Enhanced Group Sparse Beamforming for Green Cloud-RAN: A Random Matrix Approach

Transcription

1 Enhanced Group Sparse Beamforming for Green Coud-RAN: A Random Matrix Approach Yuanming Shi, Member, IEEE, Jun Zhang, Senior Member, IEEE, Wei Chen, Senior Member, IEEE, and Khaed B. etaief, Feow, IEEE arxiv: v [cs.it] 9 Nov 07 Abstract Group sparse beamforming is a genera framewor to minimize the networ power consumption for coud radio access networs Coud-RANs, which, however, suffers high computationa compexity. In particuar, a compex optimization probem needs to be soved to obtain the remote radio head RRH ordering criterion in each transmission boc, which wi hep to determine the active RRHs and the associated fronthau ins. In this paper, we propose innovative approaches to reduce the compexity of this ey step in group sparse beamforming. Specificay, we first deveop a smoothed p-minimization approach with the iterative reweighted- agorithm to return a Karush-Kuhn-Tucer KKT point soution, as we as enhancing the capabiity of inducing group sparsity in the beamforming vectors. By everaging the agrangian duaity theory, we obtain cosed-form soutions at each iteration to reduce the computationa compexity. The we-structured soutions provide the opportunities to appy the arge-dimensiona random matrix theory to derive deterministic approximations for the RRH ordering criterion. Such an approach heps to guide the RRH seection ony based on the statistica channe state information CSI, which does not require frequent update, thereby significanty reducing the computation overhead. Simuation resuts sha demonstrate the performance gains of the proposed p-minimization approach, as we as the effectiveness of the arge system anaysis based framewor for computing RRH ordering criterion. Index Terms Coud-RAN, green communications, sparse optimization, smoothed p-minimization, agrangian duaity, and random matrix theory. I. INTRODUCTION Networ densification [], [3], [4] has been proposed as a promising way to provide utra-high data rates, achieve ow atency, and support ubiquitous connectivity for the upcoming 5G networs [5]. However, to fuy harness the benefits of dense wireess networs, formidabe chaenges arise, incuding interference management, radio resource aocation, mobiity management, as we as high capita expenditure and operating expenditure. Coud-RAN emerges as a disruptive technoogy to depoy cost-effective dense wireess networs Y. Shi is with the Schoo of Information Science and Technoogy, ShanghaiTech University, Shanghai, China e-mai: shiym@shanghaitech.edu.cn. J. Zhang is with the Department of Eectronic and Computer Engineering, Hong Kong University of Science and Technoogy, Hong Kong e-mai: eejzhang@ust.h. W. Chen is with the Department of Eectronic Engineering, Tsinghua University, Beijing, China e-mai: wchen@tsinghua.edu.cn. K. B. etaief is with Hamad bin Khaifa University e-mai: etaief@hbu.edu.qa and Hong Kong University of Science and Technoogy e-mai: eehaed@ust.h. Part of this wor was presented at the 06 IEEE Internationa Symposium on Information Theory ISIT [], Barceona, Spain, Ju. 06. This wor was party supported by the Hong Kong Research Grant Counci under Grant No [6], [7]. It can significanty improve both energy and spectra efficiency, by everaging recent advances in coud computing and networ function virtuaization [8]. With shared computation resources at the coud data center and distributed ow-cost ow-power remote radio heads RRHs, Coud-RAN provides an idea patform to achieve the benefits of networ cooperation and coordination [9]. There is a unique characteristic in Coud-RANs, namey, the high-capacity fronthau ins are required to connect the coud center and RRHs [0], []. Such ins wi consume power comparabe to that of each RRH, and thus brings new chaenges to design green Coud- RANs. To address this issue, a new performance metric, i.e., the networ power consumption, which consists of both the fronthau in power consumption and RRH transmit power consumption, has been identified in [7] for the green Coud- RAN design. Unfortunatey, the networ power minimization probem turns out to be a mixed-integer noninear programming probem [], which is highy intractabe. Specificay, the combinatoria composite objective function invoves a discrete component, indicating which RRHs and the corresponding fronthau ins shoud be switched on, and a continuous component, i.e., beamformers to reduce RRH transmit power. To address this chaenge, a unique group sparsity structure in the optima beamforming vector has been identified in [7] to unify RRH seection and beamformer optimization. Accordingy, a nove three-stage group sparse beamforming framewor was proposed to promote group sparsity in the soution. Specificay, a mixed / -minimization approach was first proposed to induce group sparsity in the soution, thereby guiding the RRH seection via soving a sequence of convex feasibiity probems, foowed by coordinated beamforming to minimize the transmit power for the active RRHs. Athough the group sparse beamforming framewor provides poynomia-time compexity agorithms via convex optimization, it has the imited capabiity to enhance group sparsity compared to the non-convex approaches [3], [4]. In particuar, the smoothed p -minimization supported by the iterative reweighted- agorithm was deveoped in [5] to enhance sparsity for muticast group sparse beamforming. However, the computation burden of this iterative agorithm is sti prohibitive in dense wireess networs with a arge number of RRHs and mobie users. In [6], a generic two-stage approach was proposed to sove arge-scae convex optimization probems in dense wireess cooperative networs via matrix stuffing and operator spitting, i.e., the aternating direction method of mutipiers ADMM [7], [8]. This approach

2 aso enabes parae computing and infeasibiity detection. However, as the proposed soutions need to be accompished for each channe reaization, and aso due to the iterative procedures of the group sparse beamforming framewor, it is sti computationay expensive. In this paper, we improve the performance of group sparse beamforming [7] via p -minimization, and a specia emphasis is on reducing the computationa compexity, with two ingredients: cosed-form soutions for each iteration, and the deterministic equivaents of the optima agrange mutipiers. Specificay, we first deveop an iterative reweighted- agorithm with cosed-form soutions at each iteration via everaging the principes of the majorization-minimization MM agorithm [9] and the agrangian duaity theory [0]. It turns out that the proposed iterative reweighted- agorithm can find a KKT point for the smoothed non-convex p - minimization probem. Furthermore, this reveas the expicit structures of the optima soutions to each subprobem in the iterative reweighted- agorithm, whie the convex optimization approach in [], [7], [5] fais to obtain the cosed-form soutions. Thereafter, the we-structured cosedform soutions provide opportunities to everage the argedimensiona random matrix theory [], [3], [4] to perform the asymptotic anaysis in the arge system regimes [5], [6]. Specificay, the deterministic equivaents of the optima agrange mutipiers are derived based on the recent resuts of arge random matrix anaysis [7] to decoupe the dependency of system parameters. These resuts are further used to perform an asymptotic anaysis for computing the RRH ordering criterion, which ony depends on ong-term channe attenuation, thereby significanty reducing the computation overhead compared with previous agorithms that heaviy depends on instantaneous CSI [8], [9], [30], [3], [7], [], [5]. Based on the above proposa, we provide a three-stage enhanced group sparse beamforming framewor for green Coud-RAN via random matrix theory. Specificay, in the first stage, we compute the enhanced RRH ordering criterion ony based on the statistica CSI. This thus avoids frequent updates, thereby significanty reducing computationa compexity, which is a sharp difference compared to the origina proposa in [8]. This new agorithm is based on the principes of the MM agorithm and the agrangian duaity theory, foowed by the random matrix theory. With the obtained deterministic equivaents of RRH ordering criterion, a two-stage arge-scae convex optimization framewor [6] is adopted to sove a sequence of convex feasibiity probems to