4920 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 63, NO. 12, DECEMBER 2015

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1 4920 IEEE TRASACTIOS O COMMUICATIOS, VOL. 63, O. 12, DECEMBER 2015 Energy Efficient Coordinated Beaforing for Multicell Syste: Duality-Based Algorith Design and Massive MIMO Transition Shiwen He, Meber, IEEE, Yonging Huang, Meber, IEEE, Luxi Yang, Meber, IEEE, Björn Ottersten, Fellow, IEEE, and Wei Hong, Fellow, IEEE Abstract In this paper, we investigate joint beaforing and power allocation in ulticell ultiple-input single-output MISO) downlink networks. Our goal is to axiize the utility function defined as the ratio between the syste weighted su rate and the total power consuption subject to the users quality of service requireents and per-base-station BS) power constraints. The considered proble is nonconvex and its objective is in a fractional for. To circuvent this proble, we first resort to an virtual uplink forulations of the the prial proble by introducing an auxiliary variable and applying the uplinkdownlink duality theory. By exploiting the analytic structure of the optial beaforers in the dual uplink proble, an efficient algorith is then developed to solve the considered proble. Furtherore, to reduce further the exchange overhead between coordinated BSs in a large-scale antenna syste, an effective coordinated power allocation solution only based on statistical channel state inforation is reached by deriving the asyptotic optiization proble, which is used to obtain the power allocation in a long-ter tiescale. uerical results validate the effectiveness of our proposed schees and show that both the spectral efficiency and the energy efficiency can be siultaneously iproved over traditional downlink coordinated schees, especially in the iddle-high transit power region. Index Ters Coordinated energy-efficient transission, uplink-downlink duality, assive ultiple-input-single-output syste, beaforing and power allocation. Manuscript received February 26, 2015; revised August 28, 2015 and October 23, 2015; accepted October 23, Date of publication oveber 2, 2015; date of current version Deceber 15, This work was supported in part by the 863 Progra of China under Grant 2015AA01A703, in part by the ational Basic Research Progra of China under Grant 2013CB329002, in part by the ational atural Science Foundation of China under Grant , Grant , Grant , and Grant , in part by the atural Science Foundation of Jiangsu Province under Grant BK The associate editor coordinating the review of this paper and approving it for publication was Z. Zhang. Corresponding author: Yonging Huang.) S. He is with the School of Inforation Science and Engineering, Southeast University, anjing , China, and also with the State Key Laboratory of Millieter Waves, Southeast University, anjing , China e-ail: shiwenhe@seu.edu.cn). Y. Huang is with the School of Inforation Science and Engineering, Southeast University, anjing , China e-ail: huangy@seu.edu.cn). L. Yang and W. Hong are with the School of Inforation Science and Engineering, Southeast University, anjing , China e-ail: lxyang@ seu.edu.cn; weihong@seu.edu.cn). B. Ottersten is with the Interdisciplinary Centre for Security Reliability and Trust SnT), University of Luxebourg, Luxebourg, and also with the Royal Institute of Technology KTH), Stockhol, Sweden e-ail: bjorn.ottersten@uni.lu; bjorn.ottersten@ee.kth.seg). Color versions of one or ore of the figures in this paper are available online at Digital Object Identifier /TCOMM I. ITRODUCTIO M ULTIPLE-IPUT ultiple-output MIMO) technologies have attracted uch research interest in the past decade due to the potential of significantly enhancing the link reliability and iproving the spectral efficiency SE) without requiring additional transit power and bandwidth resources [1]. Aong those, ultiuser MIMO and coordinated ultipoint transission and reception CoMP) have been intensively studied for advanced cellular systes [2]. The cooperation between the base stations BSs) offers great advantages in suppressing the inter-cell interference and enabling efficient resource allocation to significantly increase the SE by exploiting the spatial diversity and ultiplexing gain [3] [5]. Recently, besides the objective of iproving the SE, energy efficiency EE) has becoe a ain focus in obile counications, due to the explosive growth of energy consuption resulting fro the exponential increase in obile ultiedia data traffic and obile terinals [6], [7]. Motivated by these, advanced technologies aiing at uch iproved EE have eerged as an iportant topic both in industry and acadeia [8]. EE is usually defined as the ratio of the syste weighted su rate WSR) to the syste total power consuption. Recently energy-efficient resource allocation has been widely studied for orthogonal frequency division ultiple access OFDMA) downlink networks with single antenna transceiver or fixed transit beaforers [9] [13]. On the other hand, there have been treendous research efforts to iprove the syste EE by using ultiple antenna technologies [14] [19]. In [14] the authors studied the power allocation proble in the uplink of ultiuser MIMO systes and showed that the EE is axiized by adaptively switching off soe of the user antennas. The EE of ultiuser MIMO downlink systes was also studied in [15] [19]. In particular, in [15], the energy-efficient ultiuser beaforing optiization proble was solved by approxiating the objective with a convex and tight upper bound function. In [16] a zero-gradient-based energy-efficient iterative approach was developed which was guaranteed to converge to a local axiu for MIMO interference channel. The authors in [17] addressed the EE optiization proble by solving three subprobles and choosing the best one. Focusing on the downlink of ulticell ultiuser systes, we addressed in [18], [19] the energy-efficient beaforing proble with per- BS power constraints by using jointly fractional prograing IEEE. Personal use is peritted, but republication/redistribution requires IEEE perission. See for ore inforation.

