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Data rate fairness cooperative beamforming for cognitive radio systems in presence of asynchronous interferences Tharranetharan, Selvakumar 2015

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Data Rate Fairness CooperativeBeamforming for Cognitive RadioSystems in Presence of AsynchronousInterferencesbySelvakumar TharranetharanB.Sc. (Eng.), University of Peradeniya, SriLanka, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)April 2015c© Selvakumar Tharranetharan, 2015AbstractCooperative beamforming for cognitive radio (CR) systems, which usesgeographically distributed transmitters to perform beamforming, can im-prove spectrum utilization. However, transmissions from geographicallydistributed cooperative transmitters introduce asynchronous interferencesat the primary receivers (PRs) and the secondary receivers (SRs) due to thepropagation delays. In this thesis, we develop data rate fairness cooperativebeamforming techniques for a CR system with multiple SRs and multiplePRs in presence of asynchronous interferences. In particular, the optimalbeamforming design is formulated as an optimization problem to maximizethe minimum data rate of the SRs subject to transmission power constraintsof the secondary cooperative transmitters (SCTs) as well as asynchronousinterference power constraints at the PRs. The optimal beamforming prob-lem is a quadratically constrained quadratic program (QCQP) max-minoptimization problem, which is non-convex and non-linear. Therefore, wereformulate the optimal beamforming problem as a quasi-convex problemusing the semidefinite program (SDP) relaxation, that can be solved usingthe standard SDP solvers and bisection method.We study important properties of the optimization problem. By exploit-ing these properties, we also develop low complexity suboptimal beamform-iiAbstracting techniques. Further, we extend the beamforming techniques to incor-porate the uncertainties in the channel and propagation delay estimationbetween the SCTs and the PRs. We present numerical results in order tocompare the complexity, the data rate fairness performance and the robust-ness of the developed beamforming techniques. Presented numerical resultsshow that the developed suboptimal beamforming techniques provide trade-offs between data rate fairness performance and computational complexity.Moreover, the developed optimal and suboptimal beamforming techniquesalways guarantee the target interference threshold at the PRs.iiiPrefaceThis research work has been done under the supervision of Dr. JahangirHossain at the School of Engineering, The University of British Columbia,Kelowna, BC, Canada. I declare that I am the first author of this thesis. Ideveloped all the analytic models and performed all the simulation resultspresented in this thesis.A part of this thesis has been submitted for a journal publication:− S. Tharranetharan and M. J. Hossain, “Data rate fairness coopera-tive beamforming techniques for cognitive radio systems in presenceof asynchronous interferences,” Submitted to IEEE Trans. Signal Pro-cess. Available at part of this thesis has been submitted for a conference publication:− S. Tharranetharan and M. J. Hossain, “Cooperative beamforming forcognitive radio systems with multiple primary and secondary receiversin presence of asynchronous interferences,” Submitted to IEEE GlobalTelecommunications Conference (Globecom), San Diego, CA, USA,Dec. 2015.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overlay Spectrum Access . . . . . . . . . . . . . . . . . . . . 31.2 Underlay Spectrum Access . . . . . . . . . . . . . . . . . . . . 31.3 Cooperative Beamforming for CR Systems . . . . . . . . . . . 41.4 Asynchronous Interference . . . . . . . . . . . . . . . . . . . . 61.5 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 10vTABLE OF CONTENTS1.8 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Chapter 2: Background . . . . . . . . . . . . . . . . . . . . . . . 132.1 Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . 132.1.1 Introduction to Convex Optimization . . . . . . . . . 132.1.2 Optimization Problems . . . . . . . . . . . . . . . . . 162.1.3 Semidefinite Program (SDP) . . . . . . . . . . . . . . 192.1.4 Semidefinite Relaxation . . . . . . . . . . . . . . . . . 21Chapter 3: Data Rate Fairness Cooperative Beamforming forCR Systems in Presence of Asynchronous Inter-ferences . . . . . . . . . . . . . . . . . . . . . . . . . 243.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.1 Signal-to-Interference plus Noise Ratio at the SRs . . 263.1.2 Asynchronous Interferences at the PRs . . . . . . . . . 313.2 Problem Formulation and Optimal Beamforming Design . . . 343.2.1 Optimal Beamforming Technique . . . . . . . . . . . . 353.2.2 Complexity of the Optimal Beamforming Technique . 403.3 Suboptimal Beamforming Techniques . . . . . . . . . . . . . . 413.3.1 Suboptimal 1 Beamforming Technique . . . . . . . . . 413.3.2 Suboptimal 2 Beamforming Technique . . . . . . . . . 453.3.3 Suboptimal 3 Beamforming Technique . . . . . . . . . 473.4 Extension of Beamforming with Estimation Uncertainty . . . 503.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 52Chapter 4: Conclusion and Future Works . . . . . . . . . . . . 65viTABLE OF CONTENTSBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Appendix I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Appendix II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79viiList of TablesTable 3.1 Coordinates of SCTs, SRs and PRs. . . . . . . . . . . 52viiiList of FiguresFigure 3.1 An example of cooperative beamforming for CR system. 25Figure 3.2 An example of timing of received signals over a sym-bol duration at the jth SR, where sˆRj [z(k)TiRj ] = f(t−zTiRjTs − τTiRj + ∆(k)Ti )hTiRjg(k)i sRj [z(k)TiRj ]. . . . . . . 28Figure 3.3 Complexity of the optimal and suboptimal beamform-ing techniques. . . . . . . . . . . . . . . . . . . . . . . 53Figure 3.4 Average minimum data rate of the SRs versus trans-mission power constraint at the SCTs for various beam-forming techniques. . . . . . . . . . . . . . . . . . . . 55Figure 3.5 CDFs of instantaneous minimum data rate of the SRswith perfect CSI for various beamforming techniques. 56Figure 3.6 CDFs of the interference at the PRs with variousbeamforming techniques. . . . . . . . . . . . . . . . . 57Figure 3.7 Performance of extended optimal beamforming tech-nique under different amount of estimation uncertain-ties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 3.8 Performance of extended optimal and suboptimal beam-forming techniques for a given amount of estimationuncertainty (ζ = −10 dB). . . . . . . . . . . . . . . . 59ixLIST OF FIGURESFigure 3.9 CDFs of minimum data rate of the system with im-perfect CSI with uncertainty of ζ = −10 dB for vari-ous beamforming techniques. . . . . . . . . . . . . . . 60Figure 3.10 CDF of interference at the PRs using various beam-forming techniques. . . . . . . . . . . . . . . . . . . . 61Figure 3.11 Comparison of average minimum data rate of SRswith the max-min criteria based beamforming and theweighted sum criteria based beamforming techniques. 62Figure 3.12 CDFs of minimum data rates of SRs with the max-min criteria based beamforming and the weighted sumcriteria based beamforming techniques (10 dBW trans-mission power constraint). . . . . . . . . . . . . . . . 63Figure 3.13 Comparison of average data rate with the max-mincriteria based beamforming and the weighted sum cri-teria based beamforming techniques. . . . . . . . . . . 64xList of Acronyms4G Fourth generation5G Fifth generationAWGN Additive white Gaussian noiseCoMP Cooperative multi-point transmissionCR Cognitive radioCSI Channel state informationDSA Dynamic spectrum allocationPR Primary receiverQCQP Quadratically constrained quadratic programSCT Secondary cooperative transmitterSDP Semidefinite programSINR Signal to interference plus noise ratioSLPR Signal to leakage power ratioSMINR Signal minus interference to noise ratioSR Secondary receiverZFBF Zero forcing beamformingxiAcknowledgementsI would like to express my deepest gratitude to my supervisor, Dr. Ja-hangir Hossain for his continuous advice and guidance throughout my MAScprogram. I have been very lucky to have had the opportunity to be Dr. Hos-sains student. He has always been actively involved in my research and wehad discussions in various aspects of my research. I can not describe howmuch I have learnt from him during my studies. This thesis would not havebeen possible without his tremendous support.I would like to thank all the members of my research group for theirencouragement. Last, but not the least, I would like to express my sincerethanks and appreciation to my family and friends. Their never ending loveand support has always been the source of any motivation in my life.xiiChapter 1IntroductionEfficient spectrum usage in wireless communications is an importanttopic for a long period of time. Traditionally, a fixed spectrum allocationpolicy has been adopted to allocate radio spectrum among various wirelessservices. According to this policy, a particular band of the spectrum isassigned to a specific wireless service or a group of users. The assignedspectrum bands (e.g., radio and television bands, cellular and satellite bands,and the air traffic control bands) are dedicated to particular group of usersfor all the time, regardless of whether there are being used or not.The proliferation of wireless services in last two decades resulted in spec-trum scarcity problem in order to support the existing as well as emergingwireless services. On the other hand, the Federal Communications Com-mission’s report has revealed that a large portion of the assigned spectrumremains underutilized in temporal and geographic dimensions [1–3]. As aresult, a dynamic spectrum access (DSA) policy [4] has been proposed as analternative to facilitate efficient and flexible spectrum allocation.According to the DSA, a particular band of the spectrum which is pri-marily allocated to a particular group of users usually referred to as primaryusers, can be used dynamically by another potential group of users usually1Chapter 1. Introductionreferred to as secondary users. Primary users have the priority right of us-ing the assigned bands exclusively. However, the secondary users can accessthe spectrum band as long as the proper function of the primary users isguaranteed. The DSA requires advance features such as spectrum sensingand transmit power control [5].The cognitive radio (CR) is a potential technology that can enable DSA.Joseph Mitola III adopted the term CR to describe a self configurable soft-ware defined radio [6]. Now a days, CR has become a common terminologyfor a range of technologies including ubiquitous wireless access, automatedradio resource optimization, and dynamic spectrum access for a device cen-tric interference management [7]. Based on the CR concept, the IEEE 802.22standard has already developed physical layer and medium access controllayer of wireless regional area network for unlicensed users in the spectrumallocated to television bands [8, 9]. Fourth generation (4G) wireless sys-tems, e.g., 802.16e/h/m standards have also developed solutions for thecoexistence of CR and WiMAX systems in unlicensed bands [10]. Further,CR is a highly promising technology to tackle the challenges in the upcom-ing fifth generation (5G) wireless communication systems such as handlingthe peak-to-mean data traffic ratio and high-volume delay-insensitive dataservices [11]. In general, the DSA is categorized as overlay spectrum ac-cess [12, 13] and underlay spectrum access [14, 15]. The following subsec-tions provide a brief summary of these spectrum access mechanisms.21.1. Overlay Spectrum Access1.1 Overlay Spectrum AccessIn the overlay dynamic spectrum access, secondary users are only al-lowed to access the spectrum bands if these bands are not being used bythe primary users. Therefore, an accurate spectrum sensing [16] needs to beperformed to avoid possible collision between primary and secondary trans-missions. The portion of the spectrum that is sensed as idle is called thespectrum hole. When a spectrum hole is sensed, the secondary users can useit. However, once the primary users return