determine the active RRHs in the second stage, as we as soving the transmit power minimization probem for the active RRHs in the fina stage. Note that goba instantaneous CSI is required for these two stages. Simuation resuts are provided to demonstrate the improvement of the enhanced group sparse beamforming framewor. Moreover, the deterministic approximations turn out to be accurate even in the finite-sized systems. A. Reated Wors Sparse Optimization in Wireess Networs: The sparse optimization paradigm has recenty been popuar for compicated networ optimization probems in wireess networ design, e.g., the group sparse beamforming framewor for green Coud-RAN design [7], [30], [3], [3], [5], wireess caching networs [33], [34], user admission contro [5], [35], as we as computation offoading [36]. In particuar, the convex reaxation approach provides a principed way to induce sparsity via the -minimization [35] for individua sparsity inducing and the mixed / -minimization for group sparsity inducing [7]. The reweighted- agorithm [3] and the reweighted- agorithm [5] were further deveoped to enhance sparsity. To enabe parae and distributed computing, the first-order method ADMM agorithm was adopted to sove the group sparse beamforming probems [37]. A generic argescae convex optimization framewor was further proposed to sove genera arge-scae convex programs in dense wireess networs to enabe scaabiity, parae computing and infeasibiity detection [6]. However, a the above agorithms need to be computed for each channe reaization, which is computationay expensive. In this paper, we adopt arge system anaysis to compute the RRH ordering criterion for networ adaptation ony based on statistica CSI. arge System Anaysis via Random Matrix Theory: Random matrix theory [] has been proven powerfu for performance anaysis, and the understanding and improving agorithms in wireess communications [38], [3], signa processing [39], and machine earning [40], [4], especiay in arge dimensiona regimes for appications in the era of big data [4]. In dense wireess networs, random matrix theory provides a powerfu way for performance anaysis and agorithm design. In particuar, the arge system anaysis was performed for simpe precoding schemes, e.g., reguarized zero-forcing in MISO broadcast channes with imperfect CSI [4]. A random matrix approach to the optima coordinated mutice beamforming for massive MIMO was presented in [43] without cose-form expressions for the optima agrange mutipiers. A simpe channe mode with two ces under different coordination eves was considered in [5]. However, a of the above resuts cannot be directy appied in the group sparse beamforming framewor in dense Coud- RAN due to the genera channe modes with heterogenous pathoss and the compicated beamformer structures with fuy cooperative transmission. To determine the RRH ordering criterion based on statistica CSI, we adopt the technique in [7], [6] to compute the cosed-forms for the optima agrange mutipiers for each iteration in the procedure of reweighted- minimization for sparsity inducing. B. Organization The remainder of the paper is organized as foows. Section II presents the system mode and probem formuation, foowed by performance anaysis with the proposed threestage enhanced group sparse beamforming framewor for green Coud-RAN. In Section III, the iterative reweighted- agorithm for group sparse beamforming is presented via the principe of MM agorithm and duaity theory. The arge system anaysis for RRH ordering is performed in Section IV.

3 3 Simuation resuts wi be demonstrated in Section V. Finay, concusions and discussions are presented in Section VI. To eep the main text cean and free of technica detais, we divert most of the proofs to the Appendix. Notations: Throughout this paper, p is the p -norm. stands for either the size of a set or the absoute vaue of a scaar, depending on the context. M is the spectra radius of the Hermitian matrix M. Bodface ower case and upper case etters represent vectors and matrices, respectivey., T, H and Tr denote the inverse, transpose, Hermitian and trace operators, respectivey. We use C to represent compex domain. E[ ] denotes the expectation of a random variabe. We denote A = diag{x,..., x N } and I N as a diagona matrix of order N and the identity matrix of order N, respectivey. II. SYSTEM MODE AND PROBEM FORMUATION A. System Mode Consider a Coud-RAN with RRHs and K singe-antenna mobie users MUs, each RRH is equipped with N antennas. In Coud-RANs, the BBU poo wi perform the centraized signa processing and is connected to a the RRHs via high-capacity and ow-atency fronthau ins. In this paper, we wi focus on the downin signa processing. Specificay, et v C N be the transmit beamforming vector from the -th RRH to the -th MU. The received signa y C at MU is given by y = = h H v s + i h H v i s i + n, h C N is the channe propagation between MU and RRH, s C with E[ s ] = is the encoded transmission information symbo for MU, and n CN 0, σ is the additive Gaussian noise at MU. We assume that s s and n s are mutuay independent and a the MUs appy singe-user detection. The signa-tointerference-pus-noise ratio SINR for MU is given by = h H sinr = v i hh v i + σ, h = [h T,..., ht ]T C is the vector consisting of the channe coefficients from a the RRHs to MU, and v = [v T,..., vt ]T C is the aggregative beamforming vector for the MU from a the RRHs. B. Probem Formuation In this paper, we aim at designing a green Coud-RAN by minimizing the networ power consumption, which consists of the fronthau ins power consumption, as we as the RRH transmit power consumption [7]. Specificay, et v = [v ] C K be the aggregative beamforming vector from a the RRHs to a the MUs. Define the support of the beamforming vector v as T v = {i v i 0}, v = [v i ] is indexed by i V with V = {,..., K}. The reative fronthau networ power consumption is given by f v = = P c IT v V, 3 P c 0 is the reative fronthau in power consumption [7] i.e., the static power saving when both the fronthau in and the corresponding RRH are switched off, V = { KN +,..., KN},, is a partition of V, and IT V is an indicator function that taes vaue if T V and 0 otherwise. Note that f is a non-convex combinatoria function. Furthermore, the tota transmit power consumption is given by f v = K = = ζ v, 4 ζ > 0 is the drain inefficiency coefficient of the radio frequency power ampifier [7]. Therefore, the networ power consumption is represented by the combinatoria composite function fv = f v + f v. 5 Given the QoS threshods γ = γ,..., γ K for a the MUs, in this paper, we aim at soving the foowing networ power consumption minimization probem with the QoS constraints: P : minimize v subject to f v + f v h H v i hh v i + σ γ,. 6 et ṽ = [v] V = [v T,..., vt K ]T C KN form the beamforming coefficient group from RRH to a the MUs. Note that, when the RRH is switched off, a the beamforming coefficients in ṽ wi be set to zeros simutaneousy. Observing that there may be mutipe RRHs being switched off to minimize the networ power consumption, the optima beamforming vector v = [ṽ T,..., ṽ T ]T C K shoud have a group-sparsity structure. Therefore, probem P is caed a group sparse beamforming probem [7]. Note that, to simpify the presentation, we ony impose the QoS constraints in probem P. However, the proposed instantaneous CSI based iterative reweighted- agorithm can be extended to the scenario with per-rrh transmit power constraints, foowing the principes in [0]. More technica efforts are required to derive the asymptotic resuts using the random matrix method with more compicated structures of the optima beamformers. C. Probem Anaysis Athough the constraints in probem P can be reformuated as convex second-order cone constraints, the non-convex objective function maes it highy intractabe. To address this chaenge, a weighted mixed / -norm minimization approach was proposed in [7] to convexify the objective and induce the group sparsity in beamforming vector v, thereby

