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Cooperative beamforming for cognitive radio systems in presence of asynchronous interference Hassan, Mai Mohamed 2015

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Cooperative Beamforming forCognitive Radio Systems in Presenceof Asynchronous InterferencebyMai Mohamed HassanB.Sc., Faculty of Engineering, Cairo University, 2009M.Sc., Faculty of Engineering, Cairo University, 2011A 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)August 2015c© Mai Mohamed Hassan 2015AbstractCognitive radio (CR) is considered a key enabling technology to exploit the underutilized and non-utilized radio spectrum bands. On the other hand, cooperative communication among nodes in CRnetworks, can improve the overall performance of CR systems, in terms of increasing data rates,attainable coverage range and overall energy efficiency, providing some diversity against shadowfading, and having low deployment costs.In a CR network, cooperative transmit beamforming can be achieved via a number of singleantenna-based CR nodes organizing themselves in a virtual antenna array and focusing their trans-mission in the direction of intended CR receiver. However, deploying the beamforming in such acooperative manner faces several implementation challenges. Therefore, in this thesis, we tacklesome of these critical challenges facing the design and implementation of cooperative CR networks.The first challenge is referred to as asynchronous interference, that results from asynchronousarrival of the same signal from the set of cooperating CR nodes at primary receivers. Next, weaddress the problem of feedback overhead needed for cooperative beamforming. Specifically, eachcooperating CR node requires knowing global information including other nodes’ locations, inaddition to accurate and instantaneous knowledge of their channel state information (CSI). Wealso tackle the problem of imperfect CSI estimation.Another important aspect of implementing cooperative CR networks is studied in this thesis,which is described as follows. Since the cooperating nodes can be located in different locations, theycontribute differently to received signals at the CR receivers, as well as to interference signals atthe primary receivers. Therefore, we propose different cooperating CR node selection strategies, tobe applied in conjunction with cooperative beamforming. Finally, we study different participationdecision making strategies that enable each CR user to independently decide whether to participatein the cooperative transmission or not, based on an offered incentive for cooperation and estimatedcost of participation in cooperative transmission represented in transmit power.iiAbstractAt the end of each chapter, we present some numerical examples to show the implications ofignoring different implementation challenges in the design of cooperative CR networks, and to assessthe performance of the proposed solutions.iiiPrefaceIn what follows, we present a list of the significant contributions to research and development thathave resulted from work presented in this thesis.• The work presented in Chapter 2 has resulted in the following list of publications:– M. H. Hassan and M. J. Hossain, ”Cooperative beamforming for cognitive radio sys-tems with asynchronous interference to primary user,” IEEE Transactions on WirelessCommunications, vol. 12, pp. 5468-5479, November 2013.– M. H. Hassan and M. J. Hossain, ”Cooperative beamforming for CR systems with asyn-chronous interference to primary user,” IEEE International Conference on Communica-tions (ICC), pp. 4272-4276, June 2013, Budapest, Hungary.• The work covered in Chapter 3 has resulted in the following list of publications:– M. H. Hassan and M. J. Hossain, ”Cooperative beamforming in CR systems with asyn-chronous interferences to multiple primary users,” IEEE Wireless Communications andNetworking Conference (WCNC), pp. 1200-1205, April 2014, Istanbul, Turkey.– M. H. Hassan, M. J. Hossain, and V. K. Bhargava, ”Cooperative Beam-Forming forCognitive Radio Based Broadcasting Systems in Presence of Asynchronous Interference,”Submitted for possible publication.• The work proposed in Chapter 4 has resulted in the following list of publications:– M. H. Hassan, M. J. Hossain, and V. K. Bhargava, ”Distributed beamforming and au-tonomous participation decision making in cooperative CR systems,” IEEE InternationalConference on Communications (ICC), in press, June 2015, London, UK.ivPreface– M. H. Hassan, M. J. Hossain, and V. K. Bhargava, ”Distributed beamforming andautonomous participation decision making in cooperative CR systems in presence ofasynchronous interference,” Submitted for possible publication.In all the research contributions made in this thesis, I was the primary researcher. I came upwith the idea of the research independently. My contributions included conducting the literaturereview and identifying the research problems. In addition, I formulated the research problems,and carried out the mathematical analysis. My contributions also included designing the proposedschemes, simulating the network performance, and analyzing the results. I also wrote the associatedmanuscripts for publication.The co-authors of the resulting manuscripts included my Ph.D. supervisors, Prof. V. K. Bhar-gava, and Dr. Md. J. Hossain, who provided directions on identifying the research problems. Theyalso provided valuable comments on my research progress, and helped me by providing technicaland editorial feedback during the preparation of these manuscripts.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivations, Objectives, and Contributions . . . . . . . . . . . . . . . . . . . . . . 51.1.1 Asynchronous Interference at Primary Receivers . . . . . . . . . . . . . . . . 61.1.2 Imperfect Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.3 Cooperating CR Node Selection . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.4 Feedback Overhead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.5 Participation Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Cooperative Beamforming with Single Primary Receiver and Single CR Re-ceiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17viTable of Contents2.1.1 Operating Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.2 Operating Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Modeling of Asynchronous Interference . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Problem Formulation and Optimal Beamforming Design . . . . . . . . . . . . . . . 232.4 Robust Beamforming Method with Imperfect Channel Knowledge . . . . . . . . . . 272.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Cooperative Beamforming with Multiple Primary Receivers and Multiple CRReceivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.1 Operating Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.2 Operating Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Modeling of Asynchronous Interferences in CR-Based Broadcasting Systems . . . . 413.2.1 Asynchronous Interference at Primary Receiver, pj . . . . . . . . . . . . . . 413.2.2 Asynchronous Interference at CR Receiver, dk . . . . . . . . . . . . . . . . . 423.3 Beamforming Design with Perfect Channel Knowledge . . . . . . . . . . . . . . . . . 443.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.2 Development of Sub-Optimal Cooperative LBF Technique . . . . . . . . . . 453.3.3 Special Case of Single PR and Single CR Receiver . . . . . . . . . . . . . . 503.3.4 Low Complexity Power Allocation Scheme . . . . . . . . . . . . . . . . . . . 513.4 Extension of Beamforming with Statistical Channel Knowledge . . . . . . . . . . . . 523.5 Joint CCRN Selection and Cooperative Beamforming . . . . . . . . . . . . . . . . . 553.5.1 Development of Optimal CCRN Selection Scheme . . . . . . . . . . . . . . . 563.5.2 Development of Sub-Optimal CCRN Selection Schemes . . . . . . . . . . . . 583.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 Distributed Beamforming and Autonomous Participation Decision Making inCooperative CR Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.1.1 Operating Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.1.2 Operating Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70viiTable of Contents4.2 Proposed Distributed Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3 Participation Decision Making Strategies . . . . . . . . . . . . . . . . . . . . . . . . 764.3.1 Regret Testing-Based Strategy (RTS) . . . . . . . . . . . . . . . . . . . . . . 794.3.2 Learning-Based Strategy (LBS) . . . . . . . . . . . . . . . . . . . . . . . . . 854.4 Relay Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.5 Extension for Multiple CR Users Requiring Cooperation . . . . . . . . . . . . . . . 884.5.1 RTS in Case of Multiple Simultaneous Cooperation Requests . . . . . . . . . 894.5.2 LBS in Case of Multiple Simultaneous Cooperation Requests . . . . . . . . . 914.6 Operation Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945 Summary, Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 1045.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110viiiList of Tables3.1 Distances between nodes in the simulated CR-based broadcasting network. . . . . . 614.1 Simulated network topology in case of one CR user requiring cooperation. . . . . . . 944.2 Simulated network topology in case of two CR users requiring cooperation. . . . . . 100ixList of Figures2.1 System model for cooperative beamforming with L cooperating CR users acting asrelays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 An example of the asynchronous interference at PR arising from a vector of Msymbols, xs [n], transmitted by relays r and f . Ts is the symbol duration and Tslotis the time slot duration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Simulated network topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Asynchronous interference signal power at the PR with perfect channel information. 312.5 Received signal power at the CR receiver with perfect channel information. . . . . . 322.6 Outage probability of SU system versus busy probability of PU. . . . . . . . . . . . . 332.7 Outage probability of SU system versus busy probability of PU for different valuesof propagation difference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.8 Transmit power versus interference power at the primary receiver for the RLBFmethod, for different values of estimation error bound, Ωc. . . . . . . . . . . . . . . . 352.9 Transmit power versus received power at the CR receiver for the RLBF method. . . 363.1 System model for cooperative beamforming with L CCRNs, K CR receivers and Jprimary receivers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 An example of the asynchronous vector of symbols received at the CR receiver dkfrom two different CCRNs, with two different propagation delays. . . . . . . . . . . . 433.3 Comparison between the computational complexity of the optimal CCRN selectionscheme and the low complexity sub-optimal CCRN selection schemes, when appliedin conjunction with the LCPA scheme. . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Total asynchronous interference power at the PRs with different interference thresh-olds (γ1th = 0.1× 10−15 and γ2th = 0.25× 10−15). . . . . . . . . . . . . . . . . . . . . 62xList of Figures3.5 Achievable sum transmission rate with various beamforming techniques and singleCCRN-based transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.6 Achievable sum transmission rate with various power allocation schemes in low noiseplus interference from the primary transmitter power environment. . . . . . . . . . . 643.7 The probability that the total asynchronous interference at each PR is greater thanγjth with statistical CSI of the PRs at the CCRNs. . . . . . . . . . . . . . . . . . . . 653.8 The sum rate of the CR receivers with the optimal CCRN selection scheme, sub-optimal CCRN selection schemes 1 and 2 and without CCRN selection. . . . . . . . 663.9 The sum rate of the CR receivers with the optimal CCRN selection scheme, sub-optimal CCRN selection schemes 1 and 2 and without CCRN selection with statis-tical CSI of the PRs at the CCRNs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1 System model of a CR network with N CR users opportunistically accessing Ffrequency channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2 Asynchronous interference to the PR at the channel f5. . . . . . . . . . . . . . . . . 954.3 Received signal power at CR5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.4 Data rates of CR users in case of participation and no participation. . . . . . . . . . 974.5 Cost of CR users with the RTS and the LBS. . . . . . . . . . . . . . . . . . . . . . . 984.6 Convergence behavior of RTS and LBS. . . . . . . . . . . . . . . . . . . . . . . . . . 994.7 Achieved data rate at CRd by cooperative transmission, in the cases of no relayselection, relay selection with full CSI knowledge at the CBS, and relay selectionusing average CSI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.8 Data rates of CR users in case of no participation, and participation according tothe modified RTS, in case of 2 CR users requesting cooperative transmission. . . . . 1014.9 Cost of CR users when using modified RTS, in case of 2 CR users requesting coop-erative transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.10 Data rates of CR users in case of no participation, and participation according tothe modified LBS, in case of 2 CR users requesting cooperative transmission. . . . . 1034.11 Cost of CR users when using modified LBS, in case of 2 CR users requesting coop-erative transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103xiGlossaryAWGN Additive White Gaussian NoiseBS Base StationCBS Cognitive Base StationCCRN Cooperating Cognitive Radio NodeCR Cognitive RadioCSI Channel State InformationDSA Dynamic Spectrum AccessJLS Joint Leakage SuppressionLBF Leakage BeamFormingLBS Learning-Based StrategyLCPA Low Complexity Power AllocationMILP Mixed-Integer Linear ProblemMINLP Mixed-Integer Non Linear ProblemMRT Maximum Ratio TransmissionNE Nash EquilibriumNLP Non Linear ProblemOSAM Overlay Spectrum Access MechanismxiiGlossaryPU Primary UserPR Primary ReceiverQPSK Quadrature Phase Shift KeyingRF Radio FrequencyRLBF Robust Leakage BeamFormingRTS Regret Testing-based StrategySDMA Space Devision Multiple AccessSINR Signal to Interference plus Noise RatioSLNR Signal to Leakage plus Noise RatioSOPA Sub-Optimal Power AllocationSU Secondary UserUSAM Underlay Spectrum Access MechanismZFBF Zero Forcing BeamFormingxiiiAcknowledgmentsI would like to express my sincere gratitude to my supervisors, Prof. Vijay K. Bhargava and Dr.Md. Jahangir Hossain for their continuous support throughout my PhD program, for their patience,motivation, and immense knowledge. I am particularly thankful for their invaluable suggestions,helpful advice, and critical comments. Without their precious support it would not have beenpossible to conduct this research. Their guidance helped me in all the time of research and writingof this thesis. I could not have imagined having better mentors for my PhD study.Besides my supervisors, I would like to thank Prof. David Michelson and Prof. Z. Jane Wangfor serving as examining committee members during my PhD qualifying examination and my PhDdepartmental exam. I am very grateful for their insightful comments and suggestions that encour-aged me to widen my research from various perspectives. My sincere thanks also go to Prof. VictorLeung and Dr. Ryozo Nagamune for serving as the University examiners during my PhD Universityexam, for their valuable suggestions and editorial corrections.I would also like to express my heartfelt gratitude to my wonderful friend and colleague in theElectrical and Computer Engineering department in UBC, who also happens to be my husband,Ahmed ElTantawy. I am very fortunate to have him by my side. His unconditional support hasalways been my greatest asset. In many occasions, he has provided me with the technical supportand innovative ideas that helped me carry out my research.Last but not least, I would like to thank my colleagues in the “Information Theory and Systems”laboratory, for the stimulating discussions, also for all the fun we had in the last four years. I amespecially thankful to Roya Arab Loodaricheh and Sudha Lohani for their friendship and genuinesupport during my PhD research. Many thanks go to Shankhanaad Mallick and RindranirinaRamamonjison for their feedback and critical suggestions. I am also grateful to all my colleaguesin the Electrical and Computer Engineering department in UBC for making my PhD study such agreat experience.xivDedicationI dedicate this dissertation to my family and friends. A special feeling of gratitude to my lovingparents, Hoda Ali and Hassan Mohamed, whose words of encouragement have always sustained methroughout my life. Many thanks to my sister, Amal, my niece, Mai, and my nephew, Ahmed, fortheir constant and unconditional love. I give my deepest expression of love and appreciation to myhusband, Ahmed, for the encouragement that he gave me and all the sacrifices he made throughoutmy PhD program. This thesis is also dedicated to my baby boy, Mazin, who was born one weekafter my PhD departmental exam. His birth was the biggest encouragement and greatest blessingthat I have ever received.xvChapter 1IntroductionOne of the main challenges facing wireless communications is the scarcity of the available radiospectrum. With the ever-increasing demand of wireless services and the limited nature of radiospectrum, the need to use the radio spectrum more effectively has become a significant aspect inthe design of wireless networks. Therefore, the wireless industry has called on governments andregulators to refarm the underutilized spectrum [1]. In particular, refarming of spectrum resourcesrefers to reassigning the government-regulated radio spectrum for services with higher value. Usersof the existing spectrum are forced out, although they may be compensated in some manner. Thefrequency bands are then assigned to communications services that yield greater economic or socialbenefit. However, refarming the underutilized spectrum is both expensive and time-consuming.An attractive and less expensive alternative to spectrum refarming is to maximize the use ofunderutilized and/or non-utilized bands through spectrum sharing, which is known as dynamicspectrum access (DSA) [2], or opportunistic spectrum access. Recently, DSA has been made possi-ble by the availability of software-defined radio, thanks to the development of fast enough processorsboth at servers and at terminals that can enable the DSA capabilities. Cognitive radio (CR) hasbeen proposed in the literature as one of the key enabling technologies of DSA to exploit boththe underutilized and non-utilized radio spectrum bands [3], [4]. In particular, a group of poten-tial users, referred to as CR users or secondary users (SUs), can be given an opportunistic accessto the underutilized/non-utilized spectrum. In addition, CR has been considered as a promisingtechnology to meet many of the challenges in future 5G networks, including exploding traffic vol-umes, emerging traffic types and data services to support applications such as smart grid, andmachine-to-machine communications [5–7].Different DSA techniques have been envisioned and studied in the literature [8]. To exploitthe unused and/or underused spectrum bands, mainly two different approaches have been widely1Chapter 1. Introductionconsidered in the literature, namely, underlay spectrum access mechanism (USAM) and overlayspectrum access mechanism (OSAM) [8, Chapter 3]. According to the OSAM, the spectrum uti-lization can be increased by granting SUs to opportunistically exploit the unused frequency bandsof the primary users (PUs), for whom the spectrum was originally assigned [9], [10]. On the otherhand, as per the USAM, the primary and the CR users can co-exist in the same spectral band(see for example, [11]). In other words, the USAM allows simultaneous sharing of underutilizedfrequency bands by the SUs along with the PUs. For this scenario, the interference introducedby the SUs to the primary receivers should be kept below a certain threshold specified by the PUsystem or the regulatory authority, see for example [12] for details.On the other hand, cooperative communication improves the overall performance of wirelessnetworks, including CR networks [13–15]. The driving motivation of cooperative communicationsis to simultaneously and drastically increase data rates, attainable coverage range and overallenergy efficiency. Cooperative communication can be realized through network densification bythe deployment of small cells, resulting in a heterogeneous architecture [16], or coordination of thetransmission of multiple individual base stations [17]. In a cost-efficient CR network, cooperativetransmission can also be realized through a set of CR devices, that support one antenna elementat their terminals. This set of CR devices cooperate to create a virtual antenna array and act ascooperating nodes1. Such a distributed antenna array could offer substantial path loss gains andwould also provide some diversity against shadow fading, in addition to having low deploymentcosts.Moreover, the distributed antenna array of CR devices can allow concurrent transmission of boththe CR users and the PUs at a given channel [18, 19], which further improves the performance ofCR systems in terms of data rates. In particular, the cooperating nodes emulate a large highlydirectional antenna array which is referred to as cooperative transmit beamforming. Then, theCR system can make use of such cooperative beamforming to mitigate the interference to primaryreceivers (PRs). Hence, in order to send a common message, a number of single antenna-based CRnodes organize themselves in a virtual antenna array and focus their transmission in the directionof the intended CR receiver.1In general, a cooperating node is a node (either a CR relay or a CR source) that participates in cooperativetransmission.2Chapter 1. IntroductionIn fact, cooperative transmit beamforming can be very effective for an USAM scenario, due tothe severe constraint on the transmission power of CR system [2]. For example, if a CR transmitterwants to broadcast a common message to a group of CR receivers, the transmitter may not beallowed to transmit sufficient power to cover all the receivers due to the interference restrictionimposed by the nearby primary system. As such, cooperative transmit beamforming allows CRusers to access a given frequency channel, with a limited or no interference to the PR [20, 21].Another example where cooperative transmit beamforming proves to be very effective is when thedirect links between a CR transmitter and its receivers are unavailable due to path loss and/orshadowing. For such situations, many cooperative transmit beamforming techniques have beenproposed in the literature to be used in CR networks, and a flourish of works has been done in thisarea, see for example [19, 22–25], and the references therein.In [26], it was shown that the transmit beamforming scheme that can achieve the channelcapacity limit in a CR network involves dirty-paper coding, which is a non-linear scheme. Theauthors in [26] considered the extreme situation where the introduction of the CR should haveno effect whatsoever on the PU’s operation and performance, i.e. the primary system should beoblivious to the presence of the CR system. Despite the optimality of the dirty-paper codingtechnique, it is difficult to be implemented in practical systems due to its high computationalburden, since it requires iterative nonlinear methods for successive encodings and decodings. Onthe other hand, zero forcing beamforming (ZFBF) [20] is a suboptimal linear strategy that canenable concurrent transmission of the CR and the PU systems on the same frequency channel,but with reduced complexity relative to DPC. In ZFBF, each CR user in the set of cooperatingnodes encodes the transmitted data stream by a specific weighting factor. Careful selection of theweighting factors can eliminate the introduced interference towards the PR, by taking advantageof spatial separation between the PR and the intended CR receiver. This type of communication,which supports multiple users simultaneously, is called space-division multiple access (SDMA) [27].In ZFBF, the beamforming weights are found by inverting the composite channel matrix of theusers. Using an innovative orthogonal projection technique, the authors in [20] obtained the ZFBFweights of the cooperating CR nodes to null the interference at all PRs in the network. Using theZFBF technique, the authors in [24] proposed a cross-layer optimization of the transmission rateand scheduling scheme of the data packets at the CR source and at the cooperating CR nodes3Chapter 1. Introductionin the CR network. In [25], power allocation for cooperative CR networks was studied, alongwith user selection under imperfect spectrum sensing. Despite its reduced complexity, the ZFBFtechnique has been shown to achieve a fairly large fraction of DPC capacity, and the sum rate ofthe beamformer approaches that of DPC as the number of users goes to infinity [28, 29]. Suchbeamforming technique has also been proposed and studied for traditional wireless communicationnetworks e.g., wireless sensor networks, as it potentially offers large increases in energy efficiency,attainable range and transmission rate, see for example [15] and the references therein. However,the ZFBF technique suffers from a design limitation, in which it requires the number of antennasin the transmitter side to be greater than that in the receiver side. Otherwise, the interference toother users in the network cannot be theoretically forced to zero. So in the context of cooperativeCR networks, the spatial diversity order of the ZFBF technique proposed in [20] equals the numberof cooperating CR nodes minus that of PRs.From a purely fundamental perspective, the ultimate limiting factor of the performance of anywireless network appears to be the availability of good enough channel state information (CSI)to facilitate the processing of data at the multiple antennas of the transmit beamformer [30].Considering factors like mobility, Doppler shifts, phase noise, and clock synchronization, acquiringhigh-quality CSI seems to be easier with a collocated antenna array than in a system where theantennas are distributed over a large geographical area [31]. However, the deployment costs of acollocated antenna array is likely to be much higher than that of a distributed antenna systemwith single-antenna elements. Moreover, a distributed antenna system provides an increase in theattainable coverage range, in addition to offering substantial path loss gains and providing somediversity against shadow fading. In conclusion, the achievable data rate gain, with such cooperativebeamforming, is quite compelling in spite of the costs associated with it, including synchronizingthe cooperating nodes and the local exchange of cooperating nodes’ information [32].Therefore, in this thesis, we tackle some of the critical challenges facing the design and implemen-tation