Robust Multicast Beamforming for Spectrum Sharing-based Cognitive Radios

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1 1 Robust Multicast Beamforg for Spectrum Sharing-based Cognitive Radios Yongei Huang, Qiang Li, Wing-Kin Ma, Shuzhong Zhang Abstract We consider a robust donlink beamforg optimization problem for secondary multicast transmission in a multiple-input multiple-output (MIMO) spectrum sharing cognitive radio (CR) netork. The imization of transmit poer is formulated both qualityof-service (QoS) constraints on the secondary receivers and interference temperature constraints on the primary users, under the assumption of imperfect channel state information (CSI). The problem is a nonconvex quadratically constrained quadratic program (QCQP), and in general it is hard to achieve the global optimality. As a compromise, e present to randomized approximation algorithms for the problem via convex optimization techniques. Apart from the general setting of the robust beamforg problem, e identify one interesting special case, the robust problem of hich can be solved efficiently. Simulation results are presented to demonstrate the performance gains of the proposed algorithms over an existing robust design. Index Terms Robust multicast beamforg, MIMO cognitive radio netorks, spectrum sharing, semidefinite programg relaxation, imperfect channel state information. I. INTRODUCTION In cognitive radio (CR) netorks, the idea of spectrum sharing using multiple transmit antennas has dran much research interest recently. Spectrum sharing allos secondary and primary users to access the same channel simultaneously, by using the spatial degree of freedom at the secondary transmitter to avoid excessive interference to the primary users. For a comprehensive coverage of the recent advances, readers are referred to the magazine paper [1]; also [2], [3] for some recent specific orks on CR transmit beamforg. In this ork e are interested in a robust multicast 1 transmit beamforg problem in the setting of a multiple-input multiple-output (MIMO) CR netork, under the assumption of imperfect channel state information (CSI). A basic and meaningful problem formulation is to transmit poer quality-of-service (QoS) constraints on the secondary receivers and interference temperature constraints (or termed as CR interference limiting constraints) on the primary receivers. To proceed, let us first discuss some related orks. The primal multicast transmit beamforg frameork, i.e., that ithout CR and ith perfect CSI, as originally developed in [4]; see also the survey paper [5]. In particular, that paper advocated to use semidefinite programg (SDP) relaxation to handle the multicast transmit optimization problems, an idea that has received groing This ork is supported by a General Research Fund aarded by Research Grant Council, Hong Kong (Project No. CUHK415908) and by a Direct Grant aarded by the Chinese University of Hong Kong (Project Code ). Part of this ork as presented at the Thirty-Sixth International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Prague, Czech Republic, May 22-27, Y. Huang is ith the Department of Systems Engineering and Engineering Management, the Chinese University of Hong Kong, Shatin, Hong Kong ( yhuang@se.cuhk.edu.hk). Q. Li and W.-K. Ma are ith the Department of Electronic Engineering, the Chinese University of Hong Kong, Shatin, Hong Kong ( qli@ee.cuhk.edu.hk, kma@ieee.org). S. Zhang is ith the Program in Industrial and Systems Engineering, the University of Minnesota, Minneapolis, MN USA ( zhangs@umn.edu). 