Resource Optimization in Relay Based CooperativeWireless Systems Under Channel UncertaintybyShankhanaad MallickM.Sc., Bangladesh University of Engineering and Technology, 2008B.Sc., Bangladesh University of Engineering and Technology, 2006A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2014c© Shankhanaad Mallick 2014AbstractOver the last decade, the demand for wireless resources has been rising exponentially withthe increase of the number of new users and services. The primary objective of wirelesscommunication research is finding solutions to meet this increasing demand with limited radioresources. Relay based cooperative systems greatly enhance the performance of resource-constrained wireless networks. By allowing cooperation via relays, it is possible to improvethe transmission quality and energy efficiency and get similar benefits to those of multiple-input-multiple-output (MIMO) systems.However, the benefits of a relay based cooperation often depend on the channel stateinformation (CSI) between various nodes. The CSI of such multi-hop systems is often im-perfect and outdated due to channel fluctuations, limited feedback capacity, and channelestimation and quantization errors. To maximize the utilization of the radio resources, itis imperative to devise optimal resource allocation schemes that are robust under imper-fect CSI. In this thesis, we consider different resource optimization problems for relay basedcooperative systems and propose solutions that are robust and computationally efficient.First, we develop power allocation and relay selection schemes for a decode-and-forward(DF) cooperative wireless network under bounded channel uncertainty. We propose worstcase optimization based approach to provide guaranteed quality-of-service (QoS) to the users.Next, we design a joint power allocation and admission control algorithm considering channelestimation error as unbounded Gaussian random variable. We propose a probabilistically-constrained optimization approach for QoS provisioning in slow fading channels. Finally,we propose to utilize user cooperation technique to establish communication among the sec-iiAbstractondary users (SUs) of a cognitive radio network (CRN). Power allocation and relay selectionschemes that maximize the system goodput of the CRN are developed. We provide QoS tothe SUs while satisfying the interference constraints of the primary user bands consideringdifferent channel uncertainty models. Numerical results demonstrate the effectiveness of ourproposed schemes and implications of ignoring channel estimation error while designing re-source optimization algorithms. Results reveal that the effects of ignoring the imperfectnessamong different channels are violations of QoS and interference constraints, which ultimatelyresult in transmission failures and wastage of transmission power.iiiPrefacePublications that have been resulted during the conduction of research in this thesis are asfollows:• S. Mallick, P. Kaligineedi, M. M. Rashid, and V. K. Bhargava, “Radio resource op-timization in cooperative wireless communication networks,” in Cooperative CellularWireless Communications, E. Hossain, D.I. Kim and V. K. Bhargava, Eds., Cam-bridge University Press, 2011, pp. 205–232. (Linked to Chapter 1)• S. Mallick, M. M. Rashid, and V. K. Bhargava, “Joint relay selection and power allo-cation for decode-and-forward cellular relay network with channel uncertainty,” IEEETrans. Wireless Commun., vol. 11, no. 10, pp. 3496–3508, Oct. 2012. (Linked toChapter 2)• S. Mallick, K. Kandhway, M. M. Rashid, and V. K. Bhargava, “Power allocation fordecode-and-forward cellular relay network with channel uncertainty,” in Proc. of 2011IEEE ICC, pp. 1–5. (Linked to Chapter 2)• S. Mallick, R. Devarajan, M. M. Rashid, and V. K. Bhargava, “Resource allocationfor selective relaying based cellular wireless system with imperfect CSI,” IEEE Trans.Commun., vol. 61, no. 5, pp. 1822–1834, May 2013. (Linked to Chapter 3)• S. Mallick, R. Devarajan, M. M. Rashid, and V. K. Bhargava, “Robust power allocationfor selective relaying based DF cellular wireless system,” in Proc. of 2012 IEEE VTC(Fall), pp. 1–5. (Linked to Chapter 3)ivPreface• S. Mallick, R. Devarajan, R. Arab Loodaricheh, and V. K. Bhargava, “Robust re-source optimization for cooperative cognitive radio networks with imperfect CSI,”IEEE Trans. Wireless Commun., to appear, pp. 1–14. (Linked to Chapter 4)• S. Mallick, R. Devarajan, M. M. Rashid, and V. K. Bhargava, “Robust power allocationdesigns for cognitive radio networks with cooperative relays,” in Proc of 2012 IEEEICC, pp. 1–5. (Linked to Chapter 4)I am the primary researcher and author for all the research contributions made in thisthesis. I conducted the literature review and identified the research problems. Moreover,I formulated the research problems and carried out the mathematical analysis and simu-lations. I also wrote the associated manuscripts for publication. The contributions of theco–authors of my papers are as follows. Prof. V. K. Bhargava is my honorable supervisor. Heprovided directions on identifying research problems and valuable comments on my researchprogress, and helped me by providing technical and editorial feedback during the preparationof associated manuscripts. Dr. M. M. Rashid and Dr. P. Kaligineedi were postdocs in ourgroup; they read the drafts of the papers/book chapter and provided their valuable technicalcomments, editorial corrections and constructive suggestions. R. Arab Loodaricheh is a cur-rent student and R. Devarnajan and K. Kandhway are former students of our group. Theyprovided some important technical feedback during the formulation of the research problemsin my thesis. They also helped in developing some of the simulation setups and providededitorial corrections while writing the associated manuscripts for publication.Some other research contributions not presented in the thesis but have been publishedduring my PhD program at UBC are listed in Appendix A.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview of Resource Optimization in Cooperative Wireless Systems withPerfect CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.1 Networks with Single Source-Destination Pair: Three-Node Relay Net-work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Networks with Single Source-Destination Pair: Dual-Hop Networkswith Multiple Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.3 Multiuser Cooperation: Networks with Multiple Users and Relays . . 221.1.4 Relay Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.2 Motivations and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 34viTable of Contents1.2.1 Fixed Relay based DF Cooperative Wireless Networks . . . . . . . . 361.2.2 User Cooperation based Cognitive Radio Network (CRN) . . . . . . 391.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 Power Allocation and Relay Selection for DF Cooperative Wireless Net-works with Bounded Channel Uncertainty . . . . . . . . . . . . . . . . . . 452.1 Accomplished Works and Research Contributions . . . . . . . . . . . . . . . 452.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . 472.2.1 Case 1: Perfect CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.2.2 Case 2: Imperfect CSI . . . . . . . . . . . . . . . . . . . . . . . . . . 502.3 Proposed Optimization Framework . . . . . . . . . . . . . . . . . . . . . . . 532.3.1 Centralized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.2 Distributed Implementation . . . . . . . . . . . . . . . . . . . . . . . 552.3.3 Power Limited Networks . . . . . . . . . . . . . . . . . . . . . . . . . 592.4 Joint Relay Selection and Power Allocation . . . . . . . . . . . . . . . . . . 632.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 Admission Control and Power Allocation for Selective Relaying Based Cel-lular Wireless Systems with Imperfect CSI . . . . . . . . . . . . . . . . . . 763.1 Accomplished Works and Research Contributions . . . . . . . . . . . . . . . 773.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . 783.2.1 Problem Formulation with Imperfect CSI . . . . . . . . . . . . . . . 813.3 Proposed Optimization Framework . . . . . . . . . . . . . . . . . . . . . . . 833.4 Admission Control and Power Allocation . . . . . . . . . . . . . . . . . . . . 933.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974 Resource Optimization for Cooperative Cognitive Radio Networks withImperfect CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110viiTable of Contents4.1 Accomplished Works and Research Contributions . . . . . . . . . . . . . . . 1104.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . 1124.2.1 Baseline Scheme: Assuming Estimated CSI to be Perfect . . . . . . . 1174.2.2 Proposed Scheme: Considering Imperfect CSI . . . . . . . . . . . . 1204.3 Optimization Framework for the Proposed Scheme . . . . . . . . . . . . . . 1254.3.1 Optimization Framework for Model-1 : Probabilistic Constraints . . . 1284.3.2 Optimization Framework for Model-2 : Bounded Uncertainty Con-straints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.3.3 Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335 Conclusions and Future Research Directions . . . . . . . . . . . . . . . . . 1425.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.2.1 Resource Optimization for Overlay CRNs with Cooperative RelaysConsidering Imperfect Sensing and CSI . . . . . . . . . . . . . . . . 1455.2.2 Energy Efficient Resource Allocation Schemes with QoS Provisioningunder Channel Uncertainty for HetNets . . . . . . . . . . . . . . . . 1465.2.3 Robust Resource Allocation and QoS Provisioning for Multi-tier Cel-lular Network under Channel and Traffic Demand Uncertainty . . . . 1475.2.4 Resource Optimization for Energy Harvesting Cooperative CellularSystems with Uncertain CSI and Power . . . . . . . . . . . . . . . . 148Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149AppendicesA List of Other Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162viiiList of Tables3.1 Comparison of the proposed admission control algorithm with random userreduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108ixList of Figures1.1 A three-node relay network . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Comparison of outage probabilities of AF and DF schemes with diversity . . 121.3 Comparison of outage probabilities of AF and DF schemes without diversity 141.4 A dual-hop cooperative relay network . . . . . . . . . . . . . . . . . . . . . . 151.5 Comparison of capacities of AF scheme with i.i.d. channels . . . . . . . . . . 181.6 Comparison of capacities of AF scheme with i.n.d. channels . . . . . . . . . 191.7 Typical cooperative cellular network with multiple users . . . . . . . . . . . 231.8 Price-based distributed power allocation algorithm . . . . . . . . . . . . . . . 301.9 Comparison of the worst user rate for the weighted sum rate and max-minrate algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.10 Comparison of the sum capacity for the weighted sum rate and max-min ratealgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.1 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2 Bounded uncertainty region model . . . . . . . . . . . . . . . . . . . . . . . 522.3 Total uplink transmit power versus network throughput for bounded uncer-tainty region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.4 Performance of the distributed algorithm for bounded uncertainty region . . 702.5 Convergence behavior of the distributed algorithm for bounded uncertaintyregion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.6 Performance of power limited networks with channel estimation error radius 732.7 The effect of ignoring channel estimation errors while selecting the best relay 74xList of Figures2.8 The effect of ignoring channel estimation errors while designing resource allo-cation algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.1 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.2 Proof of the proposition that shows for |K(s)| = 1, the proposed inequalityPr.(|K(s)|+1∑j=1Xjpj ≤ γs) ≤|K(s)|+1∏j=1Pr.(Xjpj ≤ γs) holds for independent andpositive RVs, Xj ∈ [0,∞) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.3 Empirical CDF versus the approximations . . . . . . . . . . . . . . . . . . . 893.4 The effect of channel estimation error variance on average transmit power peruser and on the ratio of the number of admitted users and total number ofusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.5 Average transmit source power per user versus target data rate . . . . . . . . 1013.6 The effect of channel estimation errors on energy efficiency and admissioncontrol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.7 Cooperation ratio versus error variance ratio, σ2esr/σ2esd . . . . . . . . . . . . 1033.8 Cooperation ratio versus power ratio, pkmax/psmax . . . . . . . . . . . . . . . 1043.9 The effect of ignoring channel estimation errors while designing power allo-cation algorithms. Solid line: algorithms based on perfect CSI, dashed lines:proposed algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053.10 Performance comparison of our proposed scheme with the schemes developedbased on perfect CSI for lower values of channel estimation error variancesand probability of outage pout requirements. Solid line: algorithms based onperfect CSI, dashed lines: proposed algorithm . . . . . . . . . . . . . . . . . 1074.1 System Model: SUs are operating from small cell PU networks . . . . . . . . 1134.2 Average goodput versus different uncertainty models among the SU-PU chan-nels and SU-SU channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.3 Average goodput versus interference threshold Ith . . . . . . . . . . . . . . . 135xiList of Figures4.4 Effect of probability of satisfying the interference constraints c, bounded chan-nel uncertainty r and error variance σ2e of SU-SU channels, respectively onaverage goodput at different regions of interference threshold, Ith . . . . . . . 1364.5 Probability of violating the interference constraints versus interference thresh-old Ith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.6 The effect of ignoring estimation errors on QoS satisfaction and wastage oftransmit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.7 Probability of selecting the correct relay versus SU-SU channel estimationerror variance σ2e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140xiiList of Abbreviations3GPP : 3rd Generation Partnership Project4G : Fourth Generation of cellular networksAF : Amplify-and-ForwardAP : Access PointAWGN : Additive White Gaussian NoiseBER : Bit Error RateBS : Base StationBW : BandWidthCC : Coded CooperationCDF : Cumulative Distribution FunctionCoMP : Coordinated Multi-Point reception and transmissionCR : Cognitive RadioCRN : Cognitive Radio NetworkCSCG : Circularly Symmetric Complex GaussianCSI : Channel State InformationDF : Decode-and-ForwardeNB : evolved Node BFDD : Frequency Division DuplexGP : Geometric ProgrammingHetNet : Heterogenous Networksi.i.d. : Independent and Identically DistributedxiiiList of Abbreviationsi.n.d. : Independent but Non-identically DistributedKKT : Karush-Kuhn-TuckerLP : Linear ProgrammingLTE : Long Term EvolutionLTE-Advanced : Long Term Evolution-AdvancedMAC : Multiple Access ChannelMIMO : Multiple-Input-Multiple-OutputMRC : Maximal-Ratio CombiningNP-hard : Non deterministic Polynomial time -hardPDF : Probability Density FunctionPU : Primary UserQoS : Quality of ServiceRV : Random VariableSC : Selection CombiningSDP : Semi Definite ProgrammingSNR : Signal-to-Noise RatioSOCP : Second Order Cone ProgrammingSU : Secondary UserTDD : Time Division DuplexUE : User EquipmentxivAcknowledgementsI would like to sincerely thank a number of people who have been involved in my journey ofPhD and in the development of my thesis. First, I would like to offer my deepest gratitudeto my supervisor Professor Vijay K. Bhargava who has supported me throughout my PhDprogram with his patience and knowledge. I must thank him sincerely for providing mewith an excellent atmosphere for doing research. This thesis would not have been possiblewithout his presence, guidance, directions, and constant encouragement.Second, I would like to thank Dr. Mohammad Mamunur Rashid, a senior colleague ofmy lab, for his invaluable suggestions, helpful advice, critical comments and contributionsduring my research.I would also like to thank Professor Robert Schober and Professor Lutz Lampe for servingas the committee members during my PhD qualifying examination and providing valuablefeedback, critical comments and suggestions. I am thankful to Professor Vincent Wongfor serving as a committee member during my PhD departmental exam and providing hisvaluable suggestions and editorial corrections.I am very fortunate and grateful to have a wonderful laboratory (Information Theory andSystems) and great colleagues, who offered me genuine and friendly support and assurance tocarry out my research. I am especially thankful to Rajiv Devarajan and Kundan Kandhwayfor their friendship, generosity, and support during my PhD research. Also many thanks goto Roya Arab Loodaricheh for her contributions, feedback and critical suggestions.I would like to extend my gratitude to my old friend Imtiaz Ahmed and my lab mateHamidreza Boostanimehr for all the fruitful discussions during my PhD program. I am alsoxvAcknowledgementsthankful to them for proofreading my thesis.I am thankful to all my friends at UBC and Vancouver for providing me a great life inVancouver during my PhD program.I am profoundly indebted to my parents, my sisters, and my brother for their love andblessings. They were always supporting me and encouraging me with their best wishes.My heartfelt thanks go to my loving wife, Arno Kamolika, for her continued encourage-ment, empathy, care and sacrifice. She was always there cheering me up and stood by methrough the good times and bad. Without her unconditional love and support, I could nothave completed this thesis.xviChapter 1IntroductionWireless networks have to be designed and deployed with unavoidable constraints on thelimited radio resources such as bandwidth and transmit power. With the booming of thenumber of new wireless users and with the introduction of new wireless services that requirea large bandwidth or data rate, the demand for the limited radio resources is rising exponen-tially. Finding a solution to meet this increasing demand with the available resources is achallenging research problem. The primary objective of such research is to find solutions thatimprove the capacity and utilization of the radio resources that are available to the serviceproviders. Based on the concept of relay channels, cooperative communication1 has beenfound to greatly enhance the performance of a resource-constrained wireless network [2]-[7].Multiple-input-multiple-output (MIMO) systems increase the data rate by increasing thenumber of antennas at the transmitters and receivers. However, the antennas have to beplaced in certain distances to realize spectral diversity of MIMO, which may not be feasiblefor mobile user equipment and small device applications such as sensor networks [8]. Co-operative communication can achieve similar benefits to those of MIMO without the needof multiple antennas at each node. By allowing users to cooperate and relay each other’smessage to destination, cooperative communication also improves the transmission qualityand the coverage [9]. In general, cooperative communications combat destructive channelimpairments, such as multi-path fading, path loss and shadowing [5]-[7]. Because of the lim-ited power and bandwidth resources of the cellular networks and also the multipath fadingnature of the wireless channels, the idea of cooperation is particularly attractive in wireless1The term cooperation was introduced in [1], where the capacity of a three-node relay network is analyzed.1Chapter 1. Introductioncellular networks [10].Cooperative communication also improves energy efficiency by allowing low poweredshort distance multi-hop communication instead of the traditional high powered long dis-tance direct communication. It can save the installation and maintenance cost of the largebase stations (BSs) and lead the way of enabling energy-efficient “green” communication byintroducing low-powered communication, prolonging the battery life of portable devices andreducing the net carbon footprint [11]. For these features, cooperative communication hasbeen considered as one of state-of-the-art technologies for wireless networks and standardssuch as 3GPP Long Term Evolution-Advanced (LTE-A) and IEEE 802.16 [10],[12].Among different cooperation techniques, such as relay based cooperation, BS cooperationand Coordinated Multiple Point (CoMP) transmission and reception, we focus only on relaybased cooperative communication in this thesis. Common protocols or strategies for relaybased cooperation, such as decode and forward (DF), amplify and forward (AF) and codedcooperation (CC) [4],[13]-[15], usually involve two steps of operation. In the first step, auser (called the source node) transmits its message to both the assigned partner (calledthe relay node) and the destination. If the relay node employs a DF scheme, it will firstdecode and regenerate the message and then transmit the message to the destination inthe second step. When the regenerated message is encoded to provide additional errorprotection to the original message, it is referred to as CC. If an AF scheme is employed, therelay simply amplifies the received message and forwards it to the destination in the nextstep. At the destination, the signals from both the source and the relay are then combinedfor detection, using well-known methods such as maximal-ratio-combining (MRC) [16] or byselection combining (SC).The benefits of a relay based cooperation often depend on the channel quality betweenthe source, relay, and destination involved in the transmission. This is because the trans-mission quality of a relayed message is limited by the channel signal-to-noise ratio (SNR).For example, a relay may fail to forward a message reliably to the destination if the received2Chapter 1. Introductionmessage from the source is already corrupted due to poor channel quality. Therefore, coop-eration may not be always useful. In general, when the complete channel state information(CSI) among the source, relay, and destination is known to the system, relays should beselected to forward the source message only if the quality of the channels between source-relay and relay-destination meet certain criteria. Whenever cooperation is beneficial, thenext goal is to allocate the limited resources (e.g., power and bandwidth) appropriately be-tween the source and relay. Therefore, resource allocation among the cooperative nodes canbe formulated as an optimization problem to achieve the maximum possible advantage ofcooperation [17]-[18].In the next section, we present an overview of resource optimization for different typesof cooperative networks assuming availability of perfect CSI. Our objective is to show howthe usage of the available radio resources in these networks can be maximized by usingdifferent optimization techniques. We start with networks with a single source-destinationpair. First, the optimal power allocation problems are studied for the most basic three-node topology using different cooperation schemes, and the capacity or throughput gain isinvestigated. Then, we consider the dual-hop relay networks, which consist of multiple relays,and study the relevant resource optimization problems. Afterwards, we study the resourceallocation problem of a cooperative network with multiple source-destination pairs. Solutionapproaches using both centralized and distributed optimization algorithms are presented.In a cooperative network, relay nodes play a vital role in achieving the intended benefit ofcooperative communication. The performance of the whole network depends on how theserelays are selected for individual communications between source and destination nodes andhow resources are allocated on the links between the relays and the communicating nodes.The relay selection strategies for different networks are also discussed briefly at the end ofthe following section.31.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIUser 1User 2Destination (S)(R)(D)h hhFeedbackF e e d b ac kFigure 1.1: A three-node relay network1.1 Overview of Resource Optimization inCooperative Wireless Systems with Perfect CSIIn this section, our strategy is to start with the problem of resource optimization in asimple cooperative architecture involving one source and one destination node (a singlesource-destination pair) and then progress towards a more general system architecture. Thesimplest topology for a single source-destination pair is the three-node relay network shownin Fig. 1.1; a more generic topology is the dual-hop networks which consist of multiple relays,and is presented in Section 1.1.2.1.1.1 Networks with Single Source-Destination Pair:Three-Node Relay NetworkConsider the three-node relay network shown in Fig. 1.1, where we let user 1 be the sourcenode (S) that intends to transmit a message to the destination (D) while user 2 serves asthe relay node (R).Cooperation requires two time slots to send a message to destination D. In the first time41.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIslot, source S transmits a message XS to both R and D with power PS. The received signalsat R and D can be expressed asXR =√PShSRXS + ZR,XD1 =√PShSDXS + ZD1,(1.1)respectively, where hSR and hSD are the complex channel coefficients for S − R and S −Dlinks, and ZR and ZD1 denote the independent and identically distributed (i.i.d) circularlysymmetric additive white Gaussian noise (AWGN) with zero mean and variance N0.In the second time slot, relay R transmits the message XR received in the first slot withpower PR either by decoding or simply amplifying it. Depending on the cooperative scheme(DF or AF), let Y = f(XR) be the transmitted message in the second slot, where f(·)denotes a function of (·). The received signal at D in the second time slot can be written asXD2 =√PRhRDf (XR) + ZD2, (1.2)where hRD is the channel coefficient between the relay - destination (R−D) pair and ZD2 isthe received AWGN with zero mean and variance N0. In the following analysis, we assumethat these channel coefficients are known (i.e., full CSI is available) to S, R, and D and theAWGN noise has unit variance, i.e., N0 = 1.Quality of Service (QoS), a measure of performance, reflects the transmission quality andthe service availability of a transmission system. For example, channel capacity, SNR, outageprobability or bit-error rate (BER) can be one of the QoS measures. Our objective is toformulate the optimization problem so that the QoS performance of the network is optimalsubject to some resource constraints. In general, the source and the relays have independentpower supplies and therefore, separate power constraint should be used. However, separatesource and relay power budget constraints do not provide useful insight of the total powerconsumption of the system. To understand how the total system or link power is distributed51.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIamong the source and the relay node, we use total power budget as the constrained resourceof the network in this section. The problems can also be solved similarly using separate powerbudget constraints. We consider channel capacity (or, maximum achievable data rate) asthe QoS measure. In the following, the optimization problem is to maximize the channelcapacity given total power constraint, and the solution will provide us the optimal allocationof PS and PR among the cooperative nodes.The solution for optimal power allocation depends on whether the direct S − D link istaken into account. If the messages from both the source and the relay are combined at thedestination for detection, it is referred to as the case with diversity in [19]. For the casewithout direct link or diversity, only the message from the relay (i.e., XD2) is considered. Inthe following, we study optimal power allocation for both the cases.Case 1 - With Direct LinkDF Scheme:First, let us formulate the capacity maximization problem taking the direct link intoaccount to examine the optimal power allocation for the three-node network with DF co-operative strategy. In a two time slot operation, the capacity of the S − R link is givenbyChop1 =12 log2(1 + |hSR|2 PS). (1.3)In the second time slot, the destination combines the messages from both the S − D andR−D links using MRC technique, and the capacity is given by [20]Chop2 =12 log2(1 + |hRD|2 PR + |hSD|2 PS). (1.4)The channel capacity in the DF cooperative scheme is limited by the minimum of the two61.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIhopsCDF, diversity = min{12 log2(1 + |hSR|2 PS),12 log2(1 + |hRD|2 PR + |hSD|2 PS)}. (1.5)Therefore the optimization problem becomes a standard max-min problemmaximize{PS ,PR}min{12 log2(1 + |hSR|2 PS),12 log2(1 + |hSD|2 PS + |hRD|2 PR)}subject to :C1 : PS + PR ≤ P0C2 : PS ≥ 0 , PR ≥ 0.(1.6)If |hRD|2 > |hSD|2 and |hSR|2 > |hSD|2, the capacity is maximized with equal capacityfor the two hops, i.e.,12 log2(1 + |hSR|2 PS) =12 log2(1 + |hSD|2 PS + |hRD|2 PR). (1.7)Since the objective is to maximize the capacity, the first inequality constraint of (1.6) shouldbe met at equality. Using PS + PR = P0 in (1.7), we get the optimal power allocation asPS∗ = P0|hRD|2|hSR|2 + |hRD|2 − |hSD|2(1.8)andPR∗ = P0|hSR|2 − |hSD|2|hSR|2 + |hRD|2 − |hSD|2. (1.9)From (1.8) and (1.9), we see that more power is allocated to the source than to the relay,which is well justified by the fact that PS contributes both to the direct path and to therelay path.However, if |hSD|2 ≥ |hRD|2, we see that the optimal power allocation that maximizesthe objective function in (1.6) is PS = P0 and PR = 0. This result indicates that if the direct71.