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Novel protocols for the two-hop half-duplex relay network Zlatanov, Nikola 2015

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NOVEL PROTOCOLS FOR THE TWO-HOP HALF-DUPLEX RELAYNETWORKbyNikola ZlatanovM. Si., Ss. Cyril and Methodius University, 2010Dipl. Eng., Ss. Cyril and Methodius University, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faulty of Graduate and Postdotoral Studies(Eletrial and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vanouver)July 2015© Nikola Zlatanov, 2015AbstratWireless ommuniation has enabled people to be onneted from anywhere and atany time. This has had a profound impat on human soiety. Currently, wirelessommuniation is performed using the ommuniation protools developed for ellu-lar and wireless loal area networks. Although these protools support a broad rangeof mobile servies, they do not fully exploit the apaity of the underlying networksand annot satisfy the exponential growth in demand for higher data rates and morereliable onnetions. Therefore, new ommuniation protools have to be developedfor general wireless networks in order to meet this demand. Ultimately, these pro-tools will have to be able to reah the fundamental limits of information ow inwireless networks, i.e., the network apaity. However, due to the omplexity of theproblem, it is urrently not known how to design suh protools for general wirelessnetworks. Therefore, in order to get insight into this problem, as a rst step, ommu-niation protools for very simple wireless networks have to be devised. Later, thegained knowledge an be exploited to design protools for more omplex networks.In this thesis, we propose new ommuniation protools for the simplest half-duplex relay network, whih is also the most basi building blok of any wirelessnetwork, the two-hop half-duplex relay network. This network is omprised of asoure, a half-duplex relay, and a destination where a diret soure-destination linkis not available. For the onsidered relay network, we propose three novel ommuni-ation protools. The rst proposed protool ahieves the apaity of the onsiderediiAbstratnetwork when fading on the soure-relay and relay-destination links is not present.The seond and third protools signiantly improve the average data rate and theoutage probability, respetively, of the onsidered network when both the soure-relayand the relay-destination links are aeted by fading.iiiPrefaeChapters 24 of this thesis are based on works performed under the supervision ofProf. Robert Shober and the ollaboration with Prof. Petar Popovski, AalborgUniversity, Denmark, and Vahid Jamali, Friedrih-Alexander-Universität Erlangen-Nürnberg, Germany.Unless otherwise stated, for all hapters and orresponding papers, I ondutedthe literature survey on related topis, identied the hallenges, and performed theanalyses and simulations. I wrote all paper drafts for whih I am the rst author.My supervisor guided the researh, validated the analyses, and gave omments onimproving the manusripts. The ollaborators' ontributions are listed below:1. Vahid Jamali gave omments for improving the paper related to Chapter 2 andvalidated the analyses.2. Prof. Petar Popovski suggested the system model for the paper related toChapter 3.Two papers related to Chapter 2 have been submitted for publiation:• N. Zlatanov, V. Jamali, and R. Shober, Capaity of the Two-Hop Half-DuplexRelay Channel, Submitted for publiation.• N. Zlatanov, V. Jamali, and R. Shober, On the Capaity of the Two-HopHalf-Duplex Relay Channel, Pro. of IEEE Globeom, San Diego, CA, USA,De. 2015.ivPrefaeTwo papers related to Chapter 3 have been published:• N. Zlatanov, R. Shober, and P. Popovski, Buer-Aided Relaying with Adap-tive Link Seletion, IEEE Journal on Seleted Areas in Communiations, vol.31, no. 8, pp. 1530-1542, Aug. 2013.• N. Zlatanov, R. Shober, and P. Popovski, Throughput and Diversity Gainof Buer-Aided Relaying, Pro. of IEEE Globeom 2011, Houston, TX, De.2011.Two papers related to Chapter 4 have been published:• N. Zlatanov and R. Shober, Buer-Aided Relaying with Adaptive Link Sele-tion - Fixed and Mixed Rate Transmission, IEEE Transations on InformationTheory, vol. 59, no. 5, pp. 2816-2840, May 2013.• N. Zlatanov and R. Shober, Buer-Aided Relaying with Mixed Rate Trans-mission, Pro. of IEEE IWCMC 2012, Limassol, Cyprus, Aug. 2012 (InvitedPaper).I have also o-authored other researh works whih have been published or sub-mitted for publiation during my time as a Ph.D. student at UBC. These works arelisted in Appendix D.vTable of ContentsAbstrat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPrefae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvList of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiDediation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Cooperative Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The Two-Hop HD Relay Channel . . . . . . . . . . . . . . . . . . . . 41.3 Motivation of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.1 The Two-Hop HD Relay Channel Without Fading . . . . . . 51.3.2 The Two-Hop HD Relay Channel With Fading . . . . . . . . 71.4 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 121.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 15viTable of Contents2 Capaity of the Two-Hop Half-Duplex Relay Channel Without Fad-ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.2 Mathematial Modelling of the HD Constraint . . . . . . . . 222.2.3 Mutual Information and Entropy . . . . . . . . . . . . . . . . 242.3 Capaity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.1 The Capaity . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.2 Ahievability of the Capaity . . . . . . . . . . . . . . . . . . 292.3.3 Simpliation of Previous Converse Expressions . . . . . . . . 362.4 Capaity Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4.1 Binary Symmetri Channels . . . . . . . . . . . . . . . . . . . 382.4.2 AWGN Channels . . . . . . . . . . . . . . . . . . . . . . . . . 422.5 Numerial Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.5.1 BSC Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.5.2 AWGN Links . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.6 Conlusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 Buer-Aided RelayingWith Adaptive Reeption-Transmission: Adap-tive Rate Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3 Preliminaries and Benhmark Shemes . . . . . . . . . . . . . . . . . 573.3.1 Adaptive Reeption-Transmission Protool and CSI Require-ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57viiTable of Contents3.3.2 Transmission Rates and Queue Dynamis . . . . . . . . . . . 583.3.3 Ahievable Average Rate . . . . . . . . . . . . . . . . . . . . 603.3.4 Conventional Relaying . . . . . . . . . . . . . . . . . . . . . . 603.4 Optimal Adaptive Reeption-Transmission Protool for Fixed Powers 633.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 633.4.2 Optimal Adaptive Reeption-Transmission Protool . . . . . 643.4.3 Deision Threshold . . . . . . . . . . . . . . . . . . . . . . . 693.4.4 Rayleigh Fading . . . . . . . . . . . . . . . . . . . . . . . . . 693.4.5 Real-Time Implementation . . . . . . . . . . . . . . . . . . . 713.5 Optimal Adaptive Reeption-Transmission and Optimal Power Allo-ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.5.1 Problem Formulation and Optimal Power Alloation . . . . . 723.5.2 Finding λ and ρ . . . . . . . . . . . . . . . . . . . . . . . . . 743.5.3 Rayleigh Fading . . . . . . . . . . . . . . . . . . . . . . . . . 763.5.4 Real-Time Implementation . . . . . . . . . . . . . . . . . . . 763.6 Delay-Limited Transmission . . . . . . . . . . . . . . . . . . . . . . . 773.6.1 Average Delay . . . . . . . . . . . . . . . . . . . . . . . . . . 773.6.2 Buer-Aided Protool for Delay Limited Transmission . . . . 793.7 Numerial and Simulation Results . . . . . . . . . . . . . . . . . . . 813.7.1 Delay-Unonstrained Transmission . . . . . . . . . . . . . . . 813.7.2 Delay-Constrained Transmission . . . . . . . . . . . . . . . . 843.8 Conlusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864 Buer-Aided RelayingWith Adaptive Reeption-Transmission: Fixedand Mixed Rate Transmission . . . . . . . . . . . . . . . . . . . . . . 874.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87viiiTable of Contents4.2 System Model and Channel Model . . . . . . . . . . . . . . . . . . . 894.3 Preliminaries and Benhmark Shemes . . . . . . . . . . . . . . . . . 904.3.1 Adaptive Reeption-Transmission and CSI Requirements . . 914.3.2 Transmission Rates and Queue Dynamis . . . . . . . . . . . 934.3.3 Link Outages and Indiator Variables . . . . . . . . . . . . . 954.3.4 Performane Metris . . . . . . . . . . . . . . . . . . . . . . . 964.3.5 Performane Benhmarks for Fixed Rate Transmission . . . . 974.3.6 Performane Benhmarks for Mixed Rate Transmission . . . . 994.4 Optimal Buer-Aided Relaying for Fixed Rate Transmission WithoutDelay Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 1024.4.2 Throughput Maximization . . . . . . . . . . . . . . . . . . . 1034.4.3 Performane in Rayleigh Fading . . . . . . . . . . . . . . . . 1124.5 Buer-Aided Relaying for Fixed Rate Transmission With Delay Con-straints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.5.1 Average Delay . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.5.2 Adaptive Reeption-Transmission Protool for Delay LimitedTransmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.5.3 Throughput and Delay . . . . . . . . . . . . . . . . . . . . . 1174.5.4 Outage Probability . . . . . . . . . . . . . . . . . . . . . . . . 1224.6 Mixed Rate Transmission . . . . . . . . . . . . . . . . . . . . . . . . 1254.6.1 Optimal Adaptive Reeption-Transmission Protool WithoutPower Alloation . . . . . . . . . . . . . . . . . . . . . . . . . 1264.6.2 Optimal Adaptive Reeption-Transmission Poliy With PowerAlloation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129ixTable of Contents4.6.3 Mixed Rate Transmission With Delay Constraints . . . . . . 1354.6.4 Conventional Relaying With Delay Constraints . . . . . . . . 1364.7 Numerial and Simulation Results . . . . . . . . . . . . . . . . . . . 1374.7.1 Fixed Rate Transmission . . . . . . . . . . . . . . . . . . . . 1384.7.2 Mixed Rate Transmission . . . . . . . . . . . . . . . . . . . . 1424.8 Conlusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455 Summary of Thesis and Future Researh Topis . . . . . . . . . . . 1475.1 Summary of the Results . . . . . . . . . . . . . . . . . . . . . . . . . 1475.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151AppendiesA Proofs for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159A.1 Proof That the Probability of Error at the Relay Goes to Zero When(2.25) Holds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159A.2 Proof That the Probability of Error at the Destination Goes to ZeroWhen (2.26) Holds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165A.3 Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 167B Proofs for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169B.1 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 169B.2 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 170B.3 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 172B.4 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 174xTable of ContentsB.5 Proof of Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 176C Proofs for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178C.1 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 178C.2 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 181C.3 Proof of Lemma 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 186C.4 Proof of Lemma 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 187C.5 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 187C.6 Proof of Lemma 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 190C.7 Proof of Lemma 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 191C.8 Proof of Theorem 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 191C.9 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 192C.10 Proof of Theorem 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . 194C.11 Proof of Theorem 4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . 197C.12 Proof of Theorem 4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . 199D Other Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201xiList of Figures1.1 The two-hop relay network omprised of a soure (S), a relay (R), anda destination (D). Sine there is no diret soure-destination link, thesoure transmits a message to the destination only via the relay. . . . 42.1 Two-hop relay hannel. . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 To ahieve the apaity in (2.18), transmission is organized in N + 1bloks and eah blok omprises k hannel uses. . . . . . . . . . . . . 292.3 Example of generated swithing vetor along with input/output ode-words at soure, relay, and destination. . . . . . . . . . . . . . . . . 352.4 Blok diagram of the proposed hannel oding protool for time sloti. The following notations are used in the blok diagram: C1|r and C2are enoders, D1 and D2 are deoders, I is an inserter, S is a seletor,B is a buer, and w(i) denotes the message transmitted by the sourein blok i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5 Comparison of rates for the BSC as a funtion of the error probabilityPε1 = Pε2 = Pε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.6 Example of proposed input distributions at the relay pV (x2). . . . . . 492.7 Soure-relay and relay destination links are AWGN hannels with P1/σ21 =P2/σ22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50xiiList of Figures2.8 Soure-relay and relay destination links are AWGN hannels with P1/σ21/10 =P2/σ22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1 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 inthe ith time slot, respetively. . . . . . . . . . . . . . . . . . . . . . . 563.2 Average rates ahieved with buer-aided relaying (BAR) with adap-tive reeption-transmission and with onventional relaying with andwithout buer for ΩSR = 0.9 and ΩRD = 1.1. . . . . . . . . . . . . . . 823.3 Estimated ρe(i) as a funtion of the time slot i. . . . . . . . . . . . . 833.4 Average rate with buer-aided relaying with adaptive reeption-transmissionwith and without power alloation for Ω¯S = 0.1 and Ω¯R = 1.9 . . . . 833.5 Average rate of BAR with adaptive reeption-transmission for dierentaverage delay onstraints. . . . . . . . . . . . . . . . . . . . . . . . . 843.6 Average delay until time slot i for T0 = 5 and γ = 20 dB . . . . . . . . 854.1 Ratio of the throughputs of buer-aided relaying and ConventionalRelaying 1, τ/τfixedconv,1, vs. γ. Fixed rate transmission without delayonstraints. γS = γR = γ, S0 = R0 = 2 bits/symb, and Ω¯R = 1. . . . . 1384.2 Outage probability of buer-aided (BA) relaying and ConventionalRelaying 1 vs. γ. Fixed rate transmission without delay onstraints.γS = γR = γ, S0 = R0 = 2 bits/symb, and Ω¯R = 1. . . . . . . . . . . . 1394.3 Throughputs of buer-aided (BA) relaying and Conventional Relaying2 vs. γ. Fixed rate transmission with delay onstraints. γS = γR = γ,S0 = R0 = 2 bits/symb, Ω¯R = 1, and Ω¯S = 1. . . . . . . . . . . . . . . 140xiiiList of Figures4.4 Outage probability of buer-aided (BA) relaying and ConventionalRelaying 2 vs. γ. Fixed rate transmission with delay onstraints. γS =γR = γ, S0 = R0 = 2 bits/symb, Ω¯R = 1, and Ω¯S = 1. . . . . . . . . . 1414.5 Throughput of buer-aided relaying with adaptive reeption-transmissionand Conventional Relaying 1 vs. Γ. Mixed rate transmission withoutdelay onstraints. Ω¯S = 10, Ω¯R = 1, and S0 = 2 bits/symb. . . . . . . 1434.6 Throughput of buer-aided relaying with adaptive reeption-transmissionand onventional relaying vs. Γ. Mixed rate and xed rate transmis-sion with delay onstraint. E{T} = 5 time slots, γS = γR = Γ, S0 = 2bits/symb, and Ω¯S = Ω¯R = 1. . . . . . . . . . . . . . . . . . . . . . . 144C.1 Markov hain for the number of pakets in the queue of the buer ifthe link seletion variable di is given by (4.59). . . . . . . . . . . . . . 188C.2 Markov hain for the number of pakets in the queue of the buer ifthe link seletion variable di is given by (4.61) or (4.63). . . . . . . . 190xivList of AbbreviationsAWGN Additive White Gaussian NoiseBA Buer-AidedBAR Buer-Aided RelayingBSC Binary Symmetri ChannelCDF Cumulative Distribution FuntionCSI Channel State InformationCSIT Channel State Information at TransmitterDF DeodeandForwardi.i.d. Independent and Identially DistributedFD Full-DuplexHD Half-DuplexLTE Long Term EvolutionPMF Probability Mass FuntionPDF Probability Density FuntionRV Random VariableSNR SignaltoNoise RatioWiMAX Worldwide Interoperability for Mirowave AessxvList of NotationE{·} Statistial expetationPr{·} Probability of an eventI(·; ·) Mutual informationH(·) EntropyxviAknowledgmentsFirst, I would like to express my deep and sinere gratitude to my advisor, Prof.Robert Shober, for his endless support and invaluable advie during my Ph.D. study.I am extraordinarily luky to have had Prof. Shober as my advisor. This thesis wouldnot have been possible without him. I am forever indebted to Prof. Shober.I am grateful to my o-supervisor, Prof. Lutz Lampe, for his help and onstantsupport. I would also like to speially thank Prof. Zoran Hadzi-Velkov for his overallguidane, onstant support, and invaluable advie. I have also greatly beneted fromthe ollaboration and support of Prof. George Karagiannidis, Prof. Petar Popovski,Prof. Ljupo Koarev, Vahid Jamali, and Dr. Derrik Wing Kwan Ng.Endless gratitude and admiration goes to my mother, Gua Zlatanova, my father,Todor Zlatanov, and my sister, Zoria Zlatanova, for their relentless support andinredible sarie. Without their love and wisdom, I would not be where I amtoday.Finally, I would like to thank my wife, Ljupka Zlatanova, who has always beenthere for me. I am grateful for her unonditional support, her endless love, andexeptional sarie. Without her, none of this would be possible.This work was supported nanially by The University of British Columbia, TheKillam Trusts, the Natural Sienes and Engineering Researh Counil of Canada(NSERC), and the German Aademi Exhange Servie (DAAD).xviiDediationTo My Family.xviiiChapter 1IntrodutionWireless ommuniation has enabled people to be onneted from anywhere andat any time. This has had a profound impat on human soiety. Currently, wirelessommuniation is performed using the ommuniation protools developed for ellularand wireless loal area networks. Although these protools support a broad rangeof mobile servies, they do not fully exploit the apaity of the underlying networksand annot satisfy the exponential growth in demand for higher data rates and morereliable onnetions. Therefore, new ommuniation protools have to be developedfor general wireless networks in order to meet this demand. Ultimately, these protoolwill have to be able to reah the fundamental limits of information ow in wirelessnetworks, i.e., the network apaity. Unfortunately, our urrent understanding ofthe network apaity is very poor even for very simple networks [1℄. Therefore, weare urrently unable to design protools whih reah the apaity of general wirelessnetworks. Hene, in order to get insight into this problem, we rst have to deviseommuniation protools for very simple wireless networks, e.g., networks omprisedof one soure, one relay, and one destination, and then use the gained knowledgeto design protools for more omplex networks, e.g., networks omprised of multiplesoures, relays, and destinations. These protools should take into aount pratiallimitations suh as half-duplex (HD) reeption and transmission in HD relaying andself-interferene in full-duplex (FD) relaying, and ultimately, should be able the reahthe apaity of the underlaying networks. In the following, we briey review the state1Chapter 1. Introdutionof the art in relay networks and motivate the work in this thesis.This hapter is organized as follows. In Setion 1.1, we briey introdue the on-ept of ooperative networks. In Setion 1.2, we desribe the simplest HD ooperativenetwork, the two-hop HD relay network. In Setion 1.3, we motivate this thesis byreviewing some of the best known protools for the two-hop HD relay network andpresent their orresponding ahievable data rates and outage probabilities. In Se-tion 1.4, we summarize the ontributions made in this thesis. The thesis organizationis provided in Setion 1.5.1.1 Cooperative NetworksFor improving our understanding of the network apaity, it has long been realizedthat we have to move away from analyzing networks as a olletion of disonnetedpoint-to-point ommuniations, and fous our attention on analyzing networks asone system in whih all nodes mutually ooperate in order for the information owin the network to reah its fundamental limit, i.e., to reah the network apaity[1℄. Thereby, the network nodes must assist eah other by willingly ating as relaysand use their own resoures to forward the information of other nodes [2℄, [3℄. Inpartiular, when a soure (e.g., mobile phone) transmits a paket to a destination(e.g., base station) wirelessly, all surrounding nodes (e.g., other mobile phones) whihoverhear this paket, should proess it, and retransmit the proessed paket to theintended destination, thus helping in the transmission proess. It has been shownthat ooperation among the nodes of a network signiantly improves the data rateand/or reliability of the network, and as a result, a host of ooperative tehniqueshave been proposed [2℄-[51℄. Due to the benets of ooperation, it is expeted thatfuture wireless ommuniation systems will inlude some form of ooperation between2Chapter 1. Introdutionnetwork nodes. In fat, simple relaying shemes have been/are being inluded in re-ent/future wireless standards suh as the Worldwide Interoperability for MirowaveAess (WiMAX) and Long Term Evolution (LTE) Advaned standards [6℄, [49℄, [50℄.In ooperative wireless networks, eah relay node an perform reeption and trans-mission either in FD or HD mode [7℄. In the FD mode, the relay transmits and reeivesat the same time and in the same frequeny band, whereas in the HD mode, trans-mission and reeption our in the same frequeny band but not at the same timeor at the same time but in dierent frequeny bands. Although ideal FD relayingahieves a higher data rate than HD relaying, given the limitations of urrent radioimplementations, ideal FD relaying is not possible due to strong self-interferene [7℄.More preisely, the transmit signal of an FD node onstitutes strong interferene forthe reeived signal at the same node, thus potentially preventing the FD node fromsuessfully deoding the reeived messages and thereby severely degrading its per-formane. Although reently there has been a lot of eort in developing FD nodeswhih an redue self-interferene [52℄, [53℄, it is still not possible to attenuate theself-interferene to a level whih makes it negligible [54℄. Therefore, HD relaying isstill a preferred hoie in pratie due to the muh simpler hardware implementationand the absene of self-interferene.In urrent HD relaying protools, reeption and transmission at the HD relaysis organized in two suessive time slots. In the rst time slot, the relay reeivesdata transmitted by a soure, and in the seond time slot the relay forwards thereeived data to a destination. Suh xed sheduling of reeption and transmissionin HD relaying has beome a ommonly aepted priniple, i.e., it has almost beomean axiom. Researhers have long thought that the apaity limits of ooperative HDrelay networks an be obtained with xed sheduling of reeption and transmission at3Chapter 1. IntrodutionS R DFigure 1.1: The two-hop relay network omprised of a soure (S), a relay (R), and adestination (D). Sine there is no diret soure-destination link, the soure transmitsa message to the destination only via the relay.the relays [18℄. However, as we will show in this thesis, xed sheduling of reeptionand transmission at HD relays is not optimal and results in signiant performanelosses.1.2 The Two-Hop HD Relay ChannelGiven our urrent knowledge, we are still not able to design ommuniation pro-tools whih reah the apaity of general HD relay networks [1℄. Hene, in orderto inrease our knowledge, we have to rst investigate ommuniation protools forvery simple HD relay networks. As a onsequene, in this thesis, we will investigateommuniation protools for the simplest HD relay network, shown in Fig. 1.1, whihwe refer to as the two-hop HD relay hannel or as the two-hop HD relay network,interhangeably. The two-hop HD relay hannel onsists of a soure, a HD relay, anda destination, and there is no diret link between the soure and the destination.Due to the HD onstraint, the relay annot transmit and reeive at the same time.Moreover, sine there is no diret soure-destination link, the soure has to transmitits information to the destination via the relay. The network shown in Fig. 1.1 isnot only the simplest relay network, but is also the most basi building blok of anyooperative network. Therefore, by understanding how to improve the performaneof this network, we will get insight into how to improve the performane of generalHD relay networks. In the following, we motivate this thesis by providing a brief4Chapter 1. Introdutionoverview of previous results for the two-hop HD relay hannel.1.3 Motivation of the ThesisAlthough extensively investigated, the apaity of the two-hop HD relay hannel isnot fully known nor understood. In partiular, a apaity expression whih an beevaluated is not available and an expliit oding sheme whih ahieves the apaityis not known either. Hene, only oding shemes whih ahieve rates stritly lowerthan the apaity are known. To motivate this thesis, in the following, we brieyreview previous results for the data rate and the outage probability of the two-hopHD relay hannel in the absene and presene of fading.1.3.1 The Two-Hop HD Relay Channel Without FadingConsider the system model in Fig. 1.1. Assume that both the soure-relay and relay-destination links are general memoryless hannels whih are not aeted by fading,i.e., the hannels' statistis do not hange with time. For this relay hannel, the sourewants to transmit a message via the HD relay to the destination in n hannel uses1with the largest possible data rate for whih the destination an reliably deode thetransmitted message. Currently, a oding sheme whih ahieves the largest knowndata rate is desribed in [18℄ and [48℄. In partiular, the ommuniation is performedin n → ∞ hannel uses and is organized in two suessive time slots. In the rstand seond time slot the hannel is used nξ and n(1 − ξ) times, respetively, where0 < ξ < 1. In the rst time slot, the soure transmits to the relay a odewordomprised of nξ symbols and with a rate equal to the apaity of the soure-relayhannel denoted by CSR. The relay deodes the reeived data, re-enodes it into a1A hannel use is equivalent to the duration of one symbol.5Chapter 1. Introdutionodeword omprised of n(1 − ξ) symbols and with a rate equal to the apaity ofthe relay-destination hannel, denoted by CRD, and transmits it to the destination.Thereby, by optimizing ξ for rate maximization, the following data rate is ahievedRconv =CSRCRDCSR + CRD. (1.1)As an be seen from the disussion above, this ommuniation protool has axed sheduling of reeption and transmission at the relay. However, suh xedsheduling of reeption and transmission at the relay was shown to be suboptimalin [8℄. In partiular, in [8℄, it was shown that if the xed sheduling of reeptionand transmission at the HD relay is abandoned, then additional information anbe enoded in the relay's reeption and transmission swithing pattern whih wouldyield a data rate larger than (1.1). Moreover, it was argued in [8℄ that the data rateahieved with the enoding of information in the relay's reeption and transmissionswithing pattern would be the apaity of the two-hop HD relay hannel in theabsene of fading. However, the results for the apaity of the two-hop HD relayhannel in [8℄, as well in the literature, are inomplete. In partiular, a apaityexpression whih an be evaluated still has not been provided and an expliit odingsheme whih ahieves the apaity rate, or any rate larger than (1.1), is still notknown. Therefore, although expliit upper bounds on the apaity exists [8℄, it isstill unknown exatly how muh larger the apaity is ompared to the rate in (1.1).Motivated by the above disussion, in Chapter 2, we derive a new easy-to-evaluateexpression for the apaity of the two-hop HD relay hannel based on simplifyingpreviously derived onverse expressions. In ontrast to previous results, this apaityexpression an be easily evaluated. Moreover, we propose a oding sheme whih anahieve the apaity. In partiular, we show that ahieving the apaity requires the6Chapter 1. Introdutionrelay to swith between reeption and transmission in a symbol-by-symbol manner.Thereby, the relay does not only send information to the destination by transmittinginformation-arrying symbols but also with the zero symbol resulting from the relay'ssilene during reeption. Furthermore, we show that the apaity is signiantlyhigher than the rate in (1.1).1.3.2 The Two-Hop HD Relay Channel With FadingFor the onsidered HD relay network in Fig. 1.1, assume that both soure-relay andrelay-destination links are additive white Gaussian noise (AWGN) hannels aetedby slow fading. Assume that the fading is a stationary and ergodi random proess.Moreover, assume that time is divided into N → ∞ time slots suh that duringone time slot the fading on both soure-relay and relay-destination links remainsonstant and hanges from one time slot to the next. Let CSR(i) and CRD(i) denotethe apaities of the soure-relay and relay-destination hannels in the i-th time slot,respetively. Furthermore, let C¯SR and C¯RD denote the average apaities of thesoure-relay and relay-destination hannels, respetively, given byC¯SR = E{CSR(i)}(a)= limN→∞1NN∑i=1CSR(i) (1.2)C¯RD = E{CRD(i)}(a)= limN→∞1NN∑i=1CRD(i), (1.3)where E{·} denotes expetation and (a) follows from the assumed ergodiity.For this network, in the following, we briey review ommuniation protoolswhih ahieve the best known average data rate and the best known outage proba-bility, respetively.7Chapter 1. IntrodutionBest Known Data RateA ommuniation protool whih ahieves the highest known data rate for this net-work was proposed in [18℄. In partiular, the ommuniation is performed in N →∞time slots. During one time slot, the hannel is used n → ∞ times. The proposedprotool in [18℄ is as follows. In the rst Nξ time slots, the soure transmits to theHD relay a odeword omprised of Nnξ symbols, where 0 < ξ < 1, and with rateequal to the average apaity of the soure-relay hannel C¯SR. The relay deodesthe reeived data, re-enodes it into a odeword omprised on Nn(1 − ξ) symbolsand with rate equal to the average apaity of the relay-destination hannel C¯RD andtransmits it to the destination. Thereby, by optimizing ξ for rate maximization, thefollowing data rate is ahievedR¯conv,1 =C¯SRC¯RDC¯SR + C¯RD. (1.4)In order to ahieve the rate in (1.4), the destination has to wait for N →∞ timeslots before it an deode the reeived odeword. This may not be pratial for ahost of appliations. To redue the delay, and yet ahieve the same rate as (1.4), thefollowing protool an be used [51℄. Both soure and relay transmit odewords whihspan one time slot and are omprised of n → ∞ symbols. The relay is equippedwith an innite size buer. The ommuniation is performed in N → ∞ time slots,and is as follows. In eah time slot i, where 1 ≤ i ≤ ξN , the soure transmits tothe relay a odeword with rate CSR(i). The relay deodes the reeived odewords,stores the information in its buer, and then sends the aumulated information tothe destination in the following (1 − ξ)N slots. In partiular, in eah time slot i,where ξN + 1 ≤ i ≤ N , the relay transmits to the destination a odeword with rateCRD(i). By optimizing ξ for rate maximization, the ahieved data rate is idential8Chapter 1. Introdutionto the one in (1.4). In this protool, the destination has to wait for ξN time slotsbefore it an start deoding the rst reeived odeword. However, sine N →∞, thisprotool may also be unpratial.To redue the delay even further, the following protool an be used [18℄. Theommuniation is performed in N →∞ time slots. In time slot i, the soure and relaytransmit odewords whih span ξ(i) and 1− ξ(i) frations of time slot i, respetively,and are omprised of ξ(i)n and (1 − ξ(i))n