1 Here multicast, in a physical layer sense, refers to common information broadcast to a multitude of intended receivers. attention recently. Its robust version under imperfect CSI as later studied in [6], here an interesting connection beteen non-robust and robust beamforg problems is revealed. More recently, the frameork is extended to the CR scenario [3]. There, the robust CR multicast beamforg problem (our problem of interest) is also considered; the idea is to apply a conservative bound on both the QoS constraints and the interference suppressing constraints, thereby obtaining a quadratically constrained quadratic program (QCQP) formulation hich is subsequently approximated by SDP relaxation. In this paper, e propose to randomized approximation algorithms for the robust CR multicast donlink beamforg problem that can provide better approximation accuracies than the previous method [3]. Specifically, in one algorithm, e take into account an equivalent QCQP reformulation of the robust problem, of hich e obtain a parameterized SDP relaxation problem. The parameterized SDP can be solved by searching a one-dimensional parameter over an interval, and a feasibility checking routine using SDP; a Gaussian randomization procedure is presented to give approximate solutions of the robust problem in a neat ay, based on the solution of the parameterized SDP. In contrast, e herein improve both the problem reformulation (17) in [3] and the randomization procedure (cf. (30) in [3]), leading to better robust performance. In the other algorithm, e consider a convex (SDP) relaxation of the robust optimal beamforg problem resorting to S-lemma; e should note that S-lemma has been used in some other transmit beamforg contexts, e.g., [8], [9]. It turns out that the resulting SDP relaxation is looser than the previous parameterized SDP, giving rise to the possibility of returning loer transmit poer at a small cost of feasibility rate. In addition, e identify one particular, interesting, scenario, in hich the global optimum of the robust beamforg problem can be found efficiently. The particular case is hen there are not too many primary and secondary receivers involved, in hich e sho that the parameterized SDP relaxation is tight. The numerical simulation results sho the outperformance of the proposed beamformers over the robust design in [3]. The paper is organized as follos. In Section II, e introduce the system model and formulate the robust optimal beamforg problem. In Section III, e propose the randomized approximation algorithms, and point out one solvable scenario of the robust beamforg problem. In Section IV, e present numerical examples shoing the performance of three different algorithms. Finally, the paper is concluded in Section V. II. SYSTEM MODEL AND PROBLEM FORMULATION Consider a CR netork that has a secondary transmitter using N antennas to transmit common information to M secondary receivers, and that there are K primary users operating in the same spectrum. Let H m C N Nm be the MIMO channel from the secondary transmitter to the mth secondary receiver, here the number of receive antennas is denoted by N m. Similarly, let G k C N N k be the MIMO channel from the secondary transmitter to the kth primary user, here the number of receive antennas is denoted byn k. The signal received by secondary receiver m is given by x m(t) =

2 2 H H my(t)+n m(t), here y(t) C N is the secondary transmit signal vector, and n m(t) C Nm is Gaussian noise vector, assumed to have zero mean and covariance σmi. 2 2 The secondary transmitter employs the multicast transmit beamforg scheme, in hich e have y(t) = s(t) here C N is the beamformer eight and s(t) C is the information signal. We assume s(t) to be ith zero mean and unit variance, ithout loss of generality. Moreover, assug maximum ratio combining (MRC) receive beamforg for all the secondary receivers, the received SNR of secondary user m is SNR m = HH m 2, here denotes the Euclidean norm or σm 2 the Frobenius norm throughout the paper. The multicast beamformer design may be formulated as (cf. [3], [5]): H H H m 2 σ 2 mτ m, m = 1,...,M, G H k 2 η k, k = 1,...,K, here τ m specifies the imal QoS of the secondary user m, in terms of SNR, and η k specifies the maximal alloable interference level from the secondary transmitter to primary user k. Problem (1) has been considered in [3], here an effective approximation method called SDP relaxation has been applied. Herein e consider the imperfect CSI case. In practice, one may not have perfect knoledge of the CSI especially for the links of primary users. The CSI errors may be caused by inaccurate channel estimation, quantization in channel