Group Sparse Beamforming for Green Cloud-RAN

Transcription

1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 5, MAY Group Sparse Beamforming for Green Coud-RAN Yuanming Shi, Student Member, IEEE, Jun Zhang, Member, IEEE, and Khaed B. Letaief, Feow, IEEE Abstract A coud radio access network (Coud-RAN) is a network architecture that hods the promise of meeting the exposive growth of mobie data traffic. In this architecture, a the baseband signa processing is shifted to a singe baseband unit (BBU) poo, which enabes efficient resource aocation and interference management. Meanwhie, conventiona powerfu base stations can be repaced by ow-cost ow-power remote radio heads (RRHs), producing a green and ow-cost infrastructure. However, as a the RRHs need to be connected to the BBU poo through optica transport inks, the transport network power consumption becomes significant. In this paper, we propose a new framework to design a green Coud-RAN, which is formuated as a joint RRH seection and power minimization beamforming probem. To efficienty sove this probem, we first propose a greedy seection agorithm, which is shown to provide nearoptima performance. To further reduce the compexity, a nove group sparse beamforming method is proposed by inducing the group-sparsity of beamformers using the weighted 1/ 2-norm minimization, where the group sparsity pattern indicates those RRHs that can be switched off. Simuation resuts wi show that the proposed agorithms significanty reduce the network power consumption and demonstrate the importance of considering the transport ink power consumption. Index Terms Coud-RAN, green communications, coordinated beamforming, greedy seection, group-sparsity. I. INTRODUCTION MOBILE data traffic has been growing enormousy in recent years, and it is expected that ceuar networks wi have to offer a 1000x increase in capacity in the foowing decade to meet this demand [1]. Massive MIMO [2] and heterogeneous and sma ce networks (HetSNets) [1] are regarded as two most promising approaches to achieve this goa. By depoying a arge number of antennas at each base station (BS), massive MIMO can expoit spatia mutipexing gain in a arge scae and aso improve energy efficiency. However, the performance of massive MIMO is imited by correated scattering with the antenna spacing constraints, which aso brings high depoyment cost to maintain the minimum spacing [1]. HetSNets expoit the spatia reuse by depoying more and more access points (APs). Meanwhie, as stated in [3], pacing APs based on the traffic demand is an effective way for compensating path-oss, resuting in energy efficient ceuar networks. However, efficient interference Manuscript received September 26, 2013; revised January 9, 2014; accepted January 31, The associate editor coordinating the review of this paper and approving it for pubication was K. Huang. The authors are with the Department of EectronicandComputerEngineering, Hong Kong University of Science and Technoogy (e-mai: {yshiac, eejzhang, eekhaed}@ust.hk). This work was supported by the Hong Kong Research Grant Counci under Grant No The work of J. Zhang was supported by the Hong Kong RGC Direct Aocation Grant DAG11EG03. Part of this work was presented at the IEEE Goba Communications Conference (GLOBECOM), Atanta, GA, Dec Digita Object Identifier /TWC /14$31.00 c 2014 IEEE management is chaenging for dense sma-ce networks. Moreover, depoying more and more sma-ces wi cause significant cost and operating chaenges for operators. Coud radio access network (Coud-RAN) has recenty been proposed as a promising network architecture to unify the above two technoogies in order to jointy manage the interference (via coordinated mutipe-point process (CoMP)), increase network capacity and energy efficiency (via network densification), and reduce both the network capita expenditure (CAPEX) and operating expense (OPEX) (by moving baseband processing to the baseband unit (BBU) poo) [4], [5]. A arge-scae distributed cooperative MIMO system wi thus be formed. Coud-RAN can therefore be regarded as the utimate soution to the spectrum crunch probem of ceuar networks. There are three key components in a Coud-RAN: (i) a poo of BBUs in a datacenter coud, supportedbythereatime virtuaization and high performance processors, where a the baseband processing is performed; (ii) a high-bandwidth ow-atency optica transport network connecting the BBU poo and the remote radio heads (RRHs); and (iii) distributed transmission/reception points (i.e., RRHs). The key feature of Coud-RAN is that RRHs and BBUs are separated, resuting acentraizedbbupoo,whichenabesefficientcooperation of the transmission/reception among different RRHs. As a resut, significant performance improvements through joint scheduing and joint signa processing such as coordinated beamforming or muti-ce processing[6] can be achieved. With efficient interference suppression, a network of RRHs with a very high density can be depoyed. This wi aso reduce the communication distance to the mobie terminas and can thus significanty reduce the transmission power. Moreover, as baseband signa processing is shifted to the BBU poo, RRHs ony need to support basic transmission/reception functionaity, which further reduces their energy consumption and depoyment cost. The new architecture of Coud-RAN aso indicates a paradigm shift in the network design, which causes some technica chaenges for impementation. For instance, as the data transmitted between the RRHs and the BBU poo is typicay oversamped rea-time I/Q digita data streams in the order of Gbps, high-bandwidth optica transport inks with ow atency wi be needed. To support CoMP and enabe computing resource sharing among BBUs, new virtuaization technoogies need to be deveoped to distribute or group the BBUs into a centraized entity [4]. Another important aspect is the energy efficiency consideration, due to the increased power consumption of a arge number of RRHs and aso of the transport inks. Conventionay, the transport network (i.e., backhau inks

2 2810 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 5, MAY 2014 between the core network and base stations (BSs)) power consumption can be ignored as it is negigibe compared to the power consumption of macro BSs. Therefore, a the previous works investigating the energy efficiency of ceuar networks ony consider the BS power consumption [7], [8]. Recenty, the impact of the backhau power consumption in ceuar networks was investigated in [9], where it was shown through simuations that the backhau power consumption wi affect the energy efficiency of different ceuarnetworkdepoyment scenarios. Subsequenty, Rao et a. in [10] investigated the spectra efficiency and energy efficiency tradeoff in homogeneous ceuar networks when taking the backhau power consumption into consideration. In Coud-RAN, the transport network power consumption wi have a more significant impact on the network energy efficiency. Hence, aowing the transport inks and the corresponding RRHs to support the seep mode wi be essentia to reduce the network power consumption for the Coud- RAN. Moreover, with the spatia and tempora variation of the mobie traffic, it woud be feasibe to switch off some RRHs whie sti maintaining the quaity of service (QoS) requirements. It wi be aso practica to impement such an idea in the Coud-RAN with the hep of centraized signa processing at the BBU poo. As energy efficiency is one of the major objectives for future ceuar networks [5], in this paper we wi focus on the design of green Coud-RAN by jointy considering the power consumption of the transport network and RRHs. A. Contributions The main objective of this paper is to minimize the network power consumption of Coud-RAN, incuding the transport network and radio access network power consumption, with aqosconstraintateachuser.specificay,weformuatethe design probem as a joint RRH seection and power minimization beamforming probem, where the transport network power consumption is determined by the set of active RRHs, whie the transmit power consumption of the active RRHs is minimized through coordinated beamforming. This is a mixedinteger non-inear programming (MINLP) probem, which is NP-hard. We wi focus on designing ow-compexity agorithms for practica impementation. The major contributions of the paper are summarized as foows: 1) We formuate the network power consumption minimization probem for the Coud-RAN by enabing both the transport inks and RRHs to support the seep mode. In particuar, we provide a group sparse beamforming (GSBF) formuation of the design probem, which assists the probem anaysis and agorithm design. 2) We first propose a greedy seection (GS) agorithm, which seects one RRH to switch off at each step. It turns out that the RRH seection rue is critica, and we propose to switch off the RRH that maximizes the reduction in the network power consumption at each step. From the simuations, the proposed GS agorithm often yieds optima or near-optima soutions, but its compexity may sti be prohibitive for a arge-sized network. 3) To further reduce the compexity, we propose a threestage group sparse beamforming (GSBF) framework, by adopting the weighted mixed 1 / p -norm to induce the group sparsity for the beamformers. In contrast to a the previous works appying the mixed 1 / p -norm to induce group sparsity, we expoit the additiona prior information (i.e., transport ink power consumption, power ampifier efficiency, and instantaneous effective channe gains) to design the weights for different beamformer coefficient groups, resuting in a significant performance gain. Two GSBF agorithms with different compexities are proposed: namey, a bi-section GSBF agorithm and an iterative GSBF agorithm. 4) We sha show that the GS agorithm aways provides near-optima performance. Hence, it woud be a good option if the number of RRHs is reativey sma, such as in custered depoyment. With a very ow computationa compexity, the bi-section GSBF agorithm is an attractive option for a arge-scae Coud-RAN. The iterative GSBF agorithm provides a good tradeoff between the compexity and performance, which makes it a good candidate for a medium-size network. B. Reated Works Amaindesigntooappiedinthispaperisoptimizationwith the group sparsity induced norm. With the recent theoretica breakthrough in compressed sensing [11], [12], the sparsity patterns in different appications in signa processing and communications have been expoited for more efficient system design, e.g., for piot aided sparse channe estimation [13]. The sparsity inducing norms have been widey appied in highdimensiona statistics, signa processing, and machine earning in the ast decade [14]. The 1 -norm reguarization has been successfuy appied in compressed sensing [11], [12]. More recenty, mixed 1 / p -norms are widey investigated in the case where some variabes forming a group wi be seected or removed simutaneousy, where the mixed 1 / 2 -norm [15] and mixed 1 / -norm [16] are two commony used ones to induce group sparsity for their computationa and anaytica convenience. In Coud-RAN, one RRH wi be switched off ony when a the coefficients in its beamformer are set to zeros. In other words, a the coefficients in the beamformer at one RRH shoud be seected or ignored simutaneousy, which requires group sparsity rather than individua sparsity for the coefficients as commony used in compressed sensing. In this paper, we wi adopt the mixed 1 / p -norm to promote group sparsity for the beamformers instead of 1 -norm, which ony promotes individua sparsity. Recenty, there are some works [17] [19] adopting the mixed 1 / p -norm to induce groupsparsity in a arge-scae cooperative wireess ceuar network. Specificay, Hong et a. [17] adopted the mixed 1 / 2 -norm and Zhao et a. [18] used the 2 -norm to induce the group sparsity of the beamformers, which reduce the amount of the shared user data among different BSs. The squared mixed 1 / -norm was investigated in [19] for antenna seection. A of the above works simpy adopted the un-weighted mixed 1 / p -norms to induce group-sparsity, in which, no