of cooperative CR networks. The first challenge is referred to as asynchronous interference,that results from the asynchronous arrival of the same signal from the set of cooperating CR nodesat the PRs. To address this problem, we develop innovative beamforming techniques that canaccount for the effects of the asynchronous interferences at the PRs. Next, we address the problemof feedback overhead needed for cooperative beamforming. Specifically, each cooperative CR node41.1. Motivations, Objectives, and Contributionsrequires to know global information including other cooperating nodes’ locations, in addition toaccurate and instantaneous information about the states of their channels towards the PR andtowards the CR receiver. To address this problem we provide a distributed beamforming techniquethat requires minimal amount of feedback overhead. We also tackle the problem of imperfect CSIestimation, through considering the case of erroneous channel estimation, and the case of havingonly statistical knowledge of the CSI.Another important aspect of implementing cooperative CR networks is studied in this thesis,which is described as follows. Since the cooperating CR nodes can be located in different geograph-ical locations, they contribute differently to the received signal at the CR receivers, as well as tothe interference signals introduced at the PRs. Therefore, we propose different cooperating nodeselection strategies, to be applied in conjunction with the cooperative beamforming, to enhancethe overall performance of the CR network. Finally, we propose the use of a suitable incentive forthe CR users to participate in the cooperative transmission. Moreover, we study different partici-pation decision strategies that enable each CR user to independently decide whether to participatein the cooperative transmission or not, based on the offered incentive and the estimated cost ofparticipation in the cooperative transmission.In Section 1.1, we present further details of each of the design problems addressed in this thesis.We define the objectives of the proposed techniques and the motivation behind each of them.Next, in Section 1.2, we provide an overview of the presented chapters and an outline of the thesisorganization.1.1 Motivations, Objectives, and ContributionsAlthough cooperative beamforming in CR network can improve the radio spectrum utilization andenhance the network performance, it faces a number of challenging issues. In this thesis, we tacklefive critical problems facing the implementation of cooperative CR networks. We propose differenttechniques to solve these problems and we evaluate the performance of the proposed techniques.These problems are introduced as follows.51.1. Motivations, Objectives, and Contributions1.1.1 Asynchronous Interference at Primary ReceiversAs mentioned earlier, implementation of beamforming in a cooperative manner can lead to variousdesign implications that are addressed throughout this thesis. The first challenge facing cooper-ative beamforming is presented as follows. Given the fact that in practice different cooperatingnodes are usually located in different geographical locations, signals from different transmitting CRnodes arrive with different propagation delays at each PR and at each CR receiver in the network.While the received signal from different cooperating CR nodes can be synchronized at the intendedCR receiver by using a timing advance mechanism, which is currently employed in GSM and 3Gcellular networks (see for example [33]), the received signals at other CR receivers and at the PRsfrom different transmitting nodes can be not synchronized simultaneously. As such, simultaneoustransmissions from the cooperating CR nodes can cause asynchronous interference at the differentPRs which is discussed in details in Section 2.2. Such asynchronous interference can highly degradethe performance of existing cooperative beamforming techniques, as we will see later in Chapters2 and 3.The asynchronous interference issue has been studied in [33] for conventional cooperative multi-cell mobile networks, where multiple base stations (BS’s) cooperate together for down-link simul-taneous transmission of information to each mobile user in the network. The proposed leakagesuppression approaches in [33] aim to maximize the signal to interference plus noise ratio (SINR)of each of the active mobile users concurrently without any restriction on the amount of inducedinterference to a particular mobile user from different BS’s. However, in cooperative beamformingfor CR systems, the design objective is different which is to maximize the SNR at the CR receiverwhile keeping the interference at the PRs below a certain threshold specified by the PU systemor the regulatory authority. To the best of our knowledge, the asynchronous interference problemhas not been considered in CR networks in the literature except in this thesis and our publishedpapers.The shortcomings in the existing literature motivate us to pursue the following objectives inthe context of applying cooperative beamforming in CR networks.• Formulating a mathematical model of the asynchronous interference problem in cooperativeCR networks.61.1. Motivations, Objectives, and Contributions• Developing a closed form expression for an innovative beamforming technique that accountfor the effect of the asynchronous interference at the PR, in case of a cooperative CR networkwith a single CR receiver and a nearby primary system with single PR.• Generalizing this beamforming technique to the case of having multiple PRs and multiple CRreceivers in the case of broadcast-based CR networks.• Proposing a lower complexity cooperative beamforming scheme to be used in the case ofhaving multiple interference constraints due to the presence of multiple PRs in the network.The first two objectives are tackled in Chapter 2. The last two objectives are addressed inChapter 3. The contributions made regarding these objectives are clarified as follows. In thisthesis, we address the asynchronous interference issue in two different settings.• First, we consider a cooperative CR network with a single CR receiver and a nearby primarysystem with single PR in Chapter 2. In particular, each CR node in the set of cooperatingnodes encodes the transmitted data stream to the intended CR receiver by a specific beam-forming weight, to forward the information to the CR receiver irrespective of whether thePU is silent or not. We formulate the asynchronous interference model for this scenario, andwe develop a convex optimization problem which aims to maximize the signal power at theCR receiver while it maintains the interference power at the PR below a target threshold,which in turn decreases the service interruption/outage of the SU. We provide a closed formsolution for the cooperative beamforming vector, where we call the beamforming techniquein this case, leakage beamforming (LBF) method.• In Chapter 3, we consider a more generalized setup where a group of cooperating CR nodesuses cooperative beamforming to broadcast a common message to multiple CR receivers, usinga wireless communication channel assigned to a primary transmitter to transmit informationto multiple PRs simultaneously. It is also considered that these PRs have different interferenceconstraints in general. For this generalized setup with multiple primary and CR receivers,transmission to a specific CR receiver introduces asynchronous interferences, not only atthe PRs, but also at all other CR receivers. Due to these asynchronous interferences, theoptimal beamforming technique developed in the first setting cannot be directly extended for71.1. Motivations, Objectives, and Contributionsthis generalized system. Therefore, we formulate the cooperative beamforming design as anoptimization problem that maximizes the weighted sum achievable transmission rate of theCR receivers while it maintains the interference thresholds at the PRs below their respectivetarget interference thresholds. However, the optimal beamforming technique for this caseis indeed intractable due to the non-convexity and non-linearity of the problem. Hence, weuse an approximation to transform the optimization problem into a convex and linear one.Then sub-optimally we obtain the beamforming directions and allocate power among differentbeamforming directions. We also show that the LBF technique developed in the first scenariois considered as a special case of the generalized beamforming technique developed in thissetting. Due to the multiple interference constraints in the case of having multiple PRs inthe network, the power allocation scheme can be of a high complexity as will be discussedlater in Chapter 3. Therefore, we also propose a low complexity power allocation scheme tobe used in the cooperative beamforming design of the generalized system.1.1.2 Imperfect Channel EstimationThe proposed cooperative beamforming methods can be implemented if the channel fading gainsbetween the cooperating CR nodes and the PRs as well as the channel fading gains between thecooperating CR nodes and the CR receivers are all known at the cooperating nodes. Differentpossible scenarios have been considered in the literature in order to estimate the CSI between thecooperating CR nodes and the PR (see for examples, [34], [35]). One possible scenario is that whenthe PR periodically transmits pilot signal to its own transmitter. If the cooperating CR nodes knowthe pilot signals, the channel between the cooperating CR nodes and the PR can be estimated bythe CR system assuming the reciprocity of the channel. This scenario is considered in [35].However in many scenarios, the cooperating CR nodes may not have the perfect CSI from thePRs in the network. During the design process of the cooperative beamforming technique, wemust account for the effects of such imperfect CSI estimation, to ensure a robust protection of theprimary system. Robust protection means that the resulting interference at the PR remains belowa predefined threshold even for the worst case estimate of the CSI towards that PR. This motivatesus to pursue the following objectives.81.1. Motivations, Objectives, and Contributions• Developing a robust cooperative beamforming method that account for the effect of havingan erroneous estimation of the CSI between the cooperating CR nodes and the PR.• Developing a robust cooperative beamforming method that account for the effect of havingonly statistical CSI of the channels between the PRs and the cooperative CR nodes at eachcooperating node.Therefore in Chapters 2 and 3, we develop robust cooperative beamforming methods thatconsider two forms of having imperfect CSI estimation, which are presented as follows.• Chapter 2 considers the case of having an erroneous estimation of the CSI between thecooperating CR nodes and the PR. The robust cooperative beamforming method, developedin this case, instantaneously meets the interference threshold at the PR despite the presenceof estimation error in the CSI between the PR and each cooperating CR node. Howeverthis robust design of the beamforming technique comes at the expense of having decreasedreceived signal power at the CR receiver, as shown later in Chapter 2.• Chapter 3 considers having only statistical CSI of the channels between the PRs and the co-operative CR nodes at each cooperating node. In this case, the asynchronous interferences atthe PRs are guaranteed in a statistical sense [11], [36], as shown in Chapter 3. In the absenceof a mathematically tractable expression of the distribution of the random interference powersat the PRs, we develop an upper bound on the probability of introducing asynchronous in-terference power at a given PR beyond a given threshold. Then this developed upper boundis used to design a robust cooperative beamforming technique. The proposed beamform-ing technique can protect the primary network’s functionality by satisfying the probabilisticinterference constraints at all PRs in the network.1.1.3 Cooperating CR Node SelectionAnother important aspect of applying cooperative beamforming has been studied in this thesis,which is described as follows. Since the cooperating CR nodes can be located in different geo-graphical locations, it is intuitive that different cooperating nodes will contribute differently tothe asynchronous interferences at the primary receivers. Also, different cooperating nodes may91.1. Motivations, Objectives, and Contributionscontribute differently towards the received signal power at the CR receivers. The joint design ofcooperative beamforming and cooperating node selection has been studied before for conventionalcooperative networks, see for example [37] and the references therein. In the case of CR systems,the cooperating node selection scheme should include the constraint of keeping the interference atthe PRs below their respective desired interference thresholds. This motivates us to pursue thefollowing design objectives.• Proposing an optimal cooperating node selection strategy, to be applied in conjunction withthe cooperative beamforming methods, and study its performance.• Proposing a lower complexity sub-optimal cooperating node selection strategy with shorterconvergence time for a more practical design of the cooperative CR network.These objectives are tackled in Chapter 3, where three different cooperating CR node selectionstrategies are proposed to be used in conjunction with the cooperative beamforming techniques.As will be shown later in Chapter 3, an improved performance can be achieved using cooperativebeamforming when applying these cooperating CR nodes selection strategies compared to the casewhen all the cooperating nodes participate in beamforming. The contributions made in Chapter 3regarding these design objectives are summarized as follows.• We formulate the optimization problem in case of joint cooperating node selection and co-operative beamforming design as a mixed-integer non linear problem (MINLP). The firstproposed scheme optimally solves this problem through exhaustive search. Therefore, thisproposed optimal cooperating node selection scheme yields the best performance when ap-plied in conjunction with the cooperative beamforming technique, compared to the other twoproposed cooperating node selection schemes.• Despite the optimality of the first proposed scheme, it suffers from high implementationcomplexity that grows exponentially with the number of cooperative nodes in the network.Therefore, we propose a suboptimal scheme that limits the maximum number of cooperatingCR nodes which are allowed to participate in the cooperative transmission to the CR receiver,while still applying exhaustive search among this set of cooperating nodes. This solution101.1. Motivations, Objectives, and Contributionsdecreases the computational complexity compared to the optimal scheme, at the expense oflosing some received signal power at the intended CR receiver.• We propose another simple low complexity cooperating CR node selection scheme, in whichthe set of cooperating CR nodes that are selected for cooperative transmission are heuristi-cally chosen based on their channel fading coefficients towards the CR receivers and towardsthe PRs. The numerical results in Chapter 3 show that the proposed selection scheme inconjunction with beamforming can further increase the sum transmission rate of the CRssignificantly compared to the case when all the cooperating CR nodes participate in thecooperative beamforming.1.1.4 Feedback OverheadCooperative beamforming requires sharing of instantaneous CSI and location information amongcooperative CR nodes or requires a master node that knows the global instantaneous CSI andlocation information. Both cases require a huge amount of feedback overhead. Exchanging suchlarge amount of control traffic between the cooperating nodes requires additional bandwidth, andcauses excessive power dissipation from the CR devices [38]. In addition, CSI estimation errorscan become a bottle neck to the potential performance gain from CR nodes cooperation [39]. Thismotivates us to pursue the following objective.• Proposing a distributed beamforming scheme that requires only minimal information sharingbetween the cooperating nodes, to enable the beamforming design in presence of asynchronousinterference.The contributions made in the thesis regarding this objective are clarified as follows.• In Chapter 4, we propose a distributed beamforming method to be used in cooperative CRnetworks that requires only information sharing between cooperating CR nodes about theirlocations. Assuming each cooperating CR node knows its own CSI towards the CR receiverand towards the PR, and using the shared information about the other cooperating nodes’locations, each node can independently design its own beamforming weight. The location111.1. Motivations, Objectives, and Contributionssharing is essential for each cooperating node to account for the effect of asynchronous inter-ference it can cause, along with other cooperating nodes, to the PR in the network.1.1.5 Participation Decision MakingThe final problem of cooperative CR networks, addressed in this thesis, is finding a suitable incentivefor CR users to participate in the cooperative transmission towards an intended CR receiver. Inreturn for their cooperation, we propose that each assisted CR user can lease its own channel, fora certain amount of time, to the cooperating CR nodes for their opportunistic access. However,a particular CR user spends a certain amount of its limited battery power when acting as a relayfor another CR user. Therefore, it becomes a critical decision for each user to decide whether toparticipate in the cooperative beamforming or not. We assume there is no cooperation amongthe participating CR users in the decision making, so each CR user does not know other users’decisions, and hence it cannot assess its own reward in case of participating in the cooperativetransmission beforehand. Hence, this problem is considered as an example of unknown games, thatcan be tackled using a Bayesian game theoretic approach [40]. This motivates us to pursue thefollowing design objectives.• Proposing an optimal autonomous participation decision making strategy to help each CRuser in deciding whether to participate in the cooperative transmission or not.• Proposing a lower complexity sub-optimal autonomous participation decision making strategythat has a shorter convergence time, yet a good performance.Therefore, in Chapter 4, we presented the following contributions in order to achieve these twoobjectives.• Chapter 4 proposes an autonomous participation making strategy known as regret testing-based strategy (RTS) and is based on the well-known regret testing procedure [41]. Wemodify the regret testing procedure based on our model of CR networks. We prove that theproposed RTS can asymptotically achieve an approximate Nash equilibrium (-NE) state ofthe network within a certain convergence time. The NE state of a system is achieved when121.2. Thesis Outlineeach CR user has chosen a strategy of participation decision making, and no CR user canbenefit by changing its strategy as long as other CR users keep theirs unchanged [42].• Despite the optimality of the proposed RTS, its complexity and slow convergence time standagainst its practical implementation. Therefore, we propose a low complexity autonomousdecision making strategy named the learning-based strategy (LBS), which has a shorter con-vergence time.• The network model considered in Chapter 4 assumes that different frequency channels, origi-nally owned by the primary system, are assigned each for one CR user only during the currentscheduling frame. When more than one CR user are requesting cooperative transmission overtheir different scheduled frequency channels, it becomes an added challenge for the CR usersthat are willing to act as relays. Therefore, we extend the two proposed autonomous decisionmaking strategies, namely the RTS and the LBS, to handle the case of having multiple CRusers requesting cooperation simultaneously. Since each CR user can only participate in thecooperative transmission towards one CR receiver over its scheduled channel at a time, itbecomes a critical decision for each CR user, not only to decide whether to participate inthe cooperative transmission or not, but also to select which CR receiver to assist among thesimultaneous requests that it receives. The decision making in this case is based on the bestincentive provided by each CR user requesting assistance. The modified RTS and LBS areshown in Chapter 4 to provide enhanced performance of the CR network, despite the lack ofcooperation among the participating CR users in making their decisions.1.2 Thesis OutlineThe rest of the thesis consists of four chapters, which are organized as follows.• In Chapter 2, we describe the asynchronous interference effect, and provide its mathematicalmodeling. We also present the motivation for new beamforming techniques to be used inthis context. We develop an innovative cooperative beamforming technique, named the LBFmethod, that enables the cooperating CR nodes to transmit data to the CR receiver with acertain limit on the interference introduced at the PR when the PU is active. Next, we address131.2. Thesis Outlinethe effect of having imperfect CSI estimation on the performance of the proposed beamformingmethod. We also propose a robust cooperative beamforming method to account for the effectof having error in the channel estimation between the PR and each cooperating CR node.At the end of Chapter 2, we present some numerical examples to assess the performance ofthe proposed cooperative beamforming methods, and show the implications of ignoring theasynchronous interference problem in the design of cooperative CR networks.• Chapter 3 considers a generalized scenario of a CR network with multiple PRs and multipleCR receivers. The cooperative beamforming design is formulated as an optimization problemof constrained weighted sum rate maximization. Due to the non-convexity and non-linearityof the formulated optimization problem, we use an approximation to convert it into a con-vex and linear one. Then sub-optimally we obtain the beamforming directions and allocatepower among different beamforming directions. However, due to the multiple interferenceconstraints, the power allocation scheme is computationally expensive as discussed in Chap-ter 3. Therefore, we also propose a low complexity power allocation scheme.Next, we extend the proposed cooperative beamforming technique for the case of havingonly statistical CSI of the channels between the PRs and the cooperating CR nodes at thecooperating nodes. In this case, the asynchronous interferences at the PRs are guaranteed ina statistical sense. In the absence of a mathematically tractable expression of the distributionof the random interference powers at the PRs, we develop an upper bound on the probabilityof introducing asynchronous interference power at a given PR beyond a given threshold. Thenthis developed upper bound is used to design a robust leakage beamforming technique. Theproposed robust beamforming technique can protect the primary network’s functionality bysatisfying the probabilistic interference constraints at all PRs in the network.Moreover, we develop an optimal cooperating CR node selection scheme to be used in con-junction with the beamforming technique. Because of the exhaustive search, the optimalcooperating CR node selection scheme can be computationally expensive. Therefore, we alsopropose two sub-optimal selection schemes. Finally, we conduct comprehensive simulationexperiments to show the performance of the different proposed beamforming schemes and thecooperating CR node selection schemes.141.2. Thesis Outline• As a solution for the problem of huge feedback overhead required for cooperative beamforming,in Chapter 4 we propose a distributed beamforming method to be used in cooperative CRnetworks with minimal amount of feedback overhead. Then, we define a suitable incentive forthe CR users to participate in the cooperative transmission. In this chapter, we also proposetwo autonomous participation decision making strategies, namely the RTS and the LBS, tohelp each CR user in deciding whether to participate in the cooperative transmission or not,without any coordination among the participating CR users in decision making. The decisionmaking strategies are based on the proposed incentive and the estimated cost of participationin the cooperative transmission.However, different cooperating CR nodes have different path loss values towards the intendedCR receiver, as well as towards the primary receiver. In addition, as more CR nodes partici-pate in the cooperative transmission, the individual reward value becomes smaller. Therefore,in case of having multiple CR users that are willing to participate in the cooperative trans-mission, a cooperating CR node selection scheme is important. As such, the received signalpower value at the intended CR receiver is maximized. Therefore, we propose a cooperatingCR node selection method in Chapter 4.Also in Chapter 4, we extend the two participation decision making strategies, the RTS andLBS, to handle the case of having multiple CR users requesting cooperation simultaneously.The numerical results in Chapter 4 reveal the effectiveness of our proposed distributed beam-forming scheme and autonomous decision making strategies.