h systems is the outage

probability. Depending on the availability of CSIT at the transmitting nodes (and

their apability of using more than one modulation/ oding s heme), we onsider two

dierent modes of transmission for the two-hop HD relay network: Fixed rate trans-

mission and mixed rate transmission. In xed rate transmission, the node sele ted

for transmission (sour e or relay) does not have CSIT and transmits with xed rate.

In ontrast, in mixed rate transmission, the relay has CSIT knowledge and exploits

it to transmit with variable rate so that outages are avoided. However, the sour e

87

Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission

still transmits with xed rate to avoid the need for CSIT a quisition.

To explore the performan e limits of the proposed xed rate and mixed rate

adaptive re eption-transmission s hemes, we onsider rst transmission without de-

lay onstraints and derive the orresponding optimal buer-aided relaying proto ols

whi h maximize the throughput of the onsidered two-hop HD relay network. Fur-

thermore, we show that in Rayleigh fading the optimal buer-aided relaying proto ol

with adaptive re eption-transmission a hieves a diversity gain of two and a diversity-

multiplexing tradeo of DM(r) = 2(1− 2r), where r denotes the multiplexing gain.

In ontrast, onventional relaying a hieves a diversity gain of one. For mixed rate

transmission, we show that a multiplexing gain of one an be a hieved with buer-

aided relaying with and without adaptive re eption-transmission implying that there

is no multiplexing gain loss ompared to ideal FD relaying. Sin e it turns out that

these optimal buer-aided proto ols introdu e innite delay, in order to limit the de-

lay, we also introdu e modied buer-aided relaying proto ols for delay onstrained

transmission. In parti ular, for xed rate and mixed rate transmission with delay on-

straints, in order to ontrol the average delay, we introdu e appropriate modi ations

to the buer-aided relaying proto ols for the delay un onstrained ase. Surprisingly,

for xed rate transmission, the full diversity gain is preserved as long as the tolerable

average delay ex eeds three time slots. For mixed rate transmission with an average

delay of ET time slots, a multiplexing gain of r = 1− 1/(2ET) is a hieved.

The remainder of this hapter is organized as follows. In Se tion 4.2, the system

model of the onsidered two-hop HD relay network is presented. In Se tion 4.3 we

introdu e the buer-aided relaying proto ols for xed and mixed rate transmission. In

Se tions 4.4 and 4.5, we analyze the proposed buer-aided relaying proto ols for delay

un onstrained and delay onstrained xed rate transmission, respe tively. Proto ols

88


for delay un onstrained and delay onstrained mixed rate transmission are proposed

and analyzed in Se tion 4.6. The proposed proto ols and the derived analyti al

results are veried and illustrated with numeri al examples in Se tion 4.7, and some

on lusions are drawn in Se tion 4.8.

4.2 System Model and Channel Model

We onsider the two-hop HD relay network shown in Fig. 3.1, where both S-R and

R-D links are AWGN hannels ae ted by slow fading. We assume that the relay

is equipped with a buer. The sour e sends odewords to the relay, whi h de odes

these odewords, possibly stores the de oded information in its buer, and eventually

sends it to the destination. We assume that time is divided into slots of equal lengths,

that the fading is onstant during one time slot, and that every odeword spans one

time slot. Throughout this hapter, we assume that the sour e node has always

data to transmit. Hen e, the total number of time slots, denoted by N , satises

N → ∞. Furthermore, unless spe ied otherwise, we assume that the buer at the

relay is not limited in size. The ase of limited buer size will be investigated in

Se tions 4.5 and 4.6.4 when we investigate delay limited transmission. In the ith

time slot, the transmit powers of sour e and relay are denoted by PS(i) and PR(i),

respe tively, and the instantaneous squared hannel gains of the S-R and R-D links

are denoted by hS(i) and hR(i), respe tively. hS(i) and hR(i) are modeled as mutually

independent, non-negative, stationary, and ergodi random pro esses with expe ted

values ΩS , EhS(i) and ΩR , EhR(i). We assume that the hannel gains are

onstant during one time slot but hange from one time slot to the next due to,

e.g., the mobility of the involved nodes and/or frequen y hopping. We note that

for most results derived in this hapter, we only require hS(i) and hR(i) to be not

89


fully temporally orrelated, respe tively. However, in some ases, we will assume that

hS(i) and hR(i) are temporally un orrelated, respe tively, to fa ilitate the analysis.

The instantaneous SNRs of the S-R and R-D hannels in the ith time slot

are given by s(i) , γS(i)hS(i) and r(i) , γR(i)hR(i), respe tively. Here, γS(i) ,

PS(i)/σ2nR

and γR(i) , PR(i)/σ2nD

denote the transmit SNRs of the sour e and the

relay without fading, respe tively, and σ2nR

and σ2nD

are the varian es of the omplex

AWGN at the relay and the destination, respe tively. The average re eived SNRs at

relay and destination are denoted by ΩS , Es(i) and ΩR , Er(i), respe tively.

Throughout this hapter, we assume transmission with apa ity a hieving odes.

Hen e, the transmitted odewords by sour e and relay span one time slot, are om-

prised of n → ∞ omplex symbols whi h are generated independently a ording to

the zero-mean omplex ir ular-invariant Gaussian distribution. In time slot i, the

varian e of sour e's and relay's odewords are PS(i) and PR(i), and their date rate

will be determined in the following se tion.

For on reteness, we spe ialize some of the derived results to Rayleigh fading.

In this ase, the PDFs of s(i) and r(i) are given by fs(s) = e−s/ΩS/ΩS and fr(r) =

e−r/ΩR/ΩR, respe tively. Similarly, the PDFs of hS(i) and hR(i) are given by fhS(hS) =

e−hS/ΩS/ΩS and fhR(hR) = e−hR/ΩR/ΩR, respe tively.

In the following, we outline the general buer-aided adaptive re eption-transmission

proto ol for transmission with xed and mixed rates.

4.3 Preliminaries and Ben hmark S hemes

In this se tion, we des ribe the general buer-aided adaptive re eption-transmission

proto ol for transmission with xed and mixed rates. In parti ular, we outline the

transmission rates of the sour e and relay in ea h time slot, the CSI requirements,

90


the dynami s of the queue, and the throughput re eived at the destination.

4.3.1 Adaptive Re eption-Transmission and CSI

Requirements

The general buer-aided adaptive re eption-transmission proto ol is as follows. At

the beginning of ea h time slot, the relay de ides to either re eive a odeword from

the sour e or to transmit a odeword to the destination, i.e., de ides whether to sele t

the S-R or R-D link for transmission in a given time slot i. To this end, the relay is

assumed to know the statisti s of the S-R and R-D hannels. On e the relay makes

the de ision, it broad asts its de ision ( ontaining one bit of information) to the other

nodes before transmission in time slot i begins. If they are sele ted for transmission,

the sour e and the relay transmit odewords spanning one time slot and with rates

whi h will be determined below. For sele tion of the re eption and transmission, the

nodes require CSI knowledge as will be detailed in the following.

CSI for Fixed Rate Transmission

For xed rate transmission, neither the sour e nor the relay have full CSIT, i.e.,

sour e and relay do not know hS(i) and hR(i), respe tively. Therefore, both nodes

an transmit only with predetermined xed rates S0 and R0, respe tively, and annot

perform power allo ation, i.e., the transmit powers are a priori xed as PS(i) = PS

and PR(i) = PR, ∀i. The relay and destination are assumed to know the CSI of their

re eiving links, whi h is needed for oherent dete tion. For the relay to be able to

de ide whether it should re eive or transmit, it requires knowledge of the outage states

of the S-R and R-D links. The relay an determine whether or not the S-R link is in

outage based on S0, PS, σ2nR, and hS(i). The destination an do the same for the R-D

91


link based on R0, PR, σ2nD, and hR(i), and inform the relay whether or not the R-D

link is in outage using one bit of feedba k

8

information. Based on the outage states

of the S-R and R-D links in a given time slot i and the statisti s of both links, the

relay sele ts the transmitting node a ording to the adaptive re eption-transmission

proto ols introdu ed in Se tions 4.4 and 4.5, and informs the sour e and destination

about its de ision using one bit of feedba k information.

CSI for Mixed Rate Transmission

For this mode of transmission, we assume that the relay has full CSIT, i.e., it knows

hR(i), and an therefore adjust its transmission rate and transmit power PR(i) to

avoid outages on the R-D link. However, the sour e still does not have CSIT and

therefore has to transmit with xed rate S0 and xed power PS as it does not know

hS(i). Similar to the xed rate ase, the relay an determine the outage state of the

S-R link based on S0, PS, σ2nR, and hS(i). However, dierent from the xed rate ase,

in the mixed rate transmission mode, the relay also has to estimate hR(i), e.g., based

on pilot symbols emitted by the destination. Based on the outage state of the S-R link

and hR(i), and on the statisti s of both links, the relay sele ts the transmitting node

a ording to the adaptive re eption-transmission proto ols proposed in Se tion 4.6,

and informs the sour e and destination about its de ision using one bit of feedba k

information.

For both modes of transmission, the relay knows the outage state of the S-R and

the R-D links. Hen e, if the relay is sele ted for transmission but the R-D link is in

outage, the relay remains silent and an outage event o urs. Whereas, if the sour e is

8

We note that in onventional relaying, feedba k of few bits of information does not improve the

outage performan e of the two-hop HD relay network. On the ontrary, the two bits of feedba k

(one from D to R and the other from R to S or D), along with the buer at the relay have a pivotal

role in the proposed proto ol.

92


sele ted for transmission and the S-R link is in outage, the relay does not transmit a

feedba k signal so that the sour e remains silent, i.e., again an outage event o urs.

On e the de ision regarding the transmitting node has been made, and the relay has

informed the transmitting node (sour e or relay) a ordingly, transmission in time

slot i begins.

Remark 4.1. We note that all the derivations and results for mixed rate transmission

in this hapter also hold for the ase when the sour e transmits with an adaptive rate

and the relay transmits with a xed rate. The only dieren e is that in the derived

results, the subs ripts S and R should swit h positons.

Remark 4.2. We note that xed rate transmission requires only two emissions of pi-

lot symbols (by sour e and relay). In ontrast, mixed rate transmission requires three

emissions of pilot symbols (by sour e, relay, and destination). Thus, the CSI re-

quirements and feedba k overhead of the buer-aided adaptive re eption-transmission

proto ols proposed in this hapter are similar to those of existing relaying proto ols,

su h as the opportunisti proto ol proposed in [14. Namely, the proto ol proposed in

[14 requires the relays to a quire the instantaneous CSI of the S-R and R-D links.

Furthermore, a few bits of information are fed ba k from the relays to both the sour e

and the destination.

4.3.2 Transmission Rates and Queue Dynami s

In the following, we dis uss the transmission rates and the state of the buer when

sour e and relay transmit in a given time slot i for both xed and mixed rate trans-

mission.

If the sour e is sele ted for transmission in time slot i and an outage does not

o ur, i.e., log2(

1 + s(i))

≥ S0, it transmits one odeword with rate SSR(i) = S0.

93


Hen e, the relay re eives S0 bits/symb from the sour e and appends them to the

queue in its buer. The number of bits/symb in the buer of the relay at the end of

the i-th time slot is denoted by Q(i) and given by

Q(i) = Q(i− 1) + S0. (4.1)

If the sour e is sele ted for transmission but the S-R link is in outage, i.e., log2(

1 +

s(i))

< S0, the sour e remains silent, i.e., SSR(i) = 0, and the queue in the buer

remains un hanged, i.e., Q(i) = Q(i− 1).

For xed rate transmission, if the relay is sele ted for transmission in time slot i

and transmits one odeword with rate R0, an outage does not o ur if log2(

1+r(i))

≥

R0. In this ase, the number of bits/symb transmitted by the relay is given by

RRD(i) = minR0, Q(i− 1), (4.2)

where we take into a ount that the maximum number of bits/symb that an be

send by the relay is limited by the number of bits/symb in the buer. The number

of bits/symb remaining in the buer at the end of time slot i is given by

Q(i) = Q(i− 1)− RRD(i), (4.3)

whi h is always non-negative be ause of (4.2). If the relay is sele ted for transmission

in time slot i but an outage o urs, i.e., log2(

1 + r(i))

< R0, the relay remains

silent, i.e., RRD(i) = 0, while the queue in the buer remains un hanged, i.e., Q(i) =

Q(i− 1).

For mixed rate transmission, the relay is able to adapt its rate to the apa ity

of the R-D hannel, log2(1 + r(i)), and outages are avoided. If the relay is sele ted

94


for transmission in time slot i, the number of bits/symb transmitted by the relay is

given by

RRD(i) = minlog2(1 + r(i)), Q(i− 1). (4.4)

In this ase, the number of bits/symb remaining in the buer at the end of time slot

i is still given by (4.3) where RRD(i) is now given by (4.4).

Furthermore, be ause of the HD onstraint, for both xed and mixed rate trans-

mission, we have RRD(i) = 0 and SSR(i) = 0 if sour e and relay are sele ted for

transmission in time slot i, respe tively.

4.3.3 Link Outages and Indi ator Variables

In order to model the outages on the S-R and R-D links, we introdu e the binary

link outage indi ator variables OS(i) ∈ 0, 1 and OR(i) ∈ 0, 1 dened as

OS(i) ,

0 if s(i) < 2S0 − 1

1 if s(i) ≥ 2S0 − 1(4.5)

and

OR(i) ,

0 if r(i) < 2R0 − 1

1 if r(i) ≥ 2R0 − 1, (4.6)

respe tively. In other words, OS(i) = 0 indi ates that for transmission with rate S0,

the S-R link is in outage, i.e., log2(1 + s(i)) < S0, and OS(i) = 1 indi ates that the

transmission over the S-R hannel will be su essful. Similarly, OR(i) = 0 indi ates

that for transmission with rate R0, the R-D link is in outage, i.e., log2(1+r(i)) < R0,

95


and OR(i) = 1 means that an outage will not o ur. Furthermore, we denote the

outage probabilities of the S-R and R-D hannels as PS and PR, respe tively. These

probabilities are dened as

PS , limN→∞

1

N

N∑

i=1

(

1− OS(i)) (a)= Pr

s(i) < 2S0 − 1

(4.7)

and

PR , limN→∞

1

N

N∑

i=1

(

1− OR(i)) (a)= Pr

r(i) < 2R0 − 1

, (4.8)

respe tively, where (a) follows from the assumed ergodi ity of the fading.

4.3.4 Performan e Metri s

In this hapter, we adopt the throughput and the outage probability as performan e

metri s.

Assuming the sour e has always data to transmit, for both xed and mixed rate

transmission, the average number of bits/symb that arrive at the destination per time

slot is given by

τ = limN→∞

1

N

N∑

i=1

RRD(i), (4.9)

i.e., τ is the throughput of the onsidered ommuni ation system.

The outage probability is dened in the literature as the probability that the

instantaneous hannel apa ity is unable to support some predetermined xed trans-

mission rate. In the onsidered system, an outage does not ause information loss

sin e the relay knows in advan e whether or not the sele ted link an support the

hosen transmission rate and data is only transmitted if the orresponding link is not

96


in outage. Nevertheless, outages still ae t the a hievable throughput negatively. In

this ase, the outage probability an be interpreted as the fra tion of the throughput

lost due to outages. Thus, denoting the maximum throughput of a system in the

absen e of outages by τ0 and the throughput in the presen e of outages by τ , the

outage probability, Fout, an be expressed as

Fout = 1−τ

τo. (4.10)

Note that maximizing the throughput is equivalent to minimizing the outage proba-

bility.

4.3.5 Performan e Ben hmarks for Fixed Rate Transmission

For xed rate transmission, two onventional relaying s hemes serve as performan e

ben hmarks for the proposed buer-aided relaying s heme with adaptive re eption-

transmission. In ontrast to the proposed s heme, the ben hmark s hemes employ a

predetermined s hedule for when sour e and relay transmit whi h is independent of

the instantaneous link SNRs.

In the rst s heme, referred to as Conventional Relaying 1, the sour e transmits

in the rst ξN time slots, where 0 < ξ < 1 and ea h odeword spans one time slot.

The relay tries to de ode these odewords and, if the de oding is su essful, it stores

the orresponding information in its buer. In the following (1− ξ)N time slots, the

relay transmits the stored information to the destination, transmitting one odeword

per time slot. Assuming that for the ben hmark s hemes sour e and relay transmit

odewords having the same rate, i.e., S0 = R0, the throughput of Conventional

97


Relaying 1 is obtained as

τfixedconv,1 = limN→∞

1

Nmin

ξN∑

i=1

R0OS(i) ,N∑

i=ξN+1

R0OR(i)

= R0min ξ(1− PS) , (1− ξ)(1− PR) . (4.11)

The throughput is maximized if ξ(1 − PS) = (1 − ξ)(1 − PR) holds or equivalently

if ξ = (1 − PR)/(2 − PS − PR). Inserting ξ into (4.11) we obtain the maximized

throughput as

τfixedconv,1 = R0(1− PS)(1− PR)

2− PS − PR

. (4.12)

The maximum throughput in the absen e of outages is τ0 = R0/2. Hen e, using

(4.10), the orresponding outage probability is obtained as

F fixedout,conv,1 = 1− 2

(1− PS)(1− PR)

2− PS − PR. (4.13)

In the se ond s heme, referred to as Conventional Relaying 2, see [12, in the

rst time slot, the sour e transmits one odeword and the relay re eives and tries

to de ode the odeword. If the de oding is su essful, in the se ond time slot, the

relay retransmits the information to the destination, otherwise it remains silent. The

throughput of Conventional Relaying 2 is obtained as

τfixedconv,2 = limN→∞

1

N

N/2∑

i=1

R0OS(2i− 1)OR(2i) =R0

2(1− PS)(1− PR). (4.14)

Based on (4.10) the orresponding outage probability is given by

F fixedout,conv,2 = 1− (1− PS)(1− PR), (4.15)

98


whi h is identi al to the outage probability obtained in [12 using the standard deni-

tion for the outage probability. We note that τfixedconv,1 ≥ τfixedconv,2 (Ffixedout,conv,1 ≤ F fixed

out,conv,2)

always holds. However, in order for Conventional Relaying 1 to realize this gain, an

innite delay is required, whereas Conventional Relaying 2 requires a delay of only

one time slot.

For the spe ial ase of Rayleigh fading, we obtain from (4.7) and (4.8) PS =

1− e− 2R0−1

ΩSand PR = 1− e

− 2R0−1ΩR

, respe tively. The orresponding throughputs and

outage probabilities for Conventional Relaying 1 and 2 an be obtained by applying

these results in (4.12)-(4.15). In parti ular, in the high SNR regime, when γS = γR =

γ → ∞, we obtain τfixedconv,1 → R0/2, τfixedconv,2 → R0/2, and

F fixedout,conv,1 →

2R0 − 1

2

ΩS + ΩR

ΩSΩR

1

γ, as γ → ∞, (4.16)

F fixedout,conv,2 → (2R0 − 1)

ΩS + ΩR

ΩSΩR

1

γ, as γ → ∞. (4.17)

Hen e, for xed rate transmission, the diversity gain of Conventional Relaying 1 and

2 is one as expe ted.

Note that due to the xed s heduling of re eption and transmission at the HD

relay for Conventional Relaying 1 and 2, feedba k of few bits of information from D

to R and R to S or D, annot improve the outage peforman e. On ontrary, as will be

shown, in buer-aided relaying with adaptive re eption-transmission, the feedba k of

few bits of information is essential and signi antly improves the outage peforman e.