2 HE et al.: EERGY EFFICIET COORDIATED BEAMFORMIG FOR MULTICELL SYSTEM 4921 [20] [22] and the relationship between the user rate and the iniu ean square error MMSE). The research results in the above literature have shown that in any cases, the syste EE is iproved at the cost of decreasing syste SE. This eans that it is crucial to achieve a reasonable tradeoff between EE and SE in the design of a wireless counications syste. Recently, assive MIMO technology has eerged as a proising solution to iprove both SE and EE and is well analyzed especially for the uncorrelated channel odel [23] [25]. Particularly its achievable perforance in etrics such as the SIR and su rate SR) has been derived with closedfor expressions for soe special cases. For instance, the deterinistic quantity of the SIR for zero-forcing ZF) and regularized ZF RZF) precoding were derived under channel uncertainty for the ultiple-input single-output MISO) broadcast syste [26], showing that the perforance of assive MIMO systes ainly depends on the channel statistics and the channel sall fading has little effect. By exploiting the asyptotic behavior of the achievable perforance of assive MIMO syste, recently a nuber of coordinated beaforing algoriths have been developed for ulticell systes, which usually have reduced inter-cell counication burden. In [27], [28], the signal to interference plus noise ratio SIR) balancing proble for the ulticell assive MIMO syste was solved by deriving the asyptotic expression of the achievable SIR based on large-diension rando atrix theory RMT) [29] [32]. Furtherore, the asyptotic properties of the SIR were also used to address the power iniization proble subject to soe predefined SIR constraints [33] [35]. More recently, energy-efficient transission design for assive MIMO systes has also drawn uch attention. In [36], the energy-efficient resource allocation proble was studied in OFDMA downlink network with a large nuber of transit antennas and fixed axiu ratio transission MRT) beaforing. The tradeoff between SE and EE was investigated in a assive MIMO uplink syste with fixed receiver, such as axiu-ratio cobining MRC) or zero-forcing ZF) [37]. More recently, the authors in [38] [40] investigated the energy efficiency of assive MIMO systes with non-ideal hardware. The research results illustrated that the axial EE is achieved at soe finite nuber of transceiver antennas when there is a non-zero circuit power per antenna. Siilarly, the studies in [41] also deonstrated that it is optial fro the perspective of EE to have a few transceiver antennas when the transceiver design is power inefficient. In this paper, we investigate joint beaforing optiization and power allocation JBOPA) for coordinated ulticell downlink systes. First, the proble of interest is forulated as axiizing the network utility function defined as the ratio of syste weighted su rate to total power consuption subject to predefined user rate requireents and per-bs power constraints. Due to the non-convex nature of the user rate and the fractional for in the objective, the proble is hard to solve directly. To circuvent this proble, we first resort to an virtual uplink forulations of the the prial proble by introducing an auxiliary variable and the uplink-downlink duality theory. By exploiting the property that the dual proble has an analytic beaforing solution, we propose to further solve the power allocation using a convex approxiation. By this eans, the dual proble can be efficiently solved and its solution is converted into the downlink using the conventional uplink-downlink theory. We show that the developed algorith has provable convergence. Then, in order to further reduce the exchange of signalling overhead between coordinated BSs, especially in a large-scale syste, a new algorith used to update the downlink power based on the statistical channel inforation is developed by exploiting the property of a assive MIMO syste. uerical results validate the effectiveness of our proposed schees and show that both the spectral efficiency and the energy efficiency can be siultaneously iproved traditional downlink coordinated schees, especially in the iddle-high transit power region. The reainder of this paper is organized as follows. The syste odel is described in Section II. Section III reveals the dual uplink for of the prial optiization proble via soe necessary optiization transforations. An effective optiization algorith based on instantaneous channel inforation is proposed in Section IV by exploiting the newly derived uplinkdownlink duality. An extended effective optiization algorith based on statistical channel inforation is further proposed in Section V to reduce the overhead of inforation exchange between coordinated BSs. In Section VI, nuerical evaluations of these algoriths are carried out and conclusions are finally drawn in Section VII. The following notations are used throughout this paper. Bold lowercase and uppercase letters represent colun vectors and atrices, respectively. The superscripts ) H, and ) 1 represent the conjugate transpose operator, and the atrix inverse, respectively. A denotes the Euclidean nor for vectors and the Frobenius nor of atrix A. The probability density function pdf) of a circular coplex Gaussian rando vector with ean μ and covariance atrix is denoted as C μ, ). The function ceil x) rounds the eleents of x to the nearest integers greater than or equal to x. The function floorx) rounds the eleents of x to the nearest integers less than x. x, y) + = ax x, y). 1l denotes diensions colun vector with all eleents equal one and a.s. denotes alost sure convergence. II. SYSTEM MODEL Consider a K -cell MU-MISO downlink syste where the BS in cell j j = 1,...,K ) is equipped with M j transit antennas and serves j single-antenna user. We assue M j j, j, thus, user scheduling is not taken into account. Let the k-th user in cell j be User- j, k) and the BS in cell be BS, respectively. Then, the received signal of the User- j, k) is denoted as y j,k = K j h, H j,k n=1 p,n w,n x,n + z j,k, 1) where h, j,k C M 1 = σ, j,k h, j,k denotes the flat fading channel coefficient fro BS- to User- j, k) 1, in which σ, j,k 1 For tie division duplex TDD) syste, the transitters can estiate the channels fro the sounding signals received in the reverse link. For frequency division duplex FDD) syste, the transitters are provided with quantized CSI via feedback.

3 4922 IEEE TRASACTIOS O COMMUICATIOS, VOL. 63, O. 12, DECEMBER 2015 represents the large-scale channel inforation fro BS- to User- j, k), and h, j,k has i.i.d. coplex entries of zero ean 1 and variance M [26], [28], w j,k C M 1 denotes the unitnor beaforing vector for User- j, k), p j,k and x j,k denote respectively the transit power and the inforation signal intended for User- j, k), and z j,k is a zero-ean circularly syetric coplex Gaussian rando noise with power spectru density ω 2 j,k. Without loss of generality, in the rest of our paper, we assue that each BS serves the sae nuber of users in each cell, i.e., j =, j 2. For notational convenience, define h n, = h n,, as the noralized channel flat fading Wω, coefficient fro BS- n to User,, where W represents the occupied bandwidth, and denote respectively the function ceil ) and the function floor ), n, = 1,...,, = K. Then, the instantaneous rate of User-, is calculated as R = W log 1 + ) γ, 2) where and γ denotes the SIR of User-, and is given as h H p, w 2 γ =, 3) h H p n n=1,n = n, w n with p = p, and w = w, denoting the transit power and transit beaforing vector for User-,, respectively. In this paper, the EE of interest is defined as the ratio of the weighted su rate to the total power consuption, given by f w, p) = ξ α R p + K ), 4) M j P c + P 0 where w denotes the collection of all beaforing vectors, p = [ ] T p1,...,p,...,p, ξ 1 is a constant which accounts for the inefficiency of the power aplifier, P c is the constant circuit power consuption per antenna which includes the power dissipations in the transit filter, the ixer, the frequency synthesizer, and the digital-to-analog converter, P 0 is the basic power consued at the BS which is independent of the nuber of transit antennas 3 [36], α = α, denotes the 2 The developed algoriths can be ipleented in the case where different BSs serve different nuber of users in each cell, i.e., i = j, i = j. 3 ote that in counication syste, a higher R iplies a higher code rate or higher order odulation being used, thereby ay increase the power consuption of the encoder and decoder circuits. This leads to that the power consuption in the baseband is a function of R, which is usually nonlinear and akes the EE optiization proble extreely coplex. For siplicity, siilar to the existing literature [7] [17], here this fact is ignored and we assue that the power consuption at the baseband is a constant and is also included in P 0. This is reasonable due to the fact that the variation of power consuption for different levels of code rates or odulations is in general arginal in contrast to the basic power consuption. weighting factor that is used to represent the priority of User-, in the syste, which is typically included to achieve a certain fairness index aong the users. For exaple, to obtain the proportional fairness, we can set α = 1 R, where R is the average data rate of User-, in the previous tie slots [42], [43]. In order to balance the SE and the EE, our goal is to axiize the EE of the K -cell MU-MISO downlink transission syste subject to soe predefined user rate requireents and per-bs transit power constrains, i.e., ax f w, p) w,p s.t. j = j 1)+1 p P j, j K, γ γ, w = 1, p 0, U, 5) where K = {1,...,K}, U = { 1,..., }, P j is the transit power constraint of BS- j, and γ represents the target SIR of User-,. ote that, different fro the EE optiization proble considered in the existing literature [14] [19], the per-user iniu rate requireents are considered in our proble forulation aking ore challenging. In what follows, we attept to find its solution fro the perspective of uplink-downlink duality and then design an effective algorith. Also, it is worth entioning that though there is a feasibility issue in the above optiization proble 5), it is siilar to the power iniization proble subject to a given iniu target rate deand set as in [34]. Throughout this paper, we assue that the set of per-user target rate requireents is feasible and only focus on finding the solution to the proble. III. PROBLEM TRASFORMATIOS It is easy to understand that proble 5) is non-convex in the optiization variables and therefore it is difficult to find the globally optial solution, since the user rate is non-convex in general for the MISO interference channel. Furtherore, the objective function in a fractional for brings ore obstacles to solve the proble directly. In what follows, we focus on finding an efficient optiization ethod to solve proble 5) by extending the conventional uplink-downlink duality with reforulating it into the following for based on the per-bs constraints. ax g w, p,β) w,p,β s.t. j = j 1)+1 p P j, j K, γ γ, w = 1, p 0, U, p β K P j,β β 1, 6)