3 SHI et a.: GROUP SPARSE BEAMFORMING FOR GREEN CLOUD-RAN 2811 BBU Poo Remote Radio Head Mobie User BBU1 BBU2 BBU3 BBU4 BBU5 Transport Links Fig. 1. The architecture of Coud-RAN, in which, a the RRHs are connected to a BBU poo through transport inks. prior information of the unknown signa is assumed other than the fact that it is sufficienty sparse. By expoiting the prior information in terms of system parameters, the weights for different beamformer coefficient groups can be more rigorousy designed and performance can be enhanced. We demonstrate through simuations that the proposed threestage GSBF framework, which is based on the weighted mixed 1 / p -norm minimization, outperforms the conventiona unweighted mixed 1 / p -norm minimization based agorithms substantiay. C. Organization The remainder of the paper is organized as foows. Section II presents the system and power mode. In Section III, the network power consumption minimization probem is formuated, foowed by some anaysis. Section IV presents the GS agorithm, which yieds near-optima soutions. The threestage GSBF framework is presented in Section V. Simuation resuts wi be presented in Section VI. Finay, concusions and discussions are presented in Section VII. Notations: p is the p -norm. Bodface ower case and upper case etters represent vectors and matrices, respectivey. ( ) T, ( ), ( ) H and Tr( ) denote the transpose, conjugate, Hermitian and trace operators, respectivey. R( ) denotes the rea part. A. System Mode II. SYSTEM AND POWER MODEL We consider a Coud-RAN with L remote radio heads (RRHs), where the -th RRH is equipped with N antennas, and K singe-antenna mobie users (MUs), as shown in Fig. 1. In this network architecture, a the base band units (BBUs) are moved into a singe BBU poo, creating a set of shared processing resources, and enabing efficient interference management and mobiity management. With the baseband signa processing functionaity migrated to the BBU poo, the RRHs can be depoyed in a arge scae with ow-cost. The BBU poo is connected to the RRHs using the common pubic radio interface (CPRI) transport technoogy via a high-bandwidth, ow-atency optica transport network [4]. In order to enabe fu cooperation among RRHs, it is assumed that a the user data are routed to the BBU poo from the core network through the backhau inks [4], i.e., a users can access a the RRHs. The digitized baseband compex inphase (I) and quadrature (Q) sampes of the radio signas are transported over the transport inks between the BBUs and RRHs. The key technica and economic issue of the Coud-RAN is that this architecture requires significant transport network resources. As the focus of this paper is on network power consumption, we wi assume a the transport inks have sufficienty high capacity and negigibe atency 1. Due to the high density of RRHs and the joint transmission among them, the energy used for signa transmission wi be reduced significanty. However, the power consumption of the transport network becomes enormous and cannot be ignored. Therefore, it is highy desirabe to switch off some transport inks and the corresponding RRHs to reduce the network power consumption based on the data traffic requirements, which forms the main theme of this work. Let L = {1,..., L} denote the set of RRH indices, A L denote the active RRH set, Z denote the inactive RRH set with A Z = L, ands = {1,..., K} denote the index set of schedued users. In a beamforming design framework, the baseband transmit signas are of the form: x = K w k s k, A, (1) where s k is a compex scaar denoting the data symbo for user k and w k C N is the beamforming vector at RRH for user k. Withoutossofgeneraity,weassumethatE[ s k 2 ]=1and s k s are independent with each other. The baseband signas x s wi be transmitted to the corresponding RRHs, but not the data information s k s [4], [21]. The baseband received signa at user k is given by y k = h H kw k s k + h H kw i s i + z k,k S, (2) A i k A where h k C N is the channe vector from RRH to user k, andz k CN(0, σk 2 ) is the additive Gaussian noise. We assume that a the users are empoying singe user detection (i.e., treating interference as noise), so that they can use the receivers with a ow-compexity and energy-efficient structure. Moreover, in the ow interference region, treating interference as noise can be optima [22]. The corresponding signa-to-interference-pus-noise ratio (SINR) for user k is hence given by A SINR k = hh k w k 2 i k A hh k w i 2 + σk 2, k S. (3) 1 The impact of imited-capacity transport inks on compression in Coud- RAN was recenty investigated in [20], [21], and its impact in our setting is eft to future work.

4 2812 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 5, MAY 2014 Each RRH has its own transmit power constraint B. Power Mode K w k 2 2 P, A. (4) The network power mode is critica for the investigation of the energy efficiency of Coud-RAN, which is described as foows. 1) RRH Power Consumption Mode: We wi adopt the foowing empirica inear mode [23] for the power consumption of an RRH: { P P rrh rrh a, = + 1 η P out, if P out > 0, Ps, rrh, out if P =0. where Pa, rrh is the active power consumption, which depends on the number of antennas N, Ps, rrh is the power consumption in the seep mode, P out is the transmit power, and η is the drain efficiency of the radio frequency (RF) power ampifier. For the Pico-BS, the typica vaues are Pa, rrh rrh =6.8W, Ps, = 4.3W,andη =1/4 [23]. Based on this power consumption mode, we concude that it is essentia to put the RRHs into seep whenever possibe. 2) Transport Network Power Consumption Mode: Athough there is no superior soution to meet the ow-cost, highbandwidth, ow-atency requirement of transport networks for the Coud-RAN, the future passive optica network (PON) can provide cost-effective connections between the RRHs and the BBU poo [24]. PON comprises an optica ine termina (OLT) that connects a set of associated optica network units (ONUs) through a singe fiber. Impementing a seep mode in the optica network unit (ONU) has been considered as the most cost-effective and promising power-saving method [25] for the PON, but the OLT cannot go into the seep mode and its power consumption is fixed[25]. Hence, the tota power consumption of the transport network is given by [25] P tn = P ot + =1 (5) P t, (6) where P ot is the OLT power consumption, P t = Pa, t and P t = Ps, t denote the power consumed by the ONU (or the transport ink ) intheactivemodeandseepmode,re- spectivey. The typica vaues are P ot =20W, Pa, t =3.85W and Ps, t =0.75W [25]. Thus, we concude that putting some transport inks into the seep mode is a promising way to reduce the power consumption of Coud-RAN. 3) Network Power Consumption: Based on the above discussion, we define P a Pa, rrh t +Pa, (P s Ps, rrh t +Ps, )asthe active (seep) power consumption when both the RRH and the corresponding transport ink are switched on (off). Therefore, the network power consumption of the Coud-RAN is given by ˆp(A) = 1 P out + P a + P s + P ot η A A Z = 1 P out + (P a P s η )+ P s + P ot A A L = K 1 w k 2 η P s + P ot, (7) A A L where P out = K w k 2 2 and P c = P a P s, and the second equaity in (7) is based on the fact Z P s = L P s A P s.givenacoud-ranwiththerrh set L, the term ( L P s + P ot ) in (7) is a constant. Therefore, minimizing the tota network power consumption ˆp(A) (7) is equivaent to minimizing the foowing re-defined network power consumption by omitting the constant term ( L P s + P ot ): p(a, w) = A K 1 w k 2 η 2 +, (8) A where w = [w11 T,...,wT 1K,...,wT L1,...,wT LK ]T.Theadvantage of introducing the term P c is that we can rewrite the network power consumption mode (7) in a more compact form as in (8) and extract the reevant parameters for our system design. In the foowing discussion, we refer to P c as the reative transport ink power consumption for simpification. Therefore, the first part of (8) is the tota transmit power consumption and the second part is the tota reative transport ink power consumption. Note 1: The re-defined network power consumption mode (8) reveas two key design parameters: the transmit power consumption ( 1 K η w k 2 2 ) and the reative transport ink power consumption.withthetypicavauesprovided in Section II-B1 and Section II-B2, the maximum transmit 1 power consumption, i.e., η P out =4W,iscomparabewith the reative transport ink power consumption, i.e., = P a P s = (Pa, rrh + Pa, t rrh ) (Ps, + Ps, t ) = 5.6W.This impies that a joint RRH seection (and the corresponding transportink seection) and power minimization beamforming is required to minimize the network power consumption. III. PROBLEM FORMULATION AND ANALYSIS Based on the power consumption mode, we wi formuate the network power consumption minimization probem in this section. A. Power Saving Strategies and Probem Formuation The network power consumption mode (8) indicates the foowing two strategies to reduce the network power consumption: Reduce the transmission power consumption; Reduce the number of active RRHs and the corresponding transport inks. However, the two strategies confict with each other. Specificay, in order to reduce the transmission power consumption, more RRHs are required to be active to expoit a