4.3.6 Performan e Ben hmarks for Mixed Rate Transmission

We also provide two performan e ben hmarks with a priori xed re eption-transmission

s hedule for mixed rate transmission. The two ben hmark proto ols are analogous to

99


the orresponding proto ols in the xed rate ase. Thus, for Conventional Relaying

1, the sour e transmits in the rst ξN time slots with xed rate S0 and the relay

transmits in the remaining (1− ξ)N time slots with rate R(i) = log2(1+ r(i)). Thus,

the throughput is given by

τmixedconv,1= lim

N→∞

1

Nmin

ξN∑

i=1

S0OS(i),

N∑

i=ξN+1

log2(1 + r(i))

= min ξ(1−PS)S0, (1− ξ)Elog2(1 + r(i)). (4.18)

The throughput is maximized if ξ satises

ξS0(1− PS) = (1− ξ)Elog2(1 + r(i)) . (4.19)

From (4.19), we obtain ξ as

ξ =Elog2(1 + r(i))

S0(1− PS) + Elog2(1 + r(i)). (4.20)

Inserting ξ into (4.18) leads to the throughput of mixed rate transmission under the

Conventional Relaying 1 proto ol

τmixedconv,1 =

S0(1− PS)Elog2(1 + r(i))

S0(1− PS) + Elog2(1 + r(i)). (4.21)

Assuming Rayleigh fading links Elog2(1 + r(i)) is obtained as

Elog2(1 + r(i)) =e1/ΩR

ln(2)E1

(

1

ΩR

)

(4.22)

100


for xed transmit powers. If adaptive power allo ation is employed, Elog2(1+r(i))

be omes

Elog2(1 + r(i)) =1

ln(2)E1

(

λcΩR

)

, (4.23)

where λc is found from the power onstraint

(1− PS)γS +

∫ ∞

λc

(

1

λc−

1

hR

)

fhR(hR)dhR = 2Γ. (4.24)

Here, Γ denotes the average transmit power. In the high SNR regime, where γS =

γR = γ → ∞, Elog2(1 + r(i)) ≫ S0(1−PS) holds. Thus, the throughput in (4.21)

onverges to

τmixedconv,1 → S0 , as γ → ∞ , (4.25)

whi h leads to the interesting on lusion that mixed rate transmission a hieves a

multiplexing rate of one even if suboptimal onventional relaying is used.

For Conventional Relaying 2, the performan e of mixed rate transmission is iden-

ti al to that of xed rate transmission. Sin e the relay does not employ a buer

for Conventional Relaying 2, even with mixed rate transmission, the relay an only

transmit su essfully all of the re eived information if S0 ≤ log2(1 + r(i)) and has to

remain silent otherwise.

101


4.4 Optimal Buer-Aided Relaying for Fixed Rate

Transmission Without Delay Constraints

In this se tion, we investigate buer-aided relaying with adaptive re eption-transmis-

sion for xed rate transmission without delay onstraints, i.e., the transmission rates

of the sour e and the relay are xed. We derive the optimal adaptive re eption-

transmission proto ol and analyze the orresponding throughput and outage prob-

ability. The obtained results onstitute performan e upper bounds for xed rate

transmission with delay onstraints, whi h will be onsidered in Se tion 4.5.

4.4.1 Problem Formulation

In order to model the re eption and transmission of the relay, again we introdu e

the binary adaptive re eption-transmission variable di ∈ 0, 1. Here, again di = 1

indi ates that the R-D link is sele ted for transmission in time slot i, i.e., the relay

transmits and the destination re eives. Similarly, if di = 0, the S-R link is sele ted

for transmission in time slot i, i.e., the sour e transmits and the relay re eives.

Based on the denitions of OS(i), OR(i), and di, the number of bits/symb sent

from the sour e to the relay and from the relay to the destination in time slot i an

be written in ompa t form as

SSR(i) = (1− di)OS(i)S0 (4.26)

and

RRD(i) = diOR(i)minR0, Q(i− 1), (4.27)

102


respe tively. Consequently, the throughput in (4.9) an be rewritten as

τ = limN→∞

1

N

N∑

i=1

diOR(i)minR0, Q(i− 1). (4.28)

In the following, we maximize the throughput by optimizing the adaptive re eption-

transmission variable di, whi h represents the only degree of freedom in the onsidered

problem. In parti ular, as already mentioned in Se tion 4.3, sin e both transmitting

nodes do not have the full CSI of their respe tive transmit hannels, power allo ation

is not possible and we assume xed transmit powers PS(i) = PS and PR(i) = PR, ∀i.

4.4.2 Throughput Maximization

Let us rst dene the average arrival rate of bits/symb per slot into the queue of the

buer, A, and the average departure rate of bits/symb per slot out of the queue of

the buer, D, as [83

A , limN→∞

1

N

N∑

i=1

(1− di)OS(i)S0 (4.29)

and

D , limN→∞

1

N

N∑

i=1

diOR(i)minR0, Q(i− 1), (4.30)

respe tively. We note that the departure rate of the queue is equal to the throughput.

The queue is said to be an absorbing queue if A > D = τ , in whi h ase a fra tion

of the information sent by the sour e is trapped in the unlimited size buer and an

never be extra ted from it. The following theorem provides a useful ondition for the

optimal poli y whi h maximizes the throughput.

103


Theorem 4.1. The adaptive re eption-transmission poli y that maximizes the through-

put of the onsidered buer-aided relaying system an be found in the set of adaptive

re eption-transmission poli ies that satisfy

limN→∞

1

N

N∑

i=1

(1− di)OS(i)S0= limN→∞

1

N

N∑

i=1

diOR(i)R0. (4.31)

When (4.31) holds, the queue is non-absorbing but is at the edge of absorption, i.e.,

a small in rease of the arrival rate will lead to an absorbing queue. Moreover, when

(4.31) holds the throughput is given by

τ = limN→∞

1

N

N∑

i=1

(1− di)OS(i)S0= limN→∞

1

N

N∑

i=1

diOR(i)R0. (4.32)

Proof. Please refer to Appendix C.1.

Remark 4.3. A queue that meets ondition (4.31) is rate-stable sin e there is no loss

of information in the unlimited buer, i.e., the information that goes in the buer

eventually leaves the buer without any loss.

Remark 4.4. The min(·) fun tion in (4.28) is absent in the throughput in (4.31),

whi h is ru ial for nding a tra table analyti al expression for the optimal adaptive

re eption-transmission poli y. In parti ular, as shown in Appendix C.1, ondition

(4.31) automati ally ensures that for N → ∞,

τ =1

N

N∑

i=1

diOR(i)minR0, Q(i− 1) =1

N

N∑

i=1

diOR(i)R0

is valid, i.e., the impa t of event R0 > Q(i−1), i = 1, . . . , N , is negligible. Hen e, for

the optimal adaptive re eption-transmission poli y, the queue is non-absorbing but is

almost always lled to su h a level that the number of bits/symb in the queue ex eed

104


the number of bits/symb that an be transmitted over the R-D hannel, i.e., the buer

is pra ti ally always fully ba klogged. This result is intuitively pleasing. Namely, if

the queue would be rate-unstable, it would absorb bits/symb and the throughput ould

be improved by having the relay transmit more frequently. On the other hand, if the

queue was not (pra ti ally) fully ba klogged, the ee t of the event R0 > Q(i − 1)

would not be negligible and the system would loose out on transmission opportunities

be ause of an insu ient number of bits/symb in the buer.

A ording to Theorem 4.1, in order to maximize the throughput, we have to sear h

for the optimal poli y only in the set of poli ies that satisfy (4.31). Therefore, the

sear h for the optimal poli y an be formulated as an optimization problem, whi h

for N → ∞ has the following form

Maximize :di

1N

∑Ni=1 diOR(i)R0


∑Ni=1(1− di)OS(i)S0 =

1N

∑Ni=1 diOR(i)R0

C2 : di ∈ 0, 1, ∀i,

(4.33)

where onstraint C1 ensures that the sear h for the optimal poli y is ondu ted only

among those poli ies that satisfy (4.31) and C2 ensures that di ∈ 0, 1. We note

that C1 and C2 do not ex lude the ase that the relay is hosen for transmission if

R0 > Q(i − 1). However, as explained in Remark 4.4, C1 ensures that the inuen e

of event R0 > Q(i− 1) is negligible. Therefore, an additional onstraint dealing with

this event is not required.

Before we solve problem (4.33), we note that, as will be shown in the following,

the optimal adaptive re eption-transmission poli y may require a oin ip. For this

purpose, let C denote the out ome of a oin ip whi h takes values from the set

0, 1, and let us denote the probabilities of the out omes by PC = PrC = 1 and

105


PrC = 0 = 1−PC , respe tively. Now, we are ready to provide the solution of (4.33),

whi h onstitutes the optimal adaptive re eption-transmission poli y maximizing the

throughput for xed rate transmission. This is onveyed in the following theorem.

Theorem 4.2. For the optimal adaptive re eption-transmission poli y maximizing the

throughput of the onsidered buer-aided relaying system for xed rate transmission,

three mutually ex lusive ases an be distinguished depending on the values of PS and

PR:

Case 1:

PS ≤S0

S0 +R0(1− PR)AND PR ≤

R0

R0 + S0(1− PS).

(4.34)

In this ase, the optimal adaptive re eption-transmission poli y is given by

di =

0 if OS(i) = 1 AND OR(i) = 0

1 if OS(i) = 0 AND OR(i) = 1

0 if OS(i) = 1 AND OR(i) = 1 AND C = 0


ε if OS(i) = 0 AND OR(i) = 0

(4.35)

where ε an be set to 0 or 1 as neither the sour e nor the relay will transmit be-

ause both links are in outage. On the other hand, if both links are not in outage,

i.e., OS(i) = 1 and OR(i) = 1, the oin ip de ides whi h node transmits and the

probability of C = 1 is given by

PC =S0(1− PS)− (1− PR)PSR0

(1− PS)(1− PR)(S0 +R0). (4.36)

106


Based on (4.35), the maximum throughput is obtained as

τ =S0R0

S0 +R0(1− PSPR). (4.37)

Case 2:

PR >R0

R0 + S0(1− PS)(4.38)

In this ase, the optimal adaptive re eption-transmission poli y is hara terized by

di =



1 if OS(i) = 0 AND OR(i) = 1

1 if OS(i) = 1 AND OR(i) = 1


(4.39)

The probability of out ome C = 1 of the oin ip is given by

PC =S0(1− PS)PR − (1− PR)R0

(1− PS)PRS0, (4.40)

and the maximum throughput an be obtained as

τ = R0(1− PR). (4.41)

Case 3:

PS >S0

S0 +R0(1− PR). (4.42)

107


In this ase, the adaptive re eption-transmission poli y that maximizes the throughput

is given by

di =

0 if OS(i) = 1 AND OR(i) = 0



0 if OS(i) = 1 AND OR(i) = 1


(4.43)

The probability of C = 1 is given by

PC =S0(1− PS)

R0(1− PR)PS, (4.44)

and the maximum throughput is

τ = S0(1− PS). (4.45)


Remark 4.5. We note that in the se ond line of (4.39), we set di = 1 although the

R-D link is in outage (OR(i) = 0) while the S-R link is not in outage (OS(i) = 1).

In other words, in this ase, neither node transmits although the sour e node ould

su essfully transmit. However, if the sour e node transmitted in this situation, the

queue at the relay would be ome an absorbing queue. Similarly, in the se ond line

of (4.43), we set di = 0 although the S-R link is in outage. Again, neither node

transmits in order to ensure that ondition (4.31) is met. However, in this ase, the

exa t same throughput as in (4.45) an be a hieved with a simpler and more pra ti al

adaptive re eption-transmission poli y than that in (4.43). This is addressed in the

108


following lemma.

Lemma 4.1. The throughput a hieved by the adaptive re eption-transmission pol-

i y in (4.43) an also be a hieved with the following simpler adaptive re eption-

transmission poli y.

If

PS >S0

S0 +R0(1− PR), (4.46)

an adaptive re eption-transmission poli y maximizing the throughput is given by

di =

0 if OS(i) = 1

1 if OS(i) = 0, (4.47)

and the maximum throughput is

τ = S0(1− PS). (4.48)

Proof. The poli y given by (4.43) has the same average arrival rate as poli y (4.47)

sin e for both poli ies the sour e always transmits when OS(i) = 1. Therefore, sin e

for both poli ies the queue is non-absorbing, by the law of onservation of ow, their

throughputs are identi al to their arrival rates. Thus, both poli ies a hieve identi al

throughputs.

Remark 4.6. Note that when PR > R0/(R0+S0(1−PS)) (PS > S0/(S0+R0(1−PR)))

holds, the throughput is given by (4.41) ((4.45)), whi h is identi al to the maximal

throughput that an be obtained in a point-to-point ommuni ation between relay and

destination (sour e and relay). Therefore, when PR > R0/(R0 + S0(1 − PS)) (PS >

109


S0/(S0 + R0(1 − PR))) holds, as far as the a hievable throughput is on erned, the

two-hop HD relay hannel is equivalent to the point-to-point R-D (S-R) hannel.

For omparison, we also provide the maximum throughput in the absen e of

outages τ0. The throughput in the absen e of outages, τ0, an be obtained by setting

OS(i) = OR(i) = 1, ∀i, whi h is equivalent to setting PS = PR = 0 in Theorem

4.2. Then, Case 1 in Theorem 4.2 always holds and the optimal adaptive re eption-

transmission poli y is

di =

0 if C = 0

1 if C = 1(4.49)

where the probability of C = 1 is given by

PC =S0

S0 +R0

. (4.50)

Based on (4.49), the maximum throughput in the absen e of outages is

τ0 =S0R0

S0 +R0

. (4.51)

The throughput loss aused by outages an be observed by omparing (4.37), (4.41),

and (4.45) with (4.51).

We now provide the outage probability of the proposed buer-aided relaying

s heme with adaptive re eption-transmission.

Lemma 4.2. The outage probability of the system onsidered in Theorem 4.2 is given

110


by

Fout =

PR − (1− PR)R0/S0 , if PR >R0

R0+S0(1−PS)

PS − (1− PS)S0/R0 , if PS >S0

S0+R0(1−PR)

PSPR , otherwise.

(4.52)


Remark 4.7. In the proof of Lemma 4.2 given in Appendix C.3, it is shown that an

outage event happens when neither the sour e nor the relay transmit in a time slot,

i.e., the number of silent slots is identi al to the number of outage events.

In the high SNR regime, when the outage probabilities of both involved links

are small, the expressions for the throughput and the outage probability an be

simplied to obtain further insight into the performan e of buer-aided relaying.

This is addressed in the following lemma.

Lemma 4.3. In the high SNR regime, γS = γR = γ → ∞, the throughput and

the outage probability of the buer-aided relaying system onsidered in Theorem 4.2

onverge to

τ → τ0 =S0R0

S0 +R0, as γ → ∞ , (4.53)

Fout = PSPR. (4.54)

Proof. In the high SNR regime, we have PS → 0 and PR → 0. Thus, ondition (4.34)

always holds and therefore Fout is given by (4.54). Furthermore, as PS → 0 and

PR → 0, (4.37) simplies to (4.53).

111


4.4.3 Performan e in Rayleigh Fading

For on reteness, we assume in this subse tion that both links of the onsidered two-

hop HD relay network are Rayleigh fading. We examine the diversity order and the

diversity-multiplexing trade-o.

Lemma 4.4. For the spe ial ase of Rayleigh fading links, the buer-aided relaying

system onsidered in Theorem 4.2 a hieves a diversity gain of two, i.e., in the high

SNR regime, when γS = γR = γ → ∞, the outage probability, Fout, de ays on a

log-log s ale with slope −2 as a fun tion of the transmit SNR γ, and is given by

Fout →2S0 − 1

ΩS

2R0 − 1

ΩR

1

γ2, as γ → ∞. (4.55)

Furthermore, the onsidered buer-aided relaying system a hieves a diversity-multiplexing

trade-o, DM(r), of

DM(r) = 2(1− 2r), 0 < r < 1/2. (4.56)


Remark 4.8. We re all that, for xed rate transmission, both onsidered onventional

relaying s hemes without adaptive re eption-transmission a hieved only a diversity

gain of one, f. (4.16), (4.17), despite the fa t that Conventional Relaying 1 also

has an unlimited buer and entails an innite delay. Thus, we expe t large gains in

terms of outage probability of the proposed buer-aided relaying proto ol with adaptive

re eption-transmission ompared to onventional relaying.

The performan e of the onsidered system an be further improved by optimizing

the transmission rates R0 and S0 based on the hannel statisti s. For Rayleigh fading

112


with given ΩS and ΩR, we an optimize R0 and S0 for minimization of the outage

probability. This is addressed in the following lemma for high SNR.

Lemma 4.5. Assuming Rayleigh fading, the optimal transmission rates S0 and R0

that minimize the outage probability in the high SNR regime, while maintaining a

throughput of τ0, are given by R0 = S0 = 2τ0.

Proof. The throughput in the high SNR regime is given by (4.53), whi h an be

rewritten as R0 = S0τ0/(S0 − τ0). Inserting this into the asymptoti expression for

Fout in (4.55) and minimizing it with respe t to S0 yields S0 = R0 = 2τ0.

Remark 4.9. For Rayleigh fading, although in the low SNR regime, the optimal S0

and R0 an be nonidenti al, in the high SNR regime, independent of the values of ΩS

and ΩR, the minimum Fout is obtained for identi al transmission rates for both links.

Furthermore, in the high SNR regime, when γS = γR → ∞, for S0 = R0, the oin

ip probability PC onverges to PC = PrC = 1 = PrC = 0 → 1/2.

4.5 Buer-Aided Relaying for Fixed Rate

Transmission With Delay Constraints

The proto ol proposed in Se tion 4.4 does not impose any onstraint on the delay

that a transmitted bit of information experien es. However, in pra ti e, most om-

muni ation servi es require delay onstraints. Therefore, in this se tion, we modify

the buer-aided relaying proto ol derived in the previous se tion to a ount for on-

straints on the average delay. Furthermore, we analyze the ee t of the applied mod-

i ation on the throughput and the outage probability. For simpli ity, throughout

this se tion, we assume S0 = R0. We note that the adaptive re eption-transmission

113


proto ols proposed in Se tion 4.5.2 are also appli able to the ase of S0 6= R0. How-

ever, sin e for S0 6= R0 the odewords transmitted by the sour e do not ontain the

same number of bits/symb as the odewords transmitted by the relay, the Markov

hain based throughput and delay analyses in Se tions 4.5.3 and 4.5.4 would be more

ompli ated. On the other hand, sin e we found in the previous se tion that, for high

SNR, identi al sour e and relay transmission rates minimize the outage probability,

we avoid these additional ompli ations here and on entrate on the ase S0 = R0.

Furthermore, to fa ilitate our analysis, throughout this se tion, we assume temporally

un orrelated fading.

4.5.1 Average Delay

We dene the delay of a bit as the time interval from its transmission by the sour e to

its re eption at the destination. Thus, assuming that the propagation delays in the

S-R and R-D links are negligible, the delay of a bit is identi al to the time that the

bit is held in the buer. As a onsequen e, we an use Little's law [85 and express

the average delay in number of time slots as

ET =EQ

A, (4.57)

where EQ = limN→∞

∑Ni=1Q(i)/N is the average length of the queue in the buer

of the relay and A is the arrival rate into the queue as dened in (4.29). From (4.57),

we observe that the delay an be ontrolled via the queue size.

114


4.5.2 Adaptive Re eption-Transmission Proto ol for Delay

Limited Transmission

As mentioned before, we modify the optimal adaptive re eption-transmission proto ol

derived in Se tion 4.4 in order to limit the average delay. However, depending on

the targeted average delay, somewhat dierent modi ations are ne essary, sin e it

is not possible to a hieve any desired delay with one proto ol. Hen e, three dierent

adaptive re eption-transmission proto ols are introdu ed in the following proposition.