4 4 guiding the RRH ordering to enabe adaptivey RRH seection. Specificay, we wi first sove the / -norm minimization probem, and denote ṽ,..., ṽ as the induced approximated group sparse beamforming vectors. Then the foowing RRH ordering criterion is adopted to determine which RRHs shoud be switched off [7]: θ = κ ṽ, =,...,, 7 κ = K = h /ν. In particuar, the group sparsity structure information for beamforming vector v is extracted from the squared -norm of the beamforming vectors ṽ s, i.e., ṽ,..., ṽ. The RRH with a smaer θ wi have a higher priority to be switched off. Based on the RRH ordering resut θ in 7, a bi-section search approach can be used to find the optima active RRHs [7] via soving a sequence of the foowing feasibiity probems: F A [i] : find v,..., v K h H subject to v i hh v i + σ γ,, 8 v = [v ] C A N and h = [h ] C A N. Probem F A [i] turns out to be convex via reformuating the QoS constraints as second-order cone constraints [7]. To further enhance the sparsity as we as to see the quadratic forms of the beamforming vectors in the muticast transmission setting, a smoothed p -minimization approach was proposed in [5]. To scae to arge probem sizes in dense Coud-RANs, a twostage arge-scae parae convex optimization framewor was deveoped in [6] with the capabiity of infeasibiity detection. However, a of the above deveoped agorithms bear high computation overhead. In particuar, whie the feasibiity probem 8 for RRH seection can be efficienty soved with the arge-scae optimization agorithm in [8], the RRH ordering criterion in 7 may be highy compicated to obtain, especiay with the non-convex formuation as in [5]. Moreover, the ordering criterion needs to be recomputed for each transmission sot, and depends on instantaneous CSI. Observing that the statistica CSI normay changes much sower than the instantaneous CSI, to reduce the computationa burden, we propose to compute the RRH ordering criterion 7 ony based on statistica CSI, i.e., the ong-term channe attenuation. This is achieved by first deveoping a group sparsity penaty with quadratic forms in the aggregative beamforming vector v, foowed by an iterative reweighted- agorithm with cosedform soutions at each iteration via agrangian duaity theory, as wi be presented in Section III. Then asymptotic anaysis is performed to obtain the RRH ordering criterion 7 based ony on statistica CSI by everaging the arge-dimensiona random matrix theory [3], [4], [7], as wi be presented in Section IV. Overa, the proposed three-stage enhanced group sparse beamforming framewor is presented in Fig.. Specificay, in the first stage, the RRH ordering criterion θ is cacuated ony based on statistica CSI using Agorithm. In the second stage, the set of active RRHs A is obtained based on instantaneous CSI via soving a sequence of convex feasibiity probems F A [i] using the arge-scae convex optimization framewor Group Sparse Optimization for RRH Ordering by Agorithm Based on Statistica CSI RRH Seection By Soving Transmit Power Minimization Stage I Stage II Stage III Fig.. The proposed three-stage enhanced group sparse beamforming framewor for dense green Coud-RAN. In the first stage, the RRH ordering criterion θ is computed ony based on the statistica CSI via arge system anaysis. The optima active RRHs A in the second stage and the optima coordinated beamforming for transmit power minimization in the third stage are computed based on the instantaneous CSI via the arge-scae convex optimization agorithm in [6]. in [6]. In the third stage, the transmit power is minimized by soving the convex program 6 with the fixed active RRHs A using the arge-scae convex optimization agorithm in [6]. Overa, the proposed enhanced group sparse beamforming framewor is scaabe to arge networ sizes. This paper wi focus on deveoping an effective agorithm for the first stage. Remar : In this paper, we assume that, in the first stage, probem P 6 is feasibe for deveoping Agorithm and Agorithm to determine the RRH ordering criterion based on the instantaneous CSI and statistica CSI, respectivey. We enabe the capabiity of handing the infeasibiity in the second stage based on the instantaneous CSI, using the argescae convex optimization agorithm [6]. Furthermore, when probem F 8 is infeasibe with a RRHs active, we adopt the user admission agorithm to find the maximum number of admitted users [5]. III. GROUP SPARSE BEAMFORMING VIA ITERATIVE REWEIGHTED- AGORITHM In this section, we deveop a group sparse beamforming approach based on the smoothed p -minimization, supported by an iterative reweighted- agorithm, thereby enhancing the group sparsity in the beamforming vectors. Instead of reformuating the QoS constraints in probem P as secondorder cone constraints [7], we use the agrangian duaity theory to revea the structures of the optima soutions at each iteration. The resuts wi assist the arge system anaysis in Section IV. A. Group Sparsity Inducing Optimization via Smoothed p - Minimization To induce the group sparsity structure in the beamforming vector v, thereby guiding the RRH ordering, the weighted mixed / -norm was proposed in [7]. However, the nonsmooth mixed / -norm fais to introduce the quadratic forms in the beamforming vector v, in order to be compatibe with the quadratic QoS constraints so that the agrangian duaity theory can be appied [0]. To address this chaenge, we adopt the foowing smoothed p -minimization 0 < p approach to induce group sparsity [4], [5]: P GSBF : minimize v subject to g p v; ɛ := ν ṽ + ɛ p/ = h H v i hh v i + σ γ,, 9

5 5 ɛ > 0 is some fixed reguarizing parameter and ν > 0 is the weight for the beamforming coefficient group ṽ by encoding the prior information of system parameters [7]. Compared with the mixed / -norm minimization approach, the p -minimization approach can induce sparser soutions based on the fact z 0 = im p 0 z p p = im p 0 i z i p. Unfortunatey, probem 9 is non-convex due to the nonconvexity of both the objective function and the QoS constraints. B. Iterative Reweighted- Agorithm We use the MM agorithm and the agrangian duaity theory to sove probem 9 with cosed-form soutions. Specificay, this approach generates the iterates {v [n] } by successivey minimizing upper bounds Qv; v [n] of the objective function g p v; ɛ. We adopt the foowing upper bounds to approximate the smoothed p -norm gv; ɛ by everaging the resuts of the expectation-maximization EM agorithm [44]. Proposition : Given the vaue of v [n] at the n-th iteration, an upper bound for the objective function g p v; ɛ can be constructed as foows: ω [n] = pν Qv; ω [n] := [ ṽ [n] = ω [n] ṽ, 0 + ɛ ] p, =,...,. Proof: The proof is mosty based on [5, Proposition ]. Therefore, at the n-th iteration, we need to sove the foowing optimization probem: P [n] GSBF : minimize v subject to = ω [n] ṽ h H v i hh v i + σ γ,, which is non-convex due to the non-convex QoS constraints. Athough the QoS constraints can be reformuated as secondorder cone constraints as in [7], in this paper, we everage the agrangian duaity theory to obtain expicit structures of the optima soution to probem P [n] GSBF, thereby reducing the computationa compexity and further aiding the arge system anaysis in next section. The success of appying the agrangian duaity approach is based on the fact that strong duaity hods for probem P [n] GSBF [45], i.e., the gap between the prima optima objective and the dua optima objective is zero. Simpe Soution Structures: As the strong duaity hods for probem P [n] GSBF [45], we sove it using duaity theory. Specificay, et λ / 0 denote the agrange mutipiers corresponding to the QoS constraints in probem P [n] GSBF. Define the agrangian function {v }, λ with λ = {λ i } as K λ {v }, λ = h H v i + σ h H v γ i = + K v H Q [n] v, 3 = Q [n] C is a boc diagona matrix with the scaed identity matrix ω [n] I N as the -th main diagona boc square matrix. To find the optima v s, we tae the gradient of the agrangian function v, λ with respect to v and set it to zero, which impies Q [n] v + i i= λ h ih H i v λ h h H v = 0. 4 γ By adding λ / h h H v to both sides of 4, we have K Q [n] λ i + h ih H i v = λ + γ h h H v, 5 which impies v = Q [n] + K i= λ i/h i h H i h λ / + /γ h H v. As λ / + /γ h H v is a scaar, the optima v must be parae to the beamforming direction v = Q [n] + K i= λ i h ih H i h,. 6 Therefore, the optima beamforming vectors v,..., v K can be written as p v v =,, 7 v p denotes the optima beamforming power. Observing that the beamforming powers p s need to satisfy the SINR constraints with equaity [0], i.e., p h H v γ v p i h H v i v i = σ,, 8 i the optima powers thus can be obtained via soving the foowing inear equation: p. = M p K [M] ij = σ. σ K h H i vi γ i v i h H i vj v j, 9, i = j,, i j. Here, [M] ij denotes the i, j-th eement of the matrix M R K K.