5 SHI et a.: GROUP SPARSE BEAMFORMING FOR GREEN CLOUD-RAN 2813 higher beamforming gain. On the other hand, aowing more RRHs to be active wi increase the power consumption of transport inks. As a resut, the network power consumption minimization probem requires a joint design of RRH (and the corresponding transport ink) seection and coordinated transmit beamforming. In this work, we assume perfect channe state information (CSI) avaiabe at the BBU poo. With target SINRs γ = (γ 1,...,γ K ),thenetworkpowerconsumptionminimization probem can be formuated as P :minimize {w k },A subject to p(a, w) A hh k w k 2 i k A hh k w i 2 + σk 2 γ k, K w k 2 2 P, A. (9) Probem P is a joint RRH set seection and transmit beamforming probem, which is difficut to sove in genera. In the foowing, we wi anayze and reformuate it. B. Probem Anaysis We first consider the case with a given active RRH set A for probem P, resutinganetworkpowerminimization probem P(A). Letw k =[wk T ]T C A N indexed by A, andh k =[h T k ]T C A N indexed by A, such that h H k w k = A hh k w k. Sincethephasesofw k wi not change the objective function and constraints of P(A) [26], the SINR constraints are equivaent to the foowing second order cone (SOC) constraints: C 1 (A) : i k hh k w i 2 + σ 2 k 1 γk R(h H k w k ),k S. (10) The per-rrh power constraints (4) can be rewritten as C 2 (A) : K A kw k 2 2 P, A, (11) where A k C A N A N is a bock diagona matrix with the identity matrix I N as the -th main diagona bock square matrix and zeros esewhere. Therefore, given the active RRH set A, thenetworkpowerminimizationprobemisgiven by ( K ) 1 P(A) :minimize A k w k 2 w 1,...,w K η 2 + P c A subject to C 1 (A), C 2 (A), (12) with the optima vaue denoted as p (A). Thisisasecondorder cone programming (SOCP) probem, and can be soved efficienty, e.g., via interior point methods [27]. Based on the soution of P(A), thenetworkpowerminimization probem an be soved by searching over a the possibe RRH sets, i.e., p =minimize Q {J,...,L} p (Q), (13) where J 1 is the minimum number of RRHs that makes the network support the QoS requirements, and p (Q) is determined by p (Q) = minimize A L, A =Q p (A), (14) where p (A) is the optima vaue of the probem P(A) in (12) and A is the cardinaity of set A. ThenumberofsubsetsA of size m is ( L m),whichcanbeveryarge.thus,ingenera,the overa procedure wi be exponentia in the number of RRHs L and thus cannot be appied in practice. Therefore, we wi reformuate this probem to deveop more efficient agorithms to sove it. C. Group Sparse Beamforming Formuation One way to sove probem P is to reformuate it as a MINLP probem [28], and the generic agorithms for soving MINLan be appied. Unfortunatey, due to the high compexity, such an approach can ony provide a performance benchmark for a simpe network setting. In the foowing, we wi pursue a different approach, and try to expoit the probem structure. We wi expoit the group sparsity of the optima aggregative beamforming vector w, whichcanbewrittenasapartition: w =[w11 T,...,wT 1K,...,wL1 T }{{},...,wt LK] T, (15) }{{} w 1 T w L T where a the coefficients in a given vector w = [w1 T,...,wT K ]T C KN form a group. WhentheRRH is switched off, the corresponding coefficients in the vector w wi be set to zeros simutaneousy. Overa there may be mutipe RRHs being switched off and the corresponding beamforming vectors wi be set to zeros. That is, w has a group sparsity structure, with the priori knowedge that the bocks of variabes in w s shoud be seected (the corresponding RRH wi be switched on) or ignored (the corresponding RRH wi be switched off) simutaneousy. Define N = K L =1 N and an index set V = {1, 2,...,N} with its power set as 2 V = {I, I V}. Furthermore, define the sets G = {K 1 i=1 N i + 1,...,K i=1 N i}, =1,...,L,asapartitionofV, such that w =[w i ] is indexed by i G.Definethesupportof beamformer w as T (w) ={i w i 0}, (16) where w =[w i ] is indexed by i V. Hence,thetotareative transport ink power consumption can be written as F (T (w)) = =1 I(T (w) G ), (17) where I(T G ) is an indicator function that takes vaue 1ifT G and 0 otherwise. Therefore, the network power minimization probem P is equivaent to the foowing group sparse beamforming (GSBF) formuation P sparse :minimize w T (w)+f(t (w)) subject to C 1 (L), C 2 (L), (18) where T (w) = L =1 K 1 η w k 2 2 represents the tota transmit power consumption. The equivaence means that if w is a soution to P sparse,then({w k }, A ) with A = { : T (w ) G } is a soution to P, andviceversa. Note that the group sparsity of w is fundamentay different from the conventiona sparsity measured by the 0 -norm of

6 2814 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 5, MAY 2014 w, whichisoftenusedincompressedsensing[11],[12].the reason is that athough the 0 -norm of w wi resut in a sparse soution for w, the zero entries of w wi not necessariy aign to a same group w to ead to switch off one RRH. As a resut, the conventiona 1 -norm reaxation [11], [12] to the 0 -norm wi not work for our probem. Therefore, we wi adopt the mixed 1 / p -norm [14] to induce group sparsity for w. The detais wi be presented in Section V. Note that the group in this work refers to the coection of beamforming coefficients associated with each RRH, but not a subset of RRHs. Since obtaining the goba optimization soutions to probem P is computationay difficut, in the foowing sections, we wi propose two ow-compexity agorithms to sove it. We wi first propose a greedy agorithm in Section IV, which can be viewed as an approximation to the iteration procedure of (13). In order to further reduce the compexity, based on the GSBF formuation P sparse,athree-stagegsbfframework wi then be deveoped based on the group-sparsity inducing norm minimization in Section V. IV. GREEDY SELECTION ALGORITHM In this section, we deveop a heuristic agorithm to sove P based on the backward greedy seection, which was successfuy appied in spare fiter design [29] and has been shown to often yied optima or near-optima soutions. The backward greedy seection agorithm iterativey seects one RRH to switch off at each step, whie re-optimizing the coordinated transmit beamforming for the remaining active RRH set. The key design eement for this agorithm is the seection rue of the RRHs to determine which one shoud be switched off at each step. A. Greedy Seection Procedure Denote the iteration number as i =0, 1, 2,...Attheith iteration, A [i] L sha denote the set of active RRHs, and Z [i] denotes the inactive RRH set with Z [i] A [i] = L. At iteration i, anadditionarrhr [i] A [i] wi be added to Z [i],resutinginanewsetz [i+1] = Z [i] {r [i] } after this iteration. We initiaize by setting Z [0] =. Inouragorithm, once an RRH is added to the set Z, itcannotberemoved. This procedure is a simpification of the exact search method described in Section III-B. At iteration i, weneedtosovethe network power minimization probem P(A [i] ) in (12) with the given active RRH set A [i]. 1) RRH Seection Rue: How to seect r [i] at the ith iteration is critica for the performance of the greedy seection agorithm. Based on our objective, we propose to seect r [i] to maximize the decrease in the network power consumption. Specificay, at iteration i, we obtain the network power consumption p (A [i] m) with A [i] m {m} = A [i] by removing any m A [i] from the active RRH set A [i].thereafter,r [i] is chosen to yied the smaest network power consumption after switching off the corresponding RRH, i.e., r [i] = arg min p (A [i] m A [i] m ). (19) We assume that p (A [i] m) = + if probem P(A [i] m) is infeasibe. The impact of switching off one RRH is reducing the transport network power consumption whie increasing the tota transmit power consumption. Thus, the proposed seection rue actuay aims at minimizing the impact of turning off one RRH at each iteration. Denote J as the set of candidate RRHs that can be turned off, the greedy seection agorithm is described as foows: Agorithm 1: The Greedy Seection Agorithm Step 0: Initiaize Z [0] =, A [0] = {1,...,L} and i =0; Step 1: Sove the optimization probem P(A [i] ) (12); 1) If (12) is feasibe, obtain p (A [i] ); If m A [i],probemp(a [i] m) is infeasibe, obtain J = {0,...,i}, go to Step 2; If m A [i] makes probem P(A [i] m) feasibe, find the r [i] according to (19) and update the set Z [i+1] = Z [i] {r [i] } and the iteration number i i +1, go to Step 1; 2) If (12) is infeasibe, when i =0, p =, go to End; when i>0, obtainj = {0, 1,...,i 1}, go to Step 2; Step 2: Obtain the optima active RRH set A [j ] with j = arg min j J p (A [j] ) and the transmit beamformers minimizing P(A [j ] ); End B. Compexity Anaysis At the i-th iteration, we need to sove A [i] SCOP probems P(A [i] m) by removing the RRH m from the set A [i] to determine which RRH shoud be seected. For each of the SOCP probem P(A), usingtheinterior-pointmethod,the computationa compexity is O((K A N ) 3.5 ) [27]. The tota number of iterations is bounded by L. As a resut, the tota number of SOCP probems required to be soved grows quadraticay with L. Athoughthisreducesthecomputationa compexity significanty compared with the mixedinteger conic programming based agorithms in [30] and [31], the compexity is sti prohibitive for arge-scae networks. Therefore, in the next section we wi propose a group sparse beamforming framework to further reduce the compexity. V. GROUP SPARSE BEAMFORMING FRAMEWORK In this section, we wi deveop two ow-compexity agorithms based on the GSBF formuation P sparse,nameya bi-section GSBF agorithm and an iterative GSBF agorithm, for which, the overa number of SOCP probems to sove grows ogarithmicay and ineary with L, respectivey.the main motivation is to induce group sparsity in the beamformer, which corresponds to switching off RRHs. In the bi-section GSBF agorithm, we wi minimize the weighted mixed 1 / 2 -norm to induce group-sparsity for the beamformer. By expoiting the additiona prior information (i.e., power ampifier efficiency, reative transport ink power consumption, and channe power gain) avaiabe in our setting, the proposed bi-section GSBF agorithm wi be demonstrated through rigorous anaysis and simuations to