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIchannel has better quality than any of the S − R or R −D links, it is intuitive to allocateall available power to S alone.AF Scheme:Now let us formulate the problem of optimal power allocation strategy for the AF co-operative scheme. In this case, the message forwarded by the relay contains an amplifiedversion of the noise along with the message transmitted originally by the source. As a result,the SNR at the destination node plays an important role in the optimal power allocationproblem [21].To compute the SNR at destination and the channel capacity for AF scheme, we re-visit(1.1) and (1.2). The received message at destination D forwarded by relay R in the secondtime slot is given byXD2 =√PRhRDf (XR) + ZD2,= (√PR/ρ)hRDXR + ZD2,= (√PR/ρ)hRD(√PShSRXS + ZR) + ZD2,(1.10)where the AF relay normalizes the received message with a normalization factor, ρ given byρ =√E{|XR|2}, (1.11)and re-transmits with power PR. Here, E{·} denotes the statistical expectation operation.The normalization factor, ρ can be obtained asρ =√E{XRXR∗},=√E{(√PShSRXS + ZR)(√PShSR∗XS∗ + ZR∗)},=√PS |hSR|2 E{|XS|2}+√PShSRE{XSZR∗}+√PShSR∗E{XS∗ZR}+ E{|ZR|2}.(1.12)81.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIConsidering E{|XS|2} = 1, E{|ZR|2} = N0 = 1 and properties of AWGN, we obtainρ =√PS |hSR|2 + 1. (1.13)Incorporating (1.13) in (1.10) we obtainXD2 =√PRPS√(PS |hSR|2 + 1)hRDhSRXS +√PR√(PS |hSR|2 + 1)hRDZR + ZD2, (1.14)where the first term is the effective signal term and the last two terms are the effective noiseterm. To obtain the SNR at the destination in the second time slot, we need to calculatethe signal and noise powers from (1.14), which are given byPsignal,D2 = E∣∣∣∣∣∣√PRPS√(PS |hSR|2 + 1)hRDhSRXS∣∣∣∣∣∣2,= PRPS(PS |hSR|2 + 1)|hSR|2 |hRD|2 ,(1.15)andPnoise,D2 = E∣∣∣∣∣∣√PR√(PS |hSR|2 + 1)hRDZR + ZD2∣∣∣∣∣∣2,= PR |hRD|2(PS |hSR|2 + 1)+ 1,(1.16)respectively. Therefore, the SNR at the destination in the second time slot can be expressedasSNRD2 =Psignal,D2Pnoise,D2,= |hSR|2 PS |hRD|2 PR1 + |hSR|2 PS + |hRD|2 PR.(1.17)91.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIThe SNR at the destination in the first time slot can be obtained similarly from (1.1) asSNRD1 =Psig,D1Pnoise,D1,= |hSD|2 PS.(1.18)Therefore, the overall SNR at the destination node and the channel capacity of the twohop AF scheme can be expressed asSNRD =|hSR|2 PS |hRD|2 PR1 + |hSR|2 PS + |hRD|2 PR+ |hSD|2 PS (1.19)andCAF,diversity =12 log2(1 + SNRD), (1.20)respectively.For the AF cooperation, optimizing the channel capacity is equivalent to SNR maximiza-tion. Therefore, the optimal power allocation problem for AF scheme can be formulatedasmaximize{PS,PR}|hSR|2 PS |hRD|2 PR1 + |hSR|2 PS + |hRD|2 PR+ |hSD|2 PSsubject to :C1 : PS + PR ≤ P0C2 : PS ≥ 0, PR ≥ 0.(1.21)Since the objective is SNR maximization, the power budget constraint should met at equalityat the optimal point. Applying Lagrange Multiplier method, the optimal power allocationof (1.21) is obtained asPS∗ =|hSR|2 |hRD|2 P0 + |hSD|2 |hRD|2 P0 + |hSD|2A+√|hSR|2P0+1|hRD|2P0+1|hSR|2 |hRD|2 A(1.22)101.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIandPR∗ =|hSR|2 |hRD|2 P0 − |hSD|2 |hSR|2 P0 − |hSD|2A+√|hRD|2P0+1|hSR|2P0+1|hSR|2 |hRD|2 A, (1.23)where A = |hSR|2 |hRD|2 + |hSD|2 |hRD|2 − |hSR|2 |hSD|2. Optimal power allocation holds ifand only if A > 0 and |hSR|2 |hRD|2 P0 > |hSD|2 |hSR|2 P0 + |hSD|2. Otherwise all availablepower should be allocated to the the source alone. This finding indicates that if S −D linkis better than S−R or R−D link, cooperation is not beneficial. When |hSR|2 ≈ |hRD|2 andboth are sufficiently larger than |hSD|2, the ratio of PS∗ and PR∗ can be approximated asPS∗PR∗≈ |hSR|2 |hRD|2 P0 + |hSD|2 |hRD|2 P0 + |hSD|2|hSR|2 |hRD|2 P0 − |hSD|2 |hSR|2 P0 − |hSD|2, (1.24)which clearly indicates that more power should be allocated to the source than to the relay.Example-1: In this example, we compare the outage probabilities of the AF and DFschemes with equal and optimal power allocation methods, as shown in Fig. 1.2. We considerthat an outage occurs when the achieved rate is less than 1 bit/sec/Hz, i.e., (C < 1) at thedestination. For simplicity, we assume that the relay node is located in the middle of thesource and the destination node and the complex channel coefficients hSR, hSD, and hRD areindependent circularly symmetric complex Gaussian (CSCG) random variables (RVs) withzero mean and variances σ2SR = 1, σ2RD = 1, and σ2SD = 1/(2α), respectively where α = 3 isthe path loss coefficient.From Fig. 1.2, we can see that the optimal power allocation method for DF scheme hasSNR gain of approximately 3 dB over the equal power allocation method at high SNR.The DF scheme performs better than AF scheme at low to moderate SNR (below 15 dB).Although the DF scheme achieves higher capacity [7], it is interesting to note that the outageprobability of the AF scheme outperforms that of the DF scheme when the SNR value issufficiently high (over 15 dB). This is because the DF scheme does not provide additionaldiversity gain since the outage depends on successful decoding at both the relay and the111.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSI0 5 10 15 20 25 3010−410−310−210−1100CapacityOutageProbabilityTotal Power (dB) Direct(No relay)DF optimal powerDF equal powerAF optimal powerAF equal powerFigure 1.2: Comparison of outage probabilities of AF and DF schemes with diversitydestination nodes.Case 2 - Without Direct LinkIn this case, we assume that the direct link does not exist due to heavy blockage and longdistance transmission. Without direct link, the destination node receives only the relayedmessage from R. For DF cooperation, the channel capacity can be written asCDF,without diversity = min{12 log2(1 + |hSR|2 PS),12 log2(1 + |hRD|2 PR)}. (1.25)The capacity maximization problem is thus a max-min problem. The maximum capacity isreached when the capacity of both hops are equal, i.e.,12 log2(1 + |hSR|2 PS) =12 log2(1 + |hRD|2 PR). (1.26)121.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIThe optimal power allocation is obtained with total power budget, i.e., using PS + PR = P0in (1.26) asPS∗ = P0|hRD|2|hSR|2 + |hRD|2(1.27)andPR∗ = P0|hSR|2|hSR|2 + |hRD|2. (1.28)For AF cooperation without diversity, the optimal power allocation is obtained by putting|hSD| = 0 in (1.22) and (1.23) asPS∗ =P01 +√|hSR|2P0+1|hRD |2P0+1(1.29)andPR∗ =P01 +√|hRD|2P0+1|hSR|2P0+1. (1.30)Thus the ratio between PS and PR is given byPS∗PR∗=√|hRD|2 P0 + 1|hSR|2 P0 + 1. (1.31)Example-2: The outage probabilities for AF and DF cooperations without direct link(or, diversity) are compared in Fig. 1.3. For each scheme, we plot the results obtained fromequal and optimal power allocation methods.It is clear that the optimal power allocation method has approximately 3 dB SNR gainover the equal power allocation method for both AF and DF schemes. Since there is nodiversity, the DF scheme performs better than the AF scheme at low to moderate SNR.However, at a very high SNR (over 30 dB), the performances of both the schemes are almostidentical.131.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSI0 5 10 15 20 25 30 3510−410−310−210−1100CapacityOutageProbabilityTotal Power (dB) DF optimal powerDF equal powerAF optimal powerAF equal powerFigure 1.3: Comparison of outage probabilities of AF and DF schemes without diversity1.1.2 Networks with Single Source-Destination Pair: Dual-HopNetworks with Multiple RelaysNow let us consider a dual-hop cooperative network (also known as parallel relay network)consisting of N relay nodes, denoted by Rk, k = 1, 2, · · · , N as shown in Fig. 1.4. Thechannel coefficients from the source S to relay Rk and from Rk to destination D are given byhSk and hkD, respectively. In such networks, S broadcasts its message in the first phase. Inthe second phase, the set of relays {Rk, k = 1, 2, · · · , N} can either transmit simultaneously,or by turns using repetition coding [5]. We assume all the channels are orthogonal and perfectCSI is available to all the nodes. The transmit powers of S and Rk are denoted by PS andPk, respectively.In this sub-section, we focus on the power allocation among the relay nodes only. Thedistribution of power between PS and PR can be determined using similar techniques de-rived in the previous sub-section. The optimization problem is formulated as a capacity141.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSISource (S) FeedbackDestination (D) Relay 1 (R)Relay N (R)Relay 2 (R)hhhhhhXXX ZXFigure 1.4: A dual-hop cooperative relay networkmaximization problem and the total power constraint is imposed on the summation of relaypowers, i.e.,N∑k=1Pk ≤ PR. In the following, we analyze the optimal power allocation for bothAF and DF cooperation.Case 1 - AF SchemeFor the AF scheme, the capacity of the dual-hop network with orthogonal channels is givenby [22]CAF =12 log2(1 + SNRD), (1.32)where the SNR at destination SNRD can be obtained similar to (1.19) asSNRD = PS(|hSD|2 +N∑k=1|hSk|2 |hkD|2 Pk|hSk|2 PS + |hkD|2 Pk + 1). (1.33)The goal is to optimize the power allocation among the relay nodes so that the capacityis maximized. Optimizing CAF is equivalent to maximizing SNRD. For a constant sourcepower PS, we defineγk =√|hSk|2 |hkD|2|hSk|2 PS + 1(1.34)151.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIso that we can writeN∑k=1|hSk|2 |hkD|2 Pk|hSk|2 PS + |hkD|2 Pk + 1=N∑k=1|hSk|2 γ2kPk|hSk|2 + γ2kPk(1.35)and formulate the following optimization problem, given bymaximizePkN∑k=1|hSk|2 γ2kPk|hSk|2 + γ2kPksubject to :C1 :N∑k=1Pk ≤ PRC2 : Pk ≥ 0. ∀k(1.36)It can be easily shown that the optimization problem is convex since the objective func-tion is now a concave increasing function of Pk and the constraints are linear. The solutionsof any convex optimization problem can be easily obtained by available convex programmingalgorithms [23]. Among these algorithms, subgradient methods are the simplest and there-fore, widely used. However, interior-point methods, bundle methods, cutting-plane methodsare also well known. The solution obtained using these algorithms does not provide anyinsight into the optimal power allocation among the relays though. Therefore, we solve theoptimization problem of (1.36) using the Lagrange-dual method [24]. The Lagrangian of theproblem is given byL(Pk,λ, η) =N∑k=1|hSk|2 γ2kPk|hSk|2 + γ2kPk+N∑k=1λkPk − η( N∑k=1Pk − PR), (1.37)where λ = λk ≥ 0, k = 1, 2...N and η are the Lagrange multipliers corresponding to theinequality constraints on the relay power. From the Karush-Kuhn-Tucker (KKT) condi-tions [23], we obtain∂L∂Pk= |hSk|2 γ2k(|hSk|2 + γ2kPk)− |hSk|2 γ4kPk(|hSk|2 + γ2kPk)2+ λk − η = 0 (1.38)161.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIandλkPk = 0. (1.39)By eliminating λk from (1.39), we get(η − |hSk|2 γ2k(|hSk|2 + γ2kPk)− |hSk|2 γ4kPk(|hSk|2 + γ2kPk)2)Pk = 0. (1.40)From (1.40) we see that if Pk 6= 0, then the expression(η − |hSk|2 γ2k(|hSk|2 + γ2kPk)− |hSk|2 γ4kPk(|hSk|2 + γ2kPk)2)must be equal to zero. After simplification, we get the optimal power allocation among therelay nodes asPk∗ =|hSk|2γk[ 1√η −1γk]+, (1.41)where [.]+ denotes the projection onto the feasible set of non-negative orthants. Therefore,the optimal power allocation problem results in the following water-filling (WF) solution [25]:Pk∗ =|hSk|2γk( 1√η −1γk) , if 1√η >1γk0, otherwise.(1.42)The Lagrange multiplier η is chosen to meet the total power constraint of the relay nodes.Also note that relay node Rk is allowed to transmit if and only if γk >√η.Example-3: The capacities of the AF scheme with different number of relays arecompared in Fig. 1.5 and Fig. 1.6. For both figures, we plot the results obtained from equaland optimal power allocation methods. In Fig. 1.5, we assume that all the S−R and R−Dchannel coefficients are i.i.d. CSCG RVs with zero mean and unit variances. Similar to171.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSI1 2 3 4 5 6 7 8 9 1000.511.522.53Capacity(Bits/Sec/Hz)Total Relay Power (Watt) 4 relay equal power4 relay optimal power2 relay optimal power2 relay equal powerFigure 1.5: Comparison of capacities of AF scheme with i.i.d. channelsthe pervious examples, the direct S − D channel coefficient, hSD is assumed to have zeromean and variance σ2SD = 1/(2α). In Fig. 1.6, we assume that S − R and R − D channelcoefficients are independent but not identically distributed (i.n.d) Gaussian RVs, i.e., somerelay channels are better than the others on average.From Fig. 1.5 and Fig. 1.6, it is clear that the achieved rates are much higher for theoptimal power allocation scheme than those of the equal power allocation schemes. As weincrease the number of relays, the capacity increases. However, for the cases where somerelay channels are better than the others (in Fig. 1.6), it is interesting to note that two relayswith optimal power allocation can achieve better rate than that with four relays with equalpower allocation.Case 2 - DF SchemeA relay node can execute the DF scheme if it is able to reliably decode the message receivedfrom the source node [26]. Considering all the N relays can decode the message received181.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSI1 2 3 4 5 6 7 8 9 1000.511.522.53Capacity(Bits/Sec/Hz)Total Relay Power (Watt) 4 relay equal power4 relay optimal power2 relay optimal power2 relay equal powerFigure 1.6: Comparison of capacities of AF scheme with i.n.d. channelsfrom the source node, the capacity of a dual-hop DF relay network with orthogonal channelsusing repetition coding can be written as [5]CDF =1N + 1 log2(1 + |hSD|2 PS +N∑k=1|hkD|2 Pk). (1.43)Let us consider a set of relays denoted by NDF , among the N available relay nodes,which are able to correctly decode the messages transmitted by the source. For optimalpower allocation of the DF scheme, the first task is to find the reliable relay nodes k ∈ NDF .We assume that a relay will be able to execute the DF strategy if it is able to reliablycommunicate with the source at a desired rate rS. Therefore, the relay nodes k ∈ NDF arereliable if the following condition is met1|NDF |+ 1log2(1 + |hSk|2 PS) ≥ rS, (1.44)where |NDF | is the cardinality of the decoding set NDF .191.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIOur goal is to find the optimal power allocation that maximizes the capacity of thenetwork. However, the problem of maximizing the capacity under the relay power constraint,N∑k=1Pk ≤ PR has its dual problem with the objective to communicate with the destinationat the desired rate, rS using minimum relay power P ∗k . The analogous solutions of both theoriginal and its dual problem become identical whenN∑k=1Pk∗ = PR. We use binary relayselection variables xk to select the reliable relays k ∈ NDF and formulate the following dualproblem.minimize{Pk,xk}N∑k=1Pkxksubject to :C1 :1|NDF |+ 1log2(1 + |hSD|2 PS +N∑k=1|hkD|2 Pkxk) ≥ rSxkC2 :1|NDF |+ 1log2(1 + |hSk|2 PS) ≥ rSxk, ∀kC3 : xk ∈ {0, 1}, ∀kC4 :N∑k=1xk = |NDF |C5 : Pk ≥ 0. ∀k(1.45)The formulated optimization problem is a mixed integer problem, which is generally difficultto solve. However, the optimization problem can be re-formulated asminimize{Pk ,xk}N∑k=1Pkxksubject to :C1 : |hSD|2 PS +N∑k=1|hkD|2 Pkxk ≥ cS,NxkC2 : |hSk|2 PS ≥ cS,Nxk, ∀kC3 : xk ∈ {0, 1}, ∀kC4 :N∑k=1xk = |NDF |C5 : Pk ≥ 0, ∀k(1.46)where cS,N = 2(|NDF |+1)rS − 1. The solution of the above optimization problem is to choose201.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIa single relay from the set NDF , i.e., |NDF | = 1. This is due to the fact that a WF solutionfinds the relay which has the best channel gain to the destination node and allocates thetotal relay power to it. This finding signifies that the selective relaying scheme is optimalfor the repetition based dual hop DF strategy with multiple relays. Therefore, the optimalpower allocation among multiple relays becomes a single relay optimization problem, givenby∀k ∈ NDF : minimize{PS ,Pk}PS + Pksubject to :C1 : |hSD|2 PS + |hkD|2 Pk ≥ cS,1C2 : |hSk|2 PS ≥ cS,1C3 : Pk ≥ 0, PS ≥ 0,(1.47)where cS,1 = 22rS − 1. Similar to the three-node case, one can show that cooperation isbeneficial with the relay k if and only if |hSk|2 > |hSD|2 and |hkD|2 > |hSD|2. Otherwise it isoptimal to allocate all the power to the source for the direct transmission. From the KKTconditions, it can be shown that when cooperation is beneficial both C1 and C2 are active2and therefore, the optimal power allocation is obtained using the first and second constraintsof (1.47) with equality, given byP ∗S =cS,1|hSk|2, ∀k ∈ NDF (1.48)andP ∗k =cS,1 − P ∗S |hSD|2|hkD|2. ∀k ∈ NDF (1.49)The optimal total power with relay k ∈ NDF is given byP ∗S + P ∗k = cS,1[1|hSk|2+ 1|hkD|2− |hSD|2|hSk|2 |hkD|2]. (1.50)2A constraint is said to be active if it meets with equality at the optimal point.211.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIThe best relay will be the one among the set of useful relays which minimizes the total powerP ∗S + P ∗k . If the set of useful relaysRU = {k ∈ NDF | |hSk|2 > |hSD|2 , |hkD|2 > |hSD|2}is non-empty, then the optimal relay selection criteria isk∗ = argmink∈RU[1|hSk|2+ 1|hkD|2− |hSD|2|hSk|2 |hkD|2]. (1.51)The capacity of the primal problem for the dual-hop DF scheme is given byCDF =12 log2(|hSD|2 P ∗S + |hk∗D|2 PR). (1.52)1.1.3 Multiuser Cooperation: Networks with Multiple Users andRelaysIn Sub-sections (1.1.1) and (1.1.2), we have analyzed the problem of resource optimizationfor a single source-destination pair. Now we examine the resource allocation problem formultiuser (or multiple source destination pairs) cooperative networks. For example, considerthe case of a cellular network shown in Fig. 1.7, where some fixed relays are deployed to assistthe users located at cell edges for both uplink and downlink transmissions. Since the numberof relays is smaller than the number of users in a practical scenario, each relay is assignedto assist multiple users.To formulate the power allocation problem for multiuser networks we choose two QoSmeasures, namely, the minimum rate of all the users (max-min fairness) or the weightedsum rates of the users (weighted-sum fairness) [27]. Therefore, the problem is to maximizeeither the minimum rate of the users or the weighted sum rates given the total relay poweras constraint. In the following, we will see that both the problems are convex, which can be221.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIRelayUser near BSUser near Cell EdgeBase Station (BS)RelayFigure 1.7: Typical cooperative cellular network with multiple userssolved using centralized or distributed algorithms.System ModelConsider a multiuser network where M source nodes Si, i ∈ {1, 2, ...,M} transmit messagesto their corresponding destination nodes Di, i ∈ {1, 2, ...,M}. There are N relay nodesRk, k ∈ {1, 2, ..., N} in the network. The set of relays assisting the transmission of Si isdenoted by R(Si). The set of sources using relay Rk is denoted by S(Rk). Thus we can writeS(Rk) = {Si |Rk ∈ R(Si)}, (1.53)which means one relay can forward the messages of several users. In this section, we considerthat no direct link exists between any source-destination (Si −Di) pair. We also assume allthe channels are orthogonal and perfect CSI is available to all the nodes.In multiuser cooperative network with the AF scheme, each source Si transmits a messageto its chosen relays in the setR(Si) in the first time slot, and each relay amplifies and forwardsits received message to Di in the second time slot. The transmit powers of the source Si and231.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIits assisting relays Rk ∈ R(Si) are denoted by PSi and P SiRk , respectively. Let us considersource Si broadcasts the message Xi to relays Rk ∈ R(Si) in the first time slot. The receivedmessage at relay Rk can be written asXSiRk =√PSihSiRkXi + ZRk , Rk ∈ R(Si) (1.54)where hSiRk denotes the channel coefficient for the Si − Rk link, and ZRk is the zero meanAWGN with unit variance.In the second time slot, relay Rk amplifies the received message XSiRk and forwards it todestination node Di. The received message at Di is given byXDiRk =√P SiRkρSiRkhDiRkXSiRk + ZDi, Rk ∈ R(Si) (1.55)where hDiRk denotes the channel coefficient for the Rk − Di links, and ZDi is the zero meanAWGN with unit variance. ρSiRk is the normalization factor which can be obtained similar to(1.13) asρSiRk =√E{∣∣XSiRk∣∣2} =√PSi∣∣hSiRk∣∣2 + 1, (1.56)and therefore, (1.55) can be expressed asXDiRk =√√√√P SiRkPSiPSi∣∣hSiRk∣∣2 + 1hDiRkhSiRkXi + ZˆDi, Rk ∈ R(Si) (1.57)where the modified AWGN at Di is denoted by ZˆDi with equivalent variance1 +(P SiRk∣∣hDiRk∣∣2)(PSi∣∣hSiRk∣∣2 + 1) .Assuming that MRC is employed at the destination node, the combined SNR at Di can be241.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIwritten similar to (1.17) and (1.33) asSNRDi =∑Rk∈R(Si)∣∣hSiRk∣∣2 ∣∣hDiRk∣∣2 P SiRkPSi∣∣hSiRk∣∣2 PSi +∣∣hDiRk∣∣2 P SiRk + 1. (1.58)Therefore, rate ri of user Si can be expressed as [28]ri =12 log2(1 + SNRDi) =12 log21 +∑Rk∈R(Si)∣∣hSiRk∣∣2 ∣∣hDiRk∣∣2 P SiRkPSi∣∣hSiRk∣∣2 PSi +∣∣hDiRk∣∣2 P SiRk + 1 . (1.59)Similar to the single source-destination pair case, it can be shown that, for a constant sourcepower PSi, the rate (ri) expression of (1.59) is a concave increasing function with respectto P SiRk , Rk ∈ R(Si). Using this fact, the optimal power allocation for both the max-minfairness and weighted-sum fairness schemes can be calculated as follows.Centralized Power AllocationMax-Min FairnessFor the rate expression given in (1.59), the power allocation problem under max-min ratecan be formulated asmaximizePSiRkminSirisubject to :C1 :∑Si∈S(Rk)P SiRk ≤ PmaxRk , k = 1, 2, ..., NC2 : P SiRk ≥ 0,(1.60)where PmaxRk is the maximum (or total) power of the relay Rk. The set of linear inequalityconstraints with positive variable in (1.60) is compact and nonempty. Hence the optimizationproblem set is always feasible. Moreover, since the objective function is an increasing functionof power, the first inequality constraint should meet with equality at the optimal point. The251.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIconvex optimization problem can be written in its standard form asminimize{PSiRk , t}−tsubject to :C1 : t− ri ≤ 0, i = 1, 2, ...,MC2 :∑Si∈S(Rk)P SiRk ≤ PmaxRk , k = 1, 2, ..., NC3 : P SiRk ≥ 0, t ≥ 0,(1.61)where t is a slack variable. The solution of this problem gives the optimal power allocationamong the relays that maximizes the worst user rate. In the special case, where all the usershare the same set of relays then the rate of all users are equal at optimal point. This iseasily understood from the max-min solution of the single source-destination pair DF case,where the capacity is maximized when the capacity of both hops are equal.Weighted-Sum FairnessThe problem of max-min rate based power allocation is that it degrades the sum ca-pacity (or total network throughput). This is because it tends to improve the performanceof the worst user rate by allocating more power to the poor links. The weighted sum ratemaximization can potentially achieve certain fairness for different users by allocating largeweights to the users in unfavorable channel conditions while maintaining good network per-formance [27]. Let wi denote the weight allocated to user i, the weighted-sum rate powerallocation problem can be formulated asmaximizePSiRkM∑i=1wirisubject to :C1 :∑Si∈S(Rk)P SiRk ≤ PmaxRk , k = 1, 2, ..., NC2 : P SiRk ≥ 0.(1.62)261.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIThe solution of this convex optimization problem gives better network throughput, whichindicates that the weighted-sum rate based power allocation favors users with good channelconditions. However, it does not severely penalize users with bad channel conditions either.In fact, because the objective function (weighted-sum rate) is concave and increasing withrespect to the allocated powers, the increment in rate is lower when the power is high.Therefore to maximize the objective function value, this scheme allocates power to ‘bad’users operating at low SNR.Distributed Power AllocationThe centralized power allocation discussed in the previous subsection may not be alwayspractical and feasible for all systems. It requires large signaling overhead since the CSI ofvarious source-relay and relay-destination links have to be estimated and transmitted toa central processing node, which allocates power based on these estimates. Moreover, thewireless channels are time varying and frequency selective. Therefore, these channel gaininformation need to be constantly updated. This process would require a large amount ofdata to be transmitted to the central processing node using the control channels. Thus,it is very important to devise distributed schemes in which each relay calculates its ownoptimal power allocation based on the information available from neighboring nodes andhence reduces the overall signaling overhead as well as large computational complexity in thecentral processing node. Several distributed power allocation schemes have been discussedin the literature for cooperative communication systems [27],[29],[30] .The centralized optimization problem for the weighted sum rate maximization basedpower allocation is given in (1.62). In the following, we solve the same problem in a dis-tributed manner. The main idea is to separate the original problem in (1.62) into indepen-dent subproblems using Lagrange dual decomposition method [24]. The Lagrangian of the271.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIproblem (1.62) is given byL(P SiRk ,λ) =M∑i=1wiri −N∑k=1λk∑Si∈S(Rk)P SiRk − PmaxRk , (1.63)where λ = [λ1, λ2, ..., λN ] represents the Lagrange multipliers corresponding to the N relaypower budget constraints of (1.62). Using (1.53) we can write the following equationN∑k=1λk∑Si∈S(Rk)P SiRk =M∑i=1∑Rk∈R(Si)λkP SiRk . (1.64)The Lagrangian in (1.63) is rewritten using (1.64) asL(P SiRk ,λ) =M∑i=1[wiri −∑Rk∈R(Si)λkP SiRk]+N∑k=1λkPmaxRk=M∑i=1Li(P SiRk ,λ) +N∑k=1λkPmaxRk ,(1.65)where Li(P SiRk ,λ) corresponds to the ith value of the Lagrangian. The corresponding dualfunction of the Lagrangian (1.65) is given byg(λ) = maximizePSiRk≥0L(P SiRk ,λ). (1.66)This dual function can be obtained by solving M separate subproblems corresponding to Mdifferent users asmaximizePSiRkLi(P SiRk ,λ) = wiri −∑Rk∈R(Si)λkP SiRksubject to :C1 : P SiRk ≥ 0. Rk ∈ R(Si)(1.67)Let the solution of the optimization problem in (1.67) be L∗i (λ), which corresponds to the281.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIoptimal value of Li(P SiRk ,λ).Since the original problem given in (1.62) is convex, strong duality holds and we canobtain the optimal power allocation by solving the dual problem given byminimizeλg(λ)subject to :C1 : λk ≥ 0, k = 1, 2, ..., N.(1.68)Using the optimal solution of (1.67), the dual problem can be rewritten asminimizeλM∑i=1L∗i (λ) +∑Nk=1 λkPmaxRksubject to :C1 : λk ≥ 0. k = 1, ..., N(1.69)The distributed power allocation algorithm aims to solve (1.67) and (1.69) sequentially.Let the Lagrange multiplier λk ≥ 0 denote the price per unit power at relay k. ThereforeλkP SiRk represent the total price that user i must pay for using PSiRk power at each relayRk ∈ R(Si). Thus each user tries to maximize the weighted rate minus the total price it hasto pay given the price coefficients at the relays.Since the dual function g(λ) is differentiable, the problem can be solved iteratively bygradient projection method. Di, the receiver of user i, finds its optimal power at eachiteration t with given λ from the assisting relays Rk ∈ R(Si) asP SiRk∗(λ[t+ 1]) = argmaxwiri −∑Rk∈R(Si)λk[t]P SiRk(λ[t]), (1.70)which is unique due to strict concavity. Di then sends this information to all the relaysallocated to user i. Based on P SiRk∗ values received from destinations, the relays update their291.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSI ff fi ff ff flffi !fiffi !"ffi !#Optimum Price Allocation to each Receiver Optimum Power Allocation to each relay 1λ( )k ikR R Sλ∈∑( )iki kSRS S RP∈∑ikSRP11( )iiSRS S RP∈∑( )iNi NSRS S RP∈∑kλNλ1( )kkR R Sλ∈∑( )k MkR R Sλ∈∑1kSRPMkSRPFigure 1.8: Price-based distributed power allocation algorithmdual variable λk as follows:λk[t+ 1] =λk[t]− δPmaxRk −∑Si∈S(Rk)P SiRk∗(λ[t])+, (1.71)where δ is the small positive step size parameter. This update equation (1.71) can beinterpreted as the price (λk) change by Rk depending on the requested power levels from itsusers. The price increases when the power allocated to the relay exceeds its maximum limit.This price-based distributed algorithm requires message passing only between each receiverand its assisting relays. The message passing continues until satisfying stopping criterion isreached. The overall structure of the distributed power allocation is shown in Fig. 1.8.Example-5: In Fig. 1.9, we compare the worst user rate for both the weighted sum301.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSI1 2 3 4 5 6 7 8 9 1000.10.20.30.40.50.60.7Total Relay Power (Watt)WorstUserRate(Bits/Sec/Hz) Weighted sum rate optimal powerEqual powerMax-min rate optimal powerFigure 1.9: Comparison of the worst user rate for the weighted sum rate and max-min ratealgorithmsand max-min fairness algorithms for a network with number of source nodes, M = 8 andnumber of relays, K = 4. The equal power allocation among the relays is also included forreference. We see that the worst user obtains the best rate under the max-min fairness andthe worst rate under weighted sum scheme with equal weight coefficients. Fig. 1.10 showsthat the weighted sum scheme results in maximum capacity or network throughput. Themax-min scheme targets to improve the performance of the worst user, which results in asignificant loss in sum capacity of the whole network.1.1.4 Relay SelectionIn the multiuser AF scheme, we have assumed that the relays are assigned a priori to powerallocation. Joint relay selection and power allocation problem is a mixed integer optimizationproblem and is in general very difficult to solve optimally. Therefore, in most of the literature,the problem is divided into two subproblems [31] and sub-optimal solutions are provided.311.