feedback, and outdated CSI effects. Let Δ m C N Nm and Δ k C N N k denote the CSI errors associated ith H m and G k, respectively. By assug that the errors Δ m and Δ k are deteristic norm-bounded, a orst-case robust version of (1) is given by (cf. problem (8) of [3]): s.t. H (1) (2a) (H m +Δ m) H 2 σmτ 2 m, m,(2b) Δ m ε m max Δ k ε k (G k +Δ k) H 2 η k, k, (2c) here ε m and ε k specify the bounds, or the orst-case magnitudes, of the CSI errors Δ m and Δ k, respectively. The robust beamforg problem (2) guarantees that for all admissible channel errors, all the secondary users must be served ith QoSs no less than the specification {τ m}, and the interferences to all the primary users must be kept belo {η k }. III. EFFICIENT ALGORITHMS FOR ROBUST BEAMFORMING PROBLEM Robust beamforg problem (2) is hard to solve in general, due to its nature of non-convexity. In this section, e ill propose to approximate solution schemes for the robust beamforg problem, and identify one interesting subclass of problem (2) that has a tight SDP relaxation. Thus that subclass of problem (2) can be solved in polynomial time, meaning that they are hidden convex programs. Let us start ith an equivalent non-convex QCQP reformulation of (2). 2 There are cases here interference from primary users to the secondary users contributes part of the noise terms n m(t). While e may not physically model n m(t) as being hite in those cases (except for N m = 1), e can transform the received model to an equivalent noise-hite model by prehitening. To describe it, suppose that the noise n m(t) has a positive definite, non-hite, covariance C m. Let x m(t) = C 1/2 m x m(t). We have x m(t) = H H my(t) + n m(t), here H m = H mc 1/2 m is the transformed channel and n m(t) = C 1/2 m n m(t) is hite. Note that in this setup, the secondary receivers can simply send back the transformed channel state information H m to the primary transmitter, rather than H m,c m hich incur a higher feedback overhead. A. An equivalent QCQP reformulation of robust optimal beamforg problem (2) Consider the first robust QoS constraint of (2b) in a slightly more general form, and set f 1() = E 1/2 1 Δ 1 ε 1 (H 1 +Δ 1) H 2, here E 1 0 governs the ellipsoid shape of the error set (or termed as the perturbation set in some robust optimization literature, e.g., [10]) and = 0. We claim that f 1() has a closed-form expression as stated in the folloing lemma (see related results in [11], [12], [13]). Lemma 3.1: It holds that f 1() = ( max { H H 1 ε 1 E 1/2 1,0}) 2. (3) Proof: See Appendix A. We remark that the optimal value f 1() remains unchanged if E 1/2 1 Δ 1 is changed to the spectral norm (the maximal singular value) from the Frobenius norm. Like the proof in Lemma 3.1, the maximization problem in the first interference limiting constraint of (2c) has the optimal value ( G H 1 +ε 1 ) 2. It thus follos that the robust beamforg problem (2) can be recast into s.t. H (4a) H H m σ m τm +ε m, m = 1,...,M, (4b) G H k η k ε k, k = 1,...,K. (4c) While problem (4) exhibits a similar form as (17) in [3], e ould point out that problem (4) admits a larger feasible region (and thus gives loer transmission poer) than (17) in [3]. To see this, e use the same configuration in [3], namely, setting H m = h m and G k = g k (i.e. N m = N k = 1, m,k). Hence, (4b) reduces to ( h H m ε m ) 2 σmτ 2 m, m. Observe that (17b) in [3] is given by H Hm := H ( h mh H m +ε m (ε m 2 h Hmh m ) ) I σmτ 2 m, m. It is straightforard to check that ( h H m ε m ) 2 H Hm, m. This implies that any fulfilling (17b) in [3] satisfies (4b). Similarly, one can sho that any satisfying (17c) in [3] fulfills (4c). It is knon that problem (4) is NP-hard [4] (in fact, problem (4) has been proved NP-hard, hen ε m = 0, m, and G k = 0, ε k = 0, k). Instead, one may resort to efficiently finding a suboptimal (or approximate) solution (e.g., see [3], [6], [4]). In the folloing, e ill propose randomized, SDP-based, methods for generating an approximate solution of the robust beamforg problem (2), as ell as presenting some efficiently solvable scenario of problem (2). B. A randomized approximation algorithm for the robust beamforg problem ith ball perturbation Problem (4) is tantamount to the folloing QCQP problem, t t 2 H H m σ m τm +ε mt, m = 1,...,M, G H k η k ε kt, k = 1,...,K, = t. (5) Note that any feasible point (,t) of (5) must satisfy λ max(h mh H m) HH m σm τm +ε m, m (6) t

