Leakage-Aware Energy-Efficient Beamforming for Heterogeneous Multicell Multiuser Systems

1268 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 6, JUNE 2014

Leakage-Aware Energy-Efficient Beamforming forHeterogeneous Multicell Multiuser Systems

Shiwen He, Yongming Huang, Member, IEEE, Haiming Wang, Member, IEEE,Shi Jin, Member, IEEE, and Luxi Yang, Member, IEEE

Abstract—Energy-efficient communications has attracted muchinterest in the research of 5G cellular systems. In this paper, westudy energy-efficient coordinated beamforming design for het-erogeneous multicell multiuser downlink systems. The consideredproblem is formulated as maximizing the weighted sum per-cellenergy efficiencies (WSPEEMax) subject to predefined per-usertarget rate demands, maximum leakage interference power con-straints, and per-BS transmit power constraints. This formulationis more general than the conventional EE optimization problemand provides a unified way to consider the EE of heterogeneousnetworks. However, it is hard to tackle due to the weightedsum-of-ratios form of the objective function and the non-convexnature of per-user target rate constraints. To address it, we pro-pose to first transform the original problem into a polynomialform optimization by introducing some auxiliary variables andthen further reveal their equivalence in finding the solution. Then,an efficient block coordinate ascent optimization algorithm is de-veloped to solve the equivalent problem by exploiting the concavenature of the considered problem with respect to each optimizationvariable. To further improve the network EE, we also develop anenergy-efficient transmission method for each small-cell network.Finally, extensive numerical results are provided to verify theeffectiveness of the proposed schemes and show that both the EEand spectral efficiency (SE) of heterogeneous network can be sig-nificantly improved by energy-efficient coordinated multiple-inputmultiple-output (MIMO) transmission.

Index Terms—Coordinated multicell beamforming, energy-efficient transmission, heterogeneous network, sum-of-ratiosoptimization.

Manuscript received December 1, 2013; revised April 25, 2014; acceptedMay 25, 2014. Date of publication June 3, 2014; date of current versionJuly 14, 2014. This work was supported by the National Science andTechnology Major Project of China under Grant 2013ZX03003006-002;by the National Basic Research Program of China under Grant 2013CB329002;by the National Natural Science Foundation of China under Grants 61271018,61372101, 61222102, and 61132003; by the Research Project of JiangsuProvince under Grants BK20130019, BK2011597, BE2012167, andBK2012021; and by the Program for New Century Excellent Talents inUniversity under Grant NCET-11-0088.

S. He is with the State Key Laboratory of Millimeter Waves, Departmentof Radio Engineering, Southeast University, Nanjing 210096, China (e-mail:[email protected]).

Y. Huang and H. Wang are with the School of Information Scienceand Engineering, Southeast University, Nanjing 210096, China (e-mail:[email protected]; [email protected]).

S. Jin is with the National Mobile Communications Research Laboratory,Southeast University, Nanjing 210096, China (e-mail: [email protected]).

L. Yang is with the Department of Radio Engineering, Southeast University,Nanjing 210096, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSAC.2014.2328142

I. INTRODUCTION

THE tremendous popularity of smart terminals and socialnetworks have spurred the explosive growth of high-

speed wireless data transfer demands. To fulfill such boomingubiquitous data demands, some advanced wireless communica-tion technologies have been developed to improve the spectralefficiency (SE) and network capacity. Particularly, multiple-input multiple-output (MIMO) technologies including point-to-point MIMO, multiuser MIMO and network MIMO have beenintensively studied in the past decades, owing to their potentialof significantly improving the SE [1]–[5]. Besides, the develop-ment of heterogeneous network and small-cell network has alsoattracted intensive research interest in industry and academia[6]–[8], which have been regarded as powerful tools to providehigh throughput for indoor and outdoor hot-spots. Meanwhile,green radio has become increasingly important in the future5G cellular system, as the explosive growth of data traffic ac-companied with rapidly increase of energy consumption, whichnecessitates the development of energy-efficient wireless com-munication technologies [9], [10]. Among those, in addition toresource allocation, energy-efficient beamforming technologiesfor MIMO systems are also of particular importance, since thescalability of a wireless network is fundamentally limited bytransmission power.