• Conclusions and future works are discussed in Chapter 5, where we provide possible futureresearch directions.15Chapter 2Cooperative Beamforming with SinglePrimary Receiver and Single CRReceiverAs mentioned earlier, the performance of CR networks can be improved using cooperation amongdifferent users in such network. In particular, in a CR network where a CR user/source wants totransmit information to a CR receiver that cannot be communicated directly by the source due tothe path loss and/or shadowing, a group of SUs can act as relays and forward the message to theCR receiver. However, in such dual-hop cooperative CR networks, the availability of spectrum ismore critical as it requires unused spectrum hole/slot for both hops’ communications. As such itcan lead to a very poor quality of communications, e.g., higher outage probability, for the CR users.In order to improve the quality of communications for such networks, beamforming methods [43]which enable concurrent transmissions of both PUs and SUs at a given channel can be exploited[22], [23], [19]. These beamforming methods require to have multiple antennas at the transmitters.However, for an implementation constraint, a CR source may not support more than one antennaelement at its terminal.For the above mentioned situations, a cooperative beamforming method as proposed in [20]can be used for the second hop i.e., CR relays to the CR receiver communication. The proposedcooperative beamforming method in [20] utilizes a virtual antenna array, which is created by a setof CR users that act as relays for the CR source and forward the information to the CR receiver. Inparticular, when the channel is not occupied by the PU, a single antenna based CR source transmitsits information to a set of single antenna based cooperating SUs that serve as relays. In the nexttime slot, the cooperating relays which have correctly detected the transmitted information, forward162.1. System Modelthis information to the CR receiver irrespective of whether the PU is silent or not. By carefullyselecting the beamforming weight in each CR relay node, the interference from the CR relays canbe efficiently suppressed or even thoroughly avoided.Implementation of beamforming in such a cooperative manner, as explained in Chapter 1, canlead to an asynchronous interference at the PR. In [20], such asynchronous interference at the PRhas been ignored. The ZFBF proposed in [20] cannot annul this asynchronous interference causedby the CR relays at the PR, except under a severe constraint, that the number of antennas in eachrelay node should be greater than the total number of antennas of all PUs [33]. This is explainedin more detail in Section 2.2. In addition, the presented numerical examples in this chapter showthat the ZFBF method introduces higher levels of interference at the PR, in such an asynchronousinterference scenario. This leads to an increased service interruption/outage of the CR system whenthere is a certain limit on the introduced interference at the PR.In this chapter, we provide the operating principles as well as the assumptions that we consider inour problem formulation. We also present a mathematical model for the asynchronous interferencepower introduced at the PR. Next, we develop a cooperative beamforming method called leakagebeamforming (LBF) method that maximizes the signal power at the CR receiver while it maintainsthe interference power at the PR below a target threshold, which in turn decreases the serviceinterruption/outage of the SU. Then, we develop a robust beamforming method that addresses theissue of having imperfect CSI estimation between each CR relay and the PR.2.1 System ModelWe consider an example of situations where cooperative beamforming proves to be very effective inCR networks. CR networks are usually low-power systems where the CR source may not be allowedto transmit enough power to cover the CR receiver due to the interference restriction imposed bythe nearby primary system. In this chapter, we consider the case when the direct link between theCR source and its CR receiver is not available, which may result from high path loss. In this case, agroup of CR users in the network can act as relays and forward the message to the CR receiver. Inwhat follows we provide the operating assumptions and principles that we consider in our problemformulation.172.1. System Model2.1.1 Operating AssumptionsWe consider a time-slotted system with a single primary link that is used to transmit informationfrom a primary transmitter to a particular PR at a given time slot with probability p. This isso-called ON-OFF behavior of PUs [11, 44]. We also consider a CR network consisting of one CRsource s, one CR receiver d and L other SUs that act as parallel relays as shown in Fig. 2.1. TheCR relay nodes are denoted by rl, l = 1, 2, · · · , L. Due to the hardware constraint, it is assumedthat the CR source s, the relays as well as the CR receiver d are equipped with single antennaeach. The CR network is employed with a reliable sensing mechanism that can accurately sensesthe channel occupancy by the PU in a given time slot.The channel path loss model used in the system is the log-distance path loss model presentedin [45]. Generally, accurate path loss models can be obtained from complex ray tracing models orempirical measurements when tight system specifications must be met. However, for simplicity, weuse a simple model that captures the essence of signal propagation without resorting to complicatedpath loss models, which are only approximations to the real channel [45]. Thus, the following pathloss model is used for system design as a function of distance Dr,d between relay r and the CRreceiver d.PLr,d = κ+ 10γ log10(Dr,dD0), (2.1)where PLr,d is the path loss in dB over the communication link between relay r and the CR receiverd, D0 is a reference distance for the antenna far-field, and γ is the path loss exponent. κ is a unit-less constant which depends on the antenna characteristics of relay r and the average channelattenuation. The value of κ is set to the free space path loss at distance D0 as followsκ = 20 log10(4piD0λc), (2.2)where λc is the RF signal wavelength. Due to scattering phenomena in the antenna near-field, themodel in eq. (2.1) is generally only valid at transmission distances Dr,d > D0. We assume D0 isequal to 10m. The value of γ depends on the propagation environment. We consider a propagationmodel that approximately follows a two-ray model, so the path loss exponent γ is set to 4.We assume that the channels between all nodes are additive white Gaussian noise (AWGN)182.1. System Modelchannels with zero mean and two sided noise power spectrum density N0/2. We also assume blockfading channel model, that was used for example in [46], in which the channel fading is assumed toremain roughly the same over a time slot, but is independent of the fading in other time slots.2.1.2 Operating PrinciplesThere are two phases of transmission for the CR system as described below. Without loss ofgenerality, let us assume that at time slot n, the communication channel is sensed as idle by theCR system. Therefore, the CR source s broadcasts its data to the CR relays. We assume that thesource s sequentially transmits M symbols2 during time slot n. At time slot n+ 1, the CR relayswhich can detect the source information correctly transmit the detected versions of the receivedsymbols to the CR receiver d. At time slot n + 1, if the channel is not occupied by the PU, therelays which have successfully detected the message, forward the message to the CR receiver d usingthe maximal ratio transmission (MRT) diversity scheme [47]. Otherwise these relays forward themessage to the CR receiver using a beamforming method, such that the interference introduced tothe PR remains below a target threshold specified by the PU system.Primary  Transmitter 1rs2rLrSource Relays Primary  Receiver dCR Receiver Figure 2.1: System model for cooperative beamforming with L cooperating CR users acting as relays.2The problem can be formulated assuming M = 1, i.e., only one symbol is transmitted in each time slot. However,we consider a practical scenario where multiple symbols are transmitted in a given time slot.192.1. System ModelWe assume that P is the average power per symbol at the source s. The binary informationbits are mapped into modulated symbols, and the data vector consisting of these M modulatedsymbols is denoted by xs. The received signal at a CR relay r at time slot n can be written asyr[n] = hs,r[n]xs[n] + zr[n], (2.3)where zr is the AWGN vector at relay r and hs,r[n] is the channel fading gain between the sources and the relay r at time slot n.Assume that at time slot n, K relays out of the L relays can successfully detect the message,where K ≤ L. In particular, a relay r is considered to have successfully detected the message,if the instantaneous Shannon capacity of the channel between the CR source and relay r exceedsthe value of a target spectral efficiency B [20]. Those successful relays are denoted by the set Kn,where |Kn| = K. Without loss of generality, the relays in set Kn can be denoted by {r1, . . . , rK}.These relays forward their detected symbols to the CR receiver at time slot n + 1. In particular,if the communication channel is occupied by the PU at time slot n+ 1, this set of relays forwardsthe message to the CR receiver d using a beamforming method which will be discussed latter. Inthis case, the received signal at d can be written asyd[n+ 1] = hgxs[n] + w + zd[n+ 1], (2.4)where xs[n] is the transmitted data vector from the source at time slot n that has been successfullydetected at the relays in Kn and w is the received interfering signal vector from the PU transmitter.g = [g1, . . . , gK ]T is the beamforming weight vector, with each element gr denoting the weight ofthe relay r. h = [h1,d[n+ 1], . . . , hK,d[n+ 1]] is the the channel vector from the transmitting relaysto the CR receiver d, where hr,d is the channel fading gain from relay r to receiver d. Our designgoal is to obtain the beamforming vector g that maximizes the received signal power at CR receiverd while keeping the interference to the PR below a certain threshold.If at time slot n+1, the PU is silent, the relays in set Kn directly forward the correctly detectedsymbols xs[n] to the CR receiver using MRT [48] diversity scheme. In this case the received signal202.2. Modeling of Asynchronous Interferenceat the CR receiver d with MRT diversity scheme can be written asyd[n+ 1] =√hh†xs[n] + zd[n+ 1]. (2.5)2.2 Modeling of Asynchronous InterferenceIn a practical scenario, the CR relays are usually located in different geographical locations3.Therefore, the received signals from different transmitting relays at the PR as well as at the CRreceiver can experience different propagation delays. Although the received signal at the CR receiverd from different relays can be synchronized by using a timing advance mechanism which is currentlyemployed in the uplink of GSM and 3G cellular networks, to compensate for the propagation delayfrom each user [33, 48], the received signals at the PR from different transmitting relays can benot synchronized simultaneously. As such, the PR will experience asynchronous interference. Theasynchronous interference issue has been studied in [33] for the conventional cellular networks wheremultiple BS’s cooperate with each other for downlink transmissions. Following this work, in whatfollows, we provide a mathematical model for the asynchronous interference introduced to the PR.In order to maintain the synchronous reception of data symbols from all the transmitting relaysin set Kn at the CR receiver, we consider that a timing advance mechanism is applied among thoserelays. In particular, the rth transmitting relay advances its signal by ∆τr,d which is calculated as∆τr,d = τr,d − τ(min), (2.6)where τr,d is the propagation delay of rth (r ∈ Kn) relay signal to the receiver d and τ (min) is theminimum signal propagation delay of all the transmitting relays in the set Kn.Let us denote τ (PU)r as the signal propagation delay from relay r (r ∈ Kn) to the PR. Then thetime delay of the received symbols at the PR from the rth relay, ∆τ (PU)r can be written as∆τ (PU)r = τ(PU)r −∆τr,d, (2.7)where the received signal from rth relay is advanced by its time advance, ∆τr,d, defined by eq. (2.6).3We are considering a cooperative CR network where a set of randomly located cooperating CR users is acting asa set of relays.212.2. Modeling of Asynchronous InterferenceLet us denote ir[n + 1] as the asynchronous vector of symbols received at the PR as shown inFig. 2.2. This asynchronous vector of symbols ir[n+ 1] is defined asir[n+ 1] = xs[nTslot −∆τ(PU)r], (2.8)where Tslot is the time slot duration.Time slots  Asynchronous                                                                              vector of symbols                                                                                         relay       Asynchronous                                                                              signal from                                                                                     relay    ......  Time slots                                 Asynchronous vector of symbols from relay f Asynchron us vector of symbols from relay r          ......   ......                  transmitted hat time slot     Figure 2.2: An example of the asynchronous interference at PR arising from a vector of M symbols, xs [n], transmittedby relays r and f . Ts is the symbol duration and Tslot is the time slot duration.The asynchronous received signal at the PR at time slot n+ 1 is given byy[n+ 1] =∑r∈Knh(PU)r [n+ 1]grir[n+ 1] + z[n+ 1], (2.9)where h(PU)r [n+ 1] is the channel fading gain from relay r to the PR at time slot n+ 1, and z[n+ 1]is the AWGN vector at the PR with zero mean and two-sided PSD N0/2.In [20], such asynchronous interference at the PR has been ignored. From eq. (2.9) it is obviousthat if ZFBF method, developed in [20], is used to force the asynchronous interference at thePR to be zero, the interference from each relay has to be annulled separately, i.e., the condition222.3. Problem Formulation and Optimal Beamforming Design∣∣∣h(PU)r gr∣∣∣ = 0 must be satisfied for every r ∈ Kn, which cannot be fulfilled for single antenna basedCR relays because it requires the number of PUs to be less than 1. Failing to satisfy this condition,the ZFBF method results in higher interference at the PR, as shown in Section 2.5. This leadsto an increased service interruption/outage of the CR system when there is a certain limit on theintroduced interference at the PR.2.3 Problem Formulation and Optimal Beamforming DesignIn this section, we develop a new cooperative beamforming method, called LBF method in orderto address the problem of asynchronous interference at the PR. In this development, we use thesame assumption as in [19, 20, 22, 23] that the channel fading gains between the CR relays andthe PR as well as the channel fading gains between the CR relays and the CR receiver are knownperfectly at the CR relays. Different possible scenarios have been considered in the literature inorder to estimate the CSI between the CR relays and the PR (see for examples, [34], [35]). Inthe next section, we consider the case when the channel between the CR relays and the PR is notknown perfectly.The objective of the developed LBF method is to keep the leakage to the PR below a desiredthreshold, where leakage is the interference caused by the CR relays to the PR. The target thresholdis imposed by the regulatory body, see for example [12] for details. The leakage signal at the PRdue to CR relays’ transmission at time slot n+ 1 can be written asΘ[n+ 1] =K∑r=1h(PU)r [n+ 1]grir[n+ 1], (2.10)where ir[n+ 1] is obtained from eq. (2.8). Now the leakage signal power at the PR can be writtenasPleak = E(Θ[n+ 1]Θ†[n+ 1]), (2.11)where E (·) is the expectation operation over the random data sequence, and Θ†[n + 1] is theconjugate transpose of Θ[n+ 1].For notational convenience, we will drop the time slot index n from now on. Using eq. (2.10) ineq. (2.11) and after some mathematical manipulations, the leakage power can finally be expressed232.3. Problem Formulation and Optimal Beamforming DesignasPleak =K∑r=1K∑f=1g†f (h(PU)f )†h(PU)r gr · E(iri†f). (2.12)Let us define a variable β(r,f) , E(iri†f), which is the correlation between the leakage symbols ofthe rth and the f th relays. Then the leakage power in eq. (2.11) can be written asPleak =K∑r=1K∑f=1g†f (h(PU)f )†h(PU)r grβ(r,f). (2.13)The correlation between asynchronous symbols, β(r,f), can be obtained from one of the followingtwo cases:• Case 1: The propagation difference between the signals of relays r and f is larger than onesymbol duration Ts. In this case, the correlation between ir and if is equal to zero, i.e.,β(r,f) = 0, if∣∣∣∆τ (PU)r −∆τ(PU)f∣∣∣ > Ts.This is because successive data symbols are assumed to be independent of each other withzero mean.• Case 2: The propagation difference between the signals of relays r and f is less than onesymbol duration Ts. In this case, the asynchronous symbols ir and if are overlapping for atime duration of Ts −∣∣∣∆τ(PU)r −∆τ(PU)f∣∣∣, and hence their correlation is equal to the part ofthe transmitted symbol power in which they intersect, i.e.,β(r,f) =Ts −∣∣∣∆τ(PU)r −∆τ(PU)f∣∣∣TsP, if 0 <∣∣∣∆τ (PU)r −∆τ(PU)f∣∣∣ < Ts.The leakage power in eq. (2.13) can be rewritten in a matrix form as followsPleak = g†Rtrueg, (2.14)242.3. Problem Formulation and Optimal Beamforming Designwhere g = [g1, . . . , gK ]T andRtrue,β(1,1)(h(PU)1 )†h(PU)1 · · · β(1,K)(h(PU)1 )†h(PU)Kβ(2,1)(h(PU)2 )†h(PU)1 · · · β(2,K)(h(PU)2 )†h(PU)K.... . ....β(K,1)(h(PU)K )†h(PU)1 · · · β(K,K)(h(PU)K )†h(PU)K. (2.15)Rtrue is the covariance matrix of the channel fading gains between the CR relays and the PR.This channel covariance matrix Rtrue can be calculated by the CR system using the known channelfading gain between each relay and the PR, h(PU)r , as well as β(r,f) which can be calculated basedon the locations of the PR and the CR receiver relative to the CR relays.The received signal power at the CR receiver is given byPsig = Pg†h†hg. (2.16)Our design goal is to maximize the received signal power at the CR receiver d, while keepingthe leakage power at the PR below a certain threshold. This design goal can be formulated as anoptimization problem, using eqs. (2.14) and (2.16), as followsg(LBF) = maxg(Pg†h†hg),subject to: g†Rtrueg ≤ γth, (2.17)where γth is the maximum allowable interference at the PR.The optimization problem in eq. (2.17) is a convex maximization problem, that maximizes aconvex quadratic function under a convex quadratic constraint. The global optimality conditionsof such optimization problem have been studied in [49]. According to [49], if the following threeconditions are satisfied, the global optimal solution of such problem can be found by solving theLagrangian dual problem, with zero duality gap. These three conditions are as follows:1. The matrix Ph†h is a positive semidefinite matrix.2. The matrix Rtrue is a positive semidefinite matrix.252.3. Problem Formulation and Optimal Beamforming Design3. There exists a vector g, for which the inequality constraint is strictly satisfied, i.e., g†Rtrueg <γth.It can be easily shown that these three conditions are satisfied in the optimization problem in eq.(2.17). Therefore, the global optimum solution can be found by the Lagrange multiplier method,as followsL(g, λ) = Pg†h†hg − λ(g†Rtrueg − γth), (2.18)where λ is a Lagrange multiplier. The critical values of the Lagrange function L(g, λ) occur whenits gradient is equal to zero. By taking the partial derivative of the Lagrange function in eq. (2.18)with respect to g, and equating it to zero, we can easily writePh†hg = λRtrueg. (2.19)Defining ψ , hg, the beamforming vector in eq. (2.19) can be written asg(LBF) =ψλPR−1trueh†. (2.20)Equating the partial derivative of L(g, λ) with respect to λ, to zero, we find thatg(LBF)†Rtrueg(LBF) = γth. (2.21)Substituting eq. (2.20) into eq. (2.21), and after some mathematical manipulations, we can writeψλas(ψλ)2=1P 2γth(R−1trueh†)†Rtrue(R−1trueh†) . (2.22)Using eq. (2.22) in eq. (2.20), the optimum beamforming vector g(LBF) can finally be expressed ina desired closed form as followsg(LBF) =√γthhR−1true†h†R−1trueh†. (2.23)262.4. Robust Beamforming Method with Imperfect Channel Knowledge2.4 Robust Beamforming Method with Imperfect ChannelKnowledgeWhen the CSI between the CR relays and the PR h(PU)r is perfectly known at the CR relays, ourdeveloped LBF method in the previous section can be used. However in some scenarios, the CRrelays may have erroneous estimation of the channel between the PR and the CR relays. Duringthe design process of the cooperative beamforming vector, we must account for the effects of sucherroneous estimation, to ensure a robust protection of the PR. Robust protection means that theresulting interference at the PR remains below the predefined threshold even if the error in CSIestimation is maximum. In order to design a robust leakage beamforming (RLBF) method for suchscenario, we adopt the following channel estimation uncertainty model.If the channel estimation of h(PU) is erroneous, the estimation error can be modeled ash(PU)true = h(PU)est + e(PU), (2.24)where h(PU)true is the actual instantaneous channel vector between the CR relays and the PR, h(PU)est isthe estimated channel vector between the CR relays and the PR, and e(PU) is the estimation errorvector. Based on the accuracy of the estimation method used, the channel estimation uncertaintycan be modeled by the so-called bounded uncertainty model4.Using the error model in eq. (2.24), the covariance matrix corresponding to h(PU)true , Rtrue can bewritten as in eq. (2.25).Rtrue=β(1,1)(h(PU)1,est†h(PU)1,est + e(PU)1†e(PU)1)· · · β(1,K)(h(PU)1,est†h(PU)K,est + e(PU)1†e(PU)K).... . ....β(K,1)(h(PU)K,est†h(PU)1,est + e(PU)K†e(PU)1)· · · β(K,K)(h(PU)K,est†h(PU)K,est + e(PU)K†e(PU)K). (2.25)4The bounded uncertainty model is a well-accepted model that has been used in [35, 50–52]. It shows that theuncertainty in the channel estimation is described by a bounded region whose shape depends on the channel estimationmethod used. However, a spherical uncertainty region gives the worst case estimation error model [51]. In this case,the estimation error vector is bounded by∥∥∥e(PU)∥∥∥2≤ .272.4. Robust Beamforming Method with Imperfect Channel KnowledgeTaking all the error terms into one matrix ∆R, we get∆R=β(1,1)(e(PU)1 )†e(PU)1 · · · β(1,K)(e(PU)1 )†e(PU)K.... . ....β(K,1)(e(PU)K )†e(PU)1 · · · β(K,K)(e(PU)K )†e(PU)K, (2.26)which is a random matrix modeling the channel estimation errors. Now we can writeRtrue = Rest + ∆R, (2.27)where Rest is the estimated covariance matrix of the channel fading gains between the set of CRrelays and the PR. This matrix Rest can be calculated using h(PU)est as well as β(r,f). ∆R is thecovariance error matrix and is bounded by ‖∆R‖ ≤ ΩR, where ΩR is the bound of the uncertaintyregion of Rest.Since Rtrue is a covariance matrix, it can be factorized using Cholesky decomposition [53].Therefore, we can write Rtrue = CtrueC†true, where Ctrue is a lower triangular matrix. Similarly, wecan write Rest = CestC†est. Then, the relation between Ctrue and Cest can be written asCtrue = Cest + ∆C , ‖∆C‖ ≤ ΩC , (2.28)where ΩC is the bound of the uncertainty region of Cest.Using eq. (2.28) in eq. (2.17), we can reformulate the leakage constraint as follows∥∥∥g†Ctrue∥∥∥2≤ γth. (2.29)However, in order to ensure a robust design of the beamforming vector using Cest, the aboveconstraint must be satisfied for the worst case estimate of Ctrue, i.e.,max‖∆C‖∥∥∥g†Ctrue∥∥∥ ≤√γth. (2.30)282.4. Robust Beamforming Method with Imperfect Channel KnowledgeUsing the triangle inequality, we can write∥∥∥g†Ctrue∥∥∥ ≤∥∥∥g†Cest∥∥∥+∥∥∥g†∆C∥∥∥ . (2.31)Now by applying Cauchy-Schwartz inequality, we can rewrite eq. (2.31) as∥∥∥g†Ctrue∥∥∥ ≤∥∥∥g†Cest∥∥∥+ ‖g‖ ‖∆C‖ . (2.32)Using the maximum value of∥∥g†Ctrue∥∥ given in eq. (2.32) and substituting it in eq. (2.30), thedesign constraint now becomes∥∥∥g†Cest∥∥∥2≤ (√γth − ‖g‖ΩC)2 . (2.33)By using the relation between Rest and Cest, the design constraint in eq. (2.33) can finally beexpressed asg†Restg ≤ (√γth − ‖g‖ΩC)2 . (2.34)Therefore, our primal optimization problem for this RLBF method can be written asg(RLBF) = maxg(g†h†hgP),subject to: g†Restg ≤ Ith, (2.35)where Ith =(√γth − ‖g‖ΩC)2. Following the same procedure as described in the previous section,we can find the optimal RLBF vector for the optimization problem in eq. (2.35). This optimalRLBF vector g(RLBF) can be written asg(RLBF) =√IthhR−1est†h†R−1esth†. (2.36)In order to find the RLBF vector g(RLBF) in eq. (2.36), we need to calculate Ith which in turndepends on ‖g(RLBF)‖. In what follows, we present the steps of finding the value of ‖g(RLBF)‖. Bysubstituting g = g(RLBF) in Ith =(√γth − ‖g‖ΩC)2and taking the norm of eq. (2.36), we can292.5. Numerical Resultsfinally write∥∥g(RLBF)∥∥ =√γthΩC +√hR−1est†h†‖R−1esth†‖. (2.37)Using eq. (2.37) in eq. (2.36), we can write the RLBF vector g(RLBF) in a desired closed form asfollowsg(RLBF) =(√γth −ΩC√γthΩC +√hR−1est†h†‖R−1esth†‖)√1hR−1est†h†R−1esth†. (2.38)2.5 Numerical ResultsCR Receiver Figure 2.3: Simulated network topology.In this section, we present some numerical examples in order to demonstrate the performancesof various beamforming methods in the presence of asynchronous interference. For all the numericalexamples presented in this section, we consider the network topology that is shown in Fig. 2.35.We assume that all the channel fading gains are identically and independently distributed withRayleigh distribution, and the log-distance path loss model with a path loss exponent value of4 is considered. A slot duration of 0.4 msec is considered. The busy probability p of the PU,5We simulated the performances of other network topologies as well. Similar performance trends have beenobtained for other network topologies.302.5. Numerical Resultsi.e., the channel occupancy probability, has a value of 0.7 unless other value is specified. Todemonstrate the effectiveness of various beamforming methods, the interference threshold at thePR γth is set to be 4.1419 × 10−19 Watts which is 100 times of the noise power spectral density.The value of the target spectral efficiency B that is used in the simulations is equal to 1 bit/sec/Hz.When the PU transmitter is active, the received normalized average interference power from theprimary transmitter to the CR receiver is assumed to be −10 dB.110 120 130 140 150 160 17010−2210−2010−1810−1610−1410−1210−10Transmit power normalized to noise power (dB)Interference power at primary receiver  ZFBF [20]JLS [34]LBF (proposed)Threshold Interference powerFigure 2.4: Asynchronous interference signal power at the PR with perfect channel information.In Fig. 2.4 we plot the normalized transmit power versus the interference signal power at thePR with our proposed LBF method. In this figure we also plot the normalized transmit powerversus the interference signal power at the PR with the ZFBF method proposed in [20], and thejoint leakage suppression (JLS) method proposed in [33]. The JLS method is proposed in [33] forconventional cooperative cellular networks. Its design goal is to maximize the signal to leakageplus noise ratio (SLNR) of each of the active mobile users in the network concurrently without anyrestrictions on the induced leakage to a particular mobile user from different BSs. In the contextof cooperative CR networks, the JLS method aims to maximize the SLNR of the CR receiver,312.5. Numerical Resultswithout any restriction on the introduced leakage to the PR. From this figure we observe that theinterference caused by the proposed LBF method increases as the transmit signal power increasesat lower values of transmit power. However, as soon as the received interference power at the PRreaches the target interference threshold, it does not increase with the transmit power. This clearlyshows that our proposed LBF method can maintain the interference threshold at the PR. On thecontrary, the interference caused by the ZFBF method increases almost linearly with the transmitsignal power and exceeds the interference target threshold for higher values of transmit power asexpected. Also the JLS method causes interference power at the PR that exceeds the interferencetarget threshold, since it aims to maximize the SINR at the CR receiver without any restriction onthe introduced interference to the PR.115 120 125 130 135 140 145 150 155 160 165 170−1001020304050  Transmit power normalized to noise power (dB)Received signal power at CR receiver normalized to noise power (dB)ZFBF [20]JLS [34]LBF (proposed)Figure 2.5: Received signal power at the CR receiver with perfect channel information.The received signal power at the CR receiver for three beamforming methods is plotted inFig. 