Group Sparse Precoding for Cloud-RAN with Multiple User Antennas

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1 entropy Artice Group Sparse Precoding for Coud-RAN with Mutipe User Antennas Zhiyang Liu ID, Yingxin Zhao * ID, Hong Wu and Shuxue Ding Tianjin Key Laboratory of Optoeectronic Sensor and Sensing Networ Technoogy, Coege of Eectronic Information and Optica Engineering, Nanai University, 38 Tongyan Road, Tianjin , China; iuzhiyang@nanai.edu.cn Z.L.); wuhong@nanai.edu.cn H.W.); sding@u-aizu.ac.jp S.D.) * Correspondence: zhaoyx@nanai.edu.cn; Te.: Received: 6 November 2017; Accepted: 19 February 2018; Pubished: 23 February 2018 Abstract: Coud radio access networ C-RAN) has become a promising networ architecture to support the massive data traffic in the next generation ceuar networs. In a C-RAN, a massive number of ow-cost remote antenna ports RAPs) are connected to a singe baseband unit BBU) poo via high-speed ow-atency fronthau ins, which enabes efficient resource aocation and interference management. As the RAPs are geographicay distributed, group sparse beamforming schemes attract extensive studies, where a subset of RAPs is assigned to be active and a high spectra efficiency can be achieved. However, most studies assume that each user is equipped with a singe antenna. How to design the group sparse precoder for the mutipe antenna users remains itte understood, as it requires the joint optimization of the mutua couping transmit and receive beamformers. This paper formuates an optima joint RAP seection and precoding design probem in a C-RAN with mutipe antennas at each user. Specificay, we assume a fixed transmit power constraint for each RAP, and investigate the optima tradeoff between the sum rate and the number of active RAPs. Motivated by the compressive sensing theory, this paper formuates the group sparse precoding probem by inducing the 0 -norm as a penaty and then uses the reweighted 1 heuristic to find a soution. By adopting the idea of boc diagonaization precoding, the probem can be formuated as a convex optimization, and an efficient agorithm is proposed based on its Lagrangian dua. Simuation resuts verify that our proposed agorithm can achieve amost the same sum rate as that obtained from an exhaustive search. Keywords: coud radio access networ; sparse beamforming; boc diagonaization; group-sparsity; antenna seection 1. Introduction Coud radio access networ C-RAN) [1] is a promising and fexibe architecture to accommodate the exponentia growth of mobie data traffic in the next-generation ceuar networ. In a C-RAN, a the base-band signa processing is shifted to a singe base-band unit BBU) poo [2]. The conventiona base-stations BSs), however, are repaced by geographicay distributed remote antenna ports RAPs) with ony antenna eements and power ampifiers, which are connected to the BBU poo via high-speed ow-atency fronthau ins by fiber. Thans to its simpe structure, it is promising to depoy utra-dense RAPs in a C-RAN with ow cost. With highy dense geographicay distributed RAPs, significant rate gains can be expected over that with the same amount of co-ocated antennas in both the singe-user and muti-user cases [3 5]. Due to the huge differences among the distance between the user and the geographicay distributed RAPs, it has been shown in [3] that the capacity in the singe-user case is cruciay determined by the access distance from the user to its cosest RAP. This motivates us to investigate whether it is possibe to achieve a significant proportion of the sum rate by using a subset of the RAPs. When the Entropy 2018, 20, 144; doi: /e

