RATE OPTIMIZATION FOR MASSIVE MIMO RELAY NETWORKS: A MINORIZATION-MAXIMIZATION APPROACH

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1 RATE OPTIMIZATION FOR MASSIVE MIMO RELAY NETWORKS: A MINORIZATION-MAXIMIZATION APPROACH Mohammad Mahdi Nahsh, Mojtaba Soltanalian, Petre Stoica +, Maryam Masjedi, and Björn Ottersten Department of Electrical and Computer Enineerin, Isfahan University of Technoloy, Isfahan, Iran Department of Electrical and Computer Enineerin, University of Illinois at Chicao + Department of Information Technoloy, Uppsala University, Uppsala, Sweden Interdisciplinary Centre for Security, Reliability and Trust SnT, University of Luxembour ABSTRACT We consider the problem of sum-rate imization in massive MIMO two-way relay networks with multiple communication operators employin the amplify-and-forward AF protocol. The aim is to desin the relay amplification matrix i.e., the relay beamformer to imize the achievable communication sum-rate throuh the relay. The desin problem for the case of sinle-antenna users can be cast as a nonconvex optimization problem, which in eneral, belons to a class of NP-hard problems. We devise a method based on the minorization-imization technique to obtain quality solutions to the problem. Each iteration of the proposed method consists of solvin a strictly convex unconstrained quadratic proram; this task can be done quite efficiently such that the suested alorithm can handle the beamformer desin for relays with up to 70 antennas within a few minutes on an ordinary PC. Such a performance lays the round for the proposed method to be employed in massive MIMO scenarios. Index Terms Beamformin, minorization-imization, massive MIMO, relay networks, sum-rate. 1. INTRODUCTION Sum-rate imization is a fundamental task arisin in sinal desin for communication, and particularly relay networks, in which relays are often used to enhance the quality of communication between pairs of users within the network. In such networks, two-way relayin is shown to achieve better spectral efficiency as compared to one-way relayin 1]. Various protocols includin decode-and-forward DF, and amplifyand-forward AF have been proposed in the literature for two-way relay networks 2, 3]. Contrary to the DF case, the AF relayin does not perform any sinal decodin at the relay, and hence enjoys a lower hardware and software complexity, as well as smaller transmission delay. As a result of such simple processin requirement, AF relayin is a more suitable scheme for lare-scale or massive MIMO systems. *Please address all the correspondence to Mohammad Mahdi Nahsh, Phone: ; mm nahsh@cc.iut.ac.ir Note that the sum-rate of a MIMO relay system depends on the amplification matrix, i.e. the beamformer of the relay. However, an optimal desin of the beamformer leads to a nonconvex in eneral NP-hard 1] optimization problem. The authors of 1] developed a polynomial-time iterative method based on a semidefinite relaxation referred to as POTDC to tackle the problem. POTDC uarantees a rank-one solution only for the special case of sinle communication operator and hence, its solution is enerally associated with a synthesis loss. Furthermore, each iteration of POTDC consists of solvin a convex MAXDET optimization that has a lare computational burden. On the other hand, POTDC results outperform those obtained by the approximate projection-based alorithm suested in 4]. Additionally, 1] includes heuristic alorithms based on one/two dimensional search for the case with sinle operator. In the case of an arbitrary number of operators, the literature does not offer efficient methods that can lead to some stron type of optimality of the obtained solutions. Furthermore, most of the proposed methods in the literature are merely suitable for small scale problems see e.. 1, 5]. In this paper, the problem is considered in a rather eneral form enablin the user to freely choose the number of operators L and the structure of the associated matrices i.e., the channel parameters. We devise an iterative method based on the minorization-imization technique to tackle the desin problem. Applyin the proposed method increases the value of the objective function at each iteration. Consequently, it can be shown that the obtained solution is a stationary point of the problem for arbitraryl. The proposed method is computationally efficient and hence can be applied to lare-scale MIMO systems 1 with M R antennas. Indeed, each iteration of the devised method consists of solvin a convex unconstrained quadratic proram QP; which can be performed efficiently for instance with an On 2.3 complexity where n is the problem dimension, n = MR 2 6]. As a result, the 1 This paper can address the beamforer desin problem in lare-scale scenarios where the near optimality of zero-forcin does not hold, e.., lowmiddle reime massive MIMO systems.