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSI1 2 3 4 5 6 7 8 9 101234567Total Relay Power (Watt)SumCapacity(Bits/Sec/Hz) Weighted sum rate optimal powerEqual powerMax-min rate optimal powerFigure 1.10: Comparison of the sum capacity for the weighted sum rate and max-min ratealgorithmsThe relays are chosen assuming fixed powers (e.g., equal power allocation) at the relays.Once the relays are selected, techniques mentioned in previous sections of this chapter areused to allocate power to the selected relays. In this section, some of the existing relayselection algorithms are presented.In spite of assuming fixed power at the relays, some of the relay selection problems arestill NP-hard and optimum solution can only be achieved through exhaustive search [28].Several greedy algorithms have been proposed in the literature for relay selection in DFsystems. The general strategy is to select the best relay or a group of relays which satisfy acertain performance improvement over the direct source to destination link [32]. In [33], arelay is selected based on the amount of time taken to deliver a fixed amount of messages.Two optimal algorithms are provided, where in the first one the relays are adaptively selectedand in the second one the optimal number of relays is pre-determined without relying on thespecific network realization. Alternatively, in [30], a relay is selected if it offers an increase in321.1. Overview of Resource Optimization in Cooperative Wireless Systems with Perfect CSIcapacity compared to source destination channel. In [34], a distributed algorithm is presentedto select the ‘best’ relay that provides the best end-to-end path between the source and thedestination. Each relay is assigned a timer with initial value inversely proportional to eitherhk = min(|hSk|2|, |hkD|2), or hk =2|hSk|2|hkD|2|hSk|2 + |hkD|2. The relay with highest value of hk isconsidered to be the one with the best end-to-end path between source and destination. Allrelays start their timers simultaneously with different initial values which depend on thechannel realizations. The relay, whose timer expires first (i.e., the one with maximum hk),starts relaying the message from the source as soon as its time expires. All the relays listento the other relays and therefore, know whether some other relay has started forwarding themessage or not. When they recognize that a particular relay has started transmission, theydo not forward the source message to the destination.The power allocation problem (1.60) for multiuser AF systems can be re-formulated asa joint relay selection and power allocation problem asmaximize{PSiRk ,xSiRk}minSi12 log2(1 +∑Rk∣∣hSiRk∣∣2 ∣∣hDiRk∣∣2 P SiRkPSixSiRk∣∣hSiRk∣∣2 PSi +∣∣hDiRk∣∣2 P SiRkxSiRk + 1)subject to :C1 :∑Si∈S(Rk)xSiRkPSiRk ≤ PmaxRk , k = 1, 2, ..., NC2 : xSiRk ∈ {0, 1}, i = 1, 2, ...,M, k = 1, 2, ..., NC3 :∑Rk∈R(Si)xSiRk = xSimax, i = 1, 2, ...,MC4 : P SiRk ≥ 0, i = 1, 2, ...,M, k = 1, 2, ..., N(1.72)where xSimax is the total number of relays allocated to user Si. This is a mixed integer non-convex optimization problem and hence a unique analytical solution cannot be achieved.Therefore, a suboptimal solution is proposed in [35] by relaxing the integrality constraintson xSiRk and allowing them to take all values in the interval [0, 1]. The relaxed optimizationproblem is convex and can be solved using convex programming techniques. A rounding331.2. Motivations and Objectivestechnique is then used to obtain sub-optimal solution for original optimization problem fromthe solution of relaxed optimization problem.Joint relay selection and resource optimization for cooperative systems is a challengingresearch problem. The research goal is to develop low-complexity schemes that treats allusers fairly and can be implemented feasibly without the requirement of significant additionalsystem resources.1.2 Motivations and ObjectivesIn the previous section, we have studied some resource optimization problems for relay-basedcooperative wireless networks assuming the availability of perfect CSI [36]. In formulatingthese optimization models, we have taken into consideration the architecture and variouscooperation schemes deployed in these networks along with the constraints on the availableradio resources such as bandwidth and transmit power. Although the bulk of the previoussection has been devoted to analyzing different optimization based solution approaches forpower allocation once relays are selected, we have also highlighted different relay selectionmethods that are proposed in the literature. Jointly optimizing relay selection and resourceallocation is in general a combinatorial problem and often handled as separate subprob-lems [31],[35]. However, computationally efficient solution for joint resource optimizationproblem in different types of cooperative wireless networks is challenging and needs furtherinvestigation.Throughout the previous section, we have assumed that perfect CSI is available forresource allocation. However, this assumption is too optimistic for cooperative wirelesssystems with multi-hop transmissions due to channel fluctuations, limited feedback capacity,and channel estimation and quantization errors [37]-[38]. The errors in channel estimationdepends on how the CSI is obtained and how precise the estimation is [39]. For example,usual error comes from the Gaussian noise in time-division duplex (TDD) systems and finite341.2. Motivations and Objectivescapacity constraint of feedback channel in frequency-division duplex (FDD) systems [40].Also, in a cooperative network, the CSI becomes outdated due to channel fluctuations [41]when CSI of different source-relay links need to be transmitted to a central processing node,such as BS, access point (AP), or Evolved Node B (eNB). Although resource allocationproblems for cooperative systems have been extensively studied in the literature [42]-[45],little research has been done under practical conditions specially considering the uncertaintyin CSI [37]-[38]. The schemes developed assuming estimated CSI to be perfect are non-robustand sensitive to errors when CSI is actually imperfect. The optimal resource allocationschemes designed assuming perfect CSI is available may fail to provide QoS to users andresult in a decrease of the overall system performance [46]-[49]. In the rest of the thesis, westudy the impact of the channel estimation error on optimal resource allocation and howQoS can be provided to the cooperative users under imperfect CSI.To provide QoS under channel uncertainty, robust resource allocation schemes need to bedesigned. The robust resource allocation schemes can be developed by applying probabilisticoptimization (stochastic optimization) or by worst-case optimization based approaches [23],depending on how the uncertainty in CSI is modeled. When some statistics of the CSIerror are known, probabilistic approach can be applied to optimize certain stochastic QoSmeasures such as outage probability. In a probabilistically-constrained approach, the QoScan be guaranteed with a certain probability. On the other hand, worst case optimizationbased approach can be applied when the uncertainty in CSI belongs to a predefined boundedregion. When no statistical knowledge is available about the error or the worst case errorvalue is known, this technique can be applied. In worst case optimization based approach,guaranteed QoS can be provided to the users. However, the resource allocation results areoften very conservative for this approach. In this thesis, we develop robust optimizationframeworks taking different channel uncertainty models into account.Our objective is to develop robust and computationally efficient resource allocationschemes with QoS provisioning for relay based cooperative systems. In this thesis, we con-351.2. Motivations and Objectivessider two types of cooperative system model: (1) fixed relay based DF cooperative wirelessnetworks and (2) user cooperation based cognitive radio network (CRN). Based on thesesystem models, the rest of this section is divided into two subsections as follows.1.2.1 Fixed Relay based DF Cooperative Wireless NetworksMost of the existing works, for example, [42]-[45] on resource allocation for fixed relay basedcooperative systems, assume that perfect CSI is available at all the nodes. We refer to [46]-[49] in motivating the importance of considering the imperfectness in CSI while allocatingresources. Ignoring the imperfectness in CSI will have undesirable effects, such as increasingthe outage leading to performance degradation of the system in terms of throughput orpower requirement [47]-[49]. Therefore, it is important to consider the imperfectness in CSIwhile designing resource optimization problems in cooperative wireless systems. The resourceoptimization schemes designed based on perfect CSI might fail to satisfy the QoS constraintin the presence of a very small error in CSI. The results of such designs are retransmissionsand wastage of power [50]. Therefore, resource allocation algorithms that can work robustlyunder channel uncertainty are highly desired for practical implementation.Taking channel uncertainty into account, centralized robust power allocation algorithmsfor AF relay networks have been developed in [37]-[38]. The robust optimization problem isformulated in [37] as a quasi-convex problem and solved using the bisection method via a se-quence of convex feasibility problems in the form of second-order cone programming (SOCP).However, with no relay selection, the spectral efficiency is reduced when orthogonal trans-mission is considered. Joint relay selection and power allocation strategy for multiuser AFnetworks assuming perfect CSI has been studied, for example, in [35]. The solution of [35]is non-robust to channel uncertainty and provides optimal relay power only when CSI isperfectly known. Since the source power is assumed to be constant, the solution gives no in-sight about the overall optimal transmit power requirement. For a single source-destinationpair, distributed power allocation for parallel regenerative DF scheme with partial CSI has361.2. Motivations and Objectivesbeen addressed in [29]. Relay selection and power allocation strategies for multiuser DFcooperative networks have been proposed in [44]-[45] assuming perfect CSI is available atall the nodes. However, relay selection and power allocation problem for DF cooperativenetworks considering imperfect CSI has not been addressed well in the literature. Besides,only centralized solutions, as pursued in [44]-[45] may not be adequate for distributed net-works and for all applications. In a centralized solution, the instantaneous wireless channelgain estimations between various source-relay links have to be transmitted to a central pro-cessing node using control channels. This process will incur a large signaling overhead andincrease processing complexity at the central node. Therefore, distributed solutions that canscale well with the size of the networks are desired along with efficient centralized schemesapplicable for centralized networks.Most of the existing literature on robust wireless relay networks assume that CSI erroris bounded and lies in an ellipsoidal uncertainty set [37]-[38] while developing resource al-location schemes based on worst-case optimization based approach. However, the boundeduncertainty region modeling is unsuitable for the cases when CSI error is unbounded, e.g.,when the CSI is obtained from a training sequence, the channel uncertainty becomes un-bounded Gaussian [55]. Moreover, the worst case optimization often leads to overly con-servative results [56]. As a result, probabilistically constrained optimization has become aneffective means to provide the robustness when channel uncertainty is modeled as a randomprocess [56]-[57].Most of the existing works in literature consider channel capacity as the QoS measure.Under imperfect CSI, the channel ergodic (Shannon) capacity is unknown in general anda lower bound of the ergodic capacity can be obtained assuming estimation error as inde-pendent Gaussian noise [48],[58]. However, lower bound of the ergodic capacity is not auseful QoS measure for systems that require QoS provision over each transmission framein slow fading channels with imperfect information [6], [59]. This is because in slow fad-ing with CSI error, a channel outage occurs whenever the transmit data rate exceeds the371.2. Motivations and Objectivesactual channel capacity. Lower bound of ergodic capacity [48] fails to capture the effect ofchannel outage events [59]. In such scenarios, the performance can be evaluated in terms ofoutage probability [6], [56]. Therefore, to provide some QoS to each user over each trans-mission frame and to capture the effect of outage-probability, probabilistic QoS constraintsare needed [56]-[57], [60].The performance of the cooperative networks also depends on how the relays are selectedfor communications between source and destination nodes [36], [44]-[45]. In most cases, afixed number of relays are chosen a priori before resource allocation is performed. However,with half-duplex relaying protocol, ‘always relaying’ may lead to certain loss in terms ofenergy efficiency or spectral efficiency, if the source-relay or relay-destination channel is verypoor [7], [51]-[54]. Selective relaying, i.e., relaying only when it is beneficial for the system,can solve this problem and improve the performance significantly [6], [51]-[54].The shortcomings in the existing literature motivate us to pursue the following objectivesin the context of resource allocation for fixed relay based DF wireless systems:1. Design bounded uncertainty model when no statistics is known about the channelestimation error.2. Develop robust power allocation and relay selection scheme with guaranteed QoS pro-visioning for DF cooperative networks considering bounded channel uncertainty.3. Develop efficient centralized as well as distributed power allocation algorithms scalablewith respect to size of the networks.4. When some statistics are known about the channel estimation error, develop robustpower allocation scheme for a selective relaying based cooperative system.5. Develop joint admission control and power allocation schemes with QoS provisioningthat are computationally efficient and suitable for implementation in a large scalenetwork.381.2. Motivations and ObjectivesThe first three objectives are tackled in Chapter 2. The last two objectives are addressedin Chapter 3. The contributions made regarding these objectives are clarified in the beginningof each chapter.1.2.2 User Cooperation based Cognitive Radio Network (CRN)In the previous sub-section, we discussed the motivation and research objectives on resourceoptimization for fixed relay based cooperative wireless network. In the following, the researchscopes on resource optimization in user cooperation based CRN are addressed.Heterogeneous networks (HetNets), where small cells are deployed inside regular macrocells, have attracted significant attention from many mobile network operators. In the smallcells, for example, micro, pico or femto cells, the licensed frequency band is efficiently used,which improves the spectral-efficiency of the overall network [61]. Small cells with low pow-ered BSs and smaller coverage area also reduce the cost and energy consumption comparedto the regular macro cells [61]-[62]. These interesting properties make HetNets suitable for3GPP LTE-A and next generation of wireless communications (4G and beyond) [62].Cognitive radio (CR) can further improve the spectrum utilization efficiency by oppor-tunistically accessing the licensed spectrum [63]. In a CRN, an unlicensed user (secondaryuser) can transmit over the licensed (primary user) band, as long as the interference receivedin the primary user (PU) band is within the limit set by the regulatory bodies [64]. IEEE802.22 standard group has been working to develop standards for CRs so that SUs canopportunistically access the spectrum allocated to the television broadcasting bands. CRsoperating from small cell PU networks impose more challenges over regular macro cells, andthese issues have not been addressed well in the literature. For example, the secondary users(SUs) can lie in two different small cell networks and can be far apart. In such a scenario,the PU receivers can be much closer than the CR receivers, and hence any transmission byCR users in their respective PU band can create significant interference. Moreover, the SUsmay not be able to transmit to each other’s PU band as the cells are geographically far apart391.2. Motivations and Objectivesand may not be reachable directly with the available power budget. The long distance andhigh power direct CR transmission can cause excessive interference to the other intermediatePU receivers reusing the same frequency in other cells. We mainly focus on such CRNs inwhich direct communication between the CR source and destination is not possible. Theresearch question is how to establish the communication links between the SUs operatingfrom different small cell PU networks keeping the interference in the PU bands within limits.To establish a connection between the SUs, user cooperation techniques can be intro-duced [64]-[67]. An intermediate SU, operating from a different PU network than those ofthe source and destination CRs, can help by operating as a relay. For example, the sourcecan transmit a message to the relay in the PU band of the destination CR without exceedingthe interference limit. The relay can forward that message to the destination in the PU bandof the source CR or destination CR depending on the received interference level. With thehelp of user cooperation, low-powered transmissions are possible in the channels used by thesmall cell PUs keeping the interference below some given threshold. However, it is necessaryto optimally allocate the resources not only to improve the performance of the CRN but alsoto keep the interference introduced in the PU bands below the prescribed limit.Most of the resource allocation schemes for cooperative CRNs are developed withoutmuch consideration of the availability of the CSI knowledge [64]-[72]. Power allocation andrelay selection schemes are proposed in [66]-[72] for relay based CRN assuming that perfectCSI is available. This assumption is too optimistic for any CRN due to channel uncertainty,fluctuations, estimation errors, outdated CSI, and quantization effects [47],[73]-[75]. Al-though the instantaneous channels between SU transmitter and receiver (SU-SU channels)can be estimated within some mean squared error, it is really difficult, if not impossible,to estimate the instantaneous channels between a SU transmitter and PU receiver (SU-PUchannels) [76]. One possible approach of the estimation can be through the knowledge ofthe PUs’ pilot symbols, provided both the PUs and SUs are sharing the same frequencyband and the PU systems are TDD [47], [76]. In [75], power allocation schemes are proposed401.2. Motivations and Objectivesfor CRN with single source-destination pair considering the uncertainty in the channels be-tween SUs and PUs. For multi-antenna CR systems, resource allocation algorithms havebeen proposed in [47] with imperfect knowledge between SU-PU channels, where a boundedchannel uncertainty model is assumed. Similar bounded uncertainty models are used in [73]-[74], where robust beamforming methods are developed for multiuser CR networks. In [55],stochastic robust beamforming methods for multi-antenna CRN are proposed, where QoS forSUs are satisfied in a probabilistic manner. However, the robust resource allocation schemesproposed in [47],[73]-[75] do not consider any relay based communication or user cooperation.Recently, some research works have been done on cooperative cognitive network consider-ing the estimation error in SU-PU channels [77]-[79]. In [77], the outage performance analysisis presented for a cognitive relay network with imperfect CSI between the SU transmitter andPU receivers. Cooperative beamforming and relay selection schemes are developed in [78]to maximize the received SNR while keeping the interference to the PU band within lim-its. The impact of imperfect CSI of the interference links is analyzed in [79]. Results showthat when CSI is imperfect, the transmission of the SU may cause harmful interferences onthe PU band. Although [47]-[74], [77]-[79] consider bounded channel uncertainties amongSU-PU channels, they ignore the estimation errors in the SU-SU relay channels. Moreover,the bounded uncertainty model requires worst case optimization, which often leads to overlyconservative results [56]. Probabilistic optimization models can provide more flexibility indesigning resource optimization schemes [56]-[57].Very few literature on CRN has considered the estimation errors in both SU-SU channelsand SU-PU channels. Considering both errors, a lower bound on ergodic sum capacityand a power allocation algorithm have been derived for a cognitive multiple access channel(MAC) in [58]. It is particularly important to consider the estimation errors in the SU-SU channels, when resources are optimized over each transmission frame [57], [59]. Robustresource allocation for CRNs operating from small cell PU networks is important for nextgeneration communication systems [63] and has not been well addressed in the literature.411.3. Thesis OutlineBesides, while allocating resources in such a network, the QoS provision for some delay-sensitive users or applications has to be taken into account.The shortcomings in the existing literature motivate us to pursue the following objectivesin the context of resource allocation for cooperative cognitive radio networks:1. Develop user cooperation based robust relay selection and power allocation scheme fora CRN, where the SUs try to communicate with each other from different small cellPU networks.2. Provide QoS to each SU while satisfying the interference constraints on the PU bandsconsidering different channel uncertainty models in the SU to SU channels and SU toPU channels.These objectives are tackled in Chapter 4. The contributions made regarding theseobjectives are clarified in the beginning of Chapter 4.1.3 Thesis OutlineThe rest of this thesis is organized as follows:• In Chapter 2, we propose power allocation and relay selection algorithms that canwork robustly under channel uncertainty for a DF cooperative wireless network. Weconsider a bounded channel uncertainty model assuming no statistical knowledge isavailable about the CSI error. We provide guaranteed QoS to all the users while mini-mizing the uplink transmit power consumption of the network by applying worst-caseoptimization approach. In this optimization framework, equivalent convex formula-tions are derived for the robust optimization problems that are often difficult to solvein their original forms. First, we develop centralized as well as distributed power al-location algorithms scalable with respect to size of the networks. In the distributedimplementation of our proposed scheme, each relay can calculate its own power based421.3. Thesis Outlineon the information available from neighboring nodes. Next, we consider the case ofpower constrained networks, where the objective is to provide QoS in the presence oflimited power budgets on source and relays. The robust optimization problem is refor-mulated accordingly and efficient solutions are provided. Finally, we propose a jointrelay selection and power allocation algorithm, which can be solved semi-distributedly.We propose best relay selection scheme and provide closed form analytical solution forrelay selection and power allocation problem. Numerical results show the effectivenessof the proposed algorithms and demonstrate the implications of ignoring channel es-timation errors while developing relay selection and power allocation algorithms withQoS provisioning.• Unlike the bounded uncertainty model developed in Chapter 2, we consider that somestatistics are known about the channel estimation error . Assuming CSI error tobe unbounded Gaussian, we propose robust power allocation and admission controlschemes for providing probabilistically constrained QoS in selective relaying based DFcooperative cellular systems. The proposed schemes are robust against imperfect CSIin slow fading while optimizing the total uplink transmit power in these networks.At first, we derive a novel closed-form power allocation solution for the optimizationproblem, where the objective is to minimize total uplink transmit power while satisfyingthe probabilistic QoS measures for a given number of admitted users. This is achievedby approximating the probabilistic optimization problem into a convex deterministicform and then by deriving closed form analytical solution for power allocation usingKKT conditions. The closed-form property of power allocation solution allows us laterto develop a very low-complexity suboptimal algorithm for joint admission controland power allocation in the presence of imperfect CSI and selective relaying. Wealso conduct comprehensive simulation experiments to demonstrate the effectivenessof our proposed schemes and to highlight the benefits gained from considering channelestimation errors in resource allocation for cooperative cellular systems.431.3. Thesis Outline• In Chapter 4, we develop robust resource allocation schemes for a CRN, where theSUs try to communicate with each other from different small cell PU networks. Usercooperation technique is considered for communication among the SUs since PUs arein close proximity and there are tight interference constraints on the PU bands. Powerallocation and relay selection schemes are optimized with the provision of QoS toeach SU considering different channel uncertainty models. We incorporate the channeloutage events, which have resulted from the imperfect CSI under slow fading channelsin our resource optimization algorithms. We maximize the system goodput of the CRNwhile satisfying the interference constraints of the PU bands both probabilisticallyand for the worst case scenario. The original probabilistic optimization problem isapproximated and transformed into a convex deterministic form and a closed formanalytical solution for power allocation is derived. The closed form power allocationsolution helps us to develop an efficient relay selection scheme based on Hungarianalgorithm. Simulation results reveal the effectiveness of our proposed schemes and theimplications of ignoring the imperfectness among different channels when developingresource allocation algorithms for CRNs.• Conclusions and possible future research directions are discussed in Chapter 5.44Chapter 2Power Allocation and Relay Selectionfor DF Cooperative WirelessNetworks with Bounded ChannelUncertaintyThe accomplished works and research contributions in this chapter are briefly described inthe following.2.1 Accomplished Works and Research Contributions(1) In this chapter, we consider a multiuser (multiple source and single destination) coopera-tive wireless network where some relays are deployed to assist the users that need cooperationusing half-duplex DF strategy [5]. At first, we propose an optimal power allocation schemethat minimizes total uplink transmit power consumption of the network while meeting thedata rate requirements of each user assuming that the available CSI is perfect.(2) Next, we propose a robust power allocation scheme with guaranteed QoS provision-ing assuming no statistical knowledge is available about the CSI error. We consider thatCSI error lies in some bounded uncertainty region and apply worst-case optimization basedapproach to obtain the robust power allocation solution. We show that the equivalence ofthe robust optimization problem is convex, which can be efficiently solved. Centralized as452.2. System Model and Problem Formulationwell as distributed power allocation algorithms scalable with respect to size of the networksare developed. In the distributed implementation of our proposed scheme, each relay cancalculate its own power based on the information available from neighboring nodes and hencecan reduce the overall signaling overhead as well as processing complexity in the central pro-cessing node. We also consider the case of power constrained networks where the objectiveis to provide QoS in the presence of limited power budgets on source and relays. The robustoptimization problem is reformulated accordingly and efficient solutions are provided.