2 Entropy 2018, 20, of 15 number of active RAPs is sma, the other RAPs wi operate in the seeping mode with very ow power consumption, which reduces the networ power consumption, and significanty improves the energy efficiency [6,7]. Moreover, it is aso abe to improve the rate performance under imited fronthau in capacity, as each RAP wi ony serve users with sma access distances [8 11]. In the singe-user case, it is straightforward to avoid distant RAPs transmitting, as they have itte contribution to improving the capacity. In the muti-user case, the probem becomes chaenging, as the beamformers of a users shoud be jointy designed. To tace this probem, a branch of sparse beamforming technoogies are therefore proposed, where the beamforming vectors are designed to be sparse with respect to the tota number of transmit antennas [8,9,12]. Intuitivey, a sparse beamforming vector impies that the number of active BS antennas is much smaer than the tota number of BS antennas, eading to a significant reduction on both active fronthau ins and the circuit power consumption. Typicay, the sparse constraint is introduced by imposing the 0 norm of the beamforming vector as a reguarization of the origina objective function. As the 0 norm is neither convex nor continuous, the sparse beamforming probem is in genera a mixed-integer optimization probem, which is difficut to be gobay optimized. Motivated by the recent theoretica breathroughs in compressive sensing [13], the sparse beamforming probem is formuated by incuding the 1 norm of the beamforming vectors as a reguarization such that the probem becomes convex. By iterativey updating the weights of the 1 norm, the sparse beamformer that minimizes the tota transmit power can be obtained by iterativey soving a second-order-conic-programming SOCP) [8] or a semi-definite-programming SDP) [12]. The probem can be further simpified to an upin beamformer design probem via upin-downin duaity [9]. Nevertheess, when each RAP incudes mutipe antennas, one RAP wi be switched off ony when a the coefficients in its beamformer are set to be zero. In other words, a antennas at a RAP shoud be seected or ignored simutaneousy, otherwise the number of active ins from the BBU poo to the RAPs and the circuit power consumption cannot be reduced. Recenty, the group sparse beamforming probem has been proposed where the antennas at the same RAP are restricted to be switched on or off simutaneousy [6,7,10,14 17], which further compicates the probem. Luong et a. [17] formuated the sum rate and power consumption tradeoff probem as a mixed-integer-second-order-conic-programming MI-SOCP) probem to obtain the goba optimum by using branch-and-reduce-and-bound BRB) agorithm, which imposes prohibitivey computationa compexity when the numbers of the RAPs and the users are arge. To reduce the compexity, convex approximations are usuay used to mae it convex, continuous, and differentiabe. For instance, Dai and Yu [14] introduces a reweighted 1 norm of the vector that identifies the transmission power at each RAP to approximate the number of active RAPs, and the non-convex weighted sum rate maximization probem is approximated by weighted minimum mean square error WMMSE) minimization and can be soved via a quadratica-constrained-quadratic-programming QCQP). A more commony used method is to repace the rate constraint by its equivaent signa-to-interference-pus-noise ratio SINR) constraint, so as to sove the probem via SOCP [6,15 17]. Compared to the individua sparse beamforming, the group sparse beamforming further reduces the networ power consumption, and the energy efficiency can be improved as we. So far, most agorithms focus on the situation where each user has a singe antenna [6,8,9,12,14 17]. As suggested by the mutipe-input-mutipe-output MIMO) theory, the capacity increases ineary with the minimum number of transmit and receive antennas [18]. In a C-RAN, it is desirabe to empoy mutipe antennas at each user to expoit the potentia mutipexing gains, which, however, further compicates the sparse precoder design. In fact, most techniques [6,8,9,12,15 17] deveoped for singe-antenna users cannot be directy appied to the mutipe-antenna user case. The difficuty originates from the fact that the rate is determined by not ony the SINR but aso by the power aocated to the mutipe sub-channes, and thus the probem cannot be transformed into an SOCP probem as in the group sparse beamformer design. Therefore, Pan et a. [19] proposed to adopt the reweighted 1 norm to mae the sparse constraint smooth and use the WMMSE method to mae