7 SHI et a.: GROUP SPARSE BEAMFORMING FOR GREEN CLOUD-RAN 2815 Minimize the weighted (or re-weighted) group-sparsity inducing norm Fig. 2. Order RRHs Fix the active RRH set and obtain transmit beamformers Stage I Stage II Stage III A three-stage GSBF framework. outperform the conventiona unweighted mixed 1 / p -norm minimization substantiay[17] [19]. By minimizing the reweighted mixed 1 / 2 -norm iterativey to enhance the group sparsity for the beamformer, the proposed iterative GSBF agorithm wi further improve the performance. The proposed GSBF framework is a three-stage approach, as shown in Fig. 2. Specificay, in the first stage, we minimize aweighted(orre-weighted)group-sparsityinducingnormto induce the group-sparsity in the beamformer. In the second stage, we propose an ordering rue to determine the priority for the RRHs that shoud be switched off, based on not ony the (approximatey) sparse beamformer obtained in the first stage, but aso some key system parameters. Foowing the ordering rue, a seection procedure is performed to determine the optima active RRH set, foowed by the coordinated beamforming. The detais wi be presented in the foowing subsections. A. Preiminaries on Group-Sparsity Inducing Norms The mixed 1 / p -norm has recenty received ots of attention and is shown to be effective to induce group sparsity [14], which is defined as foows: Definition 1: Consider the vector w =[w k ] indexed by L and k S as define in (15). Its mixed 1 / p -norm is defined as foows: R(w) = β w p, p > 1, (20) =1 where β 1, β 2,...,β L are positive weights. Define the vector r = [ w 1 p,..., w L p ] T,thenthe mixed 1 / p -norm behaves as the 1 -norm on the vector r, and therefore, inducing group sparsity (i.e., each vector w is encouraged to be set to zero) for w. Notethat,withinthe group w,the p -norm does not promote sparsity as p>1. By setting p =1,themixed 1 / p -norm becomes a weighted 1 -norm, which wi not promote group sparsity. The mixed 1 / 2 -norm and 1 / -norm are two commony used norms for inducing group sparsity. For instance, the mixed 1 / 2 - norm is used with the name group east-absoute seection and shrinkage operator (or Group-Lasso) inmachineearning [15]. In high dimensiona statistics, the mixed 1 / -norm is adopted as a reguarizer in the inear regression probems with sparsity constraints for its computationa convenience [16]. B. Bi-Section GSBF Agorithm In this section, we propose a binary search based GSBF agorithm, in which, the overa number of SOCP probems required to be soved grows ogarithmicay with L, instead of quadraticay for the GS agorithm. 1) Group-Sparsity Inducing Norm Minimization: With the combinatoria function F ( ) in the objective function p(w) = T (w)+f(t (w)), theprobemp sparse becomes computationay intractabe. Therefore, we first construct an appropriate convex reaxation for the objective function p(w) as a surrogate objective function, resuting a weighted mixed 1 / 2 - norm minimization probem to induce group sparsity for the beamformer. Specificay, we first deriveitstightestpositivey homogeneous ower bound p h (w), which has the property p h (λw) = λp h (w), 0 < λ <. Sincep h (w) is sti not convex, we further cacuate its Fenche-Legendre biconjugate p h (w) to provide a tightest convex ower bound for p h(w). We ca p h (w) as the convex positivey homogeneous ower bound (the detais can be found in [32]) of function p(w), which is provided in the foowing proposition: Proposition 1: The tightest convex positivey homogeneous ower bound of the objective function in P sparse,denotedas p(w), isgivenby P c Ω(w) =2 w 2. (21) η =1 Proof: Pease refer to Appendix A. This proposition indicates that the group-sparsityinducing norm (i.e., the weighted mixed 1 / 2 -norm) can provide a convex reaxation for the objective function p(w). Furthermore, it encapsuates the additiona prior information in terms of system parameters into the weights for the groups. Intuitivey, the weights indicate that the RRHs with a higher transport ink power consumption and ower power ampifier efficiency wi have a higher chance being forced to be switched off. Using the weighted mixed 1 / 2 -norm as a surrogate for the objective function, we minimize the weighted mixed 1 / 2 - norm Ω(w) to induce the group-sparsity for the beamformer w: P GSBF :minimize w Ω(w) subject to C 1 (L), C 2 (L), (22) which is an SOCP probem and can be soved efficienty. 2) RRH Ordering: After obtaining the (approximatey) sparse beamformer ŵ via soving the weighted group-sparsity inducing norm minimization probem P GSBF,thenextquestion is how to determine the active RRH set. We wi first give priorities to different RRHs, so that an RRH with a higher priority shoud be switched off before the one with aowerpriority.mostpreviousworks[17] [19]appyingthe idea of group-sparsity inducing norm minimization directy to map the sparsity to their appication, e.g., in [19], the transmit antennas corresponding to the smaer coefficients in the group (measured by the -norm) wi have a higher priority to be switched off. In our setting, one might be tempted to give a higher priority for an RRH with a smaer coefficient r =( K ŵ k 2 2 ) 1/2,asitmayprovideaower beamforming gain and shoud be encouraged to be turned off. It turns out that such an ordering rue is not a good option and wi bring performance degradation. To get a better performance, the priority of the RRHs shoud be determined by not ony the beamforming gain but aso other key system parameters that indicate the impact of the RRHs

