Trust Degree Based Beamforming for Multi-Antenna Cooperative Communication Systems

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1 Trust Degree Based Beamforming for Multi-Antenna Cooperative Communication Systems Mojtaba Vaezi, Hazer Inaltekin, Wonjae Sin, H. Vincent Poor, and Junsan Zang* Department of Electrical Engineering, Princeton University, Princeton, NJ 08540, USA Department of Electrical and Computer Eng., Seoul National University, Seoul , Korea * Scool of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA {mvaezi, inaltek, poor}@princeton.edu, wonjae.sin@snu.ac.kr, junsan.zang@asu.edu Abstract In tis paper, beamforming design is investigated for a multi-antenna cooperative communication system in wic bot pysical links and social connections (trust degrees) between nodes are taken into account. An optimal beamformer aims to balance between te direct link and te cooperating link as well as respecting te trust degree. Te resulting optimization problem is nontrivial to solve, even numerically, as it is not convex. Te complexity of te problem is largely reduced by sowing tat a linear combination of te direct and cooperating links cannel vectors imizes te acievable rate. Ten, a computationally efficient numerical solution is used to imize te rate. Numerical results demonstrate tat significant gains in communication rates can be obtained wit te proposed optimal beamforming design. I. INTRODUCTION A salient caracteristic of mobile data networks is te dominant operating mode of one person beind eac device, wic differentiates tem from oter tecnological networks suc as sensor networks. A direct consequence of tis feature of mobile data networks is te existence of bot pysical coupling between mobile devices troug sared communication resources suc as wireless spectrum, and virtual coupling among te users beind tese devices in te social domain. Tese virtual ties, in many ways, sape te data traffic flows and quality-of-service (QoS) requirements in te pysical domain. Social interactions could be positive (representing friendsip) or negative (representing antagonism), wic yield non-trivial impact on users decision-making processes (e.g., in terms of cooperation) for data transmission and reception. We refer to tis two-layered network structure as a mobile/social network. A number of recent works, e.g., [1] [6], ave studied te integration of mobile data networks wit social networks in a variety of settings. In [1], a game teoretic framework called social group utility imization is developed to imize a weigted sum of te individual utilities rater tan a totally selfis utility. In [2] [5], social community-aware resource allocation is studied for device-to-device (D2D) networks. A social-aware peer discovery sceme for D2D systems is proposed in [6]. It exploits social network caracteristics to assist ad oc peer discovery and enance data transmission performance. Tis researc was supported by te U.S. Army Researc Office under Grant W911NF Tx RN (a) Pysical connections g Rx Tx RN (b) Trust degrees Fig. 1. A multi-antenna cooperative communication network wit pysical and social connections (trust degrees). Most of te above works develop communication strategies based on single-antenna nodes. In a recent study, Ryu et al. [7] proposed te use of social connections for beamforming in a cooperative communication system wit a single-antenna relay. Tis sceme uses trustwortiness between nodes to design beamforming for efficient data relaying. It is proved tat a linear combination of cannel vectors makes an optimal beamformer to imize expected acievable rate. To find a closed-form solution for te optimal weigts of te linear combination, a ig signal-to-noise ratio (SNR) approximation is used wic explicitly sows te effect of trust degrees on te beamforming design. In tis paper, we improve te acievable rate of [7] and extend it to te case were te relay as multiple antennas. Te rate improvement is based on two observations. First, we sow tat imum trust degree does not always result in te best rate performance. Tus, we consider optimization over trust degree as well as te beamforming vectors at te source and relay nodes. Second, unlike [7], our design is for te exact acievable rate, not just in te ig SNR regime. We use a computationally practical numerical solution tat acieves near optimal beamformer performance. Te resulting beamformer improves te rate performance wen compared wit te closed-form solution of te approximated rate given in [7]. Anoter contribution of te current paper is te extension of te beamforming design in [7] to te case in wic te relay as multiple antennas. Te paper is organized as follows. We describe te system model in Section II and formulate an optimization problem to imize te acievable rate at te destination in Section III. Rx