to access the spectrum, the sec-ondary users should immediately stop operating over that particular band.Although the secondary users have the privilege of using the spectrum hole,the interruptive transmissions will lead to a discontinuous data service andintolerable delay to secondary users [17].1.2 Underlay Spectrum AccessIn the underlay dynamic spectrum access, primary and secondary usersare allowed to access a particular band of the spectrum simultaneously.However, the secondary users should maintain the interference at primaryreceivers (PRs) below the threshold specified by the primary network orby the regulatory authority [5]. This approach imposes a severe constrainton the transmission power of secondary transmitter. It does not requirecontinuous spectrum sensing however requires to know the channel state in-formation (CSI) between the secondary transmitter(s) and the PR(s). Dif-ferent possible scenarios have been considered in the literature in order toestimate the CSI between the secondary transmitters and the PRs [18, 19].31.3. Cooperative Beamforming for CR SystemsOne possible scenario is when the primary receiver periodically transmitspilot signals to its own transmitter. If the CR transmitter knows the pilotsignals, the channel between the CR transmitter and the primary receivercan be estimated by the CR system assuming the reciprocity of the chan-nel [19]. In underlay spectrum access, the directionality of the transmitsignal is a highly desired feature to avoid interference to the PRs. This canbe achieved via the beamforming technique [20] as described below.1.3 Cooperative Beamforming for CR SystemsIn general, beamforming is a signal processing technique that is usedto direct the reception or transmission of a signal in a desired directionby using an array of antennas. This is achieved by properly selecting thespacing between antennas and adjusting the phase of the transmit signalsacross antennas, such that the signals from different antennas combine con-structively in a desired direction and nullify each other in the unwanteddirections after the propagation. Due to its simplicity, zero forcing beam-forming (ZFBF) [21] can be used in the CR systems to nullify/reduce theinterference at the PRs. However, implementation of the ZFBF requiresmultiple antennas at the secondary transmitters [22, 23]. Implementationof an antenna array in a mobile device can be restricted by the size and theenergy storage capability of the mobile device.Cooperative communication has been shown to be very promising inorder to improve the overall performance of the conventional wireless com-munication systems. In particular, cooperative communication enhances the41.3. Cooperative Beamforming for CR Systemscapacity and transmit power efficiency [24]. Moreover, cooperative beam-forming techniques enable geographically distributed cooperative transmit-ters with a single antenna to cooperatively act as an antenna array andbeamform to improve the performance of the conventional wireless commu-nication systems [25]. Such a beamforming technique has been proposed andstudied for traditional wireless communication networks as it potentially of-fers large increases in energy efficiency, attainable range and transmissionrate, see for example [26] and the references therein. With the cooperativebeamforming technique, the achievable data rate gain is quite compelling inspite of certain costs associated with it, e.g., frequency synchronization [27–29], timing synchronization [30] and information sharing [30, 31].In order to reap the benefit of cooperative beamforming in the context ofCR systems, the concept of combining cooperative beamforming and CR isproposed in [32] where a set of secondary cooperative nodes collaborativelyassists a secondary source to transmit its message to a secondary destinationin the presence of a primary source-destination pair. In [33], a cooperativebeamforming technique is developed for a secondary system with a singlesecondary source, multiple cooperative nodes and multiple secondary desti-nations. These cooperative beamforming techniques for CR system providebetter spectrum utilization and capacity gain.Using an innovative orthogonal projection technique, the authors in [26]obtained the ZFBF weights of the cooperating CR nodes to null the inter-ference at the PRs. Using the ZFBF technique, the authors in [34] proposeda cross-layer optimization of the transmission rate and scheduling schemeof the data packets. In [35], power allocation for cooperative CR networks51.4. Asynchronous Interferencewas studied, along with user selection under imperfect spectrum sensing.Although the cooperative beamforming technique can improve the overallperformance of CR systems, the above mentioned beamforming techniquehas overlooked the asynchronous interference issue in designing the cooper-ative beamformer as described below.1.4 Asynchronous InterferenceThe asynchronous interference simply occurs due to the multiple copies ofthe received symbols, which arrive with different propagation delays, causinginterference with each other. In particular, given the fact that differentcooperating nodes are usually located in different geographical locations,their signals can arrive with different propagation delays at each PR and ateach secondary receiver (SR). This can lead to asynchronous interferences atthe PRs and the SRs which is described in Chapter 3 in detail. Asynchronousinterference can affect the system performance, if it is ignored or not properlydealt with.The impact of the asynchronous interference in the context of beamform-ing is first considered in a traditional cooperative multi-point transmission(CoMP) system, where multiple base stations cooperatively beamform to themobile stations [36]. Further, a mean square error minimization based beam-forming for conventional wireless systems in the presence of asynchronousinterference is studied for a CoMP system [37]. The goal for cooperativebeamforming in CR system is different from that of the conventional co-operative communication systems as the CR system has to guarantee the61.5. Motivationinterference constraints at the PRs. Therefore, the beamforming techniquesdeveloped in [36, 37] can not be used for CR systems in order to address theasynchronous issue. Recently, asynchronous interference was considered inthe context of cooperative beamforming for CR systems [38, 39]. In [38], thebeamforming technique maximized the received signal power at a SR whilekeeping the asynchronous interference at a PR below a target threshold.This beamforming technique is applicable to a system with only one SR.In [39], the authors considered a CR system with multiple SRs and multi-ple PRs, where the SRs receive common information simultaneously. Thebeamforming objective was formulated as an weighted sum of data ratesmaximization problem which is solved suboptimally.1.5 MotivationAlthough cooperative beamforming techniques for CR systems in thepresence of asynchronous interference have been developed, these techniquesconsidered that multiple SRs receive common information, i.e., a broadcast-ing scenario [39]. However, in a cooperative CR system different SRs canreceive different information. Examples of such a CR system include thesecond hop of a multiple peer-to-peer communication system via a com-mon set of cooperative nodes [40], where the secondary transmitters cannotcommunicate to their corresponding SRs directly, and cooperative multi-point transmission where a group of base stations cooperatively transmitsinformation to the mobile stations [36]. Moreover, the cooperative beam-forming design in [39] was formulated to maximize the weighted sum rate71.6. Contributionsof the SRs and solved suboptimally. In systems with multiple SRs, whereSRs receive their corresponding information, it is important to maintain thefairness among the SRs. Therefore, for such systems, a data rate fairnessbeamforming technique should be considered to provide fairness among theSRs [41, 42]. This motivates us to develop the data rate fairness beamform-ing techniques for CR systems with multiple SRs and multiple PRs in thepresence of asynchronous interference.In a CR system, secondary cooperative transmitters (SCTs) must main-tain the target interference threshold to facilitate the proper functionality ofthe primary network. In order to maintain the target interference thresholds,channel estimation between SCTs and PRs is a critical aspect in practice.In many scenarios, only an erroneous channel estimation can be achieved.This issue is often addressed by considering the uncertainties in channelestimation [43, 44]. Due to the importance of robustness of beamformingtechniques, we are also motivated to develop robust beamforming techniquesin the presence of asynchronous interference and estimation uncertainty.1.6 Contributions− In this thesis, we consider an underlay spectrum access based coop-erative CR system where a group of SCTs cooperatively transmitsdifferent information to different SRs simultaneously. We develop co-operative beamforming techniques to provide a data rate fairness tothe SRs while maintaining the asynchronous interference at the PRsbelow their target thresholds with limited transmission power at the81.6. ContributionsSCTs. The optimal beamforming design is formulated as a quadrat-ically constrained quadratic program (QCQP) max-min optimizationproblem. This max-min problem is non-convex and non-linear. There-fore, we reformulate the beamforming design as a quasi-convex opti-mization problem by using the semidefinite program (SDP) relaxation.Then the optimal cooperative beamforming technique is developed bysolving this quasi-convex problem using the standard SDP solvers andbisection method.− We study important properties of the optimization problem. The firstproperty shows that the optimal beamforming vectors should satisfy atleast one of the inequality constraints at the boundary, i.e., they shouldsatisfy at least one of the inequality constraints with the equality. An-other interesting property shows that the optimal beamforming vectorsshould lead to equal data rates for all the SRs. By exploiting theseproperties, we propose three low complexity suboptimal beamformingtechniques, which provide trade-offs between data rate fairness perfor-mance and the complexity. The suboptimal 1 beamforming techniqueis developed based on signal to leakage power ratio (SLPR) of theindividual transmissions and the properties of the optimization prob-lem. We develop suboptimal 2 beamforming technique by introducinga new metric called signal minus interference to noise ratio (SMINR),which enables us to directly formulate a convex optimization probleminstead of a quasi-convex in optimal beamforming technique. The sub-optimal 3 beamforming technique is developed by approximating the91.7. Thesis Outlineoptimization solution with an effective algorithm.− We incorporate uncertainties in channel gain and propagation delayestimation between SCTs and PRs in our system by using the well-known bounded uncertainty model. Then we derive robust PR inter-ference constraints with uncertainties and extend the optimal and suboptimal beamforming techniques to provide a robust protection of theSRs under such uncertainties.