symbols, respetively, where n → ∞. Inthe ξ(i) fration of time slot i, the soure sends a odeword with rate CSR(i) to theHD relay. Then, in the remaining 1− ξ(i) fration of time slot i, the relay re-enodesthe reeived information and sends it to the destination with rate CRD(i). As aresult, the overall rate transmitted from soure to destination during time slot i isR(i) = min{ξ(i)CSR(i), (1− ξ(i))CRD(i)}. By optimizing ξ(i) for rate maximization,the following maximum rate is ahieved in time slot iR(i) = CSR(i)CRD(i)CSR(i) + CRD(i). (1.5)Thereby, during N →∞ time slots, the average rate ahieved with this ommunia-tion protool is given byR¯conv,2 = E{ CSR(i)CRD(i)CSR(i) + CRD(i)}. (1.6)Ahieving (1.6) requires the odeword lengths to be variable and adopted for eahfading state, whih may not be desirable in pratie. In that ase, the above protoolan be modied by setting ξ(i) = 1/2, ∀i, and thereby the following average rate anbe ahievedR¯conv,3 =12E {min{CSR(i), CRD(i)}} . (1.7)9Chapter 1. IntrodutionComparing (1.4), (1.6), and (1.7) we observe that R¯conv,1 ≥ R¯conv,2 ≥ R¯conv,3holds. However, to realize R¯conv,1 and R¯conv,2, an innite delay and adaptive odewordlengths must be introdued, respetively.As seen from the disussion above, all four protools have a xed shedule of thereeption and transmissions at the relay. We refer to these protools as onventionalrelaying protools throughout this thesis. To inrease the average data rate, inChapter 3, we develop a HD relaying protool in whih the HD relay adaptivelyhooses whether to reeive or transmit a odeword in a given time slot based onthe instantaneous qualities of the soure-relay and relay-destination links. This newapproah requires the relay to have a buer, and therefore, the new protool is referredto as buer-aided relaying with adaptive reeption-transmission. We will show thatbuer-aided relaying with adaptive reeption-transmission ahieves rates whih aresigniantly higher than the rates in (1.4), (1.6), and (1.7). In the following, webriey desribe buer-aided relaying with adaptive reeption-transmission and reviewprevious works on this subjet.Buer-Aided Relaying With Adaptive Reeption-TransmissionBuer-aided relaying with adaptive reeption-transmission belongs to a lass of om-muniation protools for wireless HD relay networks where the HD relays use theirbuers to adaptively hoose whether to reeive or transmit a paket in a given timeslot based on the instantaneous qualities of their respetive reeiving and transmittinghannels.In onventional relaying protools, the relays employ a prexed shedule of trans-mission and reeption, independent of the quality of the transmitting and reeivinghannels. This prexed sheduling may lead to a signiant performane degrada-tion in wireless systems, where the quality of the transmitting and reeiving hannels10Chapter 1. Introdutionvaries with time, sine it may prevent the relays from exploiting the best transmittingand the best reeiving hannels. Clearly, performane ould be improved if the linkwith the higher quality ould be used in eah time slot. This an be ahieved viaa buer-aided relaying protool whih does not have a prexed shedule of reep-tion and transmission. In partiular, buer-aided relaying with adaptive reeption-transmission an exploit the stronger of the reeiving and transmitting hannels ineah time slot, and thereby improve the performane.We devised the onept of buer-aided relaying with adaptive reeption-transmis-sion in [55℄ and showed that signiant improvement of the average data rate andthe outage probability are possible ompared to onventional relaying. Later, in [56℄and [57℄ we investigated buer-aided relaying with adaptive reeption-transmissionfor the two-hop HD relay network, and these two papers onstitute the basis of Chap-ters 3 and 4, respetively. The works in [55℄-[57℄ led to other extension. For example,buer-aided relaying with adaptive reeption-transmission were also proposed for thetwo-hop HD relay network with bit interleaved oded modulation and orthogonalfrequeny division multiplexing in [58℄ and with statistial quality of servie on-straints in [59℄, for two-way relaying in [60℄-[63℄, for the HD relay hannel with adiret soure-destination link in [64℄, [65℄, for the multihop relay network in [66℄,[67℄, for two soure and two destination pairs sharing a single HD relay in [68℄, forseure ommuniation for two-hop HD relaying and HD relay seletion in [69℄ and[70℄, respetively, for amplify-and-forward relaying in [71℄, for energy harvesting in[72℄, for HD relay-seletion in [73℄-[75℄, and for hybrid FD/HD in [76℄.In the following, we review HD relay protools for xed rate transmission.11Chapter 1. IntrodutionBest Known Outage ProbabilityFor the onsidered relay network in Fig. 1.1, assume that both soure and relay donot have hannel state information at the transmitter (CSIT) and therefore have totransmit odewords with a xed data rate R0. Moreover, assume that both soureand relay transmit odewords whih span one time slot and are omprised of n→∞symbols. In this ase, a hannel apaity in the strit Shannon sense does not exist[77℄. In other words, not all transmitted odewords an be deoded at the respetivereeivers, and for the undeodable reeived odewords the system is said to be inoutage. An appropriate measure for suh systems is the outage probability whih isthe fration of undeodable odewords at the reeiver [77℄.For this senario, a ommuniation protool was proposed in [12℄ for the two-hopHD relay hannel. Thereby, the soure transmits odewords to the HD relay in oddtime slots, and the HD relay retransmits the reeived data to the destination in eventime slots. This protool, ahieves the following outage probabilityPout = Pr{CSR(i) < R0 OR CRD(i) < R0}, (1.8)where Pr{·} denotes probability. In this thesis, in Chapter 4, we will show that wean improve the outage probability in (1.8) signiantly, using a novel buer-aidedrelaying protool with adaptive reeption-transmission speially designed for xedrate transmission and improvement of the outage probability.1.4 Contributions of the ThesisThis thesis presents novel HD relaying protools for the two-hop HD relay hannel.In the following, we list the main ontributions of this thesis.12Chapter 1. Introdution1. We derive a new easy-to-evaluate expression for the apaity of the two-hopHD relay hannel in the absene of fading based on simplifying previously de-rived onverse expressions. Compared to previous results, this apaity expres-sion an be easily evaluated. Moreover, we propose a oding sheme whihan ahieve the apaity. In partiular, we show that ahieving the apaityrequires the relay to swith between reeption and transmission in a symbol-by-symbol manner. Thereby, the relay does not only send information to thedestination by transmitting information-arrying symbols but also with thezero symbol resulting from the relay's silene during reeption. As examples,we derive simplied apaity expressions for the following two speial ases: 1)The soure-relay and relay-destination links are both binary-symmetri han-nels (BSCs); 2) The soure-relay and relay-destination links are both AWGNhannels. For these two ases, we numerially ompare the apaity with therate ahieved by onventional relaying where the relay reeives and transmitsin a odeword-by-odeword fashion and swithes between reeption and trans-mission in a stritly alternating manner. Our numerial results show that theapaity is signiantly larger than the rate ahieved with onventional relayingfor both the BSC and the AWGN hannel.2. For the two-hop HD relay hannel when both soure-relay and relay-destinationlinks are aeted by fading, we propose a new relaying protool employing adap-tive reeption-transmission, i.e., in any given time slot, based on the hannelstate information of the soure-relay and the relay-destination links a deisionis made whether the relay should reeive or transmit. In order to avoid data lossat the relay, adaptive reeption-transmission requires the relay to be equippedwith a buer suh that data an be queued until the relay-destination link13Chapter 1. Introdutionis seleted for transmission. We study both delay-unonstrained and delay-onstrained transmission. For the delay-unonstrained ase, we haraterizethe optimal adaptive reeption-transmission shedule, derive the orrespondingahievable rate, and develop an optimal power alloation sheme. For the delay-onstrained ase, we propose a modied buer-aided protool whih satises apredened average delay onstraint at the expense of a lower data rate. Ouranalytial and numerial results show that buer-aided relaying with adaptivereeption-transmission with and without a delay onstraint ahieve signiantrate gains ompared to onventional relaying protools with and without buerswhere the relay employs a xed shedule for reeption and transmission.3. For the two-hop HD relay hannel when both the soure-relay and relay-destinat-ion links are aeted by fading, we propose two new buer-aided relayingshemes with dierent requirements regarding the availability of CSIT. In therst sheme, neither the soure nor the relay have full CSIT, and onsequently,both nodes are fored to transmit with xed rates. In ontrast, in the se-ond sheme, the soure does not have full CSIT and transmits with xed ratebut the relay has full CSIT and adapts its transmission rate aordingly. Inthe absene of delay onstraints, for both xed rate and mixed rate transmis-sion, we derive the throughput-optimal buer-aided relaying protools whihselet the relay to either reeive or transmit based on the instantaneous qual-ities of the soure-relay and relay-destination links. In addition, for the de-lay onstrained ase, we develop buer-aided relaying protools with adaptivereeption-transmission that ahieve a predened average delay. Compared toonventional relaying protools, the proposed buer-aided protools with adap-tive reeption-transmission ahieve large performane gains. In partiular, for14Chapter 1. Introdutionxed rate transmission, we show that the proposed protool ahieves a diver-sity gain of two as long as an average delay of more than three time slots anbe aorded. In ontrast, onventional relaying protools ahieve a diversitygain of one. Furthermore, for mixed rate transmission with an average delayof E{T} time slots, a multiplexing gain of r = 1 − 1/(2E{T}) is ahieved.As a by-produt of the onsidered adaptive reeption-transmission protools,we also develop a novel onventional relaying protool for mixed rate trans-mission whih yields the same multiplexing gain as the protool with adaptivereeption-transmission. Hene, for mixed rate transmission, for suiently largeaverage delays, buer-aided HD relaying with and without adaptive reeption-transmission does not suer from a multiplexing gain loss ompared to FDrelaying.1.5 Organization of the ThesisIn the following, we provide a brief overview of the remainder of this thesis.In Chapter 2, we derive the apaity of the two-hop HD relay hannel whenthe soure-relay and relay-destination links are not aeted by fading. Thereby, werst formally dene the hannel model. Then, we introdue a new expression forthe apaity of the onsidered relay hannel, prove that it satises the onverse,and introdue an expliit hannel oding sheme whih ahieves this apaity. Wealso investigate the apaity for the speial ases when the soure-relay and relay-destination links are both BSCs and AWGN hannels, respetively, and numeriallyompare the derived apaity expressions with the rate ahieved by onventionalrelaying.In Chapter 3, we introdue a novel relaying protool, whih we refer to as buer-15Chapter 1. Introdutionaided relaying with adaptive reeption-transmission, for improving the average datarate of the two-hop HD relay hannel when the soure-relay and relay-destinationlinks are AWGN hannels aeted by fading. Thereby, we rst formally dene theonsidered system and hannel models. Then, we formulate optimization problemsfor maximization of the ahievable average rate of buer-aided relaying with andwithout power alloation. From these optimization problems we derive the optimalbuer-aided relaying protools whih maximize the data rate. Sine these protoolsrequire unlimited delay, we also propose a heuristi buer-aided relaying protoolwhih limits the average delay.In Chapter 4, we introdue novel buer-aided relaying protools for improving theoutage probability of the two-hop HD relay hannel when the soure-relay and relay-destination links are AWGN hannels aeted by fading. Thereby, we investigate twosystem models. In the rst system, neither the soure nor the relay have full CSIT,and onsequently, both nodes are fored to transmit with xed rates. In ontrast,in the seond system model, the soure does not have full CSIT and transmits withxed rate but the relay has full CSIT and adapts its transmission rate aordingly.For both system models, we introdue buer-aided relaying protools with adaptivereeption-transmission for delay unonstrained and delay onstrained transmission,respetively. The protools for delay unonstrained and delay onstrained transmis-sions are analyzed and onlusions are drawn.Chapter 5 summarizes the ontributions of this thesis and outlines areas of futureresearh.Appendies A - C ontain proofs of theorems and lemmas used in this thesis.16Chapter 2Capaity of the Two-HopHalf-Duplex Relay Channel in theAbsene of Fading2.1 IntrodutionThroughout this hapter, we assume that the two-hop HD relay hannel is not aetedby fading, i.e., the statistis of the soure-relay and relay-destination hannels do nothange with time.The apaity of the two-hop FD relay hannel without self-interferene has beenderived in [5℄ (see the apaity of the degraded relay hannel). On the other hand,although extensively investigated, the apaity of the two-hop HD relay hannel isnot fully known nor understood. The reason for this is that a apaity expressionwhih an be evaluated is not available and an expliit oding sheme whih ahievesthe apaity is not known either. Currently, for HD relaying, detailed oding shemesexist only for rates whih are stritly smaller than the apaity, see [18℄ and [48℄. Toahieve the rates given in [18℄ and [48℄, the HD relay reeives a odeword in one timeslot, deodes the reeived odeword, and re-enodes and re-transmits the deodedinformation in the following time slot, see Setion 1.3.1 for more details. However,17Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingsuh xed swithing between reeption and transmission at the HD relay was shown tobe suboptimal in [8℄. In partiular, in [8℄, it was shown that if the xed sheduling ofreeption and transmission at the HD relay is abandoned, then additional informationan be enoded in the relay's reeption and transmission swithing pattern yieldingan inrease in data rate. In addition, it was shown in [8℄ that the HD relay hannelan be analyzed using the framework developed for the FD relay hannel in [5℄. Inpartiular, results derived for the FD relay hannel in [5℄ an be diretly applied tothe HD relay hannel. Thereby, using the onverse for the degraded relay hannel in[5℄, the apaity of the disrete 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 soure andrelay, respetively, Y1 and Y2 are the outputs at relay and destination, respetively,and p(x1, x2) is the joint probability mass funtion (PMF) of X1 and X2. Moreover, itwas shown in [8℄, [9℄, [78℄ that X2 an be represented as X2 = [X ′2, U ], where U is anauxiliary random variable with two outomes t and r orresponding to the HD relaytransmitting and reeiving, respetively. Thereby, (2.1) an be written equivalentlyasC = 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 apaity expres-sions in (2.1) and (2.2), respetively, annot be evaluated sine it is not known howX1 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 fat, the authors of [78, page 2552℄ state that: Despite knowing18Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingthe apaity expression (i.e., expression (2.2)), its atual evaluation is elusive as itis not lear what the optimal input distribution p(x1, x′2, u) is. On the other hand,for the oding sheme that would ahieve (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 deode-and-forward strategy sinethe two-hop HD relay hannel belongs to the lass of the degraded relay hannelsdened in [5℄. Thereby, the HD relay should deode any reeived odewords, mapthe deoded information to new odewords, and transmit them to the destination.Moreover, it is known from [8℄ that suh a oding sheme requires the HD relay toswith between reeption and transmission in a symbol-by-symbol manner, and notin a odeword-by-odeword manner as in [18℄ and [48℄. However, sine p(x1, x2) andp(x1, x′2, u) are not known and sine an expliit oding sheme does not exist, it isurrently not known how to evaluate (2.1) and (2.2) nor how to enode additionalinformation in the relay's reeption and transmission swithing pattern and therebyahieve (2.1) and (2.2).Motivated by the above disussion, in this hapter, we derive a new expression forthe apaity of the two-hop HD relay hannel based on simplifying previously derivedonverse expressions. In ontrast to previous results, this apaity expression an beeasily evaluated. Moreover, we propose an expliit oding sheme whih ahieves theapaity. In partiular, we show that ahieving the apaity requires the relay indeedto swith between reeption and transmission in a symbol-by-symbol manner as pre-dited in [8℄. Thereby, the relay does not only send information to the destination bytransmitting information-arrying symbols but also with the zero symbols resultingfrom the relay's silene during reeption. In addition, we propose a modied od-ing sheme for pratial implementation where the HD relay reeives and transmitsat the same time (i.e., as in FD relaying), however, the simultaneous reeption and19Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingtransmission is performed suh that the self-interferene is ompletely avoided. Asexamples, we ompute the apaities of the two-hop HD relay hannel for the aseswhen the soure-relay and relay-destination links are both binary-symmetri han-nels (BSCs) and additive white Gaussian noise (AWGN) hannels, respetively, andwe numerially ompare the apaities with the rates ahieved by onventional relay-ing where the relay reeives and transmits in a odeword-by-odeword fashion andswithes between reeption and transmission in a stritly alternating manner. Ournumerial results show that the apaities of the two-hop HD relay hannel for BSCand AWGN links are signiantly larger than the rates ahieved with onventionalrelaying.We note that the apaity of the two-hop HD relay hannel was also investigatedin [79℄ as a speial ase of the multi-hop HD relay hannel, but only for the ase whenall involved links are error-free BSCs.The rest of this hapter is organized as follows. In Setion 2.2, we present thehannel model. In Setion 2.3, we introdue a new expression for the apaity of theonsidered hannel, expliitly show the ahievability of the derived apaity, and provethat the new apaity expression satises the onverse. In Setion 2.4, we investigatethe apaity for the ases when the soure-relay and relay-destination links are bothBSCs and AWGN hannels, respetively. In Setion 2.5, we numerially evaluate thederived apaity expressions and ompare them to the rates ahieved by onventionalrelaying. Finally, Setion 2.6 onludes the hapter.2.2 System ModelThe two-hop HD relay hannel onsists of a soure, a HD relay, and a destination,and the diret link between soure and destination is not available, see Fig. 2.1. Due20Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without FadingSource Relay Destinat.X1 X2Y1 Y2Figure 2.1: Two-hop relay hannel.to the HD onstraint, the relay annot transmit and reeive at the same time. In thefollowing, we formally dene the hannel model.2.2.1 Channel ModelThe disrete 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 enodersof the soure and the relay, respetively, Y1 and Y2 are the nite output alphabetsat the deoders of the relay and the destination, respetively, and p(y1, y2|x1, x2) isthe PMF on Y1 × Y2 for given x1 ∈ X1 and x2 ∈ X2. The hannel is memorylessin the sense that given the input symbols for the i-th hannel use, the i-th outputsymbols are independent from all previous input symbols. As a result, the onditionalPMF p(yn1 , yn2 |xn1 , xn2 ), where the notation an is used to denote the ordered sequenean = (a1, a2, ..., an), an be fatorized as p(yn1 , yn2 |xn1 , xn2 ) =∏ni=1 p(y1i, y2i|x1i, x2i).For the onsidered hannel and the i-th hannel use, let X1i and X2i denote therandom variables (RVs) whih model the input at soure and relay, respetively, andlet Y1i and Y2i denote the RVs whih model the output at relay and destination,respetively.In the following, we model the HD onstraint of the relay and disuss its eet onsome important PMFs that will be used throughout this hapter.21Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fading2.2.2 Mathematial Modelling of the HD ConstraintDue to the HD onstraint of the relay, the input and output symbols of the relayannot assume non-zero values at the same time. More preisely, for eah hanneluse, if the input symbol of the relay is non-zero then the output symbol has to bezero, and vie versa, if the output symbol of the relay is non-zero then the inputsymbol has to be zero. Hene, the following holdsY1i =Y ′1i if X2i = 00 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 representthe input set of the relay X2 as the union of two sets X2 = X2R ∪ X2T , where X2Rontains only one element, the zero symbol, and X2T ontains all symbols in X2exept the zero symbol. Note that, beause of the HD onstraint, X2 has to ontainthe zero symbol. Furthermore, we introdue an auxiliary random variable, denotedby Ui, whih takes values from the set {t, r}, where t and r orrespond to the relaytransmitting a non-zero symbol and a zero symbol, respetively. Hene, Ui is denedasUi =r if X2i = 0t if X2i 6= 0.(2.4)Let us denote the probabilities of the relay transmitting a non-zero and a zero symbolfor the i-th hannel use as Pr{Ui = t} = Pr{X2i 6= 0} = PUi and Pr{Ui = r} =Pr{X2i = 0} = 1 − PUi, respetively. We now use (2.4) and represent X2i as a22Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingfuntion of the outome of Ui. Hene, we haveX2i =0 if Ui = rVi if Ui = t,(2.5)where Vi is an RV with distribution pVi(x2i) that takes values from the set X2T , orequivalently, an RV whih takes values from the set X2, but with pVi(x2i = 0) = 0.From (2.5), we obtainp(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 ofthe apaity, we will also need the onditional PMF p(x1i|x2i = 0) whih is the inputdistribution at the soure when the relay transmits a zero (i.e., when Ui = r). Aswe will see in Theorem 2.1, the distributions p(x1i|x2i = 0) and pVi(x2i) have to beoptimized in order to ahieve the apaity. Using p(x2i|Ui = r) and p(x2i|Ui = t),and the law of total probability, the PMF of X2i, p(x2i), is obtained asp(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 distributionof Y2i, p(y2i), whih, using the law of total probability, an be written asp(y2i) = p(y2i|Ui = t)PUi + p(y2i|Ui = r)(1− PUi). (2.9)23Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without FadingOn the other hand, using X2i and the law of total probability, p(y2i|Ui = r) an bewritten asp(y2i|Ui = r) =∑x2i∈X2p(y2i, x2i|Ui = r)=∑x2i∈X2p(y2i|x2i, Ui = r)p(x2i|Ui = r)(a)=∑x2i∈X2p(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 variabletwie sine if X2i = 0 then Ui = r, and vie versa. On the other hand, using X2i andthe law of total probability, p(y2i|Ui = t) an be written asp(y2i|Ui = t) =∑x2i∈X2p(y2i, x2i|Ui = t) =∑x2i∈X2p(y2i|x2i, Ui = t)p(x2i|Ui = t)(a)=∑x2i∈X2Tp(y2i|x2i, Ui = t)pVi(x2i)(b)=∑x2i∈X2Tp(y2i|x2i)pVi(x2i), (2.11)where (a) follows from (2.7) and sine Vi takes values from set X2T , and (b) followssine 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 inputX2i.2.2.3 Mutual Information and EntropyFor the apaity expression given later in Theorem 2.1, we need I(X1; Y1|X2 = 0),whih is the mutual information between the soure's input X1 and the relay's outputY1 onditioned on the relay having its input set to X2 = 0, and I(X2; Y2), whih is24Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingthe mutual information between the relay's input X2 and the destination's outputY2.The mutual information I(X1; Y1|X2 = 0) is obtained by denition asI(X1; Y1|X2 = 0)=∑x1∈X1∑y1∈Y1p(y1|x1, x2 = 0)p(x1|x2 = 0) log2(p(y1|x1, x2 = 0)p(y1|x2 = 0)),(2.12)wherep(y1|x2 = 0) =∑x1∈X1p(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 soure-relay hannel onditioned on the relay having its input set to X2 = 0, and onditionedon the input symbols at the soure X1.On the other hand, I(X2; Y2) is given byI(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 onditionedon X2. The entropy H(Y2) an be found by denition asH(Y2) = −∑y2∈Y2p(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 in25Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fading(2.10) and (2.11), respetively, into (2.15), we obtain the nal expression for H(Y2),asH(Y2) = −∑y2∈Y2[PU∑x2∈X2Tp(y2|x2)pV (x2) + p(y2|x2 = 0)(1− PU)]× log2[PU∑x2∈X2Tp(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 itsdenition asH(Y2|X2) = −∑x2∈X2p(x2)∑y2∈Y2p(y2|x2) log2(p(y2|x2))(a)= −PU∑x2∈X2TpV (x2)∑y2∈Y2p(y2|x2) log2(p(y2|x2))− (1− PU)∑y2∈Y2p(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), respetively, into (2.14), we obtain the nal expressionfor I(X2; Y2), whih is dependent on p(x2), i.e., on pV (x2) and PU . To highlight thedependene of I(X2; Y2) with respet to PU , in the following, we write I(X2; Y2) asI(X2; Y2)∣∣PU.We are now ready to present the apaity of the onsidered hannel.2.3 CapaityIn this setion, we provide an easy-to-evaluate expression for the apaity of thetwo-hop HD relay hannel, an expliit oding sheme that ahieves the apaity, andprove that the new apaity expression satises the onverse.26Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fading2.3.1 The CapaityA new expression for the apaity of the two-hop HD relay hannel is given in thefollowing theorem.Theorem 2.1. The apaity of the two-hop HD relay hannel is given byC = maxPUmin{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 apaity in (2.18) is given by P ∗U = min{P ′U , P ′′U},where P ′U is the solution ofmaxp(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 thesolution of∂(maxpV (x2)I(X2; Y2)∣∣PU)∂PU= 0. (2.20)If P ∗U = P ′U , the apaity in (2.18) simplies toC = maxp(x1|x2=0)I(X1; Y1|X2 = 0)(1− P ′U) = maxpV (x2)I(X2; Y2)∣∣PU=P ′U, (2.21)whereas, if P ∗U = P ′′U , the apaity in (2.18) simplies toC = maxpV (x2)I(X2; Y2)∣∣PU=P ′′U= maxp(x2)I(X2; Y2), (2.22)whih is the apaity of the relay-destination hannel.27Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without FadingProof. The proof of the apaity given in (2.18) is provided in two parts. In the rstpart, given in Setion 2.3.2, we show that there exists a oding sheme that ahievesa rate R whih is smaller, but arbitrarily lose to apaity C. In the seond part,given in Setion 2.3.3, we prove that any rate R for whih the probability of erroran be made arbitrarily small, must be smaller than apaity C given in (2.18). Therest of the theorem follows from solving (2.18) with respet to PU , and simplifyingthe result. In partiular, note that the rst term inside the min{·} funtion in (2.18)is a dereasing funtion with respet to PU . This funtion ahieves its maximumfor PU = 0 and its minimum, whih is zero, for PU = 1. On the other hand, theseond term inside the min{·} funtion in (2.18) is a onave funtion with respet toPU . To see this, note that I(X2; Y2) is a onave funtion with respet to p(x2), i.e.,with respet to the vetor omprised of the probabilities p(x2), for x2 ∈ X2, see [80℄.Now, sine 1 − PU is just the probability p(x2 = 0) and sine pV (x2) ontains therest of the probability onstrained parameters in p(x2), I(X2; Y2) is a jointly onavefuntion with respet to pV (x2) and PU . In [81, pp. 87-88℄, it is proven that if f(x, y)is a jointly onave funtion in both (x, y) and C is a onvex nonempty set, thenthe funtion g(x) = maxy∈Cf(x, y) is onave in x. Using this result, and noting thatthe domain of pV (x2) is speied by the probability onstraints, i.e., by a onvexnonempty set, we an onlude that maxpV (x2)I(X2, Y2)∣∣PUis onave with respet to PU .Now, the maximization of the minimum of the dereasing and onave funtionswith respet to PU , given in (2.18), has a solution PU = P ′′U , when the onavefuntion reahes its maximum, found from (2.20), and when for this point, i.e., forPU = P ′′U , the dereasing funtion is larger than the onave funtion. Otherwise,the solution is PU = P ′U whih is found from (2.19) and in whih ase P ′U < P ′′Uholds. If (2.19) has two solutions, then P ′U has to be the smaller of the two sine28Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fading...PSfrag replaements1 2 3 4 N N + 1k n = NkFigure 2.2: To ahieve the apaity in (2.18), transmission is organized in N + 1bloks and eah blok omprises k hannel uses.maxp(x1|x2=0)I(X1; Y1|X2 = 0)(1− PU) is a dereasing funtion with respet 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)maxPUI(X2; Y2) = maxp(x2)I(X2; Y2), thereby leadingto (2.22).2.3.2 Ahievability of the CapaityIn the following, we desribe a method for transferring nR bits of information inn + 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 outin N + 1 bloks, where N → ∞. In eah blok, we use the hannel k times. Thenumbers N and k are hosen suh that n = Nk holds. The transmission in N + 1bloks is illustrated in Fig. 2.2.The soure transmits messageW , drawn uniformly frommessage set {1, 2, ..., 2nR},from the soure via the HD relay to the destination. To this end, before the start oftransmission, message W is spilt into N messages, denoted by w(1), ..., w(N), whereeah w(i), ∀i, ontains kR bits of information. The transmission is arried out in thefollowing manner. In blok one, the soure sends message w(1) in k hannel uses tothe relay and the relay is silent. In blok i, for i = 2, ..., N , soure and relay sendmessages w(i) and w(i− 1) to relay and destination, respetively, in k hannel uses.In blok N + 1, the relay sends message w(N) in k hannel uses to the destination29Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingand the soure is silent. Hene, in the rst blok and in the (N + 1)-th blok therelay and the soure are silent, respetively, sine in the rst blok the relay does nothave information to transmit, and in blok N+1, the soure has no more informationto transmit. In bloks 2 to N , both soure and relay transmit, while meeting theHD onstraint in every hannel use. Hene, during the N + 1 bloks, the hannel isused k(N + 1) times to send nR = NkR bits of information, leading to an overallinformation rate given bylimN→∞limk→∞NkRk(N + 1) = R bits/use. (2.23)A detailed desription of the proposed oding sheme is given in the following,where we explain the rates, odebooks, enoding, and deoding used for transmission.Rates: The transmission rate of both soure and relay is denoted by R and givenbyR = C − ǫ, (2.24)where C is given in Theorem 2.1 and ǫ > 0 is an arbitrarily small number. Note thatR is a funtion of P ∗U , see Theorem 2.1.Codebooks: We have two odebooks: The soure's transmission odebook andthe relay's transmission odebook.The soure's transmission odebook is generated by mapping eah possible binarysequene omprised of kR bits, where R is given by (2.24), to a odeword2 x1|romprised of k(1 − P ∗U) symbols. The symbols in eah odeword x1|r are generatedindependently aording to distribution p(x1|x2 = 0). Sine in total there are 2kRpossible binary sequenes omprised of kR bits, with this mapping we generate 2kR2The subsript 1|r in x1|r is used to indiate that odeword x1|r is omprised of symbols whihare transmitted by the soure only when Ui = r, i.e., when X2i = 0.30Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingodewords x1|r eah ontaining k(1 − P ∗U) symbols. These 2kR odewords form thesoure's transmission odebook, whih we denote by C1|r.The relay's transmission odebook is generated by mapping eah possible binarysequene omprised of kR bits, where R is given by (2.24), to a transmission odewordx2 omprised of k symbols. The i-th symbol, i = 1, ..., k, in odeword x2 is generatedin the following manner. For eah symbol a oin is tossed. The oin is suh that itprodues symbol r with probability 1− P ∗U and symbol t with probability P ∗U . If theoutome of the oin ip is r, then the i-th symbol of the relay's transmission odewordx2 is set to zero. Otherwise, if the outome of the oin ip is t, then the i-th symbolof odeword x2 is generated independently aording to