3 3 and λ (G k G H k ) GH k ηk ε t k, k. (7) In fact, the first inequality in (6) follos from the basic property 2 λ (A) H A 2 λ max(a) for a Hermitian matrix A, and the second inequality in (6) is due to the feasibility in (5). λ max(h mh H m) ε m > 0 Likeise, (7) is derived. Suppose that for all m (otherise problem (5) ould be infeasible). It follos that a necessary condition for t to be feasible for (5) is t 0 t t 1, (8) here the loer bound t 0 and the upper bound t 1 are respectively given by σ m τm t 0 = max (9) 1 m M λ max(h mh H m) ε m and t 1 = 1 k K ηk λ (G k G H k )+ε. (10) k Then problem (5) indeed amounts to the folloing problem W, t t (11a) s.t. H mh H m W (σ m τm +ε mt) 2, m = 1,...,M, (11b) G k G H k W ( η k ε kt) 2, k = 1,...,K, I W = t 2, W ર 0, Rank(W) = 1, t 0 t t 1, (11c) (11d) (11e) here A B = tr(ab H ). Dropping the rank-one constraint yields the relaxation problem W, t t (11b), (11c), (11d) satisfied, W ર 0, t 0 t t 1. (12) Fixing t, problem (12) is an SDP feasibility problem. No, let g(t) be the optimal value of such a feasibility problem. In other ords, e have g(t) = t if (12) is feasible for a given t (any feasible W is thus optimal), and g(t) = + if it is infeasible at point t. Therefore, (12) amounts to the one-dimensional optimization problem: g(t) t t 0 t t 1. (13) In other ords, (12) can be solved by solving (13): fixing t, solving the SDP feasibility problem (obtaining g(t)), and reducing t iteratively. In the optimization literature there are some derivative-free methods for solving the one-dimensional optimization problem (13). One of these methods is called compass or coordinate search (cf. [14, Algorithm 3.1 and Section 8.1], [15, Algorithm 7.1]). In practice, e adopt either the uniform sampling or the Matlab function fbnd, in order to output a satisfactory solution. Once such a solution (W,t ) of (12) is obtained, e retrieve a rank-one approximate solution of (12) by making use of W. Particularly, a randomization procedure is proposed as follos: Take random vectors i, i = 1,...,I, from the complex normal distribution N C (0,W ), and compute { H H m i σ m τm ηk G H k i λ( i) =,, ε m ε k m = 1,...,M, k = 1,...,K}.(14) Clearly, if i λ( i), then ( i, i ) is feasible for (5). Summarizing, a randomized approximate solution of problem (4) (or equivalently (5)) can be generated by Algorithm 1. Algorithm 1 Gaussian randomization procedure for robust beamforg problem (4) Input: H m, G k, σ m, τ m, ε m, η k, ε k, I; Output: a randomized approximate solution of (4); 1: solve (13), and find an optimal solution (W,t ); 2: if Rank(W ) = 1, then output ith H = W and terate; 3: dra random vectors i N C (0,W ), i = 1,...,I, and compute λ( i) by (14); 4: pick up i0 such that i 0 = arg{ i : i λ( i),i = 1,...,I}. C. A convex relaxation for robust beamforg problem (2) by Slemma In this subsection, e study an SDP relaxation of the robust beamforg problem (2), resorting to S-lemma (e.g., see [10, Appendix B.2]). Let δ m = vec(δ m), h m = vec(h m), and g k and δ k are defined similarly. Problem (2) can be reformulated equivalently into the folloing problem s.t. I W δ H m(i W)δ m +2R(h H m(i W)δ m)+h H m(i W)h m σ 2 mτ m, δ m 2 ε 2 m, m, (δ k) H (I W)(δ k)+2r(g H k (I W)δ k)+ g H k (I W)g k η k, δ k 2 (ε k) 2, k, W = H. (15) Here e use the fact that tr(a H BC) = vec(a) H (I B)vec(C), and ignore the dimension of each I since the dimension is clear in the context. The so-called S-lemma (e.g., see [10]) says that the condition x H Bx+2R(b H x)+β 0, x H Ax+α 0 is equivalent to the linear matrix inequality (LMI): [ ] B λa b λ 0 : b H ર 0, β λα provided that some Slater condition holds. By applying S-lemma to the constraints of (15) (e.g., for the first constraint of (15), setting B = (I W), b H = h H 1 (I W), β = h H 1 (I W)h 1 σ 2 1τ 1, A = I, α = ε 2 1), e can reexpress the constraints respectively into LMI constraints; therefore, (15) is reformulated to the relaxed problem (removing the rank-one constraint on W ): I W,{μ m},{λ k } (16a) [ I W +μmi (I W)h m h H m(i W) h H m(i W)h m σmτ 2 m μ mε 2 m [ I W +λk I ર 0,m = 1,...,M, (16b) (I W)g k ] g H k (I W) gh k (I W)g k η k λ k (ε k) 2 W ર 0, μ m 0, m, λ k 0, k. 0,k = 1,...,K, (16c) (16d) Note that (16) is a convex relaxation of (2), hile problem (12) is a non-convex relaxation of problem (4) (or equivalently, (2)). We have shon in the last subsection that even though (12) is not a convex ]