2.5. It is observed from this figure that the received signal power at the CR receiver for ourproposed LBF method is slightly less than that of the ZFBF and the JLS methods. The ZFBF andthe JLS methods are both having higher received signal power at the Cr receiver at the expense of322.5. Numerical Resultshigher asynchronous interference at the PR.0.65 0.7 0.75 0.8 0.85 0.9 0.95 110−310−210−1100Busy probability pOutage probability  ZFBF [20]JLS[34]LBF (proposed)Figure 2.6: Outage probability of SU system versus busy probability of PU.Another important performance metric is the outage probability which corresponds to theprobability of having the instantaneous transmission capacity C in a given time slot below thetarget spectral efficiency B [54]. The outage events are determined as follows. If the number ofsuccessful relays K is zero, this is considered an outage event. Otherwise, the K CR relays eitherforward the data directly to the CR receiver when the PU is silent in the second hop or theycooperatively beamform the data stream to transmit the data to the CR receiver when the PU isactive. The outage events in these two cases occur when the instantaneous transmission capacity Cis below the target spectral efficiency B. When the PU is active, outage events also occur when thethe instantaneous interference caused by the CR relays to the PR exceeds the target interferencethreshold γth. In this case, the CR relays are not allowed to transmit their data. The outageprobability versus channel busy probability, p, is plotted in Fig. 2.6 for our proposed LBF methodand that of the ZFBF method. Although we did not include the performance of the JLS method,JLS will have a higher outage probability than the LBF method as expected. It is also observed332.5. Numerical Results0.5 0.6 0.7 0.8 0.9 110−410−310−210−1100Difference in propagation delay = 0Busy probability pOutage probability  0.4 0.5 0.6 0.7 0.8 0.9 110−410−310−210−1100Difference in propagation delay = Ts/10Busy probability pOutage probability  0.2 0.4 0.6 0.8 110−410−310−210−1100Difference in propagation delay = Ts/2Busy probability pOutage probability  0.2 0.4 0.6 0.8 110−410−310−210−1100Difference in propagation delay = TsBusy probability pOutage probability  ZFBF [20]LBF (proposed)ZFBF [20]LBF (proposed)LBF (proposed)ZFBF [20] ZFBF [20]LBF (proposed)Figure 2.7: Outage probability of SU system versus busy probability of PU for different values of propagationdifference.from Fig. 2.6 that the outage probability of the LBF method is significantly less than that of theZFBF method. This can be explained as follows. When the PU is silent in the second hop, theoutage probabilities of the ZFBF method proposed in [20] and our proposed LBF scheme are equal.However, when the PU is active, the instantaneous interference introduced to the PR by the ZFBFmethod exceeds the target threshold γth most of the time. As such the CR relays cannot forwardthe data symbols and outage events occur frequently. On the other hand, our proposed schemesatisfies the target interference threshold instantaneously. Therefore our proposed LBF method hasa lower outage probability. In particular, our proposed LBF method decreases the system outageprobability up to 75% compared to the ZFBF method, for a PU’s busy probability of 0.75.Next, we investigate the effect of the difference in propagation delays between the cooperatingCR relays on the performance of the wireless network. In Fig. 2.7, we consider having two CR342.5. Numerical Resultsrelays in the network. In this figure, we plot the outage probability of the proposed LBF methodand that of the ZFBF method in [20] versus the busy probability of the PU, p, for 4 different valuesof propagation difference between the two relays. When the propagation difference between thetwo relays at the PR is zero, the proposed LBF method has the same performance as that of ZFBFmethod. As the value of the propagation difference increases, the outage probability of the ZFBFmethod increases, while the LBF method continues to have the same outage probability values.Therefore, the performance gap between the two schemes increases with increasing the propagationdifference between the cooperating relays, until it reaches its maximum value when the propagationdifference is Ts.110 120 130 140 150 160 17010−2210−2010−1810−16Transmit power normalized to noise power (dB)Interference power at primary receiver estimation error bound = 4.1419*10−9  110 120 130 140 150 160 17010−2210−2010−1810−16Transmit power normalized to noise power (dB)Interference power at primary receiver estimation error bound = 4.1419*10−8  110 120 130 140 150 160 17010−2210−2010−1810−16estimation error bound = 4.1419*10−7Transmit power normalized to noise power (dB)Interference power at primary receiver  LBFRLBFThreshold interference power LBF RLBFThreshold interference power LBFRLBFThreshold interference powerFigure 2.8: Transmit power versus interference power at the primary receiver for the RLBF method, for differentvalues of estimation error bound, Ωc.Now we investigate the performances of the proposed RLBF and LBF methods for a transmission352.5. Numerical Resultsscenario when the CR relays have imperfect estimate of the CSI between the PR and the CR relays.In Fig. 2.8, we plot the normalized transmit power versus the interference signal power at the PRfor both the LBF and the RLBF methods when the CR relays have imperfect estimation of channelsbetween the CR relays and the PR, for different values of estimation error bound, Ωc. From thisfigure, it is obvious that the introduced interference from the RLBF method is always well belowthe interference threshold even when the channel is not known perfectly at the CR relays, howeverthe LBF method cannot meet the interference constraint at the PR for higher values of estimationerrors. This is due to the fact that the designed RLBF method is a conservative design consideringthe maximum channel estimation error i.e., the worst case estimate of the channel. For the valuesof the estimation error bound ΩC ≥ 4.1419 × 10−8, the LBF method violates the interferencethreshold of the PR. As the estimation error bound increases, the violation of the LBF method tothe interference threshold becomes more severe.110 120 130 140 150 160 170−30−20−1001020304050Transmit power normalized to noise power (dB)Received signal power at CR receivernormalized to noise power (dB)  LBFRLBFFigure 2.9: Transmit power versus received power at the CR receiver for the RLBF method.In Fig. 2.9 we plot the normalized transmit power versus the received signal power at the CRreceiver using both the LBF and the RLBF methods, using a value of estimation error bound ΩC362.5. Numerical Resultsequals to 4.1419× 10−8. Both of these methods exhibit similar behavior as the the transmit signalpower increases the received signal power at the CR receiver increases. However with the RLBFmethod, the CR receiver receives less power compared to the LBF method. Again this is expecteddue to the conservative design of the RLBF method considering the worst case channel estimationas discussed above.37Chapter 3Cooperative Beamforming withMultiple Primary Receivers andMultiple CR ReceiversAs a follow-up of our work in Chapter 2, in this chapter we consider a more generalized setupwhere a group of CR nodes which are referred to as cooperating cognitive radio nodes (CCRNs)uses cooperative beamforming technique to broadcast a common message to multiple CR receivers.The cooperative CR network uses a wireless broadcast channel assigned to a primary transmitterto transmit information to multiple PRs simultaneously. This scenario is particularly importantwhen a CR source wants to broadcast a common message to a group of CR receivers, the CR sourcemay not be allowed to transmit enough power to cover all the CR receivers due to the interferencerestriction imposed by the nearby primary system. In such situation, a group of CCRNs cancollaboratively use transmit beamforming to broadcast the common message to the CR receivers.With multiple primary and CR receivers, transmission to a specific CR receiver introducesasynchronous interferences, not only at the PRs, but also at all other CR receivers. Due to theseasynchronous interferences, the optimal beamforming technique developed in Section 2.3 cannotbe directly extended for a generalized system with multiple primary and multiple CR receivers.In particular, when there is only one CR receiver in the system, there is no cross asynchronousinterferences between CR receivers [55]. Then optimal beamforming design problem becomes aconvex problem that can be solved optimally using the dual Lagrange function, as we have seen inSection 2.3. However, for the generalized setup with multiple primary and CR receivers consideredin this chapter, the optimal beamforming technique is indeed intractable due to the non-convexityand non-linearity of the problem. In light of the intractability of the optimal beamforming design383.1. System Modelproblem, in this chapter, we propose an approximation to design the beamforming directions andto allocate power among different beamforming directions. Even then development of a subopti-mal beamforming technique is complex due to multiple interference constraints corresponding tomultiple PRs which is discussed later in Section 3.3. Therefore, we also propose a low complexitypower allocation algorithm.In what follows, we present the overall system description and model the asynchronous inter-ference signals at the PRs as well as at the CR receivers mathematically. Next, we develop theproposed beamforming techniques with perfect CSI at the CCRNs. Then, we extend the beam-formong design problem for the case of having only statistical CSI of the PRs available at theCCRNs. Finally, we propose and investigate the performance of joint CCRN selection and cooper-ative beamforming.3.1 System ModelIn what follows, we provide the operating assumptions as well as the operating principles of thesystem model that we consider in our problem formulation.3.1.1 Operating AssumptionsIn this chapter, we consider a CR-based broadcasting system as the one shown in Fig. 3.1, wherea group of L CCRNs uses a transmit beamforming technique to broadcast a common informationto a group of K CR receivers. All nodes are assumed to be equipped with single antenna each. Aswe mentioned earlier in Chapter 1 that similar type of cooperative beamforming scenario has beenconsidered for traditional wireless networks, e.g., wireless sensor networks, due to its compellinggain in the transmission rate, see for example [15], [32]. Using the USAM, the CR system shares acommunication broadcasting channel with a primary transmitter, e.g., a primary BS that transmitsinformation to J PRs simultaneously. For notational convenience, K CR receivers are denoted bydk, k = 1, · · · ,K, L cooperating CR relay nodes are denoted by cl, l = 1, · · · , L and J PRs aredenoted by pj , j = 1, · · · , J . We consider that both primary and CR systems work in a time-slottedfashion with a slot duration Tslot sec. We assume a block fading channel model, similar to the onewe considered in Section 2.1.393.1. System ModelCR Receiver 1 CR Receiver K Figure 3.1: System model for cooperative beamforming with L CCRNs, K CR receivers and J primary receivers.3.1.2 Operating PrinciplesAt CCRN, cl the information stream is mapped into modulated symbols, xs which has averagepower P and the data vector consisting of these M modulated symbols is denoted by xs. The setof cooperative CCRNs uses K different beamforming vectors to transmit the data to K differentCR receivers. The received signal at CR receiver, dk can be written asyk[n] = hsk[n]gk[n]xs[n] + Ik[n] + mk[n] + zk[n], (3.1)where xs[n] is common message symbols transmitted at time slot n and hsk[n] , [hsk1[n], . . . , hskL[n]]is the channel vector from L transmitting CCRNs to CR receiver, dk. The vector gk[n] ,[gk1[n], . . . , gkL[n]]T denotes the beamforming weight vector of the set of CCRNs correspondingto transmission to CR receiver, dk with each element gkr denoting the weight of the CCRN, cr.zk[n] is the AWGN vector at CR receiver, dk with zero mean and two-sided power spectrum densityN0/2, and mk[n] is the received interfering signal vector from the primary BS at CR receiver dk.Ik[n] is the asynchronous interference signal at CR receiver, dk resulting from the data transmis-sions to the other (K−1) CR receivers6. The mathematical model of this asynchronous interference6Even though the same message is transmitted to all CR receivers, the asynchronous arrival of the data streamintended to one CR receiver, at other CR receivers, is still considered a form of interference. This concept is similarto that of inter-symbol interference that takes place in multi-path environments.403.2. Modeling of Asynchronous Interferences in CR-Based Broadcasting Systemssignal is introduced in the following section. It is important to note that the proposed techniquesin this chapter can be applied to unicast systems as well as broadcast systems without any changein the algorithms, as the data stream intended for each CR receiver is transmitted using this user’sspecific beamforming vector.3.2 Modeling of Asynchronous Interferences in CR-BasedBroadcasting SystemsDue to the difference in path lengths between the CCRNs, the received signals from differentCCRNs at different PRs and at different CR receivers can experience different propagation de-lays. Although the received signal at a particular CR receiver e.g., d1 from different CCRNs canbe synchronized by using the timing advance mechanism, as mentioned in Section 2.2, or othermechanism [32], the received signals at the PRs pj (j = 1, · · · , J) and at the other CR receivers,dk (k = 2, · · · ,K) cannot be synchronized simultaneously. As such, the signal transmissions fromCCRNs will introduce asynchronous interferences at PRs pj (j = 1, · · · , J) and at the other CRreceivers, dk (k = 2, · · · ,K). In Section 2.2, we have modeled the asynchronous interference atthe PR with one PR and one CR receiver in the system. However for the generalized scenarioconsidered in this chapter, we need to model the asynchronous interferences not only at differentPRs but also at different CR receivers. In what follows, we model these asynchronous interferences.For notational convenience, we will drop the time slot index n.3.2.1 Asynchronous Interference at Primary Receiver, pjUsing eq. (2.13) from Chapter 2, the asynchronous interference power resulting from transmissionto CR receiver, dk at PR, pj , P(j,k)asynch, can be written mathematically in the following formP (j,k)asynch =L∑r=1L∑f=1g†kf (hpjf )†hpjrgkrβjk(r,f), (3.2)where hpjr is the channel fading gain from CCRN cr to PR, pj , βjk(r,f)is the correlation betweenthe asynchronous symbols of CCRNs, cr and cf at PR, pj corresponding to the transmission to CRreceiver, dk. The value of βjk(r,f)can be calculated for given propagation delays between CCRNs,413.2. Modeling of Asynchronous Interferences in CR-Based Broadcasting Systemscr and cf to PR, pj using the same technique described in Section 2.2. Now the total asynchronousinterference power at PR, pj can be expressed asP jasynch =K∑k=1L∑r=1L∑f=1g†kf (hpjf )†hpjrgkrβjk(r,f). (3.3)The asynchronous interference power at PR, pj in eq. (3.3) can be rewritten in a matrix form asfollowsP jasynch =K∑k=1gk†Rjkgk, (3.4)where Rjk is expressed asRjk =βjk(1,1)(hpj1)†hpj1 · · · βjk(1,L)(hpj1)†hpjLβjk(2,1)(hpj2)†hpj1 · · · βjk(2,L)(hpj2)†hpjL.... . ....βjk(L,1)(hpjL)†hpj1 · · · βjk(L,L)(hpjL)†hpjL. (3.5)The received signal power at CR receiver, dk is given byPk,signal = Pgk†(hsk)†hskgk. (3.6)3.2.2 Asynchronous Interference at CR Receiver, dkWe denote the asynchronous interference signal at CR receiver, dk resulting from the data trans-missions to the other (K − 1) CR receivers, by Ik[n]. This asynchronous interference signal can bewritten as followsIk[n] =K∑i=1,i 6=kL∑r=1hskr[n]gir[n]ikr [n], (3.7)where ikr [n] is the asynchronous vector of symbols received at the CR receiver dk from the CCRNcr, as shown in Fig. 3.2.The asynchronous interference power at CR receiver, dk resulting from transmission to CRreceiver, di is given byAIki =L∑r=1L∑f=1βki(r,f)g†if (hskf )†hskrgir, (3.8)423.2. Modeling of Asynchronous Interferences in CR-Based Broadcasting Systemsk rτ∆rck rn[]ikdsn[]xk fτ∆fcsn[]xknf[]ikdFigure 3.2: An example of the asynchronous vector of symbols received at the CR receiver dk from two differentCCRNs, with two different propagation delays.where βki(r,f)is the correlation between the asynchronous symbols of CCRNs, cr and cf at CRreceiver, di corresponding to the transmission to CR receiver, dk where k 6= i. The value of βki(r,f)can be calculated for given propagation delays between CCRNs, cr and cf to CR receivers, di anddk using the same method described in Section 2.2. Therefore, the total asynchronous interferencepower at CR receiver, dk resulting from data transmission to the other (K − 1) CR receivers, AIkcan be written asAIk =K∑i=1,i 6=kL∑r=1L∑f=1βki(r,f)g†if (hskf )†hskrgir. (3.9)Similar to eq. (3.6), AIk can be written in a matrix form as followsAIk =K∑i=1,i 6=kg†iTki gi, (3.10)where Tki is written asTki ,βki(1,1)(hsk1)†hsk1 · · · βki(1,L)(hsk1)†hskL.... . ....βki(L,1)(hskL)†hsk1 · · · βki(L,L)(hskL)†hskL.From eqs. (3.2) and (3.8), it is obvious that if ZFBF method is used to force the asynchronousinterference power at each PR pj and each CR receiver dk to be zero, the interference from eachCCRN has to be annulled separately, i.e. each CCRN must have a number of antennas > (K + J).433.3. Beamforming Design with Perfect Channel KnowledgeSo for single antenna based CCRN, failing to satisfy this condition, the ZFBF method results inhigher interference at the PRs, as shown in Section 3.6.3.3 Beamforming Design with Perfect Channel KnowledgeIn this section, we reformulate the cooperative leakage beamforming (LBF) technique in order toaddress the problem of asynchronous interferences at the PRs and other CR receivers. In thisreformulation, we use the same assumption as in Section 2.3 that the channel fading gains, i.e.,instantaneous CSI between the CCRNs and the PRs as well as the instantaneous CSI between theCCRNs and the CR receivers are known perfectly at the CCRNs. In the next section, we considerthe case when only the statistical CSI between the CCRNs and the PRs are known at the CCRNs.3.3.1 Problem FormulationLet us denote the achievable transmission rate of CR receiver dk by rk, which can be expressedusing the ideal capacity formula as followsrk = log2(1 +Pg†k(hsk)†hskgkσ2n,i,k +K∑i=1,i 6=kg†iTki gi), (3.11)where σ2n,i,k is the total interference (from the primary transmitter) and noise power at CR receiverdk. The goal is to design K different beamforming vectors corresponding to K CR receivers thatmaximize the weighted sum rate of all CR receivers while keeping the interference to the PRs belowtheir target thresholds. We consider maximizing the weighted sum rate of all the K CR receiverssince it is more generalized (see for example [56], and the references therein). The design goal can443.3. Beamforming Design with Perfect Channel Knowledgebe formulated as an optimization problem as follows7g(opt)1 ,g(opt)2 , · · · ,g(opt)K = maxg1,··· ,gKK∑k=1wk log2(1 +Pg†k(hsk)†hskgkσ2n,i,k +K∑i=1,i 6=kg†iTki gi),subject to:K∑k=1g†kRjkgk ≤ γjth, for j = 1, · · · , J. (3.12)where wk is the weighting factor of CR receiver dk, and γjth is the required interference thresholdfor PR, pj .3.3.2 Development of Sub-Optimal Cooperative LBF TechniqueThe optimization problem in eq. (3.12) is a non-linear and non-convex optimization problem dueto the presence of the interference power AIk =∑Ki=1,i 6=k g†iTki gi in CR receiver dk’s transmissionrate, rk. The design problem of the beamforming vectors in eq. (3.12) can be converted into a jointbeamforming and power allocation problem without the loss of optimality. In particular, we proposea cooperative bemforming technique that has two phases as described below. It is important tonote that if the asynchronous interference terms at the PRs, as well as the cross asynchronousinterference terms between the CR receivers, are neglected in eq. (3.12) , the ZFBF technique in[20] can provide the optimal beamforming vectors directly in a single step.Phase IIn this phase, the direction of the normalized beamforming vector, g¯k is obtained. As such thereceived signal power at CR receiver dk is maximized while minimizing the interference at all PRsand other CR receivers. This can be written as the following optimization problemg¯(LBF)k = maxg¯kg¯†k(hsk)†hskg¯kg¯†k(Rk + Tk)g¯k, for k = 1, · · · ,K, (3.13)where Tk =∑Ki=1,i 6=k Tik and Rk =∑Jj=1 Rjk. The signal-to-leakage power ratio in eq. (3.13) isin the form of a generalized Rayleigh quotient, that is maximized when g¯(LBF)k is the normalized7We do not consider transmit power constraint for the CCRNs as we develop the beamforming technique for theinterference limited scenario.453.3. Beamforming Design with Perfect Channel Knowledgeeigen vector of the matrix(Rk + Tk)−1(hsk)†hsk that corresponds to its maximum eigen value [57].Therefore the normalized beamforming vector, g¯(LBF)k , can be expressed as followsg¯(LBF)k =(Rk + Tk)−1hs†k∥∥∥(Rk + Tk)−1hs†k∥∥∥. (3.14)Phase IIIn this phase, power is allocated among different beamforming directions. As such the weightedsum rate of CR receivers is maximized while the interference thresholds at different PRs are met.In particular, given the normalized beamforming vector g¯(LBF)k obtained in phase I, we obtain itsallocated power α(LBF)k that satisfies the interference thresholds at all PRs simultaneously, whereg(LBF)k =√α(LBF)k g¯(LBF)k .Using the normalized beamforming vector, g¯(LBF)k , obtained in phase I into the optimizationproblem in eq. (3.12), the optimal power allocation problem among different beamforming direc-tions can be rewritten as followsα(LBF)1 , · · · , α(LBF)K = maxα1,··· ,αKK∑k=1wk log2(1 +Pαkg¯(LBF)†k (hsk)†hskg¯(LBF)kσ2n,i,k +K∑i=1,i 6=kαig¯(LBF)†i Tki g¯(LBF)i),subject to:K∑k=1αkg¯(LBF)†k Rjkg¯(LBF)k ≤ γjth, for j = 1, · · · , J. (3.15)The power allocation problem in eq. (3.15) is still a non-linear and non-convex optimizationproblem due to the presence of the asynchronous interference power AIk. Non-convex optimizationis significantly harder to be analyzed and solved, even by numerical methods. In particular, alocal optimum may not be a global optimum and the duality gap can be strictly positive [58].Several approximations, in different contexts, have been proposed to solve non-convex optimizationproblems, see for examples [59, 60] and the references therein. However, these approximations arenot applicable to the optimization problem in our hand. On the other hand, power control problemsfor sum rate maximization for interference channels in conventional wireless communication systemshave been studied in [61] and [62] under a sum power constraint. These power allocation schemescannot be extended for the problem in eq. (3.15) due to the asynchronous interference power463.3. Beamforming Design with Perfect Channel Knowledgeconstraints at the PRs, which is essential for underlay CR networks. Moreover, the problem in ourcase is a weighted sum rate maximization problem which requires different handling. Therefore, wepropose a different approximation for the power allocation problem in eq. (3.15) to relax it into aconvex problem. In Section 3.6, the performance of the proposed approximation is compared withthe numerical solution of eq. (3.15) obtained using active set method [63], for different noise plusinterference power levels, σ2n,i,k. In fact, it is shown that the proposed approximation has a veryclose performance to that of the active set method in terms of achievable sum rate, but with muchless computational complexity.The proposed approximation is done by initially assuming that the power values, αi (i =1, · · · ,K) are equal, i.e.,α1 = α2 = · · · = αK = α(0)EQ, (3.16)where the initial value α(0)EQ is obtained from the constraints of the interference thresholds at thePRs as followsα(0)EQ = min(γ1th∑Kk=1 g¯(LBF)†k R1kg¯(LBF)k, · · · ,γJth∑Kk=1 g¯(LBF)†k RJk g¯(LBF)k). (3.17)The asynchronous interference power AIkEQ corresponding to α(0)EQ is given byAIkEQ = (K − 1)α(0)EQg¯(LBF)†i Tki g¯(LBF)i , k = 1, 2, · · · ,K. (3.18)Then using this asynchronous interference power AIkEQ, we propose to obtain power values αk(k = 1, 2, · · · ,K) corresponding to different beamforming directions as followsα(LBF)1 , · · · , α(LBF)K = maxα1,··· ,αKK∑k=1wk log2(1 +Pαkg¯(LBF)†k (hsk)†hskg¯(LBF)kσ2n,i,k + AIkEQ),subject to:K∑k=1αkg¯(LBF)†k Rjkg¯(LBF)k ≤ γjth, for j = 1, · · · , J. (3.19)Now the approximated optimization problem in eq. (3.19) is a convex optimization problem, thatcan be solved using the dual Lagrange function, with zero duality gap. Please note that thedifference of updated AIk corresponding to α(LBF)k (k = 1, 2, · · · ,K) in eq. (3.19) and AIkEQ in473.3. Beamforming Design with Perfect Channel Knowledgeeq. (3.18) is negligible compared to σ2n,i,k. This is due to the fact that the mutual asynchronousinterference signals between the CR receivers, g¯(LBF)†i Tki g¯(LBF)i , have already been minimized byoptimizing the beamformer directions in phase I. Therefore, the weighted sum rate obtained bysolving the approximated optimization problem in eq. (3.19) is very close to the weighted sumrate that is obtained by numerically solving the original non-convex problem in eq. (3.15). Thisis verified in Section 3.6, where the values of α(LBF)k obtained by solving eq. (3.19) are shown toprovide a weighted sum rate which is very close to that of the numerically obtained value in eq.