3 Entropy 2018, 20, of 15 the rate expression convex, and a ow-compexity agorithm is proposed to sove the networ power consumption minimization probem by expoiting the specia structure of the WMMSE approximation. This motivates us to extend some we-structured precoding scheme to C-RAN, and design a group sparse precoding approach with ow computationa compexity. In particuar, we focus on designing a group sparse precoder based on a orthogona precoding scheme, boc diagonaization BD) [20], which has gained widespread popuarity thans to its ow compexity and near-capacity performance when the number of transmit antennas is arge [21 24]. With BD, the receiving beamformer can be directy cacuated from the channe gain matrix and the transmit precoder, and the design of the receive beamformer, which is mutuay couped with the transmit beamformer and difficut to be optimized in the mutipe antenna user case [25], can be further simpified. In this paper, we address the joint probem of RAP seection and joint precoder design in a C-RAN with mutipe antennas at each user and each RAP. Whereas the probem is typicay non-deterministic poynomia-time hard NP-hard), we show that the probem becomes convex by inducing the reweighted 1 norm of a vector that indicates the transmit power at each RAP as a reguarization. Based on its Lagrangian dua probem, we propose an agorithm by iterativey updating the weights of the 1 norm to generate a sparse soution. Simuation resuts verify that the proposed agorithm can achieve amost the same sum rate as that from exhaustive search. The rest of this paper is organized as foows: Section 2 introduces the system mode and formuates the probem. Section 3 proposes an iterative agorithm to sove the group sparse precoding probem. The compexity anaysis of the proposed agorithm is presented in Section 4. Simuation resuts are presented and discussed in Section 5. Section 6 concudes this paper. Notation: Itaic etters denote scaars, and bodface upper-case and ower-case etters denote matrices and vectors, respectivey. x p denotes the p norm of vector x. X T, X, Tr{X} and det{x} denote the transpose, conjugate transpose, trace and determinant of matrix X, respectivey. diaga 1,..., a N ) denotes an N N diagona matrix with diagona entries {a i }. I N denotes an N N identity matrix. 0 N M and 1 N M denote N M matrices with a entries zero and one, respectivey. X denotes the cardinaity of set X. and E[ ] denote the ceiing and expectation operators, respectivey. 2. System Mode and Probem Formuation Consider a C-RAN with a set of remote antenna ports RAPs), denoted as L, and a set of users, denoted as K, with L = L and K = K, as shown in Figure 1. Suppose that each RAP is equipped with N c antennas, and each user is equipped with N antennas. Then we have a tota number of M = N c L BS antennas. The baseband units BBUs) are moved to a singe BBU poo which are connected to the RAPs via high-speed fronthau ins, such that the BBU poo has access to the perfect channe state information CSI) between the RAPs and the users, and the signas of a RAPs can be jointy processed. With a high density of geographicay distributed RAPs, the access distances from each user to the RAPs varies significanty, and the distant RAPs have itte contribution to improve the capacity. This motivates us to find a subset of RAPs that can provide near optima sum rate performance. In particuar, et A L denote the set of active RAPs, with A = A. To utiize the mutipexing gains from the use of mutipe user antennas, we assume that AN c N. The received signa at user can be then modeed as y = H,A x,a + H,A x,a + z, 1) j = where x,a C AN c 1 and y C N 1 denote the transmit and receive signa vectors, respectivey. H,A C N AN c is the channe gain matrix between the active RAPs and user. z denotes the additive noise, which is modeed as a Gaussian random vector with zero mean and covariance σ 2 I. With inear precoding, the transmit signa vector x,a can be expressed as x,a = T,A s, 2)