2 method can handle problems with n 10 3 variables i.e., M R 70 on an ordinary PC within a few minutes. 2. PROBLEM FORMULATION We consider a MIMO AF two-way relay network consistin ofm R antennas,loperators and pairs of user terminals. We assume sinle-antenna user terminals and flat fadin channels between the k th user of the l th operator and the relay, which are denoted by{h k,l } 1]. The received sinal at the relay can be expressed as 1, 4], r = h k,l x k,l +n R 1 where x k,l is the transmitted symbol by the k th user of the l th operator with power p k,l iven by E{ x k,l 2 }, and n R denotes the circularly symmetric white Gaussian noise with covariance matrixσr 2 I at the relay. By employin the AF protocol, the transmit sinal of the relay is iven by r = Gr with G C MR MR bein the relay amplification matrix, which is to be desined. We assume reciprocal channels between the relay and users 4]; thus, the received sinal y k,l of the k th user at the l th operator becomes y k,l = h T k,l r+n k,l 2 where n k,l is the associated white noise component with varianceσ 2 k,l and.t stands for transpose. The sum-rate of the system can be formulated as 4] R sum = 1 2 lo 2 1+η k,l. 3 Herein η k,l denotes the sinal-to-interference-plus-noise ratio SINR for the k th user of the l th operator and it has the followin expression 4] H Φ k,l η k,l = H Υ k,l + k,l +σk,l 2 where = vecg that vec. operator stacks the columns of a matrix into a vector,. H stands for Hermitian transpose, and the matricesφ k,l,υ k,l, k,l are defined as Φ k,l = p k,l h T 3 k,l h T H k,l h T 3 k,l h T k,l Υ k,l = k H p k, l h T k, l h T k,l h T k, l h T k,l l l k,l = σ 2 R IMR h k,l h T k,l. 4 5 The sum-rate imization is constrained via the total available powerp R at the relay, viz. E{ r 2 2 } = tr{e{grrh G H }} 6 = p k,l Gh k,l 2 2 +σr G 2 2 F P R wherein. 2 and. F denote the Euclidean norm of the vector and the Frobenius norm of the matrix aruments, respectively. The latter equation can be expressed with respect to w.r.t. as H C P R where C = σ 2 RI M 2 R + p k,l h k,l h H k,l T I MR. 7 Therefore, the desin problem i.e., sum-rate imization in MIMO AF relay networks withloperators can be cast as 1 lo H Φ k,l H Υ k,l + k,l +σk,l 2 s. t. H C P R. 8 Note that the inequality constraint in the above problem is active i.e. satisfied with equality at the optimal point. More precisely, assume that is an optimal solution to 8 with H C = P 0 < P R. Then a scaled version of which satisfies the constraint with equality, i.e. 1 = P R /P 0, will lead to a larer objective value which is a contradiction. 3. SUM-RATE MAXIMIZATION The aim is to desin the AF amplification matrix G in order to imize the sum-rate R sum. Considerin the fact that the inequality constraint in 8 is satisfied with equality at the optimal solution, the optimization in 8 can be recast as lo H A k,l lo H B k,l ] 9 where we have used the followin definitions: B k,l = Υ k,l + k,l + σ2 k,l P R C, A k,l = B k,l +Φ k,l 10 The above optimization problem is non-convex and belons to a class of NP-hard problems in eneral 1]. Note that the objective function of 9 is invariant with respect to scalin; therefore, we can deal with the unconstrained problem and then scale the solution such that it satisfies the constraint H C = P R. In this paper, we use the minorizationimization technique to tackle the non-convex desin problem formulated in 9. Minorization-imization MaMi is an iterative technique that can be used for obtainin a solution to the eneral imization problem 7, 8]: z fz subject to cz Each iteration of MaMi consists of two steps: Minorization Step: Findin p κ z such that its imization is simpler than that of fz and p κ z minorizes fz, i.e., p κ z fz, z, p κ z κ 1 = f z κ 1

3 with z κ 1 bein the value of z at the κ 1 th iteration. Maximization Step: Solvin the followin optimization problem to obtainz κ : z p κ z subject to cz 0 where Q κ = q κ = ] B k,l κ H Bk,l +U κ k,l, 15 b k,l 2U k,l κ], Note that the followin inequality holds due to the concavity oflox for all x,x 0 R + : lox lox x 0 x x Settin x 0 = H 0 B k,l 0 and x = H B k,l leads to a minorizer for lo H B k,l. By substitutin the minorizer into 9, we have the followin imization problem at the κ+1 th iteration: lo H A k,l ] 1 κ H B k,l κ H B k,l. 13 Inspired by the rich literature on semidefinite relaxation, we note that by considerin X = H as the optimization variable in 13 and droppin the rank-1 constraint, a convex alternative of 13 can be obtained at each iteration. However, there is no uarantee for a rank-1 solutionx, and hence, this approach is associated with a synthesis loss. In addition, applyin the relaxation leads to iteratively solvin a MAXDET problem possessin a hih computational burden. Instead, in the sequel, we devise a computationally efficient method that increases the objective value at each iteration and uarantees the first-order optimality condition for the solution. To this end, we proceed by findin a minorizer for the term lo H A k,l as a function of usin the followin lemma whose proof is omitted for the sake of brevity. Lemma 1. Let sx = lox H Tx and x H Cx = P for positive-definite matrices T,C in C N N, and P R +. Then, the followin inequality holds x,x 0 : sx sx 0 +R b H x x 0 +x x 0 H Ux x 0 2 4P where b = Tx x H 0, U = I, 0 Tx0 w +ǫ w 1 H 1 is Cw1 the principal eienvector of T, and ǫ > 0 bein an arbitrary scalar. Assume that H C = P R at each iteration see Remark 1 below. Now by usin Lemma 1 for minorizin the objective of 13, the followin unconstrained QP will be obtained: min H Q κ +R q κ H 14 b k,l = U k,l = 2 A κ H k,l κ, 16 Ak,l κ 4P R w k,l H C w +ǫ I k,l and w k,l denotes the principal eienvector of A k,l. Note that B k,l 0, and also, U k,l 0 as it is a scaled version of identity matrix I with a positive scalar. Therefore, the matrix Q κ is positive-definite at each iteration. Consequently, the problem in 14 is strictly convex w.r.t.. The unique solution to this optimization is obtained by solvin the system of linear equations2q κ +q κ = 0, viz. = 1 2 Q κ+1 1 q κ. 17 Remark 1: Note that the above solution does not necessarily satisfy the constraint H C = P R of the oriinal problem 9 at each iteration. As mentioned before, we can scale the obtained solution at the converence to deal with this issue as the objective function in 9 is scale invariant. However, the derivation of the matrix U k,l in Lemma 1 requires the satisfaction of the constraint at each iteration. Therefore, we need to scale the obtained at each iteration such that H C = P R. Note also that the scalin does not affect the converence of the sequence of the objective function values. Table 1 summarizes the steps of the proposed method for relay beamformer desin to imize the communication sum-rate. The suested method improves the value of the sum-rate at each iteration. As a result, employin the proposed method will lead to the converence of the network sum-rate value due to the upper boundedness of the sum-rate metric see 7 9] and references therein for details of the converence of MaMi technique. 4. SIMULATIONS In this section, the performance of the proposed method is evaluated via Monte-Carlo simulations. An AF based bidirectional MIMO relay network with L operators and M R antennas at the relay is considered. The variances of the Gaussian noises for the relay and users are assumed to be equal, i.e., σr 2 = σ2 k,l = σ2 n. We assume that the transmit powers of the relay and users are identical, i.e., P R = p k,l = p. The SNR is defined as p/σn 2. Moreover, the normalized distance