(3.15) by using active set method [63], for different noise plus interference power levels, σ2n,i,k.The Lagrange function of the above optimization problem can be written asL =K∑k=1wk log2(1 +Pαkg¯(LBF)†k (hsk)†hskg¯(LBF)kσ2n,i,k + (K − 1)αAppg¯(LBF)†i Tki g¯(LBF)i)−J∑j=1(λj(K∑k=1αkg¯(LBF)†k Rjkg¯(LBF)k − γjth)), (3.20)where {λ1, · · · , λJ} are the Lagrange multipliers. Using KKT conditions, we can writewkln 2(αk +σ2n,i,k + (K − 1)αAppg¯(LBF)†i Tki g¯(LBF)iP g¯(LBF)†k (hsk)†hskg¯(LBF)k)−1−J∑j=1(λj g¯(LBF)†k Rjkg¯(LBF)k)= 0 for k = 1, · · · ,K, (3.21)λj(K∑k=1αkg¯(LBF)†k Rjkg¯(LBF)k − γjth)= 0, for j = 1, · · · , J, (3.22)K∑k=1αkg¯(LBF)†k Rjkg¯(LBF)k − γjth ≤ 0, for j = 1, · · · , J, (3.23)λ1, · · · , λJ ≥ 0, (3.24)According to eq. (3.21), the power allocation for beamforming direction corresponding to CR483.3. Beamforming Design with Perfect Channel Knowledgereceiver dk is given byα(LBF)k = max(0,wk(ln 2)∑Jj=1(λj g¯(LBF)†k Rjkg¯(LBF)k) −σ2n,i,k + (K − 1)αAppg¯(LBF)†i Tki g¯(LBF)iP g¯(LBF)†k (hsk)†hskg¯(LBF)k), (3.25)for k = 1, · · · ,K. The power allocation in eq. (3.25) is the cap-limited water-filling solution. Ineq. (3.25), the power allocation values α(LBF)k are expressed in terms of Lagrange multipliers λj(j = 1, · · · , J) which need to be evaluated.In order to obtain the Lagrange multipliers and consequently α(LBF)k , a recursive technique isused as described below. First, we assume that only one Lagrange multiplier is greater than zero,i.e., λj > 0, while λi = 0, for all i except i 6= j. This implies that the power allocation values α(LBF)k ,(k = 1, · · · ,K), satisfy the interference threshold with equality only at PR pj . For this case, wecan writeK∑k=1α(LBF)k g¯(LBF)†k Rjkg¯(LBF)k − γjth = 0. (3.26)Now the value of λj and the power allocation values α(LBF)k , for all k are found by solving set ofequations in (3.25) and (3.26) simultaneously. If these values of α(LBF)k satisfy the remaining (J−1)interference constraints given by the set of equations in (3.23), then α(LBF)k for all k represent theoptimum solution of (3.19). Otherwise, we set λk > 0 (k 6= j) while λi = 0, for all i except i 6= k,and so on until we find the power allocation values that satisfy all constraints simultaneously.If no power allocation values that satisfy all constraints simultaneously is found, considering oneconstraint as equality constraint we consider the case when two constraints are met with equality.In other words, we set simultaneously λj > 0 and λl > 0 while λi = 0, for all i except i 6= j, l.Then, the following two slackness conditions in eq. (3.22) are satisfied as followsK∑k=1α(LBF)k g¯(LBF)†k Rjkg¯(LBF)k − γjth = 0, (3.27)K∑k=1α(LBF)k g¯(LBF)†k Rlkg¯(LBF)k − γlth = 0. (3.28)The values of λj , λl and the power allocation values α(LBF)k for all k are found by solving the setof equations in (3.25), (3.27), and (3.28) simultaneously, where the Lagrange multipliers are found493.3. Beamforming Design with Perfect Channel Knowledgeusing the sub-gradient method. If these values of α(LBF)k satisfy the remaining (J − 2) interferenceconstraints given by the set of equations in (3.23), then α(LBF)k for all k are the optimum powerallocation values. Otherwise, we set another set of two constraints as equality constraint, i.e.,λm > 0 and λn > 0 (m,n 6= j, l) while λi = 0, for all i except i 6= m,n, and so on until we findthe values of α(LBF)k that satisfy all constraints simultaneously. The worst case scenario in terms ofcomplexity occurs when the J constraints hold with equality simultaneously.This procedure is summarized below:for i = 1→ J do- Form(Ji)different sets of λ’s, such that each set Sik for k = 1, · · · ,(Ji)is composed ofi different λ’s.for j = 1→(Ji)do-Assume that λm = 0 for λm 6∈ Sij , and that λn > 0 for λn ∈ Sij , i.e., the interferenceconstraints at i PRs are satisfied with equality simultaneously.-Substitute these λ’s in eq. (3.25), and in slackness conditions given in eqs. (3.22) to getthe optimum power allocation, α(LBF)k ∀k.-Check whether the total interference introduced due to the transmissions to K CRreceivers satisfies the other (J − i) interference constraints given in eqs. (3.23),- if yes, exit. Otherwise, continue.end forend for3.3.3 Special Case of Single PR and Single CR ReceiverIn Chapter 2, a cooperative beamforming technique is developed for a CR network with one PRand one CR receiver. In what follows, we show that in presence of a single PR and a single CRreceiver, the sub-optimal cooperative LBF algorithm proposed in this chapter provides the optimalLBF vector same as that in Section 2.3 as a special case. With only one CR receiver in the network,there are no cross asynchronous interference terms present i.e., Tk = 0 in eq. (3.14). In this case,503.3. Beamforming Design with Perfect Channel Knowledgethe optimum beamforming direction can be written as follows (c.f. eq. (3.14))g¯(LBF)k =R−1k hs†k∥∥∥R−1k hs†k∥∥∥. (3.29)When Tk = 0, the optimization problem in eq. (3.15) is already a convex one that is solvedoptimally using the dual Lagrange function. Using the power allocation for the beamformingvector given in eq. (3.25), and the slackness condition in eq. (3.22), and after some mathematicalmanipulations, the optimum power allocation value is given byα(LBF)k =γthg¯(LBF)†k Rkg¯(LBF)k. (3.30)The optimum beamforming vector, g(LBF)k in case of a single PR, single CR receiver is given by,g(LBF)k =√α(LBF)k g¯(LBF)k ,g(LBF)k =√γthhskR−1k†hs†kR−1k hs†k (3.31)which is identical to the cooperative beamforming technique developed in Section 2.3.3.3.4 Low Complexity Power Allocation SchemeThe computational complexity of the power allocation scheme among different beamforming vec-tors proposed in Section 3.3.2 can, in the worst case scenario, is in the order of O(K(2J − 1)).The sub-optimal power allocation (SOPA) scheme8 among beamforming directions obtained bysolving approximated optimization problem in eq. (3.19), jointly finds all the K allocated powervalues which can, in the worst case, require solving the J interference constraints simultaneously.Therefore, we also propose a low complexity power allocation (LCPA) scheme as described below.Rather than finding the power allocation value αk by keeping all the J interference constraintssimultaneously in eq. (3.19), we propose to find the power allocation value for only one interferenceconstraint e.g., jth interference constraint at a time. For notational convenience let us denote, the8Throughout this chapter, we refer to the solution of eq. (3.19) as sub-optimal one as we have approximated theoriginal power allocation problem in eq. (3.15).513.4. Extension of Beamforming with Statistical Channel Knowledgecorresponding power value by αj,LCPAk (k = 1, · · · ,K) which can be written as followsαj,LCPAk = max(0,wk(ln 2)λj g¯(LBF)†k Rjkg¯(LBF)k−σ2n,i,k + (K − 1)αAppg¯(LBF)†i Tki g¯(LBF)iP g¯(LBF)†k (hsk)†hskg¯(LBF)k). (3.32)The value of λj is obtained from the following complementary slackness conditionλj(K∑k=1αj,LCPAk g¯(LBF)†k Rjkg¯(LBF)k − γjth)= 0, for j = 1, · · · , J. (3.33)So, now for a given beamforming direction corresponding to a particular CR receiver dk, wehave J power values αj,LCPAk (j = 1, · · · , J) corresponding to J interference constraints. Out ofthese J power values, the minimum power value is selected as the final power allocation value forkth beamforming direction, i.e.,αLCPAk = min(α1,LCPAk , α2,LCPAk , · · · , αJ,LCPAk). (3.34)The complexity of this proposed LCPA scheme varies linearly with the number of CR receivers, aswell as with the number of PRs, given by O(KJ), compared to that of the SOPA scheme whichis in the order of O(K(2J − 1))in the worst case. This lower complexity comes at the expense ofsum transmission rate of CR receivers.3.4 Extension of Beamforming with Statistical ChannelKnowledgeIn many scenarios, the instantaneous CSI of the channels between the CCRNs and the PRs maynot be available at the CCRNs. In this section, we consider having only the statistical CSI9 ofthe channels between the primary users and the CCRNs rather than the instantaneous CSI. Forsuch scenario, the interference thresholds at the PRs can be guaranteed statistically. In absence ofinstantaneous CSI of the channel between PR and a CR transmitter, such statistical interferenceconstraint to PRs has been used in [11], [36]. According to this statistical asynchronous interference9Statistical CSI refers to the distribution of CSI which is assumed to be Rayleigh, and the corresponding parameter.523.4. Extension of Beamforming with Statistical Channel Knowledgeconstraint, interference thresholds are met probabilistically as followsPr(P jasynch ≥ γjth)≤ j , (3.35)where Pr denotes probability and j is the maximum allowable probability of violating the inter-ference threshold γjth at PR pj . Since the distribution of the random interference power Pjasynchis not available in a closed-form, the probability in the left side of eq. (3.35) cannot be writtenin a closed-form in terms of average channel gains between the CCRNs and the PRs. In whatfollows we develop an upper bound on this probability value, i.e., Pr(P jasynch ≥ γjth), using thewell-known Markov’s inequality [64], in terms of average channel fading power gains between PRpj and CCRNs.According to Markov’s inequality the probability that a nonnegative random variable X isgreater than or equal to some positive constant a is upper bounded by the ratio of expected valueof X and a, i.e., Pr(X ≥ a) ≤ E(X)a [64], where E(·) denotes an expectation operator. Sincethe asynchronous interference power P jasynch is a non-negative function of the random variableshPjr, r = 1, · · · , L, according to Markov’s inequality, the probability Pr(P jasynch ≥ γjth)is upperbounded as followsPr(P jasynch ≥ γjth)≤E(P jasynch)γjth, (3.36)which leads to a limit on the average asynchronous interference power on PR pj (c.f. eq. (3.35))E(P jasynch)≤ jγjth. (3.37)Since the total asynchronous interference power at PR pj , Pjasynch, is the summation of the inter-ference powers corresponding to the transmissions of different CR receivers, the average value ofthe total asynchronous interference power at PR pj can be written asE(P jasynch)=K∑k=1E(P (j,k)asynch). (3.38)The interference power at pj resulting from transmission to CR receiver dk, P(j,k)asynch can be533.4. Extension of Beamforming with Statistical Channel Knowledgewritten in an expanded form as follows (c.f. eq. (3.2))P (j,k)asynch =L∑r=1L∑f = 1, f 6= rg†kf (hpjf )†hpjrgkrβjk(r,f)+L∑r=1g†kr∣∣∣hpjr∣∣∣2gkrβjk(r,r). (3.39)Since the channel fading coefficients between different CCRNs and PR pj are independent and havezero mean, the average value of the first term in eq. (3.39) is equal to zero. For the second termin eq. (3.39), it can be easily shown that for a Rayleigh fading channel, the the fading power gain,∣∣∣hpjr∣∣∣2has an exponential distribution with a mean value of Ωjr, where Ωjr is the path loss over thechannel from the rth CCRN to the jth PR. The term∑Lr=1 g†kr∣∣∣hpjr∣∣∣2gkrβj(r,r) is a summation ofL independent and identically distributed (i.i.d.) exponential random variables, which is a hypo-exponential random variable, with a mean value of∑Lr=1 g†krΩjrgkrβj(r,r). Therefore the averagevalue of P (j,k)asynch is given byE(P (j,k)asynch)=L∑r=1g†krΩjrgkrβjk(r,r). (3.40)This average interference power at PR pj can be rewritten in a matrix form as followsE(P (j,k)asynch)= g†kR¯jkgk, (3.41)whereR¯jk =βjk(1,1)Ωj1 · · · 0.... . ....0 · · · βjk(L,L)ΩjL. (3.42)Using eq. (3.41), eq. (3.37) can be written asK∑k=1g†kR¯jkgk ≤ jγjth. (3.43)Now the cooperative beamforming vector that maximizes the weighted sum rate of CR receiverswhile satisfying the new interference constraint in eq. (3.43) can be formulated as an optimization543.5. Joint CCRN Selection and Cooperative Beamformingproblem as followsgˆ1, gˆ2, · · · , gˆK = maxg1,··· ,gKK∑k=1wkrk,subject to:K∑k=1g†kR¯jkgk ≤ jγjth, for j = 1, · · · , J. (3.44)The above optimization problem is again a non-linear and non-convex optimization problem.However, using the two steps procedure described in Section 3.3.2, a sub-optimal solution forthe cooperative leakage beamforming method for the statistical CSI scenario can be obtained bysubstituting for γjth by jγjth in eq. (3.12).3.5 Joint CCRN Selection and Cooperative BeamformingContributions of different CCRNs vary significantly towards the interfering signals at the PRs, aswell as the received signals at the CR receivers. This is due to the fact that they are located indifferent geographical locations, hence their signals experience different amount of path loss andfading conditions. Intuitively, a CCRN selection strategy can further improve the performance ofthe cooperative beamforming. The joint design of cooperative beamforming and relay selection hasbeen studied before for conventional cooperative networks, see for example [37] and the referencestherein. In the case of CR systems, the CCRN selection scheme should include the constraint ofkeeping the interference at the PRs below their desired interference thresholds. In this section, westudy CCRN selection strategies with multiple primary and multiple CR receivers. In particular,we develop the joint design of cooperative beamforming and optimal CCRN selection scheme. Wealso propose two sub-optimal CCRN selection schemes that have lower complexity, compared tothe optimal scheme. In this development, we assume that the channel gains of the PRs are knownperfectly at the CCRNs for simplicity of problem formulation. However, it is straightforward toextend the proposed CCRN selection schemes to the case where statistical channel knowledge ofthe channels between the PRs and the CCRNs is available at the CCRNs.553.5. Joint CCRN Selection and Cooperative Beamforming3.5.1 Development of Optimal CCRN Selection SchemeTo formulate the CCRN selection problem mathematically, we define a CCRN selection vector Sof size L × 1, where L is the number of CCRNs in the network. The elements of S, sj can takevalue of either 1 or 0, to indicate whether the CCRN cj has been selected for transmission or notrespectively. For notational convenience, we define W as a diagonal matrix having its diagonalelements equal to those of vector S, as followsW = Diag(S). (3.45)With this CCRN selection matrix W, the received signal power at CR receiver dk, Pk,sig, isgiven byPk,sig = Pgk†W†(hsk)†hskWgk. (3.46)Using this value of the received signal power in the optimization problem given in eq. (3.12), wecan formulate the joint CCRN selection and beamforming problem as followsg(S,opt)1 ,g(S,opt)2 , · · · ,g(S,opt)K = maxg1,··· ,gK ,WK∑k=1wk log2(1 +Pg†kW†(hsk)†hskWgkσ2n,i,k +K∑i=1,i 6=kg†iW†TkiWgi),subject to:K∑k=1g†kWRjkWgk ≤ γjth, for j = 1, · · · , J. (3.47)The problem of joint CCRN selection and cooperative beamforming in eq. (3.47) is considered asa mixed-integer non linear problem (MINLP), since the elements of W can only take values of 0 or1 each. Such MINLP can be solved, without losing optimality, by decoupling it into a non-linearproblem (NLP) and a mixed-integer linear problem (MILP), as in [65, 66].Using the value of the received signal power in eq. (3.46) for a given CCRN selection matrix563.5. Joint CCRN Selection and Cooperative BeamformingW, the beamforming vector g(opt)k (W) can be found from the following NLPg(opt)1 (W),g(opt)2 (W), · · · ,g(opt)K (W) = maxg1,··· ,gKK∑k=1wk log2(1 +Pg†kW†(hsk)†hskWgkσ2n,i,k +K∑i=1,i 6=kg†iW†TkiWgi),subject to:K∑k=1g†kWRjkWgk ≤ γjth, for j = 1, · · · , J. (3.48)For a given CCRN selection matrix W, the optimization problem in eq. (3.48) is non-convex andnon-linear similar to the one in eq. (3.12). Therefore, for a given CCRN selection matrix W, wecan use the same two-phase sub-optimal approach described in Section 3.3.2 to find the sub-optimalgk(LBF)(W) for all k.As explained before, having a single PR and a single CR receiver in the network is considereda special case of the more generalized optimization problem in eq. (3.48). For this special case,the optimization problem in eq. (3.48) is convex, and the beamforming vector g(S,LBF)k (W) can beoptimally found in a desirable closed form expression. After some mathematical manipulations, thebeamforming vector for a given CCRN selection matrix W, g(S,LBF)k (W), can finally be written asg(LBF)k (W) =√γthhskW(W†RkW)−1†W†hs†k· (W†RkW)−1W†hs†k . (3.49)Using the resulting beamforming vector g(LBF)k (W), the optimal CCRN selection matrix W(S,LBF)10is obtained from the optimization problem in eq. (3.47) by formulating the following optimizationMILPW(S,LBF) = maxWK∑k=1wk log2(1 +Pg(LBF)†k (W)W†(hsk)†hskWg(LBF)k (W)σ2n,i,k +K∑i=1,i 6=kg(LBF)†i W†TkiWg(LBF)i),wl,l ∈ {0, 1},L∑l=1wl,l ≤ K. (3.50)where wl,l, for l = 1, · · · , L are the diagonal elements of the CCRN selection matrix W. The10Although the cooperative beamforming design for a given CCRN selection matrix is suboptimal, but we call theCCRN selection scheme an optimal one because the selection pattern is selected optimally via exhaustive search.573.5. Joint CCRN Selection and Cooperative Beamformingoptimal solution for this problem is found via exhaustive search over all possible combinations ofall possible subsets in the set of CCRNs. Finally, the corresponding gk(S,LBF) is obtained accordingly.3.5.2 Development of Sub-Optimal CCRN Selection SchemesThe computational complexity of the optimal CCRN selection scheme when used in conjunctionwith the LCPA scheme is of the order of O(KJ∑Lr=2(Lr)), and its complexity order increases toO(K(2J − 1)∑Lr=2(Lr)), when it is used in conjunction with the SOPA method. The main com-plexity comes from the fact that CCRN selection scheme uses exhaustive search over all possiblecombinations of two or more CCRNs to find the best set of CCRNs for cooperative beamform-ing, and for every possible combination, the cooperative beamforming weight vectors need to becalculated for every combination. In particular, with L CCRNs, there are∑Lr=2(Lr)different com-binations of selecting at least two CCRNs for cooperative beamforming, then the correspondingbeamforming weight vectors have to be calculated using the method proposed in Section 3.3 forevery combination.Therefore, in this section we develop two low complexity sub-optimal CCRN selection schemesto reduce the complexity of the optimal CCRN selection scheme.Sub-Optimal CCRN Selection Scheme 1One option to reduce the complexity of the optimal CCRN selection algorithm, is to limit themaximum number of CCRNs that can forward the vector of data symbols to CR receiver dk. Thissolution decreases the computational complexity compared to the optimal one. For example, if onlyR relays are allowed to participate in the cooperative data transmission, where R ≤ L, we need tocompare∑Rr=2(Lr)possible selections only. As an example, if we have 5 CCRNs in the network,and we have to select R = 2 CCRNs for cooperative transmission, we need to search only in 10selections to determine the best 2 CCRNs, instead of having 26 possible selections to compare incase of applying the optimal CCRN selection algorithm in Section 3.5.1. The value of R, whichtrades off complexity with performance, is decided by the CR system under consideration.583.5. Joint CCRN Selection and Cooperative BeamformingSub-Optimal CCRN Selection Scheme 2Now, we develop another low complexity sub-optimal CCRN selection scheme in which R CCRNsare first selected heuristically based only on their channel fading coefficients towards the CR re-ceivers and the PRs. Then these selected R CCRNs participate in cooperative beamforming andcorresponding weight vectors for these selected CCRNs are calculated using the method describedin Section 3.3.In order to select R CCRNs out of the L cooperating CCRNs, we have NS =(LR)possiblecandidate sets and let us denote these sets C1, C2, · · · , CNS where the cardinality of Ci is R for alli = 1, 2, · · · , NS. Since the goal is to increase the received signal power at each of the CR receivers,while reducing the interference to each of the PRs, heuristically we define a metric for each CCRNin a given set based on the channel fading gains between the CCRNs and the CR receivers and thePRs. In particular, the metric of CCRN r in a particular set, Mr, is directly proportional to thechannel fading coefficients from CCRN r to all the CR receivers in the network. Also the metricof CCRN r, Mr is inversely proportional to the channel fading coefficients from CCRN r to all thePRs in the network. So, we define the metric Mr as followsMr =|hs1r|+ |hs2r|+ · · ·+ |hsKr||hp1r|+ |hp2r|+ · · ·+ |hpJr|. (3.51)Using the metrics for the CCRNs in a given set, the combined metric for that particular set is definedas the summation of the metrics of each of its constituting CCRNs. Without loss of generality, weexpress the combined metric for the first set of CCRNs, C1 as followsMC1 = M1 +M2 + · · ·+MR. (3.52)After calculating all the NS combined metrics MCi , corresponding to candidate sets Ci, (i =1, 2, · · · , NS), the set of CCRNs that has the highest combined metric is selected. Finally theCCRNs of the selected set participate in cooperative beamforming and beamforming vectors forthese selected R CCRNs are calculated using the method described in Section 3.3.In Fig. 3.3, we have plotted the computational complexity of the optimal CCRN selectionscheme and the two sub-optimal selection schemes when applied in conjunction with the LCPA593.5. Joint CCRN Selection and Cooperative Beamforming2 3 4 5 6050100150200250300350400450500550Number of transmitting relays LComputational complexity  Optimal CCRN selection schemeSub−optimal CCRN selection scheme 1Sub−optimal CCRN selection scheme 2Figure 3.3: Comparison between the computational complexity of the optimal CCRN selection scheme and the lowcomplexity sub-optimal CCRN selection schemes, when applied in conjunction with the LCPA scheme.scheme, versus the number of CCRNs L in the system, given that the maximum number of CCRNsthat can be used in case of the sub-optimal CCRN selection schemes is 2. As shown in Fig. 3.3,the sub-optimal schemes are less complex than the optimal one in Section 3.5.1, and the differencebetween their complexity increases for higher number of cooperating CCRNs in the system. It isalso shown in this figure that the sub-optimal scheme 2 has lower complexity than that of scheme 1.Note that the gap between the computational complexities of the three selection schemes increaseswhen the schemes are applied in conjunction with the SOPA method. This is because the complexitymetric plotted in this figure is to be multiplied by K(2J − 1)in case of SOPA method, instead ofmultiplying by KJ in case of LCPA method.603.6. Numerical Results3.6 Numerical ResultsIn this section, we present some numerical results in order to compare the performances of vari-ous beamforming techniques in the presence of asynchronous interference. For all the numericalexamples presented in this section, we consider a network topology with L = 4 CCRNs denoted by{c1, c2, c3, c4}, K = 3 CR receivers denoted by {d1, d2,d3}, and J = 3 PRs denoted by {p1, p2,p3}.The distances between the set of CCRNs to each CR receiver, and to each PR are given in Table3.1. The locations of the nodes in Table 3.1 are picked up arbitrarily. We assume that all thechannel fading amplitude gains are independently Rayleigh distributed. We consider a slot dura-tion Tslot = 1 msec, during which a data frame of 1000 symbols is transmitted. We also considera log-distance path loss model with a path loss exponent value of 4. Unless stated otherwise, theAWGN power used in the simulations is −110 dBm. The transmission bandwidth is 1 MHz. Thenormalized average interference power from the primary transmitter to the CR receivers, d1, d2 andd3 are assumed to be −10,−20, and −15 dB, respectively. For simplicity, we consider weightingfactors w1, w2 and w3 are equal to one.Table 3.1: Distances between nodes in the simulated CR-based broadcasting network.CCRN d1 d2 d3 p1 p2 p3c1 100m 120m 120m 2700m 2500m 1700mc2 110m 60m 43m 2400m 1800m 1400mc3 150m 110m 83m 2800m 2200m 1500mc4 170m 150m 130m 3200m 2700m 1880mIn Fig. 3.4, we plot the average normalized symbol power versus the total asynchronous inter-ference signal power introduced at the PRs using our proposed cooperative LBF technique. Weassume that interference thresholds at the PRs p1, p2, and p3 are respectively, γ1th = 0.02× 10−16,γ2th = 0.05×10−16, and γ3th = 0.2×10−16, which are in the order of the noise plus interference powerfrom the primary transmitter, σ2n,i,k, value. In this figure we also plot the asynchronous interferencesignal powers introduced at the PRs when ZFBF technique [20] is used. This figure clearly showsthat our proposed LBF technique can maintain the asynchronous interference thresholds at thePRs simultaneously. On the contrary, the interference caused by the ZFBF technique exceeds the613.6. Numerical Resultsinterference target thresholds at the PRs. This is expected as ZFBF does not take asynchronousinterferences into account in its design.110 120 130 140 150 160 17010−1910−1810−1710−16Average normalized symbol power (dB)Total asynchronous interference at primary receivers  Interference threshold at p3Total asynchronous interference at p3, ZFBF [20]Total asynchronous interference at p3, LBFInterference thershold at p2Total asynchronous interference at p2, ZFBF [20]Total asynchronous interference at p2, LBFInterference threshold at p1Total asynchronous interference at p1, ZFBF [20]Total asynchronous interference at p1, LBFFigure 3.4: Total asynchronous interference power at the PRs with different interference thresholds (γ1th = 0.1×10−15and γ2th = 0.25 × 10−15).To assess the performance of the proposed approximations to design the LBF weight vector inSection 3.3.2, their achievable average sum rate is compared to that of the active set method [63]that numerically obtains the power values for the beamforming vectors. The active set methodhas the following structure. It first finds a feasible starting point, and computes the Lagrangemultipliers for the active set, which is a set made up of the optimization constraints that aresatisfied with equality at this starting point. Then the subset of constraints that have negativeLagrange multipliers are removed and a new feasible point is found and the algorithm is repeateduntil the solution is optimal enough, i.e., the constraints are satisfied with a tolerance factor, whichwe assumed to be equal 10−20.In Fig. 3.5, we plot the achievable average sum rate of CR receivers with the proposed co-operative LBF with SOPA scheme, the LBF with LCPA scheme, the active set method [63], andthe ZFBF technique. For the sake of completeness in Fig. 3.5, we also plot the achievable sum623.6. Numerical Results110 120 130 140 150 160 17000.10.20.30.40.50.60.70.8Average normalized symbol power (dB)Sum rate of CR receivers (Mbps)  Active set method [63]Cooperative beamforming with SOPACooperative beamforming with LCPAZFBF [20]Single transmitting CCRNFigure 3.5: Achievable sum transmission rate with various beamforming techniques and single CCRN-based trans-mission.transmission rate of CR receivers when a single CCRN is used for transmission to the CR receiverswithout applying any beamforming 11. From this figure we can observe that the proposed LBF withSOPA method can achieve a sum transmission rate value that is very close to that of the activeset method. It can also achieve a higher sum rate than the well-known ZFBF technique for theCR-based broadcasting system. In particular, the increase in sum transmission rate of CR receiversis about 150% with our proposed LBF technique compared to the ZFBF technique. This can beexplained as follows. With ZFBF an outage is considered if the instantaneous interference causedby the CCRNs at any PR exceeds its corresponding target interference threshold. The ZFBF tech-nique usually cannot satisfy the interference threshold(s), which leads to a frequent transmissionoutage events. In addition, the mutual asynchronous interference signals between the CR receiversare not optimized in the ZFBF method. As such the overall transmission rate of the CR receiverswith ZFBF is degraded. From Fig. 3.5, we can also see that the proposed LCPA scheme that11In this case we also consider an underlay spectrum access mechanism, in which we select the CCRN out of LCCRNs that offers the highest sum rate of all the CR receivers. The selected CCRN uses a transmit power valuethat satisfies all the primary interference constraints.633.6. Numerical Resultshas a lower complexity suffers from a performance degradation compared to the SOPA schemeas expected, but still achieves a higher transmission rate compared to the ZFBF technique. Wecan also observe from this figure that the single CCRN-based transmission without beamformingoffers the lowest possible sum transmission rate for the CR system. This can be explained by thefact that the single CCRN-based transmission scheme does not benefit from beamforming whichimproves the received signal power at the CR receivers while minimizing the effect of asynchronousinterferences at the PRs.110 120 130 140 150 160 17033.544.555.5Average normalized symbol power (dB)Sum rate of CR receivers (Mbps)  Active set method [63]Cooperative beamforming with SOPACooperative beamforming with LCPAFigure 3.6: Achievable sum transmission rate with various power allocation schemes in low noise plus interferencefrom the primary transmitter power environment.For a thorough comparison between the proposed beamforming techniques and the numericallyobtained power values using the active set method [63], the achievable sum rate of these methodsare compared in a low noise plus interference power environment, assuming σ2n,i,k = −130 dBm. Wecan see from this figure that the performance of the LBF method with the proposed SOPA methodis very close to the active set method. The LBF with the proposed LCPA offers a lower sum rateperformance than the SOPA scheme, as expected but with much less computational complexity. Inparticular, as we mentioned earlier in this chapter, the worst case complexity of the SOPA method643.6. Numerical Resultsis linear with respect to the number of CR receivers, and exponential with respect to the numberof constraints. The complexity of the LCPA scheme is linear with respect to both the numberof CR receivers and constraints. On the other hand, the worst case complexity of the active setmethod is exponential with respect to both the number of CR receivers and constraints [67]. Theaverage computational time needed to find the beamforming weights with the LCPA method, theSOPA method, and the active set method is 0.35 msec, 0.5 msec, and 25 msec, respectively usingMATLAB on a Desktop computer with Intel Core i5-2410M processor and a clock frequency of 2.3GHz. This clearly shows that the active set method requires much higher computational time.110 120 130 140 150 160 17000.010.020.030.040.050.060.070.080.090.1Average normalized symbol power (dB)Probability of total asynchronous interference > γ thj  Pr(Pasynch3 >γth3 )Pr(Pasynch2 >γth2 )Pr(Pasynch1 >γth1 )Figure 3.7: The probability that the total asynchronous interference at each PR is greater than γjth with statisticalCSI of the PRs at the CCRNs.Next, we investigate the performance of the proposed cooperative leakage beamforming tech-nique, in case of having only statistical CSI of the channels between the PRs and the CCRNs.In Fig. 3.7, we plot the probability of having the instantaneous asynchronous interference powerat each PR greater than its target threshold. The value of the maximum allowable probabilityof violating the interference thresholds 1, 2, and 3 are assumed to be 0.1. It is obvious fromFig. 