4 Entropy 2018, 20, of 15 where T,A C ANc N is the precoding matrix. s C N 1 is the information bearing symbos. It is assumed that Gaussian codeboo is used for each user at the transmitter, [ and therefore ] s CN 0, I). The transmit covariance matrix for user can be then written as S,A = E x,a x,a. It is easy to verify that S,A = T,A T,A. The sum rate can be written from 1) as R = og 2 det I N + H,A S,A H,A σ 2 I N + j = H,A S j,a H,A ). 3) BBU Poo Fronthau Lins N c antennas per RAP N antennas per user Figure 1. Graphic iustration of a Coud radio access networ C-RAN); BBU: baseband unit; RAP: remote antenna port. Note that it is difficut to find the optima inear precoder that maximizes the sum rate R due to the non-convexity of 3), and the mutua couping of the transmit and receive beamformers maes it difficut to jointy optimize the beamformers [25]. In this paper, we assume that boc diagonaization BD) is adopted, where the desired signa is projected to the nu space of the channe gain matrices of a the other users, such that H,A x j,a = 0, or equivaenty H,A S j,a H,A = 0 for a j =. With BD, the sum rate can be obtained as R = og 2 det I N + 1 ) σ 2 H,AS,A H,A. 4) As the signas come from more than one RAP, they need to satisfy a set of per-rap power constraints, i.e., Tr B S,A P,max, A, 5) where P,max denotes the per-rap power constraint. B = diag{b } is a diagona matrix, whose diagona entries is defined as b = [0,, 0, 1,, 1, 0,, 0]. 6) }{{}}{{}}{{} N c 1) N c M N c

5 Entropy 2018, 20, of 15 This paper focuses on the tradeoff between the sum rate and the group sparsity. In particuar, the probem can be formuated as the foowing optimization probem: maximize A,{S,A } og 2 det I N + 1 ) σ 2 H,AS,A H,A η A 7) s.t. H j,a S,A H j,a = 0 j, K, j = 8) Tr B S,A P,max A 9) S,A 0 K 10) where 8) is the zero-forcing ZF) constraint, which ensures that the inter-user interference can be competey eiminated at the optimum. η 0 is the tradeoff constant, which contros the sparsity of the soution, and thus the number of active RAPs. With η = 0, the probem reduces to a BD precoder optimization probem with per-rap power constraint. The group sparsity can be improved by assigning a arger η. Note that the optimization probem defined in 7) 10) needs to jointy determine the subset A and design the transmit covariance matrices S,A for K users, which is a combinatoria optimization probem and is NP-hard. A brute-force soution to a combinatoria optimization probem ie 7) 10) is exhaustive search. Specificay, we must chec a possibe combinations of the active RAPs. For each combination, we must search for the optima {S,A } that satisfies the constraints 8) 10). In the end, we pic out the combination that maximizes the sum rate. However, the compexity grows exponentiay with L, which cannot be appied to rea-word appication. Instead, we use the concept of 0 norm to reformuate probem 7) 10). In particuar, define ω R 1 L as ω = [ Tr{B 1 ] S }, Tr{B 2 S },, Tr{B S },, Tr{B L S }, 11) where S = E [ x x ], with x C M 1 denoting the transmit signa vector from a RAPs in L to user. The N c 1) + 1)-th to the N c -th entries of x are zero if RAP is inactive. It is cear that the -th entry ω = Tr{B S } is the transmit power of RAP, which is non-zero if and ony if RAP A. It is easy to verify that ω 0 = A. We then have the foowing emma: Lemma 1. The probem defined in 7) 10) is equivaent to the foowing optimization probem: maximize {S } og 2 det I N + 1 ) σ 2 H S H η ω 0 s.t. H j S H j = 0 j, K, j = Tr B S P,max L S 0 K 12) where H C N M is the channe gain matrix from a RAPs in L to user, K. Proof. Pease refer to Appendix A for detaied proof. Lemma 1 indicates that instead of searching over the possibe combinations of A and then optimizing according to the corresponding channe gain matrices {H,A }, 7) 10) can be soved based on the channe between the users and a RAP antennas, i.e., {H }. However, the probem 12) is non-convex due to the existence of the 0 norm, maing it difficut to find the goba optima soution.