distribution pV (x2). The 2kRodewords x2 form the relay's transmission odebook denoted by C2.The two odebooks are known at all three nodes.Enoding, Transmission, and Deoding: In the rst blok, the soure mapsw(1) to the appropriate odeword x1|r(1) from its odebook C1|r. Then, odewordx1|r(1) is transmitted to the relay, whih is sheduled to always reeive and be silent(i.e., to set its input to zero) during the rst blok. However, knowing that thetransmitted odeword from the soure x1|r(1) is omprised of k(1 − P ∗U) symbols,the relay onstruts the reeived odeword, denoted by y1|r(1), only from the rstk(1− P ∗U) reeived symbols. In Appendix A.1 , we prove that odeword x1|r(1) sentin the rst blok an be deoded suessfully from the reeived odeword at the relayy1|r(1) using a typial deoder [80℄ sine R satisesR < maxp(x1|x2=0)I(X1; Y1|X2 = 0)(1− P ∗U). (2.25)In bloks i = 2, ..., N , the enoding, transmission, and deoding are performed asfollows. In bloks i = 2, ..., N , the soure and the relay map w(i) and w(i − 1) to31Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingthe appropriate odewords x1|r(i) and x2(i) from odebooks C1|r and C2, respetively.Note that the soure also knows x2(i) sine x2(i) was generated from w(i− 1) whihthe soure transmitted in the previous (i.e., (i − 1)-th) blok. The transmission ofx1|r(i) and x2(i) an be performed in two ways: 1) by the relay swithing betweenreeption and transmission, and 2) by the relay always reeiving and transmitting asin FD relaying. We rst explain the rst option.Note that both soure and relay know the position of the zero symbols in x2(i).Hene, if the rst symbol in odeword x2(i) is zero, then in the rst symbol intervalof blok i, the soure transmits its rst symbol from odeword x1|r(i) and the relayreeives. By reeiving, the relay atually also sends the rst symbol of odewordx2(i), whih 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 blok i, therelay transmits its rst symbol from odeword x2(i) and the soure is silent. Thesame proedure is performed for the j-th hannel use in blok i, for j = 1, ..., k. Inpartiular, if the j-th symbol in odeword x2(i) is zero, then in the j-th hannel useof blok i the soure transmits its next untransmitted symbol from odeword x1|r(i)and the relay reeives. With this reeption, the relay atually also sends the j-thsymbol of odeword x2(i), whih is the symbol zero, i.e., x2j = 0. On the otherhand, if the j-th symbol in odeword x2(i) is non-zero, then for the j-th hanneluse of blok i, the relay transmits the j-th symbol of odeword x2(i) and the soureis silent. Note that odeword x2(i) ontains k(1 − P ∗U) ± ε(i) symbols zeros, whereε(i) > 0. Due to the strong law of large numbers [80℄, limk→∞ε(i)/k = 0 holds, whihmeans that for large enough k, the fration of symbols zeros in odeword x2(i) is1 − P ∗U . Hene, for k → ∞, the soure an transmit pratially all3 of its k(1 − P ∗U)3When we say pratially all, we mean either all or all exept for a negligible frationlimk→∞ ε(i)/k = 0 of them.32Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingsymbols from odeword x1|r(i) during a single blok to the relay. Let y1|r(i) denotethe orresponding reeived odeword at the relay. In Appendix A.1, we prove thatthe odewords x1|r(i) sent in bloks i = 2, . . . , N an be deoded suessfully at therelay from the orresponding reeived odewords y1|r(i) using a typial deoder [80℄sine R satises (2.25). Moreover, in Appendix A.1, we also prove that, for k →∞,the odewords x1|r(i) an be suessfully deoded at the relay even though, for somebloks 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 theentire odeword x2(i), omprised of k symbols of whih a fration 1 − P ∗U are zeros,to the destination. In partiular, the relay sends the zero symbols of odeword x2(i)to the destination by being silent during reeption, and sends the non-zero symbolsof odeword x2(i) to the destination by atually transmitting them. On the otherhand, the destination listens during the entire blok and reeives a odeword y2(i).In Appendix A.2 , we prove that the destination an suessfully deode x2(i) fromthe reeived odeword y2(i), and thereby obtain w(i− 1), sine rate R satisesR < maxpV (x2)I(X2; Y2)∣∣∣PU=P ∗U. (2.26)In a pratial implementation, the relay may not be able to swith between reep-tion and transmission in a symbol-by-symbol manner, due to pratial onstraintsregarding the speed of swithing. Instead, we may allow the relay to reeive andtransmit at the same time and in the same frequeny band similar to FD relaying.However, this simultaneous reeption and transmission is performed while avoid-ing self-interferene sine, in eah symbol interval, either the input or the outputinformation-arrying symbol of the relay is zero. This is aomplished in the follow-ing manner. The soure performs the same operations as for the ase when the relay33Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingswithes between reeption and transmission. On the other hand, the relay transmitsall symbols from x2(i) while ontinuously listening. Then, the relay disards fromthe reeived odeword, denoted by y1(i), those symbols for whih the orrespondingsymbols in x2(i) are non-zero, and only ollets the symbols in y1(i) for whih theorresponding symbols in x2(i) are equal to zero. The olleted symbols from y1(i)onstitute the relay's information-arrying reeived odeword y1|r(i) whih is used fordeoding. Codeword y1|r(i) is ompletely free of self-interferene sine the symbols iny1|r(i) were reeived in symbol intervals for whih the orresponding transmit symbolat the relay was zero.In the last (i.e., the (N +1)-th) blok, the soure is silent and the relay transmitsw(N) by mapping it to the orresponding odeword x2(i) from set C2. The relaytransmits all symbols in odeword x2(i) to the destination. The destination andeode the reeived odeword in blok N + 1 suessfully, sine (2.26) holds.Finally, sine both relay and destination an deode their respetive odewordsin eah blok, the entire message W an be deoded suessfully at the destinationat the end of the (N + 1)-th blok.Coding ExampleIn Fig. 2.3, we show an example for vetors x1|r, x1, y1, y1|r, x2, and y2, for k = 8and P ∗U = 1/2, where x1 ontains all k input symbols at the soure inluding thesilenes. From this example, it an be seen that x1 ontains zeros due to silenes forhannel uses for whih the orresponding symbol in x2 is non-zero. By omparing x1and x2 it an be seen that the HD onstraint is satised for eah symbol duration.The blok diagram of the proposed oding sheme is shown in Fig. 2.4. In par-tiular, in Fig 2.4, we show shematially the enoding, transmission, and deodingat soure, relay, and destination. The ow of enoding/deoding in Fig. 2.4 is as34Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without FadingPSfrag replaementsx22 x23 x25 0 x28x11|rx11|rx12|rx12|rx13|rx13|rx14|ry11y11y14y14y16y16y17y12 y13 y15 y17 y180000000x14|rx2::x1:y1:x1|r:y1|r:y2: y21 y22 y23 y24 y25 y26 y27 y28Figure 2.3: Example of generated swithing vetor along with input/output ode-words at soure, relay, and destination.PSfrag replaementsw(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|rC2C2IS D1D2BChannel 1 Channel 2Source Relay DestinationFigure 2.4: Blok diagram of the proposed hannel oding protool for time slot i.The following notations are used in the blok diagram: C1|r and C2 are enoders, D1and D2 are deoders, I is an inserter, S is a seletor, B is a buer, and w(i) denotesthe message transmitted by the soure in blok i.35Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingfollows. Messages w(i− 1) and w(i) are enoded into x2(i) and x1|r(i), respetively,at the soure using the enoders C2 and C1|r, respetively. Then, an inserter I isused to reate the vetor x1(i) by inserting the symbols of x1|r(i) into the positionsof x1(i) for whih the orresponding elements of x2(i) are zeros and setting all othersymbols in x1(i) to zero. The soure then transmits x1(i). On the other hand, therelay, enodes w(i− 1) into x2(i) using enoder C2. Then, the relay transmits x2(i)while reeiving y1(i). Next, using x2(i), the relay onstruts y1|r(i) from y1(i) byseleting only those symbols for whih the orresponding symbol in x2(i) is zero. Therelay then deodes y1|r(i), using deoder D1, into w(i) and stores the deoded bitsin its buer B. The destination reeives y2(i), and deodes it using deoder D2, intow(i− 1).2.3.3 Simpliation of Previous Converse ExpressionsAs shown in [8℄, the HD relay hannel an be analyzed with the framework developedfor the FD relay hannel in [5℄. Sine the onsidered two-hop HD relay hannelbelongs 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 asI(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)36Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingwhere (a) follows from (2.3) sine when X2 6= 0, Y1 is deterministially zero therebyleading to I(X1; Y1|X2 6= 0)= 0. Inserting (2.28) into (2.27), we obtain that forsome p(x1, x2), the following holdsR ≤ min{I(X1; Y1|X2 = 0)(1− PU) , I(X2; Y2)}. (2.29)Now, sine I(X1; Y1|X2 = 0)(1−PU) is a funtion of p(x1|x2 = 0), and no other fun-tion inside the min{·} funtion 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 asR ≤ 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 sine the mutual information I(X1; Y1|X2 =0)is a onave funtion with respet to p(x1|x2 = 0). On the other hand, I(X2; Y2)is a funtion of p(x2) whih is given in (2.8) as a funtion of pV (x2) and PU . Hene,I(X2; Y2)is also a funtion of pV (x2) and PU . Now, sine in the right hand side of(2.30) only I(X2; Y2)is a funtion of pV (x2), and sine I(X2; Y2)is a onave funtionof pV (x2) (see proof of Theorem 2.1 for the proof of onavity), we an upper boundthe right hand side of (2.30) and obtain a new upper bound for the rate R asR ≤ 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 seond term inside the min{·} funtion in (2.31) aredependent on PU . If we maximize (2.31) with respet to PU , we obtain a new upper37Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingbound for rate R asR ≤ maxPUmin{maxp(x1|x2=0)I(X1; Y1|X2 = 0)(1− PU) , maxpV (x2)I(X2; Y2)∣∣∣PU}, (2.32)where the maximum with respet to PU exists sine the rst and the seond terms in-side the min{·} funtion in (2.32) are monotonially dereasing and onave funtionswith respet to PU , respetively (see proof of Theorem 2.1 for proof of onavity).This onludes the proof that the new apaity expression in Theorem 2.1 satisesthe onverse.2.4 Capaity ExamplesIn the following, we evaluate the apaity of the onsidered relay hannel when thesoure-relay and relay-destination links are both BSCs and AWGN hannels, respe-tively.2.4.1 Binary Symmetri ChannelsAssume that the soure-relay and relay-destination links are both BSCs, where X1 =X2 = Y1 = Y2 = {0, 1}, with probability of error Pε1 and Pε2, respetively. Now,in order to obtain the apaity for this relay hannel, aording to Theorem 2.1, werst have to nd maxp(x1|x2=0)I(X1; Y1|X2 = 0)and maxpV (x2)I(X2; Y2). For the BSC, theexpression 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)38Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingwhere H(Pε1) is the binary entropy funtion, whih for probability P is dened asH(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) =12 . (2.35)On the other hand, for the BSC, the only symbol in the set X2T is symbol 1, whihRV V takes with probability one. In other words, pV (x2) is a degenerate distribution,given by pV (x2) = δ(x2 − 1). Hene,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 thedistributions of p(y2|x2 = 0) and p(y2|x2 = 1). For the BSC with probability of errorPε2, these distributions are obtained asp(y2|x2 = 0) =1− Pε2 if y2 = 0Pε2 if y2 = 1,(2.39)39Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingandp(y2|x2 = 1) =Pε2 if y2 = 01− 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)asH(Y2)∣∣pV (x2)=δ(x2−1)= −A log2(A)− (1− A) log2(1−A), (2.41)whereA = Pε2(1− 2PU) + PU . (2.42)Inserting (2.38) and (2.41) into (2.36), we obtain maxpV (x2)I(X2; Y2) asmaxpV (x2)I(X2; Y2) = −A log2(A)− (1−A) log2(1− A)−H(Pε2). (2.43)We now have the two neessary omponents required for obtaining P ∗U from (2.18),and thereby obtaining the apaity. This is summarized in the following orollary.Corollary 2.1. The apaity of the onsidered relay hannel with BSCs links is givenbyC = maxPUmin{(1−H(Pε1))(1− PU),−A log2(A)− (1− A) log2(1−A)−H(Pε2)}(2.44)40Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingand is ahieved withpV (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 whih maximizes (2.44). If PU found from4(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 whih maximizes (2.44) is found as thesolution to (2.47), and the apaity simplies toC = (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 theoptimal P ∗U whih maximizes (2.44) is P ∗U = 1/2, and the apaity simplies toC = 1−H(Pε2). (2.49)Proof. The apaity 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). Hene, 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 whih ase the apaity is given by(2.21), whih 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 whih ase the apaity is given by (2.22), whih simplies4Solving (2.47) with respet to PU leads to a nonlinear equation, whih an be easily solvedusing e.g. Newton's method [82℄.41Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingto (2.49) for the BSC.2.4.2 AWGN ChannelsIn this subsetion, we assume that the soure-relay and relay-destination links areAWGN hannels, i.e., hannels whih are impaired by independent, real-valued, zero-mean AWGN with varianes σ21 and σ22, respetively. More preisely, the outputs atthe relay and the destination are given byYk = Xk +Nk, k ∈ {1, 2}, (2.50)where Nk is a zero-mean Gaussian RV with variane σ2k, k ∈ {1, 2}, with distributionpNk(z), k ∈ {1, 2}, −∞ ≤ z ≤ ∞. Moreover, assume that the symbols transmittedby the soure and the relay must satisfy the following average power onstraints5∑x1∈X1x21 p(x1|x2 = 0) ≤ P1 and∑x2∈X2Tx22 pV (x2) ≤ P2. (2.51)Obtaining the apaity for this relay hannel using Theorem 2.1, requires expressionsfor the funtions 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 informationmaxp(x1|x2=0)I(X1; Y1|X2 = 0)and the entropy H(Y2|X2) are well known and given bymaxp(x1|x2=0)I(X1; Y1|X2 = 0)= 12 log2(1 + P1σ21)(2.52)H(Y2|X2) =12 log2(2πeσ22), (2.53)5If the optimal distributions p(x1|x2 = 0) and pV (x2) turn out to be ontinuous, the sums in(2.51) should be replaed by integrals.42Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingwhere, 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 variane P1. On the other hand,H(Y2|X2) is just the dierential entropy of Gaussian RV N2, whih is independent ofp(x2), i.e., of pV (x2). Hene,maxpV (x2)I(X2; Y2) = maxpV (x2)H(Y2)−12 log2(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 thedistribution pV (x2) whih maximizesH(Y2). Finding an expression forH(Y2) requiresthe distribution of p(y2|x2). This distribution is found using (2.50) asp(y2|x2) = pN2(y2 − x2). (2.55)Inserting (2.55) into (2.16), we obtain H(Y2) asH(Y2) = −∫ ∞−∞[PU∑x2∈X2TpN2(y2 − x2)pV (x2) + pN2(y2)(1− PU)]× log2[PU∑x2∈X2TpN2(y2 − x2)pV (x2) + pN2(y2)(1− PU)]dy2, (2.56)where, sine p(y2|x2) is now a ontinuos probability density funtion, the summationin (2.16) with respet to y2 onverges to an integral as∑y2→∞∫−∞dy2. (2.57)We are now ready to maximizeH(Y2) in (2.56) with respet to pV (x2). Unfortunately,obtaining the optimal pV (x2) whih maximizes H(Y2) in losed form is diult, if43Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingnot impossible. However, as will be shown in the following lemma, we still anharaterize the optimal pV (x2), whih is helpful for numerial alulation of pV (x2).Lemma 2.1. For the onsidered relay hannel where the relay-destination link is anAWGN hannel and where the input symbols of the relay must satisfy the averagepower onstraint given in (2.51), the distribution pV (x2) whih maximizes H(Y2) in(2.56) for a xed PU < 1 is disrete, i.e., it has the following formpV (x2) =K∑k=1pkδ(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 satisfyK∑k=1pk = 1 andK∑k=1pkx22k = P2. (2.59)In the limiting ase when PU → 1, distribution pV (x2) onverges to the zero-meanGaussian distribution with variane 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-fore searh has to be used in orderto nd x2k and pk, ∀k.44Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without FadingNow, inserting (2.58) into (2.56) we obtain maxpV (x2)H(Y2) asmaxpV (x2)H(Y2) = −∞∫−∞(PUK∑k=1p∗kpN2(y2 − x∗2k) + (1− PU)pN2(y2))× log2(PUK∑k=1p∗kpN2(y2 − x∗2k) + (1− PU)pN2(y2))dy2,(2.60)where p∗V (x2) =∑Kk=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) andmaxpV (x2)I(X2; Y2) in Theorem 2.1, we obtain the apaity of the onsidered relay hannelwith AWGN links. This is onveyed in the following orollary.Corollary 2.2. The apaity of the onsidered relay hannel where the soure-relayand relay-destination links are both AWGN hannels with noise varianes σ21 and σ22,respetively, and where the average power onstraints of the inputs of soure and relayare given by (2.51), is given byC = 12 log2(1 + P1σ21)(1− P ∗U)(a)= −∞∫−∞(P ∗UK∑k=1p∗kpN2(y2 − x∗2k) + (1− P ∗U)pN2(y2))× log2(P ∗UK∑k=1p∗kpN2(y2 − x∗2k) + (1− P ∗U)pN2(y2))dy2 −12 log2(2πeσ22),(2.61)where the optimal P ∗U is found suh that equality (a) in (2.61) holds. The apaityin (2.61) is ahieved when p(x1|x2 = 0) is the zero-mean Gaussian distribution withvariane P1 and p∗V (x2) =∑Kk=1 p∗kδ(x2−x∗2k) is a disrete distribution whih satises45Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fading(2.59) and maximizes H(Y2) given in (2.60).Proof. The apaity 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 respet to PU . For themaximization of the orresponding apaity with respet to PU , we note that P ′U <P ′′U = 1 always holds. Hene, the apaity is given by (2.21), whih for the Gaussianase simplies to (2.61). To see that P ′′U = 1, note the relay-destination hannel isan AWGN hannel for whih the mutual information is maximized when p(x2) is aGaussian distribution. From (2.8), we see that p(x2) beomes a Gaussian distributionif and only if PU = 1 and pV (x2) also assumes a Gaussian distribution.2.5 Numerial ExamplesIn this setion, we numerially evaluate the apaities of the onsidered HD relayhannel when the soure-relay and relay-destination links are both BSCs and AWGNhannels, respetively. As a performane benhmark, we use the maximal ahievablerate of onventional relaying [48℄. Thereby, the soure transmits to the relay oneodeword with rate maxp(x1|x2=0)I(X1; Y1|X2 = 0)in 1 − PU fration of the time, where0 < PU < 1, and in the remaining fration of time, PU , the relay retransmits thereeived information to the destination with rate maxp(x2)I(X2; Y2), see [18℄ and [48℄.The optimal PU , is found suh that the following holdsRconv = maxp(x1|x2=0)I(X1; Y1|X2 = 0)(1− PU) = maxp(x2)I(X2; Y2)PU . (2.62)46Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without FadingEmploying the optimal PU obtained from (2.62), the maximal ahievable rate ofonventional relaying an be written asRconv =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 LinksFor simpliity, we assume symmetri links with Pε1 = Pε2 = Pε. As a result, P ∗U < 1/2in Corollary 2.1 and the apaity is given by (2.48). This apaity is plotted inFig. 2.5, where P ∗U is found from (2.47) using a mathematial software pakage, e.g.Mathematia. As a benhmark, in Fig. 2.5, we also show the maximal ahievablerate using onventional relaying, obtained by insertingmaxp(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 rateis obtainedRconv =12(1−H(Pε)). (2.65)As an be seen from Fig. 2.5, when both links are error-free, i.e., Pε = 0, onventionalrelaying ahieves 0.5 bits/hannel use, whereas the apaity is 0.77291, whih is 54%larger than the rate ahieved with onventional relaying. This value for the apaityan be obtained by inserting Pε1 = Pε2 = 0 in (2.47), and thereby obtainC = 1− P ∗U(a)= H(P ∗U). (2.66)47Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fading0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.8PεRate(bits/use)  CapacityConventional relayingFigure 2.5: Comparison of rates for the BSC as a funtion of the error probabilityPε1 = Pε2 = Pε.Solving (a) in (2.66) with respet to P ∗U and inserting the solution for P ∗U bak into(2.66), yields C = 0.77291. We note that this value was rst reported in [9, page327℄.2.5.2 AWGN LinksFor the AWGN ase, the apaity is evaluated based on Corollary 2.2. However, sinefor this ase the optimal input distribution at the relay p∗V (x2) is unknown, i.e., thevalues of p∗k and x∗2k in (2.61) are unknown, we have performed a brute fore searh forthe values of p∗k and x∗2k whih maximize (2.61). Two examples of suh distributions6are shown in Fig. 2.6 for two dierent values of the SNR P1/σ21 = P2/σ22. Sinewe do not have a proof that the distributions obtained via brute-fore searh areatually the exat optimal input distributions at the relay that ahieve the apaity,the rates that we obtain, denoted by CL, are lower than or equal to the atual6Note that these distributions resemble a disrete, Gaussian shaped distribution with a gaparound zero.48Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fading−20 −10 0 10 2000.10.20.30.4x2kp k  p(x2|t) for 10 dBp(x2|t) for 15 dBFigure 2.6: Example of proposed input distributions at the relay pV (x2).apaity. These rates are shown in Figs. 2.7 and 2.8, for symmetri links and non-symmetri links, respetively, where we set P1/σ21 = P2/σ22 and P1/σ21/10 = P2/σ22,respetively. We note that for the results in Fig. 2.7, for P1/σ21 = P2/σ22 = 10 dB andP1/σ21 = P2/σ22 = 15 dB, we have used the input distributions at the relay shown inFig. 2.6. In partiular, for P1/σ21 = P2/σ22 = 10 dB we have used the following valuesfor p∗k and x∗2kp∗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/σ22 = 15 dB we have usedp∗k = [0.212303, 0.142311, 8.12894×10−2, 3.95678×10−2, 1.64121×10−2, 5.80092×10−31.7472×10−3, 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, sine the values of p∗k andx∗2k when x∗2k < 0 an be found from symmetry, see Fig. 2.6.49Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fading−10 −5 0 5 10 15 20 25 3000.511.522.53P1/σ21 = P2/σ22 (in dB)Rate(bits/use)  CLRGaussRconvUnachievable bound from [8] and [78]Figure 2.7: Soure-relay and relay destination links are AWGN hannels with P1/σ21 =P2/σ22.−10 −5 0 5 10 15 20 25 3000.511.522.533.5P1/σ21/10 = P2/σ22 (in dB)  Rate(bits/use)CLRGaussRconvUnachievable bound from [8] and [78]Figure 2.8: Soure-relay and relay destination links are AWGN hannels withP1/σ21/10 = P2/σ22 .50Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without FadingIn Figs. 2.7 and 2.8, we also show the rate ahieved when instead of an optimaldisrete input distribution at the relay p∗V (x2), f. Lemma 2.1, we use a ontinuous,zero-mean Gaussian distribution with variane P2. Thereby, we obtain the followingrateRGauss =12 log2(1 + P1σ21)(1− PU)(a)= −∞∫−∞(PU pG(y2) + (1− PU)pN2(y2))× log2(PU pG(y2) + (1− PU)pN2(y2))dy2 −12 log2(2πeσ22),(2.67)where PU is found suh that equality (a) holds and pG(y2) is a ontinuous, zero-meanGaussian distribution with variane P2 +σ22 . From Figs. 2.7 and 2.8, we an see thatRGauss ≤ CL, whih was expeted from Lemma 2.1. However, the loss in performaneaused by the Gaussian inputs is moderate, whih suggests that the performanegains obtained by the proposed protool are mainly due to the exploitation of thesilent (zero) symbols for onveying information from the HD relay to the destinationrather than the optimization of pV (x2).As benhmark, in Figs. 2.7 and 2.8, we have also shown the maximal ahievablerate using onventional relaying, obtained by insertingmaxp(x1|x2=0)I(X1; Y1|X2 = 0, U = r)= 12 log2(1 + P1σ21)(2.68)andmaxpV (x2)I(X2; Y2|U = t) =12 log2(1 + P2σ22)(2.69)51Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadinginto (2.63), whih yieldsRconv =12log2(1 + P1σ21)log2(1 + P2σ22)log2(1 + P1σ21)+ log2(1 + P2σ22) . (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 ahieves 3 to 6 dB gain ompared to Rconv. Hene, large per-formane gains are ahieved using the proposed apaity protool even if suboptimalinput distributions at the relay are employed.Finally, as additional benhmark in Figs. 2.7 and 2.8, we show the unahievableupper bounds reported in [8℄ and [78℄, given byCUpper = maxPU min{12 log2(1 + P1σ21)(1− PU) ,12 log2(1 + P2σ22)PU +H(PU)}.(2.71)As an be seen from Figs. 2.7 and 2.8, this bound is loose for low SNRs but beomestight for high SNRs.2.6 ConlusionWe have derived an easy-to-evaluate expression for the apaity of the two-hop HDrelay hannel without fading based on simplifying previously derived onverse ex-pressions. Moreover, we have proposed an expliit oding sheme whih ahievesthe apaity. In partiular, we showed that the apaity is ahieved when the re-lay swithes between reeption and transmission in a symbol-by-symbol manner andwhen additional information is sent by the relay to the destination using the zerosymbol impliitly sent by the relay's silene during reeption. Furthermore, we haveevaluated the apaity for the ases when both links are BSCs and AWGN hannels,52Chapter 2. Capaity of the Two-Hop Half-Duplex Relay Channel Without Fadingrespetively. From the numerial examples, we have observed that the apaity ofthe two-hop HD relay hannel is signiantly higher than the rates ahieved withonventional relaying protools.53Chapter 3Buer-Aided Relaying With AdaptiveReeption-Transmission: AdaptiveRate Transmission3.1 IntrodutionThe apaity of the two-hop HD relay network when the soure-relay and relay-destination links are AWGN hannels aeted by fading is not known, and onlyahievable rates have been reported in the literature so far, see Setion 1.3.2. In thishapter, we present new ahievable average rates for this network whih are largerthan the best known average rates. These new average rates are ahieved with abuer-aided relaying protool with adaptive reeption-transmission.In this hapter, we onsider buer-aided relaying with adaptive reeption-transmis-sion for the two-hop HD relay network when the soure-relay and relay-destinationlinks are AWGN hannels aeted by fading. In partiular, in any given time slot,based on the hannel state information (CSI) of the soure-relay and the relay-destination link a deision is made on whether the relay transmits or reeives. Forthe two-hop HD relay network, this is equivalent to seleting either the soure-relayor relay-destination link for transmission in a given time slot, i.e., equivalent to54Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionlink seletion. We onsider the ases of delay-unonstrained and delay-onstrainedtransmission. For the delay-unonstrained ase, we optimize the adaptive reeption-transmission protool and the power alloated to the soure and the relay for datarate maximization. Interestingly, the optimal adaptive reeption-transmission poliyrequires only knowledge of the instantaneous CSI of the onsidered time slot andthe statistial CSI of the involved links. However, the instantaneous CSI of pastand future time slots and the state of the relay's buer are not required for opti-mal reeption-transmission. For the delay-onstrained ase, we propose a heuristibuer-aided protool with adaptive reeption-transmission whih limits the averagedelay and ahieves a rate lose to the rate ahieved without a delay onstraint. Thisprotool only requires the instantaneous CSI of both links, and an be easily imple-mented in real-time. Our analytial and simulation results show, in good agreement,that buer-aided relaying with adaptive reeption-transmission an ahieve signif-iant performane 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 Setion 3.2, the onsid-ered system and hannel models are presented. The proposed adaptive reeption-transmission protool for buer-aided relaying is introdued in Setion 3.3, and opti-mized for rate maximization and power alloation in Setions 3.4 and 3.5, respetively.In Setion 3.6, we propose an adaptive reeption-transmission protool that limits thedelay. Numerial results are presented in Setion 3.7, and some onlusions are drawnin Setion 3.8.55Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissions(i) r(i)S R DFigure 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 timeslot, respetively.3.2 System ModelWe onsider the two-hop HD relay network shown in Fig. 3.1. For simpliity of presen-tation, in this and the following hapter, we denote the soure, relay, and destinationby S, R, and D, respetively, and the soure-relay and relay-destination links by S-Rand R-D links, respetively. We assume that the HD relay is equipped with an unlim-ited buer. The soure sends odewords to the relay, whih deodes these odewords,possibly stores the information in its buer, and eventually sends the information tothe destination. Throughout this hapter, we assume that the S-R and R-D linksare AWGN hannels aeted by fading and that the soure has always data to trans-mit. We assume that time is divided into N → ∞ slots of equal lengths. In the ithtime slot, the transmit powers of soure and relay are denoted by PS(i) and PR(i),respetively, and the instantaneous (squared) hannel gains of the S-R and R-D linksare denoted by hS(i) and hR(i), respetively. The hannel gains hS(i) and hR(i) aremodeled as mutually independent, non-negative, stationary, ergodi, and ontinuousrandom proesses with expeted values E{hS(i)} , Ω¯SR and E{hR(i)} , Ω¯RD. Weassume slow fading suh that the hannel gains are onstant during one time slotbut hange from one time slot to the next due to e.g. the mobility of the involvednodes and/or frequeny hopping. The instantaneous link SNRs of the S-R and R-Dhannels in the ith time slot are given by s(i) , γS(i)hS(i) and r(i) , γR(i)hR(i),respetively. Here, γS(i) = PS(i)/σ2nR and γR(i) = PR(i)/σ2nD denote the transmit56Chapter 3. Buer-Aided Relaying: Adaptive Rate TransmissionSNRs without fading of the soure and the relay, respetively, and σ2nR and σ2nD arethe varianes of the omplex AWGN at the relay and the destination, respetively.The average SNRs reeived at relay and destination are denoted by ΩSR , E{s(i)}and ΩRD , E{r(i)}, respetively. Throughout this hapter, we assume transmissionwith apaity ahieving odes. Hene, the transmitted odewords by soure and relayspan one time slot, are omprised of n → ∞ omplex symbols whih are generatedindependently aording to the zero-mean omplex irular-invariant Gaussian dis-tribution. In time slot i, the variane of soure's and relay's odewords are PS(i) andPR(i), and their data rate will be determined in the following setion.In the following, we outline the general buer-aided adaptive reeption-transmis-sion protool.3.3 Preliminaries and Benhmark ShemesIn this setion, we desribe the general buer-aided adaptive reeption-transmissionprotool for the two-hop HD rely network. Later, we optimize the general buer-aidedadaptive reeption-transmission protool for average rate maximization and therebyobtain the proposed buer-aided protool.3.3.1 Adaptive Reeption-Transmission Protool and CSIRequirementsThe general buer-aided adaptive reeption-transmission protool is as follows. Atthe beginning of eah time slot, the relay deides to either reeive a odeword fromthe soure or to transmit a odeword to the destination, i.e., whether to selet theS-R or R-D link for transmission in a given time slot i. One the relay makes the57Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissiondeision, it broadasts its deision7to the other nodes before transmission in timeslot i begins. If they are seleted for transmission, the soure and the relay transmitodewords spanning one time slot with rates whih are adapted to the apaity oftheir respetive links. For seletion of the reeption-transmission and of the rateadaptation, the nodes require CSI knowledge as will be detailed in the following.CSI requirements: The relay node requires knowledge of the instantaneous hannelgains hS(i) and hR(i) in order to make the deision of whether it should reeive ortransmit. In addition, if the S-R link is seleted for transmission, the soure requiresknowledge of hS(i) in order to adapt the rate of its odeword. On the other hand, ifthe R-D link is seleted for transmission, the relay the destination requires knowledgeof hR(i) for deoding. In a given time slot i, this CSI an be obtained by three pilotsymbol transmissions, one from soure and destination, respetively, and one fromthe relay. Furthermore, we assume that the noise variane σ2nR is known at soureand relay, and that the noise variane σ2nD is known at relay and destination.3.3.2 Transmission Rates and Queue DynamisIn the following, we dene the