6 6 To find the optima λ-parameter, by mutipying both sides Q [n] + K λ i i= h ih H i in 5, we obtain h H v by h H as λ + γ h H Q [n] + which impies λ = + γ h H K i= Q [n] + λ i h ih H i K i= λ i h ih H i h h H v, 0 h. These fixed-point equations can be computed using iterative function evauation. Based on 7, 9 and, we obtain the squared -norm of the optima soution to probem P [n] GSBF as foows: ṽ [n] K = v H Q v, = Q C is a boc diagona matrix with the identity matrix I N as the -th main diagona boc square matrix and zeros ese. Note that the optima powers p s 9 and agrangian mutipiers λ s shoud depend on the weights ω [n] s at each iteration n. The instantaneous CSI based iterative reweighted- agorithm for group sparse beamforming is presented in Agorithm. We have the foowing resut for its performance and convergence. Agorithm : Instantaneous CSI based Iterative Reweighted- Agorithm for Probem P GSBF input: Initiaize ω [0] =,..., ; I the maximum number of iterations Repeat Compute the squared -norm of the soution to probem P [n] GSBF, ṽ[n],..., ṽ [n], using 7, 9, and. Update the weights ω [n+] using. Unti convergence or attain the maximum iterations and return output. output: RRH ordering criterion θ 7. Theorem : et {v [n] } n= be the sequence generated by the iterative reweighted- agorithm. Then, every imit point v of {v [n] } n= has the foowing properties: v is a KKT point of probem P GSBF 9; g p v [n] ; ɛ converges monotonicay to g p v ; ɛ for some KKT point v. Proof: Pease refer to Appendix A for detais. Compared with the agorithm for the smoothed- p minimization in [5], the main novety of the proposed iterative reweighted- agorithm is reveaing the expicit structures of the soutions at each iteration in Agorithm. Instead of using the interior-point agorithm to sove the convex subprobems at each iteration [46], Agorithm with cosed-form soutions heps reduce the computationa cost. The proposed iterative reweighted- agorithm can ony guarantee to converge to a KKT point, which may be a oca minimum or the other stationary point e.g., a sadde point and oca maximum. Remar : The main contributions of the deveoped Agorithm incude finding the cosed-form soutions in the iterations, in comparison with the wor [5] using the interiorpoint agorithm, and proving the convergence of the owcompexity cosed-form iterative agorithm for the non-convex group sparse inducing probem P GSBF 9, instead of the convex coordinated beamforming probem [45]. Unfortunatey, computing the RRH ordering criterion θ s in 7 requires to run Agorithm for each channe reaization, which brings a heavy computation burden. In next section, we wi find deterministic approximations for θ s to determine the RRH ordering ony based on statistica CSI, which changes much more sowy than instantaneous channe states, and thus requires ess frequent update. IV. GROUP SPARSE BEAMFORMING VIA ARGE SYSTEM ANAYSIS In this section, we present the arge system anaysis for the iterative reweighted- agorithm in Agorithm, thereby enabing RRH ordering ony based on statistica CSI. In this way, the ordering criterion wi change ony when the ong-term channe attenuation is updated, and thus it can further reduce the computationa compexity of group sparse beamforming. The main novety of this section is providing the cosed-forms for the asymptotic anaysis of the optima agrangian mutipiers based on the recent resuts in [7], thereby providing expicit expressions for the asymptotic RRH ordering resuts. A. Deterministic Equivaent of Optima Parameters In this subsection, we provide asymptotic anaysis for the optima beamforming parameters p s 9 and λ s when N. In a Coud-RAN with distributed RRHs, the channe can be modeed as h = Θ / g,, g CN 0, I is the sma-scae fading and Θ = diag{d,..., d } I N with d as the path-oss from RRH to MU. With this channe mode, we have the foowing resut for the deterministic equivaent of the λ-parameter. emma Asymptotic Resuts for λ-parameter: Assume 0 < im inf K/N im sup K/N <. et {d } and {γ } satisfy im sup N max, {d } < and im sup N max γ <, respectivey. We have max K λ λ 0 amost surey, λ is given by λ = γ d η. 3 = Here, η is the unique soution of the foowing set of equations: K d i γ i η = j= d + ω [n]. 4 ijη j + γ i i=

7 7 Proof: Pease refer to Appendix B for detais. Based on the above resuts, we further have the foowing asymptotic resut for the optima powers p s. emma Asymptotic Resuts for Optima Powers: et C K K be such that [ ],i := γ i ψ i + γ i ψ. 5 If and ony if im sup K <, then max p p 0 amost surey, p is given by p ψ τ = γ ψ + γ + σ. 6 Here ψ, ψ and ψ i are given as foows: ψ = d η, 7 and ψ = and ψ i = = = d η + d i d η + K j= = i λ j ψ j + γ j K j= λ j ψ j + γ j d i d j η, 8 = d i d j η,9 = respectivey; and τ = [τ,..., τ K ] T is given as δ = [δ,..., δ K ] T with τ = σ I K δ, 30 δ = K ψ i γ i. 3 i= ψi Proof: Pease refer to Appendix C for detais. B. Statistica CSI Based Group Sparse Beamforming Agorithm Based on the deterministic equivaents of the optima beamforming parameters λ s and p s in emma and emma, respectivey, we have the foowing theorem on the squared -norm of the soution to probem P [n] GSBF, i.e., ṽ [n],..., ṽ [n]. Theorem Asymptotic Resuts for the Beamformers: At the n-th iteration, for the squared -norm of the soution to probem P [n] GSBF, ṽ[n],..., ṽ [n], we have max ṽ [n] χ 0 amost surey with χ = K = p K λ j ψ j j= +γ j ψ ψ, 3 ψ = d η + = d id j η. Proof: Pease refer to Appendix D for detais. We thus have the foowing deterministic equivaent of the weights ω [n] s : ω [n] = pν [ χ + ɛ ] p, =,...,. 