8 2816 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 5, MAY 2014 on the network performance. In particuar, the channe power gain κ = K h k 2 2 shoud be taken into consideration. Specificay, by the broadcast channe (BC)-mutipe-access channe (MAC) duaity [33], we have the sum capacity of the Coud-RAN as: C sum = og det(i N + snr K h k h H k ), (23) where we assume equa power aocation to simpify the anaysis, i.e., snr = P/σ 2, k =1,...,K.Onewaytoupperbound C sum is through upper-bounding the capacity by the tota receive SNR, i.e., using the foowing reation og det(i N + snr K K h k h H k ) Tr(snr h k h H k ) = snr κ, (24) which reies on the inequaity og(1 + x) x. Therefore, from the capacity perspective, the RRH with a higher channe power gain κ contributes more to the sum capacity, i.e., it provides a higher power gain and shoud not be encouraged to be switched off. Therefore, different from the previous democratic assumptions (e.g., [17] [19]) on the mapping between the sparsity and their appications directy, we expoit the prior information in terms of system parameters to refine the mapping on the group-sparsity. Specificay, considering the key system parameters, we propose the foowing ordering criterion to determine which RRHs shoud be switched off, i.e., θ := κ η =1 ( K ) 1/2 ŵ k 2, =1,...,L, (25) where the RRH with a smaer θ wi have a higher priority to be switched off. This ordering rue indicates that the RRH with a ower beamforming gain, ower channe power gain, ower power ampifier efficiency, and higher reative transport ink power consumption shoud have a higher priority to be switched off. The proposed ordering rue wi be demonstrated to significanty improve the performance of the GSBF agorithm through simuations. 3) Binary Search Procedure: Based on the ordering rue (25), we sort the coefficients in the ascending order: θ π1 θ π2 θ πl to fix the fina active RRH set. We set the first J smaest coefficients to zero, as a resut, the corresponding RRHs wi be turned. Denote J 0 as the maximum number of RRHs that can be turned off, i.e., the probem P(A [i] ) is infeasibe if i>j 0,whereA [i] Z [i] = L with Z [i] = {π 0, π 1,...,π i } and π 0 =. Abinarysearchprocedurecan be adopted to determine J 0,whichonyneedstosovenomore than (1 + og(1 + L) ) SOCP probems. In this agorithm, we regard A [J0] as the fina active RRH set and the soution of P(A [J0] ) is the fina transmit beamformer. Therefore, the bi-section GSBF agorithm is presented as foows: Agorithm 2: The Bi-Section GSBF Agorithm Step 0: Sove the weighted group-sparsity inducing norm minimization probem P GSBF ; 1) If it is infeasibe, set p =, go to End; 2) If it is feasibe, obtain the soution ŵ, cacuate ordering criterion (25), and sort them in the ascending order: θ π1 θ πl, go to Step 1; Step 1: Initiaize J ow =0, J up = L, i =0; Step 2: Repeat 1) Set i Jow+Jup 2 ; 2) Sove the optimization probem P(A [i] ) (12): if it is infeasibe, set J ow = i; otherwise,setj up = i; Step 3: Unti J up J ow =1,obtainJ 0 = J ow and obtain the optima active RRH set A with A J = L and J = {π 1,...,π J0 }; Step 4: Sove the probem P(A ),obtaintheminimum network power consumption and the corresponding transmit beamformers; End C. Iterative GSBF Agorithm Under the GSBF framework, the main task of the first two stages is to order the RRHs according to the criterion (25), which depends on the sparse soution to P GSBF,i.e.,{ŵ k }. However, when the minimum of r =( K ŵ k 2 2 ) 1/2 > 0 is not cose to zero, it wi introduce arge bias in estimating which RRHs can be switched off. To resove this issue, we wi appy the idea from the majorization-minimization (MM) agorithm [34] (pease refer to appendix B for detais on this agorithm), to enhance group-sparsity for the beamformer to better estimate which RRHs can be switched off. The MM agorithms have been successfuy appied in the re-weighted 1 -norm (or mixed 1 / 2 -norm) minimization probem to enhance sparsity [18], [19], [35]. However, these agorithms faied to expoit the additiona system prior information to improve the performance. Specificay, they used the un-weighted 1 -norm (or mixed 1 / p -norm) minimization as the start point of the iterative agorithms and re-weighted the 1 -norm (or mixed 1 / p -norm) ony using the estimate of the coefficients obtained in the ast minimization step. Different from the above conventiona re-weighted agorithms, we expoit the additiona system priorinformationateachstep (incuding the start step) to improve the estimation on the group sparsity of the beamformer. 1) Re-weighted Group-Sparsity Inducing Norm Minimization: One way to enhance the group-sparsity compared with using the weighted mixed 1 / 2 norm Ω(w) in (21) is to minimize the foowing combinatoria function directy: P c R(w) =2 I( w 2 > 0), (26) η =1 for which the convex function Ω(w) in (21) can be regarded as an 1 -norm reaxation. Unfortunatey, minimizing R(w) wi ead to a non-convex optimization probem. In this subsection, we wi provide a sub-optima agorithm to sove (25) by adopting the idea from the MM agorithm to enhance sparsity.

9 SHI et a.: GROUP SPARSE BEAMFORMING FOR GREEN CLOUD-RAN 2817 Based on the foowing fact in [36] og(1 + xϵ 1 ) im ϵ 0 og(1 + ϵ 1 ) = { 0 if x =0, 1 if x>0, we rewrite the indicator function in (26) as (27) og(1 + w 2 ϵ 1 ) I( w 2 > 0) = im ϵ 0 og(1 + ϵ 1, L. (28) ) The surrogate objective function R(w) can then be approximated as L P c f(w) =λ ϵ og(1 + w 2 ϵ 1 ), (29) η =1 by negecting the imit in (28) and choosing an appropriate 2 ϵ > 0, whereλ ϵ = og(1+ϵ 1 ).ComparedwithΩ(w) in (21), the og-sum penaty function f(w) has the potentia to be much more sparsity-encouraging. The detaied expanations can be found in [35]. Since og(1 + x),x 0, isaconcavefunction,wecan construct a majorization function for f by the first-order approximation of og(1 + w 2 ϵ 1 ),i.e., L f(w) λ ϵ w 2 η =1 w [m] 2 + ϵ + c(w[m] ), (30) }{{} g(w w [m] ) where w [m] is the minimizer at the (m 1)-th iteration, and c(w [m] ) = og(1+ w [m] 2 ) w [m] 2 /( w [m] 2 +ϵ) is a constant provided that w [m] is aready known at the current m-th iteration. By omitting the constant part of g(w w [m] ) at the m-th iteration, which wi not affect the soution, we propose a reweighted GSBF framework to enhance the group-sparsity: where P [m] igsbf :{ w[m+1] β [m] = } L =1 =arg min L =1 β [m] w 2 subject to C 1 (L), C 2 (L), (31) P c 1, =1,...,L, (32) η ( w [m] 2 + ϵ) are the weights for the groups at the m-th iteration. At each step, the mixed 1 / 2 -norm optimization is re-weighted using the estimate of the beamformer obtained in the ast minimization step. As this iterative agorithm cannot guarantee the goba minimum, it is important to choose a suitabe starting point to obtain a good oca optimum. As suggested in [18], [19], [35], this agorithm can be initiated with the soution of the unweighted 1 -norm minimization, i.e., β [0] =1, =1,...,L. In our setting, however, the prior information on the system parameters can hep us generate a high quaity stating point for the iterative GSBF framework. Specificay, with the avaiabe channe state information, we choose the 2 -norm of the initia beamformer at the -th RRH w [0] 2 to be proportiona to its corresponding channe power gain κ,arguingthattherrh with a ow channe power gain shoud be encouraged to be switched off as justified in Section V-B. Therefore, from (32), we set the foowing weights as the initiation weights for P [0] igsbf : β [0] = η κ, =1,...,L. (33) The weights indicate that the RRHs with a higher reative transport ink consumption, ower power ampifier efficiency and ower channe power gain shoud be penaized more heaviy. As observed in the simuations, this agorithm converges very fast (typicay within 20 iterations). We set the maximum number of iterations as m max = L in our simuations. 2) Iterative Search Procedure: After obtaining the (approximatey) sparse beamformers using the above re-weighted GSBF framework, we sti adopt the same ordering criterion (25) to fix the fina active RRH set. Different from the aggressive strategy in the bi-section GSBF agorithm, which assumes that the RRH shoud be switched off as many as possibe and thus resuts a minimum transport network power consumption, we adopt a conservative strategy to determine the fina active RRH set by reaizing that the minimum network power consumption may not be attained when the transport network power consumption is minimized. Specificay, denote J 0 as the maximum number of RRHs that can be switched off, the corresponding inactive RRH set is J = {π 0, π 1,...,π J0 }.Theminimumnetworkpowerconsumption shoud be searched over a the vaues of P (A [i] ), where A [i] = L\{π 0, π 1,...,π i } and 0 i J 0.Thiscanbe accompished using an iterative search procedure that requires to sove no more than L SOCP probems. Therefore, the overa iterative GSBF agorithm is presented as Agorithm 3. Agorithm 3: The Iterative GSBF Agorithm Step 0: Initiaize the weights β [0], =1,...,L as in (33) and the iteration counter as m =0; Step 1: Sove the weighted GSBF probem P [m] igsbf (31): if it is infeasibe, set p = and go to End; otherwise, set m = m +1, go to Step 2; Step 2: Update the weights using (32); Step 3: If converge or m = m max,obtainthesoutionŵ and cacuate the seection criterion (25), and sort them in the ascending order: θ π1 θ πl, go to Step 4; otherwise, go to Step 1; Step 4: Initiaize Z [0] =, A [0] = {1,...,L}, andi =0; Step 5: Sove the optimization probem P(A [i] ) (12); 1) If (12) is feasibe, obtain p (A [i] ),updatetheset Z [i+1] = Z [i] {π i+1 } and i = i +1, go to Step 5; 2) If (12) is infeasibe, obtain J = {0, 1,...,i 1}, go to Step 6; Step 6: Obtain optima RRH set A [j ] and beamformers minimizing P(A [j ] ) with j = arg min j J p (A [j] ); End