Proposition 4.1. For xed rate transmission with delay onstraint, depending on

the targeted average delay ET and the outage probabilities PS and PR, we propose

the following adaptive re eption-transmission poli ies:

Case 1: If PR < 1/(2− PS) and the required delay ET satises

ET >1

1− PR (2− PS)+

2 (1− PS)

1− PSPR (2− PS), (4.58)

we propose the following adaptive re eption-transmission variable di to be used:

If Q(i− 1) ≤ R0 and OS(i) = 1, then di = 0,

otherwise di is given by (4.35). (4.59)

Case 2: If PR < 1/(2− PS) and the required delay ET satises

1

1− PR (2− PS)< ET ≤

1

1− PR (2− PS)+

2 (1− PS)

1− PSPR (2− PS), (4.60)

115



If Q(i− 1) = 0 and OS(i) = 1, then di = 0,


Case 3: If the required delay ET satises

1

1− PR

< ET ≤1

1− PR (2− PS), (4.62)


If Q(i− 1) = 0 and OS(i) = 1, then di = 0,


For ea h of the proposed adaptive re eption-transmission variables di, the required

delay an be met by adjusting the value of PC = PrC = 1, where the minimum and

maximum delays are a hieved with PC = 1 and PC = 0, respe tively.

Remark 4.10. The delay limits given by (4.58), (4.60), and (4.62) arise from

the analysis of the proposed proto ols with adaptive re eption-transmission variables

(4.59), (4.61), and (4.63), respe tively. We will investigate these delay limits in

Lemma 4.7 in Se tion 4.5.3 and the orresponding proof is provided in Appendix C.7.

Remark 4.11. We have not proposed a buer-aided relaying proto ol with adaptive

re eption-transmission that an satisfy a required delay smaller than 1/(1−PR). For

su h small delays, Conventional Relaying 2 without adaptive re eption-transmission

an be used. However, if retransmission of the outage odewords is taken into a ount,

then not even for Conventional Relaying 2 an a hieve a delay smaller than 1/(1 −

116


PR).

4.5.3 Throughput and Delay

In the following, we analyze the throughput, the average delay, and the probability

of having k pa kets in the queue for the modied adaptive re eption-transmission

proto ols proposed in Proposition 4.1 in the previous subse tion. The results are

summarized in the following theorem.

Theorem 4.3. Consider a buer-aided relaying system operating in temporally un-

orrelated blo k fading. Let sour e and relay transmit with rate R0, respe tively, and

let the buer size at the relay be limited to L pa kets ea h omprised of R0 bits/symb.

Assume that the relay drops newly re eived pa kets if the buer is full. Then, depend-

ing on the adopted adaptive re eption-transmission proto ol, the following ases an

be distinguished:

Case 1: If the adaptive re eption-transmission variable di is given by (4.59), the

probability of the buer having k pa kets in its queue, PrQ = kR0, is obtained as

PrQ = kR0 =

pL−1(2p+q−1)(PS−q)pL−1(2p(1−q)+q(2−q)−PS (2−PS))−(1−p−q)L−1(1−PS)2

, k = 0

pL−1(2p+q−1)(1−PS)pL−1(2p(1−q)+q(2−q)−PS (2−PS))−(1−p−q)L−1(1−PS)2

, k = 1

pL−k(2p+q−1)(1−PS )2(1−p−q)k−2

pL−1(2p(1−q)+q(2−q)−PS (2−PS))−(1−p−q)L−1(1−PS)2, k=2, ..., L

(4.64)

where p and q are given by

p = (1− PS)(1− PR)PC + PS(1− PR) ; q = PSPR. (4.65)

117


Furthermore, the average queue length, EQ, the average delay, ET, and through-

put, τ , are given by

EQ = R01− PS

2p+ q − 1

pL−1 ((2p+ q)2 − p− q − PS(3p+ q − 1))− (1− PS)(1− p− q)L−1(L(2p+ q − 1) + p)

pL−1(2p(1− q) + (2− q)q − (2− PS)PS)− (1− PS)2(1− p− q)L−1

(4.66)

ET =1

2p+ q − 1

pL−1 ((2p+ q)2 − PS(3p+ q − 1)− p− q)− (1− PS)(1− p− q)L−1(L(2p+ q − 1) + p)

pL−1(PS(p+ q − 1)− q(2p+ q) + p+ q)− (1− PS)p(1− p− q)L−1

(4.67)

τ = (1− PS)

1− PS)p(1− p− q)L−1 + pL−1(PS(1− p− q) + q(2p+ q)− p− q)

pL−1((2− PS)PS − 2p(1− q)− (2− q)q)(1− PS)2(1− p− q)L−1. (4.68)

Case 2: If adaptive re eption-transmission variable di is given by either (4.61) or

(4.63), the probability of the buer having k pa kets in its queue, PrQ = kR0, is

given by

PrQ = kR0 =

pL(2p+q−1)pL(2p+q−PS)−(1−PS)(1−p−q)L

, k = 0

(1−PS)(2p+q−1)pL−k(1−p−q)k−1

pL(2p+q−PS)−(1−PS)(1−p−q)L, k = 1, ..., L

(4.69)

where, if adaptive re eption-transmission variable di is given by (4.61), p and q are

118


given by (4.65), while if adaptive re eption-transmission variable di is given by (4.63),

p and q are given by

p = 1− PR and q = PSPR + (1− PS)PRPC . (4.70)

Furthermore, the average queue length, EQ, the average delay, ET, and through-

put, τ , are given by

EQ = R01− PS

2p+ q − 1

pL+1 − (1− p− q)L(L(2p+ q − 1) + p)

pL(2p+ q − PS)− (1− PS)(1− p− q)L, (4.71)

ET =1

2p+ q − 1

1

p

pL+1 − (1− p− q)L(L(2p+ q − 1) + p)

pL − (1− p− q)L, (4.72)

τ = R0(1− PS)ppL − (1− p− q)L

pL(2p+ q − PS)− (1− PS)(1− p− q)L. (4.73)


Due to their omplexity, the equations in Theorem 4.3 do not provide mu h insight

into the performan e of the onsidered system. To over ome this problem, we onsider

the ase L≫ 1, whi h leads to signi ant simpli ations and design insight. This is

addressed in the following lemma.

Lemma 4.6. For the system onsidered in Theorem 4.3, assume that L→ ∞. In this

ase, for a system with adaptive re eption-transmission variable di given by (4.59),

(4.61), or (4.63) to be able to a hieve a xed delay, ET, that does not grow with

L as L → ∞, the ondition 2p + q − 1 > 0 must hold. If 2p + q − 1 > 0 holds, the

following simpli ations an be made for ea h of the onsidered adaptive re eption-

transmission variables:

119



probability of the buer being empty, the average delay, ET, and throughput, τ ,

simplify to

PrQ = 0 = PS2PC(1− PR)(1− PS) + (2− PR)PS − 1

2PC(1− PS)(1− PSPR) + P 2S(1− PR)

(4.74)

ET =1

2PC(1− PR)(1− PS)− PRPS + 2PS − 1

+2PC(1− PS)

P 2S(PC(2PR − 1)− PR + 1)− 2PCPRPS + PC

(4.75)

τ = R0(1− PS)P 2S(PC(2PR − 1)− PR + 1)− 2PCPRPS + PC

2PC(1− PS)(1− PSPR) + (1− PR)P 2S

. (4.76)


probability of the buer being empty, the average delay, ET, and throughput, τ ,

simplify to

PrQ=0=2PC(1− PR)(1− PS) + PS(2− PR)− 1

(1− PR)(PS + 2PC(1− PS))(4.77)

ET =1

2PC(1− PR)(1− PS)− PRPS + 2PS − 1(4.78)

τ = R0(1− PS)PC(1− PS) + PS

2PC(1− PS) + PS

. (4.79)


probability of the buer being empty, the average delay, ET, and the throughput,

120


τ , simplify to

PrQ = 0 =1− PR(2− PS − PC(1− PS))

2− PS − PR(2− PS − PC(1− PS))(4.80)

ET =1

1− PR(2− PS − PC(1− PS))(4.81)

τ = R01 + PSPR − PR − PS

2− PS − PR(2− PS − PC(1− PS)). (4.82)

For ea h of the onsidered ases, the probability PC an be used to adjust the desired

average delay ET in (4.75), (4.78), and (4.81).


As already mentioned in Proposition 4.1, it is not possible to a hieve any desired

average delay with the proposed buer-aided adaptive re eption-transmission proto-

ols. The limits of the a hievable average delay for ea h of the proposed adaptive

re eption-transmission variables di in Proposition 4.1 are provided in the following

lemma.

Lemma 4.7. Depending on the adopted adaptive re eption-transmission variable di

the following ases an be distinguished for the average delay:

Case 1: If the adaptive re eption-transmission variable di is given by (4.59), then

if PR < 1/(2− PS) and PS < 1/(2− PR), the system an a hieve any average delay

ET ≥ Tmin,1, where Tmin,1 is given by

Tmin,1 =1

1− PR (2− PS)+

2 (1− PS)

1− PSPR (2− PS). (4.83)

On the other hand, if PR < 1/(2−PS) and PS > 1/(2−PR), the system an a hieve

121


any average delay in the interval Tmin,1 ≤ ET ≤ Tmax,1, where Tmax,1 is given by

Tmax,1 =1

PS(2− PR)− 1. (4.84)


if PR < 1/(2− PS) and PS < 1/(2− PR), the system an a hieve any average delay

ET ≥ Tmin,2, where Tmin,2 is given by

Tmin,2 =1

1− PR(2− PS). (4.85)

However, if PR < 1/(2 − PS) and PS > 1/(2 − PR), the system an a hieve any

average delay Tmin,2 ≤ ET ≤ Tmax,2, where Tmax,2 = Tmax,1.


if PR > 1/(2− PS), the system an a hieve any average delay ET ≥ Tmin,3, where

Tmin,3 is given by

Tmin,3 =1

1− PR. (4.86)

On the other hand, if PR < 1/(2 − PS), the system an a hieve any average delay

Tmin,3 ≤ ET ≤ Tmax,3, where Tmax,3 = Tmin,2.


In the following, we investigate the outage probability of the proposed buer-aided

relaying proto ol for delay onstrained xed rate transmission.

4.5.4 Outage Probability

The following theorem spe ies the outage probability.

122


Theorem 4.4. For the onsidered buer-aided relaying proto ol in Proposition 4.1, if

the required delay an be satised by using the adaptive re eption-transmission variable

di in either (4.59) or (4.61), the outage probability is given by

Fout = PSPrQ = 0+ PSPR

(

1− PrQ = 0 − PrQ = LR0)

+(

(1−PS)PR + (1−PSPR)(1−PC))

PrQ = LR0,

(4.87)

where if di is given by (4.59), PrQ = 0 and PrQ = LR0 are given by (4.64) with

p and q given by (4.65). On the other hand, if di is given by (4.61), PrQ = 0 and

PrQ = LR0 are given by (4.69) with p and q given by (4.65).

If the required delay is satised by using the adaptive re eption-transmission vari-

able di given by (4.63), then the outage probability is given by


(

1− PrQ = 0 − PrQ = LR0)

+ (1− PS)PR(1− PC)PrQ = LR0, (4.88)

where PrQ = 0 and PrQ = LR0 are given by (4.61) with p and q given by (4.70).


The expressions for Fout in Theorem 4.4 are valid for general L. However, sig-

ni ant simpli ations are possible if L ≫ 1. This is addressed in the following

lemma.

Lemma 4.8. When L → ∞, the outage probability given by (4.87) and (4.88) sim-

123


plies to


(

1− PrQ = 0)

, (4.89)

where PrQ = 0 is given by (4.74), (4.77), and (4.80) if di is given by (4.59), (4.61),

and (4.63), respe tively.

Proof. Eq. (4.89) is obtained by letting PrQ = LR0 → 0 when L → ∞ in (4.87)

and (4.88).

The expression for the outage probability in (4.89) an be further simplied in

the high SNR regime, whi h provides insight into the a hievable diversity gain. This

is summarized in the following theorem.

Theorem 4.5. In the high SNR regime, when γS = γR = γ → ∞, depending on the

required delay that the system has to satisfy, two ases an be distinguished:

Case 1: If 1 < ET ≤ 3, the outage probability asymptoti ally onverges to

Fout →PS

ET+ 1, as γ → ∞. (4.90)

Case 2: If ET > 3, the outage probability asymptoti ally onverges to

Fout →P 2S

ET − 1+ PSPR, as γ → ∞. (4.91)

Therefore, assuming Rayleigh fading, the onsidered system a hieves a diversity gain

of two if and only if ET > 3.


A ording to Theorem 4.5, for Rayleigh fading, a diversity gain of two an be

124


also a hieved for delay onstrained transmission, whi h underlines the appeal of

buer-aided relaying with adaptive re eption-transmission ompared to onventional

relaying, whi h only a hieves a diversity gain of one even in ase of innite delay

(Conventional Relaying 1).

4.6 Mixed Rate Transmission

In this se tion, we investigate buer-aided relaying proto ols with adaptive re eption-

transmission for mixed rate transmission. In parti ular, we assume that the sour e

does not have CSIT and transmits with xed rate S0 but the relay has full CSIT

and transmits with the maximum possible rate, RRD(i) = log2(1 + r(i)), that does

not ause an outage in the R-D hannel. For this s enario, we onsider rst delay

un onstrained transmission and derive the optimal adaptive re eption-transmission

buer-aided relaying proto ols with and without power allo ation. Subsequently, we

investigate the impa t of delay onstraints.

Before we pro eed, we note that for mixed rate transmission the throughput an

be expressed as

τ = limN→∞

1

N

N∑

i=1

diminlog2(1 + r(i)), Q(i− 1), (4.92)

where we used (4.4) and (4.9). For the derivation of the maximum throughput of

buer-aided relaying with adaptive re eption-transmission the following theorem is

useful.

Theorem 4.6. The adaptive re eption-transmission poli y that maximizes the through-

put of the onsidered buer-aided relaying system for mixed rate transmission an be

125


found in the set of adaptive re eption-transmission poli ies that satisfy

limN→∞

1

N

N∑

i=1

(1− di)OS(i)S0 = limN→∞

1

N

N∑

i=1

di log2(1 + r(i)) . (4.93)

Furthermore, for adaptive re eption-transmission poli ies within this set, the through-

put is given by the right (and left) hand side of (4.93).

Proof. A proof of this theorem an obtained by repla ing OR(i)R0 by log2(1 + r(i))

in the proof of Theorem 4.1 given in Appendix C.1.

Hen e, similar to xed rate transmission, for the set of poli ies onsidered in The-

orem 4.6, for N → ∞, the buer at the relay is pra ti ally always fully ba klogged.

Thus, the min(·) fun tion in (4.92) an be omitted and the throughput is given by

the right hand side of (4.93).

4.6.1 Optimal Adaptive Re eption-Transmission Proto ol

Without Power Allo ation

Sin e the relay has the instantaneous CSI of both links, it an also optimize its trans-

mit power. However, to get more insight, we rst onsider the ase where the relay

transmits with xed power. We note that power allo ation is not always desirable

as it requires highly linear power ampliers and thus, in reases the implementation

omplexity of the relay.

A ording to Theorem 4.6, the optimal adaptive re eption-transmission poli y

maximizing the throughput an be found in the set of poli ies that satisfy (4.93).

Therefore, the optimal poli y an be obtained from the following optimization prob-

126


lem

Maximize :di

1N

∑Ni=1 di log2

(

1 + r(i))


∑Ni=1(1− di)OS(i)S0 =

1N

∑Ni=1 di log2

(

1 + r(i))

C2 : di ∈ 0, 1, ∀i,

(4.94)

where N → ∞, onstraint C1 ensures that the sear h for the optimal poli y is

ondu ted only among the poli ies that satisfy (4.93), and C2 ensures that di ∈ 0, 1.

The solution of (4.94) leads to the following theorem.

Theorem 4.7. Let the PDFs of s(i) and r(i) be denoted by fs(s) and fr(r), re-

spe tively. Then, for the onsidered buer-aided relaying system in whi h the sour e

transmits with a xed rate S0 and xed power PS , and the relay transmits with an

adaptive rate RRD(i) = log2(1+r(i)) and xed power PR, two ases have to be distin-

guished for the optimal adaptive re eption-transmission variable di, whi h maximizes

the throughput:

Case 1: If

PS ≤S0

S0 +∫∞

0log2(1 + r)fr(r)dr

(4.95)

holds, then

di =

1 if OS(i) = 0

1 if OS(i) = 1 AND r(i) ≥ 2ρS0 − 1

0 if OS(i) = 1 AND r(i) < 2ρS0 − 1 ,

(4.96)

127


where ρ is a onstant whi h an be found as the solution of

S0(1− PS)

∫ 2ρS0−1

0

fr(r)dr = PS

∫ ∞

0

log2(1 + r)fr(r)dr + (1− PS)

∫ ∞

2ρS0−1

log2(1 + r)fr(r)dr .

(4.97)

In this ase, the maximum throughput is given by the right (and left) hand side of

(4.97).

Case 2: If (4.95) does not hold, then

di =

0 if OS(i) = 1

1 if OS(i) = 0 .(4.98)

In this ase, the maximum throughput is given by

τ = S0(1− PS) . (4.99)


We note that with mixed rate transmission the S-R link is used only if it is not

in outage, f. (4.96), (4.98). On the other hand, the R-D link is never in outage sin e

the transmission rate is adjusted to the hannel onditions. Furthermore, buer-

aided relaying with adaptive re eption-transmission has a larger throughput than

Conventional Relaying 1, and also a hieves a multiplexing gain of one.

To get more insight, we spe ialize the results derived thus far in this se tion to

Rayleigh fading links.

Lemma 4.9. For Rayleigh fading links, ondition (4.95) simplies to

PS = 1− exp

(

−2S0 − 1

ΩS

)

≤S0

S0 + e1/ΩRE1(1/ΩR)/ ln(2). (4.100)

128


Furthermore, (4.97) simplies to

S0 exp

(

−2S0 − 1

ΩS

)[

1− exp

(

−2ρS0 − 1

ΩR

)]

=e1/ΩR

ln(2)

[

(

1− exp

(

−2S0 − 1

ΩS

))

E1

(

1

ΩR

)

+ exp

(

−2S0 − 1

ΩS

)

E1

(

2ρS0

ΩR

)

]

+exp

(

−2ρS0 − 1

ΩR

)

exp

(

−2S0 − 1

ΩS

)

ρS0 , (4.101)

and the maximum throughput is given by the right (and left) hand side of (4.101). If

(4.100) does not hold, the throughput an be obtained by simplifying (4.99) to

τ = S0 exp

(

−2S0 − 1

ΩS

)

. (4.102)

Proof. Equations (4.100)-(4.102) are obtained by inserting the PDFs of s(i) and r(i)

into (4.95), (4.97), and (4.99), respe tively.

4.6.2 Optimal Adaptive Re eption-Transmission Poli y With

Power Allo ation

As mentioned before, sin e for mixed rate transmission the relay is assumed to have

the full CSI of both links, power allo ation an be applied to further improve perfor-

man e. In other words, the relay an adjust its transmit power PR(i) to the hannel

onditions while the sour e still transmits with xed power PS(i) = PS, ∀i. In the

following, for onvenien e, we will use the transmit SNRs without fading, γS and

γR(i), whi h may be viewed as normalized powers, as variables instead of the a tual

powers PS = γSσ2nR

and PR(i) = γR(i)σ2nD.