4 Table 1: Relay Beamformer Desin Alorithm 15 Step 0: Initialize with a random vector in C M2 R and scale it such that H C = P R ; setκ = 0. Step 1: Compute Q κ and q κ usin 15. Step 2: Solve the convex problem in 14 usin either the closed-from expression 17 or the direct methods to obtain κ+1. Step 3: Scale the obtained solution κ+1 such that κ+1 H C κ+1 = P R ; setκ κ+1. Step 5: Repeat steps 1-3 until a pre-defined stop criterion is satisfied, e.. f κ+1 f κ ξ where f denotes the objective function of the problem 9 for someξ > 0. sum-rate bit/sec/hz Proposed MR = 8 POTDC MR = 8 Proposed MR = 4 POTDC MR = 4 sum-rate bit/sec/hz Proposed MR = 8 POTDC MR = 8 Proposed MR = 4 POTDC MR = N/F ratio Fi. 2: The sum-rate values versus N/F ratio for L = 2 and SNR=20dB Proposed POTDC SNR db Fi. 1: The sum-rate values associated with the proposed method and POTDC 1] versus SNR for L = 2. Averae run-time s between k th user of the l th operator and the relay is represented by d k,l. For simplicity and without loss of enerality, we assume that d 1,l = d 1 and d 2,l = d 2 with d 1 +d 2 = 1. Therefore, the near-far N/F ratio is defined as d 1 /d 2. The Rayleih flat fadin channel vectors{h k,l } are reciprocal and spatially uncorrelated and the path loss exponent is assumed to be3in all simulations. All the results are presented considerin 100 realizations of the associated fadin channels. We bein by investiatin the effect of the SNR on the sum-rate in a symmetric scenario i.e., d 1 = d 2. The sum-rate values associated with the proposed method as well as the POTDC method of 1] which is dealt with via CVX toolbox 10] versus SNR are shown in Fi. 1 for M R = 4 and M R = 8 with L = 2. As expected, the sum-rate is increasin with respect to SNR. Furthermore, the results of the proposed method are slihtly better than those of the method in 1] because the proposed method circumvents the synthesis loss associated with POTDC. Next, we study the effect of the N/F ratio. Fi. 2 plots the sum-rate values versus different N/F ratios L = 2. The proposed method achieves better results in the whole interval of the N/F ratio. Moreover, Fi. 1 and Fi. 2 show that a larer number of antennas M R leads to a larer sumrate value of the network as expected. The computational times of both methods are investiated in Fi. 3, which plots the averae computational times by considerin 10 runs of the Fi. 3: The averae run-time s versus the number of antennasm R for the case ofl = 2. methods with random initializations on an ordinary PC with 8GB RAM and CPU CoRe i5. It can be observed that the proposed method exhibits a low computational cost compared to its rival note that the values for POTDC correspond to 10 iterations. The presented computational results illustrate the applicability of the proposed method to currently available prototypes of massive MIMO e.. Aros 11]. MR 5. CONCLUSION The problem of relay beamformer desin for sum-rate imization in massive MIMO AF relay networks was considered. An iterative method based on the minorizationimization MaMi technique was devised to deal with the desin problem. The proposed method provides quality solutions to the desin problem for an arbitrary number of operators L. Numerical examples confirmed the effectiveness of the proposed method when compared to other methods in terms of the solution quality and the computational efficiency.

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