10 2818 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 5, MAY 2014 Parameter Path-oss at distance d k (km) TABLE I SIMULATION PARAMETERS Vaue og 2 (d k ) db Standard deviation of og-norm shadowing σ s 8dB Sma-scae fading distribution g k CN(0, I) Noise power σ 2 k [1] (10 MHz bandwidth) Maximum transmit power of RRH P [1] -102 dbm 1W Power ampifier efficiency η [23] 25% Transmit antenna power gain 9 dbi D. Compexity Anaysis and Optimaity Discussion We have demonstrated that the maximum number of iterations is inear and ogarithmica to L for the Iterative GSBF Agorithm and the Bi-Section GSBF Agorithm, respectivey. Therefore, the convergence speed of the proposed GSBF agorithms scaes we for arge-scae Coud-RAN (e.g., with L =100). However, the main computationa compexity of the proposed agorithms is reated to soving an SOCP probem at each iteration. In particuar, with a arge number of RRHs, the computationa compexity of soving an SOCP probem using the interior-point method is proportiona to O(L 3.5 ). Therefore, in order to sove a arge-sized SOCP probem, other approaches need to be expored (e.g., the aternating direction method of mutipiers (ADMM) method [37]). This is an on-going research topic, and we wi eave it as our future research direction. Furthermore, the proposed group sparse beamforming agorithm is a convex reaxation to the origina combinatoria optimization probem using the group-sparsity inducing norm, i.e., the mixed 1 / 2 -norm. It is very chaenging to quantify the performance gap due to the convex reaxation, for which normay specific prior information is needed, e.g., in compressive sensing, the sparse signa is assumed to obey a power aw (see Eq.(1.8) in [12]). However, we do not have any prior information about the optima soution. This is the fundamenta difference between our probem and the existing ones in the fied of sparse signa processing. The optimaity anaysis of the group sparse beamforming agorithms wi be eft to our future work. VI. SIMULATION RESULTS In this section, we simuate the performance of the proposed agorithms. We consider the foowing channe mode h k =10 L(d k)/20 ϕ k s k g k, (34) where L(d k ) is the path-oss at distance d k,,asgivenin Tabe I, s k is the shadowing coefficient, ϕ k is the antenna gain and g k is the sma scae fading coefficient. We use the standard ceuar network parameters as showed in Tabe I. Each point of the simuation resuts is averaged over 50 randomy generated network reaizations. The network power = 4.3W and Ps, t =0.7W,, andp ot =20W. The proposed agorithms are compared to the foowing agorithms: Coordinated beamforming (CB) agorithm: In this agorithm, a the RRHs are active and ony the tota transmit power consumption is minimized [7]. consumption is given in (7). We set P rrh s, Average Network Power Consumption [W] Fig Proposed GS Agorithm Proposed Bi Section GSBF Agorithm 130 Proposed Iterative GSBF Agorithm RMINLP Based Agorithm 120 Conventiona SP Based Agorithm CB Agorithm MINLP Agorithm Target SINR [db] Average network power consumption versus target SINR. Mixed-integer noninear programming (MINLP) agorithm: Thisagorithm[30],[31]canobtainthegoba optimum. Since the compexity of the agorithm grows exponentiay with the number of RRHs L, weonyrun it in a sma-size network. Conventiona sparsity pattern (SP) based agorithm: Inthisagorithm,theunweightedmixed 1 / p - norm is adopted to induce group sparsity as in [17] and [19]. The ordering of RRHs is determined ony by the group-sparsity of the beamformer, i.e., θ = ( K ŵ k 2 ) 1/2, =1,...,L,insteadof(25).The compexity of the agorithm grows ogarithmicay with L. Reaxed mixed-integer noninear programming (RMINLP) based agorithm: In this agorithm, a defation procedure is performed to switch off RRHs one-by-one based on the soutions obtained via soving the reaxed MINLP by reaxing the integers to the unit intervas [31]. The compexity of the agorithm grows ineary with L. A. Network Power Consumption versus Target SINR Consider a network with L = 10 2-antenna RRHs and K = 15 singe-antenna MUs uniformy and independenty distributed in the square region [ ] [ ] meters. We set a the reative transport ink power consumption to be P c =(5+)W, =1,...,L,whichistoindicate the inhomogeneous power consumption on different transport inks and RRHs. Fig. 3 demonstrates the average network power consumption with different target SINRs. This figure shows that the proposed GS agorithm can aways achieve goba optimum (i.e., the optima vaue from the MINLP agorithm), which confirms the effectiveness of the proposed RRH seection rue for the greedy search procedure. With ony ogarithmic compexity, the proposed bi-section GSBF agorithm achieves amost the same performance as the RMINLP agorithm, which has a inear compexity. Moreover, with the same compexity, the gap between the conventiona SP based agorithm and the proposed bi-section GSBF agorithm is arge. Furthermore, the proposed iterative GSBF

11 SHI et a.: GROUP SPARSE BEAMFORMING FOR GREEN CLOUD-RAN 2819 agorithm aways outperforms the RMINLP agorithm, whie both of them have the same computationa compexity. These confirm the effectiveness of the proposed GSBF framework to minimize the network power consumption. Overa, this figure shows that our proposed schemes have the potentia to reduce the power consumption by 40% in the ow QoS regime, and by 20% in the high QoS regime. This figure aso demonstrates that, when the target SINR increases 2,theperformancegapbetweentheCBagorithmand the other agorithms becomes smaer. In particuar, when the target SINR is reativey high (e.g., 8 db), a the other agorithms achieve amost the same network power consumption as the CB agorithm. This impies that amost a the RRHs need to be switched on when the QoS requirements are extremey high. In the extreme case with a the RRHs active, a the agorithms wi yied the same network power consumption, as a of them wi perform coordinated beamforming with a the RRHs active, resuting in the same tota transmit power consumption. 1) Impact of Different Components of Network Power Consumption: Consider the same network setting as in Fig. 3. The corresponding average tota transmit power consumption p 1 (A) = A 1 K η w k 2 2 is demonstrated in Fig. 4, and the corresponding average tota reative transport ink power consumption p 2 (A) = A P c is shown in Fig. 5. Tabe II shows the average numbers of RRHs that are switched off with different agorithms. From Fig. 4 and Fig. 5, we see that the CB agorithm, which intends to minimize the tota transmit power consumption, achieves the owest tota transmit power consumption due to the highest beamforming gain with a the RRH active, but it has the highest tota reative transport ink power consumption. This impies that a joint RRH seection and power minimization beamforming is required to minimize the network power consumption. From Tabe II, we see that the proposed GS agorithm can switch off amost the same number of RRHs as the MINLP agorithm. Furthermore, the proposed GSBF agorithms can switch off more RRHs than the RMINLP based agorithm and the conventiona SP based agorithm on average. Overa, the proposed agorithms achieve a ower tota reative transport ink power consumption, as shown in Fig. 5. In particuar, the proposed iterative GSBF agorithm can achieve a higher beamforming gain to minimize the tota transmit power consumption, as shown in Fig. 4. Therefore, the resuts in Fig. 4, Fig. 5, and Tabe II demonstrate the effectiveness of our proposed RRH seection rue and RRH ordering rue for the GS agorithm and the GSBF agorithms, respectivey. Furthermore, the resuts in Tabe II verify the group sparsity assumption in the GSBF agorithms. 2 We wi show, in Tabe II and Fig. 4, both the number of active RRHs and the tota transmit power consumption wi increase simutaneousy to meet the QoS requirements. Fig. 4. Average Tota Transmit Power Consumption [W] Tota Reative Transport Link Power Consumption [W] Fig. 5. SINR Proposed GS Agorithm Proposed Bi Section GSBF Agorithm Proposed Iterative GSBF Agorithm RMINLP Based Agorithm Conventiona SP Based Agorithm CB Agorithm MINLP Agorithm Target SINR [db] Average tota transmit power consumption versus target SINR Proposed GS Agorithm 50 Proposed Bi Section GSBF Agorithm Proposed Iterative GSBF Agorithm RMINLP Based Agorithm 40 Conventiona SP Based Agorithm CB Agorithm MINLP Agorithm Target SINR [db] Average tota reative transport ink power consumption versus target B. Network Power Consumption versus Transport Links Power Consumption Consider a network invoving 3 L =202-antenna RRHs and K = 15 singe-antenna MUs uniformy and independenty distributed in the square region [ ] [ ] meters. We set a the reative transport ink power consumption to be the same, i.e., =, = 1,...,L and set the target SINR as 4 db. Fig. 6 presents average network power consumption with different reative transport ink power consumption. This figure shows that both the GS agorithm and the iterative GSBF agorithm significanty outperform other agorithms, especiay in the high transport ink power consumption regime. Moreover, the proposed bi-section GSBF agorithm provides better performance than the conventiona SP based agorithm and is cose to the RMINLP based agorithm, whie with a ower compexity. This resut ceary indicates the importance of considering the key system parameters when 3 In [4, Section 6.1], some fied trias were demonstrated to verify the feasibiity of Coud-RAN, in which, a BBU poo can typicay support 18 RRHs.