For the power allo ation ase, Theorem 4.6 is still appli able but it is onvenient

129


to rewrite the throughput as

τ = limN→∞

1

N

N∑

i=1

di log2(1 + γR(i)hR(i)). (4.103)

We note that (4.93) also applies to the ase of power allo ation. Furthermore, in

order to meet the average power onstraint Γ, the instantaneous (normalized) power

γR(i) and the xed (normalized) power γS have to satisfy the following ondition:

limN→∞

1

N

N∑

i=1

(1− di)OS(i)γS + limN→∞

1

N

N∑

i=1

diγR(i) ≤ Γ. (4.104)

Thus, the optimal adaptive re eption-transmission poli y for mixed rate transmission

is the solution of the following optimization problem:

Maximize :di,γR(i)

1N

∑Ni=1 di log2

(

1 + γR(i)hR(i))


∑Ni=1(1− di)OS(i)S0 =

1N

∑Ni=1 di log2

(

1 + γR(i)hR(i))

C2 : di ∈ 0, 1 , ∀i

C3 : 1N

∑Ni=1(1− di)OS(i)γS + 1

N

∑Ni=1 diγR(i) ≤ Γ,

(4.105)

where N → ∞, onstraints C1 and C3 ensure that the sear h for the optimal poli y

is ondu ted only among those poli ies that jointly satisfy (4.93) and the sour e-relay

power onstraint (4.104), respe tively, and C2 ensures that di ∈ 0, 1. The solution

of (4.105) is provided in the following theorem.

Theorem 4.8. Let the PDFs of hS(i) and hR(i) be denoted by fhS(hS) and fhR

(hR),

respe tively. Then, for the onsidered buer-aided relaying system where the sour e

130


transmits with a xed rate S0 and xed power γS and the relay transmits with adaptive

rate RRD(i) = log2(1 + r(i)) = log2(1 + γR(i)hR(i)) and adaptive power γR(i), two

ases have to be onsidered for the optimal adaptive re eption-transmission variable

di whi h maximizes the throughput:

Case 1: If

PS ≤S0

S0 +∫∞

λtlog2(hR/λt)fhR

(hR)dhR, (4.106)

holds, where λt is found as the solution to

PS

∫ ∞

λt

(

1

λt−

1

hR

)

fhR(hR)dhR + γS(1− PS) = Γ, (4.107)

then the optimal power γR(i) and adaptive re eption-transmission variable di whi h

maximize the throughput are given by

γR(i) = max

0,1

λ−

1

hR(i)

, (4.108)

and

di =

1 if OS(i) = 0 AND hR(i) ≥ λ

1 if OS(i) = 1 AND hR(i) ≥ λ AND ln(

hR(i)λ

)

+ λhR(i)

≥ ρS0 − λγS + 1

0 if OS(i) = 1 AND hR(i) < λ

0 if OS(i) = 1 AND hR(i) ≥ λ AND ln(

hR(i)λ

)

+ λhR(i)

< ρS0 − λγS + 1

ε if OS(i) = 0 AND hR(i) < λ ,

(4.109)

131


where ε is either 0 or 1 and has not impa t on the throughput. Constants ρ and λ are

hosen su h that onstraints C1 and C3 in (4.105) are satised with equality. These

two onstants an be found as the solution to the following system of equations

S0(1−PS)

∫ G

0

fhR(hR)dhR = PS

∫ ∞

λ

log2

(

hRλ

)

fhR(hR)dhR

+(1− PS)

∫ ∞

G

log2

(

hRλ

)

fhR(hR)dhR, (4.110)

PS

∫ ∞

λ

(

1

λ−

1

hR

)

fhR(hR)dhR + (1− PS)

∫ ∞

G

(

1

λ−

1

hR

)

fhR(hR)dhR

+γS(1− PS)

∫ G

0

fhR(hR)dhR = Γ, (4.111)

where the integral limit G is given by

G = −λ

W−eλγS−ρS0−1. (4.112)

Here, W· denotes the Lambert W -fun tion dened in [84, whi h is available as

built-in fun tion in software pa kages su h as Mathemati a. In this ase, the maxi-

mized throughput is given by the right (and left) hand side of (4.110).

Case 2: If (4.106) does not hold, the optimal power γR(i) and adaptive re eption-

transmission variable di are given by

γR(i) = max

0,1

λ−

1

hR(i)

, if OS(i) = 0; (4.113)

di =

0 if OS(i) = 1

1 if OS(i) = 0,(4.114)

132


where λ = λt is the solution to (4.107). In this ase, the maximum throughput is

given by

τ = S0(1− PS). (4.115)


Remark 4.12. Note that when onditions (4.95) and (4.106) do not hold, the through-

put with and without power allo ation is identi al, f. (4.99) and (4.115). If onditions

(4.95) and (4.106) do not hold, this means that the SNR in the S-R hannel is low,

whereas the SNR in the R-D hannel is high. In this ase, power allo ation is not

bene ial sin e the S-R hannel is the bottlene k link, whi h annot be improved by

power allo ation at the relay. Furthermore, the throughput in (4.99) and (4.115) is

identi al to the throughput of a point-to-point ommuni ation between the sour e and

the relay sin e the number of time slots required to transmit the information from

the relay to the destination be omes negligible. Therefore, in this ase, as far as the

a hievable throughput is on erned, the two-hop HD relay hannel is transformed into

a point-to-point hannel between the sour e and the relay.

In the following lemma, we on entrate on Rayleigh fading for illustration purpose.

Lemma 4.10. For Rayleigh fading hannels, PS is given by

PS = 1− exp

(

−2S0 − 1

γSΩS

)

.

Furthermore, ondition (4.106) simplies to

PS ≤S0

S0 + E1(λt/ΩR)/ ln(2), (4.116)

133


where λt is found as the solution to

PS

[

e−λt/ΩR

λt−

1

ΩR

E1

(

λtΩR

)

]

= Γ. (4.117)

For the ase where (4.116) holds, (4.110) and (4.111) simplify to

S0(1− PS)(

1− e−G/ΩR)

=1

ln(2)

[

PSE1

(

λ

ΩR

)

+(1− PS)

(

E1

(

G

ΩR

)

+ ln

(

G

λ

)

e−G/ΩR

)]

(4.118)

and

PS

[

e−λ/ΩR

λ−

1

ΩR

E1

(

λ

ΩR

)

]

+ (1− PS)

[

e−G/ΩR

λ

−1

ΩR

E1

(

G

ΩR

)]

+ γS(1− PS)(

1− e−G/ΩR)

= Γ, (4.119)

respe tively, where integral limit G is given by (4.112). The maximum throughput is

given by the right (and left) hand side of (4.118).

For the ase, where (4.116) does not hold, the throughput is given by τ = S0(1 −

PS).

Proof. Equations (4.116), (4.117), (4.118), and (4.119) are obtained by inserting the

PDFs of hS(i) and hR(i) into (4.106), (4.107), (4.110), and (4.111), respe tively.

Remark 4.13. Conditions (4.95) and (4.106) depend only on the long term fading

statisti s and not on the instantaneous fading states. Therefore, for xed ΩS and ΩR,

the optimal poli y for ondition (4.95) is given by either (4.96) or (4.98), but not by

both. Similarly, the optimal poli y for ondition (4.106) is given by either (4.109) or

(4.114), but not by both.

134


Remark 4.14. We note that equations (4.107), (4.110), (4.111), (4.117), (4.118),

and (4.119) an be solved using software pa kages su h as Mathemati a.

4.6.3 Mixed Rate Transmission With Delay Constraints

Now, we turn our attention to mixed rate transmission with delay onstraints. For

the delay un onstrained ase, Theorem 4.6 was very useful to arrive at the optimal

proto ol sin e it removed the omplexity of having to deal with the queue states.

However, for the delay onstrained ase, the queue states determine the throughput

and the average delay. Moreover, for mixed rate transmission, the queue states

an only be modeled by a Markov hain with ontinuous state spa e, whi h makes

the analysis ompli ated. Therefore, we resort to a suboptimal adaptive re eption-

transmission proto ol in the following.

Proposition 4.2. Let the buer size be limited to Qmax bits. For this ase, we propose

the following adaptive re eption-transmission proto ol for mixed rate transmission

with delay onstraints:

1. If OS(i) = 0, set di = 1.

2. Otherwise, if log2(1 + r(i)) ≤ Q(i − 1) ≤ Qmax − S0, sele t di as proposed in

Theorem 4.7 for the ase of transmission without delay onstraint.

3. Otherwise, if Q(i− 1) > Qmax − S0, set di = 1.

4. Otherwise, if Q(i− 1) < log2(1 + r(i)), set di = 0.

If the S-R link is in outage, the relay transmits. Otherwise, if there is enough

room in the buer to a ommodate the bits/symb possibly sent from the sour e to

the relay and there are enough bits/symb in the buer for the relay to transmit, the

135


adaptive re eption-transmission proto ol introdu ed in Theorem 4.7 is employed. On

the other hand, if there exists the possibility of a buer overow, the relay transmits

to redu e the amount of data in the buer. If the number of bits/symb in the buer

is too low, the sour e transmits. The value of Qmax an be used to adjust the average

delay while maintaining a low throughput loss ompared to the throughput without

delay onstraint.

Although on eptually simple, as pointed out before, a theoreti al analysis of

the throughput of the proposed queue size limiting proto ol is di ult be ause of

the ontinuous state spa e of the asso iated Markov hain. Thus, we will resort to

simulations to evaluate its performan e in Se tion 4.7.

4.6.4 Conventional Relaying With Delay Constraints

To have a ben hmark for delay onstrained buer-aided relaying with adaptive re eption-

transmission, we propose a orresponding onventional relaying proto ol, whi h may

be viewed as a delay onstrained version of Conventional Relaying 1.

Proposition 4.3. The sour e transmits to the relay in k onse utive time slots fol-

lowed by the relay transmitting to the destination in the following p time slots. Then,

this pattern is repeated, i.e., the sour e transmits again in k onse utive time slots,

and so on. The values of k and p an be hosen to satisfy any delay and throughput

requirements.

For this proto ol, the queue is non-absorbing if

k(1− PS)S0 ≤ pElog2(1 + r(i)). (4.120)

Assuming (4.120) holds, the average arrival rate is equal to the throughput and hen e

136


the throughput is given by

τ =k

k + p(1− PS)S0 , (4.121)

Using a numeri al example, we will show in Se tion 4.7 ( f. Fig. 4.6) that the

proto ol with adaptive re eption-transmission in Proposition 4.2 a hieves a higher

throughput than the onventional proto ol in Proposition 4.3. However, the onven-

tional proto ol is more amendable to analysis and it is interesting to investigate the

orresponding throughput and multiplexing gain for a given average delay in the high

SNR regime, γS = γR = γ → ∞. This is done in the following theorem.

Theorem 4.9. For a given average delay onstraint, ET, the maximal throughput

τ and multiplexing rate r of mixed rate transmission, for γS = γR = γ → ∞, are

given by

τ → S0

(

1−1

2ET

)

, as γ → ∞ . (4.122)

r → 1−1

2ET, as γ → ∞ . (4.123)


Remark 4.15. Theorem 4.9 reveals that, as expe ted from the dis ussion of the ase

without delay onstraints, delay onstrained mixed rate transmission approa hes a

multiplexing gain of one as the allowed average delay in reases.

4.7 Numeri al and Simulation Results

In this se tion, we evaluate the performan e of the proposed xed rate and mixed

rate transmission s hemes for Rayleigh fading. We also onrm some of our analyti al

137


−5 0 5 10 15 20 25 301

1.2

1.4

1.6

1.8

2

γ

τ/τ

fixed

conv,1

TheorySimulation

ΩS = 10

ΩS = 1

ΩS = 0.1

Figure 4.1: Ratio of the throughputs of buer-aided relaying and Conventional

Relaying 1, τ/τfixedconv,1, vs. γ. Fixed rate transmission without delay onstraints.

γS = γR = γ, S0 = R0 = 2 bits/symb, and ΩR = 1.

results with omputer simulations. We note that our analyti al results are valid for

N → ∞. For the simulations, N has to be nite, of ourse, and we adopted N = 107

in all simulations.

4.7.1 Fixed Rate Transmission

For xed rate transmission, we evaluate the proposed adaptive re eption-transmission

proto ols for transmission with and without delay onstraints. Throughout this se -

tion we assume that sour e and relay transmit with identi al rates, i.e., S0 = R0.


In Fig. 4.1, we show the ratio of the throughputs a hieved with the proposed buer-

aided relaying proto ol with adaptive re eption-transmission and Conventional Re-

laying 1 as a fun tion of the transmit SNR γS = γR = γ for ΩR = 1, S0 = R0 = 2

bits/symb, and dierent values of ΩS. The throughput of buer-aided relaying, τ ,

was omputed based on (4.37), (4.41), and (4.45) in Theorem 4.2, while the through-

138


0 10 20 30 40 5010

−8

10−6

10−4

10−2

100

γ (in dB)

F o

ut

Conventional Relaying 1BA Relaying (Theory)BA Relaying (Simulation)

ΩS = 0.1; 1; 10

ΩS = 10; 1; 0.1

Figure 4.2: Outage probability of buer-aided (BA) relaying and Conventional Re-

laying 1 vs. γ. Fixed rate transmission without delay onstraints. γS = γR = γ,S0 = R0 = 2 bits/symb, and ΩR = 1.

put of Conventional Relaying 1, τfixedconv,1, was obtained based on (4.12). Furthermore,

we also show simulation results where the throughput of the buer-aided relaying

proto ol was obtained via Monte Carlo simulation. From Fig. 4.1 we observe that

theory and simulation are in ex ellent agreement. Furthermore, Fig. 4.1 shows that

ex ept for ΩS = ΩR the proposed adaptive re eption-transmission s heme a hieves

its largest gain for medium SNRs. For very high SNRs, both links are never in outage

and thus, Conventional Relaying 1 and the adaptive re eption-transmission s heme

a hieve the same performan e. On the other hand, for very low SNR, there are very

few transmission opportunities on both links as the links are in outage most of the

time. The proposed adaptive re eption-transmission proto ol an exploit all of these

opportunities. In ontrast, for ΩS = ΩR, Conventional Relaying 1 hoses ξ = 0.5 and

will miss half of the transmission opportunities by sele ting the link that is in out-

age instead of the link that is not in outage be ause of the pre-determined s hedule

for re eption and transmission. On the other hand, if ΩS and ΩR dier signi antly,

Conventional Relaying 1 sele ts ξ lose to 0 or 1 (depending on whi h link is stronger)

139


0 5 10 15 200

0.2

0.4

0.6

0.8

1

γ (in dB)

τ(inbits/symb)

Conventional Relaying 2BA Relaying (Theory)BA Relaying (Simulation)

ET = 2.1

ET = 3.1

ET → ∞

ET = 1.1

Figure 4.3: Throughputs of buer-aided (BA) relaying and Conventional Relaying 2

vs. γ. Fixed rate transmission with delay onstraints. γS = γR = γ, S0 = R0 = 2bits/symb, ΩR = 1, and ΩS = 1.

and the loss ompared to the link adaptive s heme be omes negligible.

In Fig. 4.2, we show the outage probability, Fout, for the proposed buer-aided

relaying proto ol with adaptive re eption-transmission and Conventional Relaying 1.

The same hannel and system parameters as for Fig. 4.1 were adopted for Fig. 4.2

as well. For buer-aided relaying with adaptive re eption-transmission, Fout was

obtained from (4.52) and onrmed by Monte Carlo simulations. For onventional

relaying, Fout was obtained from (4.13). As expe ted from Lemma 4.4, buer-aided

relaying a hieves a diversity gain of two, whereas onventional relaying a hieves only

a diversity gain of one, whi h underlines the superiority of buer-aided relaying with

adaptive re eption-transmission.


In Fig. 4.3, we show the throughput of buer-aided relaying with adaptive re eption-

transmission as a fun tion of the transmit SNR γS = γR = γ for xed rate trans-

mission with dierent onstraints on the average delay ET. The theoreti al urves

for buer-aided relaying were obtained from the expressions given in Lemma 4.6 for

140


0 10 20 30 40 5010

−6

10−4

10−2

100

γ (in dB)

Fou

t

Conventional Relaying 2BA Relaying (theory)BA Relaying (simulation)

ET → ∞

ET = 1.1

ET = 2.1

ET = 3.1

Figure 4.4: Outage probability of buer-aided (BA) relaying and Conventional Relay-

ing 2 vs. γ. Fixed rate transmission with delay onstraints. γS = γR = γ, S0 = R0 = 2bits/symb, ΩR = 1, and ΩS = 1.

throughput and the average delay. For omparison, we also show the throughput

of buer-aided relaying with adaptive re eption-transmission and without delay on-

straint ( f. Theorem 4.2), and the throughput of Conventional Relaying 2 given by

(4.14). These two s hemes introdu e an innite delay, i.e., ET → ∞ as N → ∞,

and a delay of one time slot, respe tively. In the low SNR regime, the proposed buer-

aided relaying s heme with adaptive re eption-transmission annot satisfy all delay

requirements as expe ted from Lemma 4.7. Hen e, for nite delays, the throughput

urves in Fig. 4.3 do not extend to low SNRs. Nevertheless, as the aordable delay

in reases, the throughput for delay onstrained transmission approa hes the through-

put for delay un onstrained transmission for su iently high SNR. Furthermore, the

performan e gain ompared to Conventional Relaying 2 is substantial even for the

omparatively small average delays ET onsidered in Fig. 4.3.

In Fig. 4.4, we show the outage probability, Fout, for the same s hemes and pa-

rameters that were onsidered in Fig. 4.3. For buer-aided relaying with adaptive

re eption-transmission, the theoreti al results shown in Fig. 4.4 were obtained from

141


(4.87) and (4.88). These theoreti al results are onrmed by the Monte Carlo simula-

tion results also shown in Fig. 4.4. Furthermore, the urves for transmission without

delay onstraint (i.e., ET → ∞ as N → ∞) were omputed from (4.52), and

for Conventional Relaying 2, we used (4.15). Fig. 4.4 shows that even for an aver-

age delay as small as ET = 1.1 slots, the proposed buer-aided relaying proto ol

with adaptive re eption-transmission outperforms Conventional Relaying 2. Further-

more, as expe ted from Theorem 4.5, buer-aided relaying with adaptive re eption-

transmission a hieves a diversity gain of two when the average delay is larger than

three time slots (e.g., ET = 3.1 time slots in Fig. 4.4 ). This leads to a large per-

forman e gain over onventional relaying whi h a hieves only a diversity gain of one.

For example, gains around 10 dB and 20 dB are a hieved for an outage probability of

10−2and 10−3

, respe tively. Finally, note that even for ET = 3.1 the performan e

loss in dB is very small ompared to the ase of ET → ∞.

Remark 4.16. For the simulation results shown in Figs. 4.3 and 4.4, we adopted a

relay with a buer size of L = 60 pa kets whi h leads to a negligible probability of

dropped pa kets. For example, for γ = 45 dB, the probability of a full buer, PrQ =

LR0, is bounded by PrQ = LR0 < 10−60, and for lower SNRs, PrQ = LR0

is even higher. This also supports the laim in the proof of Theorem 4.5 that for

large enough buer sizes the probability of dropping a pa ket due to a buer overow

be omes negligible.

4.7.2 Mixed Rate Transmission

In this se tion, we investigate the a hievable throughput for mixed rate transmission.

For this purpose, we onsider again the delay onstrained and the delay un onstrained

ases separately.

142


0 5 10 15 20 25 30 35 40 45

0.6

0.8

1

1.2

1.4

1.6

1.8

Γ (in dB)

τ(bits/symb)

With Power AllocationWithout Power AllocationSimulation of Buffer-aided Relaying

Conventional Relaying 1

Buffer−aided Relaying

Figure 4.5: Throughput of buer-aided relaying with adaptive re eption-transmission

and Conventional Relaying 1 vs. Γ. Mixed rate transmission without delay on-

straints. ΩS = 10, ΩR = 1, and S0 = 2 bits/symb.