2 In Section IV, we prove tat optimal beamforming is acieved troug a linear combination of te cannel vectors for direct and cooperating links, respecting te trust degree. Numerical results are presented in Section V and te paper is concluded in Section VI. A. Notation We use ( ) t, ( ), ( ), and tr( ) to denote te transpose, conjugate, conjugate transpose, and trace of a matrix, respectively. Matrices are written in bold capital letters and vectors are written in bold small letters. I represents an identity matrix. For scalar x, 0 x 1, we define x = 1 x. x is te Euclidean norm of x. II. SYSTEM MODEL In tis paper, we consider a two-layered network structure, wic we call a mobile/social network. As sown in Fig. 1, te network as pysical connections (Fig. 1(a)) as well as social connections represented by trust degrees (Fig. 1(b)). We describe te details of tis two-layered network model below, starting wit te pysical system model. A. Pysical System Model Consider a multiple-input-multiple-output (MIMO) decodeand-forward (DF) full-duplex two-op relay network as sown in Fig. 1(a). We assume tat te source (S) and relay (R) ave N s and N r antennas, respectively, and te destination (D) as one antenna only. Let C Ns 1, H C Ns Nr, and g C Nr 1 represent te cannel coefficients for S-D, S-R, and R-D links, respectively. We assume tat cannel state information (CSI) is known at te transmitters, e.g., troug training/feedback and cannel reciprocity, so tat tey can steer teir beamforming vectors to imize teir transmission performance. In tis setting, information transmission can be divided into two pases, as described below. 1) Source Pase: In tis pase, te information symbol x s is first multiplied by a beamforming vector before being transmitted at S. Te complex baseband signal received at D and R can be represented as y SD = t x s + n SD, (1) y SR = H t x s + n SR, (2) were n SD CN (0, 1) and n SR CN (0, I) represent complex additive Gaussian noise at D and R, respectively. Te received SNRs at D and R are given by γ SD = t 2 P s, (3) γ SR = H t 2 P s. (4) Tus, te imum acievable rates of communication from S to D and S to R are given by C SD = log 2 (1 + γ SD ), (5) C SR = log 2 (1 + γ SR ). (6) 2) Relay Pase: In tis pase, R forwards its decoded symbol x r to D wit power P r troug transmission over N r antennas. Te complex baseband signal received at D is given by y RD = g t w r x r + n RD, (7) were n RD CN (0, 1). Te received SNR at D, in tis pase, is given by γ RD = g t w r 2 P r. (8) Ten, D can combine te signal received from S and R to improve te SNR. Te acievable rate for te composite link is given by [8] C D = log 2 (1 + γ SD + γ RD ). (9) Finally, te acievable rate for te DF system is obtained as C DF = min{c SR, C D } (10) { } = min log 2 (1 + γ SR ), log 2 (1 + γ SD + γ RD ). B. Trust Degrees Similar to [7], we model te social system by trust degrees. Defined for eac link between every two nodes, trust degree is a level of belief tat one node can elp te oter node for a specific action suc as relaying [7], [9]. Trust degree can be quantified based on te previous interactions between te nodes or can be estimated from te istory of past interactions in a social platform suc as in Facebook or Twitter [7], [9] [11]. Specifically, in our cooperative communication system (see Fig. 1), te relay would be willing to participate in cooperation wit probability α 1 and would use a fraction β 1 of its power for tis purpose, wit α 1, β 1 [0 1]. 1 Witout loss of generality, and for simplicity of presentation, we absorb β 1 in te relay power. Tat is, we assume P r is te fraction of relay power used for cooperation. Ten, te trust degree based rate will be R T = α 1 C DF + (1 α 1 )C SD { } = α 1 min log 2 (1 + γ SR ), log 2 (1 + γ SD + γ RD ) + ᾱ 1 log 2 (1 + γ SD ) (11) in wic C DF and C SD are from (10) and (5), and ᾱ 1 = 1 α 1. It is wort noting tat for α 1 = 1 we get R T = C DF and our system simplifies to a DF MIMO relay beamforming problem wic is a non-convex problem and as been extensively studied, e.g., see [8], [12], [13] and te reference terein. On te oter and, for α 1 = 0 we get R T = C SD and te system simplifies to multiple-input-single-output (MISO) cannel wic is a well-understood problem. 1 In general, for link i we can define te pair (α i, β i ), α i, β i [0 1], as a virtual (social) tie in wic α i denotes te probability of cooperation and β i denotes te fraction of power tat te node uses for collaboration [7]. Here, trust degrees are assumed to be bidirectional.