4 4 optimization problem, the global optimality of it may be achieved by solving a one-dimensional optimization problem over an interval. To discuss some relations beteen the to relaxations, e have the folloing observations. Proposition 3.2: It holds that 1) if (W, I W) is feasible for (12), then W, together ith some μ m 0 and λ k 0, is feasible for (16); 2) if ( H,{μ m},{λ k }) is feasible for (16), then ( H, ) is feasible for (12). Proof: See Appendix B. The proposition indicates that the convex relaxation (16) is not as tight as the non-convex relaxation (12) in general (namely, (16) alays gives a loer bound of (12)). As to the second argument of the proposition, e provide an alternative proof ithout using Lemma 3.1 and S-lemma, notithstanding the fact that if (16) has an optimal rank-one solution H, then problems (16) and (2) are equivalent due to S-lemma, and problems (2) and (12) are equivalent due to Lemma 3.1. Comparing to (12), SDP problem (16) is solved in a single step (unlike (12) resorting to solving one-dimension optimization problem (13)), and e take advantage of it to solve the robust beamforg problem (2). Precisely, the advantages of the relaxation (16) are threefold: (i) if e get a rank-one optimal solution H (16), then there is no gap beteen (2) and (16), and thus e do not have to iteratively solve (13); (ii) in case of getting a solution W of rank to or higher for (16), the optimal value of (16) can serve as a ne t 0 for solving (13); namely, update t 0 := max{t 0, I W }; (iii) it is possible to use W as a covariance in order to get an approximate solution for (4), according to (14). We point out that the observation in the third point is interesting. In fact, it is not obvious ho to generate, from W, a beamforg vector so that H satisfies the SDP constraints (16b), (16c). Nevertheless, according to (14), the generated beamforg vector from W fulfills (4b), (4c), thus satisfies (2b), (2c), hich in turn implies that H is feasible for (16b), (16c). As a consequence, e conclude that the randomized approximation algorithm for (4) via the SDP relaxation (16) consists of solving the SDP (16) (obtaining an optimal solution (W,{μ m},{λ k})), and steps 2-4 of Algorithm 1. Note that the complexities of to approximation algorithms are doated by solving the respective SDP relaxation problems. In Algorithm 1, the cost of outputting a solution by t-search is about 20 times empirically of solving an SDP feasibility problem, hich has orst-case complexity of O((max{M+K,N}) 4 N 0.5 log(1/ζ)) for a given accuracy ζ (see [7]); in the algorithm via (16), the computational cost is higher since the sizes of the involved SDP cones are quite large; for instance, in the particular case of N m = N k = 1 m,k, the complexity is up to of O(N 6.5 log(1/ζ)) for a small (M +K) (see [16]). D. Solvable subclass of robust beamforg problem via SDP relaxation In this subsection, e shall elaborate that robust beamforg problem (2) can be solved efficiently ith parameters such that M +K = 3 and N 3 (i.e., the number of primary and secondary receivers equals three and the number of the transmit antennas is not less than three), or ith parameter condition M +K = 2. Let t be a numerical r for g(t) over the interval [t 0,t 1] as in problem (13), and W be a corresponding feasible solution, namely, (W,t ) complies ith the constraints of (12). To proceed, let us assume M = 2 and K = 1 ithout loss of generality. Then, it follos that H mh H m W (δ m τm +ε mt ) 2, m = 1,2, G 1G H 1 W (η 1 ε 1t ) 2, I W = (t ) 2. It is verified