rates of the odewords transmitted by the soure andrelay in a given time slot i, and determine the state of the queue at the buer of therelay.Soure transmits relay reeives: If the soure is seleted for transmission intime slot i, it transmits one odeword with rateSSR(i) = log2(1 + s(i)). (3.1)7The deision ontains an information of one bit whih is: should relay reeive or transmit intime slot i.58Chapter 3. Buer-Aided Relaying: Adaptive Rate TransmissionHene, the relay reeives SSR(i) bits/symb from the soure and appends them to thequeue in its buer. The number of bits/symb in the buer of the relay at the end oftime slot i is denoted by Q(i) and given byQ(i) = Q(i− 1) + SSR(i). (3.2)Relay transmits destination reeives: If the relay transmits in time slot i,the number of bits/symb transmitted by the relay is given byRRD(i) = min{log2(1 + r(i)), Q(i− 1)}, (3.3)where we take into aount that the maximal number of bits/symb that an be sendby the relay is limited by the number of bits/symb in its buer and the instantaneousapaity of the R-D link. The number of bits/symb remaining in the buer at theend of time slot i is given byQ(i) = Q(i− 1)− RRD(i), (3.4)whih is always non-negative beause of (3.3).Beause of the HD onstraint, we have RRD(i) = 0 when the soure transmitsand the relay reeives, and we have SSR(i) = 0 when the relay transmits.In the following, we determine the average data rate reeived at the destinationduring N →∞ time slots.59Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission3.3.3 Ahievable Average RateSine we assume the soure has always data to transmit, the average number ofbits/symb that arrive at the destination per time slot is given byR¯SD = limN→∞1NN∑i=1RRD(i), (3.5)i.e., R¯SD is the ahievable average rate of the onsidered ommuniation system.The main goal in this hapter is the maximization of R¯SD by optimizing the relay'sreeption and transmission in eah time slot and the transmit power alloated tosoure and relay.3.3.4 Conventional RelayingFor omparison purpose, we provide the ahievable average rate of two baselineshemes and provide the CSI requirements. Thereby, we assume that the transmitpowers at the soure 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 reeives datafrom the soure in the rst ξN time slots, where 0 < ξ < 1, and sends this umulativeinformation to the destination in the next (1 − ξ)N slots, where N → ∞. Theorresponding ahievable average rate is given in (1.4), where after setting E{log2(1+s(i))} = C¯SR and E{log2(1 + r(i))} = C¯RD, we obtain the following rateR¯conv,1 =E{log2(1 + s(i))}E{log2(1 + r(i))}E{log2(1 + s(i))}+ E{log2(1 + r(i))}. (3.6)CSI Requirements: In order to ahieve (3.6) using onventional relaying with60Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionbuer as proposed [51℄, the soure and relay have to aquire the CSI of the soure-relay link in the rst ξN time slots, and the relay and destination have to aquirethe 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-aidedrelaying with adaptive reeption-transmission, this protool requires one pilot symboltransmission less.Conventional Relaying Without BuerThe ahievable average rate of onventional relaying without buer, where the relayreeives a odeword in ξ(i) fration of time slot i and transmits the reeived infor-mation in the remaining 1 − ξ(i) fration of time slot i, where 0 < ξ(i) < 1, is givenin (1.6). After setting log2(1+ s(i)) = CSR(i) and log2(1+ r(i)) = CRD(i), we obtainfrom (1.6) the following average data rateR¯conv,2 = E{ log2(1 + s(i))× log2(1 + r(i))log2(1 + s(i)) + log2(1 + r(i))}. (3.7)However, to ahieve (3.7), the lengths of odewords have to vary and to be adaptedto the fading state of the hannels in eah time slot, whih may not be desirable inpratie. In that ase, by setting ξ(i) = 1/2, ∀i, the odeword lengths ould be xed,and thereby, the following average rate is ahievedR¯conv,3 =12E {min{log2(1 + s(i), log2(1 + r(i))}} . (3.8)CSI Requirements: In order to ahieve (3.7) and (3.8) using onventional relayingwithout buer, the soure, relay, and destination have to aquire the CSI of both theS-R and R-D links in eah time slot. Thereby, three pilot transmissions are required61Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionper time slot. In addition, the CSI of S-R and R-D links have to be feedbakto destination and soure, respetively, in eah time slot. In omparison, buer-aided relaying with adaptive reeption-transmission also requires three pilot symboltransmissions per time slot, however, it requires only one bit of feedbak informationper time slot. On the other hand, sine the CSI of the S-R and R-D links are realnumbers, the two feedbaks required for onventional relaying without buer mustontain large (ideally innite) number of information bits.Comparing (3.7) and (3.6), we observe that R¯conv,2 ≤ R¯conv,1 holds. However, torealize this performane gain, the relay has to be equipped with a buer of innitesize and and innite delay is introdued.Rayleigh FadingFor the numerial results shown in Setion 3.7, we onsider the ase where the S-R and R-D links are both Rayleigh faded, i.e., the probability density funtions(PDFs) of s(i) and r(i) are given by fs(s) = e−s/ΩSR/ΩSR and fr(r) = e−r/ΩRD/ΩRD,respetively. In this ase, R¯conv,1, R¯conv,2, and R¯conv,3 given in (3.6), (3.7), and (3.8),respetively, an be obtained asR¯conv,1 =1ln(2)exp(1ΩSR)E1(1ΩSR)exp(1ΩRD)E1(1ΩRD)exp(1ΩSR)E1(1ΩSR)+ exp(1ΩRD)E1(1ΩRD) , (3.9)R¯conv,2 =∫ ∞0∫ ∞0log2(1 + s)× log2(1 + r)log2(1 + s) + log2(1 + r)e−s/ΩSR−r/ΩRDΩSRΩRDdsdr, (3.10)62Chapter 3. Buer-Aided Relaying: Adaptive Rate TransmissionandR¯conv,3 =12 ln(2) exp(ΩR + ΩSΩSΩR)E1(ΩR + ΩSΩSΩR), (3.11)respetively, where E1(x) =∫∞x e−t/t dt, x > 0, denotes the exponential integralfuntion.In the following, we optimize the general buer-aided protool with adaptivereeption-transmission for rate maximization.3.4 Optimal Adaptive Reeption-TransmissionProtool for Fixed PowersTo gain insight, we rst derive the optimal adaptive reeption-transmission poliyand the orresponding ahievable average rate for the ase when the soure andrelay transmit with xed powers, i.e., PS(i) = PS, PR(i) = PR, ∀i. Optimal poweralloation will be disussed in Setions 3.5.3.4.1 Problem FormulationIn order to formulate the reeption and transmission at the relay in time slot i, weintrodue a binary deision variable di ∈ {0, 1}. We set di = 1 if the R-D link isseleted for transmission in time slot i, i.e., the relay transmits and the destinationreeives. Similarly, we set di = 0 if the S-R link is seleted for transmission in timeslot i, i.e., the soure transmits and the relay reeives. Exploiting di, the number ofbits/symb send from the soure to the relay and from the relay to the destination in63Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissiontime slot i an be written in ompat form asSSR(i) = (1− di) log2(1 + s(i)) (3.12)andRRD(i) = di min{log2(1 + r(i)), Q(i− 1)}, (3.13)respetively. Consequently, the average rate in (3.5) an be rewritten asR¯SD = limN→∞1NN∑i=1di min{log2(1 + r(i)), Q(i− 1)}. (3.14)The onsidered rate maximization problem an now be stated as follows: Find the op-timal adaptive reeption-transmission poliy, i.e., the optimal sequene di, ∀i, whihmaximizes the ahievable average rate R¯SD given in (3.14).3.4.2 Optimal Adaptive Reeption-Transmission ProtoolUsing notation from queueing theory [83℄, we dene the average arrival rate ofbits/symb per time slot arriving into the queue of the buer, denoted by A, andthe average departure rate of bits/symb time per slot departing out of the queue ofthe buer, denoted by D, asA , limN→∞1NN∑i=1(1− di) log2(1 + s(i)) (3.15)andD , limN→∞1NN∑i=1di min{log2(1 + r(i)), Q(i− 1)}, (3.16)64Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionrespetively. We note that the average departure rate D is equal to the ahievableaverage rate given in (3.14). In order to derive the optimal protool, we give thefollowing denition for an absorbing and non-absorbing queue.Denition 3.1. An absorbing queue is a queue for whih A > D holds. A non-absorbing queue is a queue for whih A = D holds.In an absorbing queue a part of the arrival rate is absorbed (trapped) inside thebuer of unlimited size and therefore the departure rate D is smaller than the arrivalrate.The following theorem haraterizes the optimal adaptive reeption-transmissionpoliy in terms of the state of the queue in the buer of the relay.Theorem 3.1. A neessary ondition for the optimal adaptive reeption-transmissionpoliy whih maximizes the ahievable average rate is that the queue in the buer ofthe relay is at the edge of non-absorbtion, i.e., the queue is non-absorbing but is atthe 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 optimaladaptive reeption-transmission poliy has to fulll and a simplied expression forthe ahievable average rate. This is the subjet of the following theorem.Theorem 3.2. The ahievable average rate in (3.14) is maximized when the followingidentity holdslimN→∞1NN∑i=1(1− di) log2(1 + s(i)) = limN→∞1NN∑i=1di log2(1 + r(i)). (3.17)65Chapter 3. Buer-Aided Relaying: Adaptive Rate TransmissionMoreover, when (3.17) holds, the ahievable average rate in (3.14) is then given byR¯SD = limN→∞1NN∑i=1di log2(1 + r(i)) = E{di 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 sine there is noloss of information, i.e., the information that goes in the buer eventually leaves thebuer without any loss. Hene, all of the information sent by the soure is eventuallyreeived 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 beomes 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 timeslots in whih the buer does not have enough data for transmission, and therebymin{Q(i − 1), log2(1 + r(i)} = Q(i − 1) ours, are negligeable ompared to thenumber of time slots in whih the buer does have enough data for transmission,and thereby min{Q(i − 1), log2(1 + r(i)} = log2(1 + r(i)) ours. In partiular, asshown in Appendix B.2, ondition (3.17) automatially ensures that for N → ∞,1N∑Ni=1 di log2(1 + r(i)) = 1N∑Ni=1 di min{log2(1 + r(i)), Q(i − 1)} is valid, i.e., theimpat of event log2(1 + r(i)) > Q(i − 1), i = 1, . . . , N , is negligible. Hene, when(3.17) holds and N →∞, we an pratially onsider the relay is fully baklogged.We are now ready to derive the optimal adaptive reeption-transmission poliyfor buer-aided relaying without power alloation. Aording to Theorem 3.2, thepoliy that maximizes the ahievable average rate R¯SD in (3.18) an be found insidethe set of poliies that produe a queue whih satises (3.17). Thus, for N →∞, we66Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionformulate the following optimization problem:Maximize :di1N∑Ni=1 di log2(1 + r(i))Subject to : C1 : 1N∑Ni=1(1− di) log2(1 + s(i)) = 1N∑Ni=1 di log2(1 + r(i))C2 : di ∈ {0, 1}, ∀i,(3.19)where onstraint C1 ensures that the searh for the optimal poliy is onduted onlyamong those poliies that satisfy (3.17) and C2 ensures that di ∈ {0, 1}. We notethat C1 and C2 do not exlude the ase that the relay is hosen for transmission iflog2(1 + r(i)) > Q(i − 1). However, aording to Remark 3.2, C1 ensures that theinuene of event log2(1 + r(i)) > Q(i − 1) is negligible. Therefore, an additionalonstraint dealing with this event is not required. The solution of problem (3.19)leads to the following theorem.Theorem 3.3. The optimal poliy maximizing the ahievable average rate of buer-aided relaying with adaptive reeption-transmission is given bydi =1 if log2(1 + r(i)) ≥ ρ log2(1 + s(i))0 otherwise(3.20)where ρ is a onstant, referred to as the deision threshold, found suh that onstraintC1 in (3.19) holds. The orresponding maximum rate, denoted by R¯SD,max, is foundby 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 deision,di, at time slot i, depends only on the instantaneous SNRs, s(i) and r(i), of thattime slot. Hene, di does not depend on the state of the queue, Q(i), in any time67Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionslot nor on the instantaneous SNRs in previous or future time slots. This makesthe proposed optimal seletion poliy easy to implement. We note that the deisionthreshold, ρ, depends on the statistial CSI of both involved links as will be establishedin the next setion. The independene of the optimal adaptive reeption-transmissionpoliy from non-ausal instantaneous CSI is aused by the relay being operated at theedge of non-absorption, i.e., the relay node is pratially fully baklogged. Non-ausalknowledge would only help buer management (i.e., ensuring that there is a suientnumber of bits/symb in the buer for upoming time slots), whih is not required inthe onsidered regime.Remark 3.4. In this hapter, we assume that the transmitting nodes have perfetCSI and apply adaptive rate transmission. However, we note that this is not ne-essary for ahieving the maximum ahievable average rate in (3.18). In fat, theproposed adaptive reeption-transmission protool (3.20) also ahieves the maximumahievable average rate in (3.18) if soure and relay transmit long odewords thatspan (ideally innitely) many time slots (and onsequently innitely many fadingstates). In this ase, both the soure and the relay an transmit with onstant rateR¯SD,max = E{(1 − di) log2(1 + s(i))} = E{di log2(1 + r(i))}, where di is given in(3.20), and rate adaptation is not neessary. The rst odeword is transmitted bythe soure without adaptive reeption-transmission and deoded by the relay. For allsubsequent odewords, adaptive reeption-transmission is performed based on (3.20)and soure and relay transmit parts of a long odeword whenever they are seletedfor transmission. The disadvantage of this approah is that the long odewords inher-ently introdue (ideally innitely) long delays and the generalization of this approahto the delay-onstrained ase is diult. Therefore, in this hapter, we onsider adap-tive rate transmission and assume that one odeword spans only one time slot (and68Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissiononsequently one fading state).3.4.3 Deision ThresholdThe deision threshold ρ an be omputed based on the following lemma.Lemma 3.1. The deision 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), respetively, and G(r) = (1 +r) 1ρ − 1 and H(s) = (1 + s)ρ − 1.Proof. Due to the ergodiity, the left hand side of (3.17) is the expetation 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)), whih is equivalent tos(i) > G(r). Therefore, the domain of integration for alulating the expetation of(1− di) log2(1+ s(i)) is s(i) > G(r) and r(i) > 0, whih leads to the left hand side of(3.21). Using a similar approah, the right hand side of (3.21) is obtained from theright hand side of (3.17). This onludes the proof.Remark 3.5. Eq. (3.21) reveals that the deision threshold ρ depends indeed on thestatistial properties of both involved links as was already alluded to in Remark 3.3.3.4.4 Rayleigh FadingFor onreteness, we provide in this subsetion expressions for ρ and the orrespond-ing maximum ahievable average rate R¯SD,max for Rayleigh fading links. Thus, by69Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissioninserting fs(s) = e−s/ΩSR/ΩSR and fr(r) = e−r/ΩRD/ΩRD into (3.21), we obtain1ln(2)∫ ∞0[exp(−(r + 1)1ρ − 1ΩSR)ln((r + 1) 1ρ)+ e1ΩSRE1((r + 1) 1ρΩSR)]e−r/ΩRDΩRDdr= 1ln(2)∫ ∞0[exp(−(s + 1)ρ − 1ΩRD)ln ((s+ 1)ρ) + e1ΩRDE1((s+ 1)ρΩRD)]× 1ΩSRexp(− sΩSR)ds. (3.22)The optimal deision threshold ρ an be found numerially from (3.22). The orre-sponding maximum ahievable average rate is obtained asR¯SD,max =1ln(2)∫ ∞0[exp(−(s+ 1)ρ − 1ΩRD)× ln ((s+ 1)ρ)+e1ΩRDE1((s+ 1)ρΩRD)] 1ΩSRexp(− sΩSR)ds, (3.23)where ρ is found from (3.22).Speial ase (ΩSR = ΩRD)For the speial ase ΩSR = ΩRD = Ω, we obtain from (3.22) ρ = 1, and the orre-sponding maximal ahievable average rate isR¯SD,max =1ln(2) exp( 1Ω)E1( 1Ω)− 12 ln(2) exp( 2Ω)E1( 2Ω). (3.24)Comparing this average rate with the average rate ahieved with onventional re-laying with a buer, f. (3.9), the gain of adaptive reeption-transmission an beharaterized byR¯SD,maxR¯conv,1= 2− exp( 2Ω)E1( 2Ω)exp( 1Ω)E1( 1Ω) ≥ 1, (3.25)70Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionwhere the ratio R¯SD,max/R¯conv,1 monotonially inreases from 1 to 1.5 as Ω dereasesfrom ∞ to zero.3.4.5 Real-Time ImplementationIn order to selet whether the relay should reeive or transmit aording to theprotool in Theorem 3.3, the relay has to ompute the onstant ρ. This onstant anbe omputed using Lemma 3.1, but this requires knowledge of the PDFs of the fadinggains of the two links before the start of transmission. Suh a priori knowledge maynot be available in pratie. In this ase, the relay has to estimate ρ in real-timeusing only the CSI knowledge until time slot i. Sine ρ is atually the Lagrangemultiplier obtained by solving the optimization problem in (3.19), see Appendix B.3,an aurate estimate of ρ an be obtained using the gradient desent method [81℄. Inpartiular, using log2(1+s(i)) and log2(1+r(i)), the destination reursively omputesan estimate of ρ, denoted by ρe(i), asρe(i) =[ρe(i− 1) + ψ(i)(De(i− 1)−Ae(i− 1))]∞0, (3.26)where [x]ba = min{max{x, a}, b}, Ae(i − 1) and De(i − 1) are real-time estimates ofthe average arrival rate A and the average departure rate D, respetively, omputedasAe(i− 1) =i− 2i− 1Ae(i− 2) +1− di−1i− 1 log2(1 + s(i− 1)), i ≥ 2, (3.27)De(i− 1) =i− 2i− 1De(i− 2) +di−1i− 1 log2(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 whihontrols the speed of onvergene of ρe(i) to ρ. In partiular, the step size ψ(i) is71Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionsome properly hosen monotonially deaying funtion of i with ψ(1) < 1, see [81℄for more details.One the relay has estimated s(i) and r(i), and omputed ρe(i), it selets theative link, i.e., the value of di, aording to Theorem 3.3.3.5 Optimal Adaptive Reeption-Transmission andOptimal Power AlloationSo far, we have assumed that the soure and relay transmit powers are xed. In thissetion, we jointly optimize the power alloation and adaptive reeption-transmissionpoliies for buer-aided relaying.3.5.1 Problem Formulation and Optimal Power AlloationOur goal is to jointly optimize the link seletion variable di and the powers PS(i)and PR(i) in eah time slot i suh that the ahievable average rate is maximized.For onveniene, we optimize in the following the transmit SNRs without fadingγS(i) and γR(i), whih may be viewed as normalized powers, instead of the powersPS(i) = γS(i)σ2nR and PR(i) = γR(i)σ2nD themselves. For a fair omparison, we limitthe average power onsumed by the soure and the relay to Γ. This leads for N →∞72Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionto the following optimization problem:Maximize :γS(i)≥0,γR(i)≥0,di1N∑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 idential to the onstraints in (3.19) and C3 is thejoint soure-relay power onstraint. The solution of Problem (3.29) is summarized inthe following theorem.Theorem 3.4. The optimal (normalized) powers γS(i) and γR(i) and deision vari-able di maximizing the ahievable average rate of buer-aided relaying with adaptivereeption-transmission while satisfying an average soure-relay power onstraint aregiven byγS(i) =ρ/λ− 1/hS(i) if hS(i) > λ/ρ0 otherwise(3.30)γR(i) =1/λ− 1/hR(i) if hR(i) > λ0 otherwise(3.31)di =1 if[ln(hR(i)λ)+ λhR(i) − 1 > ρ ln( ρλhS(i))+ λhS(i) − ρAND hR(i) > λ AND hS(i) > λρ]OR[hR(i) > λ AND hS(i) ≤ λρ]0 otherwise(3.32)73Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionwhere ρ and λ are found suh 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) intoR¯SD,max = limN→∞1NN∑i=1di log2(1 + γR(i)hR(i)). (3.33)Proof. Please refer to Appendix B.4.3.5.2 Finding λ and ρThe following lemma establishes two equations from whih the optimal λ and ρ anbe 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 soure and the relay in time slot i be given by(3.30) and (3.31), respetively, and the link seletion variable di by (3.32). Then,ρ and λ maximizing the ahievable average rate of buer-aided relaying with adap-tive reeption-transmission and power alloation are found from the following twoequations∫ λ0[∫ ∞λ/ρlog2(ρhSλ)fhS(hS)dhS]fhR(hR)dhR+∫ ∞λ[∫ ∞L1(ρhSλ)fhS(hS)dhS]fhR(hR)dhR=∫ λ/ρ0[∫ ∞λlog2(hRλ)fhR(hR)dhR]fhS(hS)dhS+∫ ∞λ/ρ[∫ ∞L2log2(hRλ)fhR(hR)dhR]fhS(hS)dhS , (3.34)74Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission∫ λ0[∫ ∞λ/ρ(ρλ −1hS)fhS(hS)dhS]fhR(hR)dhR+∫ ∞λ[∫ ∞L1(ρλ −1hS)fhS(hS)dhS]fhR(hR)dhR+∫ λ/ρ0[∫ ∞λ(1λ −1hR)fhR(hR)dhR]fhS(hS)dhS+∫ ∞λ/ρ[∫ ∞L2( 1λ −1hR)fhR(hR)dhR]fhS(hS)dhS = Γ (3.35)whereL1 = −λρW (−e(hR−λ)/(ρhR)−1(λ/hR)1/ρ),L2 = −λW (−eρ−1−λ/hS (λ/(ρhS))ρ). (3.36)Here, W (·) is the Lambert W -funtion [84℄, whih is available as built-in funtionin software pakages suh as Mathematia. The maximum ahievable average rate isgiven 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 sine (3.34) and (3.35) only dependon the statistial properties of the S-R and the R-D links. Sine these statistialproperties hange on a muh slower time sale than the instantaneous hannel gains,λ and ρ an be updated with a low rate.75Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission3.5.3 Rayleigh FadingFor the speial ase of Rayleigh fading with fhS(hS) = e−hS/Ω¯SR/Ω¯SR and fhR(hR) =e−hR/Ω¯RD/Ω¯RD, (3.34) and (3.35) an be simplied to an be simplied to1ln(2)[(1− e−λ/Ω¯RD)E1( λρΩ¯SR)+∫ ∞λ{e−L1/Ω¯SR ln(ρL1λ)+ E1( L1Ω¯SR)}e−hR/Ω¯RDΩ¯RDdhR]= 1ln(2)[(1− e−λ/(ρΩ¯SR))E1( λΩ¯RD)+∫ ∞λ/ρ{e−L2/Ω¯RD ln(L2λ)+ E1( L2Ω¯RD)}e−hS/Ω¯SRΩ¯SRdhS](3.37)and(1− e−λ/Ω¯RD)ρλe−λ/(ρΩ¯SR) −E1(λρΩ¯SR)Ω¯SR+∫ ∞λρλe−L1/Ω¯SR −E1(L1Ω¯SR)Ω¯SRe−hR/Ω¯RDΩ¯RDdhR+(1− e−λ/(ρΩ¯SR))1λe−λ/Ω¯RD −E1(λΩ¯RD)Ω¯RD+∫ ∞λ/ρ1λe−L2/Ω¯RD −E1(L2Ω¯RD)Ω¯RDe−hS/Ω¯SRΩ¯SRdhS = Γ, (3.38)respetively, where L1 and L2 are given in (3.36) and the maximum ahievable averagerate is given by the left (and right) hand side of equation (3.37).3.5.4 Real-Time ImplementationSimilar to the real-time implementation for nding ρ for the ase without poweralloation desribed in Se. 3.5.1, we an also onstrut a real-time implementation76Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionfor nding ρ and λ for the ase with power alloation. In this ase, an estimate forρ 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 asλe(i) =[λe(i− 1) + φ(i)(ΓeS(i− 1) + ΓeR(i− 1)− Γ]∞0, (3.39)whereΓeS(i− 1) =i− 2i− 1ΓeS(i− 2) +1− di−1i− 1 γS(i), i ≥ 2, (3.40)ΓeR(i− 1) =i− 2i− 1ΓeR(i− 2) +di−1i− 1γR(i), i ≥ 2, (3.41)where ΓeS(0) and ΓeR(0) are set to zero. In (3.39), φ(i) is an adaptive step size whihontrols the speed of onvergene of λe(i) to λ. In partiular, the step size φ(i) issome properly hosen monotonially deaying funtion of i with φ(1) < 1, see [81℄ formore details.3.6 Delay-Limited TransmissionSo far, we have assumed that there is no delay onstraint. In pratie, there is usuallysome onstraint on the delay. In this setion, we propose a buer-aided adaptivereeption-transmission protool for delay onstrained transmission. For simpliity,we assume xed transmit powers, i.e., PS(i) = PS, PR(i) = PR, ∀i.3.6.1 Average DelaySine we assume that the soure is baklogged and has always information to transmit,for the onsidered network, the transmission delay is aused only by the buer at the77Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionrelay. Let T (i) denote the delay of a bit of information that is transmitted by thesoure in time slot i and reeived at the destination in time slot i + T (i), i.e., theonsidered bit is stored for T (i) time slots in the buer. Then, aording to Little'slaw [85℄ the average delay E{T (i)} in number of time slots is given byE{T (i)} = E{Q(i)}A , (3.42)where E{Q(i)} is the average queue length at the buer and A is the average arrivalrate into the queue.The queue size at time slot i an be obtained asQ(i) = Q(i− 1) + (1− di) log2(1 + s(i))− di min{Q(i− 1), log2(1 + s(i))}. (3.43)Due to the reursiveness of the expression in (3.43), it is diult, if not impossible,to obtain an analytial expression for the average queue size E{Q(i)} for a generalbuer-aided relaying poliy. Hene, in ontrast to the ase without delay onstraint,for the delay limited ase, it is very diult to formulate an optimization problemfor maximization of the average rate subjet to some average delay onstraint. Asa result, in the following, we develop a simple heuristi protool for delay limitedtransmission. In the proposed protool, the relay itself deides whether it shouldreeive or transmit in eah time slot suh that the average delay onstraint is satised,and informs the soure and destination about the deision. We note that the proposedprotool does not need any knowledge of the statistis of the hannels. The protoolneeds only the instantaneous CSI of the S-R and R-D links at the relay, and thedesired average delay T0. This allows for relatively easy real-time implementation ofthe proposed protool for delay-limited transmission.78Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission3.6.2 Buer-Aided Protool for Delay Limited TransmissionBefore presenting the proposed heuristi protool for delay limited transmission, werst explain the intuition behind the protool.Intuition Behind the ProtoolAssume that we have a buer-aided protool whih, when implemented in the on-sidered network, enfores the following relationE{Q(i)}A = T0, (3.44)where T0 is the desired average delay. There are many ways to enfore (3.44) atthe relay. Our preferred method for enforing (3.44) is to have the relay reeive andtransmit when Q(i)/A < T0 and Q(i)/A > T0, respetively. In this way, Q(i)/Abeomes a random proess whih exhibits utuation around its mean value T0, andthereby ahieves (3.44) in the long run. We are now ready to present the proposedprotool.The Proposed ProtoolLet T0 be the desired average delay onstraint of the system. At the beginningof time slot i, soure and destination transmit pilots in suessive pilot time slots.This enables the relay to aquire the CSI of their respetive S-R and R-D links,respetively. Using the aquired 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 itsbuer, Q(i− 1), the relay omputes a variable ω(i) as followsω(i) = ω(i− 1) + ζ(i)(T0 −Q(i− 1)Ae(i− 1)), (3.45)79Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionwhere Ae(i−1) is a real-time estimate of A, omputed using (3.27). In (3.45), ζ(i) isthe step size funtion, whih is some properly hosen monotonially deaying funtionof i with ζ(1) < 1. Now, using log2(1 + s(i)), log2(1 + r(i)), Q(i− 1), and ω(i), therelay omputes di asdi =1 if 1ω(i) min{Q(i− 1), log2(1 + r(i))} ≥ ω(i) log2(1 + s(i))0 otherwise(3.46)The relay then broadasts a ontrol paket ontaining pilot symbols and informationabout whether the relay reeives or transmits to the soure and destination. From thepaket broadasted by the relay, both soure and destination learn the S-R and R-Dlinks, respetively, and learn whether the relay is sheduled to reeive or transmit.If the relay is sheduled to transmit, then it extrats information from its buer andtransmits a odeword to the destination with rateRRD(i) = min{Q(i− 1), log2(1 + r(i))}.However, if the relay is sheduled to reeive, then the soure transmits a odewordto the relay with rate SSR(i) = log2(1 + s(i)).Remark 3.6. The required overhead of the proposed delay-limited protool is identialto the overhead of the proposed protool without delay onstraint.Remark 3.7. Although oneptually simple, a theoretial analysis of the ahievableaverage rate of the proposed delay-limited protool is diult. Thus, we will resort tosimulations to evaluate the performane of the delay limiting protool in Setion 3.7.Remark 3.8. We note that the proposed protool for the delay-onstrained ase isheuristi in nature. The searh for other protools with possibly superior perfor-80Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmissionmane is an interesting topi for future work. The proposed protools for the delay-onstrained and the delay-unonstrained ase an serve as benhmark and perfor-mane upper bound for these new protools, respetively.3.7 Numerial and Simulation ResultsIn this setion, we evaluate the performane of buer-aided relaying (BAR) withadaptive reeption-transmission and ompare it with that of onventional relaying.Throughout this setion, we assume Rayleigh fading. All results shown in this setionhave been onrmed by omputer simulations. However, the simulations are notshown in all instanes for larity of presentation. We note that the simulation resultsare independent of whether the fading is slot-by-slot orrelated or unorrelated.3.7.1 Delay-Unonstrained TransmissionFirst, we assume that there are no delay onstraints and investigate the ahievableaverage rates with and without power alloation.In Fig. 3.2, we show the ahievable average rates of buer-aided relaying withadaptive reeption-transmission, R¯SD,max, without power alloation, given in (3.23),and the ahievable average rate of onventional relaying with a buer, R¯conv,1, givenin (3.9), and without a buer with adaptive and xed odeword lengths R¯conv,2 andR¯conv,3, respetively, given in (3.10) and (3.11), respetively, for Ω¯SR = 0.9, Ω¯RD =1.1, and γs = γr = γ. Moreover, we have also shown the rate of buer-aided relayingobtained via simulations. As an be seen from Fig. 3.2, the simulated and theoretialresults math perfetly. The gure shows that buer-aided relaying with adaptivereeption-transmission leads to substantial gains ompared to onventional relaying.In partiular, for γ = 10 dB, the gain of buer-aided relaying with adaptive reeption-81Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission−10 −5 0 5 10 15 20 25 3000.511.522.533.544.55γ (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 lengthsFigure 3.2: Average rates ahieved with buer-aided relaying (BAR) with adaptivereeption-transmission and with onventional relaying with and without buer forΩSR = 0.9 and ΩRD = 1.1.transmission over onventional relaying with a buer is 3 dB, and without a buerwith adaptive and xed odeword lengths is 4 dB and 6 dB, respetively.For the parameters adopted in Fig. 3.2, we show in Fig. 3.3 the orrespondingonstant ρ obtained using Lemma 3.1, and the orresponding estimated parameterρe(i) obtained using the reursive method in (3.26) as funtions of time for γ = 0dB. As an be seen from Fig. 3.3, the estimated parameter ρe(i) onverges relativelyquikly to ρ.In Fig. 3.4, we investigate the gains ahieved with power alloation for a systemwith Ω¯S = 0.1 and Ω¯R = 1.9. Thereby, we ompare the performanes of buer-aidedrelaying with adaptive reeption-transmission with and without power alloation.For buer-aided relaying with adaptive reeption-transmission and power alloationthe average rate, power alloation, and adaptive reeption-transmission poliy wereobtained as desribed in Theorem 3.4 and Lemma 3.2 in Setion 3.5. As an be seenfrom Fig. 3.4, and as expeted, power alloation inreases the average rate.82Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission50 100 150 200 250 300 350 400 450 50000.511.52  Time slot iρρe(i)ρFigure 3.3: Estimated ρe(i) as a funtion of the time slot i.