6 Entropy 2018, 20, of 15 In compressive sensing theory, the 0 norm is usuay repaced by a 1 norm, and sparse soution can be achieved. However, simpy substituting ω 0 by ω 1 in 12) wi not necessariy produce sparse soution in genera, as ω 1 equas the sum power consumption instead of the number of non-zero entries. By repacing ω 0 by ω 1, the transmit power at a RAPs sti tend to satisfy the power constraints with equaity at the optimum, eading to a non-sparse soution. In this paper, we propose to sove 12) heuristicay by iterativey reaxing the 0 norm as a weighted 1 norm. In particuar, at the t-th iteration, the 0 norm ω t) 0 is approximated by where β t) = 1 reformuated as Tr{B S t 1) maximize ω t) 0 L =1 β t) ω t), 13), with ɛ > 0 denoting a sma positive constant. Equation 12) can be then }+ɛ og 2 det I N + 1 ) σ 2 H S H Tr Ψ t) S s.t. Constraints in 12), where Ψ t) = η β t) B. 15) ) By noting that the first item of the objective function, i.e., og 2 det I + 1 H σ 2 S H, } is concave with respect to S, and the second item Tr {Ψ t) S is affine with respect to S, we can then concude that 14) is a convex optimization probem. The probem can be soved by standard convex optimization techniques, e.g. interior point method [26], which, however, is typicay sow. In fact, by utiizing the structure of BD precoding, the probem can be efficienty soved by its Lagrangian dua. 3. Reweighted 1 Based Agorithm In this section, the agorithm to sove the group sparse inear precoding probem wi be presented. To sove 14), it is desirabe to remove the set of ZF constraints in the first pace. It has been proved in [27] that the optima soution for BD precoding with per-rap constraint is given by 14) S = Ṽ Q Ṽ, 16) where Q 0. Ṽ is given from the singuar vaue decomposition SVD) of G = [H T 1,, HT 1, HT,, HT K ]T as G = U Σ [V, Ṽ ], 17) where Ṽ C M M NK 1)) is the ast M NK 1) coumns of the right singuar matrix of G. It is easy to verify that H j S H j = 0 for a j =. Therefore, by substituting 16) into 14), the probem reduces to maximize s.t. og 2 det I N + 1 ) σ 2 H ṼQ Ṽ H Tr Q 0 B Ṽ Q Ṽ P,max { } Tr Ψ t) Ṽ Q Ṽ L K 18)

7 Entropy 2018, 20, of 15 Note that 18) is a convex probem, its Lagrangian dua can be written as L {Q }, λ) = og 2 det I N + 1 ) σ 2 H ṼQ Ṽ H Tr { } Ψ t) Ṽ Q Ṽ λ Tr ) B Ṽ Q Ṽ P,max 19) where λ 0 denoting the Lagrangian dua variabes. The Lagrangian dua function can be given as gλ) = max L {Q }, λ), 20) Q 0, where λ = [λ 1, λ 2,, λ L ]. We can then obtain the Lagrangian dua probem of 18) as min gλ). 21) λ 0 Since the probem 18) is convex and satisfies the Sater s condition, strong duaity hods. The respective prima and dua objective vaues in 18) and 21) must be equa at the goba optimum, and the compementary sacness must hod at the optimum, i.e., λ ) Tr B Ṽ Q Ṽ P,max = 0, = 1,, L, 22) where {Q } and {λ } are the optima prima and dua variabes, respectivey. For fixed λ, the Lagrangian dua function gλ) can be obtained by soving max og 2 det I N + 1 ) Q 0 σ 2 H ṼQ Ṽ H Tr ΩṼ Q Ṽ, 23) where Ω = Ψ t) + λ B. Appendix B shows that the optima Q can be obtained as Q = Ṽ ΩṼ ˆV Λ ˆV Ṽ ΩṼ, 24) where ˆV is obtained from the foowing reduced SVD: H Ṽ Ṽ ΩṼ = Û Ξ ˆV. 25) Λ = diag{ λ,1,, λ,n }, with ) + 1 λ,n = n 2 σ2 ξ,n 2, 26) where x + = maxx, 0). The optima S for given λ can be then obtained as S λ) = Ṽ Ṽ ΩṼ ˆV Λ ˆV Ṽ ΩṼ Ṽ 27) With the optima S λ) achieved for given λ, we can then find the Lagrangian dua variabes λ by the projected subgradient method. Projected subgradient methods foowing, e.g., the square summabe but not summabe step size rues, have been proven to converge to the optima vaues [28].