(3) Finally, we develop a joint relay selection and power allocation algorithm that pro-vides guaranteed QoS under channel uncertainty. An efficient semi-distributed algorithm isproposed, in which the power allocation problem is solved at the relays and the best relayis selected at the destination (central processing node). Closed form analytical solutions areprovided for the best relay selection and power allocation problem. We conduct extensivesimulation experiments to evaluate the performance of the proposed schemes considering dif-ferent system parameters and compare the results with existing algorithms. Our proposedalgorithms are designed to provide guaranteed QoS to all users under imperfect CSI in com-parison to the relay selection and power allocation algorithms derived for perfect CSI cases,for example, in [35],[44]-[45].The rest of the chapter is organized as follows. The system model, relaying protocoland the channel estimation error model are described and the power allocation problems areformulated in Section 2.2. Section 2.3 proposes the optimization framework and discussescentralized and distributed power allocation methods. Section 2.4 formulates the joint relayselection and power allocation problem and describes the solution approach. Numericalresults are presented in Section 2.5.462.2. System Model and Problem FormulationUser (m) p$%& Destination (d) Relay k (m)Relay L (m)'()*+ ,-./01 2p3%&456789p:User (1)User (M)Relay 1 Relay K ;<=>? @ABCDEF GH IJB KBLMKGHNFBIMO PFBQ RSTUVWXYZ[\]^_BCBLIBK QBCDEOMQPFBQ RFigure 2.1: System model2.2 System Model and Problem FormulationConsider a multiuser cooperative network as shown in Fig. 2.1, where a set of fixed relaysR = {1, 2, ..., k, ..., K} are deployed to assist the users that need cooperation. The set ofusers are denoted by U = {1, 2, ..., m, ...,M}, where each user m ∈ U transmits its messageto the destination (a central processing node). Depending on the architecture of the network,the central processing node can be a BS, AP, or eNB. Since the number of relays is muchsmaller than the number of users in a practical scenario due to cost and space constraints,each relay is assigned to assist multiple users. We consider the case of half-duplex repetitionbased DF cooperative diversity [5], where orthogonal transmission is assumed for all the usersand relays. Repetition based cooperation requires simpler coordination and synchronization;however, cooperation is useful in terms of power requirement or system throughput if a useris assisted by smaller number of relays [45]. The set of relays that are able to decode themessages for a user m ∈ U is denoted by R(m). A potential relay can fully decode themessage if the received SNR at the relay is greater than some threshold value [44]. The472.2. System Model and Problem Formulationquality of the channel between the source and the relay determines whether the receivedSNR is above that threshold. U(k) denotes the set of users, whose messages can be decodedby relay k. We assume that the size of the decoding sets R(m) and U(k) are chosen a priori.Repetition based DF protocol consists of two transmission phases. In the first phase, userm transmits signal xm to both destination d and to the relays with power pm. The receivedsignal at relay k and at destination d is given byyk =√pmhm,kxm + zk,yd1 =√pmhm,dxm + zd1 ,(2.1)respectively. Here hm,k and hm,d are the channel coefficients for the user m to relay k anduser m to destination d links, respectively. zk and zd1 denote the i.i.d. circularly symmetricAWGN with zero-mean and unit variance.The chosen relays k ∈ R(m) decode and forward the received signal in the second phaseto the destination with power pk,m. The received signal from a relay k at destination d isgiven byyd2 =√pk,mhk,dxk + zd2 , (2.2)where hk,d is the channel coefficient between relay k and destination d.The CSI is obtained at the receivers (central processing node and relays) usually bysending pilot symbols (a training sequence). The transmitters (sources) usually obtain theCSI via feedback channel or from previous received signals, exploiting the channel reciprocityin TDD [39]-[40].2.2.1 Case 1: Perfect CSIIn this sub-section, we assume that the available CSI is perfect. Our first goal is to providethe solution for the power allocation problem considering the idealistic scenario [44]-[45].For repetition based DF relaying, a relay k will be in the set R(m) if the instantaneous482.2. System Model and Problem Formulationmutual information between a user m and relay k is greater than some fixed target raterm [44], i.e.,1Lm + 1log2(1 + gm,kpm) ≥ rm, ∀k ∈ R(m) (2.3)where Lm is the total number of relays in the set R(m) and gm,k = |hm,k|2 is the channelpower gain between user m and relay k link. Therefore, the minimum source power pmrequired for guaranteed decoding at any relay k ∈ R(m) can be written aspmmin = argmaxk∈R(m)(2(Lm+1)rm − 1gm,k). (2.4)With cooperation, the instantaneous mutual information between user m and destination dis given by [5]Im =1Lm + 1log2(1 + gm,dpm +Lm∑k=1gk,dpk,m), (2.5)where gm,d = |hm,d|2 and gk,d = |hk,d|2 are the channel power gains between user m todestination d and relay k to destination d, respectively.Our objective is to develop an algorithm that minimizes the uplink transmit power con-sumption of the network taking each user’s target data rate as the QoS constraint. Hence,by utilizing the available power at the relays, this power efficient design can save the batterylife of the source and prolong its lifetime. The optimization problem can be written asminimize{pm,pk,m}M∑m=1(pm +∑k∈R(m)pk,m)subject to :C1 : Im ≥ rm, ∀mC2 : pm ≥ pmmin , ∀mC3 : pk,m ≥ 0, ∀k ∈ R(m), ∀m.(2.6)492.2. System Model and Problem FormulationUsing (2.5), C1 of (2.6) can be written asgTmpm ≥ cm, ∀m (2.7)where cm =(2(Lm+1)rm − 1), gm = [gm,d g1,d ... gLm,d]T and pm = [pm p1,m ... pLm,m]T . Hereg1,d and gLm,d denote the channel power gains between relay 1 to destination d and relay Lmto destination d, respectively and p1,m and pLm,m denote the transmission power of relay 1and relay Lm, respectively for user m. Thus, the optimization problem (2.6) becomesminimize{pm,pk,m}M∑m=1(pm +Lm∑k=1pk,m)subject to :C1 : gTmpm ≥ cm, ∀mC2 : pm ≥ pmmin , ∀mC3 : pk,m ≥ 0, ∀k ∈ R(m), ∀m(2.8)which is a linear program (LP). Note that cooperation is beneficial for user m with relayk ∈ R(m) if and only if gk,d > gm,d and gm,k > gm,d. Otherwise, direct transmission isoptimal for user m. Problem (2.8) is always feasible. The problem is compact and nonemptydue to the linear inequality constraints with positive variables, which can be solved efficientlyusing any LP solver [23].2.2.2 Case 2: Imperfect CSILet us consider a more practical scenario where the CSI is not known perfectly at the re-lays and at the destination. We assume that no statistical knowledge is available about thechannel estimation error [39]-[40],[46]. We consider that the channel estimation error liesin some bounded uncertainty region, where the shape and the size of the region depend onthe physical phenomena that produces the region, e.g, Gaussian noise from the estimationprocess and errors from the quantization of the channel estimate. It can be viewed as a502.2. System Model and Problem Formulationdeterministic modeling of imperfect CSI considering the worst case scenario [46]. For exam-ple, the uncertainty region associated with the Gaussian noise in the estimation process is acircle. For quantization process in channel estimation, the region becomes a polytope aroundthe estimated channel. Due to the joint effect of Gaussian noise and quantization process,the shape of the region can be found from the hyper-convolution of the two considered re-gions [39]-[40]. For simplicity, we assume that the uncertainty region is convex, however, onecan deal with the non-convex region by taking its convex hull [40]. The complex channelcoefficient with uncertainty can be written as h = hˆ + , where is the random estimationerror which determines how far the actual channel h can extend in both real and imaginarypart from the estimated value hˆ.The channel power gain, g = |h|2 is a convex function for each of the channel coefficients,h. Therefore, the channel power gain with uncertainty can be written asg = (hˆ+ )(hˆ + )∗ = gˆ + δ, (2.9)where gˆ = hˆhˆ∗ is the estimated channel power gain and δ = ∗ + 2<(hˆ∗) is the uncertaintyin estimation. Here, <(·) is the real part of (·). For each of the channel coefficients that liesin some bounded uncertainty region, the channel power gain g, lies on a set of line segmentgiven byg ∈ L = {gˆ + δv | − 1 ≤ v ≤ 1}. (2.10)To determine the worst possible case, we consider the maximum error given by || = max.Note that the line segment, L, is generally asymmetric around the estimated gain gˆ for 6= 0.This is due to the fact that ∗ is a positive random variable which makes the distributionof δ asymmetric even when has a symmetric distribution. For the worst case scenario, wetake the maximum absolute uncertainty, δmax = max∗max + 2|<(hˆ∗max)|, as the uncertainty512.2. System Model and Problem FormulationActual ChannelCoefficienthChannel Gaing=|h|`Figure 2.2: Bounded uncertainty region modelon both sides of gˆ, and re-write the channel gain asg ∈ L = {gˆ + δmaxv | − 1 ≤ v ≤ 1}, (2.11)where the estimated channel power gain gˆ is the mid point of the set of line segment L. Thebounded uncertainty model is shown in Fig. 2.2.Therefore, for bounded uncertainty region, (2.3) can be re-written as1Lm + 1log2(1 + gm,kpm) ≥ rm, ∀k ∈ R(m) (2.12)where gm,k ∈ Lm,k = {gˆm,k + δmaxm,kv | − 1 ≤ v ≤ 1} is the channel power gain with uncer-tainty between user m and relay k link. The instantaneous mutual information expressionunder channel uncertainty between user m and destination d is given byIm2 =1Lm + 1log2(1 + gm,dpm +Lm∑k=1gk,dpk,m), (2.13)522.3. Proposed Optimization Frameworkwhere gm,d ∈ Lm,d = {gˆm,d+δmaxm,dv | −1 ≤ v ≤ 1} and gk,d ∈ Lk,d = {gˆk,d+δmaxk,dv | −1 ≤v ≤ 1} are the channel power gains between user m and destination d and relay k anddestination d links, respectively.Taking this error model into account, next, we re-formulate the optimization problem(2.8) asminimize{pm,pk,m}M∑m=1(pm +Lm∑k=1pk,m)subject to :C1 : Im2 ≥ rm, ∀m, gm,d ∈ Lm,d, gk,d ∈ Lk,dC2 : pm ≥cmgm,k, ∀k ∈ R(m), ∀m, gm,k ∈ Lm,kC3 : pk,m ≥ 0, ∀k ∈ R(m), ∀m(2.14)The first constraint (C1) is the QoS constraint, which guarantees the target data rate rm ofuser m under channel uncertainty and C2 provides the minimum source power pm requiredfor guaranteed decoding at any relay k ∈ R(m).2.3 Proposed Optimization FrameworkIn this section, we develop optimization framework for the formulated optimization problemin the previous section. Our goal is to develop the equivalent convex reformulation of theoptimization problem (2.14) under channel uncertainty and then propose an efficient solutionprocedure.2.3.1 Centralized SolutionSimilar to (2.7), the first constraint of (2.14) can be re-written asgTmpm ≥ cm, ∀m (2.15)532.3. Proposed Optimization Frameworkwhere gm lies in some ellipsoid given bygm ∈ Em = {gˆm +Amu | ||u||2 ≤ 1}, (2.16)in which gˆm is the mean of gm and Am = diag(δmaxm,d, δmax1,d , ..., δmaxLm,d) is a diagonalmatrix, which determines the extension of uncertainty around the estimated channel powergain gˆm. If Am = 0, it indicates that gm is known perfectly. To satisfy (2.15), we need toguarantee the QoS at the worst case. Therefore, (2.15) can be expressed asinf{gTmpm | gm ∈ Em} ≥ cm, ∀m⇒ inf{(gˆm +Amu)Tpm | ||u||2 ≤ 1} ≥ cm, ∀m⇒ gˆTmpm + inf{uTATmpm | ||u||2 ≤ 1} ≥ cm, ∀m⇒ gˆTmpm − ||ATmpm||2 ≥ cm, ∀m.(2.17)where the last step is obtained using Cauchy-Schwarz inequality. Note that infimum of aset, inf{·} captures the essence of worst case optimization. To provide the minimum sourcepower pm required for guaranteed decoding at any relay k ∈ R(m), we can write the sourcepower constraint using (2.12) asinf{pmgm,k | gm,k ∈ Lm,k} ≥ cm, ∀k ∈ R(m), ∀m⇒ inf{(gˆm,k + δmaxm,kv)pm | − 1 ≤ v ≤ 1} ≥ cm, ∀k ∈ R(m), ∀m⇒ gˆm,kpm − δmaxm,kpm ≥ cm, ∀k ∈ R(m), ∀m(2.18)since |δmaxm,k | = δmaxm,k .Thus, under channel uncertainty, the robust optimization problem of (2.8) can be ex-542.3. Proposed Optimization Frameworkpressed as the following SOCP problem:minimize{pm,pk,m}M∑m=1(pm +Lm∑k=1pk,m)subject to :C1 : gˆTmpm − ||ATmpm||2 ≥ cm, ∀mC2 : pm ≥cmgˆm,k − δmaxm,k, ∀k ∈ R(m), ∀mC3 : pk,m ≥ 0, ∀k ∈ R(m), ∀m(2.19)which can be efficiently solved using any convex optimization algorithm [23].Unlike problem (2.8) which is always feasible, the robust problem (2.19) is feasible if andonly if gˆk,d > δmaxk,d , gˆm,d > δmaxm,d and gˆm,k > δmaxm,k . In the rest of the chapter, we assumethat the feasibility conditions are satisfied. Also note that cooperation is beneficial for userm with relay k ∈ R(m) if and only if gˆk,d − δmaxk,d > gˆm,d − δmaxm,d and gˆm,k − δmaxm,k >gˆm,d − δmaxm,d . Otherwise, no cooperation is optimal for user m.2.3.2 Distributed ImplementationNow we propose a method to distribute the computation of the centralized power allocationscheme (2.19) among the relays, so that each relay minimizes its own power and power of theusers assisted by it. The motivation behind distributed approach is to reduce the signalingoverhead and processing complexity at the central processing node. Although single (best)relay selection policy is optimal for DF system with orthogonal transmission when perfectCSI is available [26],[45], we start investigating the distributed power allocation policy underchannel uncertainty assuming that each user m can be assisted at most by two relays fromthe set R(m).For convenience of implementation, the users are subdivided into two groups: the setof users assisted by a single relay is denoted by US = {1, 2, ..., i, ..., I} and the set of usersassisted by two relays is denoted by UD = {1, 2, ..., j, ..., J}, where, US ∪ UD = U and552.3. Proposed Optimization FrameworkI +J = M . The set of users US and UD, assisted by relay k is denoted by US(k) and UD(k) ,respectively . Consider for a particular user j ∈ UD(k), the assisting set of relays is denotedby Kˆj = {k1j, k2j}, where k1j and k2j are the first and the second relay that assist user j. Fordistributed implementation, the objective of the problem (2.19) can be equivalently writtenasminimize{px}K∑k=1∑i∈US(k)(pi + pk,i) +∑j∈UD(k)(xj + pk,j). (2.20)Here pi is the source power of user i ∈ US(k), pj is the source power of user j ∈ UD(k),xj = pj for k = k1j and xj = 0 for k = k2j , and px ∈ {pi, pj, pk,i, pk,j, xj}. For the distributedimplementation, problem (2.19) can be re-written asminimize{px}K∑k=1{∑i∈US(k)(pi + pk,i) +∑j∈UD(k)(xj + pk,j)}subject to :C1 : gˆTi pi − ||ATi pi||2 ≥ ci, ∀i ∈ US(k), ∀k,C2 : gˆTj pj − ||ATj pj ||2 ≥ cj , ∀k ∈ R(j), ∀j,C3 : pi ≥cigˆi,k − δmaxi,k, ∀i ∈ US(k), ∀k,C4 : pj ≥cjgˆj,k − δmaxj,k, ∀j ∈ UD(k), ∀k,C5 : pk,i, pk,j = 0, ∀i /∈ US(k), ∀j /∈ UD(k), ∀k,(2.21)where gˆi = [gˆi,d gˆk,d]T , pi = [pi pk,i]T , Ai = diag(δmaxi,d , δmaxk,d) and ci = 22ri − 1. Similarlygˆj = [gˆj,d gˆk1j ,d gˆk2j ,d]T , pj = [pj pk1j ,j pk2j ,j]T , Aj = diag(δmaxj,d , δmaxk1j ,d , δmaxk2j ,d) andcj = 23rj − 1.The relays are coupled in the second constraint of (2.21) for each j ∈ UD, which gives a setof J coupling constraints involving the variables pk1j ,j and pk2j ,j. These coupling constrainsdetermine how much power is shared between the coupled relays. Not only the users thatuse more than one relay but also the 2-norm term in the second constraint of (2.21) make thedistributed implementation complicated. The problem can be decoupled and each relay can562.3. Proposed Optimization Frameworkindependently minimize its own power if we relax the second constraint of (2.21) by replacingthe 2-norm term with 1-norm. Since our goal is worst case optimization, the approximation||ATj pj||2 ≤ ||ATj pj ||1, (2.22)will give us an upper bound, where||ATj pj||1 = |δmaxj,dpj|+ |δmaxk1j ,dpk1j ,j|+ |δmaxk2j ,dpk2j ,j|, (2.23)and hence, the solution will be suboptimal.With the norm approximation (2.22), the problem (2.21) can be solved iteratively ina distributed manner using primal or dual decomposition algorithms [24]. Here, we usethe primal decomposition algorithm and separate the problem (2.21) into K independentsubproblems. The convergence proof of the primal decomposition approach can be foundin [24]. The distributed algorithm can be written as follows:Distributed Power Allocation Algorithm• Initialization: set iteration, t = 0 and the shared variable, q = [q1, ..., qj , ...qJ ]T = 0.The shared variable fixes the allocation of power between the coupled relays.1. Solve the subproblems for each relay k:572.3. Proposed Optimization Frameworkminimize{px}[∑i∈US(k)(pi + pk,i) +∑j∈UD(k)(xj + pk,j)]subject to :C1 : gˆTi pi − ||ATi pi||2 ≥ ci, ∀i ∈ US(k),C2 : bˆTj yj − ||ETj yj||1 ≥ dj, ∀j ∈ UD(k),C3 : pi ≥cigˆi,k − δmaxi,k, ∀i ∈ US(k),C4 : pj ≥cjgˆj,k − δmaxj,k, ∀j ∈ UD(k),C5 : pk,i ≥ 0, ∀i ∈ US(k),C6 : pk,j ≥ 0, ∀j ∈ UD(k),(2.24)where bˆj = [gˆj,d gˆk1j ,d], yj = [pj pk1j ,j], Ej = diag(δmaxj,d , δmaxk1j ,d), dj = −qj + cj fork = k1j andbˆj = [gˆk2j ,d 0], yj = [pk2j ,j 0], Ej = diag(δmaxk2j ,d, 0), dj = qj for k = k2j.The shared variable qj fixes the allocation of power between the shared relays k1j andk2j, respectively for the user j ∈ UD(k). The subproblem in (2.24) is convex and canbe solved as SOCP problem when qj is fixed [23].2. Solve the master problem at the destination:minimize{q}K∑k=1φk(q), (2.25)where φk is the optimal value of the subproblem for relay k. The master problem canbe solved iteratively using the following updates:q(t + 1) = q(t)− µ(t) [λ1(t)− λ2(t)] , (2.26)where λ1(t) = [λk11 ... λk1j ... λk1J ]T , λ2(t) = [λk21 ... λk2j ... λk2J ]T and µ(t) is anappropriate small positive step size parameter. Here λk1j and λk2j are the optimal582.3. Proposed Optimization FrameworkLagrange multipliers associated with the coupled constraints for the user j ∈ UD(k) inthe two subproblems for k = k1j and k = k2j , respectively .3. Set t = t + 1 and go to step (1) until the desired stopping criterion is achieved.From (2.24)-(2.26), we can see that the relays become coupled for the users that areserved by two relays and require coordination and message passing between them whichresults in increased overhead. Employing more than one relay increases the overhead andimplementation complexity further which negates the possible benefit of distributed imple-mentation. Therefore, even under channel uncertainty, the best strategy for DF scheme withorthogonal transmission is to select a single (best) relay for each cooperative user. The single(best) relay selection strategy is discussed in Section (2.4).2.3.3 Power Limited NetworksMost of the wireless systems are resource-limited where the source and the relays have astrict maximum power budget. In the previous section, we have assumed that the sourcesand relays are able to transmit as much power as possible to meet the rate requirementsof users. However, with the additional constraints on source and relay power, some of thetarget rates may not be achievable even with the maximum power budget of the sourceand relays. Therefore, it may be impossible to serve all the users at their desired QoSrequirements with limited power budgets. To overcome this issue, several new approachescan be used; for instance, the rate requirement, rm for some users can be relaxed or someusers can be dropped. The latter scenario requires a joint admission control and powerallocation algorithm which is combinatorially hard [43] and is addressed in Chapter (3) indetails. Alternatively, we can change the QoS constraint to power budgets and reformulatethe optimization problem as a max-min rate or weighted sum rate maximization problemdiscussed in Section 1.1.3 of Chapter 1. The power constraints for the sources and relays in592.3. Proposed Optimization Frameworka power limited network can be written aspm ≤ pmmax , ∀m∑m∈U(k)pk,m ≤ pkmax , ∀k ∈ R(m)(2.27)where pmmax and pkmax are the maximum available power at the user equipment m and relayk.Next, we reformulate the optimization problem (2.19) taking the power budgets given in(2.27) into account.Max-Min Rate Based Power AllocationAn important issue in a QoS policy is the fairness among the users. Max-min fairness is aQoS policy, where the performance of the worst user(s) in the network is guaranteed, i.e., itaims to maximize the minimum rate over all the users. Max-min rate based power allocationproblem considering imperfect channel knowledge can be formulated asmaximize{pm,pk,m}min{m=1,...,M}rmsubject to :C1 : 0 ≤ pm ≤ pmmax , ∀mC2 :∑m∈U(k)pk,m ≤ pkmax , ∀k ∈ R(m)C3 : pk,m ≥ 0, ∀k ∈ R(m), ∀m(2.28)where the instantaneous mutual information (achievable data rate) rm for user m can bewritten as a function of SNR, γm asrm =1Lm + 1log2(1 + γm). (2.29)602.3. Proposed Optimization FrameworkWith the approximation in (2.22), the SNR is expressed asγm = gˆTmpm − ||ATmpm||1. (2.30)To make sure that cooperation is indeed beneficial, we can add the following additionalconstraints:rm ≥ rdirect, ∀mpm ≥ pmin ∀k ∈ R(m), ∀m(2.31)where,rdirect = log2(1 + γdirect) (2.32)is the rate achievable by direct transmission with γdirect = gˆm,dpmmax − δmaxm,dpmmax . We setpmmin aspmmin =( γdirectgˆm,k − δmaxm,k), ∀k ∈ R(m), ∀m (2.33)i.e., we set the minimum power required for guaranteed decoding at any relay k ∈ R(m) foruser m such that cooperation is beneficial. It guarantees that a cooperative source alwaysuses less power than a non-cooperative source. If pmmax is less than pmmin , no relay ischosen since direct transmission is optimal in that scenario. However, with these additionalconstraints, the optimization problem (2.28) can become infeasible. Therefore, we solve theproblem (2.28) without the additional constraints and check the optimal power allocationsolution. If r∗m ≤ rdirect or p∗m ≤ pmmin , we remove that user from cooperating group sinceno cooperation is optimal for that user.Centralized SolutionSimilar to problem (2.19), the max-min rate based power allocation problem (2.28) isfeasible if and only if gˆk,d > δmaxk,d , gˆm,d > δmaxm,d and gˆm,k > δmaxm,k . It can be easilyshown that the formulated problem is convex since the objective function rm is a concavefunction of pm and the constraints are linear. The solution can be easily obtained using theefficient convex programming algorithms [23].612.3. Proposed Optimization FrameworkDistributed ImplementationA distributed approach to solve problem (2.28) can be implemented as follows. For theconsidered problem, maximizing rm is equivalent to maximizing γm for pm, pk,m ≥ 0. Afternorm approximation, γm becomes separable and hence, the objective function is triviallyseparable at each relay node. Since each relay has a maximum power budget, the relaypower constraint is inherently decoupled. If a user is served by two relays, the source powerof that user can be optimized by any of the two relays which can be assigned a priori.Weighted Sum Rate Maximization Based Power AllocationOne of the downsides of max-min rate based power allocation is that it degrades the totalthroughput as it tends to improve the performance of the worst user rate by allocating morepower to the poor channels. To improve the total throughput, instead of using max-min rate,we can use sum rate or weighted sum rate maximization as the objective. The weighted sumrate maximization can potentially achieve certain fairness for different users by allocatinglarge weights to the users in unfavorable channel conditions while maintaining good networkperformance [27]. Let wm denote the weight allocated to user m, the weighted-sum ratepower allocation problem with imperfect CSI can be formulated similar to (2.28) asmaximize{pm,pk,m}M∑m=1wmrmsubject to :C1 : 0 ≤ pm ≤ pmmax , ∀mC2 :∑m∈U(k)pk,m ≤ pkmax . ∀k ∈ R(m)C3 : pk,m ≥ 0, ∀k ∈ R(m), ∀m(2.34)The optimization problem (2.34) is feasible if and only if gˆk,d > δmaxk,d , gˆm,d > δmaxm,d andgˆm,k > δmaxm,k . The problem (2.34) is compact and nonempty due to the linear inequalityconstraints with positive variables. Since, the objective function is a concave and increasing622.4. Joint Relay Selection and Power Allocationfunction of pm, the problem is convex and hence can be efficiently solved.2.4 Joint Relay Selection and Power AllocationIn this section, we develop an efficient algorithm for joint relay selection and power allocationproblem. Our objective is to minimize the uplink transmit power with QoS provisioning toeach user. The joint relay selection and power allocation problem can be formulated asminimize{pm,pk,m,xk,m}M∑m=1(pm +∑k∈R(m)xk,mpk,m)subject to :C1 : gˆTmp′m − ||ATmp′m||2 ≥ xk,mcxm, ∀mC2 : pm ≥cxmxk,mgˆm,k − δmaxm,k, ∀k ∈ R(m), ∀mC3 : xk,m ∈ {0, 1}, ∀k ∈ R(m), ∀mC4 :∑k∈R(m)xk,m = xmax, ∀mC5 : 0 ≤ pk,m ≤ xk,mpmax, ∀k ∈ R(m), ∀m(2.35)where xk,m is the binary relay selection variable as indicated in C3. C4 indicates the maximumnumber of relays that a user can use from the set R(m) and is denoted by xmax, andcxm = 2(xmax+1)rm − 1. C1 is the QoS constraint, which guarantees the SNR of each userwith the selected xmax relays. In the SNR expression, p′m is obtained by incorporating therelay selection variable in pm as p′m = [pm p1,mx1,m ... pk,mxk,m ... pLm,mxLm,m]T . The lastconstraint (C5) makes pk,m = 0 if xk,m = 0, i.e., that particular relay k ∈ R(m) does notassist user m. On the other hand, if xk,m = 1, relay k ∈ R(m) is selected to assist user mwith pk,m which is less than some maximum power pmax. Note that if xmax = Lm, then all therelays in R(m) are selected to assist user m and the problem reduces to the centralized powerallocation problem (2.19). The problem (2.35) is a mixed integer non-linear optimizationproblem which is very difficult to solve in this form.632.4. Joint Relay Selection and Power AllocationTo provide an efficient solution to the joint optimization problem, we propose best (sin-gle) relay selection for each user. Note that in Section 1.1.2 of Chapter 1, it has been shownthat for repetition based DF relay networks with orthogonal transmission, best (single) relayselection scheme is optimal when perfect CSI is available. Moreover, from the distributedpower allocation algorithm given in (2.24)-(2.26), we have concluded that employing morethan one relay for each user increases the overhead and implementation complexity. There-fore, we restrict the number of relays per user to be one, i.e., xmax = 1. If a single relayselection scheme is used, each relay is able to optimize its own power independently andseparately in a distributed manner, without any message passing or coordination with theother relays. For single relay selection we use the fact thatM∑m=1pm +∑k∈R(m)xk,mpk,m =K∑k=1∑m∈U(k)xk,m (pm + pk,m) , (2.36)and re-write the optimization problem in order to distribute the computation among therelays asminimize{pm,pk,m,xk,m}K∑k=1∑m∈U(k)xk,m (pm + pk,m)subject to :C1 : gˆTmp′m − ||ATmp′m||2 ≥ xk,mc1m, ∀m ∈ U(k), ∀kC2 : pm ≥c1mxk,mgˆm,k − δmaxm,k, ∀m ∈ U(k), ∀kC3 : xk,m ∈ {0, 1}, ∀m ∈ U(k), ∀kC4 :∑k∈R(m)xk,m = 1, ∀mC5 : 0 ≤ pk,m ≤ xk,mpmax, ∀m ∈ U(k), ∀k(2.37)where, c1m = 22rm−1. The problem (2.37) is still a mixed integer non-linear problem which isvery difficult to solve in this present form. Moreover, the set of relays R(m) are coupled viaconstraint C4, which makes the distributed implementation further complicated. To solvethis problem (2.37), we ignore constraint C4 and assume xk,m = 1 for all the relays in the642.4. Joint Relay Selection and Power Allocationset R(m), and propose a semi-distributed algorithm as follows.In our proposed scheme, each relay k considers itself as the only relay in the set R(m) foruser m and therefore, independently optimizes the overall transmit power requirement forthat user and sends the information to the central processing node. Based on the optimalpower allocation solution, the central processing node selects the best relay for user m.Without constraint C4 and assuming xk,m = 1, the optimization problem (2.37) can bedistributed at each relay k as∀k ∈ R : minimize{pm,pk,m}∑m∈U(k)(pm + pk,m)subject to :C1 : gˆTmpm − ||ATmpm||2 ≥ c1m, ∀m ∈ U(k)C2 : pm ≥c1mgˆm,k − δmaxm,k, ∀m ∈ U(k)C3 : pk,m ≥ 0, ∀m ∈ U(k)(2.38)which is a SOCP problem and can be solved efficiently using convex optimization algo-rithms [23]. The optimal source and relay powers p∗m and p∗k,m obtained from (2.38) are sentto the central processing node by all the relays. The central processing node chooses onlythe best (single) relay for user m using the following metricargmink∈R(m)(p∗m + p∗k,m). (2.39)Note that this step takes care of constraint C3 and C4 of (2.37), and hence, problem (2.38)-(2.39) and (2.37) are equivalent. Therefore, the optimal solution obtained by solving (2.38)-(2.39) is identical to the original joint relay selection and power allocation problem (2.35)when best relay is selected for each user.However, the optimal solution obtained by solving (2.38)-(2.39) does not provide any in-sight into relay selection policy. In order to gain some insight and also to obtain a closed formanalytical solution for best relay selection and power allocation, we use the approximation652.4. Joint Relay Selection and Power Allocationin (2.22) and re-write the left hand side of the first constraint of (2.38) asgˆTmpm − ||ATmpm||2,≥ gˆTmpm − ||ATmpm||1,= (gˆm,d − δmaxm,d)pm + (gˆk,d − δmaxk,d)pk,m(2.40)With the approximation in (2.40), problem (2.38) can be re-written as∀k ∈ R : minimize{pm,pk,m}∑m∈U(k)(pm + pk,m)subject to :C1 : (gˆm,g − δmaxm,d)pm + (gˆk,d − δmaxk,d)pk,m ≥ c1m, ∀m ∈ U(k)C2 : pm ≥c1mgˆm,k − δmaxm,k, ∀m ∈ U(k)C3 : pk,m ≥ 0, ∀m ∈ U(k)(2.41)The Lagrangian of the optimization problem in (2.41) can be written asL(·) = ∑m∈U(k)(pm + pk,m)+∑m∈U(k)λm,1[c1m − {(gˆm,d − δmaxm,d)pm + (gˆk,d − δmaxk,d)pk,m}]+∑m∈U(k)λm,2( c1mgˆm,k − δmaxm,k− pm)− ∑m∈U(k)λk,mpk,m(2.42)where λm,1, λm,2 λk,m are the Lagrange multipliers associated with the first, second andthird inequality constraints of (2.41), respectively. The KKT conditions for the Lagrangianfunction in (2.42) are given byλ∗m,1[c1m − {(gˆm,d − δmaxm,d)p∗m + (gˆk,d − δmaxk,d)p∗k,m}]= 0, ∀m ∈ U(k) (2.43a)λ∗m,2( c1mgˆm,k − δmaxm,k− p∗m)= 0, ∀m ∈ U(k) (2.43b)λ∗k,mp∗k,m = 0, ∀m ∈ U(k) (2.43c)662.4. Joint Relay Selection and Power Allocation(gˆm,d − δmaxm,d)p∗m + (gˆk,d − δmaxk,d)p∗k,m ≥ c1m, ∀m ∈ U(k) (2.43d)p∗m ≥c1mgˆm,k − δmaxm,k, ∀m ∈ U(k) (2.43e)p∗k,m ≥ 0, ∀m ∈ U(k) (2.43f)λ∗m,1, λ∗m,2, λ∗k,m ≥ 0, ∀m ∈ U(k) (2.43g)∂L∂p∗m= 1− λ∗m,1(gˆm,d − δmaxm,d)− λ∗m,2 = 0, ∀m ∈ U(k) (2.43h)∂L∂p∗k,m= 1− λ∗m,1(gˆk,d − δmaxk,d)− λ∗k,m = 0. ∀m ∈ U(k) (2.43i)When cooperation is beneficial, p∗k,m > 0, and from (2.43c) we can conclude that λ∗k,m = 0.Therefore, from (2.43i), we obtainλ∗m,1 =1gˆk,d − δmaxk,d, (2.44)where λ∗m,1 > 0 for gˆk,d > δmaxk,d . Therefore, the first constraint of (2.41) must be active andmet with equality at optimal point.Using (2.44) in (2.43h) we obtainλ∗m,2 = 1−gˆm,d − δmaxm,dgˆk,d − δmaxk,d. (2.45)We see that λ∗m,2 > 0 if gˆk,d − δmaxk,d > gˆm,d − δmaxm,d , which is true when cooperation isbeneficial. Note that cooperation is beneficial with the relay k if and only if gˆk,d − δmaxk,d >gˆm,d− δmaxm,d and gˆm,k− δmaxm,k > gˆm,d− δmaxm,d . Therefore, when cooperation is beneficial,optimal power allocation is obtained using the first and second constraints of (2.41) withequality:p∗m =c1mgˆm,k − δmaxm,k, ∀m ∈ U(k) (2.46)andp∗k,m =c1m − p∗m(gˆm,d − δmaxm,d)gˆk,d − δmaxk,d, ∀m ∈ U(k). (2.47)672.5. Numerical ResultsThe best relay k∗ for user m will be the one among the set k ∈ R(m) which minimizesthe total power p∗m + p∗k,m. If the set of relaysK(m) = {k ∈ R(m) | (gˆk,d−δmaxk,d) > (gˆm,d−δmaxm,d), (gˆm,k−δmaxm,k) > (gˆm,d−δmaxm,d)}is non-empty, then the best relay selection criteria isk∗ = argmink∈K(m)[p∗m + p∗k,m]= argmink∈K(m)[ c1mgˆm,k − δmaxm,k+c1m − p∗m(gˆm,d − δmaxm,d)gˆk,d − δmaxk,d]= argmink∈K(m)[ 1gˆm,k − δmaxm,k+ 1gˆk,d − δmaxk,d− gˆm,d − δmaxm,d(gˆm,k − δmaxm,k)(gˆk,d − δmaxk,d)](2.48)2.5 Numerical ResultsWe have conducted extensive simulations considering different system parameters to evaluatethe performance of the proposed optimization framework. For the purpose of simulation, weconsider a cooperative wireless network, as shown in Fig. 2.1, with 10 users and 4 relays.Frequency-flat Rayleigh fading channels are assumed in all simulations. For simplicity, weassume that the channel coefficients between source to relay and relay to destination arei.i.d. CSCG RVs with zero mean and unit variance. The source to destination channelcoefficients are i.i.d. CSCG RVs and distributed as CN (0, 2−α), where α = 3 is a constant(path loss coefficient). We would like to emphasize that the assumed channel model is chosenfor demonstration purposes and is not critical to the performance of the proposed resourceallocation algorithms.Same target data rate is assumed for all users, and the uncertainty region associated witheach of the channels are considered to be circular with radius r. The results are averagedover 10,000 independent channel realizations.In Fig. 