immediately that the conditions of the specific rank-one matrix decomposition theorem 2.3 of [17] are satisfied, and thus one is able to polynomially construct a matrix H according to the rank-one decomposition theorem, such that H H mh H m = H mh H m W, m = 1,2, H G 1G H 1 = G 1G H 1 W, 2 = I W. This implies that ( H,t ) is feasible for (12); thus (,t ) is feasible for (5). Therefore, e conclude that is optimal for (5) since the problem shares the same optimal value t ith its relaxation problem (12). For the scenario ith parameters fulfilling M +K = 2, it follos from the specific rank-one theorem of [18] that a rank-one matrix H can be discovered efficiently so that (H 1H H 1 (δ1 τ1 +ε 1t ) 2 (t ) 2 I) H = 0, (G 1G H 1 (η1 ε 1t ) 2 (t ) 2 I) H = 0. Then, it is seen that = t is feasible for (5), and the objective function value is the same as the optimal value of its relaxation problem (12), hence is optimal. IV. SIMULATION RESULTS We consider a CR netork ith a three-antenna secondary transmitter, five single-antenna secondary receivers and to single-antenna primary receivers (i.e., N = 3, M = 5, K = 2, and N m = N k = 1, m, k). The elements of the channels (from the secondary transmitter to either the primary or the secondary users) are assumed to be i.i.d. complex Gaussian distributed ith mean 0 and variance 1. We fix the secondary receivers noise variance σ 2 m = 1 m. Fig. 1 (a) exaes ho the average transmit poer is affected by the radius of the channel perturbation set. In the simulation, e set τ m = 10 db m and η k = 0 db k. The same channel perturbation level is assumed for all primary and secondary channels, i.e., ε m = ε k = ε, m,k. A total of 3000 channel realizations (each ith10000 Gaussian randomizations) are tested. We compare our to proposed robust beamforg designs, namely t-search (problem (5)) and S-lemma (problem (16)) designs, ith an existing robust design provided by problem (17) of [3]. Moreover, as the robust poer imization problem ith QoS constraints could be intrinsically infeasible, the average transmit poer in Fig. 1 (a) is obtained by averaging only those channel realizations for hich all the three robust designs are feasible, i.e., at least one feasible beamforg solution can be found for each design after randomization procedure. In the legend, the beamformer stands for the result after randomization, hile SDP relaxation value means the optimal value of the SDP relaxations corresponding to the three robust designs. A practically logical result e see from Fig. 1 (a) is that higher transmit poer is required to assure larger radius of the channel error set (i.e., provide more robust beamformer). Fig. 1 (a) also shos that the average transmit poers by our proposed robust beamformers are loer than that by (17) of [3] in general. This means that the former methods are less conservative than the latter. Let us compare the performance of our to robust proposed designs. In Fig. 1 (a), for the SDP relaxation values, e note that S-lemma yields a slightly loer value than t-search, hich is consistent ith our claim in Prop. 3.2, i.e., the relaxation (16) (S-lemma) is looser than (12) (t-search). For the beamformer s poer, e see that S-lemma leads to slightly better performance than t-search. As observed, the performance gap of our to algorithms hoever is not big. This phenomenon may be caused

5 5 by the precision of the relaxed solution, the approximation procedure employed and the simulation settings. To get a better understanding of the conservativeness, Fig. 1 (b) plots the feasibility rate of the three designs ith the same setup as Fig. 1 (a). Here the feasibility rate is denoted as the ratio of the number of channel realizations, for hich e can generate a feasible beamformer via randomization, over the total 3000 channel trials. It can be seen from Fig. 1 (b) that the proposed to robust designs have much higher feasible rates than that of [3] over the hole perturbation radii tested. In Fig. 1 (b), e also observe that t-search method yields slightly higher feasibility rate than S-lemma. In contrast, as Fig. 1 (a) shos, S-lemma design has superior