4 HE et al.: EERGY EFFICIET COORDIATED BEAMFORMIG FOR MULTICELL SYSTEM 4923 where β is a newly introduced auxiliary variable, β denotes the ratio of the iniu total transit power that needs to satisfy the target SIR constraints to the total power constraints 4, and α R g w, p,β) = ξβ K P j + K ). 7) M j P c + P 0 The equivalent relation between proble 5) and proble 6) is given by the following theore. Theore 1: Probles 5) and 6) have the sae optial objective value and the sae optial solutions. Proof: Let w opt and p opt be the optial solution to proble 5) and let w opt, p opt and β opt be the optial solution to proble 6), respectively. One can easily prove by contradiction that p opt K = β opt P j. Otherwise, β opt can be scaled down to achieve a better objective value. On the one hand, it is easy to know that w opt and p opt are also a feasible solution to proble 5) due to the fact that the constraint sets of proble 5) is a subset of the constraint sets of proble 6). On the other hand, let β = 1 K P j p opt, recalling the per-bs transit power constraints, we have 0 β 1 and easily know that w opt and p opt are also a feasible solution to proble 6) in conjunction with the rate constraints. With these two observations, we can prove by contradiction that f w opt, p opt) = g w opt, p opt, β opt) ust hold and the sae objective is achieved with the sae beaforing and power solution. Otherwise, the objective with a lower value can be iproved by replacing the beaforing atrix and power vector with the solution of the other proble, which contradicts the assuption that these two objective values are both optial ones. It is easy to verify that the achievable objective value of 6) onotonically increases with the auxiliary variable β in the range β β β opt, and onotonically decreases with the auxiliary variable β in the range β opt β 1. Thus the optial value of β can be efficiently obtained through a one diension search ethod such as the siulated annealing algorith, or the hill-clibing ethod in the range β β 1 [44], [45]. Based on these analysis, in what follows, the original EE optiization proble is first transfored into a su-rate axiization proble subject to per-bs transit power constraints, per-user target rate requireents and additional su power consuption, i.e., 8), where the value of the auxiliary variable β is fixed. Then, we solve iteratively 8) by updating β via a one diension search ethod to obtain the solution to the energy efficiency optiization proble 6). 4 The value of β can be deterined by solving a power iniization proble PMP) subject to the sae per-bs transit power constraints and the sae per-user iniu rate deand constraints [34]. ote that the solution to the PMP is not necessarily the solution to proble 5) [17], [18]. ax w,p s.t. α R j = j 1)+1 p P j, j K, γ γ, w = 1, p 0, U, p β K P j. 8) In contrast to the conventional WSR axiization WSRMax) proble, proble 8) has additional total power constraint and iniu SIR requireents, which is naed as extended WSRMax proble, aking the proble ore difficult to solve due to the coupling between the optiization variables. In order to design an effective algorith, in the sequel, the uplink-downlink duality is applied to solve proble 8) which can be rewritten as the following proble. ψ λ) = ax w,p α R s.t. γ γ, w = 1, p 0, U, λ p where λ ={λ j 0, K +1 K λ j + λ K +1 β ) P j, 9) λ j = 0} are auxiliary variables, and λ = λ + λ K +1,. ote that a feasible solution to proble 8) is also feasible for proble 9). It also eans that the optial objective value of proble 9) for any given λ 0, is an upper bound on the optial objective value of proble 8). Furtherore, it has been shown in [46], [47] that the solution to proble 8) can be equivalently achieved by solving the proble in ψ λ). One can easily verify that the function ψ λ) is a λ convex function but is not necessarily differentiable 5,sowecan solve the proble in ψ λ) via the sub-gradient algorith or λ ellipsoid algorith. Furtherore, it is shown in [48] that such an algorith is guaranteed to converge to its optial solution. The sub-gradient of the function ψ λ) is given by the following proposition, that can be proved with a siilar ethod as used in [46]. Proposition 1: The sub-gradient g of the function ψ λ) with fixed paraeters λ j P j g j = β K P j = j 1)+1 p ), j = 1,...,K, p ) j = K + 1, 10) 5 ote that the interior point ethod requires the twice continuously differentiable property, however, the function ψ λ) is a convex function but is not necessarily differentiable, so the interior point ethod cannot be used to solve the proble of interest.

5 4924 IEEE TRASACTIOS O COMMUICATIOS, VOL. 63, O. 12, DECEMBER 2015 where p ) is the optial solution of proble 9) with fixed paraeters λ = λ and β. The conventional uplink-downlink duality theory in [49], [50] shows that proble 9) is dual to the following virtual uplink proble: ax w,q α R s.t. γ γ, w = 1, q 0, U, q K ) λ j + βλ K +1 Pj, 11) in which q R 1 + denotes the virtual uplink transit power vector with q = q,, and R = R, denotes the virtual uplink rate of User-, and is given by h H Adding the ter q p, w 2 on both sides of 15) and suing with respect to the subscript,wehave λ p = q. 16) The above equalities can be used to calculate the downlink power solution fro the corresponding uplink solution. IV. JBOPA ALGORITHM WITH ISTATAEOUS CHAEL IFORMATIO In this section, we focus on finding the solution to the rate constrained EE axiization proble 5). Section III has revealed that solving 5) is equivalent to iteratively solving the extended WSRMax proble 8) for fixed β and searching for the optial value of β, which can be further recast to solve the virtual uplink proble 11). R = W log 1 + γ ), 12) where γ = γ, denotes the virtual uplink SIR of User-, and is calculated as γ = n=1,n = q h H, w 2 q n h H,n w 2 + λ. 13) According to the uplink-downlink duality theory [46], [47], the uplink and the downlink have the sae achievable SIR region with the sae beaforing vectors and the sae total transit power constraint. This eans that once the uplink proble is solved, the beaforing solution to the downlink proble can be obtained at the sae tie, while the downlink power solution reains to be calculated. Using the fact that the uplink and the downlink achieve the sae SIR, we have n=1,n = = p h H, w 2 p n h H n=1,n = n, w n q h H, w 2 By rearranging the ters in 14), we have q n=1,n = = p p n h H n=1,n = q n h H,n w 2 + λ. 14) n, w n 2 + q q n h H,n w 2 + p λ. 15) A. Algorith Design It is well known that the optial receiver beaforing vector that axiizes the uplink SIR expressed in 13) is the MMSE filter, which is in general calculated as 6 [50] + λ I ) 1 h, w = + λ I ), U, 17) 1 h, where = n=1,n = q n h,n h H,n. In what follows, we focus on the axiization over q of proble 11) for fixed w and β that can be forulated as follows: in q, γ 1 + γ ) α s.t. γ γ, γ γ, q 0, U, q K ) λ j + βλ K +1 Pj, 18) 6 ote that though the constants P 0, P c and ζ which odel the power consuption in transceiver circuits and the power aplifier efficiency affect the energy efficiency through the denoinator of the energy efficiency defined in 4), P 0 and P c are independent of the expression of the noralized beaforer w, as shown in 17). In other words, they are only related with the search over β and thus has an ipact on the value of energy efficiency. It also eans that a possible isatch between the true value of P c, P 0 and the value that is used to design the proposed beaforer only ipacts the value of energy efficiency but does not ipact the expression of the beaforer. Furtherore, it is worth entioning that the considered proble 5) is ore coplex than the energy efficient design in [15] [19] where the user rate requireents is not taking into account. Different fro the, we solve the coplex energy efficiency optiization proble fro the perspective of uplink-downlink duality by iteratively updating an auxiliary variable. In addition, we would like to point out that the solution of the beaforing vector w given by 17) is the optial receiver to the dual proble 11) of the extended WSRMax proble with a fixed β, as aresultζ is not included in the structure of the noralized beaforer w.