12 2820 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 5, MAY 2014 TABLE II THE AVERAGE NUMBER OF INACTIVE RRHSWITHDIFFERENT ALGORITHMS Target SINR [db] Proposed GS Agorithm Proposed Bi-Section GSBF Agorithm Proposed Iterative GSBF Agorithm RMINLP Based Agorithm Conventiona SP Based Agorithm CB Agorithm MINLP Agorithm Average Network Power Consumption [W] Proposed GS Agorithm Proposed Bi Section GSBF Agorithm 170 Proposed Iterative GSBF Agorithm RMINLP Based Agorithm Conventiona SP Based Agorithm CB Agorithm Reative Transport Link Power Consumption [W] Fig. 6. Average network power consumption versus reative transport inks power consumption. appying the group sparsity beamforming framework. Furthermore, this figure shows that a the agorithms achieve amost the same network power consumption when the reative transport ink power consumption is reativey ow (e.g., 2W ). This impies that amost a the RRHs need to be switched on to get a high beamforming gain to minimize the tota transmit power consumption when the reative transport ink power consumption can be ignored, compared to the RRH transmit power consumption. C. Network Power Consumption versus the Number of Mobie Users Consider a network with L = 20 2-antenna RRHs uniformy and independenty distributed in the square region [ ] [ ] meters. We set a the reative transport ink power consumption to be the same, i.e., = 20W, = 1,...,L and set the target SINR as 4 db. Fig. 7 presents the average network power consumption with different numbers of MUs, which are uniformy and independenty distributed in the same region. Overa, this figure further confirms the foowing concusions: 1) With the O(L 2 ) computationa compexity, the proposed GS agorithm has the best performance among a the ow-compexity agorithms. 2) With the O(L) computationa compexity, the proposed iterative GSBF agorithm outperforms the RMINLP agorithm, which has the same compexity. Average Network Power Consumption [W] Fig. 7. users Proposed GS Agorithm 220 Proposed Bi Section GSBF Agorithm Proposed Iterative GSBF Agorithm RMINLP Based Agorithm Conventiona SP Based Agorithm Number of Mobie Users Average network power consumption versus the number of mobie 3) With O(og(L)) computationa compexity, the proposed bi-section GSBF agorithm has amost the same performance with the RMINLP agorithm and outperforms the conventiona SP based agorithm, which has the same compexity. Therefore, the bi-section GSBF agorithm is very attractive for practica impementation in arge-scae Coud-RAN. VII. CONCLUSIONS AND DISCUSSIONS In this paper, we proposed a new framework to improve the energy efficiency of ceuar networks with the new architecture of Coud-RAN. It was shown that the transport network power consumption can not be ignored when designing green Coud-RAN. By jointy seecting the active RRHs and minimizing the transmit power consumption through coordinated beamforming, the overa network power consumption can be significanty reduced, especiay in the ow QoS regime. The proposed group sparse formuation P sparse serves as a powerfu design too for deveoping ow compexity GSBF agorithms. Through rigorous anaysis and carefu simuations, the proposed GSBF framework was demonstrated to be very effective to provide near-optima soutions. Especiay, for the arge-scae Coud-RAN, the proposed bi-section GSBF agorithm wi be a prior option due to its ow compexity, whie the iterative GSBF agorithm can be appied to provide better performance in a medium-size network. Simuation aso showed that the proposed GS agorithm can aways achieve neary optima performance, which makes it very attractive in the sma-size custered depoyment of Coud-RAN. This initia investigation demonstrated the advantage of Coud-RAN in terms of the network energy efficiency. More works wi be needed to expoit the fu benefits and overcome the main chaenges of Coud-RAN. Future research directions incude theoretica anaysis of the optimaity of the proposed group sparse beamforming agorithms, more efficient beamforming agorithms for very arge-scae Coud- RAN depoyment, joint beamforming and compression when considering the imited-capacity transport inks, joint user scheduing, and effective CSI acquisition methods.

13 SHI et a.: GROUP SPARSE BEAMFORMING FOR GREEN CLOUD-RAN 2821 APPENDIX A PROOF OF PROPOSITION 1 We begin by deriving the tightest positivey homogeneous ower bound of p(w), which is given by[32],[38] p(λw) p h (w) = inf = inf λ>0 λ λt (w)+ 1 F (T (w)). (35) λ>0 λ Setting the gradient of the objective function to zero, the minimum is obtained at λ = F (T (w))/t (w). Thus,the positivey homogeneous ower bound of the objective function becomes p h (w) =2 T (w)f (T (w)), (36) which combines two terms mutipicativey. Define diagona matrices U R N N, V R N N with N = K L =1 N,forwhichthe-th bock eements are η I KN and 1 η I KN,respectivey.Next,wecacuatethe convex enveope of p h (w) via computing its conjugate: ( p h(y) = sup y T U T Vw 2 ) T (w)f (T (w)), w C N ( = sup sup yi T U T IIV II w I 2 T (w I )F (I) = I V w I C I { 0 if Ω (y) 1, otherwise. ) (37) where y I is the I -dimensiona vector formed with the entries of y indexed by I (simiary for w), and U II is the I I matrix formed with the rows and coumns of U indexed by I (simiary for V), and Ω (y) defines a dua norm of Ω(w): Ω (y) = sup I V,I y IU I 2 2 F (I) = 1 2 max η =1,...,L y G 2. (38) The first equaity in (38) can be obtained by the Cauchy- Schwarz inequaity: y T I UT II V IIw I y I U I 2 V II w I 2 = y I U I 2 T (w I ). (39) The second equaity in (38) can be justified by ( ) Ω 1 (y) sup I V,I 2 max F (I) y I G U I G 2 =1,...,L = 1 2 max =1,...,L η P c y G 2, (40) and ( ) Ω y I U I 2 (y) sup I V,I 2 min =1,...,L F (I G ) = 1 2 max =1,...,L η P c y G 2. (41) Therefore, the tightest convex positivey homogeneous ower bound of the function p(w) is Ω(w) = sup w T y Ω (y) 1 sup w G 2 y G 2 Ω (y) 1 =1 sup Ω (y) 1 =2 =1 ( L =1 ) ( P c η w G 2 max η =1,...,L y G 2 ) η w G 2. (42) This upper bound actuay hods with equaity. Specificay, we et ȳ G =2 η wg w G 2,suchthatΩ (ȳ) =1.Therefore, Ω(w) = sup Ω (y) 1 w T y wg T ȳ G =2 =1 =1 η w G 2. (43) APPENDIX B PRELIMINARIES ON MAJORIZATION-MINIMIZATION ALGORITHMS The majorization-minimization (MM) agorithm, being a powerfu too to find a oca optimum by minimizing a surrogate function that majorizes the objective function iterativey, has been widey used in statistics, machine earning, etc., [34]. We introduce the basic idea of MM agorithms, which aows us to derive our main resuts. Consider the probem of minimizing f(x) over F. The idea of MM agorithms is as foows. First, we construct a majorization function g(x x [m] ) for f(x) such that g(x x [m] ) f(x), x F, (44) and the equaity is attained when x = x [m].inanmmagorithm, we wi minimize the majorization function g(x x [m] ) instead of the origina function f(x). Letx [m+1] denote the minimizer of the function g(x x [m] ) over F at the m-th iteration, i.e., x [m+1] = arg min x F g(x x[m] ), (45) then we can see that this iterative procedure wi decrease the vaue of f(x) monotonicay after each iteration, i.e., f(x [m+1] ) g(x [m+1] x [m] ) g(x [m] x [m] )=f(x [m] ), (46) which is a direct resut from the definitions (44) and (45). The decreasing property makes an MM agorithm numericay stabe. More detais can be found in a tutoria on MM agorithms [34] and references therein. ACKNOWLEDGMENT The authors woud ike to thank anonymous reviewers and the associate editor for their constructive comments.