performance in terms of the SDP relaxation values and the transmit poer of beamformers. In other ords, there is a tradeoff beteen the to proposed robust designs. Fig. 2 investigates ho the average transmit poer changes ith the QoS of the secondary users and the interference level requirement of the primary users. We fix ε m = ε k = 0.04, k,m, τ m = 10dB, m, and η k = 0dB k, if not mentioned. Fig. 2 (a) plots the average transmit poer versus the QoS of the secondary users of the to proposed robust designs. To provide a reference for the transmit poer loer bound, e also plot the result of perfect CSI (i.e., ε m = ε k = 0, k,m). It can be seen that as τ increase, e need to use more transmit poer to support higher secondary users QoS requirements. Also there is marginal performance difference beteen the t-search and the S-lemma based robust designs. Fig. 2 (b) shos the average transmit poer versus the primary users interference level requirement. As expected, a loose interference temperature requirement leads to lo average transmit poer, and vice versa. Transmit poer (db) Feasibility rate beamformer by t search beamformer by (17) of [3] beamformer by S lemma SDP relaxation value of t search SDP relaxation value of (17) of [3] SDP relaxation value of S lemma ε (a) beamfomer by t search beamfomer by S lemma beamfomer by (17) of [3] 0.2 V. CONCLUSION We have considered a robust secondary multicast beamformer design problem for spectrum sharing in a MIMO CR netork. To efficient algorithms for the robust problem have been proposed: one algorithm includes solving a one-dimensional optimization problem, checking the feasibility of SDPs, and a Gaussian randomized procedure; the other algorithm resorts to S-lemma leading to an SDP relaxation problem and a randomization procedure. In the special case of not too many primary and secondary receivers (cf. Sec. III-D), e have also proved that the robust optimal beamforg problem can be solved efficiently. The performance of the proposed beamforg designs has been demonstrated by simulations. Future research topics may include efficiently robust design of multi-group and multicast beamformer for a CR netork. A. Proof of Lemma 3.1 APPENDIX Proof: When H H 1 ε 1 E 1/2 1, e select Δ 1 = E 1 E 1/2 1 1 H H 1. It is easily verified that E1/2 2 1 Δ 1 = H H 1 / E 1/2 1 ε 1 and (H 1 +Δ 1) H = 0. Suppose H H 1 > ε 1 E 1/2 1. It then follos that (H 1 +Δ 1) H H H 1 (E 1/2 1 Δ 1) H E 1/2 1 H H 1 E 1/2 1 Δ 1 E 1/2 1 H H 1 ε 1 E 1/2 1 > 0, and the inequality chain become equality chain hen Δ 1 = ε 1E 1 1 H H 1/( H H 1 E 1/2 1 ) (it is seen that E 1/2 Δ 1 = ε 1) ε (b) Fig. 1. (a) Average transmit poer versus the radius of channel perturbation set. (b) Feasibility rate versus perturbation radius ε. B. Proof of Proposition 3.2 Proof: (1) Since (W, I W) is feasible for (12), hence H mh H m W (σ m τm+ε m I W) 2, m, hich means that W 1/2 H m ε m W 1/2 σ m τm, m. (17) Likeise e have W 1/2 G k + ε k W 1/2 η k, k. Observe that W 1/2 (H m + Δ m) W 1/2 H m W 1/2 Δ m W 1/2 H m ε m W 1/2, for Δ m : Δ m ε m. Therefore, it follos from (17) that σmτ 2 m Δ m ε m tr((h m + Δ m) H W(H m + Δ m)). Similarly, it has η k max Δ tr(g k +Δ k) H W(G k +Δ k). k ε k By S-lemma, e conclude that (W,{μ m},{λ k }) is feasible for (16) for some μ m 0 and some λ k 0, m = 1,...,M and k = 1,...,K. (2) Let us re-denote δ m = vec(δ m), h m = vec(h m); δ k and g k are defined analogously. Suppose that ( H,{μ m},{λ k }) is feasible for (16). Let us look into the first constraint. Suppose that μ 1 > 0. It follos from the first constraint of (16) and Schur complement lemma that h H 1 (I H )h 1 σ 2 1τ 1 μ 1ε 2 1 h H 1 (I H )(I H +μ 1I) 1 (I H )h 1. (18) It is straightforard to verify that (I H +μ 1I) 1 = 1 μ 1 (I I H μ ) by noting that (I H )(I H ) = 2 (I