3.7 that with our proposed cooperative LBF technique with statistical channel knowledge, the653.6. Numerical Resultsprobability of violating the interference thresholds is maintained within the maximum allowableprobability value.110 120 130 140 150 160 17000.20.40.60.811.21.4Average normalized symbol power (dB)Sum rate of CR receivers (Mbps)  Cooperative LBF with optimal CCRN selection schemeCooperative LBF with sub−optimal CCRN selection scheme 1Cooperative LBF with sub−optimal CCRN selection scheme 2Cooperative LBF without CCRN selectionFigure 3.8: The sum rate of the CR receivers with the optimal CCRN selection scheme, sub-optimal CCRN selectionschemes 1 and 2 and without CCRN selection.Finally the sum rate performance enhancement that can be achieved by applying the CCRNselection schemes in conjunction with the cooperative LBF technique proposed in Section 3.5, isinvestigated in Fig. 3.8. In particular, in this figure we plot the normalized transmit power versusthe average achievable sum rate of the CR receivers with cooperative beamforming technique andvarious CCRN selection schemes that are developed throughout the chapter. In this figure we alsoplot the achievable sum rate of the cooperative LBF technique assuming that all the CCRNs par-ticipate in beamforming (i.e., without applying any CCRN selection strategy). From this figure, itis interesting to see that the optimal CCRN selection scheme in conjunction with the LBF tech-nique outperforms the LBF technique when no CCRN selection is employed. The increase in sumrate is about 45% and the reason can be explained intuitively as follows. When a CCRN selectionscheme is employed, the CCRNs are selected judiciously considering their contributions towardsthe achievable sum rate at the CR receivers as well as the total interference power at the PRs.The proposed low complexity sub-optimal CCRN selection schemes suffer from performance degra-663.6. Numerical Results110 120 130 140 150 160 17000.10.20.30.40.50.60.70.8Average normalized symbol power (dB)Sum rate of CR receivers (Mbps)  Cooperative LBF with optimal CCRN selection schemeCooperative LBF with sub−optimal CCRN selection scheme 1Cooperative LBF with sub−optimal CCRN selection scheme 2Cooperative LBF without CCRN selectionFigure 3.9: The sum rate of the CR receivers with the optimal CCRN selection scheme, sub-optimal CCRN selectionschemes 1 and 2 and without CCRN selection with statistical CSI of the PRs at the CCRNs.dation with respect to the optimal CCRN selection scheme as expected however they outperformthe LBF method without any CCRN selection strategy. The sub-optimal CCRN selection scheme1 has a better performance compared to that of the sub-optimal scheme 2, since its complexity ishigher than that of sub-optimal scheme 2, as discussed in Section 3.5.In Fig. 3.9, we plot the performance of the CCRN selection schemes in conjunction with theLBF method, in case of having only statistical CSI of the channels between the PRs and the CCRNs.Similar observations can be made from this figure, as we did from Fig. 3.8.67Chapter 4Distributed Beamforming andAutonomous Participation DecisionMaking in Cooperative CR SystemsIn this chapter, we address other challenges of applying cooperative beamforming in CR networks,which are the feedback overhead problem and the participation decision making issue. First, weaddress the problem of feedback overhead needed for cooperative beamforming. Cooperative beam-forming requires sharing of instantaneous channel state information (CSI) and location informationamong cooperative CR relays or requires a master node that knows the global instantaneous CSIand location information. Both cases require a huge amount of feedback overhead.Exchanging such large amount of information between the cooperating relays requires addi-tional bandwidth, and causes excessive power dissipation from the CR devices [38]. In addition,CSI estimation errors can become a bottle neck to the potential performance gain from CR relayscooperation [39], as explained earlier. Therefore, in this chapter we propose a distributed beam-forming method to be used in cooperative CR networks that requires only information sharingbetween cooperating CR relays about their locations. Assuming each CR relay knows its own CSItowards the CR receiver and towards the PR, and using the shared information about the otherrelays’ locations, each relay can independently design its own beamforming weight.As an incentive for the CR users to participate in the cooperative transmission, the assisted CRuser can lease its scheduled channel, for a certain amount of time, to the cooperating CR relaysfor their opportunistic access. On the other hand, the CR users spend a certain amount of theirlimited life-time battery power when acting as relays for another CR user. Therefore, it becomes acritical decision for each user to decide whether to participate in the distributed beamforming or68Chapter 4. Distributed Beamforming and Autonomous Participation Decision Making in Cooperative CR Systemsnot. Since no cooperation is assumed between the participating CR relays in the decision making,each CR user does not know other users’ decisions, and hence it cannot assess its own reward incase of participating in the cooperative transmission beforehand. Hence, this problem is consideredas an example of unknown games, that can be tackled using a Bayesian game theoretic approach[40]. In this chapter, we propose two autonomous participation decision making strategies to helpeach CR user in deciding whether to participate in the cooperative transmission or not.We also propose a relay selection method to enhance the performance of the cooperative CRnetwork. On one hand, different cooperating CR relays have different path loss values towards theintended CR user, as well as towards the PR. Accordingly, they have different contributions towardsthe received signal at the intended CR user, as explained before in Section 3.5. On the other hand,as more CR relays participate in the cooperative transmission, the reward value represented inthe amount of time during which the channel of the assisted CR user is leased to the cooperatingCR relays is divided among more CR users. This means smaller reward value for each of them.Therefore, in case of having multiple CR users that are willing to participate in the cooperativetransmission, we need to select some of the CR users to form the distributed beamformer. Hence,in this chapter, we also propose a relay selection method that only uses statistical CSI of the CRrelays towards the intended CR user and towards the PR, to choose the best set of CR users thatyields the maximum received signal power value at the intended destination. .Finally, we extend the two proposed autonomous decision making strategies to handle the caseof having multiple CR users requesting cooperative transmission simultaneously. Since each CRuser can only participate in the cooperative transmission towards one receiver, it becomes a criticaldecision for each CR user to select such receiver among the simultaneous cooperation requests. Thedecision making in this case is based on the best incentive provided by each CR user requestingcooperation. The modified autonomous decision making methods are shown to provide enhancedperformance of the CR network, despite the lack of cooperation among the participating CR usersin making their decisions.In what follows, we provide the system model that we consider in our problem formulation.We present the proposed distributed beamforming method, and the two proposed participationdecision making strategies. Next we present the proposed relay selection mechanism. Finally, weextend the two proposed participation decision strategies to help each CR user decide whether to694.1. System Modelparticipate in the distributed beamforming or not and to which CR receiver, in case of receivingmultiple cooperation requests simultaneously.4.1 System ModelIn what follows, we provide the operating assumptions and principles of the system model that weconsider in our problem formulation.4.1.1 Operating AssumptionsWe consider a CR network with a central cognitive base station (CBS) and N CR users, and aprimary network that has F non-overlapping frequency channels. A PU can randomly occupyits channel at a given time slot, which can be modeled by the so-called ON-OFF model (see forexample [68] and the references therein). The CBS assigns these F channels to different CR usersin the network. Usually, the scheduled CR user can use its assigned channel whenever the PU isidle. We consider a time-slotted system with slot duration Tslot. Both the primary network and theCR network operate in a time-synchronized manner. Channel assignment to the different CR usersin the network takes place every scheduling frame, that has a duration of D time slots. The choiceof the channel scheduling policy is beyond the scope of this thesis. We also assume that each ofthe F channels independently experiences a slow and frequency non-selective fading. In our systemmodel, we assume that each CR user, as well as the CBS, is having only a single antenna element.We consider a downlink transmission scenario but the uplink case can be treated similarly.4.1.2 Operating PrinciplesSometimes, due to high path loss and/or shadowing, the channel between a specific CR user andthe CBS gets weaker which degrades this user’s quality of service, in terms of its data transmissionrate. We denote such CR user by CRd that is allocated with a frequency channel fd during thecurrent scheduling frame. Without loss of generality, we assume that the primary channel fd isidle during the nth time slot. In this case, the CBS can ask other CR users in the network to actas relays and forward the message to CRd at the beginning of the nth time slot. The set of CR704.1. System ModelNo channel assigned 1 2 3 F … CR1 CR5 CRN CR4 CR2 Frequency channels Figure 4.1: System model of a CR network with N CR users opportunistically accessing F frequency channels.users that agree to act as relays inform the CBS with their decision during the same time slot 12.The CBS broadcasts the first data packet to be forwarded to CRd to all the cooperating CR relaysover the channel fd, during the nth time slot. The set of cooperating CR relays can beamform ina distributed manner to forward the data to CRd during the (n + 1)th time slot, irrespective ofwhether the primary channel fd is occupied or not. In particular, by carefully designing a complexweighting factor for relay r independently, denoted by gr, a relatively higher signal power valuecan be achieved at CRd, while limiting the interference introduced at the PR that uses channelfd. The distributed design of the beamforming weights of different CR relays reduces the requiredCSI feedback in the network in terms of sharing the channel information between the cooperatingrelays. In addition to the data packets to be forwarded to CRd, the CBS broadcasts the locationof each of the CR relays participating in the cooperative transmission. Such location informationsharing is essential to account for the effects of asynchronous interference at the PR, as will be seenlater in Section 4.2.As a suitable incentive for other CR users in the network to participate in the cooperativebeamforming to CRd, we consider leasing the scheduled channel of CRd, fd, to the cooperating CRrelays for a certain amount of time. During this time, each of the CR users gets to opportunisticallyuse either the frequency channel fd or its own scheduled frequency channel, for its own data12In this section, we only consider the case of having a single CR user requesting cooperative transmission. However,in Section 4.5, we extend the system model for the case of having multiple simultaneous cooperation requests.714.1. System Modeltransmission. As such, the probability of each of these CR users to find an idle time slot for its datatransmission increases. Also, the CR users that are not allocated any frequency channels duringthe current scheduling frame can access the frequency channel fd during such amount of leasedtime, whenever the PU in this channel is idle. Choosing a scheduling scheme for the cooperatingCR relays to use the leased channel fd is beyond the scope of this thesis.The amount of time, during which channel fd is leased for CR relay r, is proportional to the timedifference gained from the cooperative transmission compared to the case of direct transmissionfrom the CBS to CRd, and is given byLTr = ψ∆TK, (4.1)where ψ is a proportionality constant, and K is the number of CR relays participating in the dataforwarding to CRd. ∆T is the time difference gained from cooperative transmission, and is givenby∆T = δd(1Rdir−1Rcoop), (4.2)where δd is the length of the data packet in bits to be transmitted to CRd for one cooperationrequest by the CBS. Rdir is the data rate that can be achieved by direct transmission from the CBSto CRd, and Rcoop is the data rate that can be achieved by cooperative transmission. Later on inSection 4.7, we show that using this cooperation model, the overall performance of the CR systemis improved. Assuming that the additive white Gaussian noise (AWGN) has a total power of σ2n,Rdir can be calculated as [45, Chapter 4]Rdir = Bd log2(1 +Phs,dnh†s,dnσ2n))(Prd(0)|n), (4.3)where Bd is the bandwidth of frequency channel fd, hs,dn is the channel fading coefficient from theCBS to CRd during the nth time slot, and Prd(0)|n is the probability of the primary channel fdbeing idle, when used for direct transmission from the CBS to CRd, during the nth time slot. We724.2. Proposed Distributed Beamformingconsider a decode and forward relaying scheme, and Rcoop can be calculated as [69]Rcoop =Bd2×min(log2(1 +P(∑Nr=1 |grhr,d(n+1) |)2σ2n), log2(1 +Phsnh†snσ2n)(Prd(0)|n)), (4.4)where hsn is the vector of channel fading coefficients from the CBS to the set of cooperative CRrelays during the nth time slot, and hr,d(n+1) is the channel fading coefficient between relay r and CRdduring the (n + 1)th time slot. gr is designed according to the proposed distributed beamformingmechanism presented in the following section.4.2 Proposed Distributed BeamformingThe decision of every CR user in the network of whether to participate in the cooperative beam-forming or not will be discussed in Section 4.3. For now, let us assume that a group of K CR usersdecided to act as relays and participate in the cooperative beamforming for CRd. The design goalis to improve the signal power at CRd, while limiting the amount of interference introduced to thePR at channel fd. To achieve this goal, each cooperating CR relay independently calculates itsbeamforming weight defined asgr = αrg¯r, (4.5)where αr is the magnitude of the beamforming weight of relay r, and g¯r is the phase of suchbeamforming weight. For notational convenience, we will drop the time slot index n from nowon. The phase of the beamforming weight, g¯r, is designed to compensate for the phase shift of thechannel fading coefficient between relay r and CRd, hr,d. In particular, g¯r is expressed asg¯r =h†r,d|hr,d|, (4.6)where |hr,d| is the magnitude of the channel fading coefficient from relay r to CRd, and (·)† denotesthe complex conjugate.To find the magnitude of the beamforming weight, αr, there are two cases. The first case is734.2. Proposed Distributed Beamformingwhen the PU at the channel fd is idle. In this case, αr is set as followsαr =1√(K), for r = 1, 2, · · · ,K. (4.7)The second case is when the PU is active. In this case, the cooperating CR relays use distributedbeamforming, and the allocated power to each relay should result in a total asynchronous interfer-ence power at the PR below a target threshold value, γth. The total asynchronous interference powervalue at the PR at channel fd, from all the relays participating in the distributed beamforming inthe CR network can be easily derived from eq. 2.12 as followsPasynch =K∑r=1K∑f=1αfαrg¯†f(h(PU)f)†h(PU)r g¯rβ(r,f), (4.8)where h(PU)f is the channel fading gain from the CR relay r to the PR. β(r,f) is the correlationbetween the asynchronous symbols of CR relays r and f at the PR, and is calculated for givenvalues of propagation delays of the CR relays r and f to the PR, using the technique described inSection 2.3. So the value of β(r,f) only depends on the locations of the relays r and f .To find αr, the total asynchronous interference power at the PR in eq. (4.8) is set below a targetthreshold value, γth. We first consider the case of K = 2 relays for simplicity, then the general caseof any K relays will be discussed later in this section. With K = 2, eq. (4.8) can be expanded asfollowsPasynch = α21P∣∣∣h(PU)1 g¯1∣∣∣2+ α1α2g¯†1(h(PU)1)†h(PU)2 g¯2β(1,2)+ α22P∣∣∣h(PU)2 g¯2∣∣∣2+ α1α2g¯†2(h(PU)2)†h(PU)1 g¯1β(2,1), (4.9)where P is the average power of the data symbols to be transmitted to CRd. According to thedefinition of β(r,f) in Section 2.3, the values of β(1,2) and β(2,1) are equal. Therefore eq. (4.9) canbe rewritten asPasynch = α21P∣∣∣h(PU)1 g¯1∣∣∣2+ α22P∣∣∣h(PU)2 g¯2∣∣∣2+ 2α1α2β(1,2)Re{g¯†1(h(PU)1)†h(PU)2 g¯2}. (4.10)744.2. Proposed Distributed BeamformingBy rearranging the terms, eq. (4.10) can be written asPasynch = α21P∣∣∣h(PU)1 g¯1∣∣∣2+ α1α2β(1,2)Re{g¯†1(h(PU)1)†h(PU)2 g¯2}+ α22P∣∣∣h(PU)2 g¯2∣∣∣2+ α1α2β(2,1)Re{g¯†1(h(PU)1)†h(PU)2 g¯2}. (4.11)Note that the first and second terms in eq. (4.11) represent the interference power introduceddue to transmission from the 1st relay, whereas the third and fourth terms represent the interferencepower introduced due to transmission from the 2nd relay. One possible power allocation schemethat can enable the CR relays to select their beamforming weights independently, is to limit theinterference power from each relay separately to be below γth/K, where K = 2 in this case.Therefore, we can write the interference constraints for the two-relay case as followsα21P∣∣∣h(PU)1 g¯1∣∣∣2+ α1α2β(1,2)Re{g¯†1(h(PU)1)†h(PU)2 g¯2}≤γth2, (4.12)α22P∣∣∣h(PU)2 g¯2∣∣∣2+ α1α2β(2,1)Re{g¯†1(h(PU)1)†h(PU)2 g¯2}≤γth2. (4.13)The second term in eq. (4.12) and eq. (4.13), Re{g¯†1(h(PU)1)†h(PU)2 g¯2}, is a function of bothrelays’ CSI. It can be rewritten asRe{g¯†1(h(PU)1)†h(PU)2 g¯2}=∣∣∣h(PU)1 g¯1∣∣∣∣∣∣h(PU)2 g¯2∣∣∣ · cos(∠(h(PU)1 g¯1)+(∠h(PU)2 g¯2)). (4.14)From eq. (4.12) and eq. (4.13), we can find the following equalityα21∣∣∣h(PU)1 g¯1∣∣∣2= α22∣∣∣h(PU)2 g¯2∣∣∣2. (4.15)Using eq. (4.15), eq. (4.14) can be rewritten asRe{g¯†1(h(PU)1)†h(PU)2 g¯2}=α1α2∣∣∣h(PU)1 g¯1∣∣∣2cos(∠(h(PU)1 g¯1)+(∠h(PU)2 g¯2))=α2α1∣∣∣h(PU)2 g¯2∣∣∣2cos(∠(h(PU)1 g¯1)+(∠h(PU)2 g¯2)). (4.16)The value of cos(∠(h(PU)1 g¯1)+(∠h(PU)2 g¯2))is a random number ranging from −1 to +1. So, the754.3. Participation Decision Making Strategiesworst case interference value occurs when the cos(·) is equal to +1. Using eq. (4.15) and consideringthe worst case interference value, we can rewrite eq. (4.14) asRe{g¯†1(h(PU)1)†h(PU)2 g¯2}≤α1α2∣∣∣h(PU)1 g¯1∣∣∣2=α2α1∣∣∣h(PU)2 g¯2∣∣∣2. (4.17)Using eq. (4.17) in eq. (4.12) and eq. (4.13), the interference constraints can be rewritten as followsα21∣∣∣h(PU)1 g¯1∣∣∣2 (P + β(1,2))≤γth2, (4.18)α22∣∣∣h(PU)2 g¯2∣∣∣2 (P + β(1,2))≤γth2. (4.19)From eq. (4.18) and eq. (4.19), the magnitudes of the beamforming weights, α1 and α2 can becalculated.For the generalized case of K relays, using a similar approach we can write the interferenceconstraint corresponding to relay r asα2r∣∣∣h(PU)r g¯r∣∣∣2(P +K∑f=1,f 6=rβ(r,f))≤γthK. (4.20)From this relation, one can easily obtain the magnitude of the beamforming weight αr. Please notethat, the asynchronous interferences from other relays are taken into account by relay r via β(r,f),which only requires knowledge of the locations of other relays.4.3 Participation Decision Making StrategiesUpon participation in the cooperative transmission, each cooperating CR user is rewarded by acertain amount of time over the leased channel fd, as explained in Section 4.1. On the otherhand, the cooperative CR relay spends a certain amount of its limited battery power when actingas a relay for CRd. Therefore, it becomes a critical decision for each user to decide whetherto participate in cooperative beamforming or not. Therefore, in this section, we propose twoautonomous participation decision making strategies that assist each CR user to decide whether toparticipate in the cooperative beamforming to CRd.764.3. Participation Decision Making StrategiesThe participation decision is made by each CR user, considering its acquired reward as well asits paid cost. The acquired reward by CR relay r is represented by the amount of its transmitteddata over the leased channel fd, which it gets in return for the cooperative transmission to CRd.So, the reward for relay r, when assisting CRd, can be written asGrd = LTr ·Bd · log2(1 +Prhr,fdh†r,fdσ2n)(Prd(0)|LTr), (4.21)where Pr is the average symbol power of the transmitted data of relay r, and hr,fd is the channelfading coefficient of relay r to its destination on the leased channel fd. Prd(0)|LTr is the probabilityof the primary channel fd being idle during the time it is leased to CR relay r.On the other hand, CR relay r spends a certain cost for participating in the cooperative trans-mission to CRd. Let us define Pr,d as the transmission power of relay r required for cooperativetransmission to CRd. So Pr,d is given byPr,d = P · αr2. (4.22)We define the cost of relay r as the amount of data that could have been transmitted by relay r, ifit used the transmission power Pr,d for its own transmission on its own channel, fr, rather than inassisting CRd. So, the cost function of relay r, when assisting CRd, can be defined asCrd = Tcoop ·Br · log2(1 +Pr,dhr,frh†r,frσ2n)(Prr(0)|(n+1)), (4.23)where Br is the bandwidth of the channel fr allocated to CR user r during the current schedulingframe, and hr,fr is the channel fading coefficient of relay r to its destination on the channel fr.Prr(0)|(n+1) is the probability of this primary channel fr being idle during the (n+ 1)th time slot,which is the time slot during which cooperative transmission to CRd takes place. Tcoop is theamount of time used for cooperative transmission to CRd, and is given byTcoop =δdRcoop. (4.24)Based on eq. (4.21), and eq. (4.23), CR user r can decide to participate in the cooperative774.3. Participation Decision Making Strategiestransmission, if its reward value exceeds its paid cost. However, due to the lack of coordinationbetween the CR users in decision making, as well as the lack of CSI exchange between them, eachCR user does not know the other users’ rewards and costs, and it does not even know its ownreward and cost ahead. This behavior represents an unknown game, that can be solved from aBayesian game perspective [40]. The optimal participation decisions of such Bayesian game arethose that cause the system to reach its NE state. The NE state of the system is achieved wheneach CR user has chosen a strategy of participation decision making, and no CR user can benefitby changing its strategy as long as other CR users keep theirs unchanged [42].To solve this unknown game, we propose two participation decision making strategies in thissection. First, let us define some parameters that are used to describe the proposed decisionmaking strategies. The decision of the CR user r to participate in the cooperative beamformingto CRd is denoted by Ar,d, which takes value 1 if the decision is to participate, or 0 otherwise.From eq. (4.4) and eq. (4.23), the expected cost to be paid in the cooperative beamforming toCRd depends on hr,d. The better the condition of the channel state from CR user r to CRd, thelower the cost paid in data forwarding to CRd, and vice versa. Accordingly, the participationdecision variable Ar,d depends on the value of hr,d, which is a continuous random variable. Asan approximation, we divide the domain of the fading coefficient hr,d into Nr intervals, based onthe average value of the fading coefficient magnitude in each interval. Accordingly, CR user r hasNr different participation decisions, which is expressed as follows, Ar,d ∈ {A¯r,d(1), · · · , A¯r,d(Nr)},where A¯r,d(m) is the participation decision of CR user r in the mth interval of the average fadingcoefficient hr,d. We denote the vector A¯r,d = {A¯r,d(1), · · · , A¯r,d(Nr)} as the action profile of CRuser r when considering cooperative transmission to CRd.The set of all possible action profiles of the CR user r when transmitting to CRd is denoted bythe set pir,d. We define the joint set of action profiles followed by all the K CR users as the mixedstrategy profile of all the K CR users in the network that are considering either to participate or notin the cooperative transmission. We denote such mixed strategy profile by Π = {A¯1,d, · · · , A¯K,d}.The set of all possible mixed strategy profiles is denoted by Σ. It is important to note that an -NEmixed strategy profile, Π, is the one that causes the system to asymptotically converge to the -NEstate. The design goal of the following two proposed decision making strategies is to search for Π.784.3. Participation Decision Making Strategies4.3.1 Regret Testing-Based Strategy (RTS)We propose the first autonomous decision making strategy for the cooperative CR network thatis based on the well-known regret testing procedure, which has been used in different contexts ofgame theory [70]. We modify the regret testing procedure based on our model of CR networks.The regret testing procedure represents one form of exhaustive search, and it can asymptoticallyconverge to an approximate NE (-NE) state [41]. We prove that the proposed RTS converges to-NE state, within a certain convergence time.In the search for the -NE mixed strategy profile, Π, every Ttest time slots are considered as atesting period. During this testing period, all CR users test their benefits or losses resulting fromeither participation or not in the cooperative transmission. During the testing period, Ttest, CRuser r follows a certain action profile, A¯r,d, with probability p and follows its complement, 1−A¯r,d,with probability 1 − p. In either cases, each CR user calculates the profit, or loss, achieved fromits decision. If the decision implied by the action profile, A¯r,d, is to participate in the cooperativetransmission, we define an indicator of the loss encountered by following such action profile asl(A¯r,d), given byl(A¯r,d) =1, if Grd < Crd,−1, otherwise.