In Fig. 4.5, we ompare the throughputs of buer-aided relaying with adaptive

re eption-transmission and Conventional Relaying 1. In both ases, we onsider the

ases with and without power allo ation. The theoreti al results shown in Fig. 4.5 for

the four onsidered s hemes were generated based on Theorem 4.7/Lemma 4.9, The-

orem 4.8/Lemma 4.10, (4.21), (4.22), and (4.21), (4.23). The transmit SNRs of both

links are identi al, i.e., γS = γR = Γ, S0 = 2 bits/symb, ΩS = 10, and ΩR = 1. As

an be observed from Fig. 4.5, for both buer-aided relaying with adaptive re eption-

transmission and Conventional Relaying 1, power allo ation is bene ial only for low

to moderate SNRs. Both s hemes an a hieve a throughput of S0 bits/symb in the

high SNR regime. However, adaptive re eption-transmission a hieves a throughput

gain ompared to Conventional Relaying 1 in the entire onsidered SNR range.

143


0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Γ (in dB)

τ(bits/symb)

Mixed rate scheme with adaptive reception−transmissionMixed rate scheme with conventional relayingFixed rate scheme with reception−transmissionUpper bound for delay of 5 time slots and infinite transmit power

Figure 4.6: Throughput of buer-aided relaying with adaptive re eption-transmission

and onventional relaying vs. Γ. Mixed rate and xed rate transmission with delay

onstraint. ET = 5 time slots, γS = γR = Γ, S0 = 2 bits/symb, and ΩS = ΩR = 1.


In Fig. 4.6, we ompare the throughputs of various mixed rate and xed rate trans-

mission s hemes for a maximum average delay of ET = 5 time slots and S0 = 2

bits/symb. The transmit SNRs of both links are identi al, i.e., γS = γR = Γ,

ΩS = ΩR = 1. For mixed rate transmission, we simulated both the buer-aided

relaying proto ol with adaptive re eption-transmission des ribed in Proposition 4.2

and the onventional relaying proto ol des ribed in Proposition 4.3. For xed rate

transmission, we hose R0 = S0 = 2 bits/symb and in luded results for buer-

aided relaying with adaptive re eption-transmission obtained based on Lemma 4.6.

Furthermore, for mixed rate transmission, we also show the maximum a hievable

throughput of buer-aided relaying with adaptive re eption-transmission in the ab-

sen e of delay onstraints (as given by Theorem 4.7/Lemma 4.9) and the maximum

throughput a hievable for a delay onstraint of ET = 5 time slots and innite

transmit power (as given by (4.122)). Fig. 4.6 reveals that for mixed rate transmis-

144


sion the proto ol with adaptive re eption-transmission proposed in Proposition 4.2

is superior to the onventional relaying s heme proposed in Proposition 4.3, and for

high SNR, both proto ols rea h the upper bound for mixed rate transmission under

a delay onstraint given by (4.122). Furthermore, Fig. 4.6 also shows that mixed

rate transmission is superior to xed rate transmission sin e the former an exploit

the additional exibility aorded by having CSIT for the R-D link. For example, for

Γ = 30 dB, mixed rate transmission with adaptive re eption-transmission a hieves a

throughput gain of 65% ompared to xed rate transmission, and even onventional

adaptive re eption-transmission still a hieves a gain of 45%. Fig. 4.6 also shows that

even in the presen e of severe delay onstraints mixed rate transmission an signi-

antly redu e the throughput loss aused by HD relaying ompared to FD relaying,

whose maximum throughput is S0 = 2 bits/symb.

4.8 Con lusions

In this hapter, we have onsidered a two-hop HD relay network. We have investi-

gated both xed rate transmission, where sour e and relay do not have full CSIT

and are for ed to transmit with xed rate, and mixed rate transmission, where

the sour e does not have full CSIT and transmits with xed rate but the relay

has full CSIT and transmits with variable rate. For both modes of transmission,

we have derived the throughput-optimal buer-aided relaying proto ols with adap-

tive re eption-transmission and the resulting throughputs and outage probabilities.

Furthermore, we have shown that buer-aided relaying with adaptive re eption-

transmission leads to substantial performan e gains ompared to onventional re-

laying with non-adaptive re eption-transmission. In parti ular, for xed rate trans-

mission, buer-aided relaying with adaptive re eption-transmission a hieves a di-

145


versity gain of two, whereas onventional relaying is limited to a diversity gain of

one. For mixed rate transmission, both buer-aided relaying with adaptive re eption-

transmission and a newly proposed onventional relaying s heme with non-adaptive

re eption-transmission have been shown to over ome the HD loss typi al for wire-

less relaying proto ols and to a hieve a multiplexing gain of one. Sin e the proposed

throughput-optimal proto ols introdu e an innite delay, we have also proposed mod-

ied proto ols for delay onstrained transmission and have investigated the resulting

throughput-delay trade-o. Surprisingly, the diversity gain of xed rate transmission

with buer-aided relaying is also observed for delay onstrained transmission as long

as the average delay ex eeds three time slots. Furthermore, for mixed rate trans-

mission, for an average delay ET, a multiplexing gain of r = 1 − 1/(2ET) is

a hieved even for onventional relaying.

146

Chapter 5

Summary of Thesis and Future

Resear h Topi s

In this nal hapter, in Se tion 5.1, we summarize our results and highlight the

ontributions of this thesis. In Se tion 5.2, we also propose ideas for future resear h.

5.1 Summary of the Results

In this thesis, we designed new ommuni ation proto ols for the two-hop HD relay

hannel. In the following, we briey review the main results of ea h hapter.

In Chapter 2, we have derived an easy-to-evaluate apa ity expression of the two-

hop HD relay hannel when fading is not present based on simplifying previously

derived onverse expressions. Moreover, we have proposed an expli it oding s heme

whi h a hieves the apa ity. We showed that the apa ity is a hieved when the relay

swit hes between re eption and transmission in a symbol-by-symbol manner and

when additional information is sent by the relay to the destination using the zero

symbol impli itly sent by the relay's silen e during re eption. Furthermore, we have

evaluated the apa ity for the ases when both links are BSCs and AWGN hannels,

respe tively. From the numeri al examples, we have observed that the apa ity of

the two-hop HD relay hannel is signi antly higher than the rates a hieved with

onventional relaying proto ols.

147

Chapter 5. Summary of Thesis and Future Resear h Topi s

In Chapter 3, we have devised new ommuni ation proto ols for improving the

a hievable average rate of the two-hop HD relay hannel when both sour e-relay

and relay-destination links are AWGN hannels ae ted by fading, referred to as

buer-aided relaying with adaptive re eption-transmission proto ols. In ontrast to

onventional relaying, where the relay re eives and transmits a ording to a pre-

dened s hedule regardless of the hannel state, in the proposed proto ol, the relay

re eives and transmits adaptively a ording to the quality of the sour e-relay and

relay-destination links. For delay-un onstrained transmission, we derived the op-

timal adaptive re eption-transmission s hedule for the ases of xed and variable

sour e and relay transmit powers. For delay- onstrained transmission, we proposed

a buer-aided proto ol whi h ontrols the delay introdu ed by the buer at the re-

lay. This proto ol needs only instantaneous CSI and the desired average delay, and

an be implemented in real-time. Our analyti al and simulation results showed that

buer-aided relaying with adaptive re eption-transmission with and without delay

onstraints is a promising approa h to signi antly in rease the a hievable average

rate ompared to onventional HD relay-assisted transmission.

In Chapter 4, we have devised new ommuni ation proto ols for improving the

outage probability of the two-hop HD relay hannel when both sour e-relay and relay-

destination links are AWGN hannels ae ted by fading. We have investigated both

xed rate transmission, where sour e and relay do not have full CSIT and are for ed

to transmit with xed rate, and mixed rate transmission, where the sour e does not

have full CSIT and transmits with xed rate but the relay has full CSIT and transmits

with variable rate. For both modes of transmission, we have derived the throughput-

optimal buer-aided relaying proto ols with adaptive re eption-transmission and

the resulting throughputs and outage probabilities. Furthermore, we ould show

148


that buer-aided relaying with adaptive re eption-transmission leads to substantial

performan e gains ompared to onventional relaying with non-adaptive re eption-

transmission. In parti ular, for xed rate transmission, buer-aided relaying with

adaptive re eption-transmission a hieves a diversity gain of two, whereas onven-

tional relaying is limited to a diversity gain of one. For mixed rate transmission,

both buer-aided relaying with adaptive re eption-transmission and a newly pro-

posed onventional relaying s heme with non-adaptive re eption-transmission have

been shown to over ome the HD loss typi al for wireless relaying proto ols and to

a hieve a multiplexing gain of one. Sin e the proposed throughput-optimal proto ols

introdu e an innite delay, we have also proposed modied proto ols for delay on-

strained transmission and have investigated the resulting throughput-delay trade-o.

Surprisingly, the diversity gain of xed rate transmission with buer-aided relaying

with adaptive re eption-transmission is also observed for delay onstrained transmis-

sion as long as the average delay ex eeds three time slots. Furthermore, for mixed rate

transmission, for an average delay ET, a multiplexing gain of r = 1− 1/(2ET)

is a hieved even for onventional relaying.

5.2 Future Work

Future resear h dire tions may in lude the following:

• Deriving the apa ity of the two-hop HD relay hannel when both sour e-relay

and relay-destination links are ae ted by fading, and designing a oding s heme

whi h a hieves the apa ity. Intuitively, we expe t this oding s heme to be a

mix of the oding s heme introdu ed in Chapter 2 and the buer-aided proto ol

introdu ed in Chapter 3.

149


• Investigating the apa ity of the two-hop FD relay hannel with self-interferen e

and determining the amount of allowable self-interferen e beyond whi h the

relay is better of by working in the HD mode. This requires a new information-

theoreti al analysis and taking into a ount that the self-interferen e is aused

by the transmitting node itself, and therefore the relay node has some knowledge

about the self-interferen e whi h it an use to its benet in order to in rease

its data rate.

• Investigating the apa ity and/or devising new buer-aided proto ols for HD

relay networks whi h are more omplex than the two-hop HD relay hannel, e.g.

networks omprised of more than one sour e and/or relay, and/or destination.

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Appendix A

Proofs for Chapter 2

A.1 Proof That the Probability of Error at the

Relay Goes to Zero When (2.25) Holds

In order to prove that the relay an de ode the sour e's odeword in blo k b, x1|r(b),

where 1 ≤ b ≤ N , from the re eived odeword y1|r(b) when (2.25) holds, i.e., that the

probability of error at the relay goes to zero as k → ∞, we will follow the standard"

method in [80, Se . 7.7 for analyzing the probability of error for rates smaller than

the apa ity. To this end, note that the length of odeword x1|r(b) is k(1− P ∗U). On

the other hand, the length of odeword y1|r(b) is identi al to the number of zeros9

in

relay's transmit odeword x2(b). Sin e the zeros in x2(b) are generated independently

using a oin ip, the number of zeros, i.e., the length of y1|r(b) is k(1 − P ∗U) ± ε(b),

where ε(b) is a non-negative integer. Due to the strong law of large numbers, the

following holds

limk→∞

ε(b)

k= 0, (A.1)

limk→∞

k(1− P ∗U)± ε(b)

k(1− P ∗U)

= 1, (A.2)

9

For b = 1, note that the number of zeros in x2(1) is k. Therefore, for b = 1, we only take into

an a ount the rst k(1− P ∗U) zeros. As a result, the length of y1|r(b) is also k(1− P ∗

U).

159

Appendix A. Proofs for Chapter 2

i.e., for large k, the length of relay's re eived odeword y1|r(b) is approximately k(1−

P ∗U).

Now, for blo k b, we dene a set R(b) whi h ontains the symbol indi es i in blo k

b for whi h the symbols in x2(b) are zeros, i.e., for whi h X2i = 0. Note that before

the start of the transmission in blo k b, the relay knows x2(b), thereby it knows a

priori for whi h symbol indi es i in blo k b, X2i = 0 holds. Furthermore, note that

|R(b)| = k(1− P ∗U)± ε(b) (A.3)

holds, where | · | denotes ardinality of a set. Depending on the relation between

|R(b)| and k(1 − P ∗U), the relay has to distinguish between two ases for de oding

x1|r(b) from y1|r(b). In the rst ase |R(b)| ≥ k(1− P ∗U) holds whereas in the se ond

ase |R(b)| < k(1 − P ∗U) holds. We rst explain the de oding pro edure for the rst

ase.

When |R(b)| = k(1 − P ∗U) + ε(b) ≥ k(1 − P ∗

U) holds, the sour e an transmit the

entire odeword x1|r(b), whi h is omprised of k(1 − P ∗U) symbols, sin e there are

enough zeros in odeword x2(b). On the other hand, sin e for this ase the re eived

odeword y1|r(b) is omprised of k(1 − P ∗U) + ε(b) symbols, and sin e for the last

ε(b) symbols in y1|r(b) the sour e is silent, the relay keeps from y1|r(b) only the rst

k(1 − P ∗U) symbols and dis ards the remanning ε(b) symbols. In this way, the relay

keeps only the re eived symbols whi h are the result of the transmitted symbols in

x1|r(b), and dis ards the rest of the symbols in y1|r(b) for whi h the sour e is silent.

Thereby, from y1|r(b), the relay generates a new re eived odeword whi h we denote

by y∗1|r(b). Moreover, let R1(b) be a set whi h ontains the symbol indi es of the

symbols omprising odeword y∗1|r(b). Now, note that the lengths of x1|r(b) and

y∗1|r(b), and the ardinality of set R1(b) are k(1 − P ∗

U), respe tively. Having reated

160


y∗1|r(b) and R1(b), we now use a jointly typi al de oder for de oding x1|r(b) from

y∗1|r(b). In parti ular, we dene a jointly typi al set A

|R1(b)|ǫ as

A|R1(b)|ǫ =

(x1|r,y∗1|r) ∈ X

|R1(b)|1 × Y

|R1(b)|1 :

∣

∣

∣

∣

∣

∣

−1

|R1(b)|

∑

i∈R1(b)

log2 p(x1i|x2i = 0)−H(X1|X2 = 0)

∣

∣

∣

∣

∣

∣

≤ ǫ, (A.4a)

∣

∣

∣

∣

∣

∣

−1

|R1(b)|

∑

i∈R1(b)

log2 p(y1i|x2i = 0)−H(Y1|X2 = 0)

∣

∣

∣

∣

∣

∣

≤ ǫ, (A.4b)

∣

∣

∣

∣

∣

∣

−1

|R1(b)|

∑

i∈R1(b)

log2 p(x1i, y1i|x2i = 0)−H(X1, Y1|X2 = 0)

∣

∣

∣

∣

∣

∣

≤ ǫ

, (A.4 )

where ǫ is a small positive number. The transmitted odeword x1|r(b) is su essfully

de oded from re eived odeword y∗1|r(b) if and only if (x1|r(b),y

∗1|r(b)) ∈ A

|R1(b)|ǫ and

no other odeword x1|r from odebook C1|r is jointly typi al with y∗1|r(b). In order to

ompute the probability of error, we dene the following events

E0 = (x1|r(b),y∗1|r(b)) /∈ A|R1(b)|

ǫ and Ej = (x(j)1|r,y

∗1|r) ∈ A|R1(b)|

ǫ , (A.5)

where x(j)1|r is the j-th odeword in C1|r that is dierent from x1|r(b). Note that

in C1|r there are |C1|r| − 1 = 2kR − 1 odewords that are dierent from x1|r(b), i.e.,

j = 1, ..., 2kR−1. Hen e, an error o urs if any of the events E0, E1, ..., E2kR−1 o urs.

Sin e x1|r(b) is uniformly sele ted from the odebook C1|r, the average probability of

error is given by

Pr(ǫ) = Pr(E0 ∪ E1 ∪ ... ∪ E2kR−1) ≤ Pr(E0) +

2kR−1∑

j=1

Pr(Ej). (A.6)

161


Sin e |R1(b)| → ∞ as k → ∞, Pr(E0) in (A.6) is upper bounded as [80, Eq. (7.74)

Pr(E0) ≤ ǫ. (A.7)

On the other hand, sin e |R1(b)| → ∞ as k → ∞, Pr(Ej) is upper bounded as

Pr(Ej) = Pr((

x(j)1|r,y

∗1|r(b)

)

∈ A|R1(b)|ǫ

)

=∑

(x(j)1|r

,y∗1|r

(b))∈A|R1(b)|ǫ

p(x(j)1|r,y

∗1|r(b))

(a)=

∑

(x(j)1|r

,y∗1|r

(b))∈A|R1(b)|ǫ

p(x(j)1|r)p(y

∗1|r(b))

(b)

≤∑

(x(j)1|r

,y∗1|r

(b))∈A|R1(b)|ǫ

2−|R1(b)|(H(X1 |X2=0)−ǫ)2−|R1(b)|(H(Y1|X2=0)−ǫ)

= |A|R1(b)|ǫ |2−|R1(b)|(H(X1|X2=0)−ǫ)2−|R1(b)|(H(Y1|X2=0)−ǫ)

(c)

≤ 2|R1(b)|(H(X1,Y1|X2=0)+ǫ)2−|R1(b)|(H(X1 |X2=0)−ǫ)2−|R1(b)|(H(Y1|X2=0)−ǫ)

= 2−|R1(b)|(H(X1|X2=0)+H(Y1|X2=0)−H(X1,Y1|X2=0)−3ǫ)

= 2−|R1(b)|(I(X1;Y1|X2=0)−3ǫ), (A.8)

where (a) follows sin e x(j)1|r and y∗

1|r(b) are independent, (b) follows sin e

p(x(j)1|r) ≤ 2−|R1(b)|(H(X1|X2=0)−ǫ)

and p(y∗1|r(b)) ≤ 2−|R1(b)|(H(Y1|X2=0)−ǫ),

whi h follows from [80, Eq. (3.6), respe tively, and (c) follows sin e

|A|R1(b)|ǫ | ≤ 2|R1(b)|(H(X1,Y1|X2=0)+ǫ),

whi h follows from [80, Theorem 7.6.1. Inserting (A.7) and (A.8) into (A.6), we

162


odewords in C∗1|r(b) and C1|r are of the same order due to (A.2). Hen e, if the relay

an de ode x∗1|r(b) from y1|r(b), then using this unique mapping between C∗

1|r(b) and

C1|r(b), the relay an de ode x1|r(b) and thereby de ode the message w(b) sent from

the sour e.

Now, for de oding x∗1|r(b) from y1|r(b), we again use jointly typi al de oding.