− Finally, we present numerical results to demonstrate the data ratefairness performance, complexity and robustness of the developed op-timal and the suboptimal techniques, with and without estimationuncertainties.1.7 Thesis OutlineThe remainder of the thesis is organized as follows. In Chapter 2, wedescribe some basic background information on the convex optimizationand the SDP and SDP relaxation techniques. In Chapter 3, we presentthe asynchronous interference model for cooperative beamforming in CRsystem with multiple SRs and PRs, and then formulate the data rate fairnessoptimal beamforming problem. In this chapter, we also develop optimal andsuboptimal beamforming techniques for the considered system. Then, thebeamforming techniques are extended to incorporate the uncertainties. Inorder to illustrate the performance of the beamforming techniques, we alsoprovide numerical results in Chapter 3. Finally, Chapter 4 concludes the101.8. Notationsthesis.1.8 NotationsThe notations that are used in this thesis are as follows. The normallowercase letter x denotes the scaler. The bold lowercase and uppercaseletters, x and X denote the vector and metrix respectively. A space isdenoted with typeface font X. More specifically, R, Rn, Rm×n, Sn+ and Hn+represent the space of real numbers, real n−vectors, real m × n matrices,n×n symmetric positive semidefinite matrices and n×n hermitian positivesemidefinite matrices respectively. Sets are represented with calligraphicfont as X . int X denotes the interior of set X . The notation XT denotesthe transpose of X and X† denotes the hermitian transpose of X. E[x]denotes the statistical expectation of an input random entity x. |x| denotesthe absolute value of x. x mod y is the remainder on division of x by y andceil(x) is the smallest integer not less than x. Matrix of all ones with sizex×y is represented by 1x×y and identity matrix of size x is represented as Ix.Trace of a matrix X is denoted as Tr(X). A diagonal matrix generated fromthe elements of vector x is denoted as diag(x). MAT(x) returns a x × xsquare matrix of the elements of a x2-length vector x and vec(X) is a vectorformed by stacking the columns of matrix X. The element wise inequalitiesare denoted as , ≺,  and ≻. X  0 and X ≻ 0 mean that X is positivesemidefinite and positive definite respectively. rank(X) denotes the rank ofmatrix X. ‖x‖2 denotes the Euclidean norm of vector x, ‖X‖F denotes theFrobenius norm of matrix X and ⊗ denotes the Kronecker product. O(·)111.8. Notationsdenotes the big-O notation of complexity. dom f is the domain of a functionf and f : X → Y is the function on the set dom f ⊆ X into the set Y. ⊆denotes the subset. ℜ (X) and ℑ (X) are real and complex components ofcomplex matrix X.12Chapter 2BackgroundThis chapter presents some background information on convex optimiza-tion, SDP and SDP relaxation.2.1 Convex Optimization2.1.1 Introduction to Convex OptimizationConvex optimization is a special class of mathematical optimization,which has numerous applications in areas such as signal processing, com-munications, networking, control systems, data analysis, statistics and fi-nance [45, 46]. It is useful in finding global optimal solution, bounds onoptimal values and approximate solutions. Further, convex optimizationtechniques can apply to non-convex optimization problems with reformula-tions [45]. In this section we provide the basics of convex optimization.Convex setsA set C is called a convex set if the line segment between any two pointsin the set C lies in set C, i.e., for any x1,x2 ∈ C and any λ with 0 ≤ λ ≤ 1,132.1. Convex Optimizationwe haveλx1 + (1− λ)x2 ∈ C. (2.1)Example: C = {x ∈ R|x ≥ 0}, C = {(x, y) ∈ R2|x ≥ 0, y ≥ 0}.Affine setsA set C is called an affine set if the line passing through any two pointsin the set C lies in set C, i.e., for any x1,x2 ∈ C and any λ, we haveλx1 + (1− λ)x2 ∈ C. (2.2)Example: C = {x ∈ Rn|aTx = b}.Convex functionsA function f : Rn → R is called a convex function if dom f is a convexset and if for all x,y ∈ dom f , and λ with 0 ≤ λ ≤ 1, we havef(λx+ (1− λ)y) ≤ λf(x) + (1− λ)f(y). (2.3)Example: f(x) = eax, x ∈ R, a ∈ R.Affine functionsA function f : Rn → Rm is called an affine function if it is a sum of alinear function and a constant, i.e., if it has a form f(x) = Ax + b, whereA ∈ Rm×n and b ∈ Rm.142.1. Convex OptimizationQuasiconvex functionsA function f : Rn → R is called a quasi-convex function if its domainand all its sublevel sets are convex. A sublevel set of a function f(x) is givenbySα = {x ∈ dom f | f(x) ≤ α} ,where α ∈ R. It is important to mention that convex functions also have allconvex sublevel sets.Example: f(x) =√|x|, x ∈ R.Convex coneA set C is called a cone if we have λx ∈ C for every x ∈ C and λ ≥ 0. Aset C is called a convex cone if it is convex and a cone, i.e., for any x1,x2 ∈ Cand any λ1, λ2 ≥ 0, we haveλ1x1 + λ2x2 ∈ C. (2.4)Example: C = {(x, y) ∈ R2|x ≥ 0, y ≥ 0}.Proper coneA cone K ⊆ Rn is called a proper cone if it is convex, closed, solid orhas a nonempty interior and is pointed or contains no line. These are thecommon examples of proper cone:− Second-order cone; K ={(x, t) ∈ Rn+1 | ‖x‖2 ≤ t}.152.1. Convex Optimization− Cone of symmetric positive semidefinite matrices (Sn+);K = {X ∈ Rn×n | X  0}.− Nonnegative orthant cone; K = {x ∈ Rn | xi ≥ 0, i = 1, · · · , n} .2.1.2 Optimization ProblemsStandard optimization problemIn general, an optimization problem can be written in the following stan-dard form:minxf0(x)subject to fi(x) ≤ 0, i = 1, · · · ,m,hj(x) = 0, j = 1, · · · , p. (2.5)The problem (2.5) finds a x that minimizes f0(x) among all x that satisfythe conditions fi(x) ≤ 0, i = 1, · · · ,m and hj(x) = 0, j = 1, · · · , p. Thefunction f0(x) is called an objective function, while the functions fi(x), i =1, · · · ,m and hj(x), j = 1, · · · , p are called inequality constraint functionsand the equality constraint functions, respectively.A domain is the set of points, for which the objective function and con-straint functions are defined, denoted as D and defined asD =m⋂i=1dom fi ∩p⋂j=1dom hj .A point x ∈ D is called a feasible point if it satisfies the constraints fi(x) ≤162.1. Convex Optimization0, i = 1. · · · ,m and hj(x) = 0, j = 1, · · · , p. The set of all feasible points iscalled a feasible set. The problem (2.5) is said to be feasible if there existsat least one feasible point, and it is infeasible otherwise. The optimal valuep∗ of the problem (2.5) is defined asp∗ = inf {f0(x) | fi(x) ≤ 0, i = 1. · · · ,m, hj(x) = 0, j = 1, · · · , p} .We say a point x∗ is an optimal point, or solves the problem (2.5), if x∗ isfeasible and p∗ , f0(x∗) ≤ f0(x) for all feasible values of x.Convex optimization problemThe convex optimization problem can be represented asminxf0(x)subject to fi(x) ≤ 0, i = 1, · · · ,m,aTj x− βj = 0, j = 1, · · · , p, (2.6)where f0(x), · · · , fm(x) are convex functions. Here we have three additionalrequirements as the objective function must be convex, the inequality con-straint functions must be convex and the equality constraint functions mustbe affine. Convex optimization problems have an important property thata feasible set of a convex optimization problem is convex.172.1. Convex OptimizationQuasiconvex optimization problemIf f0(x) in problem (2.6) is a quasiconvex function instead of a convexfunction, then the problem (2.6) is a quasiconvex optimization problem.Since the sublevel sets of a quasiconvex function are convex, a convex opti-mization problem can be formulated corresponding to each sublevel set. Oneof these sublevel set is the optimal set and that set provides the optimal so-lution to the quasiconvex optimization problem. The optimal sublevel setcan be identified by solving feasibility problems over the range of sublevelsets by using an efficient algorithm, e.g., bisection method.Feasibility problemA feasibility problem is often used to verify whether a feasible point xexists to solve the problem in a given set. It can be represented as follows:find xsubject to fi(x) ≤ 0, i = 1, · · · ,m,hj(x) = 0, j = 1, · · · , p. (2.7)Here there is no objective function. However, optimization software oftentakes the objective function as zero. Then, the optimal value is zero if thefeasible set is nonempty, or it is ∞ if the feasible set is empty.182.1. Convex Optimization2.1.3 Semidefinite Program (SDP)Before defining the SDP, it is important to discuss generalized inequali-ties and optimization problems with generalized inequalities.Generalized inequalitiesIn the above optimization problems, inequality constraints are definedon the space Rn. A generalized inequality is defined on a more general-ized space. For example, a proper cone K ⊆ Rn can be used to define ageneralized inequality, which is the partial ordering of Rn asx K y ⇐⇒ y − x ∈ K.Similarly a strict partial ordering can be written asx ≺K y ⇐⇒ y− x ∈ int K,where int K is the set of points of K which are not belongs to the boundaryof K.Optimization problem with generalized inequalitiesThe standard form of convex optimization problem with generalized in-equality constraints can be expressed asminxf0(x)subject to fi(x) Ki 0, i = 1, · · · ,m,192.1. Convex OptimizationaTj x− βj = 0, j = 1, · · · , p, (2.8)where f0 : Rn → R,Ki ⊆ Rki are proper cones and fi : Rn → Rki areconvex. The problem (2.6) is a special case with Ki = Rn+.Importantly, the optimization problems with generalized inequalitieshold many of the properties of ordinary convex optimization problems. Thefeasibility set, sublevel set, and the optimal set of optimization problemswith generalized inequalities are convex. Also the optimum of optimizationproblems with generalized inequalities is a globally optimal. Further, convexoptimization problems with generalized inequality constraints can often besolved as easily as ordinary convex optimization problems.A SDP problem is a convex optimization problem of a linear objec-tive function subjected to the generalized inequality constraints over thecone of symmetric positive semidefinite matrices or cone of hermitian pos-itive semidefinite matrices and linear equality constraints. This is a subclass of optimization problems with generalized inequalities, where Ki ∈Sn+ or Hn+, ∀i in (2.8). The general form of the SDP problem is given byminXTr{F0X}subject to Tr{FiX} − αi ≤ 0, i = 1, · · · ,m,Tr{AjX} − βj = 0, j = 1, · · · , p,X  0, (2.9)where F0,F1, · · · ,Fm,A1, · · · ,Ap ∈ Sn+ or Hn+.To deal with complex matrices, one can formulate a complex SDP, which202.1. Convex Optimizationconsists of minimizing a real linear function subject to complex linear matrixinequalities. For any complex matrix X, we have X  0 if and only ifℜ (X) −ℑ (X)ℑ (X) ℜ (X) 0.This property enables formulation of any complex SDP problem as a realSDP problem. For the simplicity, throughout this section we use real ma-trices to explain the background theory.2.1.4 Semidefinite RelaxationSemidefinite relaxation is a powerful technique to handle non convexQCQPs, which has many applications in the field of signal processing andcommunications [47]. A QCQP problem can be represented asminxxTF0xsubject to xTFix− αi ≤ 0, i = 1, · · · ,m,xTAjx− βj = 0, j = 1, · · · , p, (2.10)where the given matrices F0,F1, · · · ,Fm,A1, · · · ,Ap ∈ Sn are assumed tobe real symmetric matrices, possibly indefinite, x ∈ Rn and α1, · · · , αm,β1, · · · , βp ∈ R. The semi definite relaxation converts the nonconvex QCQPto a convex SDP, which is directly solvable with the standard convex op-timization software. The procedure of semidefinite relaxation [47] can besummarized as follows.212.1. Convex OptimizationThe quadratic forms in (2.10) can be equivalently represented asxTFix = Tr{xTFix} = Tr{FixxT}xTAjx = Tr{xTAjx} = Tr{AjxxT}for i = 0, 1, · · · ,m and j = 1, · · · , p. Thus, the objective function and con-straints are linear in the matrix xxT. Therefore, introducing a new variableX = xxT, where X is rank one and is a symmetric positive semidefinitematrix, the problem is reformulated asminXTr{F0X}subject to Tr{FiX} − αi ≤ 0, i = 1, · · · ,m,Tr{AjX} − βj = 0, j = 1, · · · , p,X  0,rank(X) = 1. (2.11)This formulation helps to directly identify the difficulty of solving (2.10).The only nonconvex constraint in the reformulated problem (2.11) is therank constraint rank(X) = 1. The objective function and all the otherconstraints are linear in X. Hence, by dropping the rank constraint, asolvable problem is considered asminXTr{F0X}subject to Tr{FiX} − αi ≤ 0, i = 1, · · · ,m,Tr{AjX} − βj = 0, j = 1, · · · , p,222.1. Convex OptimizationX  0. (2.12)The problem in (2.12) returns an optimal solution X∗. The next step is tofind the optimal solution x∗ of (2.10) from the optimal solution X∗ of (2.12).Due to the fact that the rank constraint is dropped, the rank of X∗ mustbe investigated to find the optimal solution x∗ of the original problem. Ifrank{X∗} = 1, X∗ can be written as X∗ = x∗x∗T, by using the principaleigenvector of X∗, and x∗ is the optimal solution of (2.10). On the otherhand, a rank one solution x∗, which is a feasible solution for (2.10), mustbe extracted from X∗. It is important to mention that x∗ is not always anoptimal solution. However, x∗ is a reasonable solution from an engineeringperspective.Different methods based on the randomization technique to extract rankone solutions are