33 Note that the asymptotic resuts of the powers p s 6 and agrangian mutipiers λ s 3 shoud depend on the weights ω [n] s 33 at each iteration. Based on 7, we have the foowing asymptotic resut for the RRH ordering criterion: θ = κ χ, 34 χ is the deterministic equivaent of the squared - norm of the soution to probem P GSBF using the iterative reweighted- agorithm. Therefore, the RRH with a smaer θ wi have a higher priority to be switched off. This ordering criterion wi change ony when the ong-term channe attenuation is updated. Note that goba instantaneous CSI is sti needed in stage II to find the active RRHs, i.e., soving a sequence of convex feasibiity probems F A [i] 8. The statistica CSI based iterative reweighted- agorithm for group sparse beamforming is presented in Agorithm. Agorithm : Statistica CSI based Iterative Reweighted- Agorithm for Probem P GSBF input: Initiaize ω [0] =,..., ; I the maximum number of iterations Repeat Compute deterministic equivaent of the squared -norm of the soution to P [n] GSBF, ṽ[n],..., ṽ [n] using χ [n] 3. Update the weights ω [n+], using 33. Unti convergence or attain the maximum iterations and return output. output: Asymptotic resut of the RRH ordering criterion θ 34. Remar 3: Athough stage and stage 3 sti require instantaneous CSI, the RRH ordering criterion computed by Agorithm in stage is ony based on statistica CSI and thus wi be updated ony when the ong-term channe propagation is changed. Therefore, Agorithm serves the purpose of further reducing the computation compexity of Agorithm for RRH ordering. A promising future research direction is to seect RRHs in stage ony based on statistica CSI. However, the main chaenge is the infeasibiity issue if ony statistica CSI is avaiabe, as discussed in Remar. V. SIMUATION RESUTS In this section, we wi simuate the proposed agorithms for networ power minimization for Coud-RANs. In a the reaizations, we ony account the channe reaizations maing the origina probem P feasibe. If probem P is infeasibe, further wors on user admission are required [5]. We set p =, ɛ = 0 3 and ω [0] =,..., for a the simuations. The proposed iterative reweighted- agorithms Agorithm and Agorithm wi terminate if either the number of iterations exceeds 30 or the difference between the objective vaues of consecutive iterations is ess than 0 3.

8 Objective Vaues Average Networ Power Consumption [W] Agorithm Agorithm Group Sparse Beamforming [7] Enhanced Group Sparse Beamforming with Agorithm Enhanced Group Sparse Beamforming with Agorithm 5.5 Scenario Scenario Scenario Iteration Target SINR [db] Fig.. Convergence of the iterative reweighted- agorithm. TABE I NETWORK POWER CONSUMPTION USING AGORITHM AND AGORITHM WITH DIFFERENT PARAMETERS p Target SINR [db] GSBF [7] Ag. with p = Ag. with p = Ag. with p = Ag. with p = A. Convergence of the Iterativey Reweighted- Agorithm Consider a networ with = 5 30-antenna RRHs and 5 singe antenna MUs uniformy and independenty distributed in the square region [ 000, 000] [ 000, 000] meters. Fig. shows the convergence of the iterative reweighted- agorithm in different scenarios with different reaizations of RRH and MU positions. For each scenario, the numerica resuts are obtained by averaging over 00 sma-scae fading reaizations. The arge-scae fading i.e., statistica CSI is randomy generated and fixed during simuations. In the two curves of each scenario, Agorithm and Agorithm are appied for Stage I of group sparse beamforming, respectivey. It demonstrates that the arge system anaysis based Agorithm provides accurate approximation even in a sma system. Fast convergence is observed in the simuated setting. B. Networ Power Minimization Consider a networ with = 5 0-antenna RRHs and 6 singe antenna MUs uniformy and independenty distributed in the square region [ 000, 000] [ 000, 000] meters. The reative transport in power consumption are set to be = W, =,...,. We average over 00 smascae channe reaizations. Fig. 3 demonstrates the average networ power consumption using different agorithms. This figure shows that the proposed iterative reweighted- agorithm outperforms the weighted mixed / -norm agorithm in [7]. Furthermore, it iustrates that the arge system anaysis P c Fig. 3. Average networ power consumption versus target SINR with different agorithms. provides accurate approximations for the iterative reweighted- agorithm in finite systems with reduced computation overhead. In Tabe I, for the simuated scenarios, we see that the iterative reweighted- agorithms achieve simiar performance with different vaues of parameter p and the arge system anaysis provides accurate approximations. Athough the simuated resuts demonstrate that the performance is robust to different vaues of parameter p, it is very interesting to theoreticay identify the typica scenarios, smaer vaues of parameter p wi yied much ower networ power consumption. VI. CONCUSIONS AND DISCUSSIONS In this paper, we deveoped an enhanced three-stage group sparse beamforming framewor with reduced computation overhead in Coud-RANs. In particuar, cosed-form soutions for the group sparse optimization probem were obtained by deveoping the iterative reweighted- agorithm based on the MM agorithm and agrangian duaity theory. This is the first effort to reduce the computation cost for RRH ordering in the first stage of group sparse beamforming. Based on the deveoped structured iterative agorithm, we further provided a arge system anaysis for the optima agrangian mutipiers at each iteration via random matrix theory, thereby computing the RRH ordering criterion ony based on the statistica CSI. This is the second effort to enabe computation scaabiity for RRH ordering in the first stage. Severa future directions of interest are isted as foows: Athough computing the RRH ordering criterion in the first stage with Agorithm ony needs statistica CSI, the second stage sti requires goba instantaneous CSI to find the active RRHs by soving a sequence of convex feasibiity probems. It is thus particuary interesting to investigate efficient agorithms to seect the active RRHs ony based on the statistica CSI. It is desirabe to estabish the optimaity of the iterative reweighted- agorithm for the networ power minimization probem P. However, considering the compi-