8 Entropy 2018, 20, of 15 In particuar, a subgradient with respect to λ is P,max Tr { B Ṽ Q Ṽ }. With a step size δt, the dua variabes can be updated as λ t+1) = max { λ t) δ t { P,max Tr B Ṽ Q Ṽ } ) }, 0, = 1,, L. 28) From the compementary sacness in 22), a stopping criterion for updating 28) can be λ Tr B Ṽ Q Ṽ 2 < ε. 29) L Once the optima λ is obtained, the optima precoding matrices {T } can be achieved by using the fact that S = T T as T = Ṽ Ṽ Ω Ṽ ˆV Λ1/2. 30) The optima set A and the corresponding precoding matrices {T,A } can be obtained from {T }. The agorithm is summarized as Agorithm 1. Agorithm 1 Reweighted 1 Norm Based Sparse Precoding Design. Initiization: Set iteration counter t = 1, Lagrangian dua variabe λ 0) > 0, = 1,, L. Repeat: 1. Cacuate Ψ t) and S t), = 1,, K, according to 15) and 27), respectivey. 2. Update {λ } according to 28). 3. Update the iteration counter t t + 1 and r = λ Tr B Ṽ Q Ṽ 2. L Stop if r < ε, where ε is a pre-defined toerance threshod. 4. Compexity Anaysis In this section, we provide our compexity anaysis of Agorithm 1. Note that Ṽ can be cacuated before the iterations. The main compexity of the proposed agorithm ies in step 1 and step 2. Let us first focus on step 1. According to [29], the cacuations of matrix mutipication XY and matrix inversion Z 1 have compexities on the orders of Omnp) and Om ), respectivey, where X C m n, Y C n p and Z C m m. Therefore, the compexity to cacuate H Ṽ Ṽ ΩṼ is on the order of O M NK 1)) 2.736). On the other hand, as the compexity of the SVD in 25) is on the order of O MM NK 1)) 2) [29], the compexity to cacuate ˆV is on the order of O MM NK 1)) 2). By considering that the matrix cacuation in 27) has a compexity on the order of O MM NK 1)) 2), we can then concude that the compexity of step 1 is on the order of O KMM NK 1)) 2). In step 2, as Tr { B Ṽ Q Ṽ } = Tr {B S } and S can be cacuated before updating λ, step 2 has a compexity on the order of O LM 3). By noting that the number of RAPs is typicay arger than the number of users, i.e., L > K, and that the tota number of BS antennas M = N c L, the overa compexity of the proposed agorithm is on the order of Ot avg N 3 c L 4 ), where t avg is the average number of iterations. As we wi show in the foowing section, the agorithm is abe to converge within 15 iterations for a propery seected step size and toerance threshod.

9 Entropy 2018, 20, of Simuation Resuts In this section, simuation resuts are presented to iustrate the resuts in this paper. We assume that the channe gain matrices {H, } from RAP to user are independent over and for a L and K, and a entries of H, are independent and identicay distributed iid) compex Gaussian random variabes with zero mean and variance γ, 2. The path-oss mode from RAP to the user is PL, db) = og 10 D,, 31) L and K, where D, is the distance between user and RAP in the unit of iometer. The arge-scae fading coefficient from RAP to user can be then obtained as γ 2, = 10 PL,dB)/10. 32) The transmit power constraint at each RAP are assumed to be identica, which is set to be P,max = 40 dbm/hz, L, and the noise variance is set to be σ 2 = 162 dbm/hz. We consider the case that L = 10 RAPs with N c = 2 antennas each and K = 2 users with N = 3 antennas each. The positions of users and RAPs are generated in a circuar area with radius 1000 m foowing uniform distribution. An exampe of the randomy generated antenna and user ayout is potted in Figure 2. Figure 3 shows the convergence behavior of the proposed agorithm under the ayout given in Figure 2 under different step size δ t = 0.5, 0.1 and As we can see from Figure 3, the vaue of 29) rapidy decreases with the number of iterations. When the step size is arge, i.e., δ t = 0.5, despite that the agorithm converges in genera, the vaue increases after severa iterations. Such observation comes from the fact that we use the reweighted 1 norm to approximate the non-convex 0 norm, and a arge step in λ wi generay eading to a significant change of Ψ in 18). When the step size is smaer, Ψ changes moderatey, and a monotonic decrease can be observed in Figure 3 when δ t = 0.1 and , RAP 6 RAP 4 RAP RAP 5 RAP 8 User 1 RAP 9 RAP User 2 RAP 7 RAP 10 RAP 2-1,000-1, ,000 Figure 2. An exampe of a randomy generated system configuration. The active RAPs at the converged state with η = 0.5 are highighted in red.