6 6 Transmit poer (db) Transmit poer (db) beamformer by t search beamformer by S lemma beamformer ith perfect CSI τ (db) (a) beamformer by t search beamformer by S lemma beamformer ith perfect CSI η (db) (b) Fig. 2. (a) Average transmit poer versus the QoS of the secondary users. (b) Average transmit poer versus the interference level requirement of the primary users. H ), and to check that the right-hand side of (18) is equal to 2 μ (I H )h 1 2. It follos from (18) that (I H )h 1 2 (1+ 2 μ 1 )(σ 2 1τ 1 +μ 1ε 2 1) = ε σ 2 1τ σ 2 1τ 1 μ 1 +μ 1ε 2 1 ε σ 2 1τ 1 +2 σ 1ε 1 τ1, hich is equivalent toh 1H H 1 H (σ 1 τ1+ε 1 I H ) 2. In ords, for any μ 1 > 0, e have { [ ] I H +μ 1I (I H )h 1 : h H 1 (I H ) h H 1 (I H )h 1 σ1τ 2 1 μ 1ε 2 1 ર 0} { : H 1H H 1 H (σ 1 τ1 +ε 1 I H ) 2 }. By a limiting argument, the inclusion relation still holds for μ 1 = 0. This means that H fulfills the first constraint of (12). Similarly, e can sho that H also fulfills the second to the M-th constraints. No let us deal ith the (M+1)-th constraints of (16). Due to the feasibility of( H,{μ m},{λ k }), e see thati H +λ 1I 0, hich means H +λ 1I 0, hich in turn implies λ 1 2. In a similar ay, e can sho that H satisfies the second set of constraints of (12). REFERENCES [1] R. Zhang, Y.-C. Liang, and S. Cui, Dynamic resouce allocation in cognitive radio netorks, IEEE Signal Processing Magazine, vol. 27, no. 3, pp , May [2] Y. J. Zhang and A. M.-C. So, Optimal spectrum sharing in MIMO cognitive radio netorks via semidefinite programg, IEEE Journal on Selected Areas in Communications, vol. 29, no. 2, pp , February [3] K. T. Phan, S. A. Vorobyov, N. D. Sidiropoulos, and C. Tellambura, Spectrum sharing in ireless netorks via QoS-aare secondary multicast beamforg, IEEE Transactions on Signal Processing, vol. 57, no. 6, pp , June [4] N. Sidiropoulos, T. D. Davidson, and Z.-Q. Luo, Transmit beamforg for physical-layer multicasting, IEEE Transactions on Signal Processing, Vol. 54, No. 6, pp , June [5] A. B. Gershman, N. D. Sidiropoulos, S. Shahbazpanahi, M. Bengtsson, and B. Ottersten, Convex optimization-based beamforg: From receive to transmit and netork designs, IEEE Signal Processing Magazine, vol. 27, no. 3, pp , May [6] E. Karipidis, N.D. Sidiropoulos, and Z.-Q. Luo, Convex transmit beamforg for donlink multicasting to multiple co-channel groups, Proceedings of IEEE ICASSP, pp , [7] Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, Semidefinite relaxation of quadratic optimization problems: From its practical deployments and scope of applicability to key theoretical results, IEEE Signal Processing Magazine, vol. 27, no. 3, pp , May [8] G. Zheng, K.-K. Wong, and B. Ottersten, Robust cognitive beamforg ith bounded channel uncertainties, IEEE Transactions on Signal Processing, vol. 57, no. 12, pp , December [9] G. Zheng, K.-K. Wong, and T.-S. Ng, Robust Linear MIMO in the Donlink: A Worst-Case Optimization ith Ellipsoidal Uncertainty Regions, EURASIP Journal on Advances in Signal Processing, vol. 2008, Article ID , 15 pages, [10] A. Ben-Tal, L. E. Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, Ne Jersey, [11] S.A. Vorobyov, A.B. Gershman, and Z.-Q. Luo, Robust adaptive beamforg using orst-case performance optimization: a solution to the signal mismatch problem, IEEE Transactions on Signal Processing, vol. 51, no. 2, pp , February [12] R. G. Lorenz and S. P. Boyd, Robust imum variance beamforg, IEEE Transactions on Signal Processing, vol. 53, no. 5, pp , May [13] S.A. Vorobyov, A.B. Gershman, Z.-Q. Luo, and N. Ma, Adaptive beamforg ith joint robustness against mismatched signal steering vector and interference nonstationarity, IEEE Signal Processing Letters, vol. 11, no. 2, pp , [14] T. G. Kolda, R. M. Leis, and V. Torczon, Optimization by direct search: ne perspectives on some classical and modern methods, SIAM Revie, vol. 45, no. 3, pp , [15] A. R. Conn, K. Scheinberg, and L. N. Vicente, Introduction to Derivative-Free Optimization, MPS-SIAM Series on Optimization, Philadelphia, [16] A. Nemirovski, Lectures on Modern Convex Optimization, Class Notes, Georgia Institute of Technology, Fall [Online]. Available nemirovs/lect ModConvOpt.pdf [17] W. Ai, Y. Huang, and S. Zhang, Ne results on Hermitian matrix rankone decomposition, Mathematical Programg: Series A, vol. 128, no. 1-2, pp , June [18] Y. Huang and S. Zhang, Complex matrix decomposition and quadratic programg, Mathematics of Operations Research, vol. 32, no. 3, pp , 2007.