3 III. PROBLEM FORMULATION AND TRANSFORMATION In tis section, we formulate a beamforming design problem to imize te objective function in (11). We ten transform tis problem into some equivalent problems tat can be tackled more easier. Our goal is to jointly optimize te beamforming vectors and w r as well as α to imize (11). Matematically, tis can be expressed as,w r,α R T s.t. 2 = 1, (12) w r 2 = 1, 0 α α 1, were R T is defined in (11). Tis is a non-convex optimization problem, 2 for wic standard convex optimization tecniques cannot be applied directly. As a first step to simplifying (12), we observe tat γ SD and γ SR are independent of w r, te beamforming vector at te relay. In oter words, w r merely affects γ RD. Hence, te imal ratio transmission (MRT) beamformer [8], [13], [14] is optimal, i.e., wr = g g. (13) Wit tis, te problem reduces to [ α min{log2 (1 + γ SR ), log 2 (1 + γ SD + γrd)},α s.t. 2 = 1, + ᾱ log 2 (1 + γ SD ) ] (14) 0 α α 1, in wic γ RD = g 2 P r is obtained from (13) and (8). A. An Upper Bound Lemma 1. An upper bound on te acievable rate in (11) is given by R T α 1 log 2 (1 + γ SD + γ RD) + ᾱ 1 log 2 (1 + γ SD) (15) = α 1 log 2 (1 + 2 P s + g 2 P r ) + ᾱ 1 log 2 (1 + 2 P s ) in wic γ SD is obtained from (3) for w s =. Proof. Using (14), we can write { } R T = α min log 2 (1 + γ SR ), log 2 (1 + γ SD + γrd) + ᾱ log 2 (1 + γ SD ) α log 2 (1 + γ SD + γ RD) + ᾱ log 2 (1 + γ SD ) α 1 log 2 (1 + γ SD + γ RD) + ᾱ 1 log 2 (1 + γ SD ) α 1 log 2 (1 + γ SD + γ RD) + ᾱ 1 log 2 (1 + γ SD) (16) were te first inequality is due to te fact tat min{a, b} b, te second inequality follows from 1 + γ SD + γ RD 1 + γ SD and te fact tat α α 1, and te tird inequality is due to a 2 R T includes logaritmic functions of quadratic functions of ; tus, it is a combination of concave and convex functions wic is non-convex. simple optimization over γ SD in wic γsd is obtained from (3) for ws =. It is wort noting tat te above upper bound is acievable if te link between S and R is strong enoug so tat it is not te bottleneck of te system. Tis is expected if te relay as multiple antennas and/or is close enoug to te transmitter. More accurately, we ave te following result. Teorem 1. If H t 2 P s 2 P s + g 2 P r for ws =, ten w s = is an optimal beamformer of (14), and R T = α 1 log 2 (1 + 2 P s + g 2 P r ) + ᾱ 1 log 2 (1 + 2 P s ) is acievable for te system depicted in Fig. 1. Proof. Te proof follows from Lemma 1 because te upper bound above is acievable for ws = if Ht 2 P s 2 P s + g 2 P r. B. Transforming te Optimization Problem Te optimization problem (12) is a non-convex problem, even for a fixed α. In te following, we aim at simplifying it by converting it into more tractable problems. More specifically, we break (12) into tree optimization problems, as described below. 1) γ SR γ SD : In suc a case, te relay link is not strong enoug to be elpful. Ten, from (14), it is straigtforward to ceck tat te optimal α is zero (α = 0) and te problem simplifies to log 2 (1 + γ SD ) s.t. 2 = 1, (17) γ SR γ SD. Ten, witout te last constraint, can be easily designed as an MRT beamformer; i.e., ws =. Tus, an optimal beamformer is given by ws = provided tat te inequality γ SR γ SD olds. Te optimal is not trivial in general. 2) γ SR γ SD + γrd : Tis appens, for example, wen te relay is very close to te transmitter. In tis case, R T = α log 2 (1 + γ SD + γrd ) + ᾱ log 2(1 + γ SD ). Now, since te argument of te first logaritm is always greater tan or equal to tat of te second one, we ave α = α 1, and (14) reduces to α 1 log 2 (1 + γ SD + γrd) + ᾱ 1 log 2 (1 + γ SD ) s.t. 2 = 1, (18) γ SR γ SD + γ RD. Ten again, ws = is optimal if te inequality γ SR γ SD + γrd olds. In suc a case, we get R T = α 1 log 2 (1 + 2 P s + g 2 P r ) + ᾱ 1 log 2 (1 + 2 P s ). Note tat tis is te best acievable rate we can expect from te optimization problem in (12).