10 Number of Active RAPs Vaue of 29) Entropy 2018, 20, of t t t Number of Iterations Figure 3. Convergence behavior of the proposed agorithm under the antenna and user ayout shown in Figure 2 with randomy generated sma-scae fading coefficients. L = 10. N c = 2. K = 2. N = 3. η = 0.5. Figure 4 shows how the number of active RAPs varies with iterations in the ayout shown in Figure 3 with step size δ t = 0.1 and the toerance threshod ε = 10 4, where a RAP is said to be active if its transmit power P 10 5 P,max. The tradeoff constant η is set to be 0, 0.1 and 0.5. Figure 4 shows that the number of active RAPs A decreases with η. Specificay, with η = 0, the probem reduces to a BD precoder design with per-rap constraint, and a RAPs are active. With η = 0.5, the sparsest soution can be achieved, i.e., A = A min = KN N c = 3. We shoud mention that the vaue of η that corresponds to the sparsest soution varies with the system configuration h=0 h=0.1 h= Iterations Figure 4. Number of active RAPs under the antenna and user ayout shown in Figure 2 with randomy generated sma-scae fading coefficients. L = 10. N c = 2. K = 2. N = 3.

11 Average Sum Rate bit/s/hz) RAP Transmit Power dbm/hz) Entropy 2018, 20, of 15 As Figure 4 shows, the first severa iterations ead to the biggest improvement. As iterations go on, there is no further improvement after the 15th iteration. Compared to the fu cooperation case, i.e., η = 0, as L A min = 7 RAPs are switched off, 70% of the circuit power consumption can be saved with, however, imited rate performance oss, which wi be iustrated ater in this section. Figure 5 pots the transmit power distribution over a 10 RAPs. Intuitivey, the RAPs that have the smaer access distances contribute most to the sum rate. As shown in Figures 2 and 5, RAP 1 and RAP 9, which are cose to User 1, and RAP 7, which is cose to User 2, are seected to be active, whie a the other RAP s transmit power eventray goes to zero after 15 iterations RAP 1 RAP 2 RAP 3 RAP 4 RAP 5 RAP 6 RAP 7 RAP 8 RAP 9 RAP Iterations Figure 5. Transmit power of the RAPs under the antenna and user ayout shown in Figure 2 with randomy generated sma-scae fading coefficients. L = 10. N c = 2. K = 2. N = 3. η = N = 2 c Proposed Agorithm Exhaustive Search Fixed L N = 1 c Number of Seected RAPs Figure 6. Tradeoff between average sum rate and the number of active RAPs. L = 10. K = 2. N = 3. Figure 6 pots how the average sum rate varies with the number of active RAPs. The average sum rate is obtained by averaging over 20 reaizations of sma-scae fading and 30 reaizations of the positions of RAPs and users. The resuts of the exhaustive search is aso presented for comparison,

12 Entropy 2018, 20, of 15 which is obtained by searching over a possibe combinations of active RAPs, and computing its achievabe sum rate by the agorithm given in [27]. For the proposed agorithm, we simuate a series of different η s to get different points aong the curve. As we can see from Figure 6, our proposed agorithm achieves amost the same average sum rate as the exhaustive search, which verifies the optimaity of our proposed agorithm. Figure 6 further pots the average sum rate with a fixed number of A instead of L uniformy distributed antennas are depoyed for comparison, which is denoted as Fixed L in Figure 6. We can ceary see that the proposed agorithm can achieve much better rate performance over that with the same amount of transmit RAP. With six seected antennas, for instance, the group sparse precoder reduces the average sum rate for ony 3 bit/s/hz, whereas if ony 6 antennas were instaed instead of L = 10, an additiona 9 bit/s/hz rate oss can be observed compared to the proposed