6 HE et al.: EERGY EFFICIET COORDIATED BEAMFORMIG FOR MULTICELL SYSTEM 4925 where γ denotes the collection of the auxiliary variables { γ }. The optiization proble 18) is a signoial optiization proble and is non-convex in general. Fortunately, note that, all constraints are onoial function in proble 18). If therefore the objective function can reforulated as a onoial function, proble 18) becoe a GP proble in standard for. Motivated by this observation, a successive convex approxiation approach is first presented in the following proposition, for the proof please see [52]. Lea 1: Let φ γ ) = κγ ϑ be a function used to approxiate function ϕ γ ) = 1 + γ, near the point γ. The paraeters κ and ϑ of the best onoial local approxiation are given by ϑ = γ 1 + γ,κ = 1 + γ γ ϑ, 19) and φ γ ) <ϕγ ), γ >0. To overcoe the non-convex nature of proble 18), we start with reforulating proble 18) by applying the local approxiation of Lea 1 in its objective function, yielding in q, γ γ α γ 1+ γ s.t. γ γ, γ γ, q 0, U, q K ) λ j + βλ K +1 Pj, 1 δ) γ γ 1 + δ) γ, U, 20) where δ is a sall constant used to control the desired approxiation accuracy, typically set to be δ = 0.1. otice that proble 20) is lower boundary proble of the original proble 18) that is difficult to solve directly. It has been proven in [53] that proble 20) can be reforulated as a convex GP proble in standard for by using jointly the logarithic change of the variable and a logarithic transforation of the objective function and constraints, respectively. In other words, proble 20) can be easily solved by using the powerful GP optiization tool packets [54]. Furtherore, the results in [52] have shown that by starting fro an initial point, we can search for a close local optiu by solving a sequence of GPs that locally approxiate the original proble 18). Based on the above analysis, an effective JBOPA optiization algorith is developed to solve the virtual uplink proble 8) with fixed β and is suarized as Algorith 1 7. According to the uplink-downlink duality, it is known that the optial uplink beaforing vectors w also serve as the optial downlink ones, while the optial downlink power allocation p can be calculated fro the uplink solution. The details are discussed in the following subsection. Reark 1: ote that when the value of β equals to one, Algorith 1 is equivalent to solving the conventional WSRMax proble. In order to solve the original EE axiization 7 The optial value of β can be efficiently obtained through a search ethod such as the siulated annealing algorith or hill-clibing researching in the range of β β 1 [44], [45]. Algorith 1. JBOPA Algorith With Instantaneous Channel Inforation 1: Initialize the uplink transit power q 0) and the beaforers w 0) such that the SIR constraints and the power constraint are satisfied, and calculate the approxiation point γ 0) with q 0) and w 0). 2: Initialize the auxiliary variable λ ) 0, q ) = 0, R ) = 0, γ ) = γ 0), and w ) = w 0). 3: Calculate the approxiation constants with 19) and γ ). Solve proble 20) with γ ), w ), λ ), and β, then obtain q ) and γ ). 4: If ax γ ) γ ) ζ, then let γ ) = γ ), q ) = q ), and go to step 6, otherwise, let γ ) = γ ), and go to step 3. 5: Update the beaforers w with q ), λ ) and 17), then obtain w ). Calculate the objective value of proble 18) with w ) and q ), then obtain R ).If R ) R ) ζ,letw ) = w ), and go to step 6, otherwise let w ) = w ), and go to step 3. 6: Judge whether the stop criterion of the ellipsoid ethod is satisfied or not. If yes, then stop the loop, otherwise update the auxiliary variables λ with w ), q ) and γ ) by using the ellipsoid ethod, then obtain the updated λ ) and go to step 3. proble, we need to iteratively run Algorith 1 to obtain the solution of β by using a one diension search ethod that usually only requires a liited nuber of iterations. Let C be the coputational coplexity of Algorith 1, the coputational coplexity of our proposed EE axiization algorith is O β C ) where β is the required nuber of iterations in searching for an optial β [44], [45]. B. Downlink Transit Power Coputation It is necessary to point out that the sub-gradient given in 10) is calculated based on a given downlink transit power p. Therefore, it is iportant to copute the dowlink transit power fro the obtained uplink transit power q and transit beaforing vector w. Cobining 14) with 16), we have γ = γ, U, 21a) λ p = P ax, 21b) where P ax = have q. After soe basic anipulations, we p = DGp + D1l, λ p = P ax, 22a) 22b)

7 4926 IEEE TRASACTIOS O COMMUICATIOS, VOL. 63, O. 12, DECEMBER 2015 where atrices G and D are given by, respectively 0, = n, [G],n = h H n, w n 2 = n. γ [D],n = h H = n,, w 2 0, = n. 23) 24) Multiplying both sides of 22a) by λ T = ] T [ λ 1,..., λ,..., λ and using 22), we have 1 = 1 P ax λ T DGp + 1 P ax λ T D1l. 25) [ ] p Defining an extended power vector p = and an extended 1 coupling atrix [ ] Q w, λ, ) DG D1l γ, P ax = 1 P ax λ T DG P ax λ 1 T. 26) D1l Thus, we have the following eigensyste by cobining 22a) and 26): p = Q w, λ, ) γ, P ax p, with p +1 = 1. 27) According to the conclusions in [51], we can easily obtain the optial power vector p as the first coponents of the doinant eigenvector of Q w, λ, ) γ, P ax, which can be scaled so that its last coponent equals one. C. Algorith Convergence and Coplexity Analysis otice that at each iteration the achieved virtual uplink energy efficiency is onotonically non-decreasing due to the fact that steps 3 and 5 in Algorith 1 all ai to axiize the objective function in proble 7), i.e., g 0) g 1)... g n)... Since the achievable rate region under the su power constraint is bounded, the convergence of Algorith 1 is guaranteed by the onotonic convergence theore [56]. Meanwhile, the studies in [46] have revealed that the convergence of the proposed update ethod based on the sub-gradient ellipsoid ethod for the auxiliary variables λ is guaranteed to iniize ψ λ). Thus, the convergence of the whole algorith is guaranteed and the convergence speed of our proposed algorith is siilar with that of the proposed algorith developed in [46]. It is easy to see that the ajor coputational load involes steps 3 and 5 in Algorith 1. The GP operation at step 3 fortunately has very fast convergence speed and low coputational coplexity [53]. The atrix inversion at step 5 can be coputed efficiently with K O M ) [55]. V. JBOPA ALGORITHM WITH STATISTICAL CHAEL IFORMATIO In this section, the joint beaforing optiization and power allocation are studied based on the statistical channel inforation to reduce the frequency of power updates and the overhead of signal exchange between coordinated BSs 8. The results in [26] [28] have shown that the SIR of each user converges to different deterinistic values when BS is equipped with a large nuber of transit antennas. In what follows, these observations can be used to obtain an approxiated power allocation proble related only to the statistical channel inforation which is used to optiize the transit power. By plugging 17) in the expression of the virtual uplink SIR of User-,, the asyptotic approxiation for γ is given in the following Theore 2. Theore 2: The instantaneous rando variable γ can be approxiated by a deterinistic quantity ϒ such that ϒ a.s. γ 0 as the syste diension M. ϒ is given by: where = ϒ = q σ 2, ω 2 n=1,n = M λ ), 28) q n h,n h H,n, and M z) denotes the Stieltjes transfor of the atrix evaluated at the point z. Also, ϒ can be obtained by solving the following fixed-point equation ϒ = 1 ω 2 λ + 1 M n=1, n = q σ 2, q σ 2, q nσ 2,n q σ 2, ω2 n +q nσ 2,n ω2 ϒ. 29) Proof: Please refer to Appendix A. In order to reduce the overhead of signal exchange between coordinated BSs and apply the conclusion obtained in Theore 2, in the sequel, we resort to achieve the power allocation solution by solving an approxiated new power allocation proble directly defined as 30) which only relies on the large scale fading coefficients in place of solving proble 11) which is related with the instantaneous CSI 9. ax q α log ϒ ) s.t. ϒ γ, q 0, U, q n=1 K ) λ j + βλ K +1 Pj. 30) 8 In this subsection, both the nuber of transit antennas M j and the nuber of serving users per cell go to infinity while the ratio load factor) li M j reains bounded, i.e., the notation M j denotes that both M j and becoe large, while li inf M > 0 and li sup j M <, j [26] [28]. j 9 In what follows, we assue that the target SIR γ, are achievable in the counication syste with statistical CSI.