4 3) γ SD < γ SR < γ SD + γrd : In tis case, since γ SD < γ SR, it can be cecked from (14) tat α = α 1 is optimal. Pysically, tis means tat te relay is strong enoug and te social connection is fully exploited. Furter, in view of γ SR < γ SD + γrd, (14) simplifies to α 1 log 2 (1 + γ SR ) + ᾱ 1 log 2 (1 + γ SD ) s.t. 2 = 1, (19) γ SD < γ SR, γ SR < γ SD + γ RD. Te new optimization problems defined by (17), (18), and (19) are simpler tan (14) in tat te optimization over te variable α is already carried out. Wit tis divide and conquer approac, we readily obtain te following useful lemmas. Lemma 2. Te optimal value of α in (14) is eiter zero or α 1. Lemma 3. Let R T1, R T2 and R T3 be te solutions to te optimization problems in (17), (18) and (19), respectively. Ten, te optimum rate acieved by te solution of (14) is equal to {R T1, R T2, R T3 }. Furter, te optimum beamforming vector at te source is te one corresponding to te imum R Ti for i = 1, 2, 3 C. Semidefinite Relaxation By converting te original optimization problem into tree subproblems, we fixed te value of α in te previous subsection. However, te new optimization problems are still nonconvex and difficult to solve. A computationally efficient way to tackle te optimization problem in (17) is to use semidefinite relaxation (SDR) [15], as described below. It can be cecked tat te same tat imizes (17) is also optimal for te following problem: min γ SD s.t. 2 = 1, (20) γ SR γ SD. Since γ SD and γ SR are quadratic in, tis new problem is a quadratically constrained quadratic program (QCQP), and can be solved by te SDR tecnique [15]. To see tis, let us write γ SR = H 2 P = w sh H P = tr(h H w s)p = P tr(aw), (21) were we ave defined A = H H and W = w s. Ten, we can transform te above optimization problem to min s.t. t P tr(aw) t tr(w) = 1 (22) rank(w) = 1, W 0, in wic W 0 means tat W is a positive semidefinite matrix. Now, te objective function is obviously a convex function. Also, te trace constraints are linear in W. However, te rank constraint is non-convex. 3 We can relax tis optimization problem by dropping te rank constraint, wic is called SDR. Te resulting optimization problem is convex and can be solved efficiently using interior point metods, in polynomial time. We note tat SDR converts an NP-ard problem to a polynomial-time solvable problem. But tis analytical convenience comes at te expense of conversion of te globally optimal solution of te relaxed problem into a feasible solution of (22). Randomization is an effective metod for converting te solution of te relaxed problem to an approximate solution of te original problem [15], [16]. Unfortunately, even SDR is not applicable to (18) and (19), as teir objective functions can be neiter quadratic nor convex. IV. AN EFFICIENT HEURISTIC APPROACH FOR BEAMFORMING In te previous section, we explained tat te optimization problem (14) is ard to solve even wen te trust degree is fixed. Te main problem lies in te fact tat te resulting optimization formulation is not convex, wic does not lend itself to an analytically or numerically efficient solution. Likewise, te SDR metod is not applicable to tis problem in general. In order to alleviate tis problem, we propose an efficient euristic metodology to construct te beamforming vector and reduce te dimension of te optimum beamforming problem in tis section. Tis approac will lead to a numerically efficient solution for trust degree based beamforming in cooperative mobile data and social networks. Te optimization problem formulated in (14) clearly indicates tat te beamforming vector sould