9 9 cated probem structures of probem P, this becomes chaenging. It is aso interesting to appy the deveoped statistica CSI based iterative reweighted- agorithm for more compicated networ optimization probems, e.g., the wireess caching probem [33], [34], the computation offoading probem [36], and beamforming probems with CSI uncertainty [3]. APPENDIX A PROOF OF THEOREM Observing that the phases of v wi not change the objective function and constraints of probem P GSBF, the QoS constraints thus can be equivaenty transformed { to the second-order cone constraints: } C = v : i hh v i + σ RhH v /γ,, which are convex cones. The KKT points of probem P GSBF shoud satisfy: 0 v g p v; ɛ + N C v, 35 N C v is the norma cone of the second-order cone at point v [47]. We sha show that any convergent subsequence {v [n ] } = of {v[n] } n= satisfies 35. Specificay, et v [n ] v be one such convergent subsequence with im v[n +] = im v[n ] = v. 36 Based on the strong duaity resut for probem P [n] GSBF, the foowing KKT condition hods at v [n +] : 0 v Qv [n +] ; ω [n ] + N C v [n +]. 37 Based on 36 and, we have im v Qv [n +] ; ω [n ] = v g p v; ɛ. Furthermore, based on [47, Proposition 6.6] and 36, we have im sup v [n +] v N Cv [n +] = N C v. Therefore, by taing in 37, we have 0 v Q v; ω + N C v, which indicates that v is a KKT point of probem P GSBF. We thus compete the proof. We first concude the foowing fact: g p v [n+] ; ɛ = Qv [n+] ; ω [n] + g p v [n+] ; ɛ Qv [n+] ; ω [n] Qv [n+] ; ω [n] + g p v [n] ; ɛ Qv [n] ; ω [n] Qv [n] ; ω [n] + g p v [n] ; ɛ Qv [n] ; ω [n] = g p v [n] ; ɛ, 38 the first inequaity is based on the fact that function g p v; ɛ Qv; ω [n] attains its maximum at v = v [n] [5, Proposition ], and the second inequaity foows from 37. Furthermore, as g p v; ɛ is continuous and C is compact, the imit of the sequence g p v [n] ; ɛ is finite. Based on the convergence resuts in, we thus compete the proof. APPENDIX B PROOF OF EMMA The proof technique here is mainy based on the wor [6]. However, we have to modify their proof as we have different weighting matrices Q [n] in vectors v 6 at the n-th iteration and such modification is non-trivia. Based on the foowing matrix inversion emma [48], x H U+cxx H = xh U +cx H U x, can be rewritten as γ = λ hh λ i h i h H i + Q h, 39 i which can be further rewritten as γ ρ = λ hh λ i h i h H i + Q ρ i i h, 40 ρ = λ /λ. Assume that 0 ρ ρ ρ K. From 40, repacing ρ i with ρ K and using monotonicity arguments, we have γ K ρ K λ K hh K or, equivaenty γ K λ K hh K λ i h i h H i + Q ρ K i K λ i h i h H i + ρ K Q i K h K, 4 h K. 4 Assume now that ρ K is infinitey often arger than + with being some positive vaue [7]. et us restrict ourseves to such a subsequence. From 4, using monotonicity arguments we obtain γ K λ K hh K Define m = hh λ i h i h H i + + Q h K. 43 i K λ i h i h H i + + Q h. 44 i Now we investigate the deterministic equivaents for m. We first introduce the foowing important emma: emma 3: [3, emma 4.] et A, A,..., with A N C N N be a series of random matrices generated by the probabiity space Ω, F, P such that, for ω A Ω, with P A =, A N ω < Kω <, uniformy on N. et x, x,..., with x N C N, be random vectors of i.i.d. entries with zero mean, variance /N, and the eighth-order moment of order O/N 4, independent of A N. Then x H NA N x N N TrA N 0, 45 amost surey. Therefore, based on emma 3, we have m K TrΘ K λ i h i h H i + + Q i K We further introduce the foowing important emma to derive the deterministic equivaent for the m.

10 0 emma 4: [4, Theorem ] et B N = X H N X N + S N with S N C N N Hermitian nonnegative definite and X N C n N random. The i-th coumn x i of X H N is x i = Ψ i y i, the entries of y i C ri are i.i.d. of zero mean, variance /N and have eighth-order moment of order O/N 4. The matrices Ψ i C N ri are deterministic. Furthermore, et Θ i = Ψ i Ψ H i C N N and define Q N C N N deterministic. Assume im sup sup i n Θ i < and et Q N have uniformy bounded spectra norm with respect to N. Define m BN,Q N z := N TrQ N B N zi N. 47 Then, for z C\R +, as n, N grow arge with ratios β N,i := N/r i and β N := N/n such that 0 < im inf N β N im sup N β N < and 0 < im inf N β N,i im sup N β N,i <, we have that m BN,Q N z m B N,Q N z 0 48 amost surey, with m B N,Q N z given by m B N,Q N z = N TrQ N N n Θ j + S N zi N + e N,jz j=,49 the functions e N, z,..., e N,n z from the unique soution of e N,i z = N TrΘ n Θ j i N + e N,j z + S N zi N,50 j= which is the Stietjes transform of a nonnegative finite measure on R +. Moreover, for z < 0, the scaars e N, z,..., e N,n z are unique nonnegative soutions to 50. Therefore, based on emma 4, we have ψ K ψ K 0 5 amost surey with ψk being the unique positive soution to ψk = TrΘ KA 5 A = K λ i Θ i + λ i ψi + + Q. 53 i= From 5, recaing 43 yieds im inf K ψ K γ K /λ K 54 Using the fact that ψk is a decreasing function of, it can be proved [3] that for any > 0, we have im sup K ψ K < γ K /λ K. 55 However, this is against the former condition and creates a contradiction on the initia hypothesis that ρ K < + infinitey often. Therefore, we must admit that ρ K + for a arge vaues of K. Reverting a inequaities and using simiar arguments yieds ρ K for a arge vaues of K. We eventuay obtain that ρ K + from which we may write max ρ K for a arge vaues of K. Taing a countabe sequence of going to zero yieds max ρ K 0 from which using ρ K = λ K /λ K and assuming im K sup γ K <, we obtain max λ K λ K 0, 56 amost surey. From 40, we have γ K /λ K = ψ K 0 = ψ K. Therefore, we obtain λ K = γ K/ψK, ψ K is the soution of the foowing equation: ψk = TrΘ K Θ i γ i K + Q γ i ψ i= K Foowing the same steps for =,..., K yieds the foowing desired resut: λ = γ /ψ,. 58 As Θ = diag{d,..., d } I N, we have ψ = TrΘ A = d η, 59 A = A0 in 53 and {η } is the unique positive soution to the foowing set of equations K d i η = i= / γ i j= d + ω [n]. 60 jη j + γ i We thus compete the proof for the asymptotic resuts of the optima agrange mutipiers in emma. = APPENDIX C PROOF OF EMMA The SINR for MU can be rewritten as sinr = K i p h H ṽ ṽ p i h H ṽ ṽi i + σ, 6 λ ṽ = i i= h ih H i + Q/ h. The interference part of the denominator of Γ in 6 can be rewritten as h H p ṽi i ṽ i = p i h H ṽi ṽ H i i ṽ i h i = hh V H[] P [] H []H Vh 6 K, with V = i= λ i h ih H i + Q H [] := [h,.{.., h, h +,..., h K ] C K and P [] := } p diag p,..., p ṽ, + p ṽ,..., K ṽ +. ṽ K In order to eiminate the dependence between h and V, rewrite 6 as hh V H[] P [] H []H Vh = hh V [] H[] P [] H []H Vh + hh V V [] H[] P [] H []H Vh, 63