Traditional coordinated multiple-point transmission and re-ception (CoMP) for homogeneous networks mainly focus onimproving the effective power to the target receiver whilereducing the intra-cell and inter-cell interference. Particularly,the classical weighted sum rate maximization (WSRMax) prob-lem has been extensively studied in recent years [11]–[15].In [11], [12] the WSRMax problem was investigated for theMIMO broadcast and interference channels by exploiting therelationship between the WSRMax and the weighted mini-mum mean square error (WMMSE). Alternatively, an iterativecoordinated beamforming algorithm was proposed based onthe Karush-Kuhn-Tucker (KKT) condition in [13]. However,this algorithm is not provably convergent. In [14], [15] thesequential convex approximation method was used to addressthe non-convex WSRMax problem for the multicell multiple-input single-output (MISO) downlink system. Note that theseworks all focus on maximizing the SE by properly designingthe transmit beamforming vectors and allocating the transmitpower.

In addition to improving the SE, increasing the energy effi-ciency (EE) in unit of bit-per-Joule or Joule-per-bit has attractedmuch research interest in the past few years [16]–[23], which

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HE et al.: ENERGY-EFFICIENT BEAMFORMING FOR HETEROGENEOUS MULTICELL MULTIUSER SYSTEMS 1269

has become an important metric for the future 5G cellularsystem. Energy-efficient MIMO precoding was investigated in[16] for point-to-point MIMO channels. In [17], an energy-efficient MIMO transmission scheme was proposed for MIMOinterference channels, which was developed based on zero-gradient condition but without considering any constraints.In [18], the energy-efficient precoding for broadcast channelswas designed by solving three independent subproblems andchoosing the best solution subject to the corresponding transmitpower and individual user rate constraints. Meanwhile, energy-efficient resource allocation has been widely studied. Particu-larly, in [19] D. Ng et al. investigated the resource allocationproblem with fixed transmit beamformers for energy-efficientcommunication in the single-cell multiuser MISO orthogo-nal frequency division multiple access (OFDMA) downlinknetwork. In [20], they further studied the energy-efficient re-source allocation problem in single-input single-output (SISO)OFDMA downlink network with an energy harvesting basestation (BS). More recently, the EE for massive MIMO andcoordinated MIMO have received much interest. In [21], thetradeoff between the spectral and energy efficiencies was in-vestigated for multiuser massive single-input multiple-output(SIMO) uplink with fixed receivers. A new fairness-basedenergy-efficient beamforming was proposed for multicell mul-tiuser joint transmission systems [22]. In [23], the authorsdeveloped an energy-efficient beamforming and power allo-cation algorithm for multicell multiuser coordinated downlinksystems by exploiting fractional programming. The researchresults in [23] also illustrated that the system EE increasesat the cost of the reduction of system throughput at the highSNR region. In other words, we need to consider the tradeoffbetween the EE and the SE during in designing energy-efficienttransmission.

Note that the above energy-efficient transmission technolo-gies focused on the homogeneous cellular network, by far littleresearch has considered energy-efficient beamforming designfor the heterogeneous network that has been regarded as apromising solution to deal with the increasing wireless trafficdemands in the 5G system. Though it is well recognized thatthe heterogeneous network realized by overlaying a numberof small-cell BSs in the macro-cell has a great potential ofsignificantly improving both the SE and the EE [24]–[26], thedense and random deployment of small-cell nodes leads tounprecedented challenges in terms of mobility and interferencemanagement, resulting in inefficient use of spectrum and en-ergy. The interference power generated by the macro-cell BSto its overlaid small-cell users needs to be controlled in orderto guarantee the quality of service (QoS). At the same time,different types of BSs may have different EE demands andminimum target rate requirements in heterogeneous multicellnetworks [27]–[30]. Furthermore, unlike the conventional ho-mogeneous macro-cell network, there are no dedicated, high-capacity backhaul between the macro-cell BS and small-cellBSs. This means that inter-cell coordination for heterogeneousnetwork needs to fully consider the backhaul capacity limitation[31]. These factors make the interference coordination prob-lem in the heterogeneous network more complex. Therefore,we need to find new ways to efficiently address both the

cross-layer and inter-layer interference, and further reveal thetradeoff between the achievable EE and SE in the heteroge-neous network.