alter te values of γ SD and γ SR in a way to imize te transmission rates. On te oter and, from (3) and (4), we observe tat γ SD and γ SR are functions of and H, respectively. Tus, it is intuitive to expect an efficient construction of te beamforming vector based on te knowledge of and H. More concretely, te optimal sould be a vector in te span of te space generated by te columns of and H. One immediate solution to tis end would be te normalization of c + N r i=1 c i i, were i is te it column of H. However, suc a leads to an intractable form wen substituted back into (14), as and te i s are not necessarily ortogonal. A more amenable form would be obtained if an ortogonal basis of te subspace defined by tese vectors is used. Tis is te approac we follow below. Te ortogonal projection of H onto te column space of is given by Π H, were Π ( ) 1 is te standard ortogonal projection matrix [17]. 4 Since Π is a projection Π onto, it is easy to ceck tat eac column of H Π H is equal to. Also, te ortogonal projection of H onto te 3 Te sum of two rank-one matrices as generic rank two. 4 Let Π X X(X X) 1 X be te ortogonal projection onto te column space of X. Ten, Π X I Π X represents te ortogonal projection onto te ortogonal complement of te column space of X [17].

5 ortogonal complement column space of is given by Π H, were Π I Π. Next, let Π H [ 1 H 1 ] were 1 is te first column of Π H and H 1 contains te remaining N r 1 columns of Π H. We can find te ortogonal projection of H 1 onto te column space of 1 by Π 1 H 1, were Π 1 1 ( 1 1 ) 1 1. Likewise, te ortogonal projection of H 1 onto te ortogonal complement column space of 1 is given by Π 1 H 1. We now denote Π 1 H 1 [ 2 H 2 ] and find ortogonal projections of H 2 onto te column space and ortogonal complement column space of 2. We repeat tis procedure until H i becomes empty. Note tat i N r as Π H as N r (independent) columns at most. 5 At tis point, te ortogonalization process is completed and we can write w (23) w i i i, i {1,..., N r}. (24) By definition, we ave w wi, i.e., w.wi = 0 i {1,..., N r }. Te optimal structure of te beamforming vector is ten given by te following lemma. Teorem 2. Te transmit beamformer tat imizes te acievable rate in (10) can be represented as = N r γ 0 w + γi w i (25) i=1 in wic w and te w i s are ortonormal bases spanning te column space of and H, respectively, as defined in (23) and (24), and γ 0 + γ γ Nr = 1. Proof. We prove tis lemma by contradiction, similar to [7]. To simplify te proof, let us rewrite (25) as = γ 0 w + γ 0 w (26) in wic we define w 1 γ0 Nr i=1 γi w i. Next, suppose tat C Ns 1 is an optimal beamformer. Obviously, can be represented as = η 1 f 1 + η 2 f η Ns f Ns were te f i s form an ortonormal basis. Since w and w are ortonormal vectors, witout loss of generality, let f 1 = w and f 2 = w. Ten, we get = η 1 w + η 2 w + η 3 f η Ns f Ns (27) in wic f i w = 0 and f i w = 0 for i 3. Next, we argue tat η i sould be zero for any i 3. To tis end, we sow tat using as a beamformer, γ SD and γ SR do not depend on η i, i 3. Te former is rater simple as γ SD = t 2 P s = η 1 2 P s, in view of (27) and (23). It is also easy to verify tat γ SR = H t 2 P is independent of η i, i 3. To see tis, recall from te ortogonalization process in (24) tat te w i s form an ortonormal basis spanning te column space of H. Hence, eac column of H is a linear combination of te w i s, and tus, is ortogonal to f i for i 3 5 Here, we ave assumed N r N s. In general, i min{n r, N s}. due to te construction of te f i s in (27) and tat of w in (26). Tis means tat ŵ s = η 1 