discussed in [41]. These methods generate a set of solution{xl} from X∗, from which the best solution is selected as x∗. What followsis an example of such a randomization method. This method uses eigen-decomposition of X∗ as X∗ = UΣUT and from that, it chooses xl as xl =UΣ1/2el, where el is a vector of uncorrelated Gaussian random variableswith zero mean and unit variance on a real or complex circularly symmetricplane.23Chapter 3Data Rate FairnessCooperative Beamformingfor CR Systems in Presenceof AsynchronousInterferencesIn this chapter, we provide a detailed description on the considered CRsystem with multiple SRs and PRs where a group of SCTs cooperativelybeamform to transmit different information to different SRs. We present themathematical model for the asynchronous interferences at the PRs and SRs.Then we formulate the beamforming design problem and develop the optimaland suboptimal beamforming techniques. We discuss the complexities ofthe developed beamforming techniques. We also extend the beamformingtechniques to incorporate the channel and the propagation delay estimationuncertainties. Finally, numerical results are provided to evaluate the datarate performance, complexity and robustness in terms of maintaining the243.1. System ModelPR 1PR LSCT 1SCT 2SCT 3SCT NSR 1SR 2SR M·········Figure 3.1: An example of cooperative beamforming for CR system.PRs target interference thresholds of various beamforming techniques.3.1 System ModelWe consider a cooperative CR system with N SCTs, M SRs and L PRsas shown in Fig. 3.1. The secondary system uses the primary system’s spec-trum using the underlay spectrum access mechanism. The SCTs transmitdifferent information to different SRs simultaneously using a cooperativebeamforming technique. As we mentioned earlier, examples of such sys-tem include the second hop of a multiple peer-to-peer communications viaa common set of cooperative nodes and a system where a group of basestations cooperatively transmits information to the mobile stations. We as-sume that each node in the system is equipped with single antenna and thelinks between nodes experience independent frequency flat Rayleigh fading.Given the fact that the SCTs are located in different geographical lo-253.1. System Modelcations, their transmissions experience different propagation delays at theSRs as well as at the PRs. A timing advance mechanism [48] can be usedto synchronize signals from multiple SCTs only at one SR. It is assumedthat the timing advance mechanism ensures the perfect synchronization ofall the signals intended for a particular SR from different SCTs at that SR.The timing advance, ∆(x)Ti , for the xth SR’s signal transmitted from the ithSCT can be calculated as∆(x)Ti = τTiRx − τ(min)Tx , (3.1)where τ (min)Tx , min(τT1Rx , · · · , τTNRx), and τTiRx is the propagation timefrom the ith SCT to the xth SR.For the conventional cooperative base station systems, a mathematicalmodel for the asynchronous interference signal at the mobile stations hasbeen developed in [36]. Leveraging the model [36], we develop the expres-sions of the signal to interference plus noise ratio (SINR) at the SRs andinterference power at the PRs for the considered CR system.3.1.1 Signal-to-Interference plus Noise Ratio at the SRsAs shown in Fig. 3.2, the jth SR receives N synchronized signals trans-mitted from N SCTs for the jth SR. It also receives N×(M−1) asynchronoussignals transmitted from N SCTs for other (M − 1) SRs. Thus, at the jthSR, the received baseband signal at the time t, rRj (t), when a linear mod-ulation with unit energy baseband waveform f(t) of duration Ts is used, is263.1. System Modelgiven byrRj (t) =∞∑m=0{ N∑i=1f(t−mTs − τTiRj + ∆(j)Ti )hTiRjg(j)Ti sRj [m]}+ nRj (t)+∞∑m=0N∑i=1M∑k=1k 6=jf(t−mTs − τTiRj + ∆(k)Ti )hTiRjg(k)Ti sRk [m],(3.2)where hTiRj is the channel fading gain from the ith SCT to the jth SR,g(x)Ti is the beamforming coefficient of the ith SCT corresponding to the xthSR, sRx [m] is the symbol to be transmitted to the xth SR during the mthsymbol duration with E[sRi [m]sRj [m]†] = 1 when i = j and zero otherwise.nRj (t) is the total of additive white Gaussian noise (AWGN) and the primaryinterference1 component at the jth SR.At the jth SR, the received signal rRj (t) is passed through a filter thatis matched to f(t−mTs − τ (min)Rj ) and the filtered signal is sampled at thetime instants t = (mTs + τ (min)Rj ) for m = 1, 2, · · · . Using (3.2) and a vectornotation, the sampled filtered signal at the jth SR can be written asYRj [m] =N∑i=1hTiRjg(j)Ti sRj [m] + nRj [m] +N∑i=1M∑k=1k 6=jhTiRjg(k)Ti a(k)TiRj [m],=(g(j))ThRjsRj [m] + nRj [m] +M∑k=1k 6=j(g(k))ThRja(k)Rj [m], (3.3)where g(x) ,[g(x)T1 , · · · , g(x)TN]T, hRj , diag([hT1Rj , · · · , hTNRj]), sRj [m] ,1The total interference due to primary transmission is assumed to be additive whiteGaussian process.273.1. System Model1stSR′ssymbolsjthSR′ssymbolsM thSR′ssymbols........................TssˆR1 [z(1)T1Rj ] sˆR1 [z(1)T1Rj + 1]sˆR1 [z(1)TNRj ] sˆR1 [z(1)TNRj + 1]sˆRj [m]sˆRj [m]sˆRM [z(M)T1Rj ] sˆRM [z(M)T1Rj + 1]sˆRM [z(M)TNRj ] sˆRM [z(M)TNRj + 1]From SCT 1From SCT 1From SCT 1From SCT NFrom SCT NFrom SCT Naˆ(M)TNRj [m]aˆ(M)T1Rj [m]aˆ(1)T1Rj [m]aˆ(1)TNRj [m]Figure 3.2: An example of timing of received signals over a symbolduration at the jth SR, where sˆRj [z(k)TiRj ] = f(t − zTiRjTs − τTiRj +∆(k)Ti )hTiRjg(k)i sRj [z(k)TiRj ].sRj [m]1N×1 and a(k)Rj [m] ,[a(k)T1Rj [m], · · · , a(k)TNRj [m]]T, and nRj [m] is thesampled value of filtered noise and interference component at the jth SRduring the mth symbol duration.As an example shown in Fig. 3.2, aˆ(k)TiRj [m] is the asynchronous interfer-ing signal at the jth SR due to two consecutive symbols, say with indicesz(k)TiRj and z(k)TiRj + 1, that are transmitted by the ith SCT for the kth SR.The sampled matched filter output of the asynchronous interfering signal283.1. System Modela(k)TiRj [m] can be expressed as [36]a(k)TiRj [m] = ρ(δ(k)TiRj − Ts)sk[z(k)TiRj ] + ρ(δ(k)TiRj )sk[z(k)TiRj + 1], (3.4)where ρ(ν) =∫ Ts0 f(t)f(t − ν)dt with ρ(0) = 1 is the sampled output oftime delayed signal f(t − ν), δ(k)TiRj = (τTiRj −∆(k)Ti − τ(min)Rj ) mod Ts, andz(k)TiRj = m− ceil(τTiRj−∆(k)Ti−τ (min)RjTs).From (3.3), the desired signal power (PDRj ) at the jth SR can be expressedasPDRj = E[((g(j))ThRjsRj [m])((g(j))ThRjsRj [m])†],=(g(j))†(hRj)† E[sRj [m]sRj [m]†]hRjg(j),=(g(j))†H(j)Rjg(j), (3.5)whereH(j)Rj =(hRj)† E[sRj [m]sRj [m]†]hRj , and E[sRj [m]sRj [m]†]= 1N×N .Similarly the interfering signal power at the jth SR can be expressed asP IRj = EM∑k1=1k1 6=j(g(k1))ThRja(k1)Rj [m]M∑k2=1k2 6=j(g(k2))ThRja(k2)Rj [m]†,=M∑k1=1k1 6=jM∑k2=1k2 6=j(g(k2))† (hRj)† E[a(k1)Rj [m](a(k2)Rj [m])†]hRjg(k1). (3.6)Using E[sRi [m]sRj [m]†] = 0 when i 6= j provides E[a(k1)Rj [m](a(k2)Rj [m])†]= 0 for k1 6= k2. Therefore, the interference power at the jth SR can be293.1. System Modelsimplified asP IRj =M∑k=1k 6=j(g(k))†(hRj)† E[a(k)Rj [m](a(k)Rj [m])†]hRjg(k),=M∑k=1k 6=j(g(k))†H(k)Rj g(k), (3.7)where H(k)Rj =(hRj)† E[a(k)Rj [m](a(k)Rj [m])†]hRj .Using (3.4), E[a(k)Rj [m](a(k)Rj [m])†]can be expressed asE[a(k)Rj [m](a(k)Rj [m])†]=β(k)Rj (1, 1) β(k)Rj (1, 2) ... β(k)Rj (1, N)β(k)Rj (2, 1) β(k)Rj (2, 2) ... β(k)Rj (2, N)... ... . . . ...β(k)Rj (N, 1) β(k)Rj (N, 2) ... β(k)Rj (N,N),(3.8)where β(k)Rj (i1, i2) is the correlation between the asynchronous interferingsignals a(k)Ti1Rj [m] and a(k)Ti2Rj[m] and it is calculated as [36]β(k)Rj (i1, i2) =0, if |z(k)Ti1Rj − z(k)Ti2Rj| ≥ 1;ρ(δ(k)Ti1Rj )ρ(δ(k)Ti2Rj− Ts), if z(k)Ti2Rj = z(k)Ti1Rj+ 1;ρ(δ(k)Ti1Rj )ρ(δ(k)Ti2Rj)+ρ(δ(k)Ti1Rj − Ts)ρ(δ(k)Ti2Rj− Ts), if z(k)Ti2Rj = z(k)Ti1Rj;ρ(δ(k)Ti1Rj − Ts)ρ(δ(k)Ti2Rj), if z(k)Ti2Rj = z(k)Ti1Rj− 1.(3.9)Finally, the signal to interference plus noise power ratio (SINR) at the303.1. System Modeljth SR can be written asSINRRj =(g(j))†H(j)Rjg(j)∑Mk=1k 6=j(g(k))†H(k)Rj g(k) + σ2Rj, (3.10)where σ2Rj is the total noise plus interference due to the primary transmis-sions power at the jth SR.3.1.2 Asynchronous Interferences at the PRsAt the lth PR, there are N ×M asynchronous signals, which are trans-mitted by N SCTs to M SRs. Thus, the total received baseband interferingsignal iPl(t) at the lth PR can be written asiPl(t) =∞∑m=0{ N∑i=1M∑k=1f(t−mTs − τTiPl + ∆(k)Ti )hTiPlg(k)Ti sRk [m]}, (3.11)where hTiPl is the channel fading gain from the ith SCT to the lth PR, τTiPlis the propagation delay from the ith SCT to lth PR.We assume that the lth PR uses a matched filter to match f(t−mTs −τPl), where τPl is the time delay used in lth PR to match its signal withrespect to primary transmitter. This assumption enables us to calculate themaximum possible interference to the PRs. In particular, if the PR uses amatched filter to match a different base band signal, the matched filteredoutput interference will be less. Hence, the sampled filtered asynchronous313.1. System Modelinterference signal at the lth PR can be written asYPl [m] =N∑i=1M∑k=1hTiPlg(k)Ti a(k)TiPl [m]=M∑k=1(g(k))ThPla(k)Pl [m], (3.12)where hPl , diag ([hT1Pl , · · · , hTNPl ]) and the vector a(k)Pl [m] ,[a(k)T1Pl [m] ,· · · , a(k)TNPl [m]]T, whose element a(k)TiPl [m] denotes the sampled matched filteroutput of asynchronous interfering signal, aˆ(k)TiPl [m] at the lth PR and can bewritten asa(k)TiPl [m] = ρ(δ(k)TiPl − Ts)sk[z(k)TiPl ] + ρ(δ(k)TiPl)sk[z(k)TiPl + 1], (3.13)where δ(k)TiPl = (τTiPl − ∆(k)Ti − τPl) mod Ts and the index z(k)TiPl = m −ceil(τTiPl−τPl−∆(k)TiTs).The interference power at the lth PR can be written asP IPl = EM∑k1=1(g(k1))ThPla(k1)Pl [m]M∑k2=1(g(k2))ThPla(k2)Pl [m]†,=M∑k1=1M∑k2=1(g(k2))†(hPl)† E[a(k1)Pl [m](a(k2)Pl [m])†]hPlg(k1). (3.14)By using E[sRi [m]sRj [m]†] = 0 when i 6= j, E[a(k1)Pl [m](a(k2)Pl [m])†]= 0 fork1 6= k2. Therefore, the interference power at the lth PR can be simplified323.1. System ModelasP IPl =M∑k=1(g(k))†(hPl)† E[a(k)Pl [m](a(k)Pl [m])†]hPlg(k),=M∑k=1(g(k))†H(k)Pl g(k), (3.15)where channel matrixH(k)Pl = (hPl)† E[a(k)Pl [m](a(k)Pl [m])†]hPl .2 From (3.13),E[a(k)Pl [m](a(k)Pl [m])†]can be written asE[a(k)Pl [m](a(k)Pl [m])†]=β(k)Pl (1, 1) β(k)Pl (1, 2) ... β(k)Pl (1, N)β(k)Pl (2, 1) β(k)Pl (2, 2) ... β(k)Pl (2, N)... ... . . . ...β(k)Pl (N, 1) β(k)Pl (N, 2) ... β(k)Pl (N,N),(3.16)where β(k)Pl (i1, i2) = E[a(k)Ti1Pl [m](a(k)Ti2Pl [m])†]is the correlation betweenthe asynchronous interfering signals a(k)Ti1Pl [m] and a(k)Ti2Pl[m], and it can becalculated asβ(k)Pl (i1, i2) =0, if |z(k)Ti1Pl − z(k)Ti2Pl| ≥ 1;ρ(δ(k)Ti1Pl)ρ(δ(k)Ti2Pl− Ts), if z(k)Ti2Pl = z(k)Ti1Pl+ 1;ρ(δ(k)Ti1Pl)ρ(δ(k)Ti2Pl)+ρ(δ(k)Ti1Pl − Ts)ρ(δ(k)Ti2Pl− Ts), if z(k)Ti2Pl = z(k)Ti1Pl;ρ(δ(k)Ti1Pl − Ts)ρ(δ(k)Ti2Pl), if z(k)Ti2Pl = z(k)Ti1Pl− 1.