14 2822 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 5, MAY 2014 REFERENCES [1] I. Hwang, B. Song, and S. Soiman, A hoistic view on hyper-dense heterogeneous and sma ce networks, IEEE Commun. Mag., vo. 51, pp , June [2] F. Rusek, D. Persson, B. K. Lau, E. Larsson, T. Marzetta, O. Edfors, and F. Tufvesson, Scaing up MIMO: opportunities and chaenges with very arge arrays, IEEE Signa Process. Mag., vo. 30, no. 1, pp , [3] J. Hoydis, M. Kobayashi, and M. Debbah, Green sma-ce networks, IEEE Veh. Techno. Mag., vo.6,pp.37 43,Mar [4] China Mobie, C-RAN: the road towards green RAN, White Paper, ver. 2.5, Oct [5] J. Wu, Green wireess communications: from concept to reaity [industry perspectives], IEEE Wireess Commun., vo. 19, pp. 4 5, Aug [6] D. Gesbert, S. Hany, H. Huang, S. Shamai Shitz, O. Simeone, and W. Yu, Muti-ce MIMO cooperative networks: a new ook at interference, IEEE J. Se. Areas Commun., vo. 28, pp , Sep [7] H. Dahrouj and W. Yu, Coordinated beamforming for the mutice muti-antenna wireess system, IEEE Trans. Wireess Commun., vo.9, pp , Sep [8] C. Li, J. Zhang, and K. Letaief, Energy efficiency anaysis of sma ce networks, in Proc IEEE Int. Conf. Commun., pp , June [9] S. Tombaz, P. Monti, K. Wang, A. Vastberg, M. Forzati, and J. Zander, Impact of backhauing power consumption on the depoyment of heterogeneous mobie networks, in Proc IEEE Goba Commun. Conf., pp.1 5. [10] J. Rao and A. Fapojuwo, On the tradeoff between spectra efficiency and energy efficiency of homogeneous ceuar networks with outage constraint, IEEE Trans. Veh. Techno., vo. 62, pp , May [11] D. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, vo. 52, pp , Apr [12] E. Candes and T. Tao, Near-optima signa recovery from random projections: Universa encoding strategies? IEEE Trans. Inf. Theory, vo. 52, pp , Dec [13] C. Berger, Z. Wang, J. Huang, and S. Zhou, Appication of compressive sensing to sparse channe estimation, IEEE Commun. Mag., vo.48, pp , Nov [14] F. Bach, R. Jenatton, J. Maira, and G. Obozinski, Optimization with sparsity-inducing penaties, Foundations Trends Mach. Learning,vo.4, pp , Jan [15] M. Yuan and Y. Lin, Mode seection and estimation in regression with grouped variabes, J. R. Statist. Soc. B, vo. 68, no. 1, pp , [16] S. Negahban and M. Wainwright, Simutaneous support recovery in high dimensions: benefits and peris of bock 1 / -reguarization, IEEE Trans. Inf. Theory, vo.57,pp ,June2011. [17] M. Hong, R. Sun, H. Baigh, and Z.-Q. Luo, Joint base station custering and beamformer design for partia coordinated transmission in heterogeneous networks, IEEE J. Se. Areas Commun., vo. 31, pp , Feb [18] J. Zhao, T. Q. Quek, and Z. Lei, Coordinated mutipoint transmission with imited backhau data transfer, IEEE Trans. Wireess Commun., vo. 12, pp , June [19] O. Mehanna, N. Sidiropouos, and G. Giannakis, Joint muticast beamforming and antenna seection, IEEE Trans. Signa Process., vo. 61, pp , May [20] S.-H. Park, O. Simeone, O. Sahin, and S. Shamai, Robust and efficient distributed compression for coud radio access networks, IEEE Trans. Veh. Techno.,vo.62,pp ,Feb [21] S.-H. Park, O. Simeone, O. Sahin, and S. Shamai, Joint precoding and mutivariate backhau compression for the downink of coud radio access networks, IEEE Trans. Signa Process., vo. 61, pp , Nov [22] V. Cadambe and S. Jafar, Interference aignment and degrees of freedom of the K-user interference channe, IEEE Trans. Inf. Theory, vo. 54, pp , Aug [23] G. Auer, V. Giannini, C. Desset, I. Godor, P. Skiermark, M. Osson, M. Imran, D. Sabea, M. Gonzaez, O. Bume, and A. Fehske, How much energy is needed to run a wireess network? IEEE Wireess Commun., vo.18,pp.40 49,Oct [24] J. Kani and H. Nakamura, Recent progress and continuing chaenges in optica access network technoogies, in Proc IEEE Int. Conf. Photonics, pp [25] A. Dhaini, P.-H. Ho, G. Shen, and B. Shihada, Energy efficiency in TDMA-based next-generation passive optica access networks, IEEE/ACM Trans. Netw., vo.pp,no.99,p.1,2013. [26] A. Wiese, Y. Edar, and S. Shamai, Linear precoding via conic optimization for fixed MIMO receivers, IEEE Trans. Signa Process., vo. 54, pp , Jan [27] S. P. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, [28] Y. Shi, J. Zhang, and K. Letaief, Group sparse beamforming for green coud radio access networks, in Proc IEEE Goba Commun. Conf., pp [29] T. Baran, D. Wei, and A. Oppenheim, Linear programming agorithms for sparse fiter design, IEEE Trans. Signa Process., vo. 58, pp , Mar [30] S. Leyffer, Mixed Integer Noninear Programming, vo Springer, [31] Y. Cheng, M. Pesavento, and A. Phiipp, Joint network optimization and downink beamforming for CoMP transmissions using mixed integer conic programming, IEEE Trans. Signa Process., vo. 61, pp , Aug [32] G. Obozinski and F. Bach, Convex reaxation for combinatoria penaties, arxiv preprint arxiv: , [33] S. Vishwanath, N. Jinda, and A. Godsmith, Duaity, achievabe rates, and sum-rate capacity of gaussian MIMO broadcast channes, IEEE Trans. Inf. Theory, vo.49,pp ,Oct [34] D. R. Hunter and K. Lange, A tutoria on MM agorithms, Amer. Statistician, vo.58,no.1,pp.30 37,2004. [35] E. J. Candes, M. B. Wakin, and S. P. Boyd, Enhancing sparsity by reweighted 1 minimization, J. Fourier Ana. App., vo.14,pp , Dec [36] B. K. Sriperumbudur, D. A. Torres, and G. R. Lanckriet, A majorization-minimization approach to the sparse generaized eigenvaue probem, Mach. Learning, vo. 85, pp. 3 39, Oct [37] S. Boyd, N. Parikh, E. Chu, B. Peeato, and J. Eckstein, Distributed optimization and statistica earning via the aternating direction method of mutipiers, Foundations Trends Mach. Learning, vo. 3, pp , Juy [38] R. T. Rockafear, Convex Anaysis, vo. 28. PrincetonUniversityPress, Yuanming Shi (S 13) received his B.S. degree in eectronic engineering from Tsinghua University, Beijing, China, in He is currenty working towards the Ph.D. degree in the Department of Eectronic and Computer Engineering at the Hong Kong University of Science and Technoogy (HKUST). His research interests incude 5G wireess communication networks, Coud-RAN, optimization theory, and arge-scae optimization and its appications. Jun Zhang (S 06-M 10) received the B.Eng. degree in eectronic engineering from the University of Science and Technoogy of China in 2004, the M.Phi. degree in information engineering from the Chinese University of Hong Kong in 2006, and the Ph.D. degree in eectrica and computer engineering from the University of Texas at Austin in He is currenty a Visiting Assistant Professor in the Department of Eectronic and Computer Engineering at the Hong Kong University of Science and Technoogy (HKUST). Dr. Zhang is co-author of the book Fundamentas of LTE (Prentice-Ha, 2010). His research interests incude MIMO communications, heterogeneous networks, cognitive radio, and green communications. He has served on TPCs of different internationa conferences incuding IEEE ICC, VTC, Gobecom, WCNC, PIMRC, etc. He served as a MAC track co-chair for IEEE WCNC 2011.

15 SHI et a.: GROUP SPARSE BEAMFORMING FOR GREEN CLOUD-RAN 2823 Khaed B. Letaief (S 85-M 86-SM 97-F 03) received the B.S. degree with distinction in eectrica engineering (1984) from Purdue University, USA. He aso received the M.S. and Ph.D. degrees in eectrica engineering from Purdue University in 1986 and 1990, respectivey. From January 1985 and as a Graduate Instructor at Purdue, he taught courses in communications and eectronics. From 1990 to 1993, he was a facuty member at the University of Mebourne, Austraia. Since 1993, he has been with the Hong Kong University of Science and Technoogy (HKUST) where he is currenty Chair Professor and the Dean of Engineering, with expertise in wireess communications and networks. In these areas, he has over 470 journa and conference papers and has given invited keynote taks as we as courses a over the word. He has 13 patents incuding 11 US patents. Dr. Letaief serves as a consutant for different organizations and is the founding Editor-in-Chief of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. Hehasservedontheeditoria boardofotherprestigious journas incuding the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS-WIRELESS SERIES (as Editor-in-Chief). He has been invoved in organizing a number of major internationa conferences. These incude WCNC 07 in Hong Kong; ICC 08 in Beijing; ICC 10 in Cape Town; TTM 11 in Hong Kong; and ICCC 12 in Beijing. Professor Letaief has been a dedicated teacher committed to exceence in teaching and schoarship. He received the Mangoon Teaching Award from Purdue University in 1990; the HKUST Engineering Teaching Exceence Award; and the Michae Gae Meda for Distinguished Teaching (Highest University-wide Teaching Award at HKUST). He is aso the recipient of many other distinguished awards incuding the 2007 IEEE Communications Society Pubications Exempary Award; the 2009 IEEE Marconi Prize Award in Wireess Communications; the 2010 Purdue University Outstanding Eectrica and Computer Engineer Award; the 2011 IEEE Communications Society Harod Sobo Award; the 2011 IEEE Wireess Communications Technica Committee Recognition Award; and 10 IEEE Best Paper Awards. Dr. Letaief is a Feow of IEEE and a Feow of HKIE. He has served as an eected member of the IEEE Communications Society (ComSoc) Board of Governors, as an IEEE Distinguished ecturer, IEEE ComSoc Treasurer, and IEEE ComSoc Vice-President for Conferences. He is currenty serving as the IEEE ComSoc Vice-President for Technica Activities as a member of the IEEE Product Services and Pubications Board, and is a member of the IEEE Feow Committee. He is aso recognized by Thomson Reuters as an ISI Highy Cited Researcher.