Thereby, we dene a jointly typi al set B|R|ǫ as

B|R|ǫ =

(x∗1|r,y1|r) ∈ X

|R|1 × Y

|R|1 :

∣

∣

∣

∣

∣

−1

|R|

∑

i∈R

log2 p(x1i|x2i = 0)−H(X1|X2 = 0)

∣

∣

∣

∣

∣

≤ ǫ, (A.11a)

∣

∣

∣

∣

∣

−1

|R|

∑

i∈R

log2 p(y1i|x2i = 0)−H(Y1|X2 = 0)

∣

∣

∣

∣

∣

≤ ǫ, (A.11b)

∣

∣

∣

∣

∣

−1

|R|

∑

i∈R

log2 p(x1i, y1i|x2i = 0)−H(X1, Y1|X2 = 0)

∣

∣

∣

∣

∣

≤ ǫ

. (A.11 )

Again, the transmitted odeword x∗1|r(b) is su essfully de oded from re eived

odeword y1|r(b) if and only if (x∗1|r(b),y1|r(b)) ∈ B

|R|ǫ and no other odeword x∗

1|r

from odebook C∗1|r is jointly typi al with y1|r(b). In order to ompute the probability

of error, we dene the following events

E0 = (x∗1|r(b),y1|r(b)) /∈ B|R|

ǫ and Ej = (x∗(j)1|r ,y1|r(b)) ∈ B|R|

ǫ , (A.12)

where x∗(j)1|r is the j-th odeword in C∗

1|r that is dierent from x∗1|r(b). Note that in

C∗1|r there are |C∗

1|r| − 1 = 2kR − 1 odewords that are dierent from x∗1|r(b), i.e.,

j = 1, ..., 2kR − 1. Hen e, an error o urs if any of the events E0, E1, ..., E2kR−1

o urs. Now, using a similar pro edure as for ase when |R(b)| ≥ k(1 − P ∗U), it an

164


be proved that if

R <|R(b)|

kI(X1; Y1|X2 = 0)− 3(1− P ∗

U)ǫ

= (1− P ∗U)I(X1; Y1|X2 = 0)−

ε(b)

kI(X1; Y1|X2 = 0)− 3(1− P ∗

U)ǫ, (A.13)

then limǫ→0

limk→∞

Pr(ǫ) = 0. In (A.13), note that

limk→∞

ε(b)

kI(X1; Y1|X2 = 0) = 0 (A.14)

holds due to (A.1). This on ludes the proof for the ase when |R(b)| < k(1− P ∗U).

A.2 Proof That the Probability of Error at the

Destination Goes to Zero When (2.26) Holds

In order to prove that the destination an de ode the relay's odeword su essfully

when (2.26) holds, i.e., that the probability of error at the destination goes to zero, we

will again follow the standard" method in [80, Se . 7.7 for analyzing the probability

of error for rates smaller than the apa ity. To this end, we again use a jointly typi al

165


de oder. In parti ular, we dene a jointly typi al set Dkǫ as follows

Dkǫ = (x2,y2) ∈ X k

2 × Yk2 :

∣

∣

∣

∣

∣

−1

k

k∑

i=1

log2 p(x2i)−H(X2)

∣

∣

∣

∣

∣

≤ ǫ (A.15a)

∣

∣

∣

∣

∣

−1

k

k∑

i=1

log2 p(y2i)−H(Y2)

∣

∣

∣

∣

∣

≤ ǫ (A.15b)

∣

∣

∣

∣

∣

−1

k

k∑

i=1

log2 p(x2i, y2i)−H(X2, Y2)

∣

∣

∣

∣

∣

≤ ǫ

, (A.15 )

where p(x2) and p(y2) are given in (2.8)-(2.11) The re eived odeword y2 is su ess-

fully de oded as the transmitted odeword x2 if and only if (x2,y2) ∈ Dkǫ , and no

other odeword x2 from odebook C2 is jointly typi al with y2. In order to ompute

the probability of error, we dene the following events

E0 = (x2,y2) /∈ Dkǫ and Ej = (x2(j),y2) ∈ Dk

ǫ , (A.16)

where x2(j) is the j-th odeword in C2 that is dierent from x2. Note that in C2 there

are |C2| − 1 = 2kR − 1 odewords whi h are dierent from x2, i.e., j = 1, ..., 2kR − 1.

An error o urs if at least one of the events E0, E1,..., E2kR−1 o urs. Sin e x2 is

uniformly sele ted from odebook C2, the average probability of error is given by

Pr(ǫ) = Pr(E0 ∪ E1 ∪ ... ∪ E2kR−1) ≤ Pr(E0) +

2kR−1∑

j=1

Pr(Ej). (A.17)

In (A.17), Pr(E0) is upper bounded as [80, Eq. (7.74)

Pr(E0) ≤ ǫ, (A.18)

166


whereas Pr(Ej) is bounded as

Pr(Ej) = Pr((x2(j),y2) ∈ Dkǫ ) =

∑

(x2(j),y2)∈Dkǫ

p(x2(j),y2)

(a)=

∑

(x2(j),y2)∈Dkǫ

p(x2(j))p(y2)

(b)

≤∑

(x2(j),y2)∈Dkǫ

2−k(H(X2)−ǫ)2−k(H(Y2)−ǫ) = |Dkǫ |2

−k(H(X2)−ǫ)2−k(H(Y2)−ǫ)

(c)

≤ 2k(H(X2,Y2)+ǫ)2−k(H(X2)−ǫ)2−k(H(Y2)−ǫ) = 2−k(H(Y2)−H(Y2|X2)−3ǫ), (A.19)

where (a) follows sin e x2(j) and y2 are independent, (b) follows sin e p(x2(j)) ≤

2−k(H(X2)−ǫ)and p(y2) ≤ 2−k(H(Y2)−ǫ)

, whi h follow from [80, Eq. (3.6), respe tively,

and (c) follows sin e |Dkǫ | ≤ 2k(H(X2,Y2)+ǫ)

, whi h follows from [80, Theorem 7.6.1.

Inserting (A.18) and (A.19) into (A.17), we obtain

Pr(ǫ) ≤ ǫ+

2kR−1∑

j=1

2−k(H(Y2)−H(Y2|X2)−3ǫ) ≤ ǫ+ (2kR − 1)2−k(H(Y2)−H(Y2|X2)−3ǫ)

≤ ǫ+ 2kR2−k(H(Y2)−H(Y2|X2)−3ǫ) = ǫ+ 2−k(H(Y2)−H(Y2|X2)−R−3ǫ). (A.20)

Hen e, if R < H(Y2) − H(Y2|X2) − 3ǫ = I(X2; Y2) − 3ǫ, then limǫ→∞

limk→∞

Pr(ǫ) = 0.

This on ludes the proof.

A.3 Proof of Lemma 2.1

To prove Lemma 2.1, we use results from [86. Hen e, we rst assume that pV (x2) is

a ontinuous distribution and then see that this leads to a ontradi tion. If pV (x2) is

a ontinuous distribution, then the distribution of X2 is also ontinuous. Now, our

167


goal is to obtain the solution of the following optimization problem

Maximize :pV (x2)

H(Y2)

Subject to :∫

x2x22pV (x2)dx2 ≤ P2,

(A.21)

where Y2 = X2 + N2, N2 is a zero mean Gaussian distributed RV with varian e σ22,

and X2 has a ontinuous distribution with an average power onstraint. However, it

is proved in [86 that the only possible solution for maximizing H(Y2) as in (A.21)

is the distribution pV (x2) that yields a Gaussian distributed Y2. In our ase, the

only possible solution that yields a Gaussian distributed Y2 is if X2 is also Gaussian

distributed. On the other hand, the distribution of X2 an be written as

p(x2) = pV (x2)PU + δ(x2)(1− PU). (A.22)

Hen e, we have to nd a pV (x2) su h that p(x2) in (A.22) is Gaussian. However,

as an be seen from (A.22), for PU < 1, a distribution for pV (x2) that makes p(x2)

Gaussian does not exist. Therefore, as proved in [86, the only other possibility is

that pV (x2) is a dis rete distribution. Only in the limiting ase when PU → 1, p(x2)

be omes a Gaussian distribution by setting pV (x2) to be a Gaussian distribution.


168

Appendix B


B.1 Proof of Theorem 3.1

We rst note that, be ause of the law of the onservation of ow, A ≥ RSD is always

valid and equality holds if and only if the queue is non-absorbing. Assume rst

we have an adaptive re eption-transmission poli y with average arrival rate A and

a hievable average rate RSD with A > RSD, i.e., the queue is absorbing. For this

poli y, we denote the set of indi es with di = 1 by I and the set of indi es with di = 0

by I, and for N → ∞ we have

A =1

N

∑

i∈I

(1− di) log2(1 + s(i)) > RSD =1

N

∑

i∈I

di minlog2(1 + r(i)), Q(i− 1).

(B.1)

From (B.1) we observe that the onsidered proto ol annot be optimal as it an be

improved by moving some of the indi es i in I to I whi h leads to an in rease of RSD

at the expense of a de rease of A. However, on e the point A = RSD is rea hed,

moving more indi es i from I to I will de rease both A and RSD be ause of the

onservation of ow. Thus, a ne essary ondition for the optimal poli y is that the

queue is non-absorbing but the queue is at the edge of non-absorption, i.e., the queue

is at the boundary of a non-absorbing and an absorbing queue. This ompletes the

proof.

169

Appendix B. Proofs for Chapter 3


We denote the sets of indi es i for whi h di = 1 and di = 0 holds by I and I,

respe tively. ǫ denotes a subset of I and | · | is the ardinality of a set. Throughout

the remainder of this proof N → ∞ is assumed.

If the queue in the buer of the relay is absorbing, A > RSD holds and on average

the number of bits/symb arriving at the queue ex eed the number of bits leaving

the queue. Thus, log2(1 + r(i)) ≤ Q(i − 1) holds almost always and as a result the

average rate an be written as

RSD =1

N

∑

i∈I

minlog2(1 + r(i)), Q(i− 1) =1

N

∑

i∈I

log2(1 + r(i)). (B.2)

Now, we assume that the queue is at the edge of non-absorption. That isA = RSD

holds but moving a small fra tion ǫ, where |ǫ|/N → 0, of indi es from I to I will make

the queue an absorbing queue with A > RSD. For this ase, we wish to determine

whether or not

1

N

∑

i∈I

log2(1 + r(i)) > RSD =1

N

∑

i∈I

minlog2(1 + r(i)), Q(i− 1)

= A =1

N

∑

i∈I

log2(1 + s(i)) (B.3)

holds. To test this, we move a small fra tion ǫ, where |ǫ|/N → 0, of indi es from I

to I, thus making the queue an absorbing queue. As a result, (B.2) holds, and (B.3)

170


be omes

1

N

∑

i∈I\ǫ

log2(1 + r(i)) = RSD =1

N

∑

i∈I\ǫ

minlog2(1 + r(i)), Q(i− 1)

< A =1

N

∑

i∈I∪ǫ

log2(1 + s(i)). (B.4)

From the above we on lude that if (B.2) holds, then based on (B.3) and (B.4), for

|ǫ|/N → 0, we must have

1

N

∑

i∈I

log2(1 + r(i)) >1

N

∑

i∈I

log2(1 + s(i)) (B.5)

and

1

N

∑

i∈I\ǫ

log2(1 + r(i)) <1

N

∑

i∈I∪ǫ

log2(1 + s(i)). (B.6)

However, for (B.5) and (B.6) to jointly hold, we require that the parti ular onsidered

moving of indi es from I to I has aused a dis ontinuity in

1N

∑

i∈I log2(1 + r(i))

or/and a dis ontinuity in

1N

∑

i∈I log2(1 + s(i)) as |ǫ|/N → 0 is assumed. Sin e the

apa ities of the S-R and R-D links are su h that limN→∞

∑

i∈ǫ log2(1+s(i))/N → 0

and limN→∞

∑

i∈ǫ log2(1 + r(i))/N → 0, ∀i, su h dis ontinuities are not possible.

Therefore, at the edge of non-absorption (B.3) is not true and we must have instead

1

N

∑

i∈I

log2(1 + r(i)) = RSD =1

N

∑

i∈I

minlog2(1 + r(i)), Q(i− 1)

= A =1

N

∑

i∈I

log2(1 + s(i)) (B.7)

Using (B.7), the average rate an be written as (3.18). This on ludes the proof.

171



To solve (3.19), we rst relax the binary onstraints di ∈ 0, 1 in (3.19) to 0 ≤ di ≤

1, ∀i. Thereby, we transform the original problem (3.19) into the following linear

optimization problem

Maximize :di

1N

∑Ni=1 di log2(1 + r(i))


∑Ni=1(1− di) log2(1 + s(i)) = 1

N

∑Ni=1 di log2(1 + r(i))

C2 : 0 ≤ di ≤ 1, ∀i,

(B.8)

In the following, we solve the relaxed problem (B.8) and then show that the optimal

values of di, ∀i, are at the boundaries, i.e., di ∈ 0, 1, ∀i. Therefore, the solution of

the relaxed problem (B.8) is also the solution to the original maximization problem

in (3.19).

Sin e (B.8) is a linear optimization problem, we an solve it by using the method

of Lagrange multipliers. The Lagrangian fun tion for maximization problem (B.8) is

given by

L =1

N

N∑

i=1

di log2(1 + r(i))− µ1

N

N∑

i=1

[di log2(1 + r(i))− (1− di) log2(1 + s(i))]

+1

N

N∑

i=1

βidi −1

N

N∑

i=1

αi(di − 1), (B.9)

where µ, βi/N , and αi/N are Lagrange multipliers. The Lagrange multipliers βi/N

and αi/N have to satisfy

βi/N ≥ 0, αi/N ≥ 0, diβi/N = 0, (di − 1)αi/N = 0. (B.10)

172


Dierentiating L with respe t to di and setting the result to zero leads to

(1− µ) log2(1 + r(i))− µ log2(1 + s(i)) + βi − αi = 0. (B.11)

If we assume that 0 < di < 1, i.e., di is not at the boundary, then βi = αi = 0 holds,

and from (B.11) we obtain that the following must hold

(1− µ) log2(1 + r(i))− µ log2(1 + s(i)) = 0. (B.12)

However, sin e r(i) and s(i) are random, i.e., hange values for dierent i, (B.12)

annot hold for all i. Therefore, di has to be at the boundary, i.e., di ∈ 0, 1. Now,

assuming di = 0 leads βi ≥ 0 and αi = 0, whi h simplies (B.11) to

di = 0 ⇒ βi = µ log2(1 + s(i))− (1− µ) log2(1 + r(i)) ≥ 0. (B.13)

Whereas, assuming di = 1 leads βi = 0 and αi ≥ 0, whi h simplies (B.11) to

di = 1 ⇒ αi = (1− µ) log2(1 + r(i))− µ log2(1 + s(i)) ≥ 0. (B.14)

(B.13) and (B.14) an be written equivalently as

di =

1 if (1− µ) log2(1 + r(i))− µ log2(1 + s(i)) ≥ 0

0 if (1− µ) log2(1 + r(i))− µ log2(1 + s(i)) ≤ 0,(B.15)

whi h is identi al to (3.20) if we set ρ = µ/(1− µ) and note that the probability of

(1 − µ) log2(1 + r(i)) = µ log2(1 + s(i)) happening is zero due to s(i) and r(i) being

ontinuous random variables. µ or equivalently ρ are hosen su h that onstraint C1

of problem (3.19) is met. This ompletes the proof.

173



To solve (3.29), we rst relax the binary onstraints di ∈ 0, 1 in (3.29) to 0 ≤ di ≤ 1,

∀i. Thereby, we transform the original problem (3.29) into the following on ave


Maximize :γS(i)≥0, γR(i)≥0, di

1N



∑Ni=1(1− di) log2(1 + γS(i)hS(i))

= 1N


C2 : 0 ≤ di ≤ 1

C3 : 1N

∑Ni=1(1− di)γS(i) +

1N

∑Ni=1 diγR(i) ≤ Γ

(B.16)

In the following, we solve the relaxed problem (B.16) and then show that the optimal

values of di, ∀i, are at the boundaries, i.e., di ∈ 0, 1, ∀i. Therefore, the solution of

the relaxed problem (B.16) is also the solution to the original maximization problem

in (3.29).

Sin e (B.16) is a on ave optimization problem, we an solve it by using the

method of Lagrange multipliers. The Lagrangian fun tion for maximization problem

(B.16) is given by

L =1

N

N∑

i=1

di log2(1 + γR(i)hR(i))

−µ1

N

N∑

i=1

[

di log2(1 + γR(i)hR(i))− (1− di) log2(1 + γS(i)hS(i))]

−ν1

N

N∑

i=1

[

(1− di)γS(i) + diγR(i)− Γ]

+1

N

N∑

i=1

βidi −1

N

N∑

i=1

αi(di − 1),

(B.17)

174


where the Lagrange multipliers µ and ν are hosen su h that C1 and C3 are satised,

respe tively. On the other hand, the Lagrange multipliers βi/N and αi/N have to

satisfy (B.10).

By dierentiating L with respe t to γS(i), γR(i), and di, and setting the results

to zero, we obtain the following three equations

−ν(1 − di) + µ(1− di)hS(i)

(1 + γS(i)hS(i)) ln(2)= 0, (B.18)

−νdi +dihR(i)

(1 + γR(i)hR(i)) ln(2)− µ

dihR(i)

(1 + γR(i)hR(i)) ln(2)= 0, (B.19)

−αi + βi − ν(γR(i)− γS(i)) + (1− µ) log2(1 + γR(i)hR(i))

−µ log2(1 + γS(i)hS(i)) = 0. (B.20)


and from (B.20) we obtain that the following must hold

−ν(γR(i)− γS(i)) + (1− µ) log2(1 + γR(i)hR(i))− µ log2(1 + γS(i)hS(i)) = 0.

(B.21)

However, sin e hR(i) and hR(i) are random, (B.21) annot hold for all i. Therefore,

di has to be at the boundary, i.e., di ∈ 0, 1. Now, assuming di = 0 leads βi ≥ 0

and αi = 0, whi h simplies (B.20) to

βi = ν(γR(i)− γS(i))− (1− µ) log2(1 + γR(i)hR(i)) + µ log2(1 + γS(i)hS(i)) ≥ 0.

(B.22)

175


Whereas, assuming di = 1 leads βi = 0 and αi ≥ 0, whi h simplies (B.20) to

αi = −ν(γR(i)− γS(i)) + (1− µ) log2(1 + γR(i)hR(i))− µ log2(1 + γS(i)hS(i)) ≥ 0.

(B.23)

From (B.22) and (B.23), we obtain the following solution for di

di =

1 if (1− µ) log2(1 + γR(i)hR(i))− νγR(i) ≥ µ log2(1 + γS(i)hS(i))− νγS(i)

0 if (1− µ) log2(1 + γR(i)hR(i))− νγR(i) ≤ µ log2(1 + γS(i)hS(i))− νγS(i)

(B.24)

Now, inserting the solution for di in (B.24) into (B.18) and (B.19), and solving the

system of two equations with respe t to γS(i) and γR(i), we obtain (3.30), (3.31),

and (3.32) after letting ρ = µ/(1−µ) and λ = ν ln(2)/(1−µ), whi h are hosen su h

that onstraints C1 and C3 are meet with equality. This ompletes the proof.

B.5 Proof of Lemma 3.2

Sin e s(i) and r(i) are ergodi random pro esses, for N → ∞, the normalized sums

in C1 and C3 in (3.29) an be repla ed by expe tations. Therefore, the left hand

side of C1 is the expe tation of variable (1 − di) log2(1 + γS(i)hS(i)). This variable

is nonzero only when both (1 − di) and γS(i) are nonzero. The domain over whi h

(1− di) and γS(i) are jointly nonzero an be obtained from (3.30) and (3.32) and is

given by

(hS(i) > λ/ρ AND hR(i) < λ)

OR (hS(i) > L1 AND hR(i) > λ) (B.25)

176


where L1 is given by (3.36). Variable (1−di) log2(1+γS(i)hS(i)) has to be integrated

over domain (B.25) to obtain its average. This leads to the left side of (3.34).

Similarly, the right hand side of C1 is the expe tation of the variable di log2(1 +

γR(i)hR(i)). This variable is nonzero only when both di and γR(i) are nonzero. The

domain over whi h di and γR(i) are jointly nonzero an be obtained from (3.31) and

(3.32) and is given by

(hR(i) > λ AND hS(i) < λ/ρ)

OR (hR(i) > L2 AND hS(i) > λ/ρ) (B.26)

where L2 is given by (3.36). Variable di log2(1+γR(i)hR(i)) has to be integrated over

domain (B.26) to obtain its average. This leads to the right side of (3.34).