11 V [] = i λ i h ih H i + Q. Using the resovent identity [48] i.e., U V = U U VV with U and V as two invertibe compex matrices of size N N, we have V V [] = VV V [] V []. 64 Then, observing that V V [] = λ h h H. From 63, one gets hh V H[] P [] H []H Vh = hh V [] H[] P [] H []H Vh [ ] λ hh Vh hh V [] H[] P [] H []H Vh. 65 emma 5: [4, emma 7] et U, V, Θ C N N be of uniformy bounded spectra norm with respect to N and et V be invertibe. Further, define x := Θ / z and y := Θ / q, z, q C N have i.i.d. compex entries of zero mean, variance /N, and finite eighth-order moment and be mutuay independent as we as independent of U, V. Define c 0, c, c R + such that c 0 c c 0 and et µ := N TrΘV and µ := N TrΘUV. Then, we have x H UV + c 0 xx H + c yy H + c xy H + c yx H x µ + c µ c 0 c c 0 66 µ + c 0 + c µ + amost surey. Furthermore, x H UV + c 0 xx H + c yy H + c xy H + c yx H y c µµ c 0 c c 0 67 µ + c 0 + c µ + amost surey. Therefore, appying emma 5, we obtain that hh V [] H[] P [] H []H µ Vh + λ µ 0 68 amost surey, µ = Tr Θ V [] and µ = Tr P[] H []H V [] Θ V [] H []. Furthermore, we have λ [ ] hh Vh hh V [] H[] P [] H []H Vh λ µµ + λ 0 69 µ amost surey. Based on the resuts in 65, 68, 69, we have hh V H[] P [] H []H µ Vh + λ 0 µ amost surey. From emma, we have µ ψ 0 amost surey. 70 Appying [4, emma 6], we have Tr P[] H []H V [] Θ V [] H [] Tr P[] H []H VΘ VH [] amost surey. We further rewrite Tr P[] H []H VΘ VH [] = i 0 7 p i hh i VΘ Vh i hh i V h i. 7 Appying [4, emma, 4 and 6], we obtain amost surey hh i VΘ Vh i TrΘ ivθ V [ + λ i TrΘ iv ] To derive a deterministic equivaent for tr Θ ivθ V, we write Tr Θ ivθ V = z Tr Θ i V zθ z=0. Observe now that [4, Theorem ] Tr Θ i V zθ ψ i z 0, 75 amost surey, ψ i z is given by ψ i z = TrΘ it z and T z is computed as T z = K λ j Θ j + λ j ψ jz + Q zθ j=. 76 By differentiating aong z, we have ψ i z = TrΘ it z, T z = T z z is given by T z = T z K λ j ψ j zθ j + λ ψ jz + Θ T z. Setting z = 0 yieds T 0 = T 0 j= 77 ψ i0 = TrΘ it 0, 78 K λ j ψ j 0Θ + γ + Θ T 0.79 j= In writing the above resut, we have taen into account that A = T j 0, ψ j 0 = TrΘ jt 0 = TrΘ ja = ψj and γ j = λ j ψ j. Pugging 79 into 78 and negecting the 74

12 functiona dependence from z = 0, ψ = [ψ..., ψ K ]T is found as the unique soution of ψ i = Tr Θ i A K λ j ψ j Θ j + γ j + Θ A j= = TrΘ iaθ A + K λ j ψ j + γ j Tr Θ iaθ j A. 80 j= Observing that TrΘ iaθ j A = d i d j η. 8 et ψ = [ψ,..., ψ K ]T C K and J = [J ij ] C K K with λ j J ij = + γ j d i d j η. 8 We can rewrite the above system of equations in the compact form as = = ψ = c + Jψ, 83 c = [c i ] with c i = = d id j η. On the basis of above resuts, using λ i trθ ia γ i 0, we eventuay obtain that hh i VΘ Vh i ψ i + γ i 0 84 amost surey. Foowing simiar arguments of above yieds hh i V ψ i h i + γ i 0 85 amost surey with ψ = [ψ,..., ψ K ]T = I K J c c C K has eements [c] i = = d iη. Putting the resuts in 84 and 85 together, we obtain amost surey i p hh i VΘ Vh i i hh i V h i i p i ψ i ψ i 0.86 The deterministic equivaent of the numerator of the SINR in 6 is now easiy obtained as p hh v p ψ hh V h ψ +γ 0, 87 since hh v ψ 0 and hh V h ψ +γ 0 amost surey. Therefore, the deterministic equivaent SINR is given by I := sinr = ψ ψ + γ p I +, 88 σ K i= p i ψ i ψ i. 89 From 88, it foows that p such that sinr = γ is obtained as p ψ = γ I + σ, 90 ψ which can be further rewritten as p ψ ψ γ = I + σ = + γ K i= p i ψ i ψ i + σ9. Therefore, the deterministic equivaent for the powers is given by p. p K [M ] ij = = M ψ i γ i ψ i σ. σ K,, i = j, ψ ji +γi ψ, i j. j We thus compete the proof for the asymptotic resuts for optima powers in emma. We rewrite ṽ as ṽ = K = = APPENDIX D PROOF OF THEOREM v H Q v = K = K = p ṽ H Q ṽ ṽ p h H VQ Vh h H V h, 9 Q C is a boc diagona matrix with the identify matrix I N as the -th main diagona boc square matrix and zeros ese. Based on the simiar arguments in Appendix C, we have the deterministic equivaents h H VQ Vh ψ + γ 0, 93 ψ = TrΘ AQ A = K λ j ψ j + γ j Tr Θ iaθ j A. 94 Observing that j= TrΘ AQ A = d η, 95 we obtain the fina resut. We thus compete the proof for the asymptotic resuts for ṽ [n] s in Theorem.

13 3 REFERENCES [] Y. Shi, J. Zhang, and K. B. etaief, Statistica group sparse beamforming for green coud-ran via arge system anaysis, in Proc. IEEE Int. Symp. Inform. Theory ISIT, pp , Ju. 06. [] N. Bhushan, J. i, D. Maadi, R. Gimore, D. Brenner, A. Damnjanovic, R. Suhavasi, C. Pate, and S. Geirhofer, Networ densification: the dominant theme for wireess evoution into 5G., IEEE Commun. Mag., vo. 5, pp. 8 89, Feb. 04. [3] Y. Shi, J. Zhang, K. etaief, B. Bai, and W. Chen, arge-scae convex optimization for utra-dense Coud-RAN, IEEE Wireess Commun. Mag., vo., pp. 84 9, Jun. 05. [4] J. G. Andrews, X. Zhang, G. D. Durgin, and A. K. Gupta, Are we approaching the fundamenta imits of wireess networ densification?, IEEE Commun. Mag., vo. 54, pp , Oct. 06. [5] J. Andrews, S. Buzzi, W. Choi, S. Hany, A. ozano, A. Soong, and J. Zhang, What wi 5G be?, IEEE J. Se. Areas Commun., vo. 3, pp , Jun. 04. [6] M. Peng, Y. i, J. Jiang, J. i, and C. Wang, Heterogeneous coud radio access networs: a new perspective for enhancing spectra and energy efficiencies, IEEE Wireess Commun. Mag., vo., pp. 6 35, Dec. 04. [7] Y. Shi, J. Zhang, and K. B. etaief, Group sparse beamforming for green Coud-RAN, IEEE Trans. Wireess Commun., vo. 3, pp , May 04. [8] D. Wubben, P. Rost, J. Bartet, M. aam, V. Savin, M. Gorgogione, A. Deorsy, and G. Fettweis, Benefits and impact of coud computing on 5G signa processing: Fexibe centraization through Coud-RAN, IEEE Signa Process. Mag., vo. 3, pp , Nov. 04. [9] D. Gesbert, S. Hany, H. Huang, S. Shamai Shitz, O. Simeone, and W. Yu, Muti-ce MIMO cooperative networs: A new oo at interference, IEEE J. Se. Areas Commun., vo. 8, pp , Sep. 00. [0] M. Peng, C. Wang, V. au, and