−5 0 5 10 15 2000.511.522.5Γ (in dB)Averagerate(inbits/symb)  With power allocationWithout power allocationFigure 3.4: Average rate with buer-aided relaying with adaptive reeption-transmission with and without power alloation for Ω¯S = 0.1 and Ω¯R = 1.983Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission−5 0 5 10 15 20 25 3000.511.522.533.544.55γ (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 = 1Figure 3.5: Average rate of BAR with adaptive reeption-transmission for dierentaverage delay onstraints.3.7.2 Delay-Constrained TransmissionWe now turn our attention to delay-limited transmission and investigate the perfor-mane of the proposed buer-aided protool for this ase. Furthermore, we assumexed transmit powers for the soure and the relay.In Fig. 3.5, we plot the ahievable average rate for buer-aided relaying withadaptive reeption-transmission without and with a delay onstraint, as a funtion ofγ, for Ω¯SR = Ω¯RD = 1. This numerial example shows that for an average delay of 5,3, and 2 time slots, the rate of the delay onstrained protool is within 0.75, 1.5, and2.5 dB from the rate of the protool without a delay onstraint (i.e., T0 → ∞). Foromparison, we have also plotted the average rate of onventional relaying withoutbuer with adaptive and xed odeword lengths, respetively, whih require a delayof 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 protool is within 3, 2.5, and 1.5 dB from the rateahieved with onventional relaying with adaptive odeword lengths, and within 5,84Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission0 100 200 300 400 500123456Time slot iAveragedelayuntiltimesloti T0=5Figure 3.6: Average delay until time slot i for T0 = 5 and γ = 20 dB .4.5, and 3.5 dB from the rate ahieved with onventional relaying with xed odewordlengths. Considering the more stringent feedbak requirements for CSI aquisition ofonventional relaying without buer, see Setion 3.3.4, this example learly shows thepotential of buer-aided relaying for pratial delay-limited transmission senarios.Furthermore, for the parameters adopted in Fig. 3.5, we have plotted the averagedelay of the proposed delay-limited protool until time slot i in Fig. 3.6, for the asewhen T0 = 5 time slots, and γ = 15 dB. The average delay until time slot i, denotedby T¯ (i) is omputed asT¯ (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 thei-th time slot. Fig. 3.6 shows that with the protool proposed for delay limitedtransmission, the average delay until time slot i onverges to the desired delay T0relatively fast. Moreover, after the average delay has reahed T0, it exhibits relativelysmall utuations around T0.85Chapter 3. Buer-Aided Relaying: Adaptive Rate Transmission3.8 ConlusionsIn this hapter, we proposed a novel adaptive reeption-transmission protool forrelays with buers. In ontrast to onventional relaying, where the soure and therelay transmit aording to a pre-dened shedule regardless of the fading state, inthe proposed sheme, always the node with the relatively stronger link is seleted fortransmission. For delay-unonstrained transmission, we derived the optimal adap-tive reeption-transmission poliy for the ases of xed and variable soure and relaytransmit powers. In both ases, the optimal poliy for a given time slot only dependson the instantaneous CSI of that time slot and the statistial CSI of the involvedlinks. For delay-onstrained transmission, we proposed a buer-aided protool whihontrols the delay introdued by the buer at the relay. This protool needs onlyinstantaneous CSI and does not need statistial CSI of the involved links, and anbe implemented in real-time. Our analytial and simulation results showed thatbuer-aided relaying with adaptive reeption-transmission with and without delayonstraints is a promising approah to inrease the ahievable average data rate om-pared to onventional relaying.86Chapter 4Buer-Aided Relaying With AdaptiveReeption-Transmission: Fixed andMixed Rate Transmission4.1 IntrodutionIn this hapter, we onsider the two-hop HD relay network where the soure-relayand the relay-destination links are AWGN hannels aeted by fading, and assumethat that the soure 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 transmittedodewords span one fading state. In this ase, a hannel apaity in the strit Shan-non sense does not exist and an appropriate measure for suh systems is the outageprobability. Depending on the availability of CSIT at the transmitting nodes (andtheir apability of using more than one modulation/oding sheme), we onsider twodierent modes of transmission for the two-hop HD relay network: Fixed rate trans-mission and mixed rate transmission. In xed rate transmission, the node seletedfor transmission (soure or relay) does not have CSIT and transmits with xed rate.In ontrast, in mixed rate transmission, the relay has CSIT knowledge and exploitsit to transmit with variable rate so that outages are avoided. However, the soure87Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionstill transmits with xed rate to avoid the need for CSIT aquisition.To explore the performane limits of the proposed xed rate and mixed rateadaptive reeption-transmission shemes, we onsider rst transmission without de-lay onstraints and derive the orresponding optimal buer-aided relaying protoolswhih maximize the throughput of the onsidered two-hop HD relay network. Fur-thermore, we show that in Rayleigh fading the optimal buer-aided relaying protoolwith adaptive reeption-transmission ahieves 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 ahieves a diversity gain of one. For mixed ratetransmission, we show that a multiplexing gain of one an be ahieved with buer-aided relaying with and without adaptive reeption-transmission implying that thereis no multiplexing gain loss ompared to ideal FD relaying. Sine it turns out thatthese optimal buer-aided protools introdue innite delay, in order to limit the de-lay, we also introdue modied buer-aided relaying protools for delay onstrainedtransmission. In partiular, for xed rate and mixed rate transmission with delay on-straints, in order to ontrol the average delay, we introdue appropriate modiationsto the buer-aided relaying protools for the delay unonstrained ase. Surprisingly,for xed rate transmission, the full diversity gain is preserved as long as the tolerableaverage delay exeeds three time slots. For mixed rate transmission with an averagedelay of E{T} time slots, a multiplexing gain of r = 1− 1/(2E{T}) is ahieved.The remainder of this hapter is organized as follows. In Setion 4.2, the systemmodel of the onsidered two-hop HD relay network is presented. In Setion 4.3 weintrodue the buer-aided relaying protools for xed and mixed rate transmission. InSetions 4.4 and 4.5, we analyze the proposed buer-aided relaying protools for delayunonstrained and delay onstrained xed rate transmission, respetively. Protools88Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionfor delay unonstrained and delay onstrained mixed rate transmission are proposedand analyzed in Setion 4.6. The proposed protools and the derived analytialresults are veried and illustrated with numerial examples in Setion 4.7, and someonlusions are drawn in Setion 4.8.4.2 System Model and Channel ModelWe onsider the two-hop HD relay network shown in Fig. 3.1, where both S-R andR-D links are AWGN hannels aeted by slow fading. We assume that the relayis equipped with a buer. The soure sends odewords to the relay, whih deodesthese odewords, possibly stores the deoded information in its buer, and eventuallysends 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 onetime slot. Throughout this hapter, we assume that the soure node has alwaysdata to transmit. Hene, the total number of time slots, denoted by N , satisesN → ∞. Furthermore, unless speied otherwise, we assume that the buer at therelay is not limited in size. The ase of limited buer size will be investigated inSetions 4.5 and 4.6.4 when we investigate delay limited transmission. In the ithtime slot, the transmit powers of soure and relay are denoted by PS(i) and PR(i),respetively, and the instantaneous squared hannel gains of the S-R and R-D linksare denoted by hS(i) and hR(i), respetively. hS(i) and hR(i) are modeled as mutuallyindependent, non-negative, stationary, and ergodi random proesses with expetedvalues Ω¯S , E{hS(i)} and Ω¯R , E{hR(i)}. We assume that the hannel gains areonstant 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 frequeny hopping. We note thatfor most results derived in this hapter, we only require hS(i) and hR(i) to be not89Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionfully temporally orrelated, respetively. However, in some ases, we will assume thathS(i) and hR(i) are temporally unorrelated, respetively, to failitate the analysis.The instantaneous SNRs of the S-R and R-D hannels in the ith time slotare given by s(i) , γS(i)hS(i) and r(i) , γR(i)hR(i), respetively. Here, γS(i) ,PS(i)/σ2nR and γR(i) , PR(i)/σ2nD denote the transmit SNRs of the soure and therelay without fading, respetively, and σ2nR and σ2nD are the varianes of the omplexAWGN at the relay and the destination, respetively. The average reeived SNRs atrelay and destination are denoted by ΩS , E{s(i)} and ΩR , E{r(i)}, respetively.Throughout this hapter, we assume transmission with apaity ahieving odes.Hene, the transmitted odewords by soure and relay span one time slot, are om-prised of n → ∞ omplex symbols whih are generated independently aording tothe zero-mean omplex irular-invariant Gaussian distribution. In time slot i, thevariane of soure's and relay's odewords are PS(i) and PR(i), and their date ratewill be determined in the following setion.For onreteness, we speialize 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, respetively. Similarly, the PDFs of hS(i) and hR(i) are given by fhS(hS) =e−hS/Ω¯S/Ω¯S and fhR(hR) = e−hR/Ω¯R/Ω¯R, respetively.In the following, we outline the general buer-aided adaptive reeption-transmissionprotool for transmission with xed and mixed rates.4.3 Preliminaries and Benhmark ShemesIn this setion, we desribe the general buer-aided adaptive reeption-transmissionprotool for transmission with xed and mixed rates. In partiular, we outline thetransmission rates of the soure and relay in eah time slot, the CSI requirements,90Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionthe dynamis of the queue, and the throughput reeived at the destination.4.3.1 Adaptive Reeption-Transmission and CSIRequirementsThe general buer-aided adaptive reeption-transmission protool is as follows. Atthe beginning of eah time slot, the relay deides to either reeive a odeword fromthe soure or to transmit a odeword to the destination, i.e., deides whether to seletthe S-R or R-D link for transmission in a given time slot i. To this end, the relay isassumed to know the statistis of the S-R and R-D hannels. One the relay makesthe deision, it broadasts its deision (ontaining one bit of information) to the othernodes before transmission in time slot i begins. If they are seleted for transmission,the soure and the relay transmit odewords spanning one time slot and with rateswhih will be determined below. For seletion of the reeption and transmission, thenodes require CSI knowledge as will be detailed in the following.CSI for Fixed Rate TransmissionFor xed rate transmission, neither the soure nor the relay have full CSIT, i.e.,soure and relay do not know hS(i) and hR(i), respetively. Therefore, both nodesan transmit only with predetermined xed rates S0 and R0, respetively, and annotperform power alloation, i.e., the transmit powers are a priori xed as PS(i) = PSand PR(i) = PR, ∀i. The relay and destination are assumed to know the CSI of theirreeiving links, whih is needed for oherent detetion. For the relay to be able todeide whether it should reeive or transmit, it requires knowledge of the outage statesof the S-R and R-D links. The relay an determine whether or not the S-R link is inoutage based on S0, PS, σ2nR, and hS(i). The destination an do the same for the R-D91Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionlink based on R0, PR, σ2nD , and hR(i), and inform the relay whether or not the R-Dlink is in outage using one bit of feedbak8information. Based on the outage statesof the S-R and R-D links in a given time slot i and the statistis of both links, therelay selets the transmitting node aording to the adaptive reeption-transmissionprotools introdued in Setions 4.4 and 4.5, and informs the soure and destinationabout its deision using one bit of feedbak information.CSI for Mixed Rate TransmissionFor this mode of transmission, we assume that the relay has full CSIT, i.e., it knowshR(i), and an therefore adjust its transmission rate and transmit power PR(i) toavoid outages on the R-D link. However, the soure still does not have CSIT andtherefore has to transmit with xed rate S0 and xed power PS as it does not knowhS(i). Similar to the xed rate ase, the relay an determine the outage state of theS-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., basedon pilot symbols emitted by the destination. Based on the outage state of the S-R linkand hR(i), and on the statistis of both links, the relay selets the transmitting nodeaording to the adaptive reeption-transmission protools proposed in Setion 4.6,and informs the soure and destination about its deision using one bit of feedbakinformation.For both modes of transmission, the relay knows the outage state of the S-R andthe R-D links. Hene, if the relay is seleted for transmission but the R-D link is inoutage, the relay remains silent and an outage event ours. Whereas, if the soure is8We note that in onventional relaying, feedbak of few bits of information does not improve theoutage performane of the two-hop HD relay network. On the ontrary, the two bits of feedbak(one from D to R and the other from R to S or D), along with the buer at the relay have a pivotalrole in the proposed protool.92Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionseleted for transmission and the S-R link is in outage, the relay does not transmit afeedbak signal so that the soure remains silent, i.e., again an outage event ours.One the deision regarding the transmitting node has been made, and the relay hasinformed the transmitting node (soure or relay) aordingly, transmission in timeslot i begins.Remark 4.1. We note that all the derivations and results for mixed rate transmissionin this hapter also hold for the ase when the soure transmits with an adaptive rateand the relay transmits with a xed rate. The only dierene is that in the derivedresults, the subsripts S and R should swith positons.Remark 4.2. We note that xed rate transmission requires only two emissions of pi-lot symbols (by soure and relay). In ontrast, mixed rate transmission requires threeemissions of pilot symbols (by soure, relay, and destination). Thus, the CSI re-quirements and feedbak overhead of the buer-aided adaptive reeption-transmissionprotools proposed in this hapter are similar to those of existing relaying protools,suh as the opportunisti protool proposed in [14℄. Namely, the protool proposed in[14℄ requires the relays to aquire the instantaneous CSI of the S-R and R-D links.Furthermore, a few bits of information are fed bak from the relays to both the soureand the destination.4.3.2 Transmission Rates and Queue DynamisIn the following, we disuss the transmission rates and the state of the buer whensoure and relay transmit in a given time slot i for both xed and mixed rate trans-mission.If the soure is seleted for transmission in time slot i and an outage does notour, i.e., log2(1 + s(i))≥ S0, it transmits one odeword with rate SSR(i) = S0.93Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionHene, the relay reeives S0 bits/symb from the soure and appends them to thequeue in its buer. The number of bits/symb in the buer of the relay at the end ofthe i-th time slot is denoted by Q(i) and given byQ(i) = Q(i− 1) + S0. (4.1)If the soure is seleted for transmission but the S-R link is in outage, i.e., log2(1 +s(i))< S0, the soure remains silent, i.e., SSR(i) = 0, and the queue in the buerremains unhanged, i.e., Q(i) = Q(i− 1).For xed rate transmission, if the relay is seleted for transmission in time slot iand transmits one odeword with rate R0, an outage does not our if log2(1+r(i))≥R0. In this ase, the number of bits/symb transmitted by the relay is given byRRD(i) = min{R0, Q(i− 1)}, (4.2)where we take into aount that the maximum number of bits/symb that an besend by the relay is limited by the number of bits/symb in the buer. The numberof bits/symb remaining in the buer at the end of time slot i is given byQ(i) = Q(i− 1)− RRD(i), (4.3)whih is always non-negative beause of (4.2). If the relay is seleted for transmissionin time slot i but an outage ours, i.e., log2(1 + r(i))< R0, the relay remainssilent, i.e., RRD(i) = 0, while the queue in the buer remains unhanged, i.e., Q(i) =Q(i− 1).For mixed rate transmission, the relay is able to adapt its rate to the apaityof the R-D hannel, log2(1 + r(i)), and outages are avoided. If the relay is seleted94Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionfor transmission in time slot i, the number of bits/symb transmitted by the relay isgiven byRRD(i) = min{log2(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 sloti is still given by (4.3) where RRD(i) is now given by (4.4).Furthermore, beause of the HD onstraint, for both xed and mixed rate trans-mission, we have RRD(i) = 0 and SSR(i) = 0 if soure and relay are seleted fortransmission in time slot i, respetively.4.3.3 Link Outages and Indiator VariablesIn order to model the outages on the S-R and R-D links, we introdue the binarylink outage indiator variables OS(i) ∈ {0, 1} and OR(i) ∈ {0, 1} dened asOS(i) ,0 if s(i) < 2S0 − 11 if s(i) ≥ 2S0 − 1(4.5)andOR(i) ,0 if r(i) < 2R0 − 11 if r(i) ≥ 2R0 − 1, (4.6)respetively. In other words, OS(i) = 0 indiates that for transmission with rate S0,the S-R link is in outage, i.e., log2(1 + s(i)) < S0, and OS(i) = 1 indiates that thetransmission over the S-R hannel will be suessful. Similarly, OR(i) = 0 indiatesthat for transmission with rate R0, the R-D link is in outage, i.e., log2(1+r(i)) < R0,95Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionand OR(i) = 1 means that an outage will not our. Furthermore, we denote theoutage probabilities of the S-R and R-D hannels as PS and PR, respetively. Theseprobabilities are dened asPS , limN→∞1NN∑i=1(1− OS(i)) (a)= Pr{s(i) < 2S0 − 1}(4.7)andPR , limN→∞1NN∑i=1(1− OR(i)) (a)= Pr{r(i) < 2R0 − 1}, (4.8)respetively, where (a) follows from the assumed ergodiity of the fading.4.3.4 Performane MetrisIn this hapter, we adopt the throughput and the outage probability as performanemetris.Assuming the soure has always data to transmit, for both xed and mixed ratetransmission, the average number of bits/symb that arrive at the destination per timeslot is given byτ = limN→∞1NN∑i=1RRD(i), (4.9)i.e., τ is the throughput of the onsidered ommuniation system.The outage probability is dened in the literature as the probability that theinstantaneous hannel apaity is unable to support some predetermined xed trans-mission rate. In the onsidered system, an outage does not ause information losssine the relay knows in advane whether or not the seleted link an support thehosen transmission rate and data is only transmitted if the orresponding link is not96Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionin outage. Nevertheless, outages still aet the ahievable throughput negatively. Inthis ase, the outage probability an be interpreted as the fration of the throughputlost due to outages. Thus, denoting the maximum throughput of a system in theabsene of outages by τ0 and the throughput in the presene of outages by τ , theoutage probability, Fout, an be expressed asFout = 1−ττo. (4.10)Note that maximizing the throughput is equivalent to minimizing the outage proba-bility.4.3.5 Performane Benhmarks for Fixed Rate TransmissionFor xed rate transmission, two onventional relaying shemes serve as performanebenhmarks for the proposed buer-aided relaying sheme with adaptive reeption-transmission. In ontrast to the proposed sheme, the benhmark shemes employ apredetermined shedule for when soure and relay transmit whih is independent ofthe instantaneous link SNRs.In the rst sheme, referred to as Conventional Relaying 1, the soure transmitsin the rst ξN time slots, where 0 < ξ < 1 and eah odeword spans one time slot.The relay tries to deode these odewords and, if the deoding is suessful, it storesthe orresponding information in its buer. In the following (1− ξ)N time slots, therelay transmits the stored information to the destination, transmitting one odewordper time slot. Assuming that for the benhmark shemes soure and relay transmitodewords having the same rate, i.e., S0 = R0, the throughput of Conventional97Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionRelaying 1 is obtained asτfixedconv,1 = limN→∞1N min{ ξN∑i=1R0OS(i) ,N∑i=ξN+1R0OR(i)}= R0 min {ξ(1− PS) , (1− ξ)(1− PR)} . (4.11)The throughput is maximized if ξ(1 − PS) = (1 − ξ)(1 − PR) holds or equivalentlyif ξ = (1 − PR)/(2 − PS − PR). Inserting ξ into (4.11) we obtain the maximizedthroughput asτfixedconv,1 = R0(1− PS)(1− PR)2− PS − PR. (4.12)The maximum throughput in the absene of outages is τ0 = R0/2. Hene, using(4.10), the orresponding outage probability is obtained asF fixedout,conv,1 = 1− 2(1− PS)(1− PR)2− PS − PR. (4.13)In the seond sheme, referred to as Conventional Relaying 2, see [12℄, in therst time slot, the soure transmits one odeword and the relay reeives and triesto deode the odeword. If the deoding is suessful, in the seond time slot, therelay retransmits the information to the destination, otherwise it remains silent. Thethroughput of Conventional Relaying 2 is obtained asτfixedconv,2 = limN→∞1NN/2∑i=1R0OS(2i− 1)OR(2i) =R02 (1− PS)(1− PR). (4.14)Based on (4.10) the orresponding outage probability is given byF fixedout,conv,2 = 1− (1− PS)(1− PR), (4.15)98Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionwhih is idential to the outage probability obtained in [12℄ using the standard deni-tion for the outage probability. We note that τfixedconv,1 ≥ τfixedconv,2 (F fixedout,conv,1 ≤ F fixedout,conv,2)always holds. However, in order for Conventional Relaying 1 to realize this gain, aninnite delay is required, whereas Conventional Relaying 2 requires a delay of onlyone time slot.For the speial ase of Rayleigh fading, we obtain from (4.7) and (4.8) PS =1− e−2R0−1ΩSand PR = 1− e−2R0−1ΩR, respetively. The orresponding throughputs andoutage probabilities for Conventional Relaying 1 and 2 an be obtained by applyingthese results in (4.12)-(4.15). In partiular, in the high SNR regime, when γS = γR =γ →∞, we obtain τfixedconv,1 → R0/2, τfixedconv,2 → R0/2, andF fixedout,conv,1 →2R0 − 12Ω¯S + Ω¯RΩ¯SΩ¯R1γ , as γ →∞, (4.16)F fixedout,conv,2 → (2R0 − 1)Ω¯S + Ω¯RΩ¯SΩ¯R1γ , as γ →∞. (4.17)Hene, for xed rate transmission, the diversity gain of Conventional Relaying 1 and2 is one as expeted.Note that due to the xed sheduling of reeption and transmission at the HDrelay for Conventional Relaying 1 and 2, feedbak of few bits of information from Dto R and R to S or D, annot improve the outage peformane. On ontrary, as will beshown, in buer-aided relaying with adaptive reeption-transmission, the feedbak offew bits of information is essential and signiantly improves the outage peformane.4.3.6 Performane Benhmarks for Mixed Rate TransmissionWe also provide two performane benhmarks with a priori xed reeption-transmissionshedule for mixed rate transmission. The two benhmark protools are analogous to99Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionthe orresponding protools in the xed rate ase. Thus, for Conventional Relaying1, the soure transmits in the rst ξN time slots with xed rate S0 and the relaytransmits in the remaining (1− ξ)N time slots with rate R(i) = log2(1+ r(i)). Thus,the throughput is given byτmixedconv,1= limN→∞1N min{ξN∑i=1S0OS(i),N∑i=ξN+1log2(1 + r(i))}= min {ξ(1−PS)S0, (1− ξ)E{log2(1 + r(i))}}. (4.18)The throughput is maximized if ξ satisesξS0(1− PS) = (1− ξ)E{log2(1 + r(i))} . (4.19)From (4.19), we obtain ξ asξ = E{log2(1 + r(i))}S0(1− PS) + E{log2(1 + r(i))}. (4.20)Inserting ξ into (4.18) leads to the throughput of mixed rate transmission under theConventional Relaying 1 protoolτmixedconv,1 =S0(1− PS)E{log2(1 + r(i))}S0(1− PS) + E{log2(1 + r(i))}. (4.21)Assuming Rayleigh fading links E{log2(1 + r(i))} is obtained asE{log2(1 + r(i))} =e1/ΩRln(2)E1( 1ΩR)(4.22)100Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionfor xed transmit powers. If adaptive power alloation is employed, E{log2(1+r(i))}beomesE{log2(1 + r(i))} =1ln(2)E1( λcΩ¯R), (4.23)where λc is found from the power onstraint(1− PS)γS +∫ ∞λc( 1λc− 1hR)fhR(hR)dhR = 2Γ. (4.24)Here, Γ denotes the average transmit power. In the high SNR regime, where γS =γR = γ →∞, E{log2(1 + r(i))} ≫ S0(1−PS) holds. Thus, the throughput in (4.21)onverges toτmixedconv,1 → S0 , as γ →∞ , (4.25)whih leads to the interesting onlusion that mixed rate transmission ahieves amultiplexing rate of one even if suboptimal onventional relaying is used.For Conventional Relaying 2, the performane of mixed rate transmission is iden-tial to that of xed rate transmission. Sine the relay does not employ a buerfor Conventional Relaying 2, even with mixed rate transmission, the relay an onlytransmit suessfully all of the reeived information if S0 ≤ log2(1 + r(i)) and has toremain silent otherwise.101Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission4.4 Optimal Buer-Aided Relaying for Fixed RateTransmission Without Delay ConstraintsIn this setion, we investigate buer-aided relaying with adaptive reeption-transmis-sion for xed rate transmission without delay onstraints, i.e., the transmission ratesof the soure and the relay are xed. We derive the optimal adaptive reeption-transmission protool and analyze the orresponding throughput and outage prob-ability. The obtained results onstitute performane upper bounds for xed ratetransmission with delay onstraints, whih will be onsidered in Setion 4.5.4.4.1 Problem FormulationIn order to model the reeption and transmission of the relay, again we introduethe binary adaptive reeption-transmission variable di ∈ {0, 1}. Here, again di = 1indiates that the R-D link is seleted for transmission in time slot i, i.e., the relaytransmits and the destination reeives. Similarly, if di = 0, the S-R link is seletedfor transmission in time slot i, i.e., the soure transmits and the relay reeives.Based on the denitions of OS(i), OR(i), and di, the number of bits/symb sentfrom the soure to the relay and from the relay to the destination in time slot i anbe written in ompat form asSSR(i) = (1− di)OS(i)S0 (4.26)andRRD(i) = diOR(i)min{R0, Q(i− 1)}, (4.27)102Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionrespetively. Consequently, the throughput in (4.9) an be rewritten asτ = limN→∞1NN∑i=1diOR(i)min{R0, Q(i− 1)}. (4.28)In the following, we maximize the throughput by optimizing the adaptive reeption-transmission variable di, whih represents the only degree of freedom in the onsideredproblem. In partiular, as already mentioned in Setion 4.3, sine both transmittingnodes do not have the full CSI of their respetive transmit hannels, power alloationis not possible and we assume xed transmit powers PS(i) = PS and PR(i) = PR, ∀i.4.4.2 Throughput MaximizationLet us rst dene the average arrival rate of bits/symb per slot into the queue of thebuer, A, and the average departure rate of bits/symb per slot out of the queue ofthe buer, D, as [83℄A , limN→∞1NN∑i=1(1− di)OS(i)S0 (4.29)andD , limN→∞1NN∑i=1diOR(i)min{R0, Q(i− 1)}, (4.30)respetively. 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 whih ase a frationof the information sent by the soure is trapped in the unlimited size buer and annever be extrated from it. The following theorem provides a useful ondition for theoptimal poliy whih maximizes the throughput.103Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionTheorem 4.1. The adaptive reeption-transmission poliy that maximizes the through-put of the onsidered buer-aided relaying system an be found in the set of adaptivereeption-transmission poliies that satisfylimN→∞1NN∑i=1(1− di)OS(i)S0 = limN→∞1NN∑i=1diOR(i)R0. (4.31)When (4.31) holds, the queue is non-absorbing but is at the edge of absorption, i.e.,a small inrease of the arrival rate will lead to an absorbing queue. Moreover, when(4.31) holds the throughput is given byτ = limN→∞1NN∑i=1(1− di)OS(i)S0 = limN→∞1NN∑i=1diOR(i)R0. (4.32)Proof. Please refer to Appendix C.1.Remark 4.3. A queue that meets ondition (4.31) is rate-stable sine there is no lossof information in the unlimited buer, i.e., the information that goes in the buereventually leaves the buer without any loss.Remark 4.4. The min(·) funtion in (4.28) is absent in the throughput in (4.31),whih is ruial for nding a tratable analytial expression for the optimal adaptivereeption-transmission poliy. In partiular, as shown in Appendix C.1, ondition(4.31) automatially ensures that for N →∞,τ = 1NN∑i=1diOR(i)min{R0, Q(i− 1)} =1NN∑i=1diOR(i)R0is valid, i.e., the impat of event R0 > Q(i−1), i = 1, . . . , N , is negligible. Hene, forthe optimal adaptive reeption-transmission poliy, the queue is non-absorbing but isalmost always lled to suh a level that the number of bits/symb in the queue exeed104Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionthe number of bits/symb that an be transmitted over the R-D hannel, i.e., the bueris pratially always fully baklogged. This result is intuitively pleasing. Namely, ifthe queue would be rate-unstable, it would absorb bits/symb and the throughput ouldbe improved by having the relay transmit more frequently. On the other hand, if thequeue was not (pratially) fully baklogged, the eet of the event R0 > Q(i − 1)would not be negligible and the system would loose out on transmission opportunitiesbeause of an insuient number of bits/symb in the buer.Aording to Theorem 4.1, in order to maximize the throughput, we have to searhfor the optimal poliy only in the set of poliies that satisfy (4.31). Therefore, thesearh for the optimal poliy an be formulated as an optimization problem, whihfor N →∞ has the following formMaximize :di1N∑Ni=1 diOR(i)R0Subject to : C1 : 1N∑Ni=1(1− di)OS(i)S0 = 1N∑Ni=1 diOR(i)R0C2 : di ∈ {0, 1}, ∀i,(4.33)where onstraint C1 ensures that the searh for the optimal poliy is onduted onlyamong those poliies that satisfy (4.31) and C2 ensures that di ∈ {0, 1}. We notethat C1 and C2 do not exlude the ase that the relay is hosen for transmission ifR0 > Q(i − 1). However, as explained in Remark 4.4, C1 ensures that the inueneof event R0 > Q(i− 1) is negligible. Therefore, an additional onstraint dealing withthis event is not required.Before we solve problem (4.33), we note that, as will be shown in the following,the optimal adaptive reeption-transmission poliy may require a oin ip. For thispurpose, let C denote the outome of a oin ip whih takes values from the set{0, 1}, and let us denote the probabilities of the outomes by PC = Pr{C = 1} and105Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionPr{C = 0} = 1−PC , respetively. Now, we are ready to provide the solution of (4.33),whih onstitutes the optimal adaptive reeption-transmission poliy maximizing thethroughput for xed rate transmission. This is onveyed in the following theorem.Theorem 4.2. For the optimal adaptive reeption-transmission poliy maximizing thethroughput of the onsidered buer-aided relaying system for xed rate transmission,three mutually exlusive ases an be distinguished depending on the values of PS andPR:Case 1:PS ≤S0S0 +R0(1− PR)AND PR ≤R0R0 + S0(1− PS).(4.34)In this ase, the optimal adaptive reeption-transmission poliy is given bydi =0 if OS(i) = 1 AND OR(i) = 01 if OS(i) = 0 AND OR(i) = 10 if OS(i) = 1 AND OR(i) = 1 AND C = 01 if OS(i) = 1 AND OR(i) = 1 AND C = 1ε if OS(i) = 0 AND OR(i) = 0(4.35)where ε an be set to 0 or 1 as neither the soure 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 deides whih node transmits and theprobability of C = 1 is given byPC =S0(1− PS)− (1− PR)PSR0(1− PS)(1− PR)(S0 +R0). (4.36)106Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionBased on (4.35), the maximum throughput is obtained asτ = S0R0S0 +R0(1− PSPR). (4.37)Case 2:PR >R0R0 + S0(1− PS)(4.38)In this ase, the optimal adaptive reeption-transmission poliy is haraterized bydi =0 if OS(i) = 1 AND OR(i) = 0 AND C = 01 if OS(i) = 1 AND OR(i) = 0 AND C = 11 if OS(i) = 0 AND OR(i) = 11 if OS(i) = 1 AND OR(i) = 1ε if OS(i) = 0 AND OR(i) = 0(4.39)The probability of outome C = 1 of the oin ip is given byPC =S0(1− PS)PR − (1− PR)R0(1− PS)PRS0, (4.40)and the maximum throughput an be obtained asτ = R0(1− PR). (4.41)Case 3:PS >S0S0 +R0(1− PR). (4.42)107Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionIn this ase, the adaptive reeption-transmission poliy that maximizes the throughputis given bydi =0 if OS(i) = 1 AND OR(i) = 00 if OS(i) = 0 AND OR(i) = 1 AND C = 01 if OS(i) = 0 AND OR(i) = 1 AND C = 10 if OS(i) = 1 AND OR(i) = 1ε if OS(i) = 0 AND OR(i) = 0(4.43)The probability of C = 1 is given byPC =S0(1− PS)R0(1− PR)PS, (4.44)and the maximum throughput isτ = S0(1− PS). (4.45)Proof. Please refer to Appendix C.2.Remark 4.5. We note that in the seond line of (4.39), we set di = 1 although theR-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 soure node ouldsuessfully transmit. However, if the soure node transmitted in this situation, thequeue at the relay would beome an absorbing queue. Similarly, in the seond lineof (4.43), we set di = 0 although the S-R link is in outage. Again, neither nodetransmits in order to ensure that ondition (4.31) is met. However, in this ase, theexat same throughput as in (4.45) an be ahieved with a simpler and more pratialadaptive reeption-transmission poliy than that in (4.43). This is addressed in the108Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionfollowing lemma.Lemma 4.1. The throughput ahieved by the adaptive reeption-transmission pol-iy in (4.43) an also be ahieved with the following simpler adaptive reeption-transmission poliy.IfPS >S0S0 +R0(1− PR), (4.46)an adaptive reeption-transmission poliy maximizing the throughput is given bydi =0 if OS(i) = 11 if OS(i) = 0, (4.47)and the maximum throughput isτ = S0(1− PS). (4.48)Proof. The poliy given by (4.43) has the same average arrival rate as poliy (4.47)sine for both poliies the soure always transmits when OS(i) = 1. Therefore, sinefor both poliies the queue is non-absorbing, by the law of onservation of ow, theirthroughputs are idential to their arrival rates. Thus, both poliies ahieve identialthroughputs.