8 HE et al.: EERGY EFFICIET COORDIATED BEAMFORMIG FOR MULTICELL SYSTEM 4927 Defining ϕ = σ, 2 ) M ω 2 λ,, proble 30) is then reforulated as follows: in α log q q ϕ ) s.t. q γ ϕ, q 0, U, q n=1 K ) λ j + βλ K +1 Pj. 31) It is easy to verify that proble 31) is convex and thus can be tackled using standard convex approaches [45]. Particularly, the power constraint eets with equality in the optial solution. The corresponding Lagrange function is given by L q, ξ,ς) = α log 2 1+q ϕ ) n=1 ξ q γ ) ϕ K ) + ς q λ j + βλ K +1 Pj, 32) where ξ and ς are respectively the Lagrange ultipliers associated with the SIR constraints and the power constraint. The Karush-Kuhn-Tucker KKT) conditions are then given by q γ, q 0,ξ 0,ς 0, 33a) ϕ α ϕ 1 + q ϕ ) ln2) ξ + ς = 0, 33b) ξ q γ ) = 0, 33c) ϕ q = n=1 K ) λ j + βλ K +1 Pj. 33d) ote that ξ 0 acts as a slack variable in the last equation and therefore can be reoved. The above equations are rewritten as q γ,ς 0, 34a) ϕ α ϕ ς, 34b) 1 + q ϕ ) ln2) q γ ) ς ϕ q = n=1 ) α ϕ = 0, 34c) 1 + q ϕ ) ln2) K ) λ j + βλ K +1 Pj. 34d) Using a siilar approach as in [45, pp ], we have α ln2)ς 1 α ϕ, ς < ϕ 1 + γ ) ln2), q = 35) γ α ϕ, ς ϕ 1 + γ ) ln2), α ln2)ς 1 ϕ, γ ϕ )+. Substituting or, siply expressed as q = this expression into the condition 34d), it yields α ln2)ς 1, γ ) = ϕ ϕ + K ) λ j + βλ K +1 Pj. 36) It is seen that the power solution in the energy-efficient largescale syste is water-filling-like power allocation algorith. As noted in Proposition 1 and Algorith 1 the update of auxiliary variables λ needs to know the downlink transit power p obtained with p = Q w, λ, ) γ, P ax p. A siilar procedure for analyzing the instantaneous rando variable γ can be applied to the atrices G and D such that the asyptotic approxiation that only depends on the statistical CSI of the atrix Q w, λ, ) γ, P ax can be obtained. The asyptotic approxiation of G and D are described as the following Theore. Theore 3: The entries of the instantaneous uplinkdownlink power transforation atrices G and D can be approxiated respectively by the entries of a deterinistic uplink-dowlink power transforation atrices G and D such a.s. a.s. that [G],n [G],n 0 and [D],n [D],n 0, as the syste diension M,, n, the eleent [G],n is given by 0, = n, σ [G],n = n, 2 ) 2 = n, 37) σ 2 M n 1+ q σ n, 2 ω 2 φ n q) and the eleent [D],n is given by 38), shown at the botto of ϒ the page, where φ q) = ω 2 ) = M q σ, 2 λ,. Proof: Please refer to Appendix A. In what follows, in order to realize the ideas of power allocation based on statistical channel inforation, the extended coupling atrix Q w, λ, ) γ, P ax is approxiated by the following atrix. [ ] Q w, λ, ) DG D1l γ, P ax = 1 P ax λ T DG P ax λ 1 T. 39) D1l [D],n = σ, 2 φ q) ω 2 λ + 1 M γ l=1,l = 1+ q l σ 2,l ω 2 l q l σ,l 2 ω l 2 2 φ q) 0, = n,, = n, 38)

9 4928 IEEE TRASACTIOS O COMMUICATIOS, VOL. 63, O. 12, DECEMBER 2015 Algorith 2. JBOPA Algorith With Statistical Channel Inforation 1: Initialize the uplink transit power q 0) by solving the PMP such that the SIR constraints and the power constraint are satisfied. 2: Initialize the auxiliary variable λ ) ), let ϒ = 0 and q ) = q 0), and copute the objective values of proble 31) with q ) and λ ), then obtain ϱ ). 3: Solve proble 36) with q ), λ ) and β, and then obtain q ), and copute the objective values of proble 31) with q ) and λ ), then obtain ϱ ) and ϒ ). If ϱ ) ϱ ) η,letq ) = q ), ϒ ) = ϒ ) and then go to step 4, otherwise let q ) = q ), ϒ ) = ϒ ) and go to step 3. 4: Judge whether satisfying the stop criterion of the ellipsoid ethod or not. If yes, the stop the loop and go to step 5, otherwise update the auxiliary variables λ with q ) and ϒ ) by using the ellipsoid ethod, then obtain the updated λ ) and go to step 3. 5: Calculate the beaforers w with 17). TABLE I ACHIEVED SR COMPARISO, P j = 46 db, j = 2, j, W = 10 MHZ, P 0 = 40 db, P c = 30 db, η = 10 3, β = 1 Fig. 1. Siulation Model. difference between the achieved SR of Algorith 1 and that of Algorith 2 decreases with an increasing nuber of transit antennas indicated by the percentage between the achieved SR of Algorith 2 and that of Algorith 1. Based on the above analysis, the power allocation proble 30) can be solved via the alternating optiization approach. To be ore specific, we solve the power allocation proble 30) by sequentially fixing two of the three variables β, λ, q and updating the third. The details of the alternating optiization algorith based on duality theore that is used to solve the power allocation proble 30) with fixed β based on statistical channel inforation is suarized as Algorith 2, where ρ and ϱ denote the large-scale syste EE and the objective value of proble 31), respectively. Reark 2: In Algorith 2, the power allocation is optiized based on statistical channel inforation and thereby is carried out in a long-ter tiescale, in which only the large-scale channel inforation and a few real nubers are exchanged between the BSs. While the beaforing vector w can be updated based on local instantaneous channel state inforation, i.e., h,n, n and thereby on a short-ter basis. Copared with Algorith 1, the coputational coplexity can be significantly reduce due the long-ter tiescale power allocation update. In addition, Algorith 2 only needs one tie beaforing vector coputation for each change of the instantaneous CSI and only needs the local CSI to copute the beaforing vector. Table I lists the achieved SR in ters of 10 7 bps of Algorith 1 and Algorith 2 for several antenna configurations over 1000 rando channel realizations. It is easy to see that the VI. UMERICAL RESULTS In this section, we investigate the perforance of the proposed JBOPA algoriths via nuerical siulations. We consider a cooperative cluster of K = 3 hexagonal adjacent cells each consisting of one BS and ultiple users. In cell j, BS-j is equipped with M transit antennas and serves single antenna users. The cell radius is set to 500 and each user randoly locates in the cooperation region and has a least 400 distance fro its serving BS as shown in Fig. 1. The channel vector h, j,k fro BS- to User- j, k) is generated based on the forulation h, j,k C M 1 = σ, j,k h w, j,k = β, j,k h w, j,k, where hw, j,k denotes the sall scale fading part and is assued to be Gaussian distributed with zero ean and variance M 1, and β, j,k denotes the large scale fading factor which in decibels is given as 10 log 10 β, j,k ) = 38 log 10 d, j,k ) η j,k, where η j,k represents the lognoral shadow fading with zero ean and standard deviation 8dB,d, j,k denotes the distance between the BS- and User- j, k) [57]. The circuit power per antenna is P c = 30 db, and the basic power consued at the BS is P 0 = 40 db [36]. The noise power spectru density ω 2 j,k = 174 db/hz, j, k and the occupied bandwidth W = 10 MHz. For siply, we assue that each BS has the sae transit power budget P, i.e., P j = P, j. The noise figure is 9 db [4]. The weighting factor α and the inefficiency factor of power aplifier ξ are all set to unit. For coparison, several related existing algoriths are also evaluated in our nuerical siulations, including the