(4.25)CR user r keeps track of its encountered loss values to calculate its regret at the end of the testingperiod. We denote the regret of CR user r as Ωr,d. At the end of each testing period, if the CR userr achieves a regret value Ωr,d ≤ 0, when it follows an action profile, A¯r,d, it stays with the sameaction profile for the next testing period. Otherwise, it randomly chooses a new action profile fromthe set pir,d. However, even if the CR user r achieves a negative regret when it follows a certainaction profile, this action profile may not be the optimal one yet. So it has to keep exploring otheraction profiles, with some probability λ, known as the exploration factor.To find the set of all possible action profiles of the CR user r when transmitting to CRd, pir,d,we make the following logical assumption. If the action for a certain channel state is set to 1, theaction for all the higher channel states must be 1. Therefore, the action profile of the CR user ris in the form of A¯r,d = {0, · · · , 0, 1, · · · , 1}. Hence, there are only Nr + 1 possible action profilesof the CR user r when transmitting to CRd, constituting the set pir,d. The proposed RTS can be794.3. Participation Decision Making Strategiesdescribed for the CR user r as follows.Definitions:- tc is the cooperation inquiry slot count, incremented each time the CBS asks for cooperativetransmission.- A¯r,d is the action profile of the CR user r, initially selected randomly from pir,d.- Ω(tc)r,d is accumulated regret value when the action profile A¯r,d is followed. The initial valueΩ(0)r,d = 0.procedurefor tc = 1→ · · · do- Find the interval, m, in which the channel fading coefficient hr,d lies in.- With probability p, CR user r follows the mth entry in its action profile, A¯r,d(m).if A¯r,d(m) = 1 then- Accumulated regret when following the action profile is Ω(tc)r,d = Ω(tc−1)r,d + l(A¯r,d).elseΩ(tc)r,d = Ω(tc−1)r,d + 0.end if- With probability 1− p, CR user r follows the complement of this mth entry, 1− A¯r,d(m).if A¯r,d(m) = 0 then- Accumulated regret when not following the action profile is Ω(tc)r,d = Ω(tc−1)r,d − l(A¯r,d).elseΩ(tc)r,d = Ω(tc−1)r,d + 0.end ifif mod (tc, Ttest) = 0 then- Calculate the average regret value of following the action profile A¯r,d, Ωr,d =Ω(tc)r,dT.if Ωr,d ≤ 0 then- With probability 1− λ, choose A¯r,d = A¯r,d,- With probability λ, randomly select a new action profile, A¯r,d, from pir,d.else- Randomly select A¯r,d, from pir,d.804.3. Participation Decision Making Strategiesend if- Reset the average regret value Ωr,d = 0.end ifend forend procedureIn what follows, we provide a convergence analysis of the proposed RTS. In particular, we provethat using the proposed RTS, the mixed strategy profile at time tc, denoted by Πtc , asymptoticallyconverges to -NE state, within a certain convergence time tc = Tcon. We start by introducing thefollowing lemma.Lemma 1 The stochastic process Πtc, tc = 1, 2, · · · , defined by the RTS with 0 < λ < 1, is ahomogeneous, recurrent, and irreducible Markov chain with state space Σ. The transition probabilityof this Markov chain has a lower-bound given byp(Πtc+1 = Π′|Πtc = Π) ≥( λNr + 1)K. (4.26)Proof of Lemma 1 To prove that the process is a Markov chain, we note that at each tc = 1, 2, · · · ,Πtc depends only on Πtc−1. It is irreducible since at each tc = 1, 2, · · · , the probability of reachingsome mixed strategy profile Πtc = Π′ from any Πtc−1 = Π is strictly positive, when λ > 0. Itis recurrent since all the states in Σ are guaranteed (with probability 1) to have a finite hittingtime. The transition probability of this Markov chain is given as follows. In the proposed RTS,the probability of a CR user r choosing an action profile randomly is ≥ λ. Then the probability ofchoosing a certain action profile A¯r,d ∈ Π′ given the current action profile A¯r,d ∈ Π can be writtenas followsp(A¯r,d ∈ Π′|A¯r,d ∈ Π) ≥λNr + 1, (4.27)where Nr+1 is the number of all possible action profiles of CR user r in the set pir,d. The probabilityof choosing a different mixed strategy profile is controlled by all the K CR users in the network.814.3. Participation Decision Making StrategiesHence, the transition probability of the Markov chain is given as followsp(Πtc+1 = Π′|Πtc = Π) ≥( λNr + 1)K. (4.28)Since this Markov chain is irreducible and recurrent, therefore it has a stationary distributionQ [71]. Note that the Markov chain converges to the stationary distribution regardless of where itbegins.We define the subset in the state space of the mixed strategy profile, Σ, that leads to -NEstate, as N. We also introduce the notation N¯, where N¯ = Σ\N, to denote the complement ofthe set of -NE state. To prove the convergence of the proposed decision making strategy, we needto prove that the probability that ΠTcon is not included in the set of -NE state is at most , where ≥ 0 is a very small number. This is expressed mathematically as followsp(ΠTcon ∈ N¯) ≤ . (4.29)For certain values of λ, Ttest, we can prove that the proposed RTS asymptotically converges to an-NE state within a convergence time Tcon. This result is summarized in theorem 1.Theorem 1 For the RTS, if the testing period Ttest ≥−22logC2, and the exploration parameter0 < λ ≤ C2KC2−1 −(1−C1)K(2−)(KC2−1), the probability of the mixed strategy profile being in an non -NEstate, after a certain amount of time Tcon ≤log(/2)log(1−(λNr+1)K) , is bounded byp(ΠTcon ∈ N¯) ≤ . (4.30)Where C1, and C2 are constant values, 0 < C2 < 1, 0 ≤ C1 ≤ 1, and  ≥ 0 is a very small number.Proof of Theorem 1 Let us denote the probability distribution of the mixed strategy profile attime tc = Tcon as, p(ΠTcon). Then the probability that the mixed strategy profile ΠTcon is in a non-NE state is denoted by, p(ΠTcon ∈ N¯). As explained before in Section 4.3.1, the mixed strategyprofile at different values of tc, Πtc, represents a Markov chain with a stationary distribution Q. Letus denote the stationary distribution of the non -NE state as Q(N¯). According to theorem 16.2.4824.3. Participation Decision Making Strategiesin [72], the probability that ΠTcon is in a non -NE state is related to the stationary distribution ofsuch state through the following inequalityp(ΠTcon ∈ N¯) ≤ Q(N¯) +(1−( λNr + 1)K)Tcon. (4.31)From the definition of the stationary distribution [72], we can writeQ(N¯) = Q(N¯)p(Πtc+1 ∈ N¯|Πtc ∈ N¯) +Q(N)p(Πtc+1 ∈ N¯|Πtc ∈ N). (4.32)Since the non -NE state is the complement of the -NE state, i.e., N¯ = Σ\N, then Q(N¯) =1 − Q(N). Therefore, eq. (4.32) can be rewritten, after some mathematical manipulations, asfollows.Q(N¯) =p(Πtc+1 ∈ N¯|Πtc ∈ N)1− p(Πtc+1 ∈ N¯|Πtc ∈ N¯) + p(Πtc+1 ∈ N¯|Πtc ∈ N). (4.33)The transition probability from an -NE state to a non -NE state, p(Πtc+1 ∈ N¯|Πtc ∈ N) isequal to 1− p(Πtc+1 ∈ N|Πtc ∈ N). Hence, to calculate the value of Q(N¯) in eq. (4.33), we needto find the bound on the probability of staying in an -NE state. When the mixed strategy profilelies in the -NE state, the expected regret values Ωr,d of all K CR users is at most . Since theregret value of CR user r, Ωr,d, is the average sum of Ttest independent random variables takingvalues between [−1, 1], then we can reach the following result using the generalization of Hoeffding’sinequality [73].p(Ωr,d ≥ 0) ≤ e(−Ttest2/2) . (4.34)The probability that a CR user r keeps using the same action profile, A¯r,d, is given by A¯r,d =(1 − λ) × p(Ωr,d ≤ 0). Then the probability that a CR user r keeps using the same action profile,A¯r,d, is ≥ (1−λ)(1− e(−Ttest2/2)). The probability of keeping the same mixed strategy profile, i.e.,all K CR users keep using their action profiles, is bounded byp(Πtc+1 ∈ N|Πtc ∈ N) ≥ (1− λ)K(1− e(−Ttest2/2))K. (4.35)834.3. Participation Decision Making StrategiesGiven that λ ≤ 1, and e(−Ttest2/2) < 1, eq. (4.35) can be rewritten asp(Πtc+1 ∈ N|Πtc ∈ N) ≥ (1−Kλ)(1−K e(−Ttest2/2)). (4.36)Then the transition probability from an -NE state to a non -NE state is bounded byp(Πtc+1 ∈ N¯|Πtc ∈ N) ≤ 1− (1−Kλ)(1−K e(−Ttest2/2)). (4.37)Let us assume that the probability of staying in a non -NE state is below certain bound C1,where C1 is a constant value 0 ≤ C1 ≤ 1. This is expressed mathematically as follows p(Πtc+1 ∈N¯|Πtc ∈ N¯) ≤ C1. Using this assumption, and eq. (4.37), we can rewrite eq. (4.33) as followsQ(N¯) ≤1− (1−Kλ)(1−K e(−Ttest2/2))2− C1 − (1−Kλ)(1−K e(−Ttest2/2)) . (4.38)Let us select the value of the testing period Ttest to be Ttest ≥−22logC2, where C2 is a constantvalue, 0 < C2 < 1. Let us select the value of the exploration parameter λ to beλ ≤C2KC2 − 1−(1− C1)K(2− )(KC2 − 1). (4.39)For these selected values of λ, Ttest, we can rewrite eq. (4.38), after some mathematical manipula-tions, as followsQ(N¯) ≤2. (4.40)Let us assume that the convergence time of Tcon is bounded byTcon ≤log(/2)log(1−(λNr+1)K) . (4.41)Using eq. (4.40) and eq. (4.41) in eq. (4.31) and after some mathematical manipulations, theupper bound on the probability of ending up in a non -NE state, after the convergence time Tcon,is given as followsp(ΠTcon ∈ N¯) ≤ . (4.42)844.3. Participation Decision Making Strategies4.3.2 Learning-Based Strategy (LBS)The RTS proposed in Section 4.3.1 is one form of exhaustive search, and so it suffers from highcomplexity and slow convergence speed. It is worth noting that the convergence time Tcon increasesas the number of cooperative CR relays, K, increases (c.f. theorem 1). We propose anotherdecision making strategy, namely LBS, which has lower complexity, yet a good performance. Asstated before, for the CR user r to decide to participate in the cooperative transmission to CRd,its expected reward value should exceed its expected paid cost. This condition is mathematicallywritten asE(Grd)− E(Crd) > 0, (4.43)where E(·) denotes the expectation operator.In [74], the authors showed that for a number of observations of a random variable, the samplemean of these observations is an estimate of the true mean of the random variable. Hence, theexpected reward and cost function of the CR relay r can be estimated by keeping track of theachieved reward and cost values in previous participations in the cooperative transmission to CRd.The estimated values of the reward and cost function of relay r, G˜rd and C˜rd, are given as followsG˜rd =1NtNt∑i=1G(i)rd , (4.44)andC˜rd =1NtNt∑i=1C(i)rd . (4.45)Nt is the number of observations of the reward and cost functions, Grd and Crd, during the previousparticipations in the cooperative transmission to CRd. G(i)rd and C(i)rd are the achievable reward andcost of CR user r, respectively, when participating in the cooperative transmission during the ithtime slot. In the proposed LBS, as CR user r participates in more cooperative transmission, abetter estimate of the expected reward and cost functions of CR user r towards CRd is obtained.The proposed LBS for CR user r is described in the following algorithm.Definitions:- tc is the cooperation inquiry slot count, incremented each time the CBS asks for cooperativetransmission.854.4. Relay Selection- A¯r,d is the action profile of the CR user r, with M entries all equal to 1 initially.- G˜rd is the estimated reward of the CR user r when participating in the cooperative transmission,initially equals 0.- C˜rd is the estimated cost of the CR user r when participating in the cooperative transmission,initially equals 0.- Tp is the number of times CR user r participated in the cooperative transmission, initiallyequals 0.procedurefor tc = 1→ .. do- Find the interval, m, in which the channel fading coefficient hr,d lies in.if A¯r,d(m) = 1 then- Increment the number of times CR user r participates in the transmission towards CRd,Tp = Tp + 1.- Estimated reward of CR user r is given by G˜rd =1TpTp∑i=1G(i)rd ,- Estimated cost of CR user r is given by C˜rd =1TpTp∑i=1C(i)rd ,end ifif G˜rd > C˜rd then- Update action profile of the CR user r, A¯r,d(m) = 1.else- A¯r,d(m) = 0.end ifend forend procedure4.4 Relay SelectionIn Section 4.2, we have considered that all CR users that are willing to participate in cooperativetransmission, beamform in a distributed manner. However, since the amount of time the channelfd is leased to the cooperating CR relays is divided among these relays, individual reward of the864.4. Relay Selectionindividual cooperating CR relays becomes smaller. This can be discouraging for the CR usersfrom participating again in future requests of cooperative transmission. Moreover, different CRusers have different channel gains towards the intended CR destination, as well as towards the PR.Therefore, we propose a relay selection strategy, such that only L of the CR users that are willingto participate in the cooperative beamforming to CRd are selected for transmission, where L ≤ K.The decision of selecting the CR relays is based on their channel qualities towards the PR at thechannel fd, as well as their channel qualities towards CRd.The selection criterion of these CR users is to choose the set of relays that maximizes thereceived data rate at CRd, given by eq. (4.48), while keeping the interference power at the PR atchannel fd below its target threshold. As discussed before, the value of αr used in eq. (4.48) isgiven byαr =αr,id = 1/√(L), if the PU at channel fd is idle in the (n+ 1)th time slot,αr,act =√γth∣∣∣h(PU)r∣∣∣√L(P +∑Kf=1,f 6=r β(r,f)) ,if the PU at channel fd is active in the (n+ 1)th time slot.(4.46)The decision of the relay selection is made at nth time slot, however, the cooperative transmissionhappens at (n+ 1)th time slot. Therefore, we use the expected achieved data rate as followsR¯coop = Rid · Prd(0|0) +Ract · Prd(1|0), (4.47)where Prd(0|0) is the probability of channel fd being idle at the (n + 1)th time slot given that itis idle in the nth time slot. Similarly, Prd(1|0) is the probability of channel fd being active at the(n+1)th time slot given that it is idle in the nth time slot. Rid, and Ract are the data rates achievedby cooperative transmission to CRd when the primary user is idle or active respectively, and aregiven byRid =Bd2·min(log2(1 +P(∑Lr=1 αr,id|hr,d|)2σ2n), log2(1 +Phsh†sσ2n)Prd(0)|n), (4.48)874.5. Extension for Multiple CR Users Requiring CooperationandRact =Bd2·min(log2(1 +P(∑Lr=1 αr,act|hr,d|)2σ2n), log2(1 +Phsh†sσ2n)Prd(0)|n). (4.49)The set of L CR users that maximizes the data rate in eq. (4.47) is chosen via exhaustive searchover the set of K CR users. However, the maximization of eq. (4.47) requires the instantaneousvalues of |hr,d| and∣∣∣h(PU)r∣∣∣. Such instantaneous knowledge of the CSI requires a huge feedbackfrom CR user r to the CBS, for r = 1, · · · ,K, which is impractical. Therefore, we propose touse the average values of |hr,d| and∣∣∣h(PU)r∣∣∣ in eq. (4.48) and eq. (4.49), to calculate Rid and Ractrespectively. Later, in Section 4.7, we show that the performance of the proposed relay selectionscheme, using only the statistical knowledge of the CSI roughly yields a similar performance to thecase of full CSI knowledge at the CBS.The average values of the channel gains, |hr,d| and∣∣∣h(PU)r∣∣∣, can be calculated based on thelocations of the CR users and the PR. In particular, for a Rayleigh fading channel, |hr,d| has anaverage value of√piΩr,d4, where Ωr,d is the path loss value over the channel from CR user r toCRd [75]. Similarly, the average value of∣∣∣h(PU)r∣∣∣ is given by√piΩ(PU)r4, where Ω(PU)r is the pathloss value over the channel from CR user r to the PR at channel fd.It is important to note that the relay selection scheme proposed in this section is a centralizedone, which is of a different nature compared to the rest of this chapter. However, we want to showthe potential performance enhancement that takes place when a node selection scheme is applied inconjunction with the participation decision making strategies, previously proposed in this chapter.A fully-distributed relay selection scheme can be an interesting future research goal, which can usethe centralized scheme proposed in this section as a baseline to be compared to.4.5 Extension for Multiple CR Users Requiring CooperationSo far we have considered the case where a single CR user CRd requires assistance to receive its datamessage from the CBS. In this section, we consider the case where J CR users request cooperation.For convenience, we denote these CR users by a set Src = {CRd1 , CRd2 , · · · , CRdJ}. We assumethat these CR users are assigned with primary channels {fd1 , fd2 , · · · , fdJ} respectively. In this884.5. Extension for Multiple CR Users Requiring Cooperationcase, the other CR users in the network, that do not belong to the set Src, have to make a decisionto either participate or not in the cooperative data forwarding to one of the CR users in the setSrc. In this generalized setup, each CR user has to decide whether to participate in the cooperativetransmission or not, and which CR user from the set Src to assist. Therefore, we modify the twodecision making strategies previously proposed.Without loss of generality, we assume that the set of Kj CR relays that are willing to participatein the cooperating beamforming for CRdj in Src is denoted by the set Rj , with cardinality |Rj | =Kj . We define the amount of time during which the scheduled channel of CRdj is leased for CRrelay rj as LTrj , for rj ∈ Rj , and is calculated according to eq. (4.1). The acquired reward byCR relay rj when participating in the cooperative beamforming to CRdj is denoted by Rrdj , and isdefined according to eq. (4.21). We denote the cost of relay rj when participating in the cooperativetransmission to CRdj as Crdj , and is defined according to eq. (4.23). Based on the acquired rewardvalues and the paid cost values, CR user r can decide to participate in the cooperative transmissionto a given CR user CRdj ∈ Src.In what follows, we modify the proposed RTS and LBS to obtain the set Rj . We define somevariables that are used to describe the proposed algorithms. We denote the participation decisionvector by Br, of size J × 1. The elements brj for j = 1, 2, · · · , J of this vector take value either 0or 1. If CR relay r decides to participate in the cooperative transmission to CR user CRdj ∈ Src,brj takes value 1. Otherwise, it is 0. Since a CR relay r can only participate in the cooperativetransmission to one CR user at a given time slot, one of the elements of vector Br will be non-zero.The CR user CRdj that results in the highest accumulated difference between reward and cost isselected for cooperative transmission by CR user r.4.5.1 RTS in Case of Multiple Simultaneous Cooperation RequestsIn what follows, we present the RTS in case of having multiple cooperation requests.Definitions:- tc is the cooperation inquiry slot count, incremented each time the CBS asks for cooperativetransmission towards J CR users, where J ≥ 1.- Tprj , ∀j is the number of times CR user r participates in the cooperative transmission to CRdj ,894.5. Extension for Multiple CR Users Requiring Cooperation- A¯r,dj , ∀j is the action profile of the CR user r when considering transmission to CRdj . Initially,it is selected randomly from pir,dj .- B(0)r is the initial participation decision vector, with brj = 1, and bri = 0, for i 6= j.- Ω(tc)r,dj is accumulated regret of following the action profile A¯r,dj , with initial value Ω(0)r,dj= 0, ∀j.procedurefor tc = 1→ · · · dofor j = 1→ J doif brj = 1 then- Find the interval, m, in which the channel fading coefficient hr,dj lies in.- With probability p, CR user r follows entry m in its action profile, A¯r,dj (m).if A¯r,dj (m) = 1 then- Accumulated regret when following the action profile is Ω(tc)r,dj = Ω(tc−1)r,dj+ l(A¯r,dj ).elseΩ(tc)r,dj = Ω(tc−1)r,dj+ 0.end if- With probability 1−p, CR user r follows the complement of this entry m, 1−A¯r,dj (m).if A¯r,dj (m) = 0 then- Accumulated regret when not following the action profile, Ω(tc)r,dj = Ω(tc−1)r,dj−l(A¯r,dj ).elseΩ(tc)r,dj = Ω(tc−1)r,dj+ 0.end ifelseΩ(tc)r,dj = Ω(tc−1)r,dj+ 0.end if- Accumulated difference between reward and cost is Dj = (∇(tc)r,dj+ ∇¯(tc)r,dj )− (γ(tc)r,dj+ γ¯(tc)r,dj ).end for- Calculate the average regret values of following the action profiles A¯r,dj , Ωr,dj =Ω(tc)r,djTprj.- Find the CR user CRdj with the minimum regret value, Ωr,dj .904.5. Extension for Multiple CR Users Requiring Cooperation- For this user, set Tprj = Tprj + 1, brj = 1, and bri = 0 for i = 1, · · · , J, i 6= j.for j = 1→ J doif mod (tc, T ) = 0 thenif Ωr,dj ≤ 0 then- With probability 1− λ, choose A¯r,dj = A¯r,dj ,- With probability λ, randomly select a new action profile, A¯r,dj , from pir,dj .else- Randomly select A¯r,dj , from pir,dj .end ifend ifend forend forend procedureIt is important to note that if it is the first time a CR user CRdj requests cooperation, the CRrelays in in the network have no history of the accumulated reward and cost values of this user.So, it has to be given higher priority to benefit from cooperative transmission by setting brj equal1 during the current time slot.4.5.2 LBS in Case of Multiple Simultaneous Cooperation RequestsThe modification of the LBS in case of having multiple cooperation requests follows similar steps tothat of the RTS. In particular, the CR user CRdj that results in the highest accumulated differencebetween the reward and cost values is selected for cooperative transmission by CR user r. However,if it is the first time a CR user CRdj requests cooperation, it must be given an opportunity to benefitfrom cooperative transmission, even if simultaneous requests are received at the CR user r. Themodified LBS can be described using the following algorithm.Definitions:- tc is the cooperation inquiry slot count, incremented each time the CBS asks for cooperativetransmission towards J CR users, where J ≥ 1,- B(0)r is the initial CR users selection vector, with brj = 1, and bri = 0, for i 6= j.914.5. Extension for Multiple CR Users Requiring Cooperation- A¯r,dj , ∀j is the action profile of the CR user r when considering transmission to CRdj , with Mentries all equal 1 initially.- R˜rdj , ∀j is the estimated reward of the CR user r when participating in the cooperativetransmission to CRdj , initially equal 0.- C˜rdj , ∀j is the estimated cost of the CR user r when participating in the cooperative transmissionto CRdj , initially equal 0.- Tprj , ∀j is the number of times CR user r has participated in the cooperative transmission toCRdj , initially equal 0.-G(i)rdj , ∀j is the achievable reward of CR user r when participating in the cooperative transmissionto CRdj during the ith time slot.- C(i)rdj , ∀j is the achievable cost of CR user r when participating in the cooperative transmissionto CRdj during the ith time slot.- Dj , ∀j is the difference between estimated reward and cost caused by CRdj .procedurefor tc = 1→ .. dofor j = 1→ J doif brj = 1 then- Find the interval, m, in which the channel fading coefficient hr,dj lies in.if A¯r,dj (m) = 1 then- Increment the number of participation times of CR user r towards CRd, Tprj =Tprj + 1- Estimated reward of CR user r is given by R˜rdj =1Tprji=Tprj∑i=1G(i)rdj ,- Estimated cost of CR user r is given by C˜rdj =1Tprji=Tprj∑i=1C(i)rdj ,end ifif R˜rdj > C˜rdj then- Update action profile of the CR user r, A¯r,dj (m) = 1.else- A¯r,dj (m) = 0.end if924.6. Operation Protocolend if- Calculate Dj = R˜rdj − C˜rdj .end for- Find the CR user CRdj with the maximum value of Dj .