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)), whih is idential to the maximalthroughput that an be obtained in a point-to-point ommuniation between relay anddestination (soure and relay). Therefore, when PR > R0/(R0 + S0(1 − PS)) (PS >109Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionS0/(S0 + R0(1 − PR))) holds, as far as the ahievable throughput is onerned, thetwo-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 absene ofoutages τ0. The throughput in the absene of outages, τ0, an be obtained by settingOS(i) = OR(i) = 1, ∀i, whih is equivalent to setting PS = PR = 0 in Theorem4.2. Then, Case 1 in Theorem 4.2 always holds and the optimal adaptive reeption-transmission poliy isdi =0 if C = 01 if C = 1(4.49)where the probability of C = 1 is given byPC =S0S0 +R0. (4.50)Based on (4.49), the maximum throughput in the absene of outages isτ0 =S0R0S0 +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 relayingsheme with adaptive reeption-transmission.Lemma 4.2. The outage probability of the system onsidered in Theorem 4.2 is given110Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionbyFout =PR − (1− PR)R0/S0 , if PR > R0R0+S0(1−PS)PS − (1− PS)S0/R0 , if PS > S0S0+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 anoutage event happens when neither the soure nor the relay transmit in a time slot,i.e., the number of silent slots is idential to the number of outage events.In the high SNR regime, when the outage probabilities of both involved linksare small, the expressions for the throughput and the outage probability an besimplied to obtain further insight into the performane of buer-aided relaying.This is addressed in the following lemma.Lemma 4.3. In the high SNR regime, γS = γR = γ → ∞, the throughput andthe outage probability of the buer-aided relaying system onsidered in Theorem 4.2onverge toτ → τ0 =S0R0S0 +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 andPR → 0, (4.37) simplies to (4.53).111Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission4.4.3 Performane in Rayleigh FadingFor onreteness, we assume in this subsetion that both links of the onsidered two-hop HD relay network are Rayleigh fading. We examine the diversity order and thediversity-multiplexing trade-o.Lemma 4.4. For the speial ase of Rayleigh fading links, the buer-aided relayingsystem onsidered in Theorem 4.2 ahieves a diversity gain of two, i.e., in the highSNR regime, when γS = γR = γ → ∞, the outage probability, Fout, deays on alog-log sale with slope −2 as a funtion of the transmit SNR γ, and is given byFout →2S0 − 1Ω¯S2R0 − 1Ω¯R1γ2 , as γ →∞. (4.55)Furthermore, the onsidered buer-aided relaying system ahieves a diversity-multiplexingtrade-o, DM(r), ofDM(r) = 2(1− 2r), 0 < r < 1/2. (4.56)Proof. Please refer to Appendix C.4.Remark 4.8. We reall that, for xed rate transmission, both onsidered onventionalrelaying shemes without adaptive reeption-transmission ahieved only a diversitygain of one, f. (4.16), (4.17), despite the fat that Conventional Relaying 1 alsohas an unlimited buer and entails an innite delay. Thus, we expet large gains interms of outage probability of the proposed buer-aided relaying protool with adaptivereeption-transmission ompared to onventional relaying.The performane of the onsidered system an be further improved by optimizingthe transmission rates R0 and S0 based on the hannel statistis. For Rayleigh fading112Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionwith given Ω¯S and Ω¯R, we an optimize R0 and S0 for minimization of the outageprobability. This is addressed in the following lemma for high SNR.Lemma 4.5. Assuming Rayleigh fading, the optimal transmission rates S0 and R0that minimize the outage probability in the high SNR regime, while maintaining athroughput of τ0, are given by R0 = S0 = 2τ0.Proof. The throughput in the high SNR regime is given by (4.53), whih an berewritten as R0 = S0τ0/(S0 − τ0). Inserting this into the asymptoti expression forFout in (4.55) and minimizing it with respet to S0 yields S0 = R0 = 2τ0.Remark 4.9. For Rayleigh fading, although in the low SNR regime, the optimal S0and R0 an be nonidential, in the high SNR regime, independent of the values of Ω¯Sand Ω¯R, the minimum Fout is obtained for idential transmission rates for both links.Furthermore, in the high SNR regime, when γS = γR → ∞, for S0 = R0, the oinip probability PC onverges to PC = Pr{C = 1} = Pr{C = 0} → 1/2.4.5 Buer-Aided Relaying for Fixed RateTransmission With Delay ConstraintsThe protool proposed in Setion 4.4 does not impose any onstraint on the delaythat a transmitted bit of information experienes. However, in pratie, most om-muniation servies require delay onstraints. Therefore, in this setion, we modifythe buer-aided relaying protool derived in the previous setion to aount for on-straints on the average delay. Furthermore, we analyze the eet of the applied mod-iation on the throughput and the outage probability. For simpliity, throughoutthis setion, we assume S0 = R0. We note that the adaptive reeption-transmission113Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionprotools proposed in Setion 4.5.2 are also appliable to the ase of S0 6= R0. How-ever, sine for S0 6= R0 the odewords transmitted by the soure do not ontain thesame number of bits/symb as the odewords transmitted by the relay, the Markovhain based throughput and delay analyses in Setions 4.5.3 and 4.5.4 would be moreompliated. On the other hand, sine we found in the previous setion that, for highSNR, idential soure and relay transmission rates minimize the outage probability,we avoid these additional ompliations here and onentrate on the ase S0 = R0.Furthermore, to failitate our analysis, throughout this setion, we assume temporallyunorrelated fading.4.5.1 Average DelayWe dene the delay of a bit as the time interval from its transmission by the soure toits reeption at the destination. Thus, assuming that the propagation delays in theS-R and R-D links are negligible, the delay of a bit is idential to the time that thebit is held in the buer. As a onsequene, we an use Little's law [85℄ and expressthe average delay in number of time slots asE{T} = E{Q}A , (4.57)where E{Q} = limN→∞∑Ni=1Q(i)/N is the average length of the queue in the buerof 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.114Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission4.5.2 Adaptive Reeption-Transmission Protool for DelayLimited TransmissionAs mentioned before, we modify the optimal adaptive reeption-transmission protoolderived in Setion 4.4 in order to limit the average delay. However, depending onthe targeted average delay, somewhat dierent modiations are neessary, sine itis not possible to ahieve any desired delay with one protool. Hene, three dierentadaptive reeption-transmission protools are introdued in the following proposition.Proposition 4.1. For xed rate transmission with delay onstraint, depending onthe targeted average delay E{T} and the outage probabilities PS and PR, we proposethe following adaptive reeption-transmission poliies:Case 1: If PR < 1/(2− PS) and the required delay E{T} satisesE{T} > 11− PR (2− PS)+ 2 (1− PS)1− PSPR (2− PS), (4.58)we propose the following adaptive reeption-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 E{T} satises11− PR (2− PS)< E{T} ≤ 11− PR (2− PS)+ 2 (1− PS)1− PSPR (2− PS), (4.60)115Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionwe propose the following adaptive reeption-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 E{T} satises11− PR< E{T} ≤ 11− PR (2− PS), (4.62)we propose the following adaptive reeption-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 eah of the proposed adaptive reeption-transmission variables di, the requireddelay an be met by adjusting the value of PC = Pr{C = 1}, where the minimum andmaximum delays are ahieved with PC = 1 and PC = 0, respetively.Remark 4.10. The delay limits given by (4.58), (4.60), and (4.62) arise fromthe analysis of the proposed protools with adaptive reeption-transmission variables(4.59), (4.61), and (4.63), respetively. We will investigate these delay limits inLemma 4.7 in Setion 4.5.3 and the orresponding proof is provided in Appendix C.7.Remark 4.11. We have not proposed a buer-aided relaying protool with adaptivereeption-transmission that an satisfy a required delay smaller than 1/(1−PR). Forsuh small delays, Conventional Relaying 2 without adaptive reeption-transmissionan be used. However, if retransmission of the outage odewords is taken into aount,then not even for Conventional Relaying 2 an ahieve a delay smaller than 1/(1 −116Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionPR).4.5.3 Throughput and DelayIn the following, we analyze the throughput, the average delay, and the probabilityof having k pakets in the queue for the modied adaptive reeption-transmissionprotools proposed in Proposition 4.1 in the previous subsetion. The results aresummarized in the following theorem.Theorem 4.3. Consider a buer-aided relaying system operating in temporally un-orrelated blok fading. Let soure and relay transmit with rate R0, respetively, andlet the buer size at the relay be limited to L pakets eah omprised of R0 bits/symb.Assume that the relay drops newly reeived pakets if the buer is full. Then, depend-ing on the adopted adaptive reeption-transmission protool, the following ases anbe distinguished:Case 1: If the adaptive reeption-transmission variable di is given by (4.59), theprobability of the buer having k pakets in its queue, Pr{Q = kR0}, is obtained asPr{Q = 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 = 0pL−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 = 1pL−k(2p+q−1)(1−PS )2(1−p−q)k−2pL−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 byp = (1− PS)(1− PR)PC + PS(1− PR) ; q = PSPR. (4.65)117Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionFurthermore, the average queue length, E{Q}, the average delay, E{T}, and through-put, τ , are given byE{Q} = R01− PS2p+ q − 1pL−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)E{T} = 12p+ q − 1pL−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 reeption-transmission variable di is given by either (4.61) or(4.63), the probability of the buer having k pakets in its queue, Pr{Q = kR0}, isgiven byPr{Q = 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−1pL(2p+q−PS)−(1−PS)(1−p−q)L , k = 1, ..., L(4.69)where, if adaptive reeption-transmission variable di is given by (4.61), p and q are118Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissiongiven by (4.65), while if adaptive reeption-transmission variable di is given by (4.63),p and q are given byp = 1− PR and q = PSPR + (1− PS)PRPC . (4.70)Furthermore, the average queue length, E{Q}, the average delay, E{T}, and through-put, τ , are given byE{Q} = R01− PS2p+ q − 1pL+1 − (1− p− q)L(L(2p+ q − 1) + p)pL(2p+ q − PS)− (1− PS)(1− p− q)L, (4.71)E{T} = 12p+ q − 11ppL+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)LpL(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 muh insightinto the performane of the onsidered system. To overome this problem, we onsiderthe ase L≫ 1, whih leads to signiant simpliations and design insight. This isaddressed in the following lemma.Lemma 4.6. For the system onsidered in Theorem 4.3, assume that L→∞. In thisase, for a system with adaptive reeption-transmission variable di given by (4.59),(4.61), or (4.63) to be able to ahieve a xed delay, E{T}, that does not grow withL as L → ∞, the ondition 2p + q − 1 > 0 must hold. If 2p + q − 1 > 0 holds, thefollowing simpliations an be made for eah of the onsidered adaptive reeption-transmission variables:119Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionCase 1: If the adaptive reeption-transmission variable di is given by (4.59), theprobability of the buer being empty, the average delay, E{T}, and throughput, τ ,simplify toPr{Q = 0} = PS2PC(1− PR)(1− PS) + (2− PR)PS − 12PC(1− PS)(1− PSPR) + P 2S(1− PR)(4.74)E{T} = 12PC(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 + PC2PC(1− PS)(1− PSPR) + (1− PR)P 2S. (4.76)Case 2: If the adaptive reeption-transmission variable di is given by (4.61), theprobability of the buer being empty, the average delay, E{T}, and throughput, τ ,simplify toPr{Q=0}= 2PC(1− PR)(1− PS) + PS(2− PR)− 1(1− PR)(PS + 2PC(1− PS))(4.77)E{T} = 12PC(1− PR)(1− PS)− PRPS + 2PS − 1(4.78)τ = R0(1− PS)PC(1− PS) + PS2PC(1− PS) + PS. (4.79)Case 3: If the adaptive reeption-transmission variable di is given by (4.63), theprobability of the buer being empty, the average delay, E{T}, and the throughput,120Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionτ , simplify toPr{Q = 0} = 1− PR(2− PS − PC(1− PS))2− PS − PR(2− PS − PC(1− PS))(4.80)E{T} = 11− PR(2− PS − PC(1− PS))(4.81)τ = R01 + PSPR − PR − PS2− PS − PR(2− PS − PC(1− PS)). (4.82)For eah of the onsidered ases, the probability PC an be used to adjust the desiredaverage delay E{T} 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 ahieve any desiredaverage delay with the proposed buer-aided adaptive reeption-transmission proto-ols. The limits of the ahievable average delay for eah of the proposed adaptivereeption-transmission variables di in Proposition 4.1 are provided in the followinglemma.Lemma 4.7. Depending on the adopted adaptive reeption-transmission variable dithe following ases an be distinguished for the average delay:Case 1: If the adaptive reeption-transmission variable di is given by (4.59), thenif PR < 1/(2− PS) and PS < 1/(2− PR), the system an ahieve any average delayE{T} ≥ Tmin,1, where Tmin,1 is given byTmin,1 =11− 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 ahieve121Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionany average delay in the interval Tmin,1 ≤ E{T} ≤ Tmax,1, where Tmax,1 is given byTmax,1 =1PS(2− PR)− 1. (4.84)Case 2: If the adaptive reeption-transmission variable di is given by (4.61), thenif PR < 1/(2− PS) and PS < 1/(2− PR), the system an ahieve any average delayE{T} ≥ Tmin,2, where Tmin,2 is given byTmin,2 =11− PR(2− PS). (4.85)However, if PR < 1/(2 − PS) and PS > 1/(2 − PR), the system an ahieve anyaverage delay Tmin,2 ≤ E{T} ≤ Tmax,2, where Tmax,2 = Tmax,1.Case 3: If the adaptive reeption-transmission variable di is given by (4.63), thenif PR > 1/(2− PS), the system an ahieve any average delay E{T} ≥ Tmin,3, whereTmin,3 is given byTmin,3 =11− PR. (4.86)On the other hand, if PR < 1/(2 − PS), the system an ahieve any average delayTmin,3 ≤ E{T} ≤ 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-aidedrelaying protool for delay onstrained xed rate transmission.4.5.4 Outage ProbabilityThe following theorem speies the outage probability.122Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionTheorem 4.4. For the onsidered buer-aided relaying protool in Proposition 4.1, ifthe required delay an be satised by using the adaptive reeption-transmission variabledi in either (4.59) or (4.61), the outage probability is given byFout = PSPr{Q = 0}+ PSPR(1− Pr{Q = 0} − Pr{Q = LR0})+((1−PS)PR + (1−PSPR)(1−PC))Pr{Q = LR0},(4.87)where if di is given by (4.59), Pr{Q = 0} and Pr{Q = LR0} are given by (4.64) withp and q given by (4.65). On the other hand, if di is given by (4.61), Pr{Q = 0} andPr{Q = LR0} are given by (4.69) with p and q given by (4.65).If the required delay is satised by using the adaptive reeption-transmission vari-able di given by (4.63), then the outage probability is given byFout = PSPr{Q = 0}+ PSPR(1− Pr{Q = 0} − Pr{Q = LR0})+ (1− PS)PR(1− PC)Pr{Q = LR0}, (4.88)where Pr{Q = 0} and Pr{Q = 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-niant simpliations are possible if L ≫ 1. This is addressed in the followinglemma.Lemma 4.8. When L → ∞, the outage probability given by (4.87) and (4.88) sim-123Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionplies toFout = PSPr{Q = 0}+ PSPR(1− Pr{Q = 0}), (4.89)where Pr{Q = 0} is given by (4.74), (4.77), and (4.80) if di is given by (4.59), (4.61),and (4.63), respetively.Proof. Eq. (4.89) is obtained by letting Pr{Q = LR0} → 0 when L → ∞ in (4.87)and (4.88).The expression for the outage probability in (4.89) an be further simplied inthe high SNR regime, whih provides insight into the ahievable diversity gain. Thisis summarized in the following theorem.Theorem 4.5. In the high SNR regime, when γS = γR = γ → ∞, depending on therequired delay that the system has to satisfy, two ases an be distinguished:Case 1: If 1 < E{T} ≤ 3, the outage probability asymptotially onverges toFout →PSE{T}+ 1 , as γ →∞. (4.90)Case 2: If E{T} > 3, the outage probability asymptotially onverges toFout →P 2SE{T} − 1 + PSPR, as γ →∞. (4.91)Therefore, assuming Rayleigh fading, the onsidered system ahieves a diversity gainof two if and only if E{T} > 3.Proof. Please refer to Appendix C.9.Aording to Theorem 4.5, for Rayleigh fading, a diversity gain of two an be124Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionalso ahieved for delay onstrained transmission, whih underlines the appeal ofbuer-aided relaying with adaptive reeption-transmission ompared to onventionalrelaying, whih only ahieves a diversity gain of one even in ase of innite delay(Conventional Relaying 1).4.6 Mixed Rate TransmissionIn this setion, we investigate buer-aided relaying protools with adaptive reeption-transmission for mixed rate transmission. In partiular, we assume that the souredoes not have CSIT and transmits with xed rate S0 but the relay has full CSITand transmits with the maximum possible rate, RRD(i) = log2(1 + r(i)), that doesnot ause an outage in the R-D hannel. For this senario, we onsider rst delayunonstrained transmission and derive the optimal adaptive reeption-transmissionbuer-aided relaying protools with and without power alloation. Subsequently, weinvestigate the impat of delay onstraints.Before we proeed, we note that for mixed rate transmission the throughput anbe expressed asτ = limN→∞1NN∑i=1di min{log2(1 + r(i)), Q(i− 1)}, (4.92)where we used (4.4) and (4.9). For the derivation of the maximum throughput ofbuer-aided relaying with adaptive reeption-transmission the following theorem isuseful.Theorem 4.6. The adaptive reeption-transmission poliy that maximizes the through-put of the onsidered buer-aided relaying system for mixed rate transmission an be125Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionfound in the set of adaptive reeption-transmission poliies that satisfylimN→∞1NN∑i=1(1− di)OS(i)S0 = limN→∞1NN∑i=1di log2(1 + r(i)) . (4.93)Furthermore, for adaptive reeption-transmission poliies 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 replaing OR(i)R0 by log2(1 + r(i))in the proof of Theorem 4.1 given in Appendix C.1.Hene, similar to xed rate transmission, for the set of poliies onsidered in The-orem 4.6, for N → ∞, the buer at the relay is pratially always fully baklogged.Thus, the min(·) funtion in (4.92) an be omitted and the throughput is given bythe right hand side of (4.93).4.6.1 Optimal Adaptive Reeption-Transmission ProtoolWithout Power AlloationSine 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 relaytransmits with xed power. We note that power alloation is not always desirableas it requires highly linear power ampliers and thus, inreases the implementationomplexity of the relay.Aording to Theorem 4.6, the optimal adaptive reeption-transmission poliymaximizing the throughput an be found in the set of poliies that satisfy (4.93).Therefore, the optimal poliy an be obtained from the following optimization prob-126Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionlemMaximize :di1N∑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 searh for the optimal poliy isonduted only among the poliies 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-spetively. Then, for the onsidered buer-aided relaying system in whih the souretransmits with a xed rate S0 and xed power PS , and the relay transmits with anadaptive rate RRD(i) = log2(1+r(i)) and xed power PR, two ases have to be distin-guished for the optimal adaptive reeption-transmission variable di, whih maximizesthe throughput:Case 1: IfPS ≤S0S0 +∫∞0 log2(1 + r)fr(r)dr(4.95)holds, thendi =1 if OS(i) = 01 if OS(i) = 1 AND r(i) ≥ 2ρS0 − 10 if OS(i) = 1 AND r(i) < 2ρS0 − 1 ,(4.96)127Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionwhere ρ is a onstant whih an be found as the solution ofS0(1− PS)∫ 2ρS0−10fr(r)dr = PS∫ ∞0log2(1 + r)fr(r)dr + (1− PS)∫ ∞2ρS0−1log2(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, thendi =0 if OS(i) = 11 if OS(i) = 0 .(4.98)In this ase, the maximum throughput is given byτ = 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 notin outage, f. (4.96), (4.98). On the other hand, the R-D link is never in outage sinethe transmission rate is adjusted to the hannel onditions. Furthermore, buer-aided relaying with adaptive reeption-transmission has a larger throughput thanConventional Relaying 1, and also ahieves a multiplexing gain of one.To get more insight, we speialize the results derived thus far in this setion toRayleigh fading links.Lemma 4.9. For Rayleigh fading links, ondition (4.95) simplies toPS = 1− exp(−2S0 − 1ΩS)≤ S0S0 + e1/ΩRE1(1/ΩR)/ ln(2). (4.100)128Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionFurthermore, (4.97) simplies toS0 exp(−2S0 − 1ΩS)[1− exp(−2ρS0 − 1ΩR)]= e1/ΩRln(2)[(1− exp(−2S0 − 1ΩS))E1( 1ΩR)+ exp(−2S0 − 1ΩS)E1(2ρS0ΩR)]+exp(−2ρS0 − 1ΩR)exp(−2S0 − 1ΩS)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) toτ = S0 exp(−2S0 − 1ΩS). (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), respetively.4.6.2 Optimal Adaptive Reeption-Transmission Poliy WithPower AlloationAs mentioned before, sine for mixed rate transmission the relay is assumed to havethe full CSI of both links, power alloation an be applied to further improve perfor-mane. In other words, the relay an adjust its transmit power PR(i) to the hannelonditions while the soure still transmits with xed power PS(i) = PS, ∀i. In thefollowing, for onveniene, we will use the transmit SNRs without fading, γS andγR(i), whih may be viewed as normalized powers, as variables instead of the atualpowers PS = γSσ2nR and PR(i) = γR(i)σ2nD .For the power alloation ase, Theorem 4.6 is still appliable but it is onvenient129Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionto rewrite the throughput asτ = limN→∞1NN∑i=1di log2(1 + γR(i)hR(i)). (4.103)We note that (4.93) also applies to the ase of power alloation. Furthermore, inorder to meet the average power onstraint Γ, the instantaneous (normalized) powerγR(i) and the xed (normalized) power γS have to satisfy the following ondition:limN→∞1NN∑i=1(1− di)OS(i)γS + limN→∞1NN∑i=1diγR(i) ≤ Γ. (4.104)Thus, the optimal adaptive reeption-transmission poliy for mixed rate transmissionis 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} , ∀iC3 : 1N∑Ni=1(1− di)OS(i)γS + 1N∑Ni=1 diγR(i) ≤ Γ,(4.105)where N →∞, onstraints C1 and C3 ensure that the searh for the optimal poliyis onduted only among those poliies that jointly satisfy (4.93) and the soure-relaypower onstraint (4.104), respetively, and C2 ensures that di ∈ {0, 1}. The solutionof (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),respetively. Then, for the onsidered buer-aided relaying system where the soure130Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissiontransmits with a xed rate S0 and xed power γS and the relay transmits with adaptiverate RRD(i) = log2(1 + r(i)) = log2(1 + γR(i)hR(i)) and adaptive power γR(i), twoases have to be onsidered for the optimal adaptive reeption-transmission variabledi whih maximizes the throughput:Case 1: IfPS ≤S0S0 +∫∞λt log2(hR/λt)fhR(hR)dhR, (4.106)holds, where λt is found as the solution toPS∫ ∞λt( 1λt− 1hR)fhR(hR)dhR + γS(1− PS) = Γ, (4.107)then the optimal power γR(i) and adaptive reeption-transmission variable di whihmaximize the throughput are given byγR(i) = max{0, 1λ −1hR(i)}, (4.108)anddi =1 if OS(i) = 0 AND hR(i) ≥ λ1 if OS(i) = 1 AND hR(i) ≥ λ AND ln(hR(i)λ)+ λhR(i) ≥ ρS0 − λγS + 10 if OS(i) = 1 AND hR(i) < λ0 if OS(i) = 1 AND hR(i) ≥ λ AND ln(hR(i)λ)+ λhR(i) < ρS0 − λγS + 1ε if OS(i) = 0 AND hR(i) < λ ,(4.109)131Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionwhere ε is either 0 or 1 and has not impat on the throughput. Constants ρ and λ arehosen suh that onstraints C1 and C3 in (4.105) are satised with equality. Thesetwo onstants an be found as the solution to the following system of equationsS0(1−PS)∫ G0fhR(hR)dhR = PS∫ ∞λlog2(hRλ)fhR(hR)dhR+(1− PS)∫ ∞Glog2(hRλ)fhR(hR)dhR, (4.110)PS∫ ∞λ(1λ −1hR)fhR(hR)dhR + (1− PS)∫ ∞G(1λ −1hR)fhR(hR)dhR+γS(1− PS)∫ G0fhR(hR)dhR = Γ, (4.111)where the integral limit G is given byG = − λW{−eλγS−ρS0−1} . (4.112)Here, W{·} denotes the Lambert W -funtion dened in [84℄, whih is available asbuilt-in funtion in software pakages suh as Mathematia. 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 reeption-transmission variable di are given byγR(i) = max{0, 1λ −1hR(i)}, if OS(i) = 0; (4.113)di =0 if OS(i) = 11 if OS(i) = 0,(4.114)132Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionwhere λ = λt is the solution to (4.107). In this ase, the maximum throughput isgiven byτ = 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 alloation is idential, 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 alloation is notbeneial sine the S-R hannel is the bottlenek link, whih annot be improved bypower alloation at the relay. Furthermore, the throughput in (4.99) and (4.115) isidential to the throughput of a point-to-point ommuniation between the soure andthe relay sine the number of time slots required to transmit the information fromthe relay to the destination beomes negligible. Therefore, in this ase, as far as theahievable throughput is onerned, the two-hop HD relay hannel is transformed intoa point-to-point hannel between the soure and the relay.In the following lemma, we onentrate on Rayleigh fading for illustration purpose.Lemma 4.10. For Rayleigh fading hannels, PS is given byPS = 1− exp(−2S0 − 1γSΩ¯S).Furthermore, ondition (4.106) simplies toPS ≤S0S0 + E1(λt/Ω¯R)/ ln(2), (4.116)133Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionwhere λt is found as the solution toPS[e−λt/Ω¯Rλt− 1Ω¯RE1( λtΩ¯R)]= Γ. (4.117)For the ase where (4.116) holds, (4.110) and (4.111) simplify toS0(1− PS)(1− e−G/Ω¯R)= 1ln(2)[PSE1( λΩ¯R)+(1− PS)(E1( GΩ¯R)+ ln(Gλ)e−G/Ω¯R)](4.118)andPS[e−λ/Ω¯Rλ −1Ω¯RE1( λΩ¯R)]+ (1− PS)[e−G/Ω¯Rλ− 1Ω¯RE1( GΩ¯R)]+ γS(1− PS)(1− e−G/Ω¯R)= Γ, (4.119)respetively, where integral limit G is given by (4.112). The maximum throughput isgiven 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 thePDFs of hS(i) and hR(i) into (4.106), (4.107), (4.110), and (4.111), respetively.Remark 4.13. Conditions (4.95) and (4.106) depend only on the long term fadingstatistis and not on the instantaneous fading states. Therefore, for xed Ω¯S and Ω¯R,the optimal poliy for ondition (4.95) is given by either (4.96) or (4.98), but not byboth. Similarly, the optimal poliy for ondition (4.106) is given by either (4.109) or(4.114), but not by both.134Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate TransmissionRemark 4.14. We note that equations (4.107), (4.110), (4.111), (4.117), (4.118),and (4.119) an be solved using software pakages suh as Mathematia.4.6.3 Mixed Rate Transmission With Delay ConstraintsNow, we turn our attention to mixed rate transmission with delay onstraints. Forthe delay unonstrained ase, Theorem 4.6 was very useful to arrive at the optimalprotool sine it removed the omplexity of having to deal with the queue states.However, for the delay onstrained ase, the queue states determine the throughputand the average delay. Moreover, for mixed rate transmission, the queue statesan only be modeled by a Markov hain with ontinuous state spae, whih makesthe analysis ompliated. Therefore, we resort to a suboptimal adaptive reeption-transmission protool in the following.Proposition 4.2. Let the buer size be limited to Qmax bits. For this ase, we proposethe following adaptive reeption-transmission protool for mixed rate transmissionwith delay onstraints:1. If OS(i) = 0, set di = 1.2. Otherwise, if log2(1 + r(i)) ≤ Q(i − 1) ≤ Qmax − S0, selet di as proposed inTheorem 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 enoughroom in the buer to aommodate the bits/symb possibly sent from the soure tothe relay and there are enough bits/symb in the buer for the relay to transmit, the135Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionadaptive reeption-transmission protool introdued in Theorem 4.7 is employed. Onthe other hand, if there exists the possibility of a buer overow, the relay transmitsto redue the amount of data in the buer. If the number of bits/symb in the bueris too low, the soure transmits. The value of Qmax an be used to adjust the averagedelay while maintaining a low throughput loss ompared to the throughput withoutdelay onstraint.Although oneptually simple, as pointed out before, a theoretial analysis ofthe throughput of the proposed queue size limiting protool is diult beause ofthe ontinuous state spae of the assoiated Markov hain. Thus, we will resort tosimulations to evaluate its performane in Setion 4.7.4.6.4 Conventional Relaying With Delay ConstraintsTo have a benhmark for delay onstrained buer-aided relaying with adaptive reeption-transmission, we propose a orresponding onventional relaying protool, whih maybe viewed as a delay onstrained version of Conventional Relaying 1.Proposition 4.3. The soure transmits to the relay in k onseutive 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 soure transmits again in k onseutive time slots,and so on. The values of k and p an be hosen to satisfy any delay and throughputrequirements.For this protool, the queue is non-absorbing ifk(1− PS)S0 ≤ pE{log2(1 + r(i))}. (4.120)Assuming (4.120) holds, the average arrival rate is equal to the throughput and hene136Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionthe throughput is given byτ = kk + p(1− PS)S0 , (4.121)Using a numerial example, we will show in Setion 4.7 (f. Fig. 4.6) that theprotool with adaptive reeption-transmission in Proposition 4.2 ahieves a higherthroughput than the onventional protool in Proposition 4.3. However, the onven-tional protool is more amendable to analysis and it is interesting to investigate theorresponding throughput and multiplexing gain for a given average delay in the highSNR regime, γS = γR = γ →∞. This is done in the following theorem.Theorem 4.9. For a given average delay onstraint, E{T}, the maximal throughputτ and multiplexing rate r of mixed rate transmission, for γS = γR = γ → ∞, aregiven byτ → S0(1− 12E{T}), as γ →∞ . (4.122)r → 1− 12E{T} , as γ →∞ . (4.123)Proof. Please refer to Appendix C.12.Remark 4.15. Theorem 4.9 reveals that, as expeted from the disussion of the asewithout delay onstraints, delay onstrained mixed rate transmission approahes amultiplexing gain of one as the allowed average delay inreases.4.7 Numerial and Simulation ResultsIn this setion, we evaluate the performane of the proposed xed rate and mixedrate transmission shemes for Rayleigh fading. We also onrm some of our analytial137Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission−5 0 5 10 15 20 25 3011.21.41.61.82γτ/τfixedconv,1  TheorySimulationΩ¯S = 10Ω¯S = 1Ω¯S = 0.1Figure 4.1: Ratio of the throughputs of buer-aided relaying and ConventionalRelaying 1, τ/τfixedconv,1, vs. γ. Fixed rate transmission without delay onstraints.γS = γR = γ, S0 = R0 = 2 bits/symb, and Ω¯R = 1.results with omputer simulations. We note that our analytial results are valid forN →∞. For the simulations, N has to be nite, of ourse, and we adopted N = 107in all simulations.4.7.1 Fixed Rate TransmissionFor xed rate transmission, we evaluate the proposed adaptive reeption-transmissionprotools for transmission with and without delay onstraints. Throughout this se-tion we assume that soure and relay transmit with idential rates, i.e., S0 = R0.Transmission Without Delay ConstraintsIn Fig. 4.1, we show the ratio of the throughputs ahieved with the proposed buer-aided relaying protool with adaptive reeption-transmission and Conventional Re-laying 1 as a funtion of the transmit SNR γS = γR = γ for Ω¯R = 1, S0 = R0 = 2bits/symb, and dierent values of Ω¯S. 