Following a similar pro edure, we an obtain (3.35) from C3 in (3.29). This

ompletes the proof.

177

Appendix C


C.1 Proof of Theorem 4.1

We rst note that, be ause of the law of the onservation of ow, A ≥ τ is always

valid and equality holds if and only if the queue is non-absorbing.

We denote the set of indi es with di = 1 by I and the set of indi es with di = 0

by I. Assume that we have an adaptive re eption-transmission proto ol with arrival

rate A and throughput τ with A > τ , i.e., the queue is absorbing. Then, for N → ∞,

we have

A =1

N

∑

i∈I

(1− di)OS(i)S0 > τ =1

N

∑

i∈I

diOR(i)minR0, Q(i− 1). (C.1)

From (C.1) we observe that the onsidered proto ol annot be optimal as the through-

put an be improved by moving some of the indi es i in I to I whi h leads to an

in rease of τ at the expense of a de rease of A. As we ontinue moving indi es from

I to I we rea h a point where A = τ holds. At this point, the queue be omes non-

absorbing (but is at the boundary between a non-absorbing and an absorbing queue)

and the throughput is maximized. If we ontinue moving indi es from I to I, in

general, A will de rease and as a onsequen e of the law of onservation of ow, τ

will also de rease. We note that A does not de rease if we move only those indi es

from I to I for whi h OS(i) = 0 holds. In this ase, A will not hange, and as a

178

Appendix C. Proofs for Chapter 4

onsequen e of the law of onservation of ow, the value of τ also remains un hanged.

Note that this is used in Lemma 4.1. However, the queue is moved from the edge

of non-absorption if OR(i) = 1 holds for some of the indi es moved from I to I. As

will be seen later, if the queue of the buer operates at the edge of non-absorption,

the throughput be omes independent of the state of the queue, whi h is desirable for

analyti al throughput maximization.

In the following, we will prove that when the queue is at the edge of non-absorption

the following holds

τ = limN→∞

1

N

N∑

i=1

diOR(i)R0 = A = limN→∞

1

N

N∑

i=1

(1− di)OS(i)S0. (C.2)

Let ǫ denote a small subset of I ontaining only indi es i for whi h OS(i) = 1,

where |ǫ|/N → 0 for N → ∞ and | · | denotes the ardinality of a set. Throughout

the remainder of this proof N → ∞ is assumed.

If the queue in the buer of the relay is absorbing, A > τ holds and on average

the number of bits arriving at the queue ex eed the number of bits leaving the queue.

Thus, R0 ≤ Q(i − 1) holds almost always and as a result the throughput an be

written as

τ =1

N

∑

i∈I

OR(i)minR0, Q(i− 1) =1

N

∑

i∈I

OR(i)R0. (C.3)

Now, we assume that the queue is at the edge of non-absorption. That is A = τ

holds but moving the small fra tion of indi es in ǫ, where |ǫ|/N → 0, from I to I will

make the queue an absorbing queue with A > τ . For this ase, we wish to determine

whether or not

1

N

∑

i∈I

OR(i)R0 > τ =1

N

∑

i∈I

OR(i)minR0, Q(i− 1) = A =1

N

∑

i∈I

OS(i)S0 (C.4)

179


holds. To test this, we move a small fra tion ǫ, where |ǫ|/N → 0, of indi es from I

to I, thus making the queue an absorbing queue. As a result, (C.3) holds and (C.4)

be omes

1

N

∑

i∈I\ǫ

OR(i)R0 = τ =1

N

∑

i∈I\ǫ

OR(i)minR0), Q(i− 1) = A =1

N

∑

i∈I∪ǫ

OS(i)S0.

(C.5)

From the above we on lude that if (C.3) holds, then based on (C.4) and (C.5), for

|ǫ|/N → 0, we must have

1

N

∑

i∈I

OR(i)R0 >1

N

∑

i∈I

OS(i)S0 (C.6)

and

1

N

∑

i∈I\ǫ

OR(i)R0 <1

N

∑

i∈I∪ǫ

OS(i)S0. (C.7)

However, for (C.6) and (C.7) to jointly hold, we require that the parti ular onsid-

ered move of indi es from I to I auses a dis ontinuity in

1N

∑

i∈I OR(i)R0 or/and

a dis ontinuity in

1N

∑

i∈I OS(i)S0 as |ǫ|/N → 0 is assumed. Sin e S0 and R0

are nite, limN→∞

∑

i∈ǫ S0/N = limN→∞ S0|ǫ|/N = 0 and limN→∞

∑

i∈ǫR0/N =

limN→∞R0|ǫ|/N = 0. Hen e, su h dis ontinuities are not possible. Therefore, at the

edge of non-absorption the inequality in (C.4) annot hold and we must have

1

N

∑

i∈I

OR(i)R0 = τ =1

N

∑

i∈I

OR(i)minR0, Q(i− 1) = A =1

N

∑

i∈I

OS(i)S0. (C.8)

Eq. (C.8) an be written as (4.31). This on ludes the proof.

180



To solve (4.33), we rst relax the binary onstraints di ∈ 0, 1 in (4.33) to 0 ≤ di ≤

1, ∀i. Thereby, we transform the original problem (4.33) into the following linear


Maximize :di

1N

∑Ni=1 diOR(i)R0


∑Ni=1(1− di)OS(i)S0 =

1N

∑Ni=1 diOR(i)R0

C2 : 0 ≤ di ≤ 1, ∀i.

(C.9)

In the following, we solve the relaxed problem (C.9) and then show that the optimal

values of di, ∀i are at the boundaries, i.e., di ∈ 0, 1, ∀i. Therefore, the solution of

the relaxed problem (C.9) is also the solution to the original maximization problem

in (4.33).

Sin e (C.9) is a linear optimization problem, we an solve it by using the method

of Lagrange multipliers. The Lagrangian for Problem (4.33) is given by

L =1

N

N∑

i=1

diOR(i)R0 − µ1

N

N∑

i=1

[diOR(i)R0 − (1− di)OS(i)S0]

+1

N

N∑

i=1

βidi −1

N

N∑

i=1

αi(di − 1), (C.10)


and αi/N have to satisfy (B.10).

Dierentiating L with respe t to di and setting the result to zero leads to

(1− µ)OR(i)R0 − µOS(i)S0 + βi − αi = 0. (C.11)


181


and from (C.11) we obtain that the following must hold

(1− µ)OR(i)R0 − µOS(i)S0 = 0. (C.12)

However, sin e OR(i) and OR(i) are independent random variables, (C.12) annot

hold for all i. Therefore, di has to be at the boundary, i.e., di ∈ 0, 1. Now,

assuming di = 0 leads βi ≥ 0 and αi = 0, whi h simplies (C.11) to

di = 0 ⇒ βi = µOS(i)S0 − (1− µ)OR(i)R0 ≥ 0. (C.13)

Whereas, assuming di = 1 leads βi = 0 and αi ≥ 0, whi h simplies (C.11) to

di = 1 ⇒ αi = (1− µ)OR(i)R0 − µOS(i)S0 ≥ 0. (C.14)

respe tively. (C.13) and (C.14) an be written equivalently as

di =

1 if (1− µ)OR(i)R0 ≥ µOS(i)S0

0 if (1− µ)OR(i)R0 ≤ µOS(i)S0.(C.15)

Furthermore, 0 ≤ µ ≤ 1 has to hold sin e for µ < 0 and µ > 1 we have always

di = 1 and di = 0, respe tively, irrespe tive of any non-negative values of OS(i)S0

and OR(i)R0. In this ase, sin e OS(i) and OR(i) are dis rete random variables, whi h

take the values zero or one, the probability of (1−µ)OR(i)R0 = µOS(i)S0 happening

is non-zero, and therefore this event has to be analyzed. This is done in the following.

First, we onsider the ase 0 < µ < 1. The boundary values µ = 0 and µ = 1 will

be investigated later. From (C.15), for 0 < µ < 1, we have four possibilities:

1. If OR(i) = 1 and OS(i) = 0, then di = 1.

182


2. If OR(i) = 0 and OS(i) = 1, then di = 0.

3. If OR(i) = 0 and OS(i) = 0, then di an be hosen to be either di = 0 or di = 1

and the hoi e does not inuen e the throughput as both the sour e and the

relay remain silent.

4. If OR(i) = 1 and OS(i) = 1 and µ is hosen su h that 0 < µ < R0/(S0+R0) then

di = 1 in all time slots with OR(i) = 1 and OS(i) = 1, and as a result, ondition

C1 annot be satised. Similarly, if µ is hosen su h that R0/(S0+R0) < µ < 1,

then di = 0 in all time slots with OR(i) = 1 and OS(i) = 1, and as a result

ondition C1 an also not be satised. Thus, we on lude that µ must be set

to µ = R0/(S0+R0) sin e only in this ase an di be hosen to be either di = 0

or di = 1, whi h is ne essary for satisfying ondition C1. Sin e for OR(i) = 1

and OS(i) = 1 neither link is in outage, di an be hosen to be either zero or

one, as long as ondition C1 is satised. In order to satisfy C1, we propose to

ip a oin and the out ome of the oin toss de ides whether di = 1 or di = 0.

Let the oin have two out omes C ∈ 0, 1 with probabilities PrC = 0 and

PrC = 1. We set di = 0 if C = 0 and di = 1 if C = 1. Thus, the probabilities

PrC = 0 and PrC = 1 have to be hosen su h that C1 is satised.

Choosing the link sele tion variable as in (4.35) and exploiting the independen e of

s(i) and r(i), ondition C1 results in

S0 [(1− PS)PR + (1− PS)(1− PR)PrC = 0]

= R0 [(1− PR)PS + (1− PS)(1− PR)PrC = 1] . (C.16)

From (C.16), we an obtain the probabilities PrC = 0 and PrC = 1, whi h after

some basi algebrai manipulations leads to (4.36). The throughput is given by the

183


right (or left) hand side of (C.16), whi h leads to (4.37).

For (4.36) to be valid, PrC = 0 and PrC = 1 have to meet 0 ≤ PrC = 0 ≤ 1

and 0 ≤ PrC = 1 ≤ 1, whi h leads to the onditions

S0(1− PS)− (1− PR)PSR0 ≥ 0 (C.17)

R0(1− PR)− (1− PS)PRS0 ≥ 0. (C.18)

Solving (C.17) and (C.18), we obtain that for the link sele tion variable di given in

(4.35) to be valid, ondition (4.34) has to be fullled.

Next, we onsider the ase where µ = 0. Inserting µ = 0 in (C.15), we obtain

three possible ases:

1. If OR(i) = 1, then di = 1.


and the hoi e has no inuen e on the throughput.


as long as ondition C1 is satised. Similar to before, in order to satisfy C1,

we propose to ip a oin and the out ome of the oin ip determines whether

di = 1 or di = 0.

Choosing the link sele tion variable as in (4.39) and exploiting the independen e of

s(i) and r(i), ondition C1 an be rewritten as

S0PR(1− PS)PrC = 0 = R0(1− PR). (C.19)

After basi manipulations (C.19) simplies to (4.40). The throughput is given by the

right (or left) hand side of (C.19) and an be simplied to (4.41). Imposing again the

184


onditions 0 ≤ PrC = 0 ≤ 1 and 0 ≤ PrC = 1 ≤ 1, we nd that for µ = 0, (C.17)

still has to hold but (C.18) an be violated, whi h is equivalent to the new ondition

PR >R0

R0 + S0(1− PS). (C.20)

For the third and nal ase, letting µ = 1 and following a similar path as for

µ = 0 leads to (4.43)(4.45) and ondition (4.42).

Finally, we have to prove that the three onsidered ases are mutually ex lusive,

i.e., for any ombination of PS and PR only one ase applies. Considering (4.34),

(4.38), and (4.42) it is obvious that Cases 1 and 2 and Cases 1 and 3 are mutually

ex lusive, respe tively. For Cases 2 and 3, the mutual ex lusiveness is less obvious.

Thus, we rewrite (4.38) and (4.42) as

PR > PR,2 (C.21)

and

PR < PR,3, (C.22)

respe tively, where PR,2 = R0/(R0 + S0(1−PS)) and PR,3 = 1+ S0/R0 − S0/(R0PS).

It an be shown that PR,2 > PR,3 for any 0 ≤ PS < 1. Hen e, for 0 ≤ PS < 1, at

most one of (C.21) and (C.22) is satised and Cases 2 and 3 are mutually ex lusive.

For PS = 1 (i.e., the S-R link is always in outage), we have PR,2 = PR,3 = 1 and

Case 1 and Case 3 apply for PR = 1 and PR < 1, respe tively. Therefore, for any

ombination of PS and PR only one of the three ases onsidered in Theorem 4.2

applies. This on ludes the proof.

185


C.3 Proof of Lemma 4.2

We provide two dierent proofs for the outage probability, Fout, in (4.52). The

rst proof is more straightforward and based on (4.10). However, the se ond proof

provides more insight into when outages o ur.

Proof 1: In the absen e of outages, the maximum a hievable throughput, denoted

by τ0, is given by (4.51). Thus, when (4.34) holds, Fout is obtained by inserting (4.37)

and (4.51) into (4.10). Similarly, when (4.38) holds, Fout is obtained by inserting

(4.41) and (4.51) into (4.10). Finally, when (4.42) holds, Fout is obtained by inserting

(4.45) and (4.51) into (4.10). After basi simpli ations, (4.52) is obtained. This

on ludes the proof.

Proof 2: The se ond proof exploits the fa t that an outage o urs when both the

sour e and the relay are silent, i.e., when none of the links is used. When (4.34) holds,

from di given by (4.35), we observe that no transmission o urs only when both links

are in outage. This happens with probability Fout = PSPR. In ontrast, when (4.38)

holds, from di given by (4.39), we observe that no node transmits when both links are

in outage or when the S-R link is not in outage, while the R-D link is in outage and

the oin ip hooses the relay for transmission. This event happens with probability

Fout = PSPR+(1−PS)PRPC , whi h after inserting PC given by (4.40) leads to (4.52).

Finally, when (4.42) holds, from di, given by (4.43), we see that no node transmits

when both links are in outage or when the S-R link is in outage, while the R-D link

is not in outage and the oin ip hooses the sour e for transmission. This happens

with probability Fout = PSPR+PS(1−PR)(1−PC), whi h after introdu ing PC given

by (4.44) leads to (4.52).

186



Computing the link outages in (4.7) and (4.8) for Rayleigh fading and exploiting

(4.54), we obtain (4.55) by employing ΩS = γΩS and ΩR = γΩR in the resulting

expression and using a Taylor series expansion for γ → ∞. As an be seen from

(4.55), the transmit SNR γ has an exponent of −2. Thus, the diversity order is two.

Moreover, for ΩS = ΩR = Ω and S0 = R0, the asymptoti expression for Fout in

(4.55) simplies to

Fout →(2R0 − 1)2

Ω2γ2, as γ → ∞. (C.23)

Furthermore, for S0 = R0, the asymptoti throughput in (4.53) simplies to τ =

R0/2. Thus, letting τ = r log2(1 + γ) we obtain R0 = 2r log2(1 + γ). Inserting R0 =

2r log2(1+γ) into (C.23), the diversity-multiplexing trade-o, DM(r), is obtained as

DM(r) = − limγ→∞

log2(Fout)

log2(γ)= − lim

γ→∞

2 log2(22r log2(1+γ) − 1)− 2 log2(Ω)− 2 log2(γ)

log2(γ)

= 2− limγ→∞

2 log2((1 + γ)2r − 1)

log2(γ)= 2− 4r . (C.24)

This ompletes the proof.


Let di be given by (4.59). Then, the following events are possible for the queue in

the buer:

1. If the buer is empty, it stays empty with probability PS and re eives one pa ket

with probability 1− PS.

187


0 1 2L. . . .

1-PS 1-PS 1-p-q1-p-q

1-pPS

P -qS p pp

q q

3

1-p-q

p

q

4

1-p-q

p

q

Figure C.1: Markov hain for the number of pa kets in the queue of the buer if the

link sele tion variable di is given by (4.59).

2. If the buer ontains one pa ket, it stays in the same state with probability

PSPR, sends the pa ket with probability PS(1−PR), and re eives a new pa ket

with probability 1− PS.

3. If the buer ontains more than one pa ket but less than L pa kets, it stays in

the same state with probability PSPR, re eives a new pa ket with probability

(1− PS)PR + (1− PS)(1− PR)(1− PC), and sends one pa ket with probability

(1− PR)PS + (1− PS)(1− PR)PC .

4. If the buer ontains L pa kets, it stays in the same state with probability

PSPR + (1 − PS)PR + (1 − PS)(1 − PR)(1 − PC), and sends one pa ket with

probability (1− PR)PS + (1− PS)(1− PR)PC .

The events for the queue of the buer detailed above, form a Markov hain whose

states are dened by the number of pa kets in the queue. This Markov hain is shown

in Fig. C.1, where the probabilities p and q are given by (4.65). Let M denote the

state transition matrix of the Markov hain and let mi,j denote the element in the

i-th row and j-th olumn of M. Then, mi,j is the probability that the buer will

transition from having i− 1 pa kets in its queue in the previous time slot to having

j−1 pa kets in its queue in the following time slot. The non-zero elements of matrix

188


M are given by

m1,1 = PS , m1,2 = 1− PS , m2,1 = PS − q ,

m2,3 = 1− PS , mL+1,L+1 = 1− p

mi,i+1 = 1− p− q , mi+1,i = p , mi,i = q , for i = 1...L.

(C.25)

Let PrQ = [PrQ = 0, PrQ = R0, ...,PrQ = LR0] denote the steady state

probability ve tor of the onsidered Markov hain, where PrQ = kR0, k = 0, . . . , L,

is the probability of having k pa kets in the buer. The steady state probability ve tor

is obtained by solving the following system of equations

PrQM = PrQ

∑Lk=0 PrQ = kR0 = 1

, (C.26)

whi h leads to (4.64). Using (4.64) the average queue size EQ an be obtained

from

EQ = R0

L∑

k=0

kPrQ = kR0, (C.27)

whi h leads to (4.66). Furthermore, the average arrival rate an be found as

A = R0

[

(1− PS)(

PrQ = 0+ PrQ = R0)

+(1− p− q)(

1− PrQ = 0 − PrQ = R0 − PrQ = LR0)]

. (C.28)

Inserting the average arrival rate given by (C.28) and the average queue size given

by (4.66) into (4.57) yields the average delay in (4.67).

189


0 1 2L. . . .

1-PS 1-p-q 1-p-q1-p-q

1-pPS

p p pp

q q

3

1-p-q

p

q

4

1-p-q

p

q

Figure C.2: Markov hain for the number of pa kets in the queue of the buer if the

link sele tion variable di is given by (4.61) or (4.63).

For the ase when di is given by either (4.61) or (4.63), the queue in the buer

of the relay an be modeled by the Markov hain shown in Fig. C.2. If the link

sele tion variable di is given by (4.61), p and q are given by (4.65), and if the link

sele tion variable di is given by (4.63), p and q are given by (4.70). Following the

same pro edure as before, (4.69)-(4.73) an be obtained. This ompletes the proof.


Let us rst assume that 2p + q − 1 < 0, whi h is equivalent to p < 1 − p− q. Now,

sin e L → ∞, pL goes to zero faster than (1 − p − q)L. Thus, by using pL = 0 as

L→ ∞ in (4.67) and (4.72) , we obtain in both ases

ET =L

p−

1

1− 2p− q. (C.29)

Thus, we on lude that if 2p + q − 1 < 0, ET grows with L and is unlimited as

L→ ∞. Thus, if ET is to be limited as L→ ∞, 2p+ q − 1 > 0 has to hold.