To this end, in this paper we focus on energy-efficient co-ordinated beamforming design for the heterogeneous network.Contrary to the conventional EE criterion defined as the ratio ofthe system sum rate (SR) to the total power consumption, ourobjective is on the basis of per-cell EE which is defined as theratio of the weighted SR to the total power consumption in eachcell, such that heterogeneous requirements from different cellsin the heterogeneous network can be investigated. Our maincontribution is three-fold. Firstly, we construct the optimiza-tion problem as maximizing the weighted sum per-cell energyefficiencies (WSPEEMax) subject to multiple constraints interms of per-cell maximum transmit power constraints, per-cell maximum leakage interference power constraints, and per-user minimum data rate demands. This formulation providesa unified way to consider the EE of heterogeneous network,but has more drawbacks than the conventional problem dueto the weighted sum-of-ratios form of the objective functionand the non-convex nature of the user rate. To overcomethese difficulties, we propose to transform it into a polynomialform optimization that possesses some pleasant properties byintroducing some auxiliary variables, and further reveal theirequivalence in finding the solution. Then, we develop an effi-cient block coordinate ascent optimization algorithm to solvethe equivalent problem by exploiting its concave nature withrespect to each optimization variables. The proposed algorithmis guaranteed to converge and provides a general solution,which can be reduced to address the conventional optimizationproblems such as the WSRMax problem. Finally, in order tofurther improve the network EE subject to limited backhaulcapacity, we also propose an energy-efficient beamformingmethod for each small-cell network where the small-cell BSservers a plurality of single-antenna users. Extensive numericalresults are provided to verify the effectiveness of our developedschemes, and show that the EE of heterogeneous network canbe significantly improved by proper coordination between themacro-BS and the lower-power BS even with limited inter-BScommunication.

The rest of this paper is organized as follows. The systemmodel and problem formulations are given in Section II. InSection III, some necessary problem transformations are given.An energy-efficient beamforming algorithm and a WSRMaxalgorithm are respectively proposed in Section IV. A small-cell energy-efficient transmission algorithm is further de-veloped in Section V. The simulation results are shownin Section VI and conclusions are finally provided inSection VII.

The following notations are used throughout this paper. Boldlowercase and uppercase letters represent column vectors andmatrices, respectively. The superscript (·)H , (·)−1, and Tr(A)represent the conjugate transpose operator, and the matrix in-verse, the trace of matrix A, respectively. [A]m,n denotes the(m,n)th entry of matrix A. log(·) is the logarithm with base eand ‖A‖ represents the Frobenius norm of matrix A. E{·} andC are the statistical expectation and the complex number field,respectively.


Fig. 1. Illustration of a heterogeneous network comprising K = 3 coordinat-ing macro-cells each overlaid with T = 3 small-cells.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. System Model

Consider a heterogeneous network consisting of K macroBSs (MBSs) and multiple overlaid small-cell BSs (SBSs) usingthe same carrier frequency, as shown in Fig. 1. Without loss ofgenerality, we assume that each macro-cell overlays T small-cells and serves a plurality of single antenna users, while eachsmall-cell also includes a plurality of single antenna users.We denote the macro-cell user set in macro-cell k as Uk ={1, . . . , Nk}, where Nk denotes the number of macro-cell usersin macro-cell k, and denote the m-th macro-cell user in macro-cell k as user-(k,m), m ∈ Uk, k ∈ K = {1, . . . ,K}, where K

denotes the set of the macro-cell BSs, respectively. We assumethat macro-cell BS-k is equipped with Mk transmit antennas.The received signal of user-(k,m) is then expressed as

yk,m =∑j∈K

∑n∈Uk

hHj,k,mwj,nxj,n + zk,m, (1)

where hj,k,m ∈ CMj×1 denotes the flat fading channel vec-

tor from macro-cell BS-j to user-(k,m), including both thelarge scale fading, the shadow fading, and the small scalefading, wk,m ∈ C

Mk×1 denotes the beamforming vector foruser-(k,m), xk,m denotes the information signal intended foruser-(k,m) with zero mean and unit variance, and zk,m denotesthe additive white Gaussian noise with zero mean and varianceσ2k,m for user-(k,m). We consider a time division duplex

(TDD) system, such that each BS can directly estimate thechannel state information of all macro-cell users by exploitingthe uplink-downlink reciprocity, i.e., each BS-k has a localchannel state information (CSI) from the BS to all macro-cell users. Also, assume that the signals for different users areindependent from each other and the receiver noise.

In the above considered heterogeneous network, coordina-tion between the macro BSs in the level of beamforming designis necessary in order to suppress the inter-cell interferenceresulting from aggressive frequency reuse. Meanwhile, it isalso critical to mitigate the cross-layer interference from themacro BS to the small-cell users. Note that the small-cell BSusually has much lower transmit power and a range expansionstrategy [25] is commonly employed to improve the offloadcapacity of small-cell, the cross-layer interference from thesmall-cell BS to the macro-cell users can be ignored, while thesmall-cell users especially at the expanded region suffer severeinterference from the macro