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...
Transcript of 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.
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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
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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
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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
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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
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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
B.2 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B.3 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 172
B.4 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 174
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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.2 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 181
C.3 Proof of Lemma 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
C.4 Proof of Lemma 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
C.5 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 187
C.6 Proof of Lemma 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
C.7 Proof of Lemma 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
C.8 Proof of Theorem 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 191
C.9 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 192
C.10 Proof of Theorem 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . 194
C.11 Proof of Theorem 4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . 197
C.12 Proof of Theorem 4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . 199
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 1. Introdu tion
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
(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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
...
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
(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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
−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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
−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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 2. Capa ity of the Two-Hop Half-Duplex Relay Channel Without Fading
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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)
Proof. Please refer to Appendix B.2.
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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).
Proof. Please refer to Appendix B.3.
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
to the following optimization problem:
Maximize :γS(i)≥0,γR(i)≥0,di
1N
∑Ni=1 di log2(1 + γR(i)hR(i))
Subject to : C1 : 1N
∑Ni=1(1− di) log2(1 + γS(i)hS(i))
= 1N
∑Ni=1 di log2(1 + γR(i)hR(i))
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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)
Proof. Please refer to Appendix B.4.
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
∫ 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).
Proof. Please refer to Appendix B.5.
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
−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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
−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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Subject to : C1 : 1N
∑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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
1 if OS(i) = 1 AND OR(i) = 1 AND C = 1
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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 =
0 if OS(i) = 1 AND OR(i) = 0 AND C = 0
1 if OS(i) = 1 AND OR(i) = 0 AND C = 1
1 if OS(i) = 0 AND OR(i) = 1
1 if OS(i) = 1 AND OR(i) = 1
ε if OS(i) = 0 AND OR(i) = 0
(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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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) = 0 AND OR(i) = 1 AND C = 0
1 if OS(i) = 0 AND OR(i) = 1 AND C = 1
0 if OS(i) = 1 AND OR(i) = 1
ε if OS(i) = 0 AND OR(i) = 0
(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)
Proof. Please refer to Appendix C.2.
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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)
Proof. Please refer to Appendix C.3.
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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)
Proof. Please refer to Appendix C.4.
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
we propose the following adaptive re eption-transmission variable di to be used:
If Q(i− 1) = 0 and OS(i) = 1, then di = 0,
otherwise di is given by (4.35). (4.61)
Case 3: If the required delay ET satises
1
1− PR
< ET ≤1
1− PR (2− PS), (4.62)
we propose the following adaptive re eption-transmission variable di to be used:
If Q(i− 1) = 0 and OS(i) = 1, then di = 0,
otherwise di is given by (4.39). (4.63)
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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)
Proof. Please refer to Appendix C.5.
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
Case 1: If the adaptive re eption-transmission variable di is given by (4.59), the
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)
Case 2: If the adaptive re eption-transmission variable di is given by (4.61), the
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)
Case 3: If the adaptive re eption-transmission variable di is given by (4.63), the
probability of the buer being empty, the average delay, ET, and the throughput,
120
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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).
Proof. Please refer to Appendix C.6.
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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)
Case 2: If the adaptive re eption-transmission variable di is given by (4.61), then
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.
Case 3: If the adaptive re eption-transmission variable di is given by (4.63), then
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.
Proof. Please refer to Appendix C.7.
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Fout = PSPrQ = 0+ PSPR
(
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).
Proof. Please refer to Appendix C.8.
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
plies to
Fout = PSPrQ = 0+ PSPR
(
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.
Proof. Please refer to Appendix C.9.
A ording to Theorem 4.5, for Rayleigh fading, a diversity gain of two an be
124
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
lem
Maximize :di
1N
∑Ni=1 di log2
(
1 + r(i))
Subject to : C1 : 1N
∑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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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)
Proof. Please refer to Appendix C.10.
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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))
Subject to : C1 : 1N
∑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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
where λ = λt is the solution to (4.107). In this ase, the maximum throughput is
given by
τ = S0(1− PS). (4.115)
Proof. Please refer to Appendix C.11.
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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)
Proof. Please refer to Appendix C.12.
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
−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.
Transmission Without Delay Constraints
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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.
Transmission With Delay Constraints
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
(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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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.
Transmission Without Delay Constraints
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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.
Transmission With Delay Constraints
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission
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
Chapter 5. Summary of Thesis and Future Resear h Topi s
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
Chapter 5. Summary of Thesis and Future Resear h Topi s
• 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.
150
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158
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
Appendix A. Proofs for Chapter 2
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
Appendix A. Proofs for Chapter 2
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
Appendix A. Proofs for Chapter 2
obtain
Pr(ǫ) ≤ ǫ+2kR−1∑
j=1
2−|R1(b)|(I(X1;Y1|X2=0)−3ǫ)
≤ ǫ+ (2kR − 1)2−|R1(b)|(I(X1;Y1|X2=0)−3ǫ)
≤ ǫ+ 2kR2−|R1(b)|(I(X1;Y1|X2=0)−3ǫ)
= ǫ+ 2−k((|R1(b)|/k)I(X1;Y1|X2=0)−R−3(1−P ∗U )ǫ). (A.9)
Hen e, if
R <|R1(b)|
kI(X1; Y1|X2 = 0)− 3(1− P ∗
U)ǫ
= (1− P ∗U)I(X1; Y1|X2 = 0)− 3(1− P ∗
U)ǫ, (A.10)
then limǫ→0
limk→∞
Pr(ǫ) = 0. This on ludes the proof for ase when |R(b)| ≥ k(1 − P ∗U)
holds. We now turn to ase two when |R(b)| < k(1− P ∗U) holds.