2.3, total uplink transmit power consumption versus the network throughputis shown. We compare our proposed approach with two heuristic schemes: (1) the relay682.5. Numerical Results2 3 4 5 6 7 8 9 101520253035Network throughput (bits/sec/Hz)Totaluplinktransmitpower(dB) proposed selectionbest source-relay linkbest relay-destination linkFigure 2.3: Total uplink transmit power versus network throughput for bounded uncertaintyregionthat minimizes the source power, i.e., the relay is chosen based on the best source-relaylink considering estimation error, and (2) the relay that minimizes the relay power, i.e., therelay is chosen based on the best relay-destination link considering error in estimation. Wecan see that our proposed scheme picks the relay based on both the source-relay and relay-destination links such that it minimizes the overall transmit power requirement, and henceperforms better than both the heuristic schemes. However, heuristic (1) scheme (best source-relay link) performs better than heuristic (2) (best relay-destination link). This is becausea minimum source power is necessary for successful decoding at the relay(s). Furthermore,source power is involved in the achievable rate expression due to the availability of the directlink. Therefore, minimizing source power is more important than that of relay power.In Fig. 2.4, the performance of the distributed algorithm is compared with that of thecentralized solution. A fixed step size parameter, µ = 0.01, and as stopping criterion, a max-imum iteration number, tmax = 60, is used. The performance of the distributed algorithm692.5. Numerical Results2 3 4 5 6 7 8 9 101618202224262830Network throughput (bits/sec/Hz)Totaluplinktransmitpower(dB) cen.1 relay ²r=0dist.1 relay ²r=0cen.2 relay ²r=0dist.2 relay ²r=0cen.1 relay ²r=0.005dist.1 relay ²r=0.005cen.2 relay ²r=0.005dist.2 relay ²r=0.005Figure 2.4: Performance of the distributed algorithm for bounded uncertainty regionis very close to that of the centralized solution, when there is no channel uncertainty or if auser is allowed to be served only by one (best) relay. However, when a user is using morethan one relay, the performance of the distributed algorithm becomes an upper bound tothat of the centralized solution due to the norm approximation (2.22).Fig. 2.5 shows the typical convergence behavior of the distributed algorithm for a singlechannel realization where each user is served by at most two relays. The required number ofiterations to converge within a distance of 10−3 of the optimal power value is less than 40.For a fixed step size parameter, µ = 0.01, the difference between the centralized solution andthe primal decomposition approach is higher due to the norm approximation, when channeluncertainty is taken into account. However, the required number of iterations to convergeto an acceptable upper bound of the optimal power value with channel uncertainty is as fastas without channel uncertainty.Figs. 2.6(a) and 2.6(b) show the worst user rate and total throughput versus channel702.5. Numerical Results0 10 20 30 40 50 6010−410−310−210−1100101102iterationP distributed-Poptimal(dB) ²r=0²r=0.005Figure 2.5: Convergence behavior of the distributed algorithm for bounded uncertaintyregionestimation error r, respectively when limited power budget is considered for the source andthe relays. Here, maximum relay power, pkmax , is set to 25 dB and equal weights are used forweighted sum rate maximization algorithm. In Fig. 2.6(a), we compare the worst user ratefor both the weighted sum and max-min fairness algorithms. The equal power allocation, i.e.,the power budget pkmax is divided equally among the users assisted by each relay k, is alsoincluded for reference. We see that the transmission data rate decreases with the increaseof the error radius for all three schemes, and the worst user obtains the best rate underthe max-min fairness. However, as error variance increases, the performance of weightedsum rate gets better than equal power allocation scheme and indicates that weighted sumrate maximization does not severely penalize the users with bad channel conditions. Thisis because the objective function of equation (47) is concave and increasing with respect toSNR, γm. Therefore, to maximize the objective function value (the sum of the weightedrate of all the users), it is obvious to allocate more power to the users operating at low712.5. Numerical ResultsSNR, since the increment is higher at that region. Fig. 2.6(b) shows the weighted sum ratemaximization scheme results in higher network throughput. The max-min scheme targetsto improve the performance of the worst user rate, which results in a significant loss in totalthroughput as the error increases.Fig. 2.7 shows how important it is to take into account the channel estimation errorswhile designing the relay selection algorithms. In this experiment, we consider that the relayselection process takes place assuming perfect CSI is available while in fact there are someestimation errors associated with the CSI in the system. Fig. 2.7 shows how much extrasource power the system needs to allocate for wrong relay selection. In Fig. 2.7, we compareour proposed best relay selection scheme given in (2.38) to that of [44], proposed based onperfect CSI assumption. For r < 10−3, both the algorithms select the same relay and showsimilar performance due to very small error in the channel estimation. However, significantincrease in the source power is observed for the scheme proposed in [44] for not selecting thebest relay when error is high, for example, r > 10−2.Figs. 2.8(a) and 2.8(b) show the implications of ignoring channel estimation errors whiledeveloping power allocation algorithms with QoS provisioning. In Fig. 2.8(a) and 2.8(b), weconsider that the power allocation process takes place assuming no uncertainty in channelestimation while in fact there are some estimation errors associated with the CSI in thesystem. Fig. 2.8(a) shows the probability of violating the QoS constraint, i.e., the first con-straint in (2.19) versus channel estimation error, r, for different rate, rm, requirements. Weobserve frequent violations of the QoS constraint for not considering the channel estimationerror in the design process. The probability of violation is higher for higher QoS require-ments and increases with the increase of uncertainty boundary. This result shows that thepower optimization schemes designed based on perfect CSI, for example [44]-[45], might failto satisfy the QoS constraints which is active, even in the presence of a very small error inCSI. Note that our proposed scheme is designed based on worst case error and hence providesguaranteed QoS under imperfect CSI.722.5. Numerical Results0 0.002 0.004 0.006 0.008 0.010.20.30.40.50.60.7Channel estimation error radius,( ²r )Worstuserrate(bits/sec/Hz) max-min rateequal powerweighted sum rate(a) Worst user rate versus channel estimation error radius0 0.002 0.004 0.006 0.008 0.014.555.566.577.58Channel estimation error radius,( ²r )Totalthroughput(bits/sec/Hz) max-min rateequal powerweighted sum rate(b) Network throughput versus channel estimation error radiusFigure 2.6: Performance of power limited networks considering the bounded uncertaintyregion732.5. Numerical Results10−4 10−3 10−2 10−11015202530Channel estimation error, 0rTransmitsourcepower(dB) proposed relay selection schemerelay selection based on perfect CSIFigure 2.7: The effect of ignoring channel estimation errors while selecting the best relayIn Fig. 2.8(b), we show the wastage of transmit power per user versus bounded channeluncertainty for different QoS requirements. Power wastage is defined as the transmit powerthat is insufficient to meet the required QoS at the destination due to the assumption ofperfect CSI at the point of power allocation, even though actual CSI is imperfect. In sucha scenario, the received rate at the destination can be less than that required, which resultsin transmission failures. Due to channel estimation error, actual rate in the presence ofimperfect CSI will be less than that obtained by assuming perfect CSI. Results reveal thatpower wastage increases with the increase of the required QoS and with uncertainty boundaryfor the schemes designed based on perfect CSI assumption, for example, [44]-[45]. This isbecause, the transmit power requirement increases with the increase of the rate requirements.Higher estimation error results in lower SNR and affects the performance of the system, whichrequires more power. Our proposed scheme provides guaranteed QoS and hence no transmitpower is wasted due to unsuccessful transmission.742.5. Numerical Results10−4 10−3 10−2 10−10.50.550.60.650.70.750.80.850.9Channel estimation error (0r)ProbabilityofviolatingtheQoSconstraint rate, rm = 1rate, rm = 0.8(a) Probability of violating the QoS constraint versus channel estimation error10−4 10−3 10−2 10−11111.51212.51313.51414.515Channel estimation error (0r)Average wastage of transmit power per user (dB) rate, rm = 1rate, rm = 0.8(b) Wastage of transmit power (dB) versus channel estimation errorFigure 2.8: The effect of ignoring channel estimation errors while designing resource alloca-tion algorithms with QoS provisioning75Chapter 3Admission Control and PowerAllocation for Selective RelayingBased Cellular Wireless Systems withImperfect CSIIn Chapter 2, we have addressed the power allocation and relay selection problem underchannel uncertainty. We have considered bounded channel uncertainty model since no sta-tistical knowledge is available about the error. Efficient and robust solutions have beendeveloped for the optimization problems via worst case optimization based approach. Inthis chapter, we address admission control and power allocation problem for selective relay-ing based cellular wireless systems with imperfect CSI. We consider that channel estimationerror is unbounded Gaussian. As the distribution of the error is known, we take a probabilis-tic optimization based approach to solve the problem as the worst case optimization oftenleads to overly conservative results [56].The accomplished works and research contributions on this chapter are briefly describedin the following.763.1. Accomplished Works and Research Contributions3.1 Accomplished Works and Research Contributions(1) First, we develop a power allocation scheme for a selective relaying based DF cooperativecellular network. Our proposed scheme takes into account the effect of channel estimationerror, which is unbounded Gaussian. For selective relaying, a power efficient method isproposed, where cooperation takes place only if it is beneficial for the network in termsof total uplink transmit power minimization or source power savings. Our objective isto provide power efficiency by minimizing the power consumption of the network whilemeeting the rate requirements of the users. The formulated optimization problem considersprobabilistically constrained approach in which QoS constraint is satisfied for all users withcertain probabilities.(2) Efficient solution for a probabilistic optimization problem is in general very diffi-cult [57]. Our main contribution in the paper is to approximate the probabilistic optimiza-tion problem into a convex deterministic form and derive closed form analytical solutionsfor power allocation using KKT conditions. Due to closed form solutions, our algorithm canhandle large scale networks efficiently unlike the schemes developed in [37]-[38],[55] that re-quire solving polynomial time optimization problems, for example, SOCP and semi-definiteprogramming (SDP), and hence, applicable only for small scale problems.(3) We utilize the benefit of closed form solutions to design a suboptimal algorithm thatcan solve the joint admission control and power allocation problem under practical networkdeployments with worst case linear complexity, which is combinatorially hard to solve in theoptimal sense. The existing suboptimal algorithms for the joint admission control and powerallocation problem are developed assuming the availability of perfect CSI [80]-[81] and requiresolving optimization problems with at least polynomial-time complexity, for example, [80]requires solving SOCP and SDP and [43] requires solving geometric programming (GP)problems, which may not be suitable for implementation in a large scale network.The rest of the chapter is organized as follows. The system model and problem for-773.2. System Model and Problem FormulationBase stationRelayUserFigure 3.1: System modelmulation are described in Section 3.2. Section 3.3 describes the optimization frameworksand solution approaches for power allocation with imperfect CSI. The optimization problemand frameworks for joint admission control and power allocation are given in Section 3.4.Numerical results are presented in Section 3.5.3.2 System Model and Problem FormulationConsider a single cell of a cellular network as shown in Fig. 3.1, where a set of users, denotedby S = {1, 2, ..., s, ..., S}, communicate with a single BS and a set of fixed relays, K ={1, 2, ..., k, ..., K}, are placed in the cell to assist them. Each user is assigned an orthogonalchannel over which the uplink transmission takes place. For cooperative communication,we consider the case of half-duplex repetition based DF cooperative diversity [5], whereorthogonal transmission is assumed for all the users and relays. Without loss of generality,we can assume that in the signaling period, which is in the beginning of each transmissionframe, channel is estimated from a training sequence and resource allocation is performed783.2. System Model and Problem Formulationbased on the estimated channel. Each transmission frame consists of a block of messages andthe channel impulse response is assumed to be time invariant (slow fading) within a frame.We propose selective relaying, where cooperation takes place only if it is beneficial forthe network in terms of total transmit power minimization or source power savings. In thesignaling period, BS (for centralized scheme) or the relays (for distributed scheme) decidewhether cooperation or direct transmission is beneficial for a particular user. The decisionis taken based on the power requirement to obtain a certain rate by direct transmission andby relay transmission. Our objective is to design a power efficient scheme, where some QoSis guaranteed to the users with lowest possible power requirement. With the help of therelays, it is possible to save the source power (battery life of the cell-phone) and prolong thenetwork lifetime.The communication between a user and BS consists of two transmission phases. In thefirst phase, user s transmits the message xs to BS d and to the relays with power ps. Thereceived message at relay k and at BS d is given byyk =√pshs,kxs + zk,yd1 =√pshs,dxs + zd1 ,(3.1)respectively. Here hs,k and hs,d are the channel coefficients for user s to relay k and user s toBS d links, respectively. zk and zd1 denote the i.i.d. AWGN noise with zero-mean and unitvariance.The set of relays k ∈ K(s) will be able to decode the messages for a user s if theinstantaneous mutual information given by [44]Is,k =1|K(s)|+ 1 log2(1 + |hs,k|2 ps), ∀k ∈ K(s) (3.2)793.2. System Model and Problem Formulationis greater than some fixed target rate rs, i.e.,Is,k ≥ rs, (3.3)where |K(s)| is the cardinality of the decoding set K(s) and |hs,k|2 is the channel power gainbetween user s to relay k link.If cooperation with relay k is beneficial for user s, which is decided a priori in the signalingperiod, relay k forwards the received message in the second phase to the BS with power pk,s.The received message at the BS d forwarded by a relay k for user s is given byyd2 =√pk,shk,dxk + zd2 , (3.4)where hk,d is the channel coefficient between relay k and BS d. If cooperation is not beneficial,user s transmits directly to the BS d in both the phases.The instantaneous mutual information between user s and BS d with and without relayis given by [5]Is,drelay =1|K(s)|+ 1 log2(1 + |hs,d|2 ps +|K(s)|∑k=1|hk,d|2 pk,s),Is,ddirect = log2(1 + |hs,d|2 ps,d),(3.5)respectively, where |hs,d|2 and |hk,d|2 are the channel power gains between user s and BS dand relay k and BS d, respectively.If the required power to obtain Is,k is greater than Is,ddirect , then no cooperation is optimalin terms of total power consumption or source power savings. In such scenarios user stransmits directly to the BS with optimal power p∗s,d = (2rs −1)/ |hs,d|2 in both transmissionphases. Otherwise, cooperation can be beneficial with the relay(s) k ∈ K(s) for user s. Wesub-divide the user group into two: the user group that do not need cooperation is given bythe set S1 = {1, 2, ..., S1} and the group for which cooperation may be beneficial is given by803.2. System Model and Problem Formulationthe set S2 = {S1 + 1, S1 + 2, ..., S1 + S2} such that S1 ∪ S2 = S and S1 + S2 = S.When cooperation can be beneficial, the optimization problem is given byminimize{ps, pk,s}S2∑s=1(ps +∑k∈K(s)pk,s)subject to :C1 : Is,drelay ≥ rs, ∀s ∈ S2C2 : Is,k ≥ rs, ∀k ∈ K(s), ∀s ∈ S2C3 : ps, pk,s ≥ 0, ∀k ∈ K(s), ∀s ∈ S2(3.6)where C1 is the QoS constraint which guarantees the target data rate rs of user s. Thesecond constraint (C2) satisfies the guaranteed decoding at the relay(s) k ∈ K(s), selected apriori to forward the message for user s. The relay selection techniques for repetition basedDF system with perfect and imperfect CSI have been discussed in Section 1.1.2 of Chapter 1and in Section 2.4 of Chapter 2, respectively.For now we assume that the target rate rs is achieved for the users with power which isbelow the maximum power budget of the source and the relay. If the optimal solution of (3.6)satisfies ps∗+∑k∈K(s)p∗k,s ≤ p∗s,d, cooperation is beneficial in terms of total power minimization,otherwise, direct transmission is optimal for user s. If the cost of source power is much higherthan relays, we can use the objective that minimizes only the sum of source powers. In thatcase, cooperation is beneficial if ps∗ ≤ p∗s,d. The optimization problem (3.6) is convex whichcan be solved using efficient algorithms [23].3.2.1 Problem Formulation with Imperfect CSIWe formulate the optimization problem now in a practical scenario considering imperfectCSI. We assume that the CSI is obtained from a training sequence and the resulting channeluncertainty is unbounded Gaussian [55]. We model the imperfect channel as h = hˆ + e,where hˆ denotes the estimated channel coefficient and e is the error in CSI which is modeled813.2. System Model and Problem Formulationas i.i.d and distributed as complex normal, CN (0, σ2e). The estimated channel hˆ and theerror e are assumed to be statistically independent. The variance σ2e of error indicates thequality of CSI.Note that for the case of imperfect CSI and in slow fading, an outage occurs wheneverthe transmit data rate exceeds the actual instantaneous mutual information, even whenchannel capacity achieving coding is applied for error protection [59]. This is because theinstantaneous mutual information becomes a RV due to the imperfectness in the CSI. Usingthis imperfect channel model, the instantaneous mutual information given in (3.2) and (3.5)can be expressed asIs,k =1|K(s)|+ 1 log2(1 + χ22s,kps), (3.7)andIs,drelay =1|K(s)|+ 1 log2(1 + χ22s,dps +|K(s)|∑k=1χ22k,dpk,s),Is,ddirect = log2(1 + χ22s,dps,d),(3.8)respectively, where χ22s,k = |hˆs,k + es,k|2, χ22s,d = |hˆs,d + es,d|2, and χ22k,d = |hˆk,d + ek,d|2.Since the modeled CSI error is unbounded Gaussian, the instantaneous mutual informationexpressions (3.7)-(3.8) become RVs. Therefore, the data rate requirements of different userscan be fulfilled only in a probabilistic manner with some predetermined allowed probabilityof outage, pout, values. Our design goal should be such that the resource allocation schemeis able to fulfill the data rate requirements of all the users as well as their channel outageprobability requirements. For a target data rate rs, the outage probability of a link isdefined as Pr.(I ≤ rs), where Pr.(·) denotes the probability of the event (·) and I denotesthe instantaneous mutual information for that link. Whenever the actual instantaneousmutual information I falls below the target data rate rs, the received message cannot besuccessfully decoded with probability 1, and the system declares an outage.Our goal is now to provide the QoS in a probabilistic manner, such that the outageprobability is less than the predetermined value pout. With CSI error, no cooperation, i.e.,823.3. Proposed Optimization Frameworkdirect transmission is optimal in terms of source power savings or total power consumption,if the required power to obtain Pr.(Is,k ≤ rs) ≤ pout is greater than Pr.(Is,ddirect ≤ rs) ≤ pout.In this scenario, user s transmits directly to the BS and the optimal power p∗s,d is obtainedby solving Pr.(Is,ddirect ≤ rs) ≤ pout.Considering CSI error, the probabilistic optimization problem of (3.6) can be written asminimize{ps, pk,s}S2∑s=1(ps +∑k∈K(s)pk,s)subject to :C1 : Pr.[(χ22s,dps +|K(s)|∑k=1χ22k,dpk,s) ≤ γs]≤ pout, ∀s ∈ S2C2 : Pr.[(χ22s,kps) ≤ γs]≤ pout, ∀k ∈ K(s), ∀s ∈ S2C3 : ps, pk,s ≥ 0, ∀k ∈ K(s), ∀s ∈ S2(3.9)where γs = 2(|K(s)|+1)rs − 1. The first constraint guarantees that the SNR γs, or equivalently,the target data rate rs of user s is satisfied with a probability of 1 − pout. The secondconstraint guarantees the decoding for the selected relay(s) k ∈ K(s) with a probability of1 − pout. The third constraint is the non-negativity power constraint. Next, our goal is totransform the probabilistic constraints into deterministic ones.3.3 Proposed Optimization FrameworkTo determine the power requirement without cooperation, the probabilistic constraint Pr.(Is,ddirect ≤ rs) ≤ pout can be re-written asPr.(χ22s,dps,d ≤ 2rs − 1)≤ pout. (3.10)where χ22s,d is recognized as a non-central Chi-square distributed RV with non-centralityparameter, λ2s,d =|hˆs,d|2σ2es,d/2and degrees of freedom n = 2.833.3. Proposed Optimization FrameworkThe left-hand side of (3.10) can be evaluated as [82]Fχ22s,d (x) = 1−Q1λs,d,√2rs − 1√σ2es,d/2 , (3.11)whereQ1(a, b) = e−(a2+b2)/2∞∑k=0(ab)kIk(ab), b > a > 0 (3.12)and Ik(ab) is the kth-order modified Bessel function of the first kind.Various approximations are proposed in the literature [83]-[85] to avoid the infinite sumof (3.12). We use the approximation in [85] to obtain a closed-form result, where the non-central Chi-square distribution is approximated by the central Chi-square asPr.(χ22s,dps,d ≤ 2rs − 1)≈ Pr.(χ22s,d(0) ≤(2rs − 1) /ps,d1 + λ2s,d/2). (3.13)In [85], it has been mentioned that the approximation in (3.13) is simple enough to be usedanalytically. The approximation is most accurate for λ2s,d/2 ≤ 0.2 and yields a conservativeresult for λ2s,d/2 > 0.2 [86]. The approximation gap reduces with the increase of the errorvariance and works very well for extremely detrimental channel conditions [57].Since χ22s,d(0), the central Chi-square distribution with degrees of freedom n = 2, yieldsexponential distribution, (3.13) can be computed asPr.(χ22s,d(0) ≤(2rs − 1) /ps,d1 + λ2s,d/2)= 1− e−(2rs − 1) /ps,d2(1 + λ2s,d/2). (3.14)Using the expression of (3.14) in (3.10), the approximate minimum power for no cooper-ation case can be obtained asps,d ≥ Cs,d, ∀s ∈ S1 (3.15)843.3. Proposed Optimization FrameworkwhereCs,d ={ 1−ln(1− pout)} (2rs − 1)2(1 + λ2s,d/2) . (3.16)Using similar techniques given in (3.13)-(3.14), the required power ps for correctly decodingwith a probability of 1−pout at any relay k ∈ K(s) can be obtained from the second constraintof (3.9) asps ≥ C(k)s , ∀k ∈ K(s), ∀s ∈ S2 (3.17)whereC(k)s ={ 1−ln(1− pout)} γs2(1 + λ2s,k/2) , (3.18)and λ2s,k =|hˆs,k|2σ2es,k/2. Therefore, cooperation can be beneficial for a user s with a relay k ∈ K(s)if C(k)s < Cs,d. Next, we need to find out a deterministic representation for the first constraintof (3.9) which involves sum of non-central Chi-square distributions.The left-hand side of the first constraint of (3.9) can be re-written asPr.(|hˆs,d + es,d|2ps +|K(s)|∑k=1|hˆk,d + ek,d|2pk,s ≤ γs) = Pr.(|K(s)|+1∑j=1|hˆj + ej |2pj ≤ γs). (3.19)By exploiting the independence of the |K(s)| + 1 strictly positive RVs |hˆj + ej |2, we candecouple the power variables pj and obtain a closed-form representation for the deterministicconstraint.Proposition 1. The inequalityPr.(|K(s)|+1∑j=1Xjpj ≤ γs) ≤|K(s)|+1∏j=1Pr.(Xjpj ≤ γs), (3.20)holds for independent and positive RVs, Xj ∈ [0,∞).Proof. For |K(s)| = 1, we can see from Fig. 3.2 that the regions Pr.(X1p1 + X2p2 ≤ γs)and Pr.(X1p1 ≤ γs)Pr.(X2p2 ≤ γs) satisfy the inequality for independent and positive RVs.853.3. Proposed Optimization Frameworksγsγ 1 1pχ2 2pχ01 1 2 2Pr.( )Sp pχ χ γ+ <1 1 2 2Pr.( )Pr.( )s sp pχ γ χ γ< <Figure 3.2: Proof of the proposition that shows for |K(s)| = 1, the proposed inequalityPr.(|K(s)|+1∑j=1Xjpj ≤ γs) ≤|K(s)|+1∏j=1Pr.(Xjpj ≤ γs) holds for independent and positive RVs,Xj ∈ [0,∞)863.3. Proposed Optimization FrameworkTherefore, in general, for |K(s)| > 1, it can be shown that the event (|K(s)|+1∑j=1Xjpj ≤ γs) is asubset of the intersection of the events A1 ∩A2 ∩ ...∩A|K(s)|+1, where A1 = (X1p1 ≤ γs) andA|K(s)|+1 = (X|K(s)|+1p|K(s)|+1 ≤ γs) and hence the inequality in (3.20) holds for independentand positive RVs, Xj ∈ [0,∞).Therefore we can writePr.(|K(s)|+1∑j=1|hˆj + ej|2pj ≤ γs) ≤|K(s)|+1∏j=1Pr.(|hˆj + ej |2pj ≤ γs). (3.21)Using the techniques given in (3.13)-(3.14), the first constraint of (3.9) can be written from(3.21) as|K(s)|+1∏j=1Pr.(|hˆj + ej |2pj ≤ γs) ≤ pout ≈|K(s)|+1∏j=11− e−γspj[2(1+λ2j/2)]−1≤ pout, ∀s ∈ S2 (3.22)where λ2j =|hˆj |2σ2ej/2. However, the approximated expression in (3.22) is not a convex functionof pj . Therefore, in order to formulate a tractable problem, we use another inequality, givenby 1− e−x ≤ x, and write the left hand side of the approximation in (3.22) as|K(s)|+1∏j=11− e−γspj[2(1+λ2j/2)]−1 ≤|K(s)|+1∏j=1γspj[2(1 + λ2j/2)]−1 . (3.23)After approximations (3.21)-(3.23), the first constraint of (3.9) can be written as|K(s)|+1∏j=1γspj[2(1 + λ2j/2)]−1 ≤ pout. ∀s ∈ S2 (3.24)In the following, we demonstrate how well (3.19) is approximated by the deterministic873.3. Proposed Optimization Frameworkconstraints given in (3.22) and (3.24) using numerical methods. We compare the simulatedcumulative distribution function (CDF) of (3.19) given byF1(γs) = Pr.(χ22s,dps +|K(s)|∑k=1χ22k,dpk,s) ≤ γswith the approximate CDF obtained in (3.22) given byF2(γs) =|K(s)|+1∏j=11− e−γspj[2(1+λ2j/2)]−1,and with the approximation obtained in (3.24) given byF3(γs) =|K(s)|+1∏j=1γspj[2(1 + λ2j/2)]−1 ,respectively, using 5000 Monte-Carlo trials.Fig. 3.3 shows numerically obtained exact CDF F1(γs), approximate CDF F2(γs), andthe approximation F3(γs), respectively for different values of error variance σ2e . As expected,exact CDF is less than the approximate ones. This implies that the original QoS constraintis guaranteed by the satisfaction of the approximate CDF and verifies the bounds used forthe approximations. However, the approximation is conservative for lower values of errorvariances and becomes tight as the error variance increases. We can see the gap betweenthe exact CDF and the approximations become smaller when the channel estimation errorvariance is increased from σ2e = 0.1 to σ2e = 0.9, which demonstrates that under criticalsituations, the QoS is well controlled by the approximate deterministic constraint in (3.24).Note that the left-hand side of the constraint (3.24) is a convex function for positive ps andpk,s, which can be proved using the Hessian of the function [23], and hence the constraints areconvex. Using the deterministic constraints (3.17) and (3.24), the probabilistic optimization883.3. Proposed Optimization Framework0 5 10 15 20 25 3000.10.20.30.40.50.60.70.80.91γsCDF empirical CDF F1(γs)approx. CDF F2(γs)approximation F3(γs)(a) Empirical CDF of γs and approximations for channel estimationerror variance σ2e = 0.1 and estimated channel gain |hˆ|2 = 10 5 10 15 20 25 3000.10.20.30.40.50.60.70.80.91γsCDF empirical CDF F1(γs)approx. CDF F2(γs)approximation F3(γs)(b) Empirical CDF of γs and approximations for channel estima-tion error variance σ2e = 0.9 and estimated channel gain |hˆ|2 = 1Figure 3.3: Empirical CDF F1(γs) = Pr.[(χ22s,dps +|K(s)|∑k=1χ22k,dpk,s) ≤ γs]versus the ap-proximate CDF F2(γs) =|K(s)|+1∏j=11 − e−γspj[2(1+λ2j/2)]−1 and the approximation F3(γs) =|K(s)|+1∏j=1γspj[2(1 + λ2j/2)]−1893.3. Proposed Optimization Frameworkproblem of (3.9) can be approximated asminimize{ps, pk,s}S2∑s=1(ps +∑k∈K(s)pk,s)subject to :C1 : ps|K(s)|∏k=1pk,s ≥ Ck,s, ∀s ∈ S2C2 : ps ≥ C(k)s , ∀k ∈ K(s), ∀s ∈ S2C3 : pk,s ≥ 0, ∀k ∈ K(s), ∀s ∈ S2(3.25)where Ck,s is obtained from (3.24) asCk,s =γspoutAs,d|K(s)|∏k=1γsAk,d, (3.26)where As,d =[2(1 + λ2s,d/2)]−1 and Ak,d =[2(1 + λ2k,d/2)]−1.Since the objective function is affine and the constraints are convex and affine, problem(3.25) is convex, for which the global optimum solution can be obtained. The Lagrangian ofthe approximated optimization problem in (3.25) can be written asL(·) =S2∑s=1(ps +∑k∈K(s)pk,s)+S2∑s=1λs,1(Ck,s − ps|K(s)|∏k=1pk,s)+S2∑s=1λs,2(C(k)s − ps)−S2∑s=1|K(s)|∑k=1λk,spk,s(3.27)where λs,1, λs,2 λk,s are the Lagrange multipliers associated with the first, second and thirdinequality constraints of (3.25), respectively. The KKT conditions for the Lagrangian func-tion in (3.27) are given byλ∗s,1Ck,s − p∗s|K(s)|∏k=1p∗k,s = 0, ∀s ∈ S2 (3.28a)λ∗s,2(C(k)s − p∗s)= 0, ∀k ∈ K(s), ∀s ∈ S2 (3.28b)903.3. Proposed Optimization Frameworkλ∗k,sp∗k,s = 0, ∀k ∈ K(s), ∀s ∈ S2 (3.28c)p∗s|K(s)|∏k=1p∗k,s ≥ Ck,s, p∗s ≥ C(k)s , ∀k ∈ K(s), ∀s ∈ S2 (3.28d)p∗k,s ≥ 0, ∀k ∈ K(s), ∀s ∈ S2 (3.28e)λ∗s,1, λ∗s,2, λ∗k,s ≥ 0, ∀k ∈ K(s), ∀s ∈ S2 (3.28f)∂L∂p∗s= 1− λ∗s,1|K(s)|∏k=1p∗k,s − λ∗s,2 = 0, ∀s ∈ S2 (3.28g)∂L∂p∗k,s= 1− λ∗s,1p∗s|K(s)|∏r=1r 6=kp∗r,s − λ∗k,s = 0, ∀k ∈ K(s), ∀s ∈ S2. (3.28h)From (3.28c) and (3.28d), we can conclude that λ∗k,s = 0, otherwise, condition (3.28d) isviolated for any positive Ck,s. This also guarantees that λ∗s,1 6= 0 in (3.28h) and hence theconstraint p∗s|K(s)|∏k=1p∗k,s ≥ Ck,s must be met with equality in (3.28a). Therefore, (3.28h) canbe expressed asp∗s|K(s)|∏r=1r 6=kp∗r,s = 1/λ∗s,1. (3.29)Note that the constraint p∗s ≥ C(k)s , may or may not be active, which indicates that therecan be two cases with λ∗s,2, as explained below:Case 1: λ∗s,2 = 0In this case, we can write the following expression from (3.28g) as|K(s)|∏k=1p∗k,s = 1/λ∗s,1. (3.30)Using (3.29)-(3.30) we can writep∗s = p∗k,s =(1/λ∗s,1)1|K(s)| . (3.31)Substituting the values of p∗s and p∗k,s into (3.28d), we obtain(1/λ∗s,1)|K(s)|+1|K(s)| = Ck,s and hence913.3. Proposed Optimization Frameworkthe solution is obtained asp∗s = p∗k,s = C1|K(s)|+1k,s . (3.32)Case 2: λ∗s,2 6= 0This guarantees from (3.28b) that the constraint p∗s ≥ C(k)s must be active. Therefore,the solution is given byp∗s = C(k)sp∗k,s =(Ck,sC(k)s)1|K(s)|.