w + ˆη 2 w wit ˆη 2 = η 2 +η η Ns can strictly increase te optimum value of te data rate in (14) wen compared wit. But, tis contradicts te assumption tat is an optimal beamformer and not equal to. Terefore, any optimum beamformer must be equal to given in (26). Tis completes te proof. Remark 1. Note tat te problem as been largely simplified in Teorem 2 as only N r real numbers are to be determined wereas in te original problem we needed to find N s complex numbers. Note tat, in practical systems, N s N r and N r is usually 1, 2, or 4 as typical relay nodes are assumed to be network users rater tan being extra middle-boxes to elp for cooperative communication. Remark 2. For te MISO case (N r = 1), from (26), it is easy to see tat w = w 1. Ten, to determine in (25), we only need to find γ 0. We sould igligt tat tis case was studied in [7], were te autors found a closed-form solution for γ 0, based on an approximated acievable rate. However, te optimization problem in [7] is solved for a fixed α, i.e., α = α 1, wereas in Lemma 2 we proved tat te optimal α can be 0 or α 1. In addition, te solution obtained for γ 0 is only for an approximated acievable rate in te ig SNR regime, wic tends to lose its accuracy wen SNR values become smaller. V. NUMERICAL RESULTS We evaluate te performance of te proposed approac for generating beamforming vectors and compare it wit te existing scemes in tis section. For te purposes of simulation, we will assume tat te cannels, H and g are complex Gaussian vectors wose entries are independent and identical distributed (i.i.d.) Gaussian random variables wit zero means and variances σ 2, σ2 H, and σ2 g, respectively. Unlike [7], we optimize te exact acievable rate given by (14), not just a ig SNR approximation of tat. All graps are based on averaging over 10,000 cannel realizations. We first focus on te MISO case and compare our results wit tose of [7], for N s = 4 and (σ 2, σ2 H, σ2 g) = ( 5, 4, 10)dB. In Fig. 2, we sow te effect of optimizing over α rater tan fixing it at α = α 1, wic is te case in [7]. Recall from Lemma 2 tat α {0, α 1 }. Hence, we coose between α = 0 or α = α 1, wicever coice gives a better rate. In Fig. 2, we take α 1 = 0.7 and direct transmission refers to te case wit α = 0. Tis figure clearly sows tat using te imum trust degree (α 1 ) is not always te best approac to optimize data rates, and, depending on te cannel conditions, te transmission rate may increase by using a finetuned α smaller tan α 1. As can be seen in Fig. 3, tere is a visible gap between our sceme and te one proposed in [7] for different values of N s at all SNRs, altoug tis gap reduces as N s increases or te relay link becomes stronger. In Fig. 4, we consider te MIMO case and compare te acievable rates for different values of N r. Expectedly, as te

6 Acievable Rate (bps/hz) Optimized α Ryu et al. BF [7] Direct transmission Acievable Rate (bps/hz) Proposed BF (Nr =4) Proposed BF (Nr =2) Proposed BF (Nr =1) Direct transmission Transmission SNR (db) Fig. 2. Acievable rates for te MISO case wit N s = 4, N r = 1, α 1 = 0.7, and (σ 2, σ2 H, σ2 g ) = ( 5, 4, 10)dB Transmission SNR (db) Fig. 4. Acievable rates for te MIMO case wit α 1 = 0.7, (σ 2, σ2 H, σ2 g) = ( 5, 0, 10)dB, N s = 4, and different N r s. Acievable Rate (bps/hz) Proposed BF Ryu et al. BF [7] Direct transmission Ns=8 Ns= Transmission SNR (db) Fig. 3. Acievable rates for te MISO case (N r = 1) wit α 1 = 0.7 and (σ 2, σ2 H, σ2 g ) = ( 5, 4, 10)dB. number of antennas increases, te acievable rate goes up. VI. CONCLUSION We ave developed a trust degree based