10 HE et al.: EERGY EFFICIET COORDIATED BEAMFORMIG FOR MULTICELL SYSTEM 4929 Fig. 2. Average EE Coparison, M = 4, = 2, η = Fig. 3. Achieved WSR Coparison, M = 4, = 2, η = power iniization algorith that ais to iniize the transit power subject to the sae constraints, the conventional WSRMax algorith that ais to axiize the SE with the sae constraints, the perforance of an extension of the algorith developed in [15] that is described in Appendix D, and the noralized MRT beaforing transission with equal power allocation at each BS and subject to no rate constraints. In our siulations, the target rate of each user is set respectively to be 70% of the user rate achieved by the noralized MRT beaforing with equal power allocation. Fig. 2 illustrates the average EE of the above coordinated beaforing algoriths for finite-size syste over 1000 rando channel realizations. It can be seen that our proposed Algorith 1 achieves the best EE perforance aong all algoriths, with a considerable gain at the high transit power region. It is interesting to see that the Algorith 1 and the SRMax algorith achieve the sae EE perforance in the lower transit power region such as db. This iplies that in this region it is possible to siultaneously achieve the optial EE and SE, while in other regions a reasonable tradeoff between SE and EE is desired, which is the goal of our algorith design. The results also show that our proposed Algorith 1 outperfors the power iniization algorith in all regions. This suggests that the power iniization subject to rate constraints in general cannot achieve the optial EE, in which all the achieved user rates exactly eet with the constraints. In other words, it is very possible that achieving rates greater than the constraints also bring iproved SE. Fig. 3 illustrates the average WSR perforances of soe coordinated beaforing algoriths for finite-size syste over 1000 rando channel realizations. It can be seen that Algorith 1 and the SRMax algorith achieve the sae SE perforance in the lower transit power region. In other words, the optial EE algorith also achieves the optial syste SE in the lower transit power region, and vice versa. However, in ters of the SR, in the iddle-high transit region, the SRMax algorith achieves the best WSR perforance at the cost of a certain EE loss. Fig. 4 illustrates the average EE of the proposed algorith and the WSRMax algorith achieved with iperfect CSI, Fig. 4. Average EE Versus uber of Transit Antennas, M = 4, = 2, η = under configuration {M, } ={4, 2}. In our nuerical siulation, only an iperfect estiate ĥ, j,k of the true channel h, j,k is available at the transitter which is odeled as [26] h, j,k C M 1 = ĥ, j,k + ĥ e, j,k = β, j,k 1 δ 2 ) ĥ w, j,k + δ β, j,k h w, j,k 40) where h e, j,k denotes the estiation error, ĥ, j,k and h e, j,k have i.i.d entries of zero ean and variance M 1 and independent of each other and independent of z j,k, β, j,k denotes the large scale fading factor which in decibels is given as 10 log 10 β, j,k ) = 38 log 10 d, j,k ) η j,k, where η j,k represents the log-noral shadow fading with zero ean and standard deviation 8 db, d, j,k denotes the distance between the BS- and User- j, k) [57]. The paraeter δ [0, 1] reflects the accuracy or quality of the channel estiate ĥ, j,k, i.e., δ = 0 corresponds to perfect channel state inforation, whereas for δ = 1 the iperfect estiate ĥ, j,k is copletely uncorrelated to the true channel. uerical results show that the EE perforance achieved by the related algoriths becoe worse with an increasing value of δ. Itissee

11 4930 IEEE TRASACTIOS O COMMUICATIOS, VOL. 63, O. 12, DECEMBER 2015 over traditional downlink coordinated schees, especially in the iddle-high transit power region. APPEDIX A. Related Proof Useful Results fro Rando Matrix Theory: We reproduce the following Leas [29] [32]. Lea 2: Let x C, independent and identically distributed i.i.d.), with zero ean and variance 1, A C Heritian with bounded spectral nor whose eleents are independent of x, then Fig. 5. Average EE Versus uber of Transit Antennas, M/ = 16, η = 10 3, P c = 10 db, P 0 = 20 db. that under different levels of iperfect CSI, the EE perforance achieved by Algorith 1 is always better than that of the WSRMax Algorith at the high transit power region. Fig. 5 illustrates the perforance coparison of the two developed coordinated beaforing algoriths versus the nuber of transit antennas over 1000 rando channel realizations, where the the ratio between the nuber of transit antennas and the nuber of served users is set to be 16. uerical results show that the perforance achieved by the proposed Algorith 2 is very closed to that of the proposed Algorith 1 with lower coputational coplexity and liited scale inforation between the coordinated BSs. This suggests that the proposed Algorith 2 based on statistical state channel inforation is feasible for using the deterination approxiate expression of SIR to optiize the transit power while the beaforers are calculated based on the instantaneous channel inforation. VII. COCLUSIOS In this paper we have studied joint beaforing and power allocation in ulticell downlink systes to axiize the syste energy efficiency subject to per-bs transit power constraints and per-user target SIR requireents. The considered objective function in fractional for is non-convex and it is hard to find the globally optial solution. In order to obtain a tractable for, the prial proble was investigated fro the viewpoint of the uplink-downlink duality by introducing soe auxiliary variables. Furtherore, the uplink power allocation proble with fixed beaforers can be efficiently solved by using standard GP solvers. The convergence of the proposed JBOPA Algorith 1 has been proven with the onotonic boundary theore and the property of sub-gradient of convex proble. To reduce the overhead of the CSI collection, a new power allocation algorith based on statistical channel inforation has been further developed. uerical results have validated the effectiveness of our proposed schees and shown that both SE and EE perforance can be siultaneously iproved x H Ax 1 a.s. tr A) 0. 41) Lea ) 3: Consider an M rando atrix A =,M Ai, j i=1, whose the entries are given by: A i, j = σ i, j B M i, j, where B i, j is i.i.d., with the following assuptions hold. S1: The coplex rando variables B i, j are i.i.d. with E [ ] ] [ Bi, B i, j = 0, E [Bi, 2 ] [ j = 0, E j 2 Bi, ] = 1 and E j 8 <. S2: There exists a real nuber σ ax < such that: sup ax σ i, j σ ax M 1 1 i 1 j M There exists a deterinistic atrix-value function z) = diag 1 z),..., z)) analytic in C R + such that: 1 ) 1 tr AA H 1 a.s. zi tr z) 0, 42) whose eleents are the unique solutions of the deterinistic syste of + M equations i z) = z 1 + M 1 j z) = z 1 + M 1 1 ), 1 i, 43) M σi, 2 j j z) 1 ), 1 j M, 44) σi, 2 j i z) i=1 where 1 tr z)) is the Stieltjes transfor [31] of the probability easure. The values of i z) and j z), i, j can be obtained by initializing the to known values and iterating over equations 43) and 44) until their values converge. Furtherore, the differential of the Stieltjes transfor of the atrix AA H can be calculated with the ethod used in [33]. Proof of Theore 2: Plugging 13) in the expression of the virtual uplink SIR of User-,, and eploying Lea 2 and Lea 3, we have γ = q h H, + λ I ) 1 h, a.s. M = q σ 2, ω 2 q σ, 2 Tr + λ I ) ) 1 M ω2 M λ ) ϒ, 45)

12 HE et al.: EERGY EFFICIET COORDIATED BEAMFORMIG FOR MULTICELL SYSTEM 4931 where M z) denotes the Stieltjes transfor of the atrix evaluated at the point z which can be coputed using Lea 3. ϒ Let φ q) = ω 2, we can obtain the following fixed-point q σ, 2 equation by eploying Lea 3: φ q) = λ + 1 M 1 n=1,n = q n σ 2,n ω 2 n +q nσ 2,n φ q). 46) This copletes the proof of Theore 2. Proof of Theore 3: To itigate the ipact of the sall scale fading coefficients on the atrix G, we firstly obtain the asyptotic deterine expression of h H, w 2 and h H n, w n 2 based on the large diension rando theory. Then the asyptotic approxiation of the uplink-downlink transforation atrix G is obtained and is denoted as G. Siilar to the derivations of 45), we have the following two asyptotic deterine expressions of h H, w 2 and h H n, w n 2 [28]: h H, w 2 = h H, + λ I ) ) 1 2 h, + λ I ) 2 h, h H, a.s. σ 2 )) 2, M λ M ω 2 ) M λ where = n=1,n = = σ, 2 φ2 q) ω 2 47) φ q), q n h,n h H,n, M z) denotes the Stieltjes transfor of the atrix evaluated at the point z, and M z) denotes the differential of M z) with respect to z evaluated at the point z and φ q) is given by φ q) = λ + 1 M φ q) n=1,n = qn σ 2,n ω 2 n 1+ qn σ,n 2 ωn 2 φ q) ) 2. 48) By using jointly the Rank-1 perturbation Lea [29] and Lea 3, we have h H n, w n 2 = hh n, 1 n = = = h n,nh H n,n 1 n h n, h H n,n 2 n h n,n h H n, 1 n h n,nh H n,n 1 n h n, h H n,n 2 n h n,n a.s. M n 1 + q h H n, 1 n σ 2 n, h n, ) 2 ω 2 M n 1 + q σn, 2 ) ) 2 M ω 2 n λ n σ 2 n, ω 2 M n 1 + q σn, 2 ) ) 2 M ω 2 n λ n ω 2 M n σ 2 n, 1 + q σ 2 n, ω 2 ) 2, 49) φ n q) where n = k=1,k =n, q k h n,k h H n,k + λ n I. Thus, the eleents of the atrix G can be calculated as follows: 0, = n, σ [G],n = n, 2 ω 2 M n 1+ q σ n, 2 ) 2 = n. 50) ω 2 φ n q) γ = n, σ, [D],n = 2 φ2 q) ω 2 51) φ q) 0, = n. Plugging 48) in 51) yields the deterinistic uplink-dowlink power transforation atrix G in Theore 3. This copletes the proof of Theore 3. B. JBOPA Initialization Algorith With Instantaneous Channel Inforation Before proceeding to propose an effective ethod to solve proble 11), we would like to discuss the feasiblity of the target SIRs in the considered systes [51]. In general, the PMP subject to SIR constraints can be adopted to validate the feasiblity of soe given SIR subject to transit power constraint in wireless counications. Herein, we propose to solve the PMP that is forulated as 17) for fixed feasible target SIRs, given by in ϖ K w,ϖ s.t. P j h, w H 2 h n, w H n n=1,n = j = j 1)+1 γ, U, w 2 ϖ P j, j. 52) By using the siilar derivation ethod used in [27], [50], we can obtain the dual proble of proble 14) that is given by the following optiization proble: ax ν in w,q q s.t. γ γ, q 0, U, K ) 1 ν j Pj = 0, 53) where γ denotes the SIR of User-, and is calculated as h H q, w 2 γ =, 54) h H q n,n w 2 + ν n=1,n =