(3.17)2For convenience, we refer to H(k)Pl as the channel matrix throughout this thesis. Itconsists of channel gains and the impact of propagation delay on received symbols.333.2. Problem Formulation and Optimal Beamforming Design3.2 Problem Formulation and OptimalBeamforming DesignThe goal of the beamforming design is to improve fairness in terms ofdata rates between different SRs. Therefore, we formulate a cooperativebeamforming optimization problem to maximize the minimum data rateof the SRs subject to asynchronous interference thresholds at the PRs andmaximum transmission power constraint for each SCT.We mentioned earlierthat maximization of minimum data rates of the SRs provides better fairnessfor the SRs. This can be expressed formally asmaxg(1),··· ,g(M)minj=1,··· ,Mlog21 +(g(j))†H(j)Rjg(j)∑Mk=1k 6=j(g(k))†H(k)Rj g(k) + σ2Rjsubject toM∑k=1(g(k))†H(k)Pl g(k) ≤ P ITHPl , l = 1, · · · , L,‖gTi‖22 ≤ PTi , i = 1, · · · , N, (3.18)where gTi , [g(1)Ti , g(2)Ti , · · · , g(M)Ti ] is vector of beamforming coefficients at theith SCT, P ITHPl is the interference power threshold at the lth PR and PTiis the transmit power constraint of ith SCT. The first set of constraints in(3.18) keeps the asynchronous interference power below the target thresh-old whereas the second set of constraints maintains the maximum transmitpower of each SCT under a predefined budget. We consider different in-terference thresholds for different PRs which represents a generalized caseto reflect different quality requirements for different PRs. Since log(1 + x)is a monotonically increasing function of x, maximizing the minimum of343.2. Problem Formulation and Optimal Beamforming Designlog2(1+x) is equivalent to maximizing the minimum of x for x ≥ 0. Hence,the optimization problem in (3.18) can be rewritten asmaxg(1),··· ,g(M)minj=1,··· ,M(g(j))†H(j)Rjg(j)∑Mk=1k 6=j(g(k))†H(k)Rj g(k) + σ2Rjsubject toM∑k=1(g(k))†H(k)Pl g(k) ≤ P ITHPl , l = 1, · · · , L,‖gTi‖22 ≤ PTi , i = 1, · · · , N. (3.19)3.2.1 Optimal Beamforming TechniqueThe max-min optimization problem in (3.19) is not tractable. However,it can be reformulated in a tractable form by introducing an auxiliary vari-able u as [49]maxg(1),··· ,g(M)usubject to u ≥ 0,(g(j))†H(j)Rjg(j)∑Mk=1k 6=j(g(k))†H(k)Rj g(k) + σ2Rj≥ u, j = 1, · · · ,MM∑k=1(g(k))†H(k)Pl g(k) ≤ P ITHPl , l = 1, · · · , L,‖gTi‖22 ≤ PTi , i = 1, · · · , N. (3.20)Although the reformulated optimization problem in (3.20) has non-linearnon-convex quadratic constraints, it can be solved using the semidefiniterelaxation technique [47]. The semidefinite form of the problem in (3.20)353.2. Problem Formulation and Optimal Beamforming Designcan be written asmaxG(1),··· ,G(M)usubject to u ≥ 0,uM∑k=1k 6=jTr{H(k)Rj G(k)}+ uσ2Rj − Tr{H(j)RjG(j)} ≤ 0,j = 1, · · · ,M,M∑k=1Tr{H(k)Pl G(k)} − P ITHPl ≤ 0, l = 1, · · · , L,G(k)  0, k = 1, · · · ,M,Tr{DTi} ≤ PTi , i = 1, · · · , N,rank(G(k)) = 1, k = 1, · · · ,M, (3.21)where G(k) , g(k)(g(k))† and DTi , diag([G(1)(i, i) ,· · · , G(M)(i, i)])isa diagonal matrix generated using the ith elements of diagonal vectors ofG(k),∀k. The rank constraints in (3.21) are nonconvex. Thus, we relax therank constraints and formulate a quasi-convex optimization problem asmaxG(1),··· ,G(M)usubject to u ≥ 0,uM∑k=1k 6=jTr{H(k)Rj G(k)}+ uσ2Rj − Tr{H(j)RjG(j)} ≤ 0,j = 1, · · · ,M,M∑k=1Tr{H(k)Pl G(k)} − P ITHPl ≤ 0, l = 1, · · · , L,G(k)  0, k = 1, · · · ,M,363.2. Problem Formulation and Optimal Beamforming DesignTr{DTi} ≤ PTi , i = 1, · · · , N. (3.22)The optimal value of u, u∗ and the correspondingG(1)∗ ,G(2)∗ , · · · ,G(M)∗in (3.22) can be obtained numerically by using a combination of the SDPand bisection method [42]. Then, if the rank(G(k)∗) = 1, ∀k, the prin-cipal component of G(k)∗ provides the optimal beamforming vector g(k)∗for the original problem formulated in (3.18). On the other hand, if therank(G(k)∗) 6= 1 for any k, the principal component of G(k)∗ does not di-rectly provide optimal beamforming vector g(k)∗ . However, optimal beam-forming vectors with a reasonable accuracy can be obtained from G(k)∗ , byusing the randomization method [41]. The optimal beamforming techniqueis summarized in Algorithm 1.Algorithm 1: Optimal beamforming techniqueStep 11: Select an initial value of u.2: Define the required tolerance (a) of optimal u∗.Step 21: Solve (3.22) to check the feasibility of u.2: if (solved, i.e., feasible) then3: Redefine u = 2u.4: loop373.2. Problem Formulation and Optimal Beamforming Design5: Solve (3.22) to check the feasibility of new u.6: if (solved, i.e., feasible) then7: Redefine u = 2u.8: else9: Define uu = u.10: Define ul = u/2.11: Break loop and move to Step 3.12: end if13: end loop14: else15: Redefine u = u/2.16: loop17: Solve (3.22) to check the feasibility of new u.18: if (solved, i.e., feasible) then19: Define uu = u.20: Define ul = u/2.21: Break loop and move to Step 3.22: else23: Redefine u = u/2.24: end if25: end loop383.2. Problem Formulation and Optimal Beamforming Design26: end ifStep 31: while (uu − ul >= a) do2: Define u = (uu − ul)/2.3: Solve (3.22) to check the feasibility of new u.4: if (solved) then5: Keep uu = uu.6: Increase ul = u.7: else8: Reduce uu = u.9: Keep ul = ul.10: end if11: end while12: Define u∗ = ul.Step 41: Find the optimal values G(k)∗∀k corresponding to u∗.2: if (rank(G(k)∗) = 1,∀k) then3: Find optimal beamforming vectors g(k)∗ ,∀k from the principalcomponent of G(k)∗ ,∀k.4: else393.2. Problem Formulation and Optimal Beamforming Design5: Find optimal beamforming vectors g(k)∗ ,∀k, by usingG(k)∗ ,∀kand randomization technique [41].6: end ifEnd3.2.2 Complexity of the Optimal Beamforming TechniqueThe aforementioned method can be used to find the beamforming vectorsin polynomial time. However, it is computationally complex due to the useof a combination of the SDP and bisection method. In order to solve theSDP problem numerically, we use the standard CVX tool which has SeDuMiand SDPT3 solvers [50]. For example, when SeDuMi is used, the complexityof solving the SDP problem can be approximated as [51]O{N4M2(2M + N + L)2.5 + (2M + N + L)3.5}.On the other hand, in general, the bisection method takes log2{a0a } iter-ations on average to converge, where a0 is the initial range and a is thetolerance of the optimal solution. Further, the randomization technique hasa complexity in order of O(ILM2N3), where I is the number of random-ization iterations. Thus, the complexity of finding the optimal solution of(3.22) can be approximated asO{log2(a0a)(N4M2(2M + N + L)2.5 + (2M + N + L)3.5)+ ILM2N3}.403.3. Suboptimal Beamforming Techniques3.3 Suboptimal Beamforming TechniquesDue to the high computational complexity of obtaining the optimalbeamforming vector as described in Section 3.2.2, in what follows, we studyimportant properties of the optimization problem in (3.19). Then, we pro-pose low complexity suboptimal beamforming techniques by exploiting theseproperties.Property 1. If(g(1)∗, · · · ,g(M)∗)is the optimal solution for the problemin (3.19), then(g(1)∗, · · · ,g(M)∗)satisfies at least one of the inequalityconstraints in (3.19) with equality. In other words, with the optimal beam-forming vectors(g(1)∗, · · · ,g(M)∗)at least one of the PRs has interferenceequal to the threshold or one of the SCTs transmits with its full quota ofpower.Proof. See Appendix I.Property 2. The optimal solution(g(1)∗, · · · ,g(M)∗)satisfies the SINR-balancing, i.e., the SINRs at the SRs are equal.Proof. See Appendix I.3.3.1 Suboptimal 1 Beamforming TechniqueAlgorithmThis suboptimal beamforming technique is based on SLPR maximiza-tion and an iterative method that exploits the aforementioned properties.According to this suboptimal beamforming technique, beamforming vectorsare obtained in two steps as follows.413.3. Suboptimal Beamforming TechniquesStep 1: Beamforming vectors that maximize the SLPR of each SR’s sig-nal are obtained individually. The SLPR of the kth SR’s signal is defined asthe ratio of the received signal power at the kth SR and the total interferencepower due to the transmission of the kth SR’s signal, i.e.,SLPRRk =(g(k))†H(k)Rkg(k)∑Mj=1j 6=k(g(k))†H(k)Rj g(k) +∑Ll=1(g(k))†H(k)Pl g(k),=(g(k))†H(k)Rkg(k)(g(k))†{∑Mj=1j 6=kH(k)Rj +∑Ll=1 H(k)Pl}g(k). (3.23)The SLPR in (3.23) is in a quadratic over quadratic form. The initial beam-forming vector g(k)SL0 , which maximizes the SLPRRk in (3.23) can be ob-tained by using the maximum eigenvalue λ(k)max of the matrix H(k), where thematrix H(k) =(∑Mj=1j 6=kH(k)Rj +∑Ll=1H(k)Pl)−1H(k)Rk and corresponding eigen-vector v(k)max. In particular, g(k)SL0 can be expressed as [39]g(k)SL0 =√λ(k)maxv(k)max. (3.24)Step 2: The beamforming vectors calculated in step 1 may not satisfythe interference and power constraints of the original problem in (3.19) andthe aforementioned properties of the optimal beamforming vectors. Takingthe beamforming vectors g(k)SL0 calculated in Step 1 as initial values, we itera-tively calculate suboptimal beamforming vectors that satisfy the constraintsand the two properties. At a particular iteration, beamforming vectors ofprevious iteration are modified by two scaling factors as follows.423.3. Suboptimal Beamforming TechniquesThe first scaling factor is common for all beamforming vectors and de-noted by α. During the pth iteration, the scaling factor αp is calculated suchthat the constraints and the property 1 are satisfied. Therefore, the valueof αp can be calculated by using the following relationshipα2p = minP ITHP1∑Mk=1(g(k)SLp−1)†H(k)P1 g(k)SLp−1, · · · ,P ITHPL∑Mk=1(g(k)SLp−1)†H(k)PLg(k)SLp−1,PT1‖g1,SLp−1‖22, · · · , PTN‖gN,SLp−1‖22)(3.25)and gi,SLp−1 =[g(1)SLp−1(i), . . . ,g(M)SLp−1(i)]. Now the modified beamformingvectors at the pth iteration can be written as g(k)SLp = αpg(k)SLp−1 .The second scaling factor for the kth beamforming vector is denotedby ηk. During the pth iteration these scaling factors are calculated suchthat the differences between the SINRs, calculated using g(k)SLp , are reduced.Therefore, the value of the second scaling factor at the pth iteration ηk,p canbe calculated by using the following relationshipη2k,p =MM − 1(1−SINRRk,pSINRR1,p + SINRR2,p + · · ·+ SINRRM,p), (3.26)where SINRRk,p (k = 1, · · · ,M) are calculated using g(k)SLp . The beam-forming vector at the pth iteration is further modified as g(k)SLp = ηk,pg(k)SLp .The iteration continues until the properties are satisfied with a reasonableaccuracy.3The details of the suboptimal 1 beamforming algorithm are summarized3Property 1 is satisfied exactly whereas Property 2 is satisfied with a certain accuracy.433.3. Suboptimal Beamforming Techniquesas follows.Algorithm 2: Suboptimal 1 beamforming techniqueStep 11: Calculate beamforming vector g(k)SL0 that maximizes SLPRk byusing (3.24).Step 21: Initialize p = 1.2: loop3: Calculate scaling factor αp by using (3.25).4: Update beamforming vectors as g(k)SLp = αpg(k)SLp−1 ,∀k.5: Calculate SINRs with beamforming vectors g(k)SLp .6: if (SINR1,p ≈ SINR2,p ≈ · · · ≈ SINRNR,p) then7: g(k)sopt1 = g(k)SLp ,∀k.8: Break loop and return suboptimal beamforming vectors.9: else10: Calculate scaling factors ηk,p,∀k by using (3.26).11: Update beamforming vectors as g(k)SLp = ηk,pg(k)SLp ,∀k.12: Increase loop count (p = p + 1).13: end if14: end loopEnd443.3. Suboptimal Beamforming TechniquesComplexityIn suboptimal 1 beamforming technique, the main complexity comesfrom matrix multiplication, matrix inversion, eigenvalue computation andfinding minimum. The complexity of each of these matrix operations is equalto O(q3) for a q × q matrix and the complexity of the sorting algorithm isO(q log2 q) to find a minimum from q values. Hence, for a system with NSCTs, M SRs and L PRs, the complexity of obtaining beamforming vectorscan be approximated as O{N3M + L+[(L+N) log2(L+N)]I}, by assumingI iterations to converge. This complexity is significantly less compared tothe optimal beamforming technique as shown in Section Suboptimal 2 Beamforming TechniqueAlgorithmIn suboptimal 2 beamforming technique, rather than maximizing theminimum SINR of the SRs, we consider maximizing the minimum of theratios of signal power minus interference power to noise power (SMINR),which enables us to formulate a convex optimization problem instead of aquasi-convex problem in the optimal beamforming technique. The SMINRat the jth SR can be written asSMINRRj =(g(j))†H(j)Rjg(j) −∑Mk=1k 6=j(g(k))†H(k)Rj g(k)σ2Rj. (3.27)The optimization problem to maximize the minimum SMINR at the SRswith transmission power constraints at the SCTs and asynchronous inter-453.3. Suboptimal Beamforming Techniquesference power constraints at the PRs can be formulated as a convex opti-mization problem asmaxG(1),··· ,G(M)u′subject toM∑k=1k 6=jTr{H(k)Rj G(k)}+ u′σ2Rj − Tr{H(j)RjG(j)} ≤ 0,j = 1, · · · ,M,M∑k=1Tr{H(k)Pl G(k)} − P ITHPl ≤ 0, l = 1, · · · , L,G(k)  0, k = 1, · · · ,M,Tr{DTi} ≤ PTi , i = 1, · · · , N. (3.28)One can directly solve (3.28) with SDP solvers and obtain G˜(1)∗ , G˜(2)∗ , · · · ,G˜(M)∗ that maximize the minimum of SMINRs. However, beamformingvectors obtained using G˜(1)∗ , G˜(2)∗ , · · · , G˜(M)∗ may not satisfy the proper-ties of optimal beamforming vectors since these beamforming vectors arenot obtained based on the SINR metric. Therefore, by taking the principalcomponent of G˜(1)∗ , G˜(2)∗ , · · · , G˜(M)∗ as initial beamforming vectors, theproposed iterative method in Step 2 of Algorithm 2 can be used to obtainthe suboptimal 2 beamforming vectors.ComplexityThe complexity of the convex optimization problem in (3.28) is approx-imately equal to solving one bisection iteration of the optimal beamformingtechnique. Thus the complexity of the suboptimal 2 beamforming technique463.3. Suboptimal Beamforming Techniquesis approximately O{N4M2(2M + N + L)2.5 + (2M + N + L)3.5}.3.3.3 Suboptimal 3 Beamforming TechniqueAlgorithmAlthough the suboptimal 2 beamforming technique does not require tosolve the SDP multiple times, it does suffer from a certain data rate per-formance degradation compared to