agorithm. Moreover, the rate gap between the proposed agorithm and that with a fixed number of L = A fu cooperative RAPs further increases as the number of seected antennas A decreases. This highights the importance of group sparse precoder design in C-RAN with a arge number of distributed RAPs. 6. Concusions In this paper, we study the group sparse precoder design that maximizes the sum rate in a C-RAN. We show that the joint antenna seection and precoder design probem can be formuated into an 0 norm probem, which is, however, combinatoria and NP-hard. Inspired by the theory of compressive sensing, we propose an approach that soves the probem via reweighted 1 norm. Simuation resuts verify the optimaity of our proposed agorithm in that it achieves amost the same performance as that obtained from the exhaustive search. Compared to fu cooperation, the group sparse precoding can achieve a significant proportion of the maximum sum rate that was achieved from fu cooperation with, however, much fewer active RAPs, which highights the importance of empoying group sparse precoding in C-RAN with utra-dense RAPs. Note that in practica system, the imperfect fronthau ins between the BBU poo and the RAPs further imits the performance. It is desirabe to extend our approach to a practica scenario by incuding the fronthau in capacity constraint. Moreover, inspired by [30] which impemented the zero-forcing precoding in a distributed manner, we woud ie to further extend our agorithm in a distributed way in our future wor, as it enabes the appication of group sparse precoding in a arge-scae C-RAN system. Acnowedgments: This wor is supported by the Nationa Natura Science Foundation of China under grant Nos , and , by Tianjin Research Program of Appication Foundation and Advanced Technoogy under grant Nos. 15JCYBJC51600 and 16YFZCSF Author Contributions: Zhiyang Liu and Yingxin Zhao modeed and anayzed the probem. Hong Wu and Shuxue Ding proposed and impemented the resut. Zhiyang Liu wrote the paper. Conficts of Interest: The authors decare no confict of interest Appendix A. Proof of Lemma 1 Define the channe gain matrix and the precoding matrix between user and the antennas of RAP as H, and T,, respectivey, K and L. Let us reorder the channe gain matrices H and the precoding matrices T as H = [ ] [H, ] A, [H, ] A, A1) and T = [ [T T, ] A, [T T, ] A] T, A2)

13 Entropy 2018, 20, of 15 respectivey. We have H j S H j = H T T H a) = ) ) H, T, H, T, A A = H j,a S,A H j,a, A3) for a j, K, where a) foows from the fact that T, = 0 if RAP is not active, i.e., A. Let us then prove the positive semidefinite constraint. The reordered covariance matrix S can be written as Ŝ = ˆT ˆT [ ] T,A T,A T,A [T, ] A = [T, ] A T,A [T,] A [T, ] A [ ] b) T =,A T,A 0, 0 0 A4) where b) hods as T, = 0 for a A. By noting that T,A T,A = S,A 0, we immediatey have S 0. Equation 12) can be then proved by combining A3) A4) and 7) 10). Appendix B. Derivation of 24) We can observe from 23) that the probem is equivaent to sove K uncouped subprobems given as max og 2 det I N + 1 ) Q 0 σ 2 H ṼQ Ṽ H Tr ΩṼ Q Ṽ, A5) By noting that Tr{XY} = Tr{YX}, we have { Ṽ ) 1/2 ) } 1/2 Tr ΩṼ Q Ṽ = Tr ΩṼ Q Ṽ ΩṼ. A6) Let us define ) 1/2 ) 1/2 Q = Ṽ ΩṼ Q Ṽ ΩṼ. A7) Equation A5) can be written as max og 2 det I N + 1 ) ) 1/2 Q 0 σ 2 H Ṽ Ṽ ΩṼ Q Ṽ ΩṼ Ṽ H Tr Q. A8) Equation A8) can be soved from standard waterfiing agorithm [31]. In particuar, by introducing the reduced) SVD: H Ṽ Ṽ ΩṼ = Û Ξ ˆV, where Û C N N and ˆV C N N are unitary matrices. Ξ = diag{ξ,1,..., ξ,n }, with ξ,n denoting the n-th singuar vaue of H Ṽ Ṽ ΩṼ. The optima Q can be then obtained as: A9) Q = ˆV Λ ˆV, A10)

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