H. V. Poor, Fronthau-constrained coud radio access networs: Insights and chaenges, IEEE Wireess Commun., vo., pp. 5 60, Apr. 05. [] S. Par, O. Simeone, O. Sahin, and S. Shamai Shitz, Fronthau compression for coud radio access networs: Signa processing advances inspired by networ information theory, IEEE Signa Process. Mag., vo. 3, pp , Nov. 04. [] S. eyffer, Mixed integer noninear programming, vo. 54. Springer, 0. [3] E. J. Candes, M. B. Wain, and S. P. Boyd, Enhancing sparsity by reweighted minimization, J. Fourier Ana. App., vo. 4, pp , Dec [4] I. Daubechies, R. DeVore, M. Fornasier, and C. S. Güntür, Iterativey reweighted east squares minimization for sparse recovery, Communications on Pure and Appied Mathematics, vo. 63, no., pp. 38, 00. [5] Y. Shi, J. Cheng, J. Zhang, B. Bai, W. Chen, and K. B. etaief, Smoothed p-minimization for green Coud-RAN with user admission contro, IEEE J. Se. Areas Commun., vo. 34, pp , Apr. 06. [6] Y. Shi, J. Zhang, B. O Donoghue, and K. etaief, arge-scae convex optimization for dense wireess cooperative networs, IEEE Trans. Signa Process., vo. 63, pp , Sept. 05. [7] S. Boyd, N. Parih, E. Chu, B. Peeato, and J. Ecstein, Distributed optimization and statistica earning via the aternating direction method of mutipiers, Found. Trends in Mach. earn., vo. 3, pp., Ju. 0. [8] B. O Donoghue, E. Chu, N. Parih, and S. Boyd, Conic optimization via operator spitting and homogeneous sef-dua embedding, J. Optim. Theory App., pp. 7, Feb. 06. [9] D. R. Hunter and K. ange, A tutoria on MM agorithms, Amer. Statistician, vo. 58, no., pp , 004. [0] W. Yu and T. an, Transmitter optimization for the muti-antenna downin with per-antenna power constraints, IEEE Trans. Signa Process., vo. 55, no. 6, pp , 007. [] B. Dai and W. Yu, Sparse beamforming and user-centric custering for downin coud radio access networ, IEEE Access, vo., pp , Nov. 04. [] A. M. Tuino and S. Verdu, Random matrix theory and wireess communications, Found. Trends in Commun. and Inf. Theory, vo., no., pp. 8, 004. [3] R. Couiet and M. Debbah, Random matrix methods for wireess communications. Cambridge University Press, 0. [4] S. Wagner, R. Couiet, M. Debbah, and D. T. M. Soc, arge system anaysis of inear precoding in correated MISO broadcast channes under imited feedbac, IEEE Trans. Inf. Theory, vo. 58, pp , Ju. 0. [5] R. Zahour and S. V. Hany, Base station cooperation on the downin: arge system anaysis, IEEE Trans. Inf. Theory, vo. 58, pp , Apr. 0. [6]. Sanguinetti, R. Couiet, and M. Debbah, arge system anaysis of base station cooperation for power minimization, IEEE Trans. Wireess Commun., vo. 5, pp , Aug. 06. [7] R. Couiet and M. McKay, arge dimensiona anaysis and optimization of robust shrinage covariance matrix estimators, Journa of Mutivariate Anaysis, vo. 3, pp. 99 0, 04. [8] J. Zhao, T. Q. Que, and Z. ei, Coordinated mutipoint transmission with imited bachau data transfer, IEEE Trans. Wireess Commun., vo., pp , Jun. 03. [9] M. Hong, R. Sun, H. Baigh, and Z.-Q. uo, Joint base station custering and beamformer design for partia coordinated transmission in heterogeneous networs, IEEE J. Se. Areas Commun., vo. 3, pp. 6 40, Feb. 03. [30] S. uo, R. Zhang, and T. J. im, Downin and upin energy minimization through user association and beamforming in C-RAN, IEEE Trans. Wireess Commun., vo. 4, pp , Jan. 05. [3] B. Dai and W. Yu, Energy efficiency of downin transmission strategies for coud radio access networs, IEEE J. Se. Areas Commun., vo. 34, pp , Apr. 06. [3] Y. Shi, J. Zhang, and K. etaief, Robust group sparse beamforming for muticast green Coud-RAN with imperfect CSI, IEEE Trans. Signa Process., vo. 63, pp , Sept. 05. [33] X. Peng, J.-C. Shen, J. Zhang, and K. B. etaief, Joint data assignment and beamforming for bachau imited caching networs, in Proc. IEEE Int. Symp. on Persona Indoor and Mobie Radio Comm. PIMRC, Washington, DC, Sep. 04. [34] M. Tao, E. Chen, H. Zhou, and W. Yu, Content-centric sparse muticast beamforming for cache-enabed Coud RAN, IEEE Trans. Wireess Commun., vo. 5, pp , Sep. 06. [35] J. Zhao, T. Q. Que, and Z. ei, Heterogeneous ceuar networs using wireess bachau: Fast admission contro and arge system anaysis, IEEE J. Se. Areas Commun., pp. 8 43, Oct. 05. [36] J. Cheng, Y. Shi, B. Bai, and W. Chen, Computation offoading in coud-ran based mobie coud computing system, in IEEE Int. Conf. on Commun. ICC, Kuaa umpur, Maaysia, pp. 6, May 06. [37] W.-C. iao, M. Hong, Y.-F. iu, and Z.-Q. uo, Base station activation and inear transceiver design for optima resource management in heterogeneous networs, IEEE Trans. Signa Process., vo. 6, pp , Aug. 04. [38] F. Ruse, D. Persson, B. K. au, E. arsson, T. Marzetta, O. Edfors, and F. Tufvesson, Scaing up MIMO: Opportunities and chaenges with very arge arrays, IEEE Signa Process. Mag., vo. 30, pp , Jan. 03. [39] R. Couiet and M. Debbah, Signa processing in arge systems: A new paradigm, IEEE Signa Process. Mag., vo. 30, pp. 4 39, Jan. 03. [40] J. A. Tropp, An introduction to matrix concentration inequaities, Found. Trends in Mach. earn., vo. 8, pp. 30, May 05. [4] R. Couiet, G. Wainrib, H. T. Ai, and H. Sevi, A random matrix approach to echo-state neura networs, in Proc. Int. Conf. Mach. earn. ICM, 33th, pp , 06. [4] S. Bi, R. Zhang, Z. Ding, and S. Cui, Wireess communications in the era of big data, IEEE Commun. Mag., vo. 53, pp , Oct. 05. [43] S. ashminarayana, M. Assaad, and M. Debbah, Coordinated mutice beamforming for massive MIMO: A random matrix approach, IEEE Trans. Inf. Theory, vo. 6, pp , Jun. 05. [44] K. ange and J. S. Sinsheimer, Norma/independent distributions and their appications in robust regression, J. Comput. Graph. Stat., vo., no., pp , 993. [45] H. Dahrouj and W. Yu, Coordinated beamforming for the mutice muti-antenna wireess system, IEEE Trans. Wireess Commun., vo. 9, pp , Sep. 00. [46] S. P. Boyd and. Vandenberghe, Convex optimization. Cambridge University Press, 004. [47] R. T. Rocafear, Convex anaysis, vo. 8. Princeton university press, 997. [48] R. A. Horn and C. R. Johnson, Matrix anaysis. Cambridge university press, 0.