the optimal beamforming technique asshown in the numerical results. The main complexity of the optimal beam-forming technique described in Section 3.2 comes from finding the value ofu∗ using the combination of the SDP and bisection method. Thus in sub-optimal 3, we propose an efficient algorithm to select an approximate valueof u∗ by using the solutions of (3.28). First, we solve the SDP problem in(3.28) and calculate the corresponding SINRs using G˜(1)∗ , G˜(2)∗ , · · · , G˜(M)∗ .Without loss of generality, let us denote the set of these SINRs as SINR =[SINR1, · · · , SINRM ]. It can be easily shown that calculated SINRs musthave at least one feasible value of u in (3.22).4 Thus, we use the highestfeasible SINR value from this set as a good approximation for u∗, whichis found by using a combination of the SDP and bisection method in theoptimal beamforming technique. Then, we obtain ˜˜G(1)∗, ˜˜G(2)∗, · · · , ˜˜G(M)∗by using the approximated values of u∗ and (3.22) . This suboptimal 1beamforming technique can be summarized as follows.4Even though the solutions G˜(1)∗ , G˜(2)∗ , · · · , G˜(M)∗ are obtained from (3.28), theysatisfy the interference and transmission power constraints of the optimal beamformingproblem (3.22). Further, it is obvious that when the minimum SINR is equal to u, theother SINRs must be greater than or equal to u, which implies that the SINR constraintsin (3.22) are satisfied and u is a feasible solution of (3.22).473.3. Suboptimal Beamforming TechniquesAlgorithm 3: Suboptimal 3 beamforming techniqueStep 11: Solve problem (3.28) to obtain G˜(k)∗ ,∀k.2: Calculate SINRs for all SRs by using G˜(k)∗ ,∀k.3: Arrange the SINRs in descending order in an array (sinrO)4: Initialize a variable (index = 1)Step 21: loop2: Define u = sinrO(index)3: Check the feasibility of u using (3.22).4: If (feasible) then5: Equation (3.22) provides ˜˜G(k)∗,∀k.6: Break the loop and exit.7: else8: Increase the index (index = index + 1).9: end if10: end loopEndThe beamforming vectors can be obtained by using ˜˜G(k)∗,∀k. Howeverthese beamforming vectors may not satisfy the properties of the optimal483.3. Suboptimal Beamforming Techniquesbeamforming vectors due to the approximation of u∗ in the optimal beam-forming technique. Again taking the beamforming vectors obtained from theprincipal component of ˜˜G(k)∗,∀k as initial beamforming vectors, the itera-tive method in Step 2 of Algorithm 2 can be used to obtain the suboptimal 3beamforming vectors.ComplexityUsing this suboptimal 3 beamforming technique, one needs to solve (M+1) SDP problems in the worst case. Therefore, the worst case complexity ofthe suboptimal 3 beamforming technique isO{(M + 1)[N4M2(2M + N + L)2.5 + (2M + N + L)3.5]}for a system with M SRs. The computational complexity of the suboptimal 3beamforming technique in the worst case depends on the number of SRs. Inparticular, if M << log2{a0a }−1, the worst case complexity of this algorithmwill be less compared to the optimal beamforming technique. On the otherhand, if M > log2{a0a } − 1, the worst case complexity of this algorithmcan be higher than that of the optimal beamforming technique. However,in practice the complexity of the suboptimal 3 beamforming technique maynot be higher hence it does not require solutions to (M +1) SDPs at all theinstances.493.4. Extension of Beamforming with Estimation Uncertainty3.4 Extension of Beamforming with EstimationUncertaintyIn order to develop beamforming techniques in the previous sections,we consider that the channel gains as well as the propagation time delaysbetween the SCTs and PRs are known perfectly by the SCTs. In manyscenarios, this information may not be obtained perfectly and the SCTs canhave erroneous information. In such situations, it is important to incorpo-rate the channel estimation and propagation delay estimation uncertaintiesin beamforming design in order to ensure robust protection for the PRs.A robust coopertive beamforming technique for CR systems has been de-veloped in [43] without considering asynchronous interference, where thechannel estimation error is modelled using a bounded uncertainty model.This bounded uncertainty model is a well accepted model to incorporateuncertainty in estimation. Using the bounded uncertainty model as used in[43], in this section we extend our beamforming techniques to ensure robustprotection for PRs with estimation uncertainty.As shown in Section 3.2, in order to obtain the beamforming vectors, thechannel matrix H(k)Pl should be known by the SCTs. When there is a certainestimation error in channel gains or/and in propagation delays, using thebounded ellipsoidal uncertainty model in [43], H(k)Pl can be expressed asH(k)Pl = Hˆ(k)Pl +∆H(k)Pl , ∀l, k, (3.29)where Hˆ(k)Pl is the estimated channel matrix from the SCTs to the lth PR503.4. Extension of Beamforming with Estimation Uncertaintyand ∆H(k)Pl denotes the corresponding uncertainties that are considered tobe bounded in the setH(k)Pl ={∆H(k)Pl : ∆H(k)Pl =(∆H(k)Pl)†, Hˆ(k)Pl + ∆H(k)Pl  0,Tr(∆H(k)Pl Q(k)Pl ∆H(k)Pl)≤ 1}, ∀l. (3.30)Further, the matrices {Q(k)Pl ≻ 0} determine the quality of the estimationand are assumed to be known. In addition, they can be decomposed intoQ(k)Pl = Q˜(k)Pl Q˜(k)Pl , ∀l (3.31)with Q˜(k)Pl ≻ 0, ∀l.Under this uncertainty model, the worst case interference power con-straint corresponding to the lth PR can be written as (Proof. See Ap-pendix II).M∑k=1Tr{Hˆ(k)Pl G(k)}+M∑k=12∥∥∥∥∥Q˜(k)Pl MAT((IN ⊗Q(k)Pl +(Q(k)Pl)†⊗ IN)−1vec(G(k)))∥∥∥F≤ P ITHPl . (3.32)Now the PR interference constraints in the optimal and suboptimal beam-forming techniques can be replaced by using these modified constraints.These constraints are convex and the problem can be solved using the sameapproaches proposed in Section 3.2.513.5. Numerical Results3.5 Numerical ResultsIn this section, we present some selected numerical results in order tocompare the performance of various beamforming techniques under consid-eration. For the presented numerical results, we consider a cooperative CRsystem with four SCTs, three SRs and two PRs. The coordinates of thenodes in the network are given in Table 3.1. All the values in the tableare given in meters. It is assumed that the AWGN power spectral den-Table 3.1: Coordinates of SCTs, SRs and PRs.SCTs SRs PRsSCT1 (10, 0) SR1 (230, 20) PR1 (1100, 0)SCT2 (0, 75) SR2 (240, 100) PR2 (1120, 200)SCT3 (10, 125) SR3 (220, 175)SCT4 (0, 200)sity is −174 dBm/Hz and the total bandwidth is 25 kHz. The interferencethresholds at PR1 and PR2 are assumed to be −145 dBW and −150 dBW,respectively. To simplify the numerical results, we assume the maximumtransmission power constraint at all the SCTs are equal and the receivedtotal interference power at the SRs from the primary system is equal to−130 dBW. Channel gains are assumed to be independent frequency flatRayleigh distributed and the log-distance path loss model is considered witha path loss exponent of 4. The symbol duration is taken as 1 µs and thepropagation delays are calculated based on the distances between the nodes.First, we plot the computational complexity5, i.e., logO(·) of various5In this figure, we plot the worst case complexity of the suboptimal 3 beamforming523.5. Numerical Results2 3 4 5 6 7 8 9 10024681012  ComplexityintermsoflogO(·)valueNumber of SRs (M)OptimalSuboptimal 1Suboptimal 2Suboptimal 3Figure 3.3: Complexity of the optimal and suboptimal beamforming tech-niques.beamforming techniques with the number of SRs (M) in Fig. 3.3. We as-sumed a system with M SRs has M + 1 SCTs and three PRs. Further,the tolerance (a) and initial bound (a0) of the bisection method used forthe optimal beamforming technique are assumed as 10−4 and 10, respec-tively. Fig. 3.3 shows that the suboptimal 1 beamforming technique hasthe lowest complexity whereas the optimal beamforming technique has thehighest complexity. The suboptimal 2 beamforming technique has a highercomputational complexity than that of the suboptimal 1 beamforming tech-nique. However, it has a lower complexity than the optimal beamformingtechnique. From Fig. 3.3, we also observe that the worst case complexitytechnique.533.5. Numerical Resultsof the suboptimal 3 beamforming technique approaches to the complexityof the optimal beamforming technique as the number of SRs increases. Thereason can be explained as follows. The complexity of the suboptimal 3beamforming technique depends on the number of the SDP problems thathave to be solved. In the worst case, it can be (M + 1) SDP problems. So,as the number of SRs in the system increases, the worst case complexity ofthe suboptimal 3 beamforming technique increases.In order to compare the performance of the developed optimal and sub-optimal beamforming techniques, Fig. 3.4 plots the average minimum datarate of SRs versus the per SCTs’ transmission power constraint. In fact theoptimal beamforming provides equal data rate and the suboptimal beam-forming techniques provides approximately equal data rates to all the SRs.Therefore average minimum data rate per SR is same as the average mini-mum data rate at each SR. From this figure, we observe that the minimumdata rate of SRs with all the beamforming techniques increases as the trans-mission power constraint increases up to a certain point. Then it saturatesdue to the interference constraints from the PRs. The optimal beamform-ing technique provides the highest average minimum data rate among allthe beamforming techniques as expected. The suboptimal 1 beamformingtechnique provides the lowest average minimum data rate as it is developedbased on SLPR criteria, which ignores the impact of noise powers. However,suboptimal 1 beamforming technique has a significantly lower complexityas shown in Fig. 3.3. As such the suboptimal 1 beamforming techniqueis useful where a low computational complexity beamforming technique isrequired. From Fig. 3.4, we also observe that the suboptimal 2 beamform-543.5. Numerical Results−40 −30 −20 −10 0 1000.  Maximum transmit power per SCT in dBWAverageminimumdatarateperSRinbps/HzOptimalSuboptimal 1Suboptimal 2Suboptimal 3Figure 3.4: Average minimum data rate of the SRs versus transmissionpower constraint at the SCTs for various beamforming technique, which uses SMINR criteria, provides a higher minimum datarate at the expense of computational complexity compared to the subopti-mal 1 beamforming technique. Fig. 3.4 also shows that the suboptimal 3beamforming technique provides a minimum data rate that is close to theminimum data rate of the optimal beamforming technique. This can beexplained as follows. Although SMINR is used as an intermediate crite-ria, suboptimal 3 beamforming technique uses SINR as the final criteriato develop beamforming vectors, which is used by the optimal beamform-ing technique and directly related to the data rate of the SRs. Therefore,suboptimal 3 and optimal beamforming techniques provide almost similarperformance.553.5. Numerical Results0 0.5 1 1.5 2 2.5 3 3.500.  Instantaneous minimum data rate per SR in bps/HzCDFOptimalSuboptimal 1Suboptimal 2Suboptimal 3Figure 3.5: CDFs of instantaneous minimum data rate of the SRs withperfect CSI for various beamforming techniques.Fig. 3.4 compares the average of minimum data rates of SRS for variousbeamforming techniques. However, it is interesting to plot the distributionof instantaneous minimum data rates and the outage probabilities for a givendata rate requirement for different beamforming techniques. Fig. 3.5 com-pares the cumulative distribution functions (CDFs) of the instantaneousminimum data rates at a 10 dBW transmission power per SCT to illus-trate the outage probabilities of the developed beamforming techniques fora given minimum data rate requirement. For example, with 1.5 bps/Hzminimum data rate requirement the suboptimal 1, 2, 3 and optimal beam-forming techniques have the outage probabilities of 84%, 67%, 47% and 40%,respectively.563.5. Numerical Results00.    −220 −200 −180 −160 −14000.  −220 −200 −180 −160 −140  Interference power in dBWInterference power in dBWCDFCDFPR 1PR 1PR 1PR 1PR 2PR 2PR 2PR 2Optimal Suboptimal 1Suboptimal 2 Suboptimal 3-145dBW-145dBW-145dBW-145dBW-150dBW-150dBW-150dBW-150dBWFigure 3.6: CDFs of the interference at the PRs with various beamformingtechniques.Since one of the most important aspects in DSA is to ensure the properoperation of the PRs by maintaining the interference thresholds at the PRs,in Fig. 3.6 we plot the CDFs of instantaneous interference at the PRs. TheseCDF plots show that the asynchronous interference power at the PRs arealways maintained below their respective threshold with all the developedbeamforming techniques.In order to demonstrate the effectiveness of the extended beamformingtechniques in Section 3.4 under channel and propagation delay estimationuncertainties, in Fig 3.7 we plot the average minimum data rate of the SRswith the extended optimal beamforming technique for various amounts of573.5. Numerical Results−40 −30 −20 −10 0 1000.  Maximum transmit power per SCT in dBWAverageminimumdatarateperSRinbps/HzPerfect CSIImperfect CSI with ζ = −10 dBImperfect CSI with ζ = −6 dBImperfect CSI with ζ = −2 dBFigure 3.7: Performance of extended optimal beamforming technique underdifferent amount of estimation uncertainties.uncertainty. The amount of uncertainty is controlled by a parameter ζ. Inparticular, the uncertainty matrix is controlled by Q(k)Pl in (3.30) where weassume Q(k)Pl =1ζ2 I. As the value of ζ increases, the amount of uncertaintyincreases and vice-versa. From Fig. 3.7, it is observed that as the amountof estimation uncertainty increases, the performance of a cooperative beam-forming technique degrades. This is due to the higher protection margin inorder to protect the PRs from asynchronous interference.In Fig. 3.8, we compare the performances of extended optimal and ex-tended suboptimal beamforming techniques for a given channel estimationuncertainty of ζ = −10 dB. Although the plotted minimum data rate of SRsversus transmit power constraint shows similar behaviour as in Fig. 3.4, it isinteresting to note that the average minimum data rate of SRs with a par-583.5. Numerical Results−40 −30 −20 −10 0 1000.  Maximum transmit power per SCT in dBWAverageminimumdatarateperSRinbps/HzOptimalSuboptimal 1Suboptimal 2Suboptimal 3Figure 3.8: Performance of extended optimal and suboptimal beamformingtechniques for a given amount of estimation uncertainty (ζ = −10 dB).ticular beamforming technique is degraded when there is a certain amountof uncertainty in channel and propagation delay estimation between SCTsand PRs.Fig. 3.9 compares the CDFs of the instantaneous minimum data rates ata 10 dBW transmission power constraint per SCT to illustrate the outageprobabilities of the extended beamforming techniques for a given minimumdata rate requirement and a given channel uncertainty. For example, with a1.5 bps/Hz minimum data rate requirement the suboptimal 1, 2, 3 and opti-mal beamforming techniques have the outage probabilities of 88%, 71%, 57%and 53%, respectively. These values clarify that the extended beamformingtechniques have higher outage probabilities compared to the beamforming593.5. Numerical Results0 0.5 1 1.5 2 2.5 300.  Instantaneous minimum data rate per SR in bps/HzCDFOptimalSuboptimal 1Suboptimal 2Suboptimal 3Figure 3.9: CDFs of minimum data rate of the system with imperfect CSIwith uncertainty of ζ = −10 dB for various beamforming techniques.techniques with perfect CSIs shown in Fig. 3.5. The reason can be explainedas follows. As the protection margin increases under a higher amount of es-timation uncertainty, the date rate performance is degraded. Hence, anincreased outage probability is observed.In order to demonstrate the robustness in terms of maintaining the PRinterference constraints under channel and propagation delay estimation un-certainty, in Fig. 3.10, we plot the CDFs of the instantaneous interferenceat the PRs of the extended beamforming techniques. The CDF plot clearlyshows that the extended optimal and suboptimal beamforming techniquescan guarantee the PR interference constraints. From Fig. 3.6 and 3.10, weobserve that the probability that the instantaneous interference at the PRs603.5. Numerical Results00.    −220 −200 −180 −160 −14000.  −220 −200 −180 −160 −140  Interference power in dBWInterference power in dBWCDFCDFPR 1PR 1PR 1PR 1PR 2PR 2PR 2PR 2Optimal suboptimal 1Suboptimal 2 Suboptimal 3-145dBW-145dBW-145dBW-145dBW-150dBW-150dBW-150dBW-150dBWFigure 3.10: CDF of interference at the PRs using various close to the interference threshold is lower in Fig. 3.10 with estimationuncertainties compared to Fig. 3.6 with perfect CSI. This is due to the factthat the extended beamforming techniques are designed conservatively bytaking channel and propagation delay estimation uncertainties into account.In this thesis, we have used the max-min criteria in order to develop allthe beamforming techniques. As such, these beamforming techniques canprovide improved fairness of the cooperative CR system where different SRsreceive different information. As we mentioned earlier, the weighted sumof data rates criteria can be used to develop a beamforming technique [39].However, such a beamforming technique can degrade the performance of613.5. Numerical Results−40 −30 −20 −10 0 1000.  Maximum transmit power per SCT in dBWDatarateperSRinbps/HzWeighted sum beamformingMax-min beamformingFigure 3.11: Comparison of average minimum data rate of SRs with themax-min criteria based beamforming and the weighted sum criteria basedbeamforming techniques.the system. Therefore, for the completeness of our work, in Fig. 3.11, wecompare the minimum data rate of the SRs.For the weighted sum data rate maximization, weights are chosen suchthat all the SRs get an equal average data rate. Fig. 3.11 shows that thebeamforming technique developed based on the max-min criteria provides ahigher average minimum data rate to the SRs compared to the beamformingdeveloped based on the weighted sum criteria. Fig. 3.12 compares the CDFof instantaneous minimum data rates at the SRs with a 10 dBW transmissionpower constraint at the SCT for weighted sum maximization based beam-forming and max-min criteria based beamforming. The CDF plots showthat the max-min criteria based beamforming technique provides a lower623.5. Numerical Results0 0.5 1 1.5 2 2.5 3 3.5 400.  CDFInstantaneous minimum data rate per SR in bps/HzWeighted sum beamformingMax-min beamformingFigure 3.12: CDFs of minimum data rates of SRs with the max-min crite-ria based beamforming and the weighted sum criteria based beamformingtechniques (10 dBW transmission power constraint).outage probability for a given minimum data rate requirement comparedto the weighted sum maximization based beamforming. For example at a1 bps/Hz data rate requirement, the max-min criteria based beamformingprovides 60% outage probability whereas the weighted sum maximizationbased beamforming provides 85% outage probability. Finally Fig. 3.13 plotsthe average data rate of the max-min criteria based beamforming and theweighted sum maximization criteria based beamforming. This figure showsthat the average data rate of the max-min criteria based beamforming islower, compared to that of the weighted sum criteria based beamforming.Thus, the proposed max-min criteria based beamforming technique improvesthe outage probability at the expense of the average data rate, compared to633.5. Numerical Results−40 −30 −20 −10 0 1000.511.5  Maximum transmit power per SCT in dBWDatarateperSRinbps/HzWeighted sum beamformingMax-min beamformingFigure 3.13: Comparison of average data rate with the max-min criteriabased beamforming and the weighted sum criteria based beamforming tech-niques.the weighted sum criteria based beamforming.64Chapter 4Conclusion and FutureWorksIn this thesis, we have developed a data rate fairness optimal beam-forming technique for a cooperative CR system with multiple SRs and PRsin the presence of asynchronous interferences. We have studied importantproperties of the optimization problem and proposed low complexity subop-timal beamforming techniques by exploiting these properties. Further, wehave extended the beamforming techniques in order to incorporate the un-certainties in the channel and propagation delay estimation between SCTsand PRs. We have presented some selected numerical results in order toshow the performance, complexity and robustness in terms of maintainingthe PR interference constraints under channel propagation delay estimationuncertainties of the developed beamforming techniques. The numerical re-sults have shown that the developed suboptimal beamforming techniquesprovide trade-offs between data rate fairness performance and complexity.Further, the developed optimal and suboptimal beamforming techniques al-ways guarantee the target interference threshold at the PRs. The proposeddata rate fairness beamforming technique improved the minimum data rate,65Chapter 4. Conclusion and Future Worksi.e., the outage probability of the SRs at the expense of the average datarate of the system, compared to the beamforming technique that maximizesthe weighted sum rates of SRs.Our developed beamforming techniques can be used for CR systemswhere there is a central/master node that knows all the CSI between theSCTs and SRs as well as the CSI between the SCTs and the PRs. Thenthis central node calculates the beamforming vectors which are conveyed tothe SCTs. These beamforming techniques can also be used for CR systemswhere all SCTs know all the CSI and calculate their beamforming vectors.This is feasible if the SCTs are connected to each other via high speedbackhaul links. Development of distributed beamforming techniques for CRsystems, where a particular SCT knows only its own CSI, is quite interestingand can be pursued in a future work. In our work, we have considered thatall the SCTs participate in the transmission. However, careful selection ofthe SCTs can improve the performance. This can be investigated in futurework.66Bibliography[1] Federal Communications Commission Spectrum PolicyTask Force, “Report of the Spectrum Efficiency Work-ing Group,” Tech. Rep. 02-135, Nov. 2002. Available at 1.pdf. →pages 1[2] B. Wang and K. R. Liu, “Advances in cognitive radio networks: Asurvey,” IEEE J. Sel. Top. Sign. Proces., vol. 5, no. 1, pp. 5–23, Feb.2011. → pages[3] Y.-C. Liang, K.-C. Chen, G. Y. Li, and P. Mahonen, “Cognitive ra-dio networking and communications: An overview,” IEEE Trans. Veh.Technol., vol. 60, no. 7, pp. 3386–3407, Sep. 2011. → pages 1[4] Q. Zhao and B. M. Sadler, “A survey of dynamic spectrum access,”IEEE Signal Process. Mag., vol. 24, no. 3, pp. 79–89, May 2007. →pages 1[5] S. Haykin, “Cognitive radio: brain-empowered wireless communica-tions,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb.2005. → pages 2, 367Bibliography[6] J. Mitola and J. Maguire, G.Q., “Cognitive radio: making softwareradios more personal,” IEEE Pers. Commun., vol. 6, no. 4, pp. 13–18,Aug. 1999. → pages 2[7] A. M. Wyglinski, M. Nekovee, and T. Hou, Cognitive radio communi-cations and networks: principles and practice. Academic Press, 2009.→ pages 2[8] C. Cordeiro, K. Challapali, D. Birru, and N. Sai Shankar, “IEEE 802.22:the first worldwide wireless standard based on cognitive radios,” inProc. IEEE First International Symposium on New Frontiers in Dy-namic Spectrum Access Networks DySPAN, Nov. 2005, pp. 328–337. →pages 2[9] C. Cordeiro, K. Challapali, and D. Birru, “IEEE 802.22: An introduc-tion to the first wireless standard based on cognitive radios,” Journalof Communications, vol. 1, no. 1, pp. 38–47, Apr. 2006. → pages 2[10] M. Kaur, M. Uddin, H. Verma, and B. Ambedkar, “Role of cognitiveradio on 4g communications a review,” Journal of Emerging Trends inComputing and Information Sciences, vol. 3, no. 2, pp. 272–276, Feb.2012. → pages 2[11] X. Hong, J. Wang, C.-X. Wang, and J. Shi, “Cognitive radio in 5G:A perspective on energy-spectral efficiency trade-off,” IEEE Commun.Mag., vol. 52, no. 7, pp. 46–53, Jul. 2014. → pages 2[12] M. Oner and F. Jondral, “On the extraction of the channel allocation68Bibliographyinformation in spectrum pooling systems,” IEEE J. Sel. Areas Com-mun., vol. 25, no. 3, pp. 558–565, Apr. 2007. → pages 2[13] I. F. Akyildiz, W.-Y. Lee, M. C. Vuran, and S. Mohanty, “Next gen-eration/dynamic spectrum access/cognitive radio wireless networks: Asurvey,” Computer Networks, vol. 50, no. 13, pp. 2127–2159, May 2006.