If 2p + q − 1 > 0, as L → ∞, (1 − p − q)L goes to zero faster than pL. Hen e,

(4.74)-(4.82) are obtained by letting (1 − p − q)L = 0, as L → ∞, in the relevant

equations in Theorem 4.3 and inserting the orresponding p and q given by (4.65)

and (4.70) into the resulting expressions. This on ludes the proof.

190



The minimum and maximum possible delays that the onsidered buer-aided relaying

system an a hieve are obtained for PC = 1 and PC = 0, respe tively. If di is given

by (4.59), the delay is given by (4.75). By setting PC = 1 in (4.75) we obtain

the minimum possible delay in (4.83). However, sin e (4.75) is valid only when

2p+ q− 1 > 0, (4.83) is valid only when PR < 1/(2−PS). This ondition is obtained

by inserting PC = 1 into the expressions for p and q given by (4.65) and exploiting

2p + q − 1 > 0. On the other hand, in order to get the maximum delay given in

(4.84), we set PC = 0 in (4.75). The derived maximum delay is valid only when

PS > 1/(2 − PR), whi h is obtained from 2p + q − 1 > 0 and inserting PC = 0 into

the expressions for p and q given by (4.65).

A similar approa h an be used to derive the delay limits Tmin,2, Tmax,2, Tmin,3,

and Tmax,3 valid for the ases when di is given by (4.61) and (4.63). This on ludes

the proof.


The outage probability, Fout, an be derived based on two dierent approa hes. The

rst approa h is straightforward and based on (4.10). However, the se ond approa h

provides more insight into how and when the outages o ur and is based on ounting

the time slots in whi h no transmissions o ur. In the following, we provide a proof

based on the latter approa h.

If di is given by (4.59) or (4.61), there are four dierent ases where no node

transmits.

1. The buer is empty and the S-R link is in outage.

191


2. The buer in not empty nor full and both the S-R and R-D links are in outage.

3. The buer is full and the S-R link is not in outage while the R-D link is in

outage. In this ase, the sour e is sele ted for transmission but sin e the buer

is full, the pa ket is dropped.

4. The buer is full, both the S-R and R-D links are not in outage, and the sour e

is sele ted for transmission based on the oin ip. In this ase, sin e the buer

is full, the pa ket is dropped.

Summing up the probabilities for ea h of the above four ases, we obtain (4.87).

If di is given by (4.63), an outage o urs in three ases: Case 1 and Case 2 as

des ribed above, and a new Case 3. In the new Case 3, the buer is full, the S-R

link is not in outage while the R-D link is in outage, and the sour e is sele ted for

transmission based on the oin ip. Summing up the probabilities for ea h of the

three ases, we obtain (4.88).


For delay onstrained transmission with ET < L, the probability of dropped pa k-

ets PrQ = LR0 an be made arbitrarily small by in reasing the buer size L. Thus,

for large enough L, we an set PrQ = LR0 = 0 in (4.87) and (4.88).

In the high SNR regime, when PS → 0 and PR → 0, PR < 1/(2 − PS) and

PS < 1/(2 − PR) always hold. Using PS → 0 and PR → 0 in the delays spe ied

in Proposition 4.1, we obtain the onditions ET > 3 and 1 < ET ≤ 3 if link

sele tion variable di is given by (4.59) and (4.61), respe tively.

We rst onsider the ase ET > 3, where di is given by (4.59). Thus, the

probability of the buer being empty, PrQ = 0, is given by (4.74). Using PS → 0

192


and PR → 0 in (4.74), we obtain

PrQ = 0 = PS

(

1−1

2PC

)

. (C.30)

On the other hand, using PS → 0 and PR → 0 in the expression for ET in (4.75),

we obtain

ET =1

2PC − 1+ 2. (C.31)

Solving (C.31) for PC yields

PC =1

2

(

1 +1

ET − 2

)

. (C.32)

Inserting (C.32) into (C.30) we obtain

PrQ = 0 =PS

ET − 1. (C.33)

Finally, inserting (C.33) into (4.87) and setting PrQ = LR0 = 0, we obtain (4.91).

Now, we onsider the ase 1 < ET ≤ 3, where di is given by (4.61). Here, the

probability of the buer being empty, PrQ = 0, is given by (4.77). For PS → 0

and PR → 0, we obtain from (4.77)

PrQ = 0 = 1−1

2PrC = 1. (C.34)

Furthermore, for PS → 0 and PR → 0, we obtain from (4.78) the asymptoti delay

ET =1

2PC − 1(C.35)

193


or equivalently

PC =1

2

(

1 +1

ET

)

. (C.36)

Inserting (C.36) into (C.34) we obtain

PrQ = 0 =1

ET+ 1. (C.37)

Finally, inserting (C.37) into (4.88) and setting PrQ = LR0 = 0, we obtain (4.90).



To solve (4.94), we rst relax the binary onstraints di ∈ 0, 1 in (4.94) to 0 ≤ di ≤ 1,

∀i. Thereby, we transform the original problem (4.94) into a linear programing

problem whose Lagrangian is given by

L =1

N

N∑

i=1

di log2(

1 + r(i))

− µ1

N

N∑

i=1

[

di log2(

1 + r(i))

−(1− di)OS(i)S0

]

+1

N

N∑

i=1

βidi −1

N

N∑

i=1

αi(di − 1), (C.38)


and αi/N have to satisfy (B.10). Dierentiating L with respe t to di and setting the

result to zero leads to

(1− µ) log2(

1 + r(i))

− µOS(i)S0 + βi − αi = 0. (C.39)

194




(1− µ) log2(

1 + r(i))

− µOS(i)S0 = 0. (C.40)

However, sin e r(i) and OR(i) are independent random variables, (C.40) annot hold

for all i. Therefore, di has to be at the boundary, i.e., di ∈ 0, 1. Now, assuming

di = 0 leads βi ≥ 0 and αi = 0, whi h simplies (C.39) to

di = 0 ⇒ βi = µOS(i)S0 − (1− µ) log2(

1 + r(i))

≥ 0. (C.41)


di = 0 ⇒ αi = −µOS(i)S0 + (1− µ) log2(

1 + r(i))

≥ 0. (C.42)

Relations (C.41) and (C.42), an be written equivalently as

di =

1 if (1− µ) log2(

1 + r(i))

≥ µOS(i)S0

0 if (1− µ) log2(

1 + r(i))

≤ µOS(i)S0,(C.43)

Sin e for µ < 0 and µ > 1, we have always di = 1 and di = 0, respe tively, irrespe tive

of the (non-negative) values of log2(

1 + r(i))

and OS(i)S0, 0 ≤ µ ≤ 1 has to hold.

Let us rst onsider the ase 0 < µ < 1 and investigate the boundary values µ = 0

and µ = 1 later. For 0 < µ < 1, (C.43) an be written in the form of (4.96) after

setting ρ = µ/(1−µ), where ρ is hosen su h that onstraint C1 of problem (4.94) is

met. Denoting the PDFs of s(i) and r(i) by fs(s) and fr(r) onstraint C1 of problem

(4.94) an be rewritten as in (4.97), whi h is valid for ρ in the range of ρ = [0,∞).

195


Thus, by setting ρ = ∞ in (4.97), we obtain the entire domain over whi h (4.96) is

valid, whi h leads to ondition (4.95).

Next, we onsider the boundary values µ = 0 and µ = 1. The boundary value

µ = 0 or equivalently ρ = 0 is relevant only in the trivial ase when the S-R link is

never in outage (i.e. PS = 0) and S0 = ∞, where a trivial solution is given by d1 = 0

and di = 1 for i = 2, . . . , N and N → ∞.

The other boundary value, µ = 1, is invoked only when by using di as dened in

(4.96), onstraint C1 annot be satised even when ρ → ∞, whi h is the ase when

ondition (4.95) does not hold. Therefore, if (4.95) does not hold, we set µ = 1 in

(C.43) and obtain the following ases:

1. If OS(i) = 1, then di = 0.

2. If OS(i) = 0, then di an be hosen arbitrarily to be either zero or one as long

as onstraint C1 holds.

However, the same throughput as obtained when OS(i) = 0 and di is hosen su h that

onstraint C1 holds, an also be obtained by hoosing always di = 1 when OS(i) = 0

resulting in (4.98). The reason behind this is as follows: Assume there is a poli y for

whi h when OS(i) = 0, di is hosen su h that onstraint C1 holds. Now, we hange di

from 0 to 1 for OS(i) = 0. However, this hange does not ae t the (average) amount

of data entering the buer. Thus, be ause of the law of onservation of ow, the

average amount of data entering the buer per time slot is identi al to the average

amount of data leaving the buer per time slot (the throughput), and the throughput

is not ae ted by the hange.

196



The Lagrangian of the relaxed optimization problem of (4.105) where 0 ≤ di ≤ 1 is

assumed, is given by

L =1

N

N∑

i=1

di log2(1 + γR(i)hR(i)) +1

N

N∑

i=1

βidi −1

N

N∑

i=1

αi(di − 1)

− µ1

N

N∑

i=1

[

di log2(1 + γR(i)hR(i))− (1− di)OS(i)S0

]

− ν1

N

N∑

i=1

[

(1− di)OS(i)γS + diγR(i)]

, (C.44)

where βi/N and αi/N have to satisfy (B.10), and where the Lagrange multipliers µ

and ν are hosen su h that C1 and C3 hold, respe tively.

By dierentiating L with respe t to γR(i) and di, and setting the results to zero,

we obtain the following two equations

−νdi +dihR(i)

(1 + γR(i)hR(i)) ln(2)− µ

dihR(i)

(1 + γR(i)hR(i)) ln(2)= 0, (C.45)

−αi + βi − ν(γR(i)− OS(i)γS) + (1− µ) log2(1 + γR(i)hR(i))− µOS(i)S0 = 0. (C.46)



−ν(γR(i)−OS(i)γS) + (1− µ) log2(1 + γR(i)hR(i))− µOS(i)S0 = 0. (C.47)

However, sin e hR(i) and OS(i) are random, (C.47) annot hold for all i. Therefore,

di has to be at the boundary, i.e., di ∈ 0, 1. Now, assuming di = 0 leads βi ≥ 0

197


and αi = 0, whi h simplies (C.46) to

βi = ν(γR(i)− OS(i)γS)− (1− µ) log2(1 + γR(i)hR(i)) + µOS(i)S0 ≥ 0. (C.48)


αi = −ν(γR(i)− OS(i)γS) + (1− µ) log2(1 + γR(i)hR(i))− µOS(i)S0 ≥ 0. (C.49)

From (C.48) and (C.49), we obtain the following solution for di

di =

1 if (1− µ) log2(1 + γR(i)hR(i))− νγR(i) ≥ µOS(i)S0 − νOS(i)γS

0 if (1− µ) log2(1 + γR(i)hR(i))− νγR(i) ≤ µOS(i)S0 − νOS(i)γS.

(C.50)

Inserting (C.50) into (C.45) and solving with respe t to γR(i), and taking into a ount

that 0 < µ < 1, and ν > 0, we obtain (4.108) and (4.109) after letting ρ = ln(2)µ/(1−

µ) and λ = ln(2)ν/(1−µ), whi h are hosen su h that onstraints C1 and C3 are met

with equality. Given the PDFs fhS(hS) and fhR

(hR), onditions (4.93) and (4.104)

an be dire tly written as (4.110) and (4.111), respe tively. Setting ρ→ ∞ in (4.110)

and (4.111), we obtain ondition (4.106) whi h is ne essary for the validity of (4.98).

Similar to the xed transmit power ase, the boundary value µ = 0 is trivial. On

the other hand, for µ = 1, we obtain that di has to be set to di = 0 when OS(i) = 1

and for OS(i) = 0, di an be hosen arbitrarily. Similar to the xed power ase, we

set di = 1 when OS(i) = 1 in order to minimize the delay. Thus, the optimal power

and link sele tion variables are given by (4.113) and (4.114), respe tively, and the

throughput is given by (4.115).

198



For γS = γR = γ → ∞, the proto ol in Proposition 4.3 is optimal in the sense

that it maximizes the throughput while satisfying the average delay onstraint. In

parti ular, for high SNR in the S-R link, the probability that the link is in outage

approa hes zero and the relay re eives S0 bits per sour e transmission. On the other

hand, the number of bits transmitted by the relay in one time slot over the R-D

link in reases with the SNR. Thus, for su iently high SNR, the sour e transmits

kS0 bits in k time slots and the relay needs just p = 1 time slot to forward the

entire information to the destination. Hen e, every transmission period omprises

k + p = k + 1 time slots, where the queue length at the relay in reases from S0 to

kS0 in the rst k time slots and is redu ed to zero in the (k+ 1)th time slot. Hen e,

the average queue length, EQ, an be written as

EQ →1

k + 1(1 + 2 + ... + k + 0)S0 =

1

k + 1

k(k + 1)

2S0

=k

2S0, as γ → ∞ . (C.51)

On the other hand, the arrival rate is identi al to the throughput and given by (4.121),

and for high SNR it onverges to

A = τ → S0k

k + 1, as γ → ∞ . (C.52)

Combining (4.57), (C.51), and (C.52) the average delay is found as

ET →k + 1

2, as γ → ∞ . (C.53)

199


Finally, ombining (C.52) and (C.53) the throughput an be expressed as (4.122),

and the multiplexing gain in (4.123) follows dire tly.

200

Appendix D

Other Contributions

I have also o-authored other resear h works whi h have been published or submitted

for publi ation during my time as a Ph.D. student at UBC. In parti ular, the following

papers have been published or submitted for publi ation.

Journal Papers:

• N. Zlatanov, V. Jamali, and R. S hober, A hievable Rates for the Fading Half-

Duplex Single Relay Sele tion Network Using Buer-Aided Relaying, A epted

to IEEE Transa tions on Wireless Communi ations, 2015.

• V. Jamali, N. Zlatanov, H. Shoukry, and R. S hober, A hievable Rate of the

Half-Duplex Multi-Hop Buer-Aided Relay Channel with Blo k Fading, A -

epted to IEEE Transa tions on Wireless Communi ations, 2015.

• V. Jamali, N. Zlatanov, and R. S hober, Buer-Aided Bidire tional Relay

Networks with Fixed Rate Transmission Part I: Delay-Un onstrained Case,

IEEE Transa tions on Wireless Communi ations, vol. 14, no. 3, pp. 1323 -

1338, Mar. 2015.

• V. Jamali, N. Zlatanov, and R. S hober, Bidire tional Buer-Aided Relay

Networks with Fixed Rate Transmission Part II: Delay-Constrained Case,

IEEE Transa tions on Wireless Communi ations, vol. 14, no. 3, pp. 1339 -

1355, Mar. 2015.

201

Appendix D. Other Contributions

• Z. Hadzi-Velkov, N. Zlatanov, and R. S hober, Multiple-a ess Fading Channel

with Wireless Power Transfer and Energy Harvesting, IEEE Communi ations

Letters, vol. 52, no. 4, pp. 1863 - 1866, Sep. 2014.

• V. Jamali, N. Zlatanov, A. Ikhlef, and R. S hober, A hievable Rate Region

of the Bidire tional Buer-Aided Relay Channel with Blo k Fading, IEEE

Transa tions on Information Theory, vol. 60, no. 11, pp. 7090 - 7111, Sep.

2014.

• N. Zlatanov, A. Ikhlef, T. Islam, and R. S hober, Buer-Aided Cooperative

Communi ations: Opportunities and Challenges, IEEE Communi ations Mag-

azine, Vol. 52, no. 4, Apr. 2014.

• N. Zlatanov and R. S hober, Buer-Aided Half-Duplex Relaying Can Outper-

form Ideal Full-Duplex Relaying, IEEE Communi ations Letters, vol. 17, no.

3, pp. 479-482, Mar. 2013.

• N. Zlatanov, R. S hober, and Z. Hadzi-Velkov, Asymptoti ally Optimal Power

Allo ation for Energy Harvesting Communi ation Networks, Submitted for

publi ation.

Conferen e Papers:

• W. Wi ke, N. Zlatanov, V. Jamali, and R. S hober, Buer-Aided Relaying

with Dis rete Transmission Rates, Pro . of IEEE 14th Canadian Workshop

on Information Theory (CWIT), St. John's, NL, Canada, July 2015.

• R. Simoni, V. Jamali, N. Zlatanov, R. S hober, L. Pieru i, and R. Fanta i,

Buer-Aided Diamond Relay Network with Blo k Fading, Pro . of IEEE

International Conferen e on Communi ations (ICC), London, UK, June 2015.

202


• N. Zlatanov, V. Jamali, and R. S hober, A hievable Rates for the Fading Half-

Duplex Single Relay Sele tion Network Using Buer-Aided Relaying, Pro . of

IEEE Globe om 2014, Austin, TX, De . 2014

• V. Jamali, N. Zlatanov, and R. S hober, A Delay-Constrained Proto ol with

Adaptive Mode Sele tion for Bidire tional Relay Networks, Pro . of IEEE

Globe om 2014, Austin, TX, De . 2014

• H. Shoukry, N. Zlatanov, V. Jamali, and R. S hober, A hievable Rates for the

Fading Three-Hop Half-Duplex Relay Network using Buer-Aided Relaying,

Pro . of IEEE Globe om 2014, Austin, TX, De . 2014

• V. Jamali, N. Zlatanov, and R. S hober, Adaptive Mode Sele tion for Bidire -

tional Relay Networks - Fixed Rate Transmission, Pro . of IEEE International

Conferen e on Communi ations (ICC), Sydney, Australia, June 2014.

• N. Zlatanov, Z. Hadzi-Velkov, and R. S hober, Asymptoti ally Optimal Power

Allo ation for Point-to-Point Energy Harvesting Communi ation Systems, Pro .

of IEEE Globe om 2013, Atlanta, GA, De . 2013.

• V. Jamali, N. Zlatanov, A. Ikhlef, and R. S hober, Adaptive Mode Sele tion in

Bidire tional Buer-aided Relay Networks with Fixed Transmit Powers, Pro .


• Z. Hadzi-Velkov, N. Zlatanov, and R. S hober, Optimal Power Control for

Analog Bidire tional Relaying with Long-Term Relay Power Constraint, Pro .


• V. Jamali, N. Zlatanov, A. Ikhlef, and R. S hober, Adaptive Mode Sele tion in

Bidire tional Buer-aided Relay Networks with Fixed Transmit Powers, Pro .

203


of EUSIPCO, Marrake h, Maro o, Sep. 2013.

• Z. Hadzi-Velkov, N. Zlatanov, and R. S hober, Optimal Power Allo ation for

Three-phase Bidire tional DF Relaying with Fixed Rates, Pro . of ISWCS

2013, Ilmenau, Germany, Aug. 2013.

• N. Zlatanov, R. S hober, and L. Lampe, Buer-Aided Relaying in a Three Node

Network, Pro . of IEEE International Symposium on Information Theory

(ISIT 2012), Cambridge, MA, July 2012.

• N. Zlatanov, Z. Hadzi-Velkov, G. K. Karagiannidis, and R. S hober, Outage

Rate and Outage Duration of De ode-and-Forward Cooperative Diversity Sys-

tems, Pro . of IEEE International Conferen e on Communi ations (ICC),

Kyoto, Japan, June 2011.

• N. Zlatanov, R. S hober, G. K. Karagiannidis, and Z. Hadzi-Velkov, Aver-

age Outage and Non-Outage Duration of Sele tive De ode-and-Forward Relay-

ing, Pro . of IEEE 12th Canadian Workshop on Information Theory (CWIT),

Kelowna, Canada, May 2011.

204

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