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-138Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission0 10 20 30 40 5010−810−610−410−2100γ (in dB)F  out  Conventional Relaying 1BA Relaying (Theory)BA Relaying (Simulation)Ω¯S = 0.1; 1; 10Ω¯S = 10; 1; 0.1Figure 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 Ω¯R = 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 relayingprotool was obtained via Monte Carlo simulation. From Fig. 4.1 we observe thattheory and simulation are in exellent agreement. Furthermore, Fig. 4.1 shows thatexept for Ω¯S = Ω¯R the proposed adaptive reeption-transmission sheme ahievesits largest gain for medium SNRs. For very high SNRs, both links are never in outageand thus, Conventional Relaying 1 and the adaptive reeption-transmission shemeahieve the same performane. On the other hand, for very low SNR, there are veryfew transmission opportunities on both links as the links are in outage most of thetime. The proposed adaptive reeption-transmission protool an exploit all of theseopportunities. In ontrast, for Ω¯S = Ω¯R, Conventional Relaying 1 hoses ξ = 0.5 andwill miss half of the transmission opportunities by seleting the link that is in out-age instead of the link that is not in outage beause of the pre-determined shedulefor reeption and transmission. On the other hand, if Ω¯S and Ω¯R dier signiantly,Conventional Relaying 1 selets ξ lose to 0 or 1 (depending on whih link is stronger)139Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission0 5 10 15 2000.20.40.60.81γ (in dB)τ(inbits/symb)  Conventional Relaying 2BA Relaying (Theory)BA Relaying (Simulation)E{T} = 2.1E{T} = 3.1E{T} → ∞E{T} = 1.1Figure 4.3: Throughputs of buer-aided (BA) relaying and Conventional Relaying 2vs. γ. Fixed rate transmission with delay onstraints. γS = γR = γ, S0 = R0 = 2bits/symb, Ω¯R = 1, and Ω¯S = 1.and the loss ompared to the link adaptive sheme beomes negligible.In Fig. 4.2, we show the outage probability, Fout, for the proposed buer-aidedrelaying protool with adaptive reeption-transmission and Conventional Relaying 1.The same hannel and system parameters as for Fig. 4.1 were adopted for Fig. 4.2as well. For buer-aided relaying with adaptive reeption-transmission, Fout wasobtained from (4.52) and onrmed by Monte Carlo simulations. For onventionalrelaying, Fout was obtained from (4.13). As expeted from Lemma 4.4, buer-aidedrelaying ahieves a diversity gain of two, whereas onventional relaying ahieves onlya diversity gain of one, whih underlines the superiority of buer-aided relaying withadaptive reeption-transmission.Transmission With Delay ConstraintsIn Fig. 4.3, we show the throughput of buer-aided relaying with adaptive reeption-transmission as a funtion of the transmit SNR γS = γR = γ for xed rate trans-mission with dierent onstraints on the average delay E{T}. The theoretial urvesfor buer-aided relaying were obtained from the expressions given in Lemma 4.6 for140Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission0 10 20 30 40 5010−610−410−2100γ (in dB)F out  Conventional Relaying 2BA Relaying (theory)BA Relaying (simulation)E{T} → ∞E{T} = 1.1E{T} = 2.1E{T} = 3.1Figure 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, Ω¯R = 1, and Ω¯S = 1.throughput and the average delay. For omparison, we also show the throughputof buer-aided relaying with adaptive reeption-transmission and without delay on-straint (f. Theorem 4.2), and the throughput of Conventional Relaying 2 given by(4.14). These two shemes introdue an innite delay, i.e., E{T} → ∞ as N → ∞,and a delay of one time slot, respetively. In the low SNR regime, the proposed buer-aided relaying sheme with adaptive reeption-transmission annot satisfy all delayrequirements as expeted from Lemma 4.7. Hene, for nite delays, the throughputurves in Fig. 4.3 do not extend to low SNRs. Nevertheless, as the aordable delayinreases, the throughput for delay onstrained transmission approahes the through-put for delay unonstrained transmission for suiently high SNR. Furthermore, theperformane gain ompared to Conventional Relaying 2 is substantial even for theomparatively small average delays E{T} onsidered in Fig. 4.3.In Fig. 4.4, we show the outage probability, Fout, for the same shemes and pa-rameters that were onsidered in Fig. 4.3. For buer-aided relaying with adaptivereeption-transmission, the theoretial results shown in Fig. 4.4 were obtained from141Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission(4.87) and (4.88). These theoretial results are onrmed by the Monte Carlo simula-tion results also shown in Fig. 4.4. Furthermore, the urves for transmission withoutdelay onstraint (i.e., E{T} → ∞ as N → ∞) were omputed from (4.52), andfor Conventional Relaying 2, we used (4.15). Fig. 4.4 shows that even for an aver-age delay as small as E{T} = 1.1 slots, the proposed buer-aided relaying protoolwith adaptive reeption-transmission outperforms Conventional Relaying 2. Further-more, as expeted from Theorem 4.5, buer-aided relaying with adaptive reeption-transmission ahieves a diversity gain of two when the average delay is larger thanthree time slots (e.g., E{T} = 3.1 time slots in Fig. 4.4 ). This leads to a large per-formane gain over onventional relaying whih ahieves only a diversity gain of one.For example, gains around 10 dB and 20 dB are ahieved for an outage probability of10−2 and 10−3, respetively. Finally, note that even for E{T} = 3.1 the performaneloss in dB is very small ompared to the ase of E{T} → ∞.Remark 4.16. For the simulation results shown in Figs. 4.3 and 4.4, we adopted arelay with a buer size of L = 60 pakets whih leads to a negligible probability ofdropped pakets. For example, for γ = 45 dB, the probability of a full buer, Pr{Q =LR0}, is bounded by Pr{Q = LR0} < 10−60, and for lower SNRs, Pr{Q = LR0}is even higher. This also supports the laim in the proof of Theorem 4.5 that forlarge enough buer sizes the probability of dropping a paket due to a buer overowbeomes negligible.4.7.2 Mixed Rate TransmissionIn this setion, we investigate the ahievable throughput for mixed rate transmission.For this purpose, we onsider again the delay onstrained and the delay unonstrainedases separately.142Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission0 5 10 15 20 25 30 35 40 450.60.811.21.41.61.8Γ (in dB)τ(bits/symb)  With Power AllocationWithout Power AllocationSimulation of Buffer-aided RelayingConventional Relaying 1Buffer−aided RelayingFigure 4.5: Throughput of buer-aided relaying with adaptive reeption-transmissionand Conventional Relaying 1 vs. Γ. Mixed rate transmission without delay on-straints. Ω¯S = 10, Ω¯R = 1, and S0 = 2 bits/symb.Transmission Without Delay ConstraintsIn Fig. 4.5, we ompare the throughputs of buer-aided relaying with adaptivereeption-transmission and Conventional Relaying 1. In both ases, we onsider theases with and without power alloation. The theoretial results shown in Fig. 4.5 forthe four onsidered shemes 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 bothlinks are idential, i.e., γS = γR = Γ, S0 = 2 bits/symb, Ω¯S = 10, and Ω¯R = 1. Asan be observed from Fig. 4.5, for both buer-aided relaying with adaptive reeption-transmission and Conventional Relaying 1, power alloation is beneial only for lowto moderate SNRs. Both shemes an ahieve a throughput of S0 bits/symb in thehigh SNR regime. However, adaptive reeption-transmission ahieves a throughputgain ompared to Conventional Relaying 1 in the entire onsidered SNR range.143Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmission0 10 20 30 40 50 60 7000.20.40.60.811.21.41.61.8Γ (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 powerFigure 4.6: Throughput of buer-aided relaying with adaptive reeption-transmissionand onventional relaying vs. Γ. Mixed rate and xed rate transmission with delayonstraint. E{T} = 5 time slots, γS = γR = Γ, S0 = 2 bits/symb, and Ω¯S = Ω¯R = 1.Transmission With Delay ConstraintsIn Fig. 4.6, we ompare the throughputs of various mixed rate and xed rate trans-mission shemes for a maximum average delay of E{T} = 5 time slots and S0 = 2bits/symb. The transmit SNRs of both links are idential, i.e., γS = γR = Γ,Ω¯S = Ω¯R = 1. For mixed rate transmission, we simulated both the buer-aidedrelaying protool with adaptive reeption-transmission desribed in Proposition 4.2and the onventional relaying protool desribed in Proposition 4.3. For xed ratetransmission, we hose R0 = S0 = 2 bits/symb and inluded results for buer-aided relaying with adaptive reeption-transmission obtained based on Lemma 4.6.Furthermore, for mixed rate transmission, we also show the maximum ahievablethroughput of buer-aided relaying with adaptive reeption-transmission in the ab-sene of delay onstraints (as given by Theorem 4.7/Lemma 4.9) and the maximumthroughput ahievable for a delay onstraint of E{T} = 5 time slots and innitetransmit power (as given by (4.122)). Fig. 4.6 reveals that for mixed rate transmis-144Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionsion the protool with adaptive reeption-transmission proposed in Proposition 4.2is superior to the onventional relaying sheme proposed in Proposition 4.3, and forhigh SNR, both protools reah the upper bound for mixed rate transmission undera delay onstraint given by (4.122). Furthermore, Fig. 4.6 also shows that mixedrate transmission is superior to xed rate transmission sine the former an exploitthe additional exibility aorded by having CSIT for the R-D link. For example, forΓ = 30 dB, mixed rate transmission with adaptive reeption-transmission ahieves athroughput gain of 65% ompared to xed rate transmission, and even onventionaladaptive reeption-transmission still ahieves a gain of 45%. Fig. 4.6 also shows thateven in the presene of severe delay onstraints mixed rate transmission an signi-antly redue the throughput loss aused by HD relaying ompared to FD relaying,whose maximum throughput is S0 = 2 bits/symb.4.8 ConlusionsIn this hapter, we have onsidered a two-hop HD relay network. We have investi-gated both xed rate transmission, where soure and relay do not have full CSITand are fored to transmit with xed rate, and mixed rate transmission, wherethe soure does not have full CSIT and transmits with xed rate but the relayhas full CSIT and transmits with variable rate. For both modes of transmission,we have derived the throughput-optimal buer-aided relaying protools with adap-tive reeption-transmission and the resulting throughputs and outage probabilities.Furthermore, we have shown that buer-aided relaying with adaptive reeption-transmission leads to substantial performane gains ompared to onventional re-laying with non-adaptive reeption-transmission. In partiular, for xed rate trans-mission, buer-aided relaying with adaptive reeption-transmission ahieves a di-145Chapter 4. Buer-Aided Relaying: Fixed and Mixed Rate Transmissionversity gain of two, whereas onventional relaying is limited to a diversity gain ofone. For mixed rate transmission, both buer-aided relaying with adaptive reeption-transmission and a newly proposed onventional relaying sheme with non-adaptivereeption-transmission have been shown to overome the HD loss typial for wire-less relaying protools and to ahieve a multiplexing gain of one. Sine the proposedthroughput-optimal protools introdue an innite delay, we have also proposed mod-ied protools for delay onstrained transmission and have investigated the resultingthroughput-delay trade-o. Surprisingly, the diversity gain of xed rate transmissionwith buer-aided relaying is also observed for delay onstrained transmission as longas the average delay exeeds three time slots. Furthermore, for mixed rate trans-mission, for an average delay E{T}, a multiplexing gain of r = 1 − 1/(2E{T}) isahieved even for onventional relaying.146Chapter 5Summary of Thesis and FutureResearh TopisIn this nal hapter, in Setion 5.1, we summarize our results and highlight theontributions of this thesis. In Setion 5.2, we also propose ideas for future researh.5.1 Summary of the ResultsIn this thesis, we designed new ommuniation protools for the two-hop HD relayhannel. In the following, we briey review the main results of eah hapter.In Chapter 2, we have derived an easy-to-evaluate apaity expression of the two-hop HD relay hannel when fading is not present based on simplifying previouslyderived onverse expressions. Moreover, we have proposed an expliit oding shemewhih ahieves the apaity. We showed that the apaity is ahieved when the relayswithes between reeption and transmission in a symbol-by-symbol manner andwhen additional information is sent by the relay to the destination using the zerosymbol impliitly sent by the relay's silene during reeption. Furthermore, we haveevaluated the apaity for the ases when both links are BSCs and AWGN hannels,respetively. From the numerial examples, we have observed that the apaity ofthe two-hop HD relay hannel is signiantly higher than the rates ahieved withonventional relaying protools.147Chapter 5. Summary of Thesis and Future Researh TopisIn Chapter 3, we have devised new ommuniation protools for improving theahievable average rate of the two-hop HD relay hannel when both soure-relayand relay-destination links are AWGN hannels aeted by fading, referred to asbuer-aided relaying with adaptive reeption-transmission protools. In ontrast toonventional relaying, where the relay reeives and transmits aording to a pre-dened shedule regardless of the hannel state, in the proposed protool, the relayreeives and transmits adaptively aording to the quality of the soure-relay andrelay-destination links. For delay-unonstrained transmission, we derived the op-timal adaptive reeption-transmission shedule for the ases of xed and variablesoure and relay transmit powers. For delay-onstrained transmission, we proposeda buer-aided protool whih ontrols the delay introdued by the buer at the re-lay. This protool needs only instantaneous CSI and the desired average delay, andan be implemented in real-time. Our analytial and simulation results showed thatbuer-aided relaying with adaptive reeption-transmission with and without delayonstraints is a promising approah to signiantly inrease the ahievable averagerate ompared to onventional HD relay-assisted transmission.In Chapter 4, we have devised new ommuniation protools for improving theoutage probability of the two-hop HD relay hannel when both soure-relay and relay-destination links are AWGN hannels aeted by fading. We have investigated bothxed rate transmission, where soure and relay do not have full CSIT and are foredto transmit with xed rate, and mixed rate transmission, where the soure does nothave full CSIT and transmits with xed rate but the relay has full CSIT and transmitswith variable rate. For both modes of transmission, we have derived the throughput-optimal buer-aided relaying protools with adaptive reeption-transmission andthe resulting throughputs and outage probabilities. Furthermore, we ould show148Chapter 5. Summary of Thesis and Future Researh Topisthat buer-aided relaying with adaptive reeption-transmission leads to substantialperformane gains ompared to onventional relaying with non-adaptive reeption-transmission. In partiular, for xed rate transmission, buer-aided relaying withadaptive reeption-transmission ahieves 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 reeption-transmission and a newly pro-posed onventional relaying sheme with non-adaptive reeption-transmission havebeen shown to overome the HD loss typial for wireless relaying protools and toahieve a multiplexing gain of one. Sine the proposed throughput-optimal protoolsintrodue an innite delay, we have also proposed modied protools 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 relayingwith adaptive reeption-transmission is also observed for delay onstrained transmis-sion as long as the average delay exeeds three time slots. Furthermore, for mixed ratetransmission, for an average delay E{T}, a multiplexing gain of r = 1− 1/(2E{T})is ahieved even for onventional relaying.5.2 Future WorkFuture researh diretions may inlude the following:• Deriving the apaity of the two-hop HD relay hannel when both soure-relayand relay-destination links are aeted by fading, and designing a oding shemewhih ahieves the apaity. Intuitively, we expet this oding sheme to be amix of the oding sheme introdued in Chapter 2 and the buer-aided protoolintrodued in Chapter 3.149Chapter 5. Summary of Thesis and Future Researh Topis• Investigating the apaity of the two-hop FD relay hannel with self-interfereneand determining the amount of allowable self-interferene beyond whih therelay is better of by working in the HD mode. This requires a new information-theoretial analysis and taking into aount that the self-interferene is ausedby the transmitting node itself, and therefore the relay node has some knowledgeabout the self-interferene whih it an use to its benet in order to inreaseits data rate.• Investigating the apaity and/or devising new buer-aided protools for HDrelay networks whih are more omplex than the two-hop HD relay hannel, e.g.networks omprised of more than one soure and/or relay, and/or destination.150Bibliography[1℄ A. El Gamal and Y. Kim, Network Information Theory. Cambridge UniversityPress, 2011.[2℄ A. Nosratinia, T. Hunter, and A. Hedayat, Cooperative Communiation inWireless Networks, IEEE Communiations Magazine, vol. 42, pp. 74  80, Ot.2004.[3℄ R. 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Theory,vol. 58, pp. 53025322, Aug. 2012.158Appendix AProofs for Chapter 2A.1 Proof That the Probability of Error at theRelay Goes to Zero When (2.25) HoldsIn order to prove that the relay an deode the soure's odeword in blok b, x1|r(b),where 1 ≤ b ≤ N , from the reeived odeword y1|r(b) when (2.25) holds, i.e., that theprobability 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 thanthe apaity. To this end, note that the length of odeword x1|r(b) is k(1− P ∗U). Onthe other hand, the length of odeword y1|r(b) is idential to the number of zeros9 inrelay's transmit odeword x2(b). Sine the zeros in x2(b) are generated independentlyusing 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, thefollowing holdslimk→∞ε(b)k = 0, (A.1)limk→∞k(1− P ∗U)± ε(b)k(1− P ∗U)= 1, (A.2)9For b = 1, note that the number of zeros in x2(1) is k. Therefore, for b = 1, we only take intoan aount the rst k(1− P ∗U ) zeros. As a result, the length of y1|r(b) is also k(1− P ∗U ).159Appendix A. Proofs for Chapter 2i.e., for large k, the length of relay's reeived odeword y1|r(b) is approximately k(1−P ∗U).Now, for blok b, we dene a set R(b) whih ontains the symbol indies i in blokb for whih the symbols in x2(b) are zeros, i.e., for whih X2i = 0. Note that beforethe start of the transmission in blok b, the relay knows x2(b), thereby it knows apriori for whih symbol indies i in blok 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 deodingx1|r(b) from y1|r(b). In the rst ase |R(b)| ≥ k(1− P ∗U) holds whereas in the seondase |R(b)| < k(1 − P ∗U) holds. We rst explain the deoding proedure for the rstase.When |R(b)| = k(1 − P ∗U) + ε(b) ≥ k(1 − P ∗U) holds, the soure an transmit theentire odeword x1|r(b), whih is omprised of k(1 − P ∗U) symbols, sine there areenough zeros in odeword x2(b). On the other hand, sine for this ase the reeivedodeword y1|r(b) is omprised of k(1 − P ∗U) + ε(b) symbols, and sine for the lastε(b) symbols in y1|r(b) the soure is silent, the relay keeps from y1|r(b) only the rstk(1 − P ∗U) symbols and disards the remanning ε(b) symbols. In this way, the relaykeeps only the reeived symbols whih are the result of the transmitted symbols inx1|r(b), and disards the rest of the symbols in y1|r(b) for whih the soure is silent.Thereby, from y1|r(b), the relay generates a new reeived odeword whih we denoteby y∗1|r(b). Moreover, let R1(b) be a set whih ontains the symbol indies of thesymbols omprising odeword y∗1|r(b). Now, note that the lengths of x1|r(b) andy∗1|r(b), and the ardinality of set R1(b) are k(1 − P ∗U), respetively. Having reated160Appendix A. Proofs for Chapter 2y∗1|r(b) and R1(b), we now use a jointly typial deoder for deoding x1|r(b) fromy∗1|r(b). In partiular, we dene a jointly typial set A|R1(b)|ǫ asA|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 suessfullydeoded from reeived odeword y∗1|r(b) if and only if (x1|r(b),y∗1|r(b)) ∈ A|R1(b)|ǫ andno other odeword xˆ1|r from odebook C1|r is jointly typial with y∗1|r(b). In order toompute the probability of error, we dene the following eventsE0 = {(x1|r(b),y∗1|r(b)) /∈ A|R1(b)|ǫ } and Ej = {(xˆ(j)1|r,y∗1|r) ∈ A|R1(b)|ǫ }, (A.5)where xˆ(j)1|r is the j-th odeword in C1|r that is dierent from x1|r(b). Note thatin C1|r there are |C1|r| − 1 = 2kR − 1 odewords that are dierent from x1|r(b), i.e.,j = 1, ..., 2kR−1. Hene, an error ours if any of the events E0, E1, ..., E2kR−1 ours.Sine x1|r(b) is uniformly seleted from the odebook C1|r, the average probability oferror is given byPr(ǫ) = Pr(E0 ∪ E1 ∪ ... ∪ E2kR−1) ≤ Pr(E0) +2kR−1∑j=1Pr(Ej). (A.6)161Appendix A. Proofs for Chapter 2Sine |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, sine |R1(b)| → ∞ as k →∞, Pr(Ej) is upper bounded asPr(Ej) = Pr((xˆ(j)1|r,y∗1|r(b))∈ A|R1(b)|ǫ)=∑(xˆ(j)1|r ,y∗1|r(b))∈A|R1(b)|ǫp(xˆ(j)1|r,y∗1|r(b))(a)=∑(xˆ(j)1|r ,y∗1|r(b))∈A|R1(b)|ǫp(xˆ(j)1|r)p(y∗1|r(b))(b)≤∑(xˆ(j)1|r ,y∗1|r(b))∈A|R1(b)|ǫ2−|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 sine xˆ(j)1|r and y∗1|r(b) are independent, (b) follows sinep(xˆ(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),whih follows from [80, Eq. (3.6)℄, respetively, and (c) follows sine|A|R1(b)|ǫ | ≤ 2|R1(b)|(H(X1,Y1|X2=0)+ǫ),whih follows from [80, Theorem 7.6.1℄. Inserting (A.7) and (A.8) into (A.6), we162Appendix A. Proofs for Chapter 2obtainPr(ǫ) ≤ ǫ+2kR−1∑j=12−|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)Hene, ifR < |R1(b)|k I(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→0limk→∞Pr(ǫ) = 0. This onludes 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 soure annottransmit all of its k(1−P ∗U) symbols omprising odeword x1|r(b) sine there are notenough 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 byx∗1|r(b). Note that the length of odewords x∗1|r(b) and y1|r(b), and the ardinalityof R(b) are all idential and equal to k(1 − P ∗U) − ε(b). In addition, let the relaygenerate a odebook C∗1|r(b) by keeping only the rst k(1− P ∗U)− ε(b) symbols fromeah odeword in odebook C1|r and disarding the remaining ε(b) symbols in theorresponding odewords. Let us denote the odewords in C∗1|r(b) by x∗1|r. Note thatthere is a unique one to one mapping from the odewords in C∗1|r(b) to the odewordsin C1|r(b) sine when k →∞, k(1−P ∗U)−ε(b) →∞ also holds, i.e., the lengths of the163Appendix A. Proofs for Chapter 2odewords in C∗1|r(b) and C1|r are of the same order due to (A.2). Hene, if the relayan deode x∗1|r(b) from y1|r(b), then using this unique mapping between C∗1|r(b) andC1|r(b), the relay an deode x1|r(b) and thereby deode the message w(b) sent fromthe soure.Now, for deoding x∗1|r(b) from y1|r(b), we again use jointly typial deoding.Thereby, we dene a jointly typial set B|R|ǫ asB|R|ǫ ={(x∗1|r,y1|r) ∈ X|R|1 × Y|R|1 :∣∣∣∣∣− 1|R|∑i∈Rlog2 p(x1i|x2i = 0)−H(X1|X2 = 0)∣∣∣∣∣≤ ǫ, (A.11a)∣∣∣∣∣− 1|R|∑i∈Rlog2 p(y1i|x2i = 0)−H(Y1|X2 = 0)∣∣∣∣∣≤ ǫ, (A.11b)∣∣∣∣∣− 1|R|∑i∈Rlog2 p(x1i, y1i|x2i = 0)−H(X1, Y1|X2 = 0)∣∣∣∣∣≤ ǫ}. (A.11)Again, the transmitted odeword x∗1|r(b) is suessfully deoded from reeivedodeword y1|r(b) if and only if (x∗1|r(b),y1|r(b)) ∈ B|R|ǫ and no other odeword xˆ∗1|rfrom odebook C∗1|r is jointly typial with y1|r(b). In order to ompute the probabilityof error, we dene the following eventsE0 = {(x∗1|r(b),y1|r(b)) /∈ B|R|ǫ } and Ej = {(xˆ∗(j)1|r ,y1|r(b)) ∈ B|R|ǫ }, (A.12)where xˆ∗(j)1|r is the j-th odeword in C∗1|r that is dierent from x∗1|r(b). Note that inC∗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. Hene, an error ours if any of the events E0, E1, ..., E2kR−1ours. Now, using a similar proedure as for ase when |R(b)| ≥ k(1 − P ∗U), it an164Appendix A. Proofs for Chapter 2be proved that ifR < |R(b)|k I(X1; Y1|X2 = 0)− 3(1− P∗U)ǫ= (1− P ∗U)I(X1; Y1|X2 = 0)−ε(b)k I(X1; Y1|X2 = 0)− 3(1− P∗U)ǫ, (A.13)then limǫ→0limk→∞Pr(ǫ) = 0. In (A.13), note thatlimk→∞ε(b)k I(X1; Y1|X2 = 0) = 0 (A.14)holds due to (A.1). This onludes the proof for the ase when |R(b)| < k(1− P ∗U).A.2 Proof That the Probability of Error at theDestination Goes to Zero When (2.26) HoldsIn order to prove that the destination an deode the relay's odeword suessfullywhen (2.26) holds, i.e., that the probability of error at the destination goes to zero, wewill again follow the standard" method in [80, Se. 7.7℄ for analyzing the probabilityof error for rates smaller than the apaity. To this end, we again use a jointly typial165Appendix A. Proofs for Chapter 2deoder. In partiular, we dene a jointly typial set Dkǫ as followsDkǫ = {(x2,y2) ∈ X k2 × Yk2 :∣∣∣∣∣−1kk∑i=1log2 p(x2i)−H(X2)∣∣∣∣∣≤ ǫ (A.15a)∣∣∣∣∣−1kk∑i=1log2 p(y2i)−H(Y2)∣∣∣∣∣≤ ǫ (A.15b)∣∣∣∣∣−1kk∑i=1log2 p(x2i, y2i)−H(X2, Y2)∣∣∣∣∣≤ ǫ}, (A.15)where p(x2) and p(y2) are given in (2.8)-(2.11) The reeived odeword y2 is suess-fully deoded as the transmitted odeword x2 if and only if (x2,y2) ∈ Dkǫ , and noother odeword xˆ2 from odebook C2 is jointly typial with y2. In order to omputethe probability of error, we dene the following eventsE0 = {(x2,y2) /∈ Dkǫ } and Ej = {(xˆ2(j),y2) ∈ Dkǫ }, (A.16)where xˆ2(j) is the j-th odeword in C2 that is dierent from x2. Note that in C2 thereare |C2| − 1 = 2kR − 1 odewords whih are dierent from x2, i.e., j = 1, ..., 2kR − 1.An error ours if at least one of the events E0, E1,..., E2kR−1 ours. Sine x2 isuniformly seleted from odebook C2, the average probability of error is given byPr(ǫ) = Pr(E0 ∪ E1 ∪ ... ∪ E2kR−1) ≤ Pr(E0) +2kR−1∑j=1Pr(Ej). (A.17)In (A.17), Pr(E0) is upper bounded as [80, Eq. (7.74)℄Pr(E0) ≤ ǫ, (A.18)166Appendix A. Proofs for Chapter 2whereas Pr(Ej) is bounded asPr(Ej) = Pr((xˆ2(j),y2) ∈ Dkǫ ) =∑(xˆ2(j),y2)∈Dkǫp(xˆ2(j),y2)(a)=∑(xˆ2(j),y2)∈Dkǫp(xˆ2(j))p(y2)(b)≤∑(xˆ2(j),y2)∈Dkǫ2−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 sine xˆ2(j) and y2 are independent, (b) follows sine p(xˆ2(j)) ≤2−k(H(X2)−ǫ) and p(y2) ≤ 2−k(H(Y2)−ǫ), whih follow from [80, Eq. (3.6)℄, respetively,and (c) follows sine |Dkǫ | ≤ 2k(H(X2,Y2)+ǫ), whih follows from [80, Theorem 7.6.1℄.Inserting (A.18) and (A.19) into (A.17), we obtainPr(ǫ) ≤ ǫ+2kR−1∑j=12−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)Hene, if R < H(Y2) − H(Y2|X2) − 3ǫ = I(X2; Y2) − 3ǫ, then limǫ→∞ limk→∞Pr(ǫ) = 0.This onludes the proof.A.3 Proof of Lemma 2.1To prove Lemma 2.1, we use results from [86℄. Hene, we rst assume that pV (x2) isa ontinuous distribution and then see that this leads to a ontradition. If pV (x2) isa ontinuous distribution, then the distribution of X2 is also ontinuous. Now, our167Appendix A. Proofs for Chapter 2goal is to obtain the solution of the following optimization problemMaximize :pV (x2)H(Y2)Subject to :∫x2 x22pV (x2)dx2 ≤ P2,(A.21)where Y2 = X2 + N2, N2 is a zero mean Gaussian distributed RV with variane σ22,and X2 has a ontinuous distribution with an average power onstraint. However, itis 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, theonly possible solution that yields a Gaussian distributed Y2 is if X2 is also Gaussiandistributed. On the other hand, the distribution of X2 an be written asp(x2) = pV (x2)PU + δ(x2)(1− PU). (A.22)Hene, we have to nd a pV (x2) suh 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 isthat pV (x2) is a disrete distribution. Only in the limiting ase when PU → 1, p(x2)beomes a Gaussian distribution by setting pV (x2) to be a Gaussian distribution.This onludes the proof.168Appendix BProofs for Chapter 3B.1 Proof of Theorem 3.1We rst note that, beause of the law of the onservation of ow, A ≥ R¯SD is alwaysvalid and equality holds if and only if the queue is non-absorbing. Assume rstwe have an adaptive reeption-transmission poliy with average arrival rate A andahievable average rate R¯SD with A > R¯SD, i.e., the queue is absorbing. For thispoliy, we denote the set of indies with di = 1 by I¯ and the set of indies with di = 0by I, and for N →∞ we haveA = 1N∑i∈I(1− di) log2(1 + s(i)) > R¯SD =1N∑i∈I¯di min{log2(1 + r(i)), Q(i− 1)}.(B.1)From (B.1) we observe that the onsidered protool annot be optimal as it an beimproved by moving some of the indies i in I to I¯ whih leads to an inrease of R¯SDat the expense of a derease of A. However, one the point A = R¯SD is reahed,moving more indies i from I to I¯ will derease both A and R¯SD beause of theonservation of ow. Thus, a neessary ondition for the optimal poliy is that thequeue is non-absorbing but the queue is at the edge of non-absorption, i.e., the queueis at the boundary of a non-absorbing and an absorbing queue. This ompletes theproof.169Appendix B. Proofs for Chapter 3B.2 Proof of Theorem 3.2We denote the sets of indies i for whih di = 1 and di = 0 holds by I¯ and I,respetively. ǫ denotes a subset of I¯ and | · | is the ardinality of a set. Throughoutthe remainder of this proof N →∞ is assumed.If the queue in the buer of the relay is absorbing, A > R¯SD holds and on averagethe number of bits/symb arriving at the queue exeed the number of bits leavingthe queue. Thus, log2(1 + r(i)) ≤ Q(i − 1) holds almost always and as a result theaverage rate an be written asR¯SD =1N∑i∈I¯min{log2(1 + r(i)), Q(i− 1)} =1N∑i∈I¯log2(1 + r(i)). (B.2)Now, we assume that the queue is at the edge of non-absorption. That isA = R¯SDholds but moving a small fration ǫ, where |ǫ|/N → 0, of indies from I¯ to I will makethe queue an absorbing queue with A > R¯SD. For this ase, we wish to determinewhether or not1N∑i∈I¯log2(1 + r(i)) > R¯SD =1N∑i∈I¯min{log2(1 + r(i)), Q(i− 1)}= A = 1N∑i∈Ilog2(1 + s(i)) (B.3)holds. To test this, we move a small fration ǫ, where |ǫ|/N → 0, of indies from I¯to I, thus making the queue an absorbing queue. As a result, (B.2) holds, and (B.3)170Appendix B. Proofs for Chapter 3beomes1N∑i∈I¯\ǫlog2(1 + r(i)) = R¯SD =1N∑i∈I¯\ǫmin{log2(1 + r(i)), Q(i− 1)}< A = 1N∑i∈I∪ǫlog2(1 + s(i)). (B.4)From the above we onlude that if (B.2) holds, then based on (B.3) and (B.4), for|ǫ|/N → 0, we must have1N∑i∈I¯log2(1 + r(i)) >1N∑i∈Ilog2(1 + s(i)) (B.5)and1N∑i∈I¯\ǫlog2(1 + r(i)) <1N∑i∈I∪ǫlog2(1 + s(i)). (B.6)However, for (B.5) and (B.6) to jointly hold, we require that the partiular onsideredmoving of indies from I¯ to I has aused a disontinuity in 1N∑i∈I¯ log2(1 + r(i))or/and a disontinuity in1N∑i∈I log2(1 + s(i)) as |ǫ|/N → 0 is assumed. Sine theapaities of the S-R and R-D links are suh that limN→∞∑i∈ǫ log2(1+s(i))/N → 0and limN→∞∑i∈ǫ log2(1 + r(i))/N → 0, ∀i, suh disontinuities are not possible.Therefore, at the edge of non-absorption (B.3) is not true and we must have instead1N∑i∈I¯log2(1 + r(i)) = R¯SD =1N∑i∈I¯min{log2(1 + r(i)), Q(i− 1)}= A = 1N∑i∈Ilog2(1 + s(i)) (B.7)Using (B.7), the average rate an be written as (3.18). This onludes the proof.171Appendix B. Proofs for Chapter 3B.3 Proof of Theorem 3.3To 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 linearoptimization problemMaximize :di1N∑Ni=1 di log2(1 + r(i))Subject to : C1 : 1N∑Ni=1(1− di) log2(1 + s(i)) = 1N∑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 optimalvalues of di, ∀i, are at the boundaries, i.e., di ∈ {0, 1}, ∀i. Therefore, the solution ofthe relaxed problem (B.8) is also the solution to the original maximization problemin (3.19).Sine (B.8) is a linear optimization problem, we an solve it by using the methodof Lagrange multipliers. The Lagrangian funtion for maximization problem (B.8) isgiven byL = 1NN∑i=1di log2(1 + r(i))− µ1NN∑i=1[di log2(1 + r(i))− (1− di) log2(1 + s(i))]+ 1NN∑i=1βidi −1NN∑i=1αi(di − 1), (B.9)where µ, βi/N , and αi/N are Lagrange multipliers. The Lagrange multipliers βi/Nand αi/N have to satisfyβi/N ≥ 0, αi/N ≥ 0, diβi/N = 0, (di − 1)αi/N = 0. (B.10)172Appendix B. Proofs for Chapter 3Dierentiating L with respet 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, sine 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, whih simplies (B.11) todi = 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, whih simplies (B.11) todi = 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 asdi =1 if (1− µ) log2(1 + r(i))− µ log2(1 + s(i)) ≥ 00 if (1− µ) log2(1 + r(i))− µ log2(1 + s(i)) ≤ 0,(B.15)whih is idential 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) beingontinuous random variables. µ or equivalently ρ are hosen suh that onstraint C1of problem (3.19) is met. This ompletes the proof.173Appendix B. Proofs for Chapter 3B.4 Proof of Theorem 3.4To 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 onaveoptimization problemMaximize :γS(i)≥0, γR(i)≥0, di1N∑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 ≤ 1C3 : 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 optimalvalues of di, ∀i, are at the boundaries, i.e., di ∈ {0, 1}, ∀i. Therefore, the solution ofthe relaxed problem (B.16) is also the solution to the original maximization problemin (3.29).Sine (B.16) is a onave optimization problem, we an solve it by using themethod of Lagrange multipliers. The Lagrangian funtion for maximization problem(B.16) is given byL = 1NN∑i=1di log2(1 + γR(i)hR(i))−µ 1NN∑i=1[di log2(1 + γR(i)hR(i))− (1− di) log2(1 + γS(i)hS(i))]−ν 1NN∑i=1[(1− di)γS(i) + diγR(i)− Γ]+ 1NN∑i=1βidi −1NN∑i=1αi(di − 1),(B.17)174Appendix B. Proofs for Chapter 3where the Lagrange multipliers µ and ν are hosen suh that C1 and C3 are satised,respetively. On the other hand, the Lagrange multipliers βi/N and αi/N have tosatisfy (B.10).By dierentiating L with respet to γS(i), γR(i), and di, and setting the resultsto 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, sine 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 ≥ 0and αi = 0, whih simplies (B.20) toβi = ν(γR(i)− γS(i))− (1− µ) log2(1 + γR(i)hR(i)) + µ log2(1 + γS(i)hS(i)) ≥ 0.