→ pages 2[14] L. B. Le and E. Hossain, “Resource allocation for spectrum underlayin cognitive radio networks,” IEEE Trans. Wireless Commun., vol. 7,no. 12, pp. 5306–5315, Dec. 2008. → pages 2[15] N. Yi, Y. Ma, and R. Tafazolli, “Underlay cognitive radio with full orpartial channel quality information,” International Journal of Naviga-tion and Observation, vol. 2010, Article ID 105723, pp. 1–12, Jun. 2010.→ pages 2[16] L. Lu, X. Zhou, U. Onunkwo, and G. Y. Li, “Ten years of research inspectrum sensing and sharing in cognitive radio,” EURASIP Journal onWireless Communications and Networking, vol. 2012, no. 1, pp. 1–16,Jan. 2012. → pages 3[17] K. Letaief and W. Zhang, “Cooperative communications for cognitiveradio networks,” Proc. IEEE, vol. 97, no. 5, pp. 878–893, May 2009. →pages 3[18] L. Zhang, Y.-C. Liang, and Y. Xin, “Joint beamforming and powerallocation for multiple access channels in cognitive radio networks,”69BibliographyIEEE J. Sel. Areas Commun., vol. 26, no. 1, pp. 38–51, Jan. 2008. →pages 3[19] T. Al-Khasib, M. Shenouda, and L. Lampe, “Dynamic spectrum man-agement for multiple-antenna cognitive radio systems: Designs withimperfect csi,” IEEE Trans. Wireless Commun., vol. 10, no. 9, pp.2850–2859, Sep. 2011. → pages 3, 4[20] B. D. Van Veen and K. M. Buckley, “Beamforming: A versatile ap-proach to spatial filtering,” IEEE ASSP Mag., vol. 5, no. 2, pp. 4–24,Apr. 1988. → pages 4[21] K. Hamdi, W. Zhang, and K. Letaief, “Joint beamforming and schedul-ing in cognitive radio networks,” in Proc. IEEE Global Telecommuni-cations Conference (Globecom), Nov. 2007, pp. 2977–2981. → pages4[22] R. Zhang and Y.-C. Liang, “Exploiting multi-antennas for opportunisticspectrum sharing in cognitive radio networks,” IEEE J. Sel. Top. Sign.Proces., vol. 2, no. 1, pp. 88–102, Feb. 2008. → pages 4[23] O. Bakr, M. Johnson, R. Mudumbai, and K. Ramchandran, “Multi-antenna interference cancellation techniques for cognitive radio applica-tions,” in Proc. IEEE Wireless Communications and Networking Con-ference, WCNC, Apr. 2009, pp. 1–6. → pages 4[24] A. Nosratinia, T. E. Hunter, and A. Hedayat, “Cooperative communi-cation in wireless networks,” IEEE Commun. Mag., vol. 42, no. 10, pp.74–80, Oct. 2004. → pages 570Bibliography[25] J. Choi, “MMSE-based distributed beamforming in cooperative relaynetworks,” IEEE Trans. Commun., vol. 59, no. 5, pp. 1346–1356, May2011. → pages 5[26] J. Liu, W. Chen, Z. Cao, and Y. Zhang, “A distributed beamform-ing approach for enhanced opportunistic spectrum access in cognitiveradios,” in Proc. IEEE Global Telecommunications Conference (Globe-com), Nov. 2009, pp. 1–6. → pages 5[27] R. Mudumbai, G. Barriac, and U. Madhow, “On the feasibility ofdistributed beamforming in wireless networks,” IEEE Trans. WirelessCommun., vol. 6, no. 5, pp. 1754–1763, May 2007. → pages 5[28] D. R. Brown III and H. V. Poor, “Time-slotted round-trip carrier syn-chronization for distributed beamforming,” IEEE Trans. Signal Pro-cess., vol. 56, no. 11, pp. 5630–5643, Nov. 2008. → pages[29] D. R. Brown III, G. B. Prince, and J. A. McNeill, “A method for carrierfrequency and phase synchronization of two autonomous cooperativetransmitters,” in Proc. IEEE 6th Workshop on Signal Processing Ad-vances in Wireless Communications, Jun. 2005, pp. 260–264. → pages5[30] R. Mudumbai, D. R. Brown, U. Madhow, and H. V. Poor, “Distributedtransmit beamforming: challenges and recent progress,” IEEE Com-mun. Mag., vol. 47, no. 2, pp. 102–110, Feb. 2009. → pages 5[31] A. F. Dana and B. Hassibi, “On the power efficiency of sensory and adhoc wireless networks,” IEEE Trans. Inf. Theory. → pages 571Bibliography[32] M. A. Beigi and S. M. Razavizadeh, “Cooperative beamforming in cog-nitive radio networks,” in Proc. IEEE Wireless Days (WD), 2nd IFIP,Dec. 2009, pp. 1–5. → pages 5[33] N. Alaoui, V. Meghdadi, J.-P. Cances et al., “Cooperative beamformingfor multi-user multi-relay cognitive radio networks based on second-order statistics of channel state information,” International Journal onCommunication (IJC), vol. 2, no. 4, pp. 89–99, Dec. 2013. → pages 5[34] J. Liu, W. Chen, Z. Cao, and Y. J. A. Zhang, “Cooperative beamform-ing for cognitive radio networks: A cross-layer design,” IEEE Trans.Commun., vol. 60, no. 5, pp. 1420–1431, May 2012. → pages 5[35] R. Xie, F. Yu, and H. Ji, “Joint power allocation and beamformingwith users selection for cognitive radio networks via discrete stochasticoptimization,” in Proc. IEEE Global Telecommunications Conference(Globecom), Dec. 2011, pp. 1–5. → pages 5[36] H. Zhang, N. B. Mehta, A. F. Molisch, J. Zhang, and H. Dai, “Asyn-chronous interference mitigation in cooperative base station systems,”IEEE Trans. Wireless Commun., vol. 7, no. 1, pp. 155–165, Jan. 2008.→ pages 6, 7, 26, 29, 30[37] J. Tang, K. Zhao, C. Ni, M. Yang, and J. Shi, “Robust MMSE designwith asynchronous interference mitigation in cooperative base stationsystems,” Wireless Personal Communications, vol. 78, no. 2, pp. 889–903, Apr. 2014. → pages 6, 772Bibliography[38] M. H. Hassan and M. J. Hossain, “Cooperative beamforming for cog-nitive radio systems with asynchronous interference to primary user,”IEEE Trans. Wireless Commun., vol. 12, no. 11, pp. 5468–5479, Nov.2013. → pages 7[39] M. H. Hassan and M. J. Hossain, “Cooperative beamforming for cog-nitive radio-based broadcasting systems with asynchronous interfer-ences,” in Proc. of the IEEE Wireless Communications and Network-ing Conference (WCNC), Apr. 2014. Journal version available online: . → pages 7, 42, 61[40] S. Fazeli-Dehkordy, S. Shahbazpanahi, and S. Gazor, “Multiple peer-to-peer communications using a network of relays,” IEEE Trans. SignalProcess., vol. 57, no. 8, pp. 3053–3062, Aug. 2009. → pages 7[41] N. D. Sidiropoulos, T. N. Davidson, and Z.-Q. Luo, “Transmit beam-forming for physical-layer multicasting,” IEEE Trans. Signal Process.,vol. 54, no. 6, pp. 2239–2251, Jun. 2006. → pages 8, 23, 37, 40[42] E. Karipidis, N. D. Sidiropoulos, and Z.-Q. Luo, “Quality of service andmax-min fair transmit beamforming to multiple cochannel multicastgroups,” IEEE Trans. Signal Process., vol. 56, no. 3, pp. 1268–1279,Mar. 2008. → pages 8, 37[43] G. Zheng, K.-K. Wong, and B. Ottersten, “Robust cognitive beamform-ing with bounded channel uncertainties,” IEEE Trans. Signal Process.,vol. 57, no. 12, pp. 4871–4881, Dec. 2009. → pages 8, 50, 80, 8173Bibliography[44] S. Akin and M. C. Gursoy, “Performance analysis of cognitive radiosystems under qos constraints and channel uncertainty,” IEEE Trans.Wireless Commun., vol. 10, no. 9, pp. 2883–2895, Sep. 2011. → pages8[45] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge uni-versity press, 2009. → pages 13, 80[46] Z.-Q. Luo and W. Yu, “An introduction to convex optimization forcommunications and signal processing,” IEEE J. Sel. Areas Commun.,vol. 24, no. 8, pp. 1426–1438, Aug. 2006. → pages 13[47] Z.-Q. Luo, W.-K. Ma, A.-C. So, Y. Ye, and S. Zhang, “Semidefiniterelaxation of quadratic optimization problems,” IEEE Signal Process.Mag., vol. 27, no. 3, pp. 20–34, May 2010. → pages 21, 35[48] 3rd Generation Partnership Project (3GPP), “Technical specificationgroup GSM/EDGE, radio access network; Radio subsystem synchro-nization,” Tech. Rep. 45.010 (v6.6.0), 2005. → pages 26[49] K. Phan, S. Vorobyov, N. Sidiropoulos, and C. Tellambura, “Spectrumsharing in wireless networks via qos-aware secondary multicast beam-forming,” IEEE Trans. Signal Process., vol. 57, no. 6, pp. 2323–2335,Jun. 2009. → pages 35[50] M. Grant and S. Boyd, “The CVX users guide. CVX research,” Inc.,Cambridge, UK, vol. 190, 2012. → pages 40[51] H. Son and B. Clerckx, “Joint beamforming design for multi-74user wireless information and power transfer,” 2014. Available at . → pages 4075AppendicesAppendix IProof of Property 1 of the Optimization Problemin (3.19)Proof. Assume that an optimal solution is represented by (g˜(1), · · · , g˜(M)),where∑Mk=1(g˜(k))†H(k)Pl g˜(k) < P ITHPl ,∀l and ‖g˜Ti‖22 < PTi ,∀i. Then, thereexists a real-value c, such that c > 1, c∑Mk=1(g˜(k))†H(k)Pl g˜(k) < P ITHPl ,∀land c‖g˜Ti‖22 < PTi ,∀i. This provides a new set of solution (√cg˜(1), · · · ,√cg˜(M)) which is feasible for the above problem, i.e., the new solutionssatisfy all interference constraints and power constraints.The new SINRs can be written asc(g˜(j))†H(j)Rj g˜(j)c∑Mk=1k 6=j(g˜(k))†H(k)Rj g˜(k) + σ2Rj>(g˜(j))†H(j)Rj g˜(j)∑Mk=1k 6=j(g˜(k))†H(k)Rj g˜(k) + σ2Rj,∀j,for c > 1, (1)since(g˜(j))†H(j)Rj g˜(j)∑Mk=1k 6=j(g˜(k))†H(k)Rj g˜(k)+σ2Rjis increasing. This concludes that all the SINRsare improved with the new set of solution. This contradicts the assumption76Bibliographythat (g˜(1), · · · , g˜(M)) are the optimal solution. Thus, either∑Mk=1(g˜(k))†H(k)Pl g˜(k) = P ITHPl for at least one value of l or ‖g˜i‖22 = PTi forat least one value of i.Proof of Property 2 of the Optimization Problemin (3.19)Proof. To prove property 2, we consider two smallest SINRs at the SRsnamely, SINRa and SINRb. Assume that an optimal set of solution isrepresented by (g˜(1), · · · , g˜(M)) and assume SINRa > SINRb. This can berepresented as(g˜(a))†H(a)Ra g˜(a)∑Mk=1k 6=a(g˜(k))†H(k)Ra g˜(k) + σ2Ra>(g˜(b))†H(b)Rb g˜(b)∑Mk=1k 6=b(g˜(k))†H(k)Rb g˜(k) + σ2Rba 6= b, a, b = 1, · · · ,M. (2)Since SINRa > SINRb, there exists a real value c, such that 0 < c < 1and(g˜(b))†H(b)Rb g˜(b)c(g˜(a))†H(a)Rb g˜(a) +∑Mk=1k 6=b,a(g˜(k))†H(k)Rb g˜(k) + σ2Rb=c(g˜(a))†H(a)Ra g˜(a)∑Mk=1k 6=a(g˜(k))†H(k)Ra g˜(k) + σ2Ra. (3)This corresponds to new set of solutions as (g˜(1), · · · ,√cg˜(a), · · · , g˜(M)).Since c < 1, it is easy to show that the interference constraints and power77Bibliographyconstraints are satisfied and thus the new solutions are feasible.Let us consider the minimum SINR among those two new SINRs (bothare equal). That is greater than the previous minimum SINR for c < 1.This can be simply shown as(g˜(b))†H(b)Rb g˜(b)c(g˜(a))†H(a)Rb g˜(a) +∑Mk=1k 6=b,a(g˜(k))†H(k)Rb g˜(k) + σ2Rb>(g˜(b))†H(b)Rb g˜(b)∑Mk=1k 6=b(g˜(k))†H(k)Rb g˜(k) + σ2Rb. (4)This contradicts the assumption (g˜(1), · · · , g˜(M)) is an optimal set of solu-tions. Thus SINRa should be equal to SINRb. This implies, we shouldhave equal SINRs with a set of optimal beamforming vectors.78Appendix IIProof of the Worst Case Interference Constraintat the PRsProof. Let us consider the worst case interference constraint at the lth PR,which can be expressed asmax∆H(k)Pl ∈H(k)PlM∑k=1Tr{H(k)Pl G(k)}= max∆H(k)Pl ∈H(k)PlM∑k=1Tr{(Hˆ(k)Pl +∆H(k)Pl)G(k)}≤ P ITHPl . (5)The worst case interference constraint depends on the maximum of∑Mk=1 Tr{∆H(k)Pl G(k)}in the set H(k)Pl ,∀k. Thus the maximum value can befound by solvingmax∆H(k)PlM∑k=1Tr{∆H(k)Pl G(k)}subject to Tr{∆H(k)Pl Q(k)Pl ∆H(k)Pl}≤ 1, ∀k,∆H(k)Pl =(∆H(k)Pl)†, ∀k. (6)79Appendix IIThe above problem is convex6 and satisfies Slater’s7 condition. Thus,based on Slater’s theorem, a strong duality holds and the optimal solutioncan be found by solving the dual problem. The dual problem can be ex-pressed asmaxrinf∆H(k)PlL(∆H(k)Pl , rk)subjected to rk ≥ 0, ∀k. (7)The Lagrangian of above problem can be written asL(∆H(k)Pl , rk) =M∑k=1{−Tr{∆H(k)Pl G(k)}+ rk(Tr{∆H(k)Pl Q(k)Pl ∆H(k)Pl}− 1)},(8)where rk > 0,∀k are the dual variables. Setting the first derivative of La-grangian with regard to ∆H(k)Pl to zero and using the properties of derivativesof a trace [43], we obtain−G(k) + rk∆H(k)Pl Q(k)Pl + rkQ(k)Pl ∆H(k)Pl = 0, ∀k. (9)Equation (9) yieldsrk∆H(k)Pl = MAT((IN ⊗Q(k)Pl + (Q(k)Pl )†⊗ IN)−1vec(G(k))). (10)6Objective function is affine and inequality constraints are convex, i.e., the quadraticterm is convex for positive semidefinite Q(k)Pl . Symmetry is an affine constraint. Thus theproblem is convex.7There exists a ∆H(k)Pl ∈ H(k)Pl, which is strictly feasible. See [45] for more details.80Appendix IIBy using (9), the infimum of dual objective function can be written asM∑k=1{−Tr{∆H(k)Pl G(k)}+ rk(Tr{∆H(k)Pl Q(k)Pl ∆H(k)Pl}− 1)}=M∑k=1{−rk − rkTr{∆H(k)Pl Q(k)Pl ∆H(k)Pl}}=M∑k=1−2rk, (11)where the fact that at the optimum of Tr{∆H(k)Pl Q(k)Pl ∆H(k)Pl}= 1 is used.Thus, the solution to the dual problem is ∑Mk=1−2rk. Following the similarsteps in [43], rk can be found asrk =∥∥∥∥∥Q˜(k)Pl MAT((IN ⊗Q(k)Pl + (Q(k)Pl )†⊗ IN)−1vec(G(k)))∥∥∥∥∥F. (12)Thus the worst case interference constraint of the lth PR can be expressedasM∑k=1Tr{Hˆ(k)Pl G(k)}+M∑k=12∥∥∥∥∥Q˜(k)Pl MAT((IN ⊗Q(k)Pl +(Q(k)Pl)†⊗ IN)−1vec(G(k)))∥∥∥F≤ P ITHPl . (13)81


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