- For this user, set Tprj = Tprj + 1, brj = 1, and bri = 0 for i = 1, · · · , J, i 6= j.end forend procedure4.6 Operation ProtocolIn this section, we summarize the proposed techniques in this Chapter, by presenting a protocolthat describes the operation of the proposed cooperative CR network, which is described as follows.1. If the channel between a specific CR user, CRd, and the CBS is weak such that CRd cannotdecode the data packet from the CBS, it feeds back a negative acknowledgment (NACK)packet to the CBS.2. In this case, the CBS ask other CR users in the network to act as relays and forward themessage to CRd at the beginning of the nth time slot.3. The CR users in the network run a participation decision making strategy, proposed in Section4.3, to decide whether to participate in the cooperative data forwarding to CRd or not.4. The CR users that decide to act as relays inform the CBS with their decision during the nthtime slot.5. Using the relay selection scheme, proposed in Section 4.4, the CBS selects the final set of CRusers that can act as relays for CRd.6. The CBS broadcasts the first data packet to be forwarded to CRd to all the cooperating CRrelays over the channel fd. The CBS also broadcasts the location of each of the CR relaysparticipating in the cooperative transmission.7. The set of cooperating CR relays forward the data to CRd, irrespective of whether the primary934.7. Numerical Resultschannel fd is occupied or not, using the distributed beamforming technique, proposed inSection 4.2.4.7 Numerical ResultsIn this section, we present some numerical results to assess the performance of the proposed dis-tributed beamforming technique, and the autonomous participation decision making strategies.Unless stated otherwise, for all the numerical examples, we consider a network topology withN = 5 CR users {CR1, CR2, CR3, CR4, CR5}. There are F = 4 primary channels {f1, f2, f3, f5},allocated by the CBS to the 4 CR users {CR1, CR2, CR3, CR5} respectively for opportunistic spec-trum access. It is assumed that CR4 is not allocated any frequency channel. The CR user thatrequires cooperation from other CR users in the network is CR5. The distances between the setof CR users to CR5, and to the PR occupying the frequency channel f5 are given in Table 4.1.The occupancy probabilities of the channels by the PUs, Prr(1), for r = 1 : 4, are also given inTable 4.1. We assume that all channel fading amplitude gains are independent and Rayleigh dis-tributed. We consider a time slot duration Tslot = 10 µsec, and the scheduling frame duration is 1sec. We also consider a log-distance path loss model with a path loss exponent value of 4. Unlessstated otherwise, the AWGN power used in our simulations is −130 dBm, and the average transmitsymbol power P is 1 mWatt. The bandwidth of each frequency channel is 25 KHz, and quadraturephase shift keying (QPSK) is used. The value of the interference threshold γth = 1× 10−16 Watt.Table 4.1: Simulated network topology in case of one CR user requiring cooperation.CR user CRi distance between CRi and CR5 distance between CRi and PR at f5 Prr(1)CR1 500m 760m 0.7CR2 225m 710m 0.5CR3 225m 710m 0.3CR4 500m 760m 1The performance of the proposed distributed beamforming technique is verified in Fig. 4.2and Fig. 4.3. In order to plot these figures, the primary channel f5 is assumed to be occupiedby the PU with probability Pr(1) = 0.7. It is also assumed that all the other 4 CR users inthe network are acting as relays to forward the data message to CR5. In Fig. 4.2, we plot the944.7. Numerical Results110 120 130 140 150 160 17010−2010−1910−1810−1710−1610−15Normalized average symbol power (dB)Interference power at the PU receiver  distributed beamforming neglecting asynchronous interferencecoopeartive LBF in Ch. 2proposed distributed beamforminginterference threshold at the PU receiverFigure 4.2: Asynchronous interference to the PR at the channel f5.asynchronous interference power at the PR resulting from our proposed distributed beamformingscheme versus the average transmit symbol power, normalized with respect to noise power. Inthis figure we also plot the performance of a distributed beamforming scheme that neglects theasynchronous interference resulting from the different cooperating CR relays. We compare theperformance of both the proposed beamforming scheme and the distributed one that neglects theasynchronous interference, with the cooperative leakage beamforming (LBF) technique, proposedin Chapter 2. The LBF method requires full cooperation among all CR relays, including sharingtheir instantaneous CSI towards CR5 and towards the PR at channel f5. As shown in Fig. 4.2,the cooperative LBF technique in Chapter 2 meets the interference threshold at the PR. However,it requires a huge feedback overhead to exchange the CSI between all cooperating relays. It isalso shown in the figure that the distributed beamforming scheme that neglects the asynchronousinterference signals from different relays fails to meet the interference threshold imposed by the954.7. Numerical Resultsprimary system. On the other hand, our proposed distributed beamforming keeps the asynchronousinterference power values well below the interference threshold, due to its conservative design whichconsiders the worst case scenario of asynchronous interference from the other cooperating relays.110 120 130 140 150 160 170−20020406080100Normalized average symbol power (dB)Normalized received power at CR 5 (dB)  proposed distributed beamformingcooperative LBF in Ch. 2Figure 4.3: Received signal power at CR5.In Fig. 4.3, we plot the normalized received signal power at CR5 versus the normalized averagetransmit symbol power. We compare the performance of our proposed method to the cooperativeLBF technique in Chapter 2. As shown in the figure, full cooperation among the CR relays yieldsbetter performance compared to the distributed beamforming case, as expected. However, theperformance degradation of the distributed beamforming is traded-off for having much reducedfeedback overhead.In the simulated performance of the decision making strategies, in Fig. 4.4 and Fig. 4.5, weconsider a testing period of Ttest = 10 time slots, and an exploration probability λ = 1/3. The964.7. Numerical Results0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.0250.050.0750.10.1250.150.1750.2Probability of PU at the channel f5 being activeData rate of the CR users (Mbps)  CR1CR2CR3CR4CR5     RTS     LBS     no cooperationFigure 4.4: Data rates of CR users in case of participation and no participation.results are shown for the case where the CBS asks for cooperative transmission 100 times, during100 different time slots. In Fig. 4.4, we plot the achieved data rate by each CR user in thenetwork versus the occupancy probability of the primary channel f5, when the proposed RTS andthe LBS are followed. In this figure, we also show the data rates of all CR users in case none ofthe CR users is participating in the cooperative transmission to CR5. As shown in this figure,applying the autonomous decision making strategies results in increased values of the CR users’data rates, compared to the case of no participation at all. The increase in the CR users’ data ratesis degraded as the occupancy probability of the channel f5 increases. When Prd(1) approaches one,the data rates of the CR users approach the values of the non cooperative transmission case. Thisfigure also shows the data rate of CR5, with cooperation and without cooperation. As expected,the cooperative transmission greatly enhances the data rate of CR5 compared to the case of nocooperation, that is the direct transmission case from CBS to CRd.Figure 4.5 compares the two participation decision making strategies in terms of the paid costsby different CR users in the network. In this figure, we plot the data rate that could have beenachieved by each CR user on its scheduled channel, if it used the transmission power Pr,d for its974.7. Numerical Results0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.0250.050.0750.10.1250.15Probability of PU at the channel f5 being activeCost paid by the CR users for cooperative transmission (Mbps)  CR1 using RTSCR1 using LBSCR2 using RTSCR2 using LBSCR3 using RTSCR3 using LBSCR4 using RTSCR4 using LBSFigure 4.5: Cost of CR users with the RTS and the LBS.own data transmission. As shown in this figure, the paid cost in terms of data rate of differentCR users decreases as the occupancy probability of the channel f5 increases. Also, for a givenoccupancy probability at channel f5, the paid cost of the CR user r decreases as the value of Prr(1)increases. This can be explained as follows. As the value of Prr(1) increases, the chance of CRuser r transmitting data over its scheduled channel decreases. Therefore, the amount of powerPr,d cannot be efficiently utilized by the CR user r for its own data transmission, and its paidcost decreases. As shown in Fig. 4.4 the sum rate of the CR users is increased by up to 43% atPrd(1) = 0.3, when using the RTS, while it increases by 98% in case of LBS. We notice the increasein the users’ data rates is higher in case of LBS compared to RTS. Yet, the LBS requires highercost, in terms of data rate, as seen in Fig. 4.5.The convergence behavior of the two autonomous decision making strategies is shown in Fig. 4.6assuming the occupancy probability of channel f5 equals 0.6. In this figure, we plot the differencebetween the accumulated sum rate and the accumulated total cost of all the CR users in thenetwork, when using the RTS and the LBS versus time slots. From this figure we notice that theRTS outperforms the LBF in terms of the difference between reward and cost values. But on the984.7. Numerical Results0 500 1000 1500 2000 2500 3000 3500 40000.250.26250.2750.28750.30.31250.3250.33750.350.36250.375time slot count  tcAccumulated sum rate − accumulated total cost of the CR network (Mbps)  RTSLBSFigure 4.6: Convergence behavior of RTS and LBS.other hand, we notice that the convergence time of the RTS is longer than that of the LBS, whichis expected due to the higher complexity of the RTS.The performance of the proposed relay selection scheme is shown in Fig. 4.7. In this figure,we plot the achieved data rate at CRd versus the occupancy probability of the primary channelfd, assuming that Prd(0|0) = 0.3. As shown from the figure, applying the proposed relay selectionscheme enhances the performance of the cooperative beamforming, compared to the case of notapplying any relay selection schemes. It is also shown in the figure that the proposed relay selectionscheme that requires only channel statistics has similar performance compared to the relay selectionscheme that requires full CSI knowledge. This can be explained as follows. The statistical channelknowledge is only used by the CBS in the relay selection phase, while each CR relay designs itsown beamforming weight using its instantaneous CSI towards the PR at f5 and towards CR5. Thisresults in roughly the same average value of the data rate achieved in the two cases.The performance of the modified autonomous decision making strategies in case of having multi-ple simultaneous cooperation requests is shown in Figs. 4.8−4.11. To plot these figures, we assumea network topology with N = 6 CR users {CR1, CR2, CR3, CR4, CR5, CR6}, and F = 5 primary994.7. Numerical Results0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.01250.0250.03750.050.06250.075Occupancy probability of PU channelData rate of intended CR receiver  Relay selection with full CSIWithout relay selectionRelay selection with average CSIFigure 4.7: Achieved data rate at CRd by cooperative transmission, in the cases of no relay selection, relay selectionwith full CSI knowledge at the CBS, and relay selection using average CSI.channels {f1, f2, f3, f5, f6}, allocated by the CBS to the 5 CR users {CR1, CR2, CR3, CR5, CR6}respectively, for opportunistic spectrum access. In these figures, the occupancy probability of f6 isassumed to be 0.7. The CR users that require cooperative transmissions from the other CR usersin the network are CR5 and CR6. The distances between the set of CR users to CR6, and to thePR occupying the frequency channel f6 are given in Table 4.2.Table 4.2: Simulated network topology in case of two CR users requiring cooperation.CR user CRi distance between CRi and CR6 distance between CRi and PR at f6CR1 650m 1200mCR2 200m 850mCR3 280m 800mCR4 1000m 750mThe performance of the modified RTS is shown in Fig. 4.8 and Fig. 4.9. In Fig. 4.8, we plot theachieved data rate by each CR user in the network with the modified RTS versus the occupancyprobability of the primary channel f5. In this figure, we also show the data rates of all CR usersin case none of the CR users is participating in the cooperative transmission. As shown from the1004.7. Numerical Results0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.0250.050.0750.10.1250.150.175Probability of PU at the channel f5 being activeData rate of the CR users (Mbps)  CR1, RTSCR2, RTSCR3, RTSCR4, RTSCR5, RTSCR6, RTSCR1, no participationCR2, no participationCR3, no participationCR4, no participationCR5, direct transmissionCR6, direct transmissionFigure 4.8: Data rates of CR users in case of no participation, and participation according to the modified RTS, incase of 2 CR users requesting cooperative transmission.figure, the modified RTS results in increased values of the CR users’ data rates, compared to thecase of no participation at all. We also notice, from this figure, that even when the occupancyprobability of f5 approaches 1, the data rates of the CR users in case of cooperative transmissionare still higher than those with no cooperative transmission in the network. This is due to the factthat the occupancy probability of channel f6 is less than 1. We also notice that the data rate ofthe assisted user CR6 starts to increase as the occupancy probability of channel f5 approaches 1.This is because more cooperation opportunities are granted to user CR6 in this case.In Fig. 4.9, we plot the data rate that could have been achieved by each CR user on its scheduledchannel, if it used the transmission power Pr,d for its own data transmission, instead of cooperativetransmission using the RTS. We observe from this figure, that the cost values of different CR usersdecrease as the occupancy probability of channel f5 increases. However, the rate of this decreaseslows down as the value of this probability tends to 1, due to the fact that the occupancy probabilityof channel f6 is still less than 1. So the cooperation is now directed more towards user CR6 instead,and does not entirely stop as the case in Fig. 4.5, even when the occupancy probability of f5 is 1.1014.7. Numerical Results0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.0050.010.0150.020.0250.030.0350.040.045Cost paid by the CR users for cooperative transmission (Mbps)Probability of PU at the channel f5 being active  CR1 using RTSCR2 using RTSCR3 using RTSCR4 using RTSFigure 4.9: Cost of CR users when using modified RTS, in case of 2 CR users requesting cooperative transmission.In Fig. 4.10, we plot the achieved data rate by each CR user in the network versus the occupancyprobability of the primary channel f5, when the modified LBS is followed. In this figure, we alsoshow the data rates of all CR users in case none of the CR users is participating in the cooperativetransmission. Similar observations to those from Fig. 4.8 can be driven from this figure. InFig. 4.11, we plot the data rate that could have been achieved by each CR user on its scheduledchannel, if it used the transmission power Pr,d for its own data transmission, instead of cooperativetransmission using the modified LBS. Similar observations can be made about the performance ofthe modified LBS compared to that of the modified RTS. However, it is noticed from Fig. 4.10and Fig. 4.11 that the increase in the achieved data rates and the cost of the CR users, as theoccupancy probability of channel f5 approaches 1, is more noticeable than that in case of modifiedRTS. This is due to the fact that the convergence of the LBS is faster than that of the RTS, whichmeans that it is less immune to the changes of the system parameters.1024.7. Numerical Results0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.0250.050.0750.10.1250.150.1750.2Probability of PU at the channel f5 being activeData rate of the CR users (Mbps)  CR1, LBSCR2, LBSCR3, LBSCR4, LBSCR5, LBSCR6, LBSCR1, no participationCR2, no participationCR3, no participationCR4, no participationCR5, direct transmissionCR6, direct transmissionFigure 4.10: Data rates of CR users in case of no participation, and participation according to the modified LBS, incase of 2 CR users requesting cooperative transmission.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.01250.0250.03750.050.06250.0750.08750.1Probability of PU at the channel f5 being activeCost paid by the CR users for cooperative transmission (Mbps)  CR1 using LBSCR2 using LBSCR3 using LBSCR4 using LBSFigure 4.11: Cost of CR users when using modified LBS, in case of 2 CR users requesting cooperative transmission.103Chapter 5Summary, Conclusion and FutureWork5.1 Summary and ConclusionsIn this thesis, we have shown that even though cooperative beamforming in CR network can im-prove the radio spectrum utilization and enhance the network performance, it faces a number ofchallenging issues. Throughout the thesis, we have tackled five critical problems facing the imple-mentation of cooperative CR networks, namely the asynchronous interference issue, the imperfectchannel estimation problem, the need to apply relay selection schemes, the problem of feedbackoverhead, and the need for applying participation decision making strategies. We have proposed dif-ferent techniques to solve these problems and we have evaluated the performance of these proposedtechniques.First, in Chapter 2, we have defined the asynchronous interference problem and have provided itsmathematical modeling. We have also proposed a novel cooperative beamforming method, namedthe LBF method for cooperative CR systems. The LBF method accounts for the asynchronousinterference at the primary receiver, and enables the CR relays to transmit data to the CR receiverwith a certain limit on the interference introduced at the PR when the PU is active. In Chapter2, we have also addressed the effect of having imperfect CSI estimation on the performance ofthe proposed beamforming method. We have proposed a robust cooperative beamforming methodnamed the RLBF method to account for the effect of having an error in the channel estimationbetween the PR and each CR relay.In conclusion, presented numerical results showed that the proposed LBF method has a superiorperformance compared to the ZFBF [20] and JLS [33] methods in the presence of asynchronous1045.1. Summary and Conclusionsinterference. In particular, the LBF method causes significantly less interference at the PR thanthe ZFBF and JLS methods. As such, our proposed LBF method decreases the CR system outageprobability up to 75% compared to the ZFBF method, for a primary user’s busy probability of0.75. In the presence of an estimation error of the channel fading gains between the CR relays andthe PR, the RLBF method proposed in Chapter 2 can satisfy the target interference threshold atthe PR despite such estimation error.Next, in Chapter 3, we have considered a generalized scenario of a CR-based broadcastingnetwork with multiple PRs and multiple CR receivers. In order to address the asynchronous in-terference issue in this network model, we have proposed an innovative cooperative beamformingtechnique. In particular, the cooperative beamforming design has been formulated as an optimiza-tion problem of constrained weighted sum rate maximization. In light of the intractability of theoptimal beamforming design problem, an approximation is used to design beamforming directionsand to allocate power among different beamforming directions. Due to the multiple interferenceconstraints corresponding to multiple primary receivers, the power allocation scheme, proposedin Chapter 3, still has high complexity. Therefore, in this chapter, we have also proposed a lowcomplexity power allocation algorithm. Moreover, we have extended the proposed cooperativebeamforming technique for the case of having only statistical CSI of the channels between thePRs and the CCRNs at the CCRNs. In this case, the asynchronous interferences at the PRs areguaranteed in a statistical sense. In the absence of a mathematically tractable expression of thedistribution of the random interference powers at the PRs, we develop an upper bound on theprobability of introducing asynchronous interference power at a given PR beyond a given thresh-old. Then this developed upper bound is used to design a robust leakage beamforming technique.In Chapter 3, we have also proposed three CCRN selection strategies for a generalized scenariowith multiple primary receivers and CR receivers, to be used in conjunction with the beamformingtechnique.To conclude the work accomplished in Chapter 3, the presented numerical results have shownthat the proposed beamforming technique with SOPA can significantly reduce the interferencesignals at all PRs and can provide an increase up to 150% in the sum transmission rate of CRreceivers compared to the ZFBF technique [20]. Moreover, it has been shown in the presentednumerical results that our proposed robust design of the beamforming vector can statistically1055.1. Summary and Conclusionsmaintain the asynchronous interference constraints at the multiple PRs, when only statistical CSIknowledge is available at the CCRNs. In Chpater 3, we have shown that the optimal CCRNselection in conjunction with beamforming can further increase the sum data rate of CR receiversup to 45%.As a solution for the problem of huge feedback overhead required for cooperative beamforming,in Chapter 4 we have proposed a distributed beamforming method to be used in cooperative CRnetworks with minimal amount of feedback overhead. Next in this chapter, we have defined asuitable incentive for the CR users to participate in the cooperative transmission. Then, we haveproposed two autonomous participation decision making strategies, namely the RTS and the LBSmethods, to help each CR user in deciding whether to participate in the cooperative transmissionor not, by introducing cost and reward functions for different CR relays. We have assumed nocoordination is present among the participating CR users in the decision making process. Theperformance of the proposed participation decision making methods has also been verified throughthe numerical results, and their convergence time has been compared. To further enhance theperformance of the cooperative CR network, we proposed a relay selection method in Chapter 4to resolve the competition taking place between multiple CR users that are willing to participatein the cooperative transmission. The selection criterion is to choose the best set of CR users thatyields the maximum received signal power value at the intended CR receiver. The proposed relayselection method only requires statistical knowledge of the CR users’ CSI, to decrease the amountof feedback overhead in the network. The provided simulation results showed the performanceenhancement achieved by the proposed relay selection method. Finally in Chapter 4, we havegeneralized the proposed autonomous decision making strategies, the RTS and the LBS methods,to handle the case of receiving multiple simultaneous cooperation requests from different CR users.The decision making strategies have been designed to help each CR user, not only to decide whetherto participate in the cooperative transmission towards a CR receiver or not, but also to select whichCR receiver among the simultaneous requests that it receives. The modified autonomous decisionmaking strategies are shown to provide better overall performance of the whole network comparedto the case of no cooperative transmission.To conclude Chapter 4, the presented numerical results have shown that the distributed beam-forming scheme can meet the interference threshold at the PR, with no information exchange1065.2. Future Workbetween the cooperating CR relays regarding their channel states towards the CR receiver andtowards the PR. However this reduced feedback overhead comes at a certain expense of receivedsignal power value at the CR receiver, compared to the cooperative LBF technique, which requiresfull cooperation between the CR relays in terms of sharing channel knowledge. The two developedautonomous participation decision making strategies, namely the RTS and the LBS, can increasethe sum rate of the CR users by up to 43%, for a primary channel occupancy probability of 0.3,when using the RTS, and 98% when using the LBS, relative to the case of no cooperation.In conclusion, cooperative beamforming can enhance the performance of CR networks underdifferent operation scenarios. The work presented in Chapter 4 acts as the umbrella under whichthe proposed beamforming techniques in Chapters 2 and 3, as well as the distributed beamformingtechnique in Chapter 4, can be applied. In particular, the application of an autonomous partici-pation decision making strategy enables the creation a cooperative CR network which outperformsconventional CR networks, in terms of the achievable data rates and overall energy efficiency.5.2 Future WorkIn our modeling of cooperative CR networks, throughout the whole thesis, we have considereddifferent sources of channel impairments, including short term fading and path loss. In particular,we have considered the narrowband flat fading model of the channel and the simplified log-distancepath loss model. In addition to these impairments, a signal can typically experience shadow fadingeffect, which represents a random variation about the path loss at a given distance, due to blockagefrom objects in the signal path. Changes in reflecting surfaces and scattering objects can also causerandom variation about the path loss. Thus, a model for the random attenuation due to theseeffects is also needed for a more realistic representation of the cooperative CR network. For suchmodeling, we can use log-normal shadowing [76], which has been confirmed empirically to accuratelymodel the variation in path loss in both outdoor and indoor radio propagation environments. Inthe log-normal shadowing model, the path loss is assumed random with a log-normal distribution.Another channel impairment that can be examined in cooperative CR networks is the widebandmultipath fading. The impact of multipath on the received signal depends on whether the spreadof time delays associated with the line-of-sight and different multipath components is large or1075.2. Future Worksmall relative to the inverse signal bandwidth. If the delay spread is large, the line-of-sight andall multipath components are typically resolvable, leading to the wideband fading model. Whilethese multipath effects are captured in our modeling of narrowband fading channels throughoutthe thesis, wideband fading can be more convenient for urban areas modeling, where the delayspread takes large values. Generally, channel delay spread is highly dependent on the propagationenvironment. In indoor channels delay spread typically ranges from 10 to 1000 nanoseconds, insuburbs it ranges from 200-2000 nanoseconds, and in urban areas it ranges from 1-30 microseconds[77].In addition to incorporating, into our future work, the generalized modeling of channel impair-ments, there are other interesting research directions, that can be built on top of the proposedwork in this thesis, as described below. The first one is considering energy efficiency in the designprocess of the cooperative beamforming techniques. In the past few years, energy-aware com-munications have received a lot of attention from research and industrial communities due to therising energy costs to operate future wireless networks, ecological, and environmental reasons. Thatsaid, designing energy-aware cooperative beamforming techniques is crucial to meet such require-ments. The objective of the energy-aware cooperative beamforming techniques should consider theachievable data rates, transmit powers, and consumed circuitry powers at the transmitters. In ourbeamforming designs, we did not consider energy efficiency aspect. As a future research direction,energy efficient beamforming design in presence of asynchronous interference for CR systems canbe considered.Another interesting future research direction of the work done in this thesis could be to incorpo-rate energy harvesting techniques into the design of cooperative CR networks. The ability to harvestenergy, from ambient or dedicated sources, enables wireless charging of low-power CR devices andenhances the system design, usability, and reliability. Energy harvesting from the transmitted RFsignal can even be used as an incentive or a form of compensation to be offered to the CR users thatagree to act as relay nodes, in exchange for the energy that the CR relay node consumes for packetreception and retransmission to the intended CR receiver. However, energy causality constraintsshould be taken into account in the design process of the cooperative beamforming techniques [78],to indicate that the amount of energy that can be used in data transmission is that which has beenharvested so far by the relay node.1085.2. Future WorkMoreover, QoS is an important consideration in designing wireless communication systems. 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