13 4932 IEEE TRASACTIOS O COMMUICATIOS, VOL. 63, O. 12, DECEMBER 2015 where q represents the dual uplink power of User,, ν denotes the virtual uplink noise variance of BS-. Siilarly, it is easily known that the axiization over w in 53) can be solved explicitly for fixed power q. In particular, the optial receiver beaforing vector w k that axiizes the SIR is the MMSE filter. According to the Sheran-Morrison identity, the MMSE receiver filter can be equivalent to the following for: ) 1 + ν I h, w = ), U. 55) 1 + ν I h, Bases on this, proble 53) can be equivalent to the following optiization proble by substituting the expression of beaforers w given by 55) into the target SIR constraints: ax ν in q s.t. q q h H, γ + ν I ) 1 h,, U, K ) 1 ν j Pj = 0, q 0, U. 56) It is easy to see that the inner optiization of proble 53) over variable q can be realized by using the fixed-point equation. The optial beaforing vectors on the downlink are of the for w = p w, where p is the power allocated to the beaforing vector w [27]. Fro the downlink-uplink SIR relations for all users, we have h H p 2 h H = γ p n 2 + γ, 57a), w ν p = q. n=1, n = n, w n Siilarly, we have the following eigensyste where P ax = 57b) p = Q w, ν, γ, P ax ) p, with p+1 = 1, 58) q, ν is a diension colun vector and is defined as ν = ν,. The details of the optial uplink power allocation algorith for finite-size syste are suarized as Algorith 3 where g denotes the subgradient of variables ν given by g = [ ] g 1,...,g j,...,g K with g j = { } j p and S ν = ν : ν 0, K ) 1 ν j Pj = 0 = j 1)+1 [50]. Furtherore, the convergence of Algorith 3 can be guaranteed by the fixed function theore [49]. Algorith 3. JBOPA Initialization Algorith With Instantaneous Channel Inforation 1: Initialize the auxiliary variable ν ) > 0, q ) = 0 and q 0) = 0. 2: Update the virtual uplink transit power q, i.e., q ) γ +ν ) = ) 1 with q ), then obtain q ). h H, I h, 3: If q ) q ) ζ, then let q ) = q ) and go to step 4, otherwise let q ) = q ) and go to step 2. 4: If q ) q 0) ζ, then stop, otherwise, calculate the downlink power p, i.e., p = Q w, ν t) ), γ, P ax p with q ) and P ax = q ), then obtain p ). Update ν with subgradient ethod i.e., ν ) k { } = P Sν ν ) k + θg k with p ), and obtain ν ),letq 0) = q ), and ν ) = ν ), then go to step 2. C. JBOPA Initialization Algorith With Statistical Channel Inforation Siilar to the JBOPA Algorith 1, we also need to solve the initialization proble of the virtual uplink transit power before applying the developed power allocation optiization Algorith 2 to solve effectively proble 6). Siilar procedure for obtaining the deterinistic quantity ϒ of γ can be applied also to γ in order to exaine the asyptotically optial solution of the PMP 53) based on statistical CSI. Proposition 2: The instantaneous rando variable γ can be approxiated by a deterinistic quantity Ɣ such that Ɣ a.s. γ 0 as the syste diension M.Also,Ɣ is given by: where = Ɣ = q σ 2, ω 2 n=1,n = ) M ν, 59) q n h,n h H,n, and M z) denotes the Stieltjes transfor of the atrix evaluated at point z. ow, instead of solving directly proble 53) to obtain initial solution, we resort to solve the following proble. ax ν in q q s.t. Ɣ γ, q 0, U, K ) 1 ν j Pj = 0. 60) For fixed ν, the optial solution of proble 60) with respect to variables q can be easily obtained by using fixed equation theory. In addition, the optiization of proble 60) with respect to variables ν can also be solved by using the subgradient ethod. The details of the algorith that is used to solve proble 60) in large-scale syste is suarized as Algorith 4.

14 HE et al.: EERGY EFFICIET COORDIATED BEAMFORMIG FOR MULTICELL SYSTEM 4933 Algorith 4. JBOPA Initialization Algorith With Statistical Channel Inforation 1: Initialize the auxiliary variable ν ) > 0, q 0) = 0, and q ) = 0. 2: Update the virtual uplink transit power μ, i.e., μ = γ σ, 2 ) with ν t) and q ), then obtain q ). ω 2 M ν 3: If q ) q ) ζ, then let q ) = q ) and go to step 4, otherwise let q ) = q ) and go to step 2. 4: If q ) q 0) ζ, then stop, otherwise, calculate ) the downlink power p, i.e., p = Q w, ν t), Ɣ, P ax p with q ) and P ax = q ), then obtain p ). Update ν with { sub-gradient ethod and p ) i.e., ν ) k = P Sν ν ) k + θg k }, then obtain ν ),letq ) = q 0), q 0) = q ), ν ) = ν ) and go to step 2. D. An Extended JBOPA Optiization Algorith It is interesting to find that the authors in [15] also adopted the convex approxiation ethod to investigate the energy efficiency optiization proble for single cell ultiuser downlink broadcast channel. It is easy to see that the developed algorith in [15] can be easily extended to solve the considered energy efficiency optiization proble. To optiize the transit power, the following lea is applied to approxiate the objective function of proble 5). Lea 4: Let φ γ ) = 2κ γ 2+ϑ γ be a function used to approxiate function ϕ γ ) = α log γ ), near the point γ.the paraeters κ and ϑ of the best onoial local approxiation are given by α 2 + ϑ γ ) γ ) ln 1 + γ ) 2 γ κ =,ϑ = γ ) ln 2 γ 2 61) and φ γ ) <ϕ γ ), γ >0. By using the local approxiation given by Lea 4 and the arithetic-geoetric ean inequality in the objective function of proble 5), the energy efficiency optiization can be reforulated into the following classical GP proble for fixed transit beaforers. μ p, γ in 2 + ϑ γ ) 1 κ γ ) 1, s.t. γ γ, γ γ, p 0, U, j p P j, j K, 62) = j 1)+1 where μ = ξ p + K ) ) M j P c + P 0. Siilar to proble 23), proble 62) can also be solved by using the powerful Algorith 5. An Extended JBOPA Algorith 1: Let l = 0, initial the downlink transit power p l) and the beaforers w l) such that the SIR constraints and the power constraint are satisfied, g l) = 0, calculate the approxiation point γ l) with p l) and w l), and calculate the approxiation constants ϑ l) and κ l) with γ l),. 2: Let l = l + 1, solve proble 62) with approxiation point γ l) for fixed w l), and obtain the optial transit power p l) and the optial SIR γ ). 3: Solve proble 63) with the target user SIRs are set to be γ ), and obtain the update beaforers w l). 4: Copute g l) with p l), w l), and 7), and check the stopping criterion, i.e., g l) g l 1) ζ.ifitisnotsatisfied, let γ l) = γ ), update the approxiation constants ϑ l) and κ l) with γ l),, and go to step 2, otherwise stop. GP optiization tool packets. In order to solve the energy efficiency optiization proble with respect to the beaforers w, we resort to solve a power iniization proble PMP) that is a classical SOCP proble which preserves the SIR of each user and given by in w s.t. w 2 h, w H 2 h n, w H n n=1,n = γ ), U, p 0, U, j w 2 P j, j. 63) = j 1)+1 where w denotes the collection of the beaforers { w k }. Assue that w ) is the optial solution of proble 63), then let w l) = w ) and p ) = w ) 2, U, respectively 10. w ) Based on these, the details of the extended energy efficiency optiization algorith is suarized as Algorith 2. ote that at step 2, the objective value of proble 62) is further iniized at current approxiation point, i.e., obtaining an iproveent of the energy efficiency. The iproveent at step 3 is that less or at ost equal power is used for each bea as that at step 2, which in turn satisfies power constraints and all user rate that achieved in step 2 constraints. In other words, an increasing or equal energy efficiency can be obtained after the update of step 3 of Algorith 5. Therefore, the sequence generated by Algorith 5 is a onotonic sequence. Since the achievable SIR region under the transit power constraint is bounded, the su rate is also bounded. In other word, the achieved syste energy efficiency is also bounded. Based on 10 We would like to point out that this ethod can also be used to initialize the initial beaforers and power allocation.