(3.33)If C(k)s ≥ C1|K(s)|+1k,s , we can conclude that the constraint p∗s ≥ C(k)s is active, otherwise, weconclude that the constraint p∗s ≥ C(k)s is not active and λ∗s,2 = 0. The algorithm to obtainthe solution for the approximated problem (3.25) is given in Algorithm 1.Algorithm 1 : Optimal power allocation for the approximated problem (3.25)Require: γs, pout, |K(s)|, |hˆj|2, σ2ej wherej ∈ [{s, d}, {k, d}, {s, k}]1: Calculate Cs,d, C(k)s and Ck,s from (3.16), (3.18) and (3.26), respectively2: if C(k)s ≥ Cs,d then3: no cooperation is the optimal strategy with power p∗s,d = Cs,d4: return5: else if C(k)s ≥ C1|K(s)|+1k,s then6: optimal p∗s and p∗k,s are obtained from Case 2 : λ∗s,2 6= 0 asp∗s = C(k)s and p∗k,s =(Ck,sC(k)s)1|K(s)|7: else if C(k)s < C1|K(s)|+1k,s then8: optimal p∗s and p∗k,s are obtained from Case 1 : λ∗s,2 = 0 asp∗s = p∗k,s = C1|K(s)|+1k,s9: end if10: if p∗s +∑k∈K(s)p∗k,s > Cs,d then11: no cooperation is beneficial and optimal solution is p∗s,d = Cs,d12: else13: cooperation is beneficial and optimal p∗s and p∗k,s are obtained.14: end ifRemarks: In the case of minimizing only the sum of source powers, the constraint923.4. Admission Control and Power Allocationp∗s ≥ C(k)s is active and the solution can be obtained directly using Case 2 : λ∗s,2 6= 0 asp∗s = C(k)s and p∗k,s =(Ck,sC(k)s)1|K(s)|.3.4 Admission Control and Power AllocationComputationally efficient power allocation plays a vital role in developing admission controlalgorithms. In the previous section, we obtained closed form power allocation solutions forthe approximated problem (3.25) considering imperfect CSI, which can be utilized to developan efficient and robust admission control algorithm. The algorithm determines which userscan be served by the system with the limited power budget of the source and the relay. Ifcooperation is not beneficial for a user s, i.e., p∗s,d ≤ p∗s, then that particular user is admittedto the system only if p∗s,d ≤ psmax, where psmax is the maximum power budget of the source.When cooperation can be beneficial, i.e., p∗s,d > p∗s, the approximated optimization problemin (3.25) can be re-written with the additional constraints on source and relay power asminimize{ps, pk,s}S2∑s=1(ps +∑k∈K(s)pk,s)subject to :C1 : ps|K(s)|∏k=1pk,s ≥ Ck,s, ∀s ∈ S2C2 : ps ≥ C(k)s , ∀k ∈ K(s), ∀s ∈ S2C3 : ps ≤ psmax, ∀s ∈ S2C4 :∑s∈S(k)pk,s ≤ pkmax , ∀kC5 : pk,s ≥ 0, ∀k ∈ K(s), ∀s ∈ S2(3.34)where pkmax is the maximum power budget of a relay k. The set of users served by relay kis given by S(k). Due to limited power budget of both the source and the relay nodes, theoptimization problem in (3.34) can become infeasible for some instances, for example, whenthe required rate rs is high or the number of users S2 is large. Now, our objective is to jointly933.4. Admission Control and Power Allocationmaximize the number of admitted users and minimize the power consumption of the systemgiven the limited power budget of the source and the relay nodes. Joint admission controland power allocation problem can be formulated as a 2-stage optimization problem [43],[80],where in the first stage, the set of possible admitted users are obtained and in the secondstage, power allocation problem is solved for the set of admitted users. The first stageproblem (admission control algorithm) can be written asargmaxU⊆S2, ps, pk,s|U |subject to :C1 : ps|K(s)|∏k=1pk,s ≥ Ck,s, ∀s ∈ UC2 : ps ≥ C(k)s , ∀k ∈ K(s), ∀s ∈ UC3 : ps ≤ psmax , ∀s ∈ UC4 :∑s∈{S(k)∩U}pk,s ≤ pkmax , ∀kC5 : pk,s ≥ 0, ∀k ∈ K(s), ∀s ∈ U(3.35)where |U | denotes the cardinality of the set U . There can be one or several set {U0, U1, ...} ofpossible admitted users, however, they should have the same cardinality |U | [43]. For each setof possible admitted users, the power allocation solutions can be obtained from Algorithm1, and the optimal solution for the admission control problem is obtained for the set thatrequires the minimum total uplink power. The complexity of the optimal admission controlin (3.35) grows exponentially with the number of users and requires exhaustive search overall possible combination of users. Joint admission control and power allocation problemcan also be formulated as a single stage multi-objective optimization problem, where thepriority is given in maximizing the number of admitted users [80]. However, similar to the2-stage problem, the single stage problem is also combinatorially hard, which is not suitablefor practical implementation. Optimal solution for the joint admission control and powerallocation is very difficult to obtain, if possible. Therefore, a sub-optimal algorithm with943.4. Admission Control and Power Allocationlow-complexity is required.The problem in (3.34) becomes infeasible if either of the source or relay power exceedsthe budget. If both C(k)s and Cs,d are higher than the source power budget psmax , then thatuser must be removed due to poor channel quality. If the relay power budget, pkmax , exceedsits limit then at least one user needs to be removed from the group S(k). Since our goal ispower minimization, the criterion is to remove the user with largest relay power requirementp∗k,s from the set of users S(k) in our proposed algorithm. User s, who takes the largestamount of power from a shared relay, has the highest impact on the other users in the samegroup. Therefore, the problem becomes feasible after minimum number of user removalfrom the group S(k). Once a user is removed from the group, we check the algorithm forthe remaining users in the group and repeat the process. However, user s, who is removedfrom the group S(k), is given another chance to transmit directly if Cs,d is less than psmax ,since admission control is given higher priority than power minimization. The details of thejoint admission control and power allocation algorithm is given in Algorithm 2.953.4. Admission Control and Power AllocationAlgorithm 2 : Joint admission control and power allocationRequire: γs, pout, |K(s)|, |hˆj |2, σ2ej , psmax , pkmaxwhere j ∈ [{s, d}, {k, d}, {s, k}]1: Calculate Cs,d, C(k)s and Ck,s from (3.16), (3.18) and (3.26), respectively for all s2: if C(k)s ≥ Cs,d and Cs,d ≤ psmax then3: p∗s,d = Cs,d4: return5: else if C(k)s ≥ C1|K(s)|+1k,s and C(k)s ≤ psmax then6: p∗s = C(k)s and p∗k,s =(Ck,sC(k)s)1|K(s)|7: else if C(k)s < C1|K(s)|+1k,s and C(k)s ≤ psmax then8: p∗s = p∗k,s = C1|K(s)|+1k,s9: else10: transmission is withheld for s11: end if12: if p∗s +|K(s)|∑k=1p∗k,s > Cs,d and Cs,d ≤ psmax then13: no cooperation is beneficial and optimal solution is p∗s,d = Cs,d14: return15: else16: cooperation can be beneficial and optimal p∗s and p∗k,s are obtained.17: end if18: if∑s∈S(k)p∗k,s ≤ pkmax , ∀k and S(k) 6= ∅, ∀k then19: optimal p∗s and p∗k,s are obtained20: else if S(k) 6= ∅ then21: smax = argmaxs∈S(k){p∗k,s};S(k) = S(k) \ {smax}, ∀k ∈ K(smax)22: if Csmax,d ≤ psmax then23: user smax is admitted as a non-cooperative user24: else25: transmission of user smax is withheld26: end if27: goto step 18:28: else29: transmission is withheld for k30: end ifThe worst case complexity of the algorithm is linear with the number of users that requirecooperation, i.e, O(S2). From Algorithm 2, it can be shown that our proposed solutionis not optimal only for the case where alternative relay selection is possible. For example,963.5. Numerical Resultsconsider that a user first selects its best relay for transmission. However, due to insufficientrelay power budget, the user is removed from that relay group. The algorithm then allowsthat user to get possible admission to the system by direct transmission. If the user failsto achieve its target rate via direct transmission due to insufficient source power and/orpoor direct link, the user is removed from the system. However, if the user was allowed toselect another relay having similar performance but with fewer admitted users, then thatuser could possibly achieve its target rate and admission with the help of the new relay.Note that the number of such possible cases are insignificant for our considered network,i.e., a network with few relays and large number of users. Also, this process of re-allocationbecomes a combinatorial problem. Optimal relay selection, admission control, and powerallocation algorithm for a multiuser network with multiple relays is also combinatoriallyhard to solve [36],[45].3.5 Numerical ResultsWe have conducted extensive simulations considering different system parameters to analyzethe effect of imperfect CSI. We evaluate and compare the performances for the followingthree system models:• No cooperation: In this model, users in a cellular network communicate directly withthe BS, without taking any assistance from the relays. Users are admitted if therequired source powers do not exceed the maximum source power budget.• Always cooperation: Users, if admitted to the network, always make use of the relaysin addition to the direct path to communicate with the BS in this model. For ourconsidered cellular network, where the number of relays is much smaller than thenumber of users due to cost and space constraints, the single relay selection schemeis useful in terms of feedback and overhead reduction [7],[45]. Moreover, it has beenproved in Section 1.1.2 of Chapter 1 that for repetition based DF relay networks with973.5. Numerical Resultsorthogonal transmission, best (single) relay selection scheme is optimal. We considerthat only one of the relays from k ∈ K(s), chosen a priori, transmit in the second timeslot. For relay selection, we have used the following criterion,argmink∈K(s)C(k)s , (3.36)i.e., the relay that is able to decode the message of user s using the minimum sourcepower.• Selective cooperation: In our proposed model, users in a cellular network if admitted,communicate with the BS via the relays only if it is beneficial in terms of powerminimization or source power savings. While cooperation, relays are selected using thesame criteria as always cooperation.We have considered a cell of radius rcell with 8 relays (unless otherwise stated), where therelays are placed on a ring as shown in Fig. 1, and 1000 users uniformly distributed allover the cell. Each of the estimated channel coefficients are independent CSCG RVs anddistributed as CN(0, 1(1 + d)α)[87], where d is the normalized distance among variousnodes in the network and α = 4 is a constant (path loss coefficient). The choice of differentchannel variances represents a typical scenario, in which the receiver which is closer to atransmitter experiences a better channel on average than the one which is farther from thetransmitter. We would like to emphasize that the assumed channel model is chosen fordemonstration purpose, which is not critical to the performance of our proposed resourceallocation schemes. Our proposed schemes can be applied to other channel models too.Unless stated elsewhere, same target data rate, rs = 1 bit/sec/Hz is assumed for all users,same channel estimation error variance, σ2e = 0.1, is assumed for all channels, probability ofoutage, pout, is assumed to be 0.1, maximum source power, psmax , and relay power, pkmax ,are set at 0.4 dB and 20 dB, respectively. The results are averaged over 1000 independentchannel realizations.983.5. Numerical Results10−4 10−3 10−2 10−1 100−25−20−15−10−50510Channel estimation error variance, σ2eAverageuplinktransmitpowerperuser(dB) Selective CoopAlways CoopNo Coop(a) Average uplink transmit power per user (dB) versus channel estimationerror variance00.10.20.30.40.50.60.70.80.91Channel estimation error variance, σ2eNumberofadmittedusers/Totalnumberofusers Selective CoopAlways CoopNo Coop1005 10 110 15 10 210 25 10 310 35 10 410 4(b) The ratio of the number of admitted users and total number of users versuschannel estimation error varianceFigure 3.4: The effect of channel estimation error variance on average transmit power peruser and on the ratio of the number of admitted users and total number of users993.5. Numerical ResultsIn Figs. 3.4(a) and 3.4(b), the effect of error variance σ2e is shown on average uplinktransmit power per user and on number of admitted users, respectively, for the three systemmodels. We can see from 3.4(b) that when error variance is relatively small, for example,σ2e < 10−3, all the users are admitted for all three system models. In this scenario, selectivecooperation performs the best in terms of total power minimization and at σ2e = 10−3 ap-proximately 2.5 dB and 1.5 dB powers can be saved compared to no cooperation and alwayscooperation models, respectively. Always cooperation requires less power than no coopera-tion till σ2e = 0.005. When σ2e > 0.05, no cooperation method performs better in terms ofpower requirement than selective and always cooperation schemes. Since higher priority isgiven in admission control algorithm, both selective and always cooperation schemes admitmore users than no cooperation till σ2e = 0.05. When error variance is relatively high, forexample, σ2e = 0.1, nearly 16% more users are admitted by selective cooperation with 0.5dB more power requirement than no cooperation scheme. When error variance is very high,i.e., close to unity, no cooperation performs better than always cooperation both in terms ofpower minimization and admission control. The loss is higher because similar error varianceis assumed for all the channels and more channels are used by always cooperation schemethan no cooperation.Fig. 3.5 shows the average transmit source power per user versus rate, rs for two differentoutage values pout = 0.05 and pout = 0.1. In this experiment, our objective is to minimizeonly the sum of source powers when all the users are admitted in the system. As expected,the source power requirement increases with the increase in rs for all the three systemmodels and the proposed selective cooperation scheme outperforms the other two modelsin terms of source power minimization for the entire rate region. For rs < 1 bit/sec/Hz,always cooperation requires less source power than no cooperation and vice-versa. We seesimilar trends in the results for different outage pout requirements, however, higher QoS(lower pout) requirement results in increase of source power for all three models. At rs =0.4, approximately 2.2 dB and 1 dB source power can be saved using selective cooperation1003.5. Numerical Results0.2 0.4 0.6 0.8 1 1.2−10−8−6−4−202468Rate, rsAveragetransmitsourcepowerperuser(dB) Selective CoopAlways CoopNo CoopProbability of outage, pout = 0.05Probability of outage, pout = 0.1Figure 3.5: Average transmit source power per user versus target data ratecompared to using no cooperation and always cooperation schemes, respectively for bothconsidered pout.In Figs. 3.6(a) and 3.6(b), we show the effect of channel estimation error on energyefficiency and on the number of admitted users for the three system models. Energy efficiencymetric is defined as achievable throughput per unit power [88]. In this experiment, we useenergy efficiency as the performance metric to evaluate the system models in terms of energyawareness. From Fig. 3.6(b), we see that when error variance is small (σ2e ≤ 10−3), allthe users are admitted. In this scenario, selective cooperation outperforms always and nocooperation schemes in terms of energy efficiency. Always cooperation scheme has higherenergy efficiency than no cooperation. This result shows that cooperative schemes are moreenergy efficient than non-cooperative schemes. The energy efficiency decreases for all threemodels with the increase of error variance. When error variance is high, for example, σ2e =0.1, energy efficiency approaches to zero for all three models. In this case, the number ofadmitted users are highest for the proposed selective cooperation scheme. Since the priority is1013.5. Numerical Results10−5 10−4 10−3 10−2 10−100.511.522.533.5Channel estimation error variance, σ2eEnergyEfficiency(bits/sec/Hz/watt) Selective CoopAlways CoopNo Coop(a) Energy efficiency (bits/sec/Hz/watt) versus channel estimation error vari-ance00.10.20.30.40.50.60.70.80.91Channel estimation error variance, σ2eNo.ofadmittedusers/Totalno.ofusers Selective CoopAlways CoopNo Coop10 15 10 210 25 10 310 35 10 410 45 10 510 5(b) The ratio of the number of admitted users and total number of users versuschannel estimation error variance (expressed in dB)Figure 3.6: The effect of channel estimation errors on energy efficiency and admission control1023.5. Numerical Results0 0.5 1 1.5 2 2.5 3 3.5 400.10.20.30.40.50.60.70.8Error variance ratio, σ2esr /σ2esdCooperationratio Rate, rs = 0.2Rate, rs = 0.4Rate, rs = 0.6Rate, rs = 0.8Rate, rs = 1.0Rate, rs = 1.2Figure 3.7: Cooperation ratio versus error variance ratio, σ2esr/σ2esdgiven in the admission control, nearly 14.5% and 10.5% more users are admitted for selectivecooperation scheme than no cooperation and always cooperation schemes, respectively, whensimilar energy efficiency is obtained in all three systems.Fig. 3.7 shows the cooperation ratio versus channel estimation error variance ratio,σ2esr/σ2esd, for different rs for the proposed selective cooperation scheme. Cooperation ra-tio is defined as the ratio of number of cooperating and non-cooperating users. When theratio σ2esr/σ2esd is low, it indicates that the error variance is higher in the direct path, and thisresult indicates that the direct channel quality is poor. In this scenario, more users cooperatesince the source relay channel quality is better than the direct link. As the ratio σ2esr/σ2esdincreases, the error variance in the source-relay link increases which results in a decrement inthe cooperation ratio. We see that cooperation ratio increases with the decrease of the raterequirements. This is because cooperation requires one more time slot than direct transmis-sion and the power requirement for cooperation is proportional to 22rs − 1, which in turn,for no-cooperation is proportional to 2rs − 1.1033.5. Numerical Results10−1 100 101 102 10300.050.10.150.20.250.30.350.40.450.5Maximum relay power to source power ratio, pkmax/psmaxCooperationratio Error variance, σ2e = 0.01Error variance, σ2e = 0.05Error variance, σ2e = 0.1Error variance, σ2e = 0.5Figure 3.8: Cooperation ratio versus power ratio, pkmax/psmaxFig. 3.8 shows the cooperation ratio versus maximum relay power to source power ratio,pkmax/psmax, for different error variances, σ2e , for our proposed selective cooperation scheme.We can see that the cooperation ratio increases with the increase of pkmax/psmax till pkmaxequals psmax and after that a floor is reached. This indicates that by increasing relay powerbudget cooperation ratio cannot be improved further. However cooperation ratio can beimproved by minimizing the channel estimation error variances. This is because of the factthat the error affects similarly to both the source-relay and relay-destination links when sameerror variance is assumed in all the channels. The result also indicates cooperation is morebeneficial when the relay channels have better estimation.Figs. 3.9(a)-3.10(b) show the implications of ignoring channel estimation errors whiledesigning power allocation algorithms. In Figs. 3.9(a)-3.10(b), we consider that the powerallocation process took place assuming perfect CSI while in fact there was some estimationerror associated with the CSI in the system. Fig. 3.9(a) shows the probability of violating1043.5. Numerical Results0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.400.10.20.30.40.50.60.70.80.91channel estimation error variance, σ2eProbabilityofviolatingtheQoSconstraint pout = 0.02pout = 0.04pout = 0.06pout = 0.08pout = 0.1(a) Probability of violating the QoS constraint versus channel estimation errorvariance for different values of pout0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4012345678channel estimation error variance, σ2eAveragewastageoftransmitpowerperuser(dB) pout = 0.02pout = 0.04pout = 0.06pout = 0.08pout = 0.1(b) Average wastage of transmit power per user (dB) versus channel estima-tion error variance for different values of poutFigure 3.9: The effect of ignoring channel estimation errors while designing power allocationalgorithms. Solid line: algorithms based on perfect CSI, dashed lines: proposed algorithm1053.5. Numerical Resultsthe QoS constraint, i.e., the first constraint in (3.25) versus σ2e for different pout values. Weobserve frequent violations of the QoS constraint for not considering the channel estimationerror in the design process. The probability of violation is higher for lower pout, i.e., higherQoS requirements, and increases with the increase of error variance. Our proposed schemeguarantees the QoS while maintaining the predefined outage probability pout values.In Fig. 3.9(b), we show the wastage of transmit power per user versus channel estimationerror variance for different pout requirements. Power wastage is defined as the transmitpower that is insufficient to meet the required QoS at the destination due to the assumptionof perfect CSI at the point of power allocation, even though actual CSI is imperfect. In sucha scenario, the received rate at the destination can be less than that required, which resultsin an outage. Results reveal that the power wastage increases with the increase of both therequired QoS and the error variance for the system designed based on perfect CSI. This is dueto the fact that the transmit power requirement increases as outage probability requirementsbecome more strict. For our proposed robust scheme, we can see that the increment inpower wastage is almost negligible due to error. The wastage of power is extremely smallcompared to the schemes based on perfect CSI which depends on the outage requirements,pout and the approximations used to obtain the closed form power allocation solutions. Thisfinding signifies that CSI error severely affects the performance of the non-robust schemes,for example, [44],[45].Figs. 3.10(a)-3.10(b) show the performance comparison of our proposed scheme with theschemes developed based on perfect CSI for lower values of channel estimation error variancesand probability of outage pout requirements. Fig. 3.10(a) shows the probability of violatingthe QoS constraint versus pout for different σ2e values. We can see that the performance ofschemes developed for perfect CSI depends not only on how good the channel estimationis but also on the outage pout requirements. This result shows that the power optimizationschemes designed based on perfect CSI, for example [44],[45], may fail to satisfy the QoSconstraints which are active, even in the presence of a very small error in CSI if the outage1063.5. Numerical Results10−5 10−4 10−3 10−2 10−100.10.20.30.40.50.60.70.80.91poutProbabilityofviolatingtheQoSconstraint σ2e = 0.001σ2e = 0.01σ2e = 0.1(a) Probability of violating the QoS constraint versus pout for different channelestimation error variances σ2e10−5 10−4 10−3 10−2 10−10246810121416poutAveragewastageoftransmitpowerperuser(dB) σ2e = 0.001σ2e = 0.01σ2e = 0.1(b) Average wastage of transmit power per user (dB) versus pout for differentchannel estimation error variances σ2eFigure 3.10: Performance comparison of our proposed scheme with the schemes developedbased on perfect CSI for lower values of channel estimation error variances and probability ofoutage pout requirements. Solid line: algorithms based on perfect CSI, dashed lines: proposedalgorithm1073.5. Numerical Resultsrequirement is strict. In Fig.3.10(a), we can see that almost 20% of the times the QoS isviolated for the scheme developed based on perfect CSI for error variance of σ2e = 0.001when the outage requirement is 10−5. The schemes based on perfect CSI outperform ourproposed scheme only if both the conditions, i.e., loose outage requirement and small errorvariance, are satisfied. Our proposed scheme is robust, and from Fig.3.10(b), we can see avery small wastage in power. The power wastage of our proposed scheme increases for higher(loose) pout requirements. However, low (tight) pout values are typically chosen in practicalapplications.Table 3.1: Comparison of the proposed admission control algorithm with random user re-ductionTotal usersProposed RandomUsers Served Power/User(dB) Users Served Power/User (dB)200 82 0.33 70 −1.09400 159 −0.13 131 −1.96600 243 −0.53 203 −2.14800 313 −0.77 269 −2.161000 381 −0.91 332 −2.17Next, we study the performance of the proposed user reduction algorithm given in Algo-rithm 2 with random user reduction algorithm when relay power budget exceeds its limit,i.e., we consider the scenario when users from a relay group must be removed. The resultsare shown in Table 3.1. In random user reduction, if pkmax, exceeds its limit then a user isremoved randomly from the group S(k). In our proposed algorithm, the criteria is to removethe user with largest relay power requirement p∗k,s from the set of users S(k). As expected,the number of admitted users increases with the increase of the total users in the network forboth schemes. However, the increment is much higher for our proposed algorithm, and onthe average, approximately 16.88% more users are admitted to the system than the randomselection case with a small increment in power requirements. Optimal solution for the jointadmission control and power allocation requires exhaustive search among all the users and1083.5. Numerical Resultsrelays, which is infeasible for our considered network with 1000 users and 8 relays.109Chapter 4Resource Optimization forCooperative Cognitive RadioNetworks with Imperfect CSIIn Chapter 2 and Chapter 3, we have addressed some resource allocation problems for fixedrelay based DF cooperative wireless networks under imperfect channel knowledge. In thischapter, we focus on robust resource optimization for CRN, where the SUs are operatingfrom small cell PU networks with the help of user cooperation.The accomplished works and important contributions on this chapter are briefly describedin the following.4.1 Accomplished Works and Research Contributions1) We consider a system model, where SUs operate from densely populated small cell net-works. It is difficult for the source and destination CR, which are far apart and operatingfrom two different small cell networks, to communicate directly. This is because PUs are inclose proximity and there are tight interference constraints on the PU bands. The SUs fromother small cells can act as relays and a communication link can be established among theSUs. In this chapter, we determine who should assist whom by incorporating efficient relayassignment algorithm to our considered problem.2) First, we solve the resource allocation problem for the idealistic case assuming that1104.1. Accomplished Works and Research Contributionsthe estimated channel is perfect, ignoring the error in estimation. We use this solution asthe baseline scheme and compare its performance with our proposed scheme, in which, amore practical scenario is considered regarding the availability of CSI. Similar to Chapter 3,we assume that training sequences are used to obtain the CSI of the SU-SU channels for ourproposed scheme and the resulting estimation error is unbounded Gaussian. Since the chan-nel ergodic (Shannon) capacity is unknown in general under imperfect CSI, ergodic capacitycannot be used as the performance measure. Although a lower bound of the ergodic capacitycan be obtained assuming estimation error as independent Gaussian noise [48],[89], it is nota useful measure when resources are optimized or QoS is provided over each transmissionframe in slow fading channels [6],[59]. This is because a channel outage occurs for a slowfading channel with CSI error whenever the transmit data rate exceeds the actual mutualinformation and ergodic capacity fails to capture it [59]. We consider the effect of channeloutage events and adopt the system goodput [59], [90], i.e., the number of information bitsper second per Hertz (Hz) successfully delivered to the receiver, as a performance measurefor our considered problem.3) Since the CSI between the SU transmitters and PU receivers are very difficult toobtain [47],[74],[77],[79], we consider the bounded channel uncertainty model developed inChapter 2 while modeling SU-PU channels. However, the bounded uncertainty region mod-eling is unsuitable for the cases when the error is unbounded [55]. Therefore, in this chapter,we consider both bounded and unbounded channel uncertainties. For the unbounded uncer-tainty, we assume a probabilistic model, where only the distribution type and the distributionparameters are known about the channels. One of the constraints of the resource allocationproblems for CRNs is to keep the interference received in the PU band below a specified limit.To satisfy the interference constraints, we develop two robust schemes. For the unboundeduncertainty, we formulate probabilistic constraints, where the interference constraints aresatisfied with certain probabilities. We consider the worst-case optimization based approachfor the bounded error model, where the interference constraints are satisfied for the worst1114.2. System Model and Problem Formulationcase scenario and hence for all channels that lie in a bounded region.4) We formulate a joint probabilistic optimization problem, which is in general difficultto solve. We approximate the probabilistic problem and transform it into a deterministicconvex form and then derive a closed form analytical solution for power allocation. Theclosed form power allocation solution helps us to develop an efficient relay selection algorithmbased on Hungarian method [91]. We conduct comprehensive simulations to demonstrate theeffectiveness of the proposed scheme. We also compare the results with the existing schemesthat ignore the uncertainties among different channels when designing resource allocationalgorithms for CRNs.The remainder of this chapter is organized as follows. The system model is introducedand the optimization problem is formulated in Section 4.2. The optimization framework andthe solution approach are presented in Section 4.3. The simulation results are demonstratedin Section 4.4.4.2 