beamforming design formulation for multi-antenna cooperative communication. Observing tat imum trust degree may not result in te best rate performance, we ave optimized te acievable rate over te trust degrees as well as te beamforming vectors at te source and relay nodes. We ave proved tat te optimal beamforming vector is a linear combination of te direct and cooperating links beamforming vectors, respecting te trust degree. Unlike existing works, te proposed sceme as been developed for te MIMO setting and is applicable to any SNR regime. For te MISO case, our sceme noticeably improves te acievable rate over tose of te existing scemes. REFERENCES [1] X. Cen, X. Gong, L. Yang, and J. Zang, A social group utility imization framework wit applications in database assisted spectrum access, in Proc. IEEE INFOCOM, pp , [2] Y. Li, S. Su, and S. Cen, Social-aware resource allocation for device-to-device communications underlaying cellular networks, IEEE Wireless Communications Letters, vol. 4, no. 3, pp , [3] F. Wang, Y. Li, Z. Wang, and Z. Yang, Social-community-aware resource allocation for D2D communications underlaying cellular networks, IEEE Transactions on Veicular Tecnology, vol. 65, no. 5, pp , [4] Y. Li, T. Wu, P. Hui, D. Jin, and S. Cen, Social-aware D2D communications: Qualitative insigts and quantitative analysis, IEEE Communications Magazine, vol. 52, no. 6, pp , [5] A. Ometov et al., Toward trusted, social-aware D2D connectivity: Bridging across te tecnology and sociality realms, IEEE Wireless Communications, vol. 23, no. 4, pp , [6] B. Zang, Y. Li, D. Jin, P. Hui, and Z. Han, Social-aware peer discovery for D2D communications underlaying cellular networks, IEEE Transactions on Wireless Communications, vol. 14, no. 5, pp , [7] J. Y. Ryu, J. Lee, and T. Q. Quek, Trust degree based beamforming for MISO cooperative communication system, IEEE Communications Letters, vol. 19, no. 11, pp , [8] J. Y. Ryu and W. Coi, Balanced linear precoding in decode-andforward based MIMO relay communications, IEEE Transactions on Wireless Communications, vol. 10, no. 7, pp , [9] J. P. Coon, Modelling trust in random wireless networks, in Proc. 11t IEEE International Symposium on Wireless Communications Systems, pp , [10] X. Gong, L. Duan, X. Cen, and J. Zang, Wen social network effect meets congestion effect in wireless networks: Data usage equilibrium and optimal pricing, IEEE Journal on Selected Areas in Communications, vol. 35, no. 2, pp , [11] X. Cen, B. Proulx, X. Gong, and J. Zang, Exploiting social ties for cooperative D2D communications: A mobile social networking case, IEEE/ACM Transactions on Networking, vol. 23, no. 5, pp , [12] X. Tang and Y. Hua, Optimal design of non-regenerative MIMO wireless relays, IEEE Transactions on Wireless Communications, vol. 6, no. 4, [13] K. Xiong, P. Fan, Z. Xu, H.-C. Yang, and K. B. Letaief, Optimal cooperative beamforming design for MIMO decode-and-forward relay cannels, IEEE Transactions on Signal Processing, vol. 62, no. 6, pp , [14] T. Lo, Maximum ratio transmission, IEEE Transactions on Communications, vol. 47, no. 10, pp , [15] Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zang, Semidefinite relaxation of quadratic optimization problems, IEEE Signal Processing Magazine, vol. 27, no. 3, pp , [16] K. T. Pan, S. A. Vorobyov, N. D. Sidiropoulos, and C. Tellambura, Spectrum saring in wireless networks via QoS-aware secondary multicast beamforming, IEEE Transactions on signal processing, vol. 57, no. 6, pp , [17] G. A. F. Seber, A Matrix Handbook for Statisticians. Hoboken, NJ: Jon Wiley & Sons, 2008.