When |R(b)| = k(1 − P ∗U) − ε(b) < k(1 − P ∗
U) holds, then the sour e annot
transmit all of its k(1−P ∗U) symbols omprising odeword x1|r(b) sin e there are not
enough zeros in odeword x2(b). Instead, the relay transmits only k(1 − P ∗U) − ε(b)
symbols of odeword x1|r(b), and we denote the resulting transmitted odeword by
x∗1|r(b). Note that the length of odewords x∗
1|r(b) and y1|r(b), and the ardinality
of R(b) are all identi al and equal to k(1 − P ∗U) − ε(b). In addition, let the relay
generate a odebook C∗1|r(b) by keeping only the rst k(1− P ∗
U)− ε(b) symbols from
ea h odeword in odebook C1|r and dis arding the remaining ε(b) symbols in the
orresponding odewords. Let us denote the odewords in C∗1|r(b) by x∗
1|r. Note that
there is a unique one to one mapping from the odewords in C∗1|r(b) to the odewords
in C1|r(b) sin e when k → ∞, k(1−P ∗U)−ε(b) → ∞ also holds, i.e., the lengths of the
163
Appendix A. Proofs for Chapter 2
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
Appendix A. Proofs for Chapter 2
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
Appendix A. Proofs for Chapter 2
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
Appendix A. Proofs for Chapter 2
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
Appendix A. Proofs for Chapter 2
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.
This on ludes the proof.
168
Appendix B
Proofs for Chapter 3
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
B.2 Proof of Theorem 3.2
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
Appendix B. Proofs for Chapter 3
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
Appendix B. Proofs for Chapter 3
B.3 Proof of Theorem 3.3
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))
Subject to : C1 : 1N
∑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
Appendix B. Proofs for Chapter 3
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
Appendix B. Proofs for Chapter 3
B.4 Proof of Theorem 3.4
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
optimization problem
Maximize :γS(i)≥0, γR(i)≥0, di
1N
∑Ni=1 di log2(1 + γR(i)hR(i))
Subject to : C1 : 1N
∑Ni=1(1− di) log2(1 + γS(i)hS(i))
= 1N
∑Ni=1 di log2(1 + γR(i)hR(i))
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
Appendix B. Proofs for Chapter 3
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)
If we assume that 0 < di < 1, i.e., di is not at the boundary, then βi = αi = 0 holds,
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
Appendix B. Proofs for Chapter 3
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
Appendix B. Proofs for Chapter 3
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
Proofs for Chapter 4
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
Appendix C. Proofs for Chapter 4
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
Appendix C. Proofs for Chapter 4
C.2 Proof of Theorem 4.2
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
optimization problem
Maximize :di
1N
∑Ni=1 diOR(i)R0
Subject to : C1 : 1N
∑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)
where µ, βi/N , and αi/N are Lagrange multipliers. The Lagrange multipliers βi/N
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)
If we assume that 0 < di < 1, i.e., di is not at the boundary, then βi = αi = 0 holds,
181
Appendix C. Proofs for Chapter 4
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
Appendix C. Proofs for Chapter 4
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
Appendix C. Proofs for Chapter 4
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.
2. 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 has no inuen e on the throughput.
3. If OR(i) = 0 and OS(i) = 1, then di an be hosen to be either di = 0 or di = 1
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
Appendix C. Proofs for Chapter 4
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
Appendix C. Proofs for Chapter 4
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
Appendix C. Proofs for Chapter 4
C.4 Proof of Lemma 4.4
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.
C.5 Proof of Theorem 4.3
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
Appendix C. Proofs for Chapter 4
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
Appendix C. Proofs for Chapter 4
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
Appendix C. Proofs for Chapter 4
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.
C.6 Proof of Lemma 4.6
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
Appendix C. Proofs for Chapter 4
C.7 Proof of Lemma 4.7
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.
C.8 Proof of Theorem 4.4
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
Appendix C. Proofs for Chapter 4
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).
C.9 Proof of Theorem 4.5
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
Appendix C. Proofs for Chapter 4
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
Appendix C. Proofs for Chapter 4
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).
This on ludes the proof.
C.10 Proof of Theorem 4.7
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)
where µ, βi/N , and αi/N are Lagrange multipliers. The Lagrange multipliers βi/N
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
Appendix C. Proofs for Chapter 4
If we assume that 0 < di < 1, i.e., di is not at the boundary, then βi = αi = 0 holds,
and from (C.39) we obtain that the following must hold
(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)
Whereas, assuming di = 1 leads βi = 0 and αi ≥ 0, whi h simplies (C.39) to
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
Appendix C. Proofs for Chapter 4
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
Appendix C. Proofs for Chapter 4
C.11 Proof of Theorem 4.8
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)
If we assume that 0 < di < 1, i.e., di is not at the boundary, then βi = αi = 0 holds,
and from (C.46) we obtain that the following must hold
−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
Appendix C. Proofs for Chapter 4
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)
Whereas, assuming di = 1 leads βi = 0 and αi ≥ 0, whi h simplies (C.46) to
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
Appendix C. Proofs for Chapter 4
C.12 Proof of Theorem 4.9
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
Appendix C. Proofs for Chapter 4
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
Appendix D. Other Contributions
• 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 .
of IEEE Globe om 2013, Atlanta, GA, De . 2013.
• Z. Hadzi-Velkov, N. Zlatanov, and R. S hober, Optimal Power Control for
Analog Bidire tional Relaying with Long-Term Relay Power Constraint, 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 .
203
Appendix D. Other Contributions
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