System Model and Problem FormulationConsider a CRN where the SUs are operating from small cell, for example, micro, pico, orfemto-cell PU networks. In this research, we put emphasis on the interference created by theSUs and assume a dedicated spectrum allocation technique as discussed in [92]-[93] to avoidinterference from the macro cell. In this approach, a fraction of the available frequency bandis used by the small cells and the other fraction is used by the macro cell. Companies suchas Comcast supports this type of deployment strategy, in which a part of the spectrum isdedicated to the WiMAX femtocells [92]. As shown in Fig. 4.1(a), the total licensed spectrumbandwidth (BW) is divided into two non-overlapping parts Bmacro and Bsmall, where Bmacrois used by the macrocell layer and Bsmall is used by the small cell layer. The total small cellBW Bsmall is further divided into T non-overlapping parts and each small cell BW containsL orthogonal channels to avoid the interference among the licensed small cell users.1124.2. System Model and Problem Formulationabcde cfgg hi jkbgg cfgg hihlmnophqhrhshtu v wwx ydz{e|e}bg c{b}}fg~} fbc{~kbgg cfgg b}hlm hlmnop(a) Dedicated spectrum allocation to avoid macro cell interferencesPUSUDesired signalInterference signalPrimay small cell BSCentral coordinator for CRNSmall cell band,B Small cell band, B Small cell band, BlŁ carrier in band B lŁ carrier in band BSU s SU k SU d(b) System Model: SU s transmits a message to SU k in the lth carrier ofband Bd and SU k, acting as relay, forwards that message to SU d in the lthcarrier of band Bs. A low-powered dual hop communication link is establishedbetween SU s and SU d via relay kFigure 4.1: System Model: SUs are operating from small cell PU networks1134.2. System Model and Problem FormulationAs shown in Fig. 4.1(b), a set of SU sources S = {1, 2, ..., s, ..., S} needs to communicatewith a set of SU destinations D = {1, 2, ..., d, ..., D}, where the source and destination nodesare far apart and operating from two different small cell PU networks with frequency bandsBs and Bd, respectively. We consider that SUs can use the spectrum bands of PUs suchthat the maximum amount of interference generated at the PU receivers is below a specifiedlimit. As PU receivers are much closer than the SU receivers to the SU transmitter, anytransmission of the SUs in their respective PU bands creates significant interference to thePU receivers. For example, a SU source s may not be able to communicate in band Bs andkeep the interference to the PU receivers within the limits. Since destination node d is farapart, a direct long distance transmission in band Bd requires high power, which may notbe available at the CRs. High power CR transmission may cause excessive interference tothe other intermediate small cell PU receivers operating in the same frequency band. As aresult, SU s may not be able to transmit directly to SU d in band Bd and therefore, we lookfor low-powered CR transmission, which does not create significant interference to the PUnetworks.Consider another set of SUs R = {1, 2, ..., k, ..., K}, which is located in between thesource and destination nodes and operates from a different PU network with frequency bandBk. These intermediate group of users can assist in establishing a communication betweenthe source and the destination nodes when they have no data to transmit for themselves. ASU source group S can transmit their messages in band Bd, which can be decoded by theintermediate relay group R. This transmission in band Bd keeps the interference received inboth the near PUs operating in band Bs and the far PUs operating in band Bd within limits.The relay group R decodes the messages sent by S and forwards them to the destination inband Bs or Bd depending on their location and the interference received in the PU band.However, the frequency band switching from Bd to Bs is beneficial as it allows full-duplexcommunication for the CRN [64]. Similarly, the destination group D can transmit theirmessages in band Bs and the relay group can forward them in band Bd, and a dual hop1144.2. System Model and Problem Formulationcommunication is possible with the help of the relays. We assume that the CRs can operatein different frequency bands [64].We further assume that the CRN has dedicated control channels. During the signalingperiod, which is a small portion of a transmission frame, SU-SU channels are estimatedfrom a training sequence and resource allocation is performed. Each transmission frameconsists of a block of messages and the channel impulse response is assumed to be timeinvariant (slow fading) within a frame. A central coordinator for the CRN performs theresource allocation and scheduling. We further assume that each relay can help only oneuser during one transmission frame of the CRN. This is a reasonable assumption as relaysare user equipments (UEs) and supporting multiple users require higher power, which is onlypossible for the fixed dedicated relay stations discussed in Chapter 2 and Chapter 3. Sincethe best relay selection scheme is optimal for DF scheme with orthogonal transmission anduseful in terms of complexity and overhead reduction, we consider that each CR user can beassisted only by a single relay.The dual hop communication between a user s and destination d via relay k consists oftwo phases. In the first phase, user s transmits the message xs to the relays in carrier l ofband Bd with power ps. Without loss of generality, we assume that the lth carrier of the PUband Bd and Bs is assigned to user s and relay k, respectively, before the data transmissionphase. The received message at relay k is given byyk =√pshs,kxs + zk, (4.1)where hs,k is the channel coefficient for the user s to relay k link. zk denotes the i.i.d. AWGNat the receiver with zero-mean and unit variance.Relay k decodes the received message and forwards it in the second phase to destinationd in carrier l of band Bs with power pk. The received message at destination d forwarded by1154.2. System Model and Problem Formulationrelay k is given byyd =√pkhk,dxk + zd, (4.2)where hk,d is the channel coefficient between relay k and destination d. zd denotes AWGNat the receiver with zero-mean and unit variance.PUs pu(d) and pu(s) that use the carrier l in band Bd and Bs, respectively, will alsoreceive the transmission from the SUs as interference during the first and second phase ofthe transmission. The interference received at the PUs can be expressed asypu(d) =√pshs,pu(d)xs, ∀pu(d) ∈ B(l)dypu(s) =√pkhk,pu(s)xk, ∀pu(s) ∈ B(l)s(4.3)respectively. hs,pu(d) and hk,pu(s) are the channel coefficients between SU s and PU receiverpu(d) and SU k and PU receiver pu(s), respectively. We denote B(l)d and B(l)s as the set ofPUs using the lth carrier of band Bd and Bs in different small cells, respectively.The received signal from the CRN, ypu(d) and ypu(s) will result in added interference tothe PU receivers in band Bd and Bs. To protect the rights of the PUs as licensed owners ofthe bandwidth, interference generated at the PU receivers should be kept below a specifiedlimit. There are two types of interference constraints: (i) per-channel or carrier interferenceconstraint and (ii) total interference constraint. Since per-carrier interference constraint ismore restrictive than that of the total interference constraint [47], we consider per-carrierinterference constraint for resource allocation in our research, which is given bygs,pu(d)ps ≤ IBd(l)th , ∀pu(d) ∈ B(l)dgk,pu(s)pk ≤ IBs(l)th , ∀pu(s) ∈ B(l)s(4.4)in which IBd(l)th and IBs(l)th are the maximum allowed per-carrier interference limit for eachCR transmission in carrier l of band Bd and Bs, respectively. gs,pu(d) = |hs,pu(d)|2 andgk,pu(s) = |hk,pu(s)|2 are the channel power gains from SU transmitters to PU receivers.1164.2. System Model and Problem FormulationConsidering half-duplex DF transmission [6], the mutual information of the links betweenuser s and relay k in the first time slot and between relay k and destination d in the secondtime slot are given byC(1)s,k =12 log2(1 + |hs,k|2 ps),C(2)k,d =12 log2(1 + |hk,d|2 pk),(4.5)respectively, where the factor 12 comes from two time slot transmission. |hs,k|2 and |hk,d|2 arethe channel power gains of the links between user s and relay k, and relay k and destinationd, respectively. We assume that there is no data buffer at the UEs of the CRN and themaximum achievable data rate of the dual hop link between user s and destination d islimited by the minimum data rate of the two hops.4.2.1 Baseline Scheme: Assuming Estimated CSI to be PerfectSimilar to [66]-[72], in this section, we assume that the estimated channel is perfect. Ourgoal is to provide the solution for the resource optimization problem considering the idealisticscenario and use it as the baseline scheme for performance comparison with our proposedscheme. Let us consider that the optimization objective is to maximize the weighted sumrate of the CRN. To achieve that we need to optimally select the relays and allocate powerfor CR transmission. In addition, we need to satisfy the QoS of the SUs such that theinterference received at the PU receivers remain within limits. The optimization problem1174.2. System Model and Problem Formulationcan be formulated asmaximize{ps,pk,xs,k}S∑s=1K∑k=1wsmin(Cˆ ′(1)s,k , Cˆ′(2)k,d )subject to :C1 : Cˆ ′(1)s,k ≥ xs,krs, ∀sC2 : Cˆ ′(2)k,d ≥ xs,krs, ∀kC3 : gˆs,pu(d)xs,kps ≤ IBd(l)th , ∀pu(d) ∈ B(l)d , ∀sC4 : gˆk,pu(s)xs,kpk ≤ IBs(l)th , ∀pu(s) ∈ B(l)s , ∀kC5 : 0 ≤ ps, pk ≤ xs,kPmax, ∀s, ∀kC6 : xs,k ∈ {0, 1}, ∀s, ∀kC7 :S∑s=1xs,k = 1, ∀kC8 :K∑k=1xs,k = 1, ∀s(4.6)where xs,k is the binary relay selection variable as indicated in C6. The constant weightsws are used to prioritize the users. The mutual information expressions with relay selectionvariable are given byCˆ ′(1)s,k =12 log2(1 + |hˆs,k|2xs,kps),Cˆ ′(2)k,d =12 log2(1 + |hˆk,d|2xs,kpk),(4.7)where hˆs,k and hˆk,d are the estimated channel coefficients between user s and relay k, andrelay k and destination d, respectively. gˆs,pu(d) and gˆk,pu(s) are the estimated channel powergains from SU transmitters to PU receivers. C1 and C2 are the QoS constraints, whichguarantee the target data rate rs of each user, if selected. Constraints C3 and C4 satisfythe per-carrier interference limit to the PU receiver for CR transmission in carrier l of bandBd and Bs, respectively. C5 is the non-negative individual power budget constraint for theUEs and limits the power to a maximum of Pmax. Since source and relays are UEs withindependent power supplies, we consider a more practical individual power constraint for1184.2. System Model and Problem Formulationsource and relays instead of a total power budget constraint. Constraint C7 and C8 restricteach relay to be used by a single user and each user to have a single relay. The optimizationproblem (4.6) is a joint optimization problem of relay selection and power allocation, whichcan be infeasible for some cases, for example, if the QoS requirements are high compared tothe available power at the CR UEs after satisfying the interference constraints at the PUs.Therefore, the solution of the problem exists if the power allocation solution satisfies thefollowing conditions: (i) QoS of each CR user, (ii) interference constraint for each PU, and(iii) maximum power budget of the CRs.In the following, we assume that the problem is feasible. To obtain the solution of (4.6),we first solve the following power allocation problem for each (s, k) pair, which is given by∀(s, k) ∈ S ×K : maximize{ps,pk}wsmin(Cˆ(1)s,k , Cˆ(2)k,d)subject to :C1 : Cˆ(1)s,k ≥ rsC2 : Cˆ(2)k,d ≥ rsC3 : gˆs,pu(d)ps ≤ IBd(l)th , ∀pu(d) ∈ B(l)dC4 : gˆk,pu(s)pk ≤ IBs(l)th , ∀pu(s) ∈ B(l)sC5 : 0 ≤ ps, pk ≤ PmaxC6 : |hˆs,k|2ps = |hˆk,d|2pk(4.8)where Cˆ(1)s,k and Cˆ(2)k,d are obtained by replacing the perfect CSI with the estimated channels in(4.5). Due to per-carrier interference constraint and individual power budget, the amount ofinformation transmitted from source to relay is not identical to that transmitted from relayto destination. This feature may cause data loss and power wastage in the network. Weinclude constraint C6 in (4.8) to make identical transmission rates in both the links to avoiddata loss and power wastage [94]. Constraint C6 helps to obtain a power efficient solutionwhen no data buffer is considered at the relays.1194.2. System Model and Problem FormulationThe problem in (4.8) is convex which can be easily solved using available algorithms [23]and the solution gives us optimal power p∗s and p∗k for each feasible (s, k) pair. To obtain theoptimal relay assignment, we form a S ×K matrix A = [a(s, k)], such thata(s, k) = wsmin(Cˆ(1)∗s,k , Cˆ(2)∗k,d ).We apply Hungarian algorithm [91] on matrix A to find the optimal (s∗, k∗) pairs thatmaximize the weighted sum rate of the CRN. Hungarian algorithm can find the optimalsolution for the linear assignment problems with complexity of O(N3). It also satisfiesconstraints C6, C7 and C8 of the joint optimization problem (4.6), i.e., each relay assists onlyone user and a user is served by a single relay.4.2.2 Proposed Scheme: Considering Imperfect CSIIn the baseline approach, we consider the estimated channels to be perfect while allocatingresources. The assumption of the availability of perfect CSI is not practical for our con-sidered cooperative CRN. Estimation errors and outdated CSI make the SU-SU channelsimperfect [48],[59]. To satisfy the interference constraints in the PU bands, the channelsbetween SU-PU need to be estimated. However, as mentioned earlier, it is extremely diffi-cult to estimate these channels perfectly. In the following, we formulate the robust resourceoptimization problem considering different channel estimation error models.SU to SU channel model with imperfectnessFor relay based CR transmission, we consider that training sequences are used to obtain theCSI of SU-SU channels and the resulting channel uncertainty is unbounded Gaussian [55].SU to SU transmission channel coefficients are modeled as h = hˆ+e, where hˆ is the estimatedchannel and e denotes the error in estimation. The error e is modeled as i.i.d and distributedas complex Gaussian with zero mean and variance given by σ2e . We also assume that the1204.2. System Model and Problem Formulationestimated channel hˆ and the error e are statistically independent.Using h = hˆ+ e, the mutual information expressions of (4.5) can be re-written asC(1)s,k = 12 log2(1 +∣∣∣hˆs,k + es,k∣∣∣2ps),C(2)k,d = 12 log2(1 +∣∣∣hˆk,d + ek,d∣∣∣2pk).(4.9)Under imperfect CSI, the channel ergodic (Shannon) capacity is unknown in generaland a lower bound of the ergodic capacity can be obtained assuming estimation error asindependent Gaussian noise. Lower bound of the ergodic capacity for source s to relay klink is given as follows [48]C(1)s,k(lower) =12 log2(1 +|hˆs,k|2psσ2es,kps +N0). (4.10)However, with CSI error, a channel outage occurs whenever the actual instantaneous mutualinformation drops below the target data rate rs given bymin(C(1)s,k , C(2)k,d) < rs, (4.11)even when channel capacity achieving coding is applied for error protection [59]. This isbecause the instantaneous mutual information expression in (4.9) is a RV due to the error inestimation. Lower bound of ergodic capacity (4.10) fails to capture the outage events. Thiscan be proved with a simple example. Consider a particular transmission frame with errores,k = −hˆs,k. For this particular error value, the actual instantaneous mutual information in(4.9) is zero; however, using the lower bound of the capacity in (4.10), we can still get a nonzero value. Thus, using this lower bound (4.10) as a system performance measure may notbe a good choice since resources are optimized over each transmission frame in our problem.What will be a reasonable objective function now and how can we fulfill the QoS re-quirements? Since the instantaneous mutual information expressions are RVs, the QoS1214.2. System Model and Problem Formulationrequirements can be fulfilled using probabilistic approach given a predetermined allowedprobability of outage, pout. For a target data rate rs, the outage probability of the sourceto relay link is defined as Pr.(C(1)s,k < rs), where Pr.(·) denotes the probability of the event(·). Whenever the actual instantaneous mutual information C(1)s,k falls below target data raters, the received message cannot be successfully decoded with probability 1, and the systemdeclares an outage.Now let us define the new system performance measure that captures the effect of outageevents. To model this effect in the optimization objective function, we adopt the systemgoodput [59], [90] as the performance measure and weighted system goodput as the objectivefunction of the problem. In the following, we first define the instantaneous goodputs U insts,kand U instk,d , i.e., bits per second per Hz successfully delivered to relay k and destination d,respectively asU insts,k = rs × 1(rs ≤ C(1)s,k),U instk,d = rs × 1(rs ≤ C(2)k,d),(4.12)where 1(·) is an indicator function, i.e., the function value is 1 if (·) is true and zero otherwise.The average goodput Ugoodputs,k for each (s, k) pair can be defined as minimum of the totalaverage bits per second per Hz successfully delivered to relay k and destination d, which isgiven byUgoodputs,k = E{min(U insts,k , U instk,d )}= Ehˆs,k,hˆk,d ×{min(rs Ehˆs,k ×[1(rs ≤ C(1)s,k |hˆs,k)], rs Ehˆk,d ×[1(rs ≤ C(2)k,d|hˆk,d)])}= Ehˆs,k,hˆk,d ×{min[rs × Pr.(rs ≤ C(1)s,k |hˆs,k), rs × Pr.(rs ≤ C(2)k,d|hˆk,d)]}.(4.13)Ehˆ(·) denotes the statistical expectation with respect to the RV hˆ. Finally, by incorporatingthe channel outage events, we define a new objective function of the problem as the averageweighted system goodput,Ugoodput =S∑s=1K∑k=1ws Ugoodputs,k . (4.14)1224.2. System Model and Problem FormulationSU to PU channel models with uncertainty(a) Model-1: ProbabilisticIn the baseline approach, we assume that the instantaneous channels between a CR trans-mitter and PU receiver are estimated perfectly. However, this assumption is impractical [76].In this subsection, we assume that CR transmitters cannot estimate the instantaneous chan-nels and have information only about the distribution type and the corresponding distribu-tion parameters of the channel gains among the CR transmitter to the PU receivers. Weconsider a probabilistic model, which requires only the knowledge of channel gain statisticsinstead of instantaneous CSI. We assume that the CR transmitters have some knowledge ofthe transmit power of the PUs and hence can obtain the mean channel gains from PUs tothemselves by exploiting the pilot signals from the PUs [76]. Using the reciprocal propertyof the wireless channels, some statistics of the SU-PU channels can be obtained [74], [76].In this model, the channel uncertainty is unbounded and interference constraints C3 and C4are guaranteed with certain probabilities.Suppose the SU to PU channels hs,pu(d) and hk,pu(s) are statistically modeled as indepen-dent CSCG RVs with zero mean and variances σ2s,pu(d), and σ2k,pu(s), respectively. We assumethat only the mean and the variance of the channels are known to the CRN. Since the chan-nel coefficients are distributed as CSCG, their channel power gains gs,pu(d) and gk,pu(s) areexponentially distributed with parameters λs,pu(d) = 1/(σ2s,pu(d)) and λk,pu(s) = 1/(σ2k,pu(s)),respectively.(b) Model-2: Bounded UncertaintyIn this case, we assume that the uncertainty among the SU-PU channels is boundedand no statistical knowledge is available about it. We use the bounded uncertainty modelpresented in Chapter 2 and model the channel as h = hˆ + , where is the estimation errorwhich determines how far the actual channel h can extend in both real and imaginary partsfrom the estimated value hˆ. This model can be viewed as deterministic by considering theworst case error, i.e., || = max [73]. The interference constraints C3 and C4 are satisfied for1234.2. System Model and Problem Formulationthe worst case and hence guaranteed for all channels [95].Using similar techniques presented in Section (2.2.2) of Chapter (2), the channel powergains to the PU receiver in the lth carrier of band Bd and Bs from the CR transmitter s andk can be written asgs,pu(d) ∈ Ls,pu(d) = {gˆs,pu(d) + δmaxs,pu(d)v | |v| ≤ 1},gk,pu(s) ∈ Lk,pu(s) = {gˆk,pu(s) + δmaxk,pu(s)v | |v| ≤ 1},(4.15)wheregˆs,pu(d) = hˆs,pu(d)hˆ∗s,pu(d),gˆk,pu(s) = hˆk,pu(s)hˆ∗k,pu(s),δmaxs,pu(d) = max∗max + 2|<(hˆs,pu(d)∗max)|,δmaxk,pu(s) = max∗max + 2|<(hˆk,pu(s)∗max)|.Problem formulation for the proposed scheme with Imperfect CSIIn this section, we re-formulate the resource allocation problem given in (4.6) taking intoaccount the estimation errors in the SU-SU channels and the uncertainties among the SU-PUchannels. We follow the same solution approach as the baseline scheme, i.e., we first solve thepower allocation problem for each feasible (s, k) pair and then find the optimal assignmentpairs (s∗, k∗).Considering CSI error, power allocation problem becomes probabilistic, which can be1244.3. Optimization Framework for the Proposed Schemewritten as∀(s, k) ∈ S ×K : argmax{ps,pk}Ugoodputs,ksubject to :C1 : Pr.(rs > C(1)s,k |hˆs,k)≤ poutC2 : Pr.(rs > C(2)k,d|hˆk,d)≤ poutC3 : Pr.(gs,pu(d)ps ≤ IBd(l)th)≥ c, ∀pu(d) ∈ B(l)dC4 : Pr.(gk,pu(s)pk ≤ IBs(l)th)≥ c, ∀pu(s) ∈ B(l)sC5 : 0 ≤ ps, pk ≤ Pmax(4.16)where C1 and C2 are the QoS constraints, which guarantee the target data rate rs of user swith a probability of 1−pout. Constraints C3 and C4 incorporate Model-1 of SU-PU channeluncertainty. These constraints satisfy the interference to PU bands with a probability c, apredefined constant known by the CRN.For Model-2 that considers bounded uncertainty, constraints C3 and C4 can be writtenfrom (4.15) considering the worst case scenario asC3 : sup{gs,pu(d)ps | gs,pu(d) ∈ Ls,pu(d)} ≤ IBd(l)th , ∀pu(d) ∈ B(l)dC4 : sup{gk,pu(s)pk | gk,pu(s) ∈ Lk,pu(s)} ≤ IBs(l)th . ∀pu(s) ∈ B(l)s(4.17)Note that the supremum of a set, sup{·}, captures the essence of worst case optimization.In the following section, we provide an efficient solution approach for the probabilistic opti-mization problem given in (4.16)-(4.17).4.3 Optimization Framework for the ProposedSchemeIn this section, we develop the optimization framework for the formulated probabilistic op-timization problem (4.16)-(4.17). Our goal is to transform the probabilistic problem to1254.3. Optimization Framework for the Proposed Schemedeterministic and develop a computationally efficient power allocation algorithm. Once thepower allocation solution is obtained, we can determine the optimal relay assignment for thetransformed problem based on the Hungarian algorithm.In order to solve the problem efficiently, we incorporate the outage-probability (or, QoS)constraints C1 and C2 of (4.16) into the objective function. This can be achieved if theseconstraints are fulfilled with equality at the optimal point, which is generally the case inpractical applications with low outage probability requirement [59]. In the following, weassume that the QoS constraints are active and the resulting optimization problem can beviewed as a more constrained version of the original problem in (4.16). Therefore, the firstQoS constraint can be written asC1 : Pr.(rs > C(1)s,k |hˆs,k)= pout. (4.18)Using (4.9), C1 of (4.18) can be expressed asPr.(rs >12 log2(1 +∣∣∣hˆs,k + es,k∣∣∣2ps)|hˆs,k)= pout,Pr.(∣∣∣hˆs,k + es,k∣∣∣2≤ (22rs − 1)ps|hˆs,k)= pout,(4.19)where χ22s,k =∣∣∣hˆs,k + es,k∣∣∣2is a non-central Chi-square distributed RV with non-centralityparameter, λ2s,k =|hˆs,k|2σ2es,k/2and degrees of freedom n = 2. The left-hand side of (4.19) can beevaluated as [82]Fχ22s,k (x) = 1−Q1λs,k,√22rs − 1/ps√σ2es,k/2 , (4.20)whereQ1(i, j) = e−(i2+j2)/2∞∑k=0( ij)kIk(ij), j > i > 0. (4.21)and Ik(ij) is the kth-order modified Bessel function of the first kind.By using the inverse of the CDF of noncentral Chi-square RV, (4.19) can be written as1264.3. Optimization Framework for the Proposed Schemea deterministic constraint, which includes the infinite sum of (4.21). The inverse functionof the CDF can be evaluated using a lookup table or software packages [59]. However, theapproach does not give useful insight about the resource allocation. Also, a closed-formanalytical expression for efficient resource allocation under practical scenarios can not bederived.To obtain a closed-form result and some insight into the power allocation policy, we usethe approximation of (3.13) of Chapter (3), given byPr.(χ22s,k ≤22rs − 1ps)≈ Pr.(χ22s,k(0) ≤(22rs − 1) /ps1 + λ2s,k/2). (4.22)Since χ22s,k(0), the central Chi-square distribution with degrees of freedom n = 2 yieldsexponential distribution, (4.22) can be computed asPr.(χ22s,k(0) ≤(22rs − 1) /ps1 + λ2s,k/2)= 1− e−(22rs − 1) /ps2(1 + λ2s,k/2). (4.23)Using the expression of (4.23) in (4.19), the deterministic constraint can be written as1− e−(22rs − 1) /ps2(1 + λ2s,k/2)= pout. (4.24)From (4.24), we can obtain the guaranteed QoS considering channel estimation error in SUsource to relay links asrs =12log2(1 + psαs,k), (4.25)where αs,k = [−ln(1− pout)] × 2(1 + λ2s,k/2).Similarly, from the second constraint, i.e.,C2 : Pr.(rs > C(2)k,d|hˆk,d)= pout, (4.26)1274.3. Optimization Framework for the Proposed Schemewe obtainrs =12log2(1 + pkαk,d), (4.27)where αk,d = [−ln(1− pout)] × 2(1 + λ2k,d/2).Since the power and interference constraints are instantaneous in (4.16), the averagegoodput Ugoodputs,k maximization is equivalent to the maximization of the instantaneous good-put for each set of channel power gains, although both criteria are different in general [59].Therefore, we can writeargmax{ps,pk}Ugoodputs,k= Ehˆs,k,hˆk,d ×{min(rs × Pr.(rs ≤ C(1)s,k |hˆs,k), rs × Pr.(rs ≤ C(2)k,d|hˆk,d))}≡ argmax{ps,pk}{min(rs × Pr.(rs ≤ C(1)s,k |hˆs,k), rs × Pr.(rs ≤ C(2)k,d|hˆk,d))}= argmax{ps,pk}min{(1− pout)12log2(1 + psαs,k), (1− pout)12log2(1 + pkαk,d)},(4.28)where “≡” denotes the equivalence relation. Pr.(rs ≤ C(1)s,k |hˆs,k)= Pr.(rs ≤ C(2)k,d|hˆk,d)=(1− pout) is obtained from (4.18) and (4.26).4.3.1 Optimization Framework for Model-1 : ProbabilisticConstraintsThe probabilistic interference constraints are given in (4.16) asC3 : Pr.(gs,pu(d)ps ≤ IBd(l)th)≥ c, ∀pu(d) ∈ B(l)dC4 : Pr.(gk,pu(s)pk ≤ IBs(l)th)≥ c, ∀pu(s) ∈ B(l)s .(4.29)1284.3. Optimization Framework for the Proposed SchemeSince gs,pu(d) is exponentially distributed with parameter 1/(σ2s,pu(d)), the left hand side ofthe first constraint in (4.29) can be expressed asPr.(gs,pu(d)ps ≤ IBd(l)th)=∫ IBd(l)th01σ2s,pu(d)pse−xσ2s,pu(d)psdx, (4.30)and evaluated asPr.(gs,pu(d)ps ≤ IBd(l)th)= 1− e−IBd(l)thσ2s,pu(d)ps. (4.31)Therefore, the constraints in (4.29) becomeps ≤IBd(l)thσ2s,pu(d){−ln(1− c)}, ∀pu(d) ∈ B(l)dpk ≤IBs(l)thσ2k,pu(s){−ln(1− c)}, ∀pu(s) ∈ B(l)s ,(4.32)which are deterministic and linear in terms of ps and pk. Therefore, the problem (4.16) canbe written as a deterministic optimization problem given by∀(s, k) ∈ S ×K : argmax{ps,pk}min{(1− pout)2 log2(1 + psαs,k),(1− pout)2 log2(1 + pkαk,d)}subject to :C1 : ps ≤IBd(l)thσ2s,pu(d){−ln(1− c)}, ∀pu(d) ∈ B(l)dC2 : pk ≤IBs(l)thσ2k,pu(s){−ln(1− c)}, ∀pu(s) ∈ B(l)sC3 : 0 ≤ ps, pk ≤ Pmax.(4.33)1294.3. Optimization Framework for the Proposed Scheme4.3.2 Optimization Framework for Model-2 : BoundedUncertainty ConstraintsFor the constraints given in (4.17) considering Model-2 with bounded uncertainty region, wepropose worst case optimization approach in which the interference constraints are satisfiedfor all possible values of the channel power gains that lie in some bounded region. Therefore,the first constraint in (4.17) can be expressed assup{gs,pu(d)ps | gs,pu(d) ∈ Ls,pu(d)} ≤ IBd(l)th , ∀pu(d) ∈ B(l)d⇒ sup{(gˆs,pu(d) + δmaxs,pu(d)v)ps | |v| ≤ 1} ≤ IBd(l)th , ∀pu(d) ∈ B(l)d⇒ gˆs,pu(d)ps + |δmaxs,pu(d)|ps ≤ IBd(l)th . ∀pu(d) ∈ B(l)d(4.34)Since δmaxs,pu(d) is always positive, we can write |δmaxs,pu(d)| = δmaxs,pu(d). Similarly the second inter-ference constraint given in (4.17) can be written asgˆk,pu(s)pk + δmaxk,pu(s)pk ≤ IBs(l)th . ∀pu(s) ∈ B(l)s (4.35)4.3.3 Solution ApproachIt can be proved that the robust power optimization problem in (4.33)- (4.35) is convex,which can be solved efficiently. To obtain the solution, we write the problem in its epigraph1304.3. Optimization Framework for the Proposed Schemeform given by [23]∀(s, k) ∈ S ×K : argmax{ps,pk,zs,k}min zs,ksubject to :C1 :(1− pout)2 log2(1 + psαs,k) ≥ zs,kC2 :(1− pout)2 log2(1 + pkαk,d) ≥ zs,kC3 : 0 ≤ ps ≤ min(IBd(l)thσ2s,pu(d){−ln(1− c)}, Pmax), ∀pu(d) ∈ B(l)dC4 : 0 ≤ pk ≤ min(IBs(l)thσ2k,pu(s){−ln(1− c)}, Pmax), ∀pu(s) ∈ B(l)sC5 : psαs,k = pkαk,d.