15 4934 IEEE TRASACTIOS O COMMUICATIOS, VOL. 63, O. 12, DECEMBER 2015 this, the convergence of Algorith 5 is also guaranteed by the onotonic boundary theore [56]. In addition, proble 60) has M = 2 K M j real optiization variables, + 1 second-order-cone SOC) constraints where each of the consists of M real diensions. It is known fro [48] that the coputational coplexity of the power iniization proble in ters ) of nuber of iterations is upper bounded by O + 1, and the coplexity of each iteration is within the order of O 2 M 3 ). Thus, the total worst-case coputational ) coplexity of PMP is given by O M 3. Also, since both the Algorith 1 and the extended Algorith 5 include a GP step, we ignore this GP step in the coplexity analysis. REFERECES [1] D. Tse and P. Viswanath, Fundaentals of Wireless Counication, 1st ed. Cabridge, U.K.: Cabridge Univ. Press, [2] D. Gesbert, M. Kountouris, R. Heath Jr., C. Chae, T. Sälzer, Shifting the MIMO paradig, IEEE Signal Process. 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His research interests include ultiuser MIMO counication, cooperative counications, energy efficient counications, illieter wave counication, and optiization theory. Yonging Huang M 10) received the B.S. and M.S. degrees fro anjing University, anjing, China, and the Ph.D. degree in electrical engineering fro Southeast University, anjing, China, in 2000, 2003 and 2007, respectively. Since March 2007, he has been a Faculty of the School of Inforation Science and Engineering, Southeast University. Fro 2008 to 2009, he visited the Signal Processing Laboratory, Electrical Engineering, Royal Institute of Technology KTH), Stockhol, Sweden. His research interests include space tie wireless counications, cooperative wireless counications, energy efficient wireless counications and optiization theory. He serves as an Associate Editor for the IEEE TRASACTIOS O SIGAL PROCESSIG, EURASIP Journal on Advances in Signal Processing, andeurasip Journal on Wireless Counications and etworking. Luxi Yang M 96) received the M.S. and Ph.D. degree in electrical engineering fro the Southeast University, anjing, China, in 1990 and 1993, respectively. Since 1993, he has been with the Departent of Radio Engineering, Southeast University, where he is currently a Professor of inforation systes and counications, and the Director of Digital Signal Processing Division. He is the author or coauthor of 2 published books and ore than 100 journal papers, and holds 10 patents. His research interests include signal processing for wireless counications, MIMO counications, cooperative relaying systes, and statistical signal processing. He received the first- and second-class prizes of the Science and Technology Progress Awards of the State Education Ministry of China, in 1998 and 2002, respectively. He is currently a eber of the Signal Processing Coittee of Chinese Institute of Electronics. Björn Ottersten S 87 M 89 SM 99 F 04) was born in Stockhol, Sweden, He received the M.S. degree in electrical engineering and applied physics fro Linköping University, Linköping, Sweden, and the Ph.D. degree in electrical engineering fro Stanford University, Stanford, CA, USA, in 1986 and 1989, respectively. He has held research positions with the Departent of Electrical Engineering, Linköping University; the Inforation Systes Laboratory, Stanford University; the Katholieke Universiteit Leuven, Leuven; and the University of Luxebourg, Luxebourg. Fro 1996 to 1997, he was the Director of Research at ArrayCo Inc., a start-up in San Jose, CA, USA, based on Ottersten s patented technology. In 1991, he was appointed as a Professor of signal processing with the Royal Institute of Technology KTH), Stockhol, Sweden. Fro 1992 to 2004, he was the Head of the Departent for Signals, Sensors, and Systes with KTH, and fro 2004 to 2008, he was the Dean of the School of Electrical Engineering with KTH. Currently, he is the Director for the Interdisciplinary Centre for Security, Reliability, and Trust with the University of Luxebourg. He is a Board Meber of the Swedish Research Council and as Digital Chapion of Luxebourg, and acts as an Adviser to the European Coission. His research interests include security and trust, reliable wireless counications, and statistical signal processing. He has served as an Associate Editor for the IEEE TRASACTIOS O SIGAL PROCESSIG and on the Editorial Board of the IEEE Signal Processing Magazine. He is currently an Editor-in-Chief of EURASIP Signal Processing Journal and a eber of the Editorial Boards of EURASIP Journal of Applied Signal Processing and Foundations and Trends in Signal Processing. He is a Fellow of the EURASIP and a eber of the IEEE Signal Processing Society Board of Governors. He has coauthored journal papers that received the IEEE Signal Processing Society Best Paper Award in 1993, 2001, 2006, and 2013, and the 3 IEEE conference papers receiving Best Paper Awards. He was the recipient of the IEEE Signal Processing Society Technical Achieveent Award in He is the first recipient of the European Research Council advanced research grant. Wei Hong M 92 SM 07 F 12) received the B.S. degree fro the University of Inforation Engineering, Zhengzhou, China, and the M.S. and Ph.D. degrees fro Southeast University, anjing, China, all in radio engineering, in 1982, 1985, and 1988, respectively. Since 1988, he has been with the State Key Laboratory of Millieter Waves and has served for the director of the laboratory since He is currently a Professor and the Dean of the School of Inforation Science and Engineering, Southeast University. In 1993, 1995, 1996, 1997, and 1998, he was a short-ter Visiting Scholar at the University of California at Berkeley, Berkeley, CA, USA, and at the University of California, Santa Cruz, CA, USA, respectively. He has been engaged in nuerical ethods for electroagnetic probles, illieter wave theory and technology, antennas, RF technology for wireless counications etc. He has authored and coauthored over 300 technical publications and 2 books. He was the recipient of the first-class Science and Technology Progress Prizes thrice) issued by the Ministry of Education of China and Jiangsu Province Governent, etc. He was also the recipient of the Foundations for China Distinguished Young Investigators and for Innovation Group" issued by SF of China. He is a Fellow of CIE, MTT-S AdCo Meber, a Vice-President of Microwave Society and Antenna Society of CIE, the Chairperson of the IEEE MTT-S/AP-S/EMC-S Joint anjing Chapter, and has served as an Associate Editor of the IEEE TRASACTIOS O MICROWAVE THEORY AD TECHIQUES fro 2007 to 2010, and an Editor Board eber for International Journal of Antennas and Propagation, China Counications, and Chinese Science Bulletin, aong others.