(B.22)175Appendix B. Proofs for Chapter 3Whereas, assuming di = 1 leads βi = 0 and αi ≥ 0, whih simplies (B.20) toαi = −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 didi =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 thesystem of two equations with respet 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), whih are hosen suhthat onstraints C1 and C3 are meet with equality. This ompletes the proof.B.5 Proof of Lemma 3.2Sine s(i) and r(i) are ergodi random proesses, for N → ∞, the normalized sumsin C1 and C3 in (3.29) an be replaed by expetations. Therefore, the left handside of C1 is the expetation of variable (1 − di) log2(1 + γS(i)hS(i)). This variableis nonzero only when both (1 − di) and γS(i) are nonzero. The domain over whih(1− di) and γS(i) are jointly nonzero an be obtained from (3.30) and (3.32) and isgiven by(hS(i) > λ/ρ AND hR(i) < λ)OR (hS(i) > L1 AND hR(i) > λ) (B.25)176Appendix B. Proofs for Chapter 3where L1 is given by (3.36). Variable (1−di) log2(1+γS(i)hS(i)) has to be integratedover 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 expetation of the variable di log2(1 +γR(i)hR(i)). This variable is nonzero only when both di and γR(i) are nonzero. Thedomain over whih 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 overdomain (B.26) to obtain its average. This leads to the right side of (3.34).Following a similar proedure, we an obtain (3.35) from C3 in (3.29). Thisompletes the proof.177Appendix CProofs for Chapter 4C.1 Proof of Theorem 4.1We rst note that, beause of the law of the onservation of ow, A ≥ τ is alwaysvalid and equality holds if and only if the queue is non-absorbing.We denote the set of indies with di = 1 by I¯ and the set of indies with di = 0by I. Assume that we have an adaptive reeption-transmission protool with arrivalrate A and throughput τ with A > τ , i.e., the queue is absorbing. Then, for N →∞,we haveA = 1N∑i∈I(1− di)OS(i)S0 > τ =1N∑i∈I¯diOR(i)min{R0, Q(i− 1)}. (C.1)From (C.1) we observe that the onsidered protool annot be optimal as the through-put an be improved by moving some of the indies i in I to I¯ whih leads to aninrease of τ at the expense of a derease of A. As we ontinue moving indies fromI to I¯ we reah a point where A = τ holds. At this point, the queue beomes non-absorbing (but is at the boundary between a non-absorbing and an absorbing queue)and the throughput is maximized. If we ontinue moving indies from I to I¯, ingeneral, A will derease and as a onsequene of the law of onservation of ow, τwill also derease. We note that A does not derease if we move only those indiesfrom I to I¯ for whih OS(i) = 0 holds. In this ase, A will not hange, and as a178Appendix C. Proofs for Chapter 4onsequene of the law of onservation of ow, the value of τ also remains unhanged.Note that this is used in Lemma 4.1. However, the queue is moved from the edgeof non-absorption if OR(i) = 1 holds for some of the indies moved from I to I¯. Aswill be seen later, if the queue of the buer operates at the edge of non-absorption,the throughput beomes independent of the state of the queue, whih is desirable foranalytial throughput maximization.In the following, we will prove that when the queue is at the edge of non-absorptionthe following holdsτ = limN→∞1NN∑i=1diOR(i)R0 = A = limN→∞1NN∑i=1(1− di)OS(i)S0. (C.2)Let ǫ denote a small subset of I¯ ontaining only indies i for whih OS(i) = 1,where |ǫ|/N → 0 for N → ∞ and | · | denotes the ardinality of a set. Throughoutthe remainder of this proof N →∞ is assumed.If the queue in the buer of the relay is absorbing, A > τ holds and on averagethe number of bits arriving at the queue exeed the number of bits leaving the queue.Thus, R0 ≤ Q(i − 1) holds almost always and as a result the throughput an bewritten asτ = 1N∑i∈I¯OR(i)min{R0, Q(i− 1)} =1N∑i∈I¯OR(i)R0. (C.3)Now, we assume that the queue is at the edge of non-absorption. That is A = τholds but moving the small fration of indies in ǫ, where |ǫ|/N → 0, from I¯ to I willmake the queue an absorbing queue with A > τ . For this ase, we wish to determinewhether or not1N∑i∈I¯OR(i)R0 > τ =1N∑i∈I¯OR(i)min{R0, Q(i− 1)} = A =1N∑i∈IOS(i)S0 (C.4)179Appendix C. Proofs for Chapter 4holds. To test this, we move a small fration ǫ, where |ǫ|/N → 0, of indies from I¯to I, thus making the queue an absorbing queue. As a result, (C.3) holds and (C.4)beomes1N∑i∈I¯\ǫOR(i)R0 = τ =1N∑i∈I¯\ǫOR(i)min{R0), Q(i− 1)} = A =1N∑i∈I∪ǫOS(i)S0.(C.5)From the above we onlude that if (C.3) holds, then based on (C.4) and (C.5), for|ǫ|/N → 0, we must have1N∑i∈I¯OR(i)R0 >1N∑i∈IOS(i)S0 (C.6)and1N∑i∈I¯\ǫOR(i)R0 <1N∑i∈I∪ǫOS(i)S0. (C.7)However, for (C.6) and (C.7) to jointly hold, we require that the partiular onsid-ered move of indies from I¯ to I auses a disontinuity in 1N∑i∈I¯ OR(i)R0 or/anda disontinuity in1N∑i∈I OS(i)S0 as |ǫ|/N → 0 is assumed. Sine S0 and R0are nite, limN→∞∑i∈ǫ S0/N = limN→∞ S0|ǫ|/N = 0 and limN→∞∑i∈ǫR0/N =limN→∞R0|ǫ|/N = 0. Hene, suh disontinuities are not possible. Therefore, at theedge of non-absorption the inequality in (C.4) annot hold and we must have1N∑i∈I¯OR(i)R0 = τ =1N∑i∈I¯OR(i)min{R0, Q(i− 1)} = A =1N∑i∈IOS(i)S0. (C.8)Eq. (C.8) an be written as (4.31). This onludes the proof.180Appendix C. Proofs for Chapter 4C.2 Proof of Theorem 4.2To 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 linearoptimization problemMaximize :di1N∑Ni=1 diOR(i)R0Subject to : C1 : 1N∑Ni=1(1− di)OS(i)S0 = 1N∑Ni=1 diOR(i)R0C2 : 0 ≤ di ≤ 1, ∀i.(C.9)In the following, we solve the relaxed problem (C.9) and then show that the optimalvalues of di, ∀i are at the boundaries, i.e., di ∈ {0, 1}, ∀i. Therefore, the solution ofthe relaxed problem (C.9) is also the solution to the original maximization problemin (4.33).Sine (C.9) is a linear optimization problem, we an solve it by using the methodof Lagrange multipliers. The Lagrangian for Problem (4.33) is given byL = 1NN∑i=1diOR(i)R0 − µ1NN∑i=1[diOR(i)R0 − (1− di)OS(i)S0]+ 1NN∑i=1βidi −1NN∑i=1αi(di − 1), (C.10)where µ, βi/N , and αi/N are Lagrange multipliers. The Lagrange multipliers βi/Nand αi/N have to satisfy (B.10).Dierentiating L with respet 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,181Appendix C. Proofs for Chapter 4and from (C.11) we obtain that the following must hold(1− µ)OR(i)R0 − µOS(i)S0 = 0. (C.12)However, sine OR(i) and OR(i) are independent random variables, (C.12) annothold 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, whih simplies (C.11) todi = 0 ⇒ βi = µOS(i)S0 − (1− µ)OR(i)R0 ≥ 0. (C.13)Whereas, assuming di = 1 leads βi = 0 and αi ≥ 0, whih simplies (C.11) todi = 1 ⇒ αi = (1− µ)OR(i)R0 − µOS(i)S0 ≥ 0. (C.14)respetively. (C.13) and (C.14) an be written equivalently asdi =1 if (1− µ)OR(i)R0 ≥ µOS(i)S00 if (1− µ)OR(i)R0 ≤ µOS(i)S0.(C.15)Furthermore, 0 ≤ µ ≤ 1 has to hold sine for µ < 0 and µ > 1 we have alwaysdi = 1 and di = 0, respetively, irrespetive of any non-negative values of OS(i)S0and OR(i)R0. In this ase, sine OS(i) and OR(i) are disrete random variables, whihtake the values zero or one, the probability of (1−µ)OR(i)R0 = µOS(i)S0 happeningis 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 willbe investigated later. From (C.15), for 0 < µ < 1, we have four possibilities:1. If OR(i) = 1 and OS(i) = 0, then di = 1.182Appendix C. Proofs for Chapter 42. 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 = 1and the hoie does not inuene the throughput as both the soure and therelay remain silent.4. If OR(i) = 1 and OS(i) = 1 and µ is hosen suh that 0 < µ < R0/(S0+R0) thendi = 1 in all time slots with OR(i) = 1 and OS(i) = 1, and as a result, onditionC1 annot be satised. Similarly, if µ is hosen suh that R0/(S0+R0) < µ < 1,then di = 0 in all time slots with OR(i) = 1 and OS(i) = 1, and as a resultondition C1 an also not be satised. Thus, we onlude that µ must be setto µ = R0/(S0 +R0) sine only in this ase an di be hosen to be either di = 0or di = 1, whih is neessary for satisfying ondition C1. Sine for OR(i) = 1and OS(i) = 1 neither link is in outage, di an be hosen to be either zero orone, as long as ondition C1 is satised. In order to satisfy C1, we propose toip a oin and the outome of the oin toss deides whether di = 1 or di = 0.Let the oin have two outomes C ∈ {0, 1} with probabilities Pr{C = 0} andPr{C = 1}. We set di = 0 if C = 0 and di = 1 if C = 1. Thus, the probabilitiesPr{C = 0} and Pr{C = 1} have to be hosen suh that C1 is satised.Choosing the link seletion variable as in (4.35) and exploiting the independene ofs(i) and r(i), ondition C1 results inS0 [(1− PS)PR + (1− PS)(1− PR)Pr{C = 0}]= R0 [(1− PR)PS + (1− PS)(1− PR)Pr{C = 1}] . (C.16)From (C.16), we an obtain the probabilities Pr{C = 0} and Pr{C = 1}, whih aftersome basi algebrai manipulations leads to (4.36). The throughput is given by the183Appendix C. Proofs for Chapter 4right (or left) hand side of (C.16), whih leads to (4.37).For (4.36) to be valid, Pr{C = 0} and Pr{C = 1} have to meet 0 ≤ Pr{C = 0} ≤ 1and 0 ≤ Pr{C = 1} ≤ 1, whih leads to the onditionsS0(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 seletion 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 obtainthree 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 = 1and the hoie has no inuene on the throughput.3. If OR(i) = 0 and OS(i) = 1, then di an be hosen to be either di = 0 or di = 1as long as ondition C1 is satised. Similar to before, in order to satisfy C1,we propose to ip a oin and the outome of the oin ip determines whetherdi = 1 or di = 0.Choosing the link seletion variable as in (4.39) and exploiting the independene ofs(i) and r(i), ondition C1 an be rewritten asS0PR(1− PS)Pr{C = 0} = R0(1− PR). (C.19)After basi manipulations (C.19) simplies to (4.40). The throughput is given by theright (or left) hand side of (C.19) and an be simplied to (4.41). Imposing again the184Appendix C. Proofs for Chapter 4onditions 0 ≤ Pr{C = 0} ≤ 1 and 0 ≤ Pr{C = 1} ≤ 1, we nd that for µ = 0, (C.17)still has to hold but (C.18) an be violated, whih is equivalent to the new onditionPR >R0R0 + S0(1− PS). (C.20)For the third and nal ase, letting µ = 1 and following a similar path as forµ = 0 leads to (4.43)(4.45) and ondition (4.42).Finally, we have to prove that the three onsidered ases are mutually exlusive,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 mutuallyexlusive, respetively. For Cases 2 and 3, the mutual exlusiveness is less obvious.Thus, we rewrite (4.38) and (4.42) asPR > PR,2 (C.21)andPR < PR,3, (C.22)respetively, 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. Hene, for 0 ≤ PS < 1, atmost one of (C.21) and (C.22) is satised and Cases 2 and 3 are mutually exlusive.For PS = 1 (i.e., the S-R link is always in outage), we have PR,2 = PR,3 = 1 andCase 1 and Case 3 apply for PR = 1 and PR < 1, respetively. Therefore, for anyombination of PS and PR only one of the three ases onsidered in Theorem 4.2applies. This onludes the proof.185Appendix C. Proofs for Chapter 4C.3 Proof of Lemma 4.2We provide two dierent proofs for the outage probability, Fout, in (4.52). Therst proof is more straightforward and based on (4.10). However, the seond proofprovides more insight into when outages our.Proof 1: In the absene of outages, the maximum ahievable throughput, denotedby 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 simpliations, (4.52) is obtained. Thisonludes the proof.Proof 2: The seond proof exploits the fat that an outage ours when both thesoure 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 ours only when both linksare 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 arein outage or when the S-R link is not in outage, while the R-D link is in outage andthe oin ip hooses the relay for transmission. This event happens with probabilityFout = PSPR+(1−PS)PRPC , whih 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 transmitswhen both links are in outage or when the S-R link is in outage, while the R-D linkis not in outage and the oin ip hooses the soure for transmission. This happenswith probability Fout = PSPR+PS(1−PR)(1−PC), whih after introduing PC givenby (4.44) leads to (4.52).186Appendix C. Proofs for Chapter 4C.4 Proof of Lemma 4.4Computing the link outages in (4.7) and (4.8) for Rayleigh fading and exploiting(4.54), we obtain (4.55) by employing ΩS = γΩ¯S and ΩR = γΩ¯R in the resultingexpression 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 Ω¯S = Ω¯R = Ω¯ and S0 = R0, the asymptoti expression for Fout in(4.55) simplies toFout →(2R0 − 1)2Ω¯2γ2 , 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 asDM(r) = − limγ→∞log2(Fout)log2(γ)= − limγ→∞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 4.3Let di be given by (4.59). Then, the following events are possible for the queue inthe buer:1. If the buer is empty, it stays empty with probability PS and reeives one paketwith probability 1− PS.187Appendix C. Proofs for Chapter 40 1 2 L. . . .1-PS 1-PS 1-p-q 1-p-q1-pPSP -qS p ppq q31-p-qpq41-p-qpqFigure C.1: Markov hain for the number of pakets in the queue of the buer if thelink seletion variable di is given by (4.59).2. If the buer ontains one paket, it stays in the same state with probabilityPSPR, sends the paket with probability PS(1−PR), and reeives a new paketwith probability 1− PS.3. If the buer ontains more than one paket but less than L pakets, it stays inthe same state with probability PSPR, reeives a new paket with probability(1− PS)PR + (1− PS)(1− PR)(1− PC), and sends one paket with probability(1− PR)PS + (1− PS)(1− PR)PC .4. If the buer ontains L pakets, it stays in the same state with probabilityPSPR + (1 − PS)PR + (1 − PS)(1 − PR)(1 − PC), and sends one paket withprobability (1− PR)PS + (1− PS)(1− PR)PC .The events for the queue of the buer detailed above, form a Markov hain whosestates are dened by the number of pakets in the queue. This Markov hain is shownin Fig. C.1, where the probabilities p and q are given by (4.65). Let M denote thestate transition matrix of the Markov hain and let mi,j denote the element in thei-th row and j-th olumn of M. Then, mi,j is the probability that the buer willtransition from having i− 1 pakets in its queue in the previous time slot to havingj−1 pakets in its queue in the following time slot. The non-zero elements of matrix188Appendix C. Proofs for Chapter 4M are given bym1,1 = PS , m1,2 = 1− PS , m2,1 = PS − q ,m2,3 = 1− PS , mL+1,L+1 = 1− pmi,i+1 = 1− p− q , mi+1,i = p , mi,i = q , for i = 1...L.(C.25)Let Pr{Q} = [Pr{Q = 0}, Pr{Q = R0}, ...,Pr{Q = LR0}] denote the steady stateprobability vetor of the onsidered Markov hain, where Pr{Q = kR0}, k = 0, . . . , L,is the probability of having k pakets in the buer. The steady state probability vetoris obtained by solving the following system of equationsPr{Q}M = Pr{Q}∑Lk=0 Pr{Q = kR0} = 1, (C.26)whih leads to (4.64). Using (4.64) the average queue size E{Q} an be obtainedfromE{Q} = R0L∑k=0kPr{Q = kR0}, (C.27)whih leads to (4.66). Furthermore, the average arrival rate an be found asA = R0[(1− PS)(Pr{Q = 0}+ Pr{Q = R0})+(1− p− q)(1− Pr{Q = 0} − Pr{Q = R0} − Pr{Q = LR0})]. (C.28)Inserting the average arrival rate given by (C.28) and the average queue size givenby (4.66) into (4.57) yields the average delay in (4.67).189Appendix C. Proofs for Chapter 40 1 2 L. . . .1-PS 1-p-q 1-p-q 1-p-q1-pPSp p ppq q31-p-qpq41-p-qpqFigure C.2: Markov hain for the number of pakets in the queue of the buer if thelink seletion 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 buerof the relay an be modeled by the Markov hain shown in Fig. C.2. If the linkseletion variable di is given by (4.61), p and q are given by (4.65), and if the linkseletion variable di is given by (4.63), p and q are given by (4.70). Following thesame proedure as before, (4.69)-(4.73) an be obtained. This ompletes the proof.C.6 Proof of Lemma 4.6Let us rst assume that 2p + q − 1 < 0, whih is equivalent to p < 1 − p− q. Now,sine L → ∞, pL goes to zero faster than (1 − p − q)L. Thus, by using pL = 0 asL→∞ in (4.67) and (4.72) , we obtain in both asesE{T} = Lp −11− 2p− q . (C.29)Thus, we onlude that if 2p + q − 1 < 0, E{T} grows with L and is unlimited asL→∞. Thus, if E{T} 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. Hene,(4.74)-(4.82) are obtained by letting (1 − p − q)L = 0, as L → ∞, in the relevantequations in Theorem 4.3 and inserting the orresponding p and q given by (4.65)and (4.70) into the resulting expressions. This onludes the proof.190Appendix C. Proofs for Chapter 4C.7 Proof of Lemma 4.7The minimum and maximum possible delays that the onsidered buer-aided relayingsystem an ahieve are obtained for PC = 1 and PC = 0, respetively. If di is givenby (4.59), the delay is given by (4.75). By setting PC = 1 in (4.75) we obtainthe minimum possible delay in (4.83). However, sine (4.75) is valid only when2p+ q− 1 > 0, (4.83) is valid only when PR < 1/(2−PS). This ondition is obtainedby inserting PC = 1 into the expressions for p and q given by (4.65) and exploiting2p + 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 whenPS > 1/(2 − PR), whih is obtained from 2p + q − 1 > 0 and inserting PC = 0 intothe expressions for p and q given by (4.65).A similar approah 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 onludesthe proof.C.8 Proof of Theorem 4.4The outage probability, Fout, an be derived based on two dierent approahes. Therst approah is straightforward and based on (4.10). However, the seond approahprovides more insight into how and when the outages our and is based on ountingthe time slots in whih no transmissions our. In the following, we provide a proofbased on the latter approah.If di is given by (4.59) or (4.61), there are four dierent ases where no nodetransmits.1. The buer is empty and the S-R link is in outage.191Appendix C. Proofs for Chapter 42. 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 inoutage. In this ase, the soure is seleted for transmission but sine the bueris full, the paket is dropped.4. The buer is full, both the S-R and R-D links are not in outage, and the soureis seleted for transmission based on the oin ip. In this ase, sine the bueris full, the paket is dropped.Summing up the probabilities for eah of the above four ases, we obtain (4.87).If di is given by (4.63), an outage ours in three ases: Case 1 and Case 2 asdesribed above, and a new Case 3. In the new Case 3, the buer is full, the S-Rlink is not in outage while the R-D link is in outage, and the soure is seleted fortransmission based on the oin ip. Summing up the probabilities for eah of thethree ases, we obtain (4.88).C.9 Proof of Theorem 4.5For delay onstrained transmission with E{T} < L, the probability of dropped pak-ets Pr{Q = LR0} an be made arbitrarily small by inreasing the buer size L. Thus,for large enough L, we an set Pr{Q = LR0} = 0 in (4.87) and (4.88).In the high SNR regime, when PS → 0 and PR → 0, PR < 1/(2 − PS) andPS < 1/(2 − PR) always hold. Using PS → 0 and PR → 0 in the delays speiedin Proposition 4.1, we obtain the onditions E{T} > 3 and 1 < E{T} ≤ 3 if linkseletion variable di is given by (4.59) and (4.61), respetively.We rst onsider the ase E{T} > 3, where di is given by (4.59). Thus, theprobability of the buer being empty, Pr{Q = 0}, is given by (4.74). Using PS → 0192Appendix C. Proofs for Chapter 4and PR → 0 in (4.74), we obtainPr{Q = 0} = PS(1− 12PC). (C.30)On the other hand, using PS → 0 and PR → 0 in the expression for E{T} in (4.75),we obtainE{T} = 12PC − 1+ 2. (C.31)Solving (C.31) for PC yieldsPC =12(1 + 1E{T} − 2). (C.32)Inserting (C.32) into (C.30) we obtainPr{Q = 0} = PSE{T} − 1 . (C.33)Finally, inserting (C.33) into (4.87) and setting Pr{Q = LR0} = 0, we obtain (4.91).Now, we onsider the ase 1 < E{T} ≤ 3, where di is given by (4.61). Here, theprobability of the buer being empty, Pr{Q = 0}, is given by (4.77). For PS → 0and PR → 0, we obtain from (4.77)Pr{Q = 0} = 1− 12Pr{C = 1} . (C.34)Furthermore, for PS → 0 and PR → 0, we obtain from (4.78) the asymptoti delayE{T} = 12PC − 1(C.35)193Appendix C. Proofs for Chapter 4or equivalentlyPC =12(1 + 1E{T}). (C.36)Inserting (C.36) into (C.34) we obtainPr{Q = 0} = 1E{T}+ 1 . (C.37)Finally, inserting (C.37) into (4.88) and setting Pr{Q = LR0} = 0, we obtain (4.90).This onludes the proof.C.10 Proof of Theorem 4.7To 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 programingproblem whose Lagrangian is given byL = 1NN∑i=1di log2(1 + r(i))− µ 1NN∑i=1[di log2(1 + r(i))−(1− di)OS(i)S0]+ 1NN∑i=1βidi −1NN∑i=1αi(di − 1), (C.38)where µ, βi/N , and αi/N are Lagrange multipliers. The Lagrange multipliers βi/Nand αi/N have to satisfy (B.10). Dierentiating L with respet to di and setting theresult to zero leads to(1− µ) log2(1 + r(i))− µOS(i)S0 + βi − αi = 0. (C.39)194Appendix C. Proofs for Chapter 4If 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, sine r(i) and OR(i) are independent random variables, (C.40) annot holdfor all i. Therefore, di has to be at the boundary, i.e., di ∈ {0, 1}. Now, assumingdi = 0 leads βi ≥ 0 and αi = 0, whih simplies (C.39) todi = 0 ⇒ βi = µOS(i)S0 − (1− µ) log2(1 + r(i))≥ 0. (C.41)Whereas, assuming di = 1 leads βi = 0 and αi ≥ 0, whih simplies (C.39) todi = 0 ⇒ αi = −µOS(i)S0 + (1− µ) log2(1 + r(i))≥ 0. (C.42)Relations (C.41) and (C.42), an be written equivalently asdi =1 if (1− µ) log2(1 + r(i))≥ µOS(i)S00 if (1− µ) log2(1 + r(i))≤ µOS(i)S0,(C.43)Sine for µ < 0 and µ > 1, we have always di = 1 and di = 0, respetively, irrespetiveof 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 = 0and µ = 1 later. For 0 < µ < 1, (C.43) an be written in the form of (4.96) aftersetting ρ = µ/(1−µ), where ρ is hosen suh that onstraint C1 of problem (4.94) ismet. 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), whih is valid for ρ in the range of ρ = [0,∞).195Appendix C. Proofs for Chapter 4Thus, by setting ρ = ∞ in (4.97), we obtain the entire domain over whih (4.96) isvalid, whih leads to ondition (4.95).Next, we onsider the boundary values µ = 0 and µ = 1. The boundary valueµ = 0 or equivalently ρ = 0 is relevant only in the trivial ase when the S-R link isnever in outage (i.e. PS = 0) and S0 = ∞, where a trivial solution is given by d1 = 0and 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 → ∞, whih is the ase whenondition (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 longas onstraint C1 holds.However, the same throughput as obtained when OS(i) = 0 and di is hosen suh thatonstraint C1 holds, an also be obtained by hoosing always di = 1 when OS(i) = 0resulting in (4.98). The reason behind this is as follows: Assume there is a poliy forwhih when OS(i) = 0, di is hosen suh that onstraint C1 holds. Now, we hange difrom 0 to 1 for OS(i) = 0. However, this hange does not aet the (average) amountof data entering the buer. Thus, beause of the law of onservation of ow, theaverage amount of data entering the buer per time slot is idential to the averageamount of data leaving the buer per time slot (the throughput), and the throughputis not aeted by the hange.196Appendix C. Proofs for Chapter 4C.11 Proof of Theorem 4.8The Lagrangian of the relaxed optimization problem of (4.105) where 0 ≤ di ≤ 1 isassumed, is given byL = 1NN∑i=1di log2(1 + γR(i)hR(i)) +1NN∑i=1βidi −1NN∑i=1αi(di − 1)− µ 1NN∑i=1[di log2(1 + γR(i)hR(i))− (1− di)OS(i)S0]− ν 1NN∑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 µand ν are hosen suh that C1 and C3 hold, respetively.By dierentiating L with respet 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, sine 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 ≥ 0197Appendix C. Proofs for Chapter 4and αi = 0, whih simplies (C.46) toβi = 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, whih simplies (C.46) toαi = −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 didi =1 if (1− µ) log2(1 + γR(i)hR(i))− νγR(i) ≥ µOS(i)S0 − νOS(i)γS0 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 respet to γR(i), and taking into aountthat 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), whih are hosen suh that onstraints C1 and C3 are metwith equality. Given the PDFs fhS(hS) and fhR(hR), onditions (4.93) and (4.104)an be diretly written as (4.110) and (4.111), respetively. Setting ρ→∞ in (4.110)and (4.111), we obtain ondition (4.106) whih is neessary for the validity of (4.98).Similar to the xed transmit power ase, the boundary value µ = 0 is trivial. Onthe other hand, for µ = 1, we obtain that di has to be set to di = 0 when OS(i) = 1and for OS(i) = 0, di an be hosen arbitrarily. Similar to the xed power ase, weset di = 1 when OS(i) = 1 in order to minimize the delay. Thus, the optimal powerand link seletion variables are given by (4.113) and (4.114), respetively, and thethroughput is given by (4.115).198Appendix C. Proofs for Chapter 4C.12 Proof of Theorem 4.9For γS = γR = γ → ∞, the protool in Proposition 4.3 is optimal in the sensethat it maximizes the throughput while satisfying the average delay onstraint. Inpartiular, for high SNR in the S-R link, the probability that the link is in outageapproahes zero and the relay reeives S0 bits per soure transmission. On the otherhand, the number of bits transmitted by the relay in one time slot over the R-Dlink inreases with the SNR. Thus, for suiently high SNR, the soure transmitskS0 bits in k time slots and the relay needs just p = 1 time slot to forward theentire information to the destination. Hene, every transmission period omprisesk + p = k + 1 time slots, where the queue length at the relay inreases from S0 tokS0 in the rst k time slots and is redued to zero in the (k+ 1)th time slot. Hene,the average queue length, E{Q}, an be written asE{Q} → 1k + 1(1 + 2 + ... + k + 0)S0 =1k + 1k(k + 1)2 S0= k2S0, as γ →∞ . (C.51)On the other hand, the arrival rate is idential to the throughput and given by (4.121),and for high SNR it onverges toA = τ → S0kk + 1 , as γ →∞ . (C.52)Combining (4.57), (C.51), and (C.52) the average delay is found asE{T} → k + 12 , as γ →∞ . (C.53)199Appendix C. Proofs for Chapter 4Finally, ombining (C.52) and (C.53) the throughput an be expressed as (4.122),and the multiplexing gain in (4.123) follows diretly.200Appendix DOther ContributionsI have also o-authored other researh works whih have been published or submittedfor publiation during my time as a Ph.D. student at UBC. In partiular, the followingpapers have been published or submitted for publiation.Journal Papers:• N. Zlatanov, V. Jamali, and R. Shober, Ahievable Rates for the Fading Half-Duplex Single Relay Seletion Network Using Buer-Aided Relaying, Aeptedto IEEE Transations on Wireless Communiations, 2015.• V. Jamali, N. Zlatanov, H. Shoukry, and R. Shober, Ahievable Rate of theHalf-Duplex Multi-Hop Buer-Aided Relay Channel with Blok Fading, A-epted to IEEE Transations on Wireless Communiations, 2015.• V. Jamali, N. Zlatanov, and R. Shober, Buer-Aided Bidiretional RelayNetworks with Fixed Rate Transmission  Part I: Delay-Unonstrained Case,IEEE Transations on Wireless Communiations, vol. 14, no. 3, pp. 1323 -1338, Mar. 2015.• V. Jamali, N. Zlatanov, and R. Shober, Bidiretional Buer-Aided RelayNetworks with Fixed Rate Transmission  Part II: Delay-Constrained Case,IEEE Transations on Wireless Communiations, vol. 14, no. 3, pp. 1339 -1355, Mar. 2015.201Appendix D. Other Contributions• Z. Hadzi-Velkov, N. Zlatanov, and R. Shober, Multiple-aess Fading Channelwith Wireless Power Transfer and Energy Harvesting, IEEE CommuniationsLetters, vol. 52, no. 4, pp. 1863 - 1866, Sep. 2014.• V. Jamali, N. Zlatanov, A. Ikhlef, and R. Shober, Ahievable Rate Regionof the Bidiretional Buer-Aided Relay Channel with Blok Fading, IEEETransations on Information Theory, vol. 60, no. 11, pp. 7090 - 7111, Sep.2014.• N. Zlatanov, A. Ikhlef, T. Islam, and R. Shober, Buer-Aided CooperativeCommuniations: Opportunities and Challenges, IEEE Communiations Mag-azine, Vol. 52, no. 4, Apr. 2014.• N. Zlatanov and R. Shober, Buer-Aided Half-Duplex Relaying Can Outper-form Ideal Full-Duplex Relaying, IEEE Communiations Letters, vol. 17, no.3, pp. 479-482, Mar. 2013.• N. Zlatanov, R. Shober, and Z. Hadzi-Velkov, Asymptotially Optimal PowerAlloation for Energy Harvesting Communiation Networks, Submitted forpubliation.Conferene Papers:• W. Wike, N. Zlatanov, V. Jamali, and R. Shober, Buer-Aided Relayingwith Disrete Transmission Rates, Pro. of IEEE 14th Canadian Workshopon Information Theory (CWIT), St. John's, NL, Canada, July 2015.• R. Simoni, V. Jamali, N. Zlatanov, R. Shober, L. Pierui, and R. Fantai,Buer-Aided Diamond Relay Network with Blok Fading, Pro. of IEEEInternational Conferene on Communiations (ICC), London, UK, June 2015.202Appendix D. Other Contributions• N. Zlatanov, V. Jamali, and R. Shober, Ahievable Rates for the Fading Half-Duplex Single Relay Seletion Network Using Buer-Aided Relaying, Pro. ofIEEE Globeom 2014, Austin, TX, De. 2014• V. Jamali, N. Zlatanov, and R. Shober, A Delay-Constrained Protool withAdaptive Mode Seletion for Bidiretional Relay Networks, Pro. of IEEEGlobeom 2014, Austin, TX, De. 2014• H. Shoukry, N. Zlatanov, V. Jamali, and R. Shober, Ahievable Rates for theFading Three-Hop Half-Duplex Relay Network using Buer-Aided Relaying,Pro. of IEEE Globeom 2014, Austin, TX, De. 2014• V. Jamali, N. Zlatanov, and R. Shober, Adaptive Mode Seletion for Bidire-tional Relay Networks - Fixed Rate Transmission, Pro. of IEEE InternationalConferene on Communiations (ICC), Sydney, Australia, June 2014.• N. Zlatanov, Z. Hadzi-Velkov, and R. Shober, Asymptotially Optimal PowerAlloation for Point-to-Point Energy Harvesting Communiation Systems, Pro.of IEEE Globeom 2013, Atlanta, GA, De. 2013.• V. Jamali, N. Zlatanov, A. Ikhlef, and R. Shober, Adaptive Mode Seletion inBidiretional Buer-aided Relay Networks with Fixed Transmit Powers, Pro.of IEEE Globeom 2013, Atlanta, GA, De. 2013.• Z. Hadzi-Velkov, N. Zlatanov, and R. Shober, Optimal Power Control forAnalog Bidiretional Relaying with Long-Term Relay Power Constraint, Pro.of IEEE Globeom 2013, Atlanta, GA, De. 2013.• V. Jamali, N. Zlatanov, A. Ikhlef, and R. Shober, Adaptive Mode Seletion inBidiretional Buer-aided Relay Networks with Fixed Transmit Powers, Pro.203Appendix D. Other Contributionsof EUSIPCO, Marrakeh, Maroo, Sep. 2013.• Z. Hadzi-Velkov, N. Zlatanov, and R. Shober, Optimal Power Alloation forThree-phase Bidiretional DF Relaying with Fixed Rates, Pro. of ISWCS2013, Ilmenau, Germany, Aug. 2013.• N. Zlatanov, R. Shober, and L. Lampe, Buer-Aided Relaying in a Three NodeNetwork, Pro. of IEEE International Symposium on Information Theory(ISIT 2012), Cambridge, MA, July 2012.• N. Zlatanov, Z. Hadzi-Velkov, G. K. Karagiannidis, and R. Shober, OutageRate and Outage Duration of Deode-and-Forward Cooperative Diversity Sys-tems, Pro. of IEEE International Conferene on Communiations (ICC),Kyoto, Japan, June 2011.• N. Zlatanov, R. Shober, G. K. Karagiannidis, and Z. Hadzi-Velkov, Aver-age Outage and Non-Outage Duration of Seletive Deode-and-Forward Relay-ing, Pro. of IEEE 12th Canadian Workshop on Information Theory (CWIT),Kelowna, Canada, May 2011.204

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