(4.36)For Model-2, interference constraints C3 and C4 of (4.36) are given byC3 : 0 ≤ ps ≤ min(IBd(l)thgˆs,pu(d) + δmaxs,pu(d), Pmax), ∀pu(d) ∈ B(l)dC4 : 0 ≤ pk ≤ min(IBs(l)thgˆk,pu(s) + δmaxk,pu(s), Pmax), ∀pu(s) ∈ B(l)sSimilar to constraint C6 in (4.8), constraint C5 in (4.36) ensures identical data rate onboth links and hence, avoids any loss of data and power wastage. The algorithm to obtainthe solution for the problem (4.36) is given in Algorithm 3.1314.3. Optimization Framework for the Proposed SchemeAlgorithm 3 : Power allocation for problem (4.36)Require: αs,k, αk,d, IBd(l)th , IBs(l)th , c, Pmax, σ2s,pu(d),σ2k,pu(s), gˆs,pu(d), gˆk,pu(s), δmaxs,pu(d), δmaxk,pu(s).1: Calculate Pows and PowkFor Model-1 :Pows = min(IBd(l)thσ2s,pu(d){−ln(1− c)}, Pmax);Powk = min(IBs(l)thσ2k,pu(s){−ln(1− c)}, Pmax);For Model-2 :Pows = min(IBd(l)thgˆs,pu(d) + δmaxs,pu(d), Pmax);Powk = min(IBs(l)thgˆk,pu(s) + δmaxk,pu(s), Pmax);2: if(Powsαs,kαk,d≤ Powk)then3: p∗s = Pows , p∗k =Powsαs,kαk,d4: else5: p∗k = Powk , p∗s =Powkαk,dαs,k6: end ifIn Algorithm 3, we obtain the closed form analytical expression for optimal p∗s and p∗kfor user-relay (s, k) pair for problem (4.36).Similar to the baseline approach, the optimal relay assignment can be obtained by forminga S ×K matrix B = [b(s, k)], such thatb(s, k) = ws[(1− pout)12log2(1 + p∗sαs,k)]= ws[(1− pout)12log2(1 + p∗kαk,d)].By applying Hungarian algorithm [91] on matrix B, we obtain the optimal (s∗, k∗) pairs thatmaximize the weighted goodput of the CRN.1324.4. Numerical Results4.4 Numerical ResultsWe have conducted extensive simulations to show the effectiveness of our proposed algo-rithm and compare its performance with that of the baseline scheme, which ignores theimperfectness in CSI. We have considered different system parameters to analyze the influ-ence of different channel estimation errors on the performance of the system. Frequency-flatRayleigh fading channels are assumed in all simulations. Each of the channel coefficientsis an independent CSCG RV and distributed as CN(0, 1(1 + d)α)[87], where d is the nor-malized distance among various nodes in the network and α = 4 is a constant (path losscoefficient). The bounded uncertainty region for Model-2 is considered to be rectangularwith four equal sides. The uncertainty distance, i.e., the distance from the center of therectangle to the center of each side (or, equivalently half length of a side) is given by r.Same channel estimation error variance σ2e is assumed for all SU to SU channels. We haveconsidered that the maximum per-carrier interference limit in both band Bd and Bs is thesame, i.e., IBd(l)th = IBs(l)th = Ith. Since both the source and the relays are UEs, the maximumpowers available are assumed to be the same. Unless stated otherwise, probability of outage,pout, is assumed to be 0.1, equal priority is given to all the users by setting ws = 1, and themaximum power, Pmax and interference threshold limit, Ith, are set at 1 Watt and 0.1 Watt,respectively. The results are averaged over 10,000 independent channel realizations.Figs. 4.2(a), 4.2(b), and 4.2(c) show the average system goodput versus the uncertaintydistance r, probability of satisfying the interference constraints c, and the channel estimationerror variance σ2e of the SU channels, respectively, for different number of users S. We seethat average goodput decreases with the increase of r, c, and σ2e and increases with theincrease of the number of users S. The effect of error variance σ2e of the SU channels is moresignificant at the lower values of r and c and at higher number of users. This is becauselow r values indicate better estimation of the channels between SU transmitters and PUreceivers. Lower c values allow higher transmission capability of the CRN as the interference1334.4. Numerical Results10−3 10−2 10−1 1000123456Bounded channe l uncertainty, 0 rAveragegoodput(bits/sec/Hz) S=1S=2S=4S=8Solid line s : σ 2e=0.02Dashed line s : σ 2e=0.01(a) Average goodput versus uncertainty distance, r amongthe channels of SU transmitters to PU receivers0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.9501234567Probability of sat is fy ing the inte r f e rence constraints , cAveragegoodput(bits/sec/Hz) S=1S=2S=4S=8Solid line s : σ 2e=0.02Dashed line s : σ 2e=0.01(b) Average goodput versus probability of satisfying theinterference constraints c10−3 10−2 10−10246810Channe l e st imat ion error var iance , σ 2eAveragegoodput(bits/sec/Hz) S=1S=2S=3S=4S=5S=6c = 0.8(c) Average goodput versus channel estimation error vari-ance σ2e of SU-SU channelsFigure 4.2: Average goodput versus different uncertainty models among the SU-PU channelsand SU-SU channels1344.4. Numerical Results−10 −8 −6 −4 −2 0 2 4123456I nte r f e rence there shold, I t h (dB)Averagegoodput(bits/sec/Hz) pmax=1, S=2pmax=2, S=2pmax=1, S=4pmax=2, S=4Figure 4.3: Average goodput versus interference threshold Iththreshold margin becomes more flexible. Similarly, with the increase of users, more poweris transmitted by the CRN. Since the average goodput is proportional to the transmissionpower and inversely proportional to σ2e , the effect of estimation error in the SU-SU channelsbecomes more detrimental at high power regimes.Fig. 4.3 shows the average goodput versus interference threshold Ith for Model-1 thatconsiders probabilistic uncertainty model among SU-PU channels. Results show the impactof power budget Pmax and number of users S on system goodput, considering c = 0.8 andσ2e = 0.01. We see that average goodput increases with the increase of both Pmax and S.For Pmax = 1 Watt and S = 2, average goodput increases with the increase of Ith untila maximum goodput floor is reached. This is because the interference constraints becomeinactive in that region, for example, beyond Ith = −2 dB for this case. The active region ofthe interference constraints increases with the increase of both Pmax and S. For example,with Pmax = 2W,S = 2 and Pmax = 1W,S = 4 the goodput floor is reached approximatelyat 0 dB and 2 dB, respectively.In Figs. 4.4(a), 4.4(b), and 4.4(c), average goodput versus interference threshold Ith areshown by varying the probability c, uncertainty distance r, and error variance σ2e , respec-1354.4. Numerical Results−10 −8 −6 −4 −2 0 2 400.511.522.5I nte r f e rence there shold, I t h (dB)Averagegoodput(bits/sec/Hz) c=0.9c=0.8c=0.7c=0.6S=2S=1(a) Effect of probability of satisfying the interference con-straints c on goodput at different regions of Ith−10 −8 −6 −4 −2 0 2 400.511.522.5I nte r f e rence there shold, I t h (dB)Averagegoodput(bits/sec/Hz) 0 r=0.20 r=0.10 r=.01S=1S=2(b) Effect of bounded channel uncertainty r on goodput atdifferent regions of Ith−10 −8 −6 −4 −2 0 2 400.20.40.60.811.21.4I nte r f e rence there shold, I t h (dB)Averagegoodput(bits/sec/Hz) σ 22 = 0.2σ 22 = 0.1σ 22 = 0.05S=1S=2(c) Effect of error variance σ2e of SU-SU channels on good-put at different regions of IthFigure 4.4: Effect of probability of satisfying the interference constraints c, bounded channeluncertainty r and error variance σ2e of SU-SU channels, respectively on average goodput atdifferent regions of interference threshold, Ith1364.4. Numerical Results−5 0 5 10 15 2000.20.40.60.81I nte r f e rence there shold, I t h (dB)Probabilityofviolatingtheinterferenceconstraints S=1S=2S=3S=4S=5Dashed l i ne : Base l i ne me thodSol i d l i ne : P roposed methodFigure 4.5: Probability of violating the interference constraints versus interference thresholdIthtively, considering Pmax = 2 Watt. In Figs. 4.4(a) and 4.4(b), we see that c and r havemore impact on the goodput for lower values of Ith where the interference constraints areactive. In that region, transmit powers are calculated from interference constraints whichdepend largely on c and r. Also, we observe that with increase of the number of users S, theactive region of Ith increases. As expected, the average goodput decreases with the increaseof c and r. In Fig. 4.4(c), we can see that when interference constraints become inactive,goodput depends largely on the estimation error σ2e of the SU-SU channels. Increasing num-ber of cooperating users can improve the system goodput in the region where interferenceconstraints are inactive.Fig. 4.5 shows the probability of violating the interference constraints given in (4.8) and(4.33) versus interference threshold Ith for different number of users. In this simulation, weconsider that some uncertainty is associated with the channels of SU transmitters to PUreceivers and compare the performance of our proposed scheme with the baseline schemewhich assumes the estimated channel to be perfect. We observe frequent violations of theconstraints for the baseline scheme in the active region of Ith for ignoring the channel un-certainty in the design process. Since increasing the number of users use more power to1374.4. Numerical Resultsimprove the system goodput, the probability of violation increases with the increase of thenumber of users. Similar trends of interference violation can be observed for the resourceallocation schemes proposed in [68]-[72]. For our proposed probabilistic interference scheme,i.e., Model-1, we set c = 0.9, that means we allow the interference constraints to be violatedwith a probability of 0.1 when the constraints are active. As a result, we see that the max-imum value of interference violation is 1 − c = 0.1, which is much lower than the baselinescheme. Model-2, i.e., the bounded uncertainty model is designed to satisfy the interferenceconstraints for the worst case scenario. As a result, no violation is observed for our proposedscheme when the uncertainty lies in some bounded region.Figs. 4.6(a) and 4.6(b) show the implications of ignoring estimation errors in the SU-SUchannels while providing QoS over each transmission frame. In Figs. 4.6(a) and 4.6(b), wecompare our proposed scheme to the baseline scheme. Fig. 4.6(a) shows the probabilityof satisfying the QoS versus the error variance σ2e , of the SU-SU channels. For our pro-posed scheme, we consider probability of QoS outage pout is 0.1. Since we satisfy the QoSconstraints with equality, we obtain a QoS satisfaction probability of 1 − pout = 0.9, whichremains constant throughout the considered values of σ2e . For the baseline scheme, an outageoccurs whenever the transmission rate with the estimated channel exceeds the actual mu-tual information C, i.e., Pr.(min(Cˆ(1)s,k , Cˆ(2)k,d) > C), where C = min(C(1)s,k ,C(2)k,d). We observefrequent violations of the QoS constraints C1 and C2 in (4.8) for not considering the estima-tion errors among the SU-SU channels in the design process. The probability of violationincreases with the increase of error variance. This result shows that the power allocationalgorithms designed based on perfect CSI of the SU-SU channels, for example [68]-[76], mightfail to satisfy any QoS constraint, even in the presence of a small error in the CSI.In Fig. 4.6(b), we show the wastage of transmit power per user versus the error variance ofthe SU-SU channels for different sizes of uncertainty regions, i.e., different r. Power wastageis defined as the transmit power that is insufficient to meet the required QoS at the desti-nation. The simulation is performed at the region where Ith is active. Therefore, the power1384.4. Numerical Results10−4 10−3 10−2 10−1 100 10100.20.40.60.81Channe l e st imat ion error var iance , σ 2eProbabilityofsatisfyingtheQoS Base l i ne me thodP roposed method(a) Probability of satisfying the QoS versus channel estimation error varianceσ2e10−4 10−3 10−2 10−1 100 10100.40.8Channe l e st imat ion error var iance , σ 2ePowerwastageperuser(Watt) 0 r = 0 .010 r = 0 .1Dashed l i ne : Base l i ne methodSol i d l i ne : P roposed method(b) Wastage of transmit power per user versus channel estimation error vari-ance σ2eFigure 4.6: The effect of ignoring estimation errors on QoS satisfaction and wastage oftransmit power1394.4. Numerical Results10−3 10−2 10−1 100 10100.20.40.60.81Channe l e st imat ion error var iance , σ 2eProbabilityofselectingthecorrectrelay Proposed methodBase l i ne me thodFigure 4.7: Probability of selecting the correct relay versus SU-SU channel estimation errorvariance σ2ewastage of our proposed scheme can be written as[Ithgˆs,pu(d) + δmaxs,pu(d)+ Ithgˆk,pu(s) + δmaxk,pu(s)]×pout. Since, pout = 0.1 is constant for our scheme, the power wastage due to outage re-mains constant throughout. With the increase of uncertainty between the SU-PU chan-nels, our proposed scheme becomes more conservative in power allocation, as a result, thepower wastage decreases. For the baseline scheme alongside the other schemes [68]-[72]developed ignoring the estimation error in SU-SU channels, power wastage is calculatedas[ Ithgˆs,pu(d)+ Ithgˆk,pu(s)]× Pr.(min(Cˆ(1)s,k , Cˆ(2)k,d) > C). We see significant wastage of transmitpower for the baseline scheme, which increases with the increase of the estimation error inthe SU-SU channels. As error variance increases, the probability of failure to provide theQoS, i.e., Pr.(min(Cˆ(1)s,k , Cˆ(2)k,d) > C)increases. Since the baseline scheme also ignores theuncertainty between the channels of SU and PU, the performance degrades further with theincrease in r.Fig. 4.7 shows how important it is to take into account the estimation errors in theSU-SU channels while designing the relay selection algorithms for cooperative CRN. In thisexperiment, we consider a single source-destination pair with two relays equidistant from1404.4. Numerical Resultsboth the source and the destination for a particular channel realization. For the channel torelay 1, the error variance is kept constant at 10−3 while the error variance for the channelto relay 2 is increased from 10−3 to 10. Fig. 4.7 shows that our proposed method selects thecorrect relay almost all the time. For the baseline scheme, it selects the correct relay onlywhen the error is very small, i.e., σ2e < 10−2. However, we see that the probability of correctselection decreases from 100% to 50% when the error variance reaches from σ2e = 10−3 toσ2e = 1.141Chapter 5Conclusions and Future ResearchDirections5.1 ConclusionsIn this thesis, we have studied resource optimization problems for different relay-based co-operative wireless systems including fixed-relay based DF cooperative wireless network anduser cooperation based CRN. We have addressed the problem of power allocation, relay se-lection, and admission control with QoS provisioning for users in a practical scenario underchannel uncertainty. While developing robust resource allocation schemes, we have consid-ered both bounded channel uncertainty and unbounded Gaussian channel estimation errormodels. Our proposed schemes are not only robust but also efficient in terms of algorith-mic complexity and are applicable to large scale networks. In the following, we present theconcluding remarks on the contributions made in this thesis.First, in Chapter 2, we have proposed power allocation and relay selection frameworksthat can work robustly under imperfect channel knowledge in DF based cooperative wirelessnetworks. We have considered a bounded uncertainty region model assuming that no statis-tical knowledge about the error is available. We have applied worst case optimization basedapproach to the resource allocation problems, where the objective has been minimizing trans-mit power consumption of the system with guaranteed QoS provisioning to the users. In thisoptimization framework, we have successfully derived equivalent convex formulation for therobust optimization problem. We have then proposed both centralized as well as distributed1425.1. Conclusionsschemes that can efficiently solve these optimization problems in practical network deploy-ments. From distributed power allocation solution, we have concluded that employing morethan one relay to assist a user increases the overhead and implementation complexity whichnegates the possible benefit of distributed implementation. We have also considered thecase of power constrained networks, where the objective has been max-min rate or weightedsum rate maximization in the presence of limited power budgets on source and relays andprovided efficient solution approaches. Finally, a robust relay selection and power allocationalgorithm has been developed with guaranteed QoS provisioning. The proposed algorithm issemi-distributed, in which the power allocation problem is solved at the relays and the bestrelay for each user is selected at the central processing node. Closed form analytical solutionsfor the best relay selection and power allocation problem have been provided. Numerical re-sults demonstrate the effectiveness of our proposed schemes. Simulation results are broughtin Chapter 2 to show how important it is to take into account the channel estimation errorswhile designing the relay selection and power allocation algorithms. Numerical results revealthat the optimal resource allocation schemes based on perfect CSI, for example [44]-[45],might fail to choose the correct relay and satisfy the QoS constraints in the presence of avery small error in CSI. The results of such non-robust solutions are retransmissions andwastage of transmit power.Second, in Chapter 3, we have proposed joint admission control and power allocationscheme for selective relaying based DF cooperative cellular systems in the presence of imper-fect CSI. Unlike the bounded channel uncertainty model proposed in Chapter 2, we considerthat some statistics are known about the channel estimation error. Assuming estimation er-ror as unbounded Gaussian, we have designed resource allocation schemes using probabilisticoptimization based approach. This approach is more flexible and less conservative than theworst case optimization approach proposed in Chapter 2.In Chapter 3, we have derived closed form approximate solutions to the problem ofmeeting probabilistic QoS guarantees to admitted users while minimizing the overall uplink1435.1. Conclusionstransmit power consumption in cooperative wireless networks. Our proposed solution is novelsince closed form analytical solutions for a probabilistic optimization problem taking the un-certainty arising from CSI errors into account are missing in the literature. Due to closedform analytical power allocation solutions, our algorithm can tackle large scale problemsefficiently unlike the schemes developed in [37]-[38],[55] that require solving polynomial timeoptimization problems, for example, SOCP and SDP, and hence, applicable only for smallscale networks. We have utilized the benefit of this closed form power allocation solution todevelop an algorithm that can solve the joint admission control and power allocation prob-lem very efficiently under practical network deployments with worst case linear complexity.This is usually combinatorially hard problem to solve in the optimal sense. However, ourproposed suboptimal approach is robust and can handle large scale networks easily unlikethe existing solutions that are sensitive to channel estimation errors [80]-[81] and applicableonly for small scale networks [43],[80]. Finally, we have demonstrated the effectiveness of theproposed algorithms through extensive simulation experiments and the importance of con-sidering channel estimation error while developing resource allocation algorithms. Resultsshow that our proposed scheme guarantees the QoS while maintaining a predefined outageprobability, whereas frequent violations are observed for the schemes that do not considerthe channel estimation error in their design process. Results also show that the wastage ofpower due to retransmissions is extremely small for our proposed scheme compared to thenon-robust schemes, where the availability of perfect CSI is assumed [44],[45].Finally, in Chapter 4, we have developed robust power allocation and relay selectionschemes for a CRN, where the SUs communicate with each other from different small cell PUnetworks. The communication among the SUs can be established via user cooperation in sucha CRN, where interference constraints in the PU band are tight, and PU receivers are closerthan SU receivers. We have developed efficient power allocation and relay selection schemeswith the provision of QoS to each SU considering different channel uncertainty models inthe network. We have considered not only the uncertainties in the channels between SU1445.2. Future Research Directionsand PU, but also the estimation errors in the SU-SU relay channels. Channel outage events,which have resulted from the imperfect CSI under slow fading channels, have been taken intoaccount in our algorithm. System goodput of the CRN has been maximized while satisfyingthe interference constraints for the PU bands both probabilistically and for the worst casescenario. The joint optimization problem is probabilistic and non-convex, which is verydifficult to solve in its original form. The problem has been approximated and transformedinto a convex deterministic form and a closed form analytical solution for power allocationhas been derived. The closed form power allocation solution helped us to develop an efficientrelay selection scheme based on Hungarian algorithm. Simulation results demonstrate theeffectiveness of our proposed scheme. Our results reveal that the implications of ignoringthe imperfectness among different channels are violations of interference constraints in thePU bands and failure to provide the QoS to the SUs, which ultimately result in transmissionfailure and wastage of power.5.2 Future Research DirectionsIn this section, we propose some possible research directions that can be followed from thisthesis and possible extension of our proposed robust resource allocation schemes in nextgeneration of wireless communication systems.5.2.1 Resource Optimization for Overlay CRNs withCooperative Relays Considering Imperfect Sensing and CSIA challenging task for overlay CRNs with cooperative relays is spectrum sensing. Channelavailability uncertainty is an important issue which needs to be taken into account for robustdesigns in such CRNs. The performance of the sensing systems can be drastically impairedwhen various wireless channels, e.g., detecting, reporting, and inter-user channels have un-certainty [96]. Spectrum sensing becomes further complicated for the CRNs operating in1455.2. Future Research Directionsmultiple bands. Providing robustness in the presence of imperfect sensing is, therefore, atask of significant practical interest [97]. Imperfect sensing can have two effects: the prob-ability of missed detection of PUs would determine the amount of interference caused tothe PU network by the CR system and the probability of false alarm would determine thethroughput loss of the CR system. The interference introduced due to errors in spectrumsensing needs to be incorporated in a probabilistic manner while developing the dynamicresource allocation algorithms for such systems. As a possible future work, we plan to de-velop joint admission control, power allocation, and relay selection for overlay CRNs withcooperative relays taking imperfect sensing and CSI into account.5.2.2 Energy Efficient Resource Allocation Schemes with QoSProvisioning under Channel Uncertainty for HetNetsEnergy efficiency is becoming an important QoS measure for the next generation of wirelesscommunication systems [98]. According to some recent statistics [99] more than 50% ofthe total energy of cellular wireless network is consumed by the radio access part, in which50 − 80% is spent by the power amplifiers. Therefore, maximizing the energy efficiencyperformance of the wireless network is very crucial [100]. HetNets, where not only relays butalso small cells are deployed inside the macro cells, can improve the utilization of the wirelessresources more efficiently. Small cells can reduce the energy consumption compared to thetraditional macro cell networks which require a large number of high-powered BSs [101].Low-cost small cell BSs can offer high-speed wireless access to their users because of theclose proximity among themselves and their own BSs. It requires very limited signaling datawhich can be transmitted over the backhaul networks via residential wireline broadbandaccess links without significant investment. Relays can improve energy awareness of HetNetswith minimum modification of the existing network infrastructure. By providing alternativelinks from source to destination, it has been shown that relays can improve network coverage1465.2. Future Research Directionsand energy efficiency.The energy efficiency metric is a fractional and nonlinear function [102], which is quitedifferent from other QoS measures that exist in the literature. Most of the works on energyefficiency maximization schemes assume that perfect and full CSI is available [102]-[104]which is often inaccurate for heterogeneous networks. The interference among the users andBSs of different cells also become imperfect in such networks, and hence robust schemes needto be designed. Energy efficiency maximization schemes with guaranteed QoS provisioningunder channel uncertainty for HetNets have not been addressed well in the literature andcan be a possible future research direction.5.2.3 Robust Resource Allocation and QoS Provisioning forMulti-tier Cellular Network under Channel and TrafficDemand UncertaintyDeployment of a multi-tier cellular network with small cells will be a key technique to achieveenhanced spectral efficiency, network coverage, and energy efficiency in next generation ofwireless networks [105]-[107]. In most of the existing works in multi-tier network, the aver-age channel gains are used to calculate the optimal number of resource blocks (RBs) [108]for each user or application. However, this RB-based resource allocation policy is designedwithout considering the actual instantaneous channel gains. The result of such optimizationschemes are improper power allocation and transmission failures, which in turn increases thetraffic demand of users. When the femto cells operate on the same frequency band as thatof macro cells to improve spectrum utilization, co-channel interference affects the overallperformance of a multi-tier network [106]. The estimation of aggregated interference to amacro cell receiver becomes imperfect due to channel variations, and this feature complicatesthe problem further. As a possible future work, robust admission control and power allo-cation schemes with guaranteed QoS provisioning for users can be developed for spectrum1475.2. Future Research Directionssharing between macro cell and femto cell users under channel variations and traffic demanduncertainty.5.2.4 Resource Optimization for Energy Harvesting CooperativeCellular Systems with Uncertain CSI and PowerEnergy harvesting (EH) is a new and promising technology for powering the relays and BSsin heterogeneous cellular wireless systems [94],[104],[109]. It enables the wireless nodes toharvest energy, such as wind, solar and thermal energy from the environment. However,environmental energy resources are not reliable for wireless communication, since they arenot available at all times and locations. Therefore, the available power at the relays and BSsmay or may not be enough to satisfy the QoS of all the users completely. 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Mallik, “Power allocation for conventionaland buffer-aided link adaptive relaying systems with energy harvesting nodes,” IEEETrans. Wireless Commun., vol. 13, no. 3, pp. 1182–1195, Mar. 2014.161Appendix AList of Other PublicationsSome other research contributions not presented in the dissertation but have been pub-lished/accepted during my PhD program at UBC are as follows:• R. Arab Loodaricheh, S. Mallick, and V. K. Bhargava, “Energy efficient resource al-location for OFDMA cellular networks with user cooperation and QoS provisioning,”IEEE Trans. Wireless Commun., vol. 13, no. 11, pp. 6132–6146, Nov. 2014.• R. Arab Loodaricheh, S. Mallick, and V. K. Bhargava, “Resource allocation for OFDMAsystems with selective relaying and energy harvesting,” in Proc. of 2014 IEEE VTC(Fall), pp. 1–5.• R. Arab Loodaricheh, S. Mallick, and V. K. Bhargava, “Joint resource optimizationfor OFDMA cellular networks with user cooperation and QoS provisioning,” in Proc.of 2014 IEEE WCNC, pp. 1–5.• R. Arab Loodaricheh, S. Mallick, and V. K. Bhargava, “Distributed subcarrier pairingand relay selection for OFDM based cooperative relay networks,” in Proc. of 2013IEEE WCNC, pp. 3557–3562.• R. Devarajan, S. Mallick, and V. K. Bhargava, “Power allocation schemes for relay-based heterogeneous networks,” in Proc. of 13th Canadian Workshop on InformationTheory, CWIT 2013, pp. 122–126.• S. Mallick, M. M. Rashid, and V. K. Bhargava, “Joint relay selection and power allo-cation for decode-and-forward cellular relay network with imperfect CSI,” in Proc. of162Appendix A. List of Other Publications2011 IEEE GLOBECOM, pp. 1–5.163
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Resource optimization in relay based cooperative wireless systems under channel uncertainty Mallick, Shankhanaad 2014
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Title | Resource optimization in relay based cooperative wireless systems under channel uncertainty |
Creator |
Mallick, Shankhanaad |
Publisher | University of British Columbia |
Date Issued | 2014 |
Description | Over the last decade, the demand for wireless resources has been rising exponentially with the increase of the number of new users and services. The primary objective of wireless communication research is finding solutions to meet this increasing demand with limited radio resources. Relay based cooperative systems greatly enhance the performance of resource-constrained wireless networks. By allowing cooperation via relays, it is possible to improve the transmission quality and energy efficiency and get similar benefits to those of multiple-input-multiple-output (MIMO) systems. However, the benefits of a relay based cooperation often depend on the channel state information (CSI) between various nodes. The CSI of such multi-hop systems is often imperfect and outdated due to channel fluctuations, limited feedback capacity, and channel estimation and quantization errors. To maximize the utilization of the radio resources, it is imperative to devise optimal resource allocation schemes that are robust under imperfect CSI. In this thesis, we consider different resource optimization problems for relay based cooperative systems and propose solutions that are robust and computationally efficient. First, we develop power allocation and relay selection schemes for a decode-and-forward (DF) cooperative wireless network under bounded channel uncertainty. We propose worst case optimization based approach to provide guaranteed quality-of-service (QoS) to the users. Next, we design a joint power allocation and admission control algorithm considering channel estimation error as unbounded Gaussian random variable. We propose a probabilistically-constrained optimization approach for QoS provisioning in slow fading channels. Finally, we propose to utilize user cooperation technique to establish communication among the secondary users (SUs) of a cognitive radio network (CRN). Power allocation and relay selection schemes that maximize the system goodput of the CRN are developed. We provide QoS to the SUs while satisfying the interference constraints of the primary user bands considering different channel uncertainty models. Numerical results demonstrate the effectiveness of our proposed schemes and implications of ignoring channel estimation error while designing resource optimization algorithms. Results reveal that the effects of ignoring the imperfectness among different channels are violations of QoS and interference constraints, which ultimately result in transmission failures and wastage of transmission power. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2014-12-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NoDerivs 2.5 Canada |
DOI | 10.14288/1.0166084 |
URI | http://hdl.handle.net/2429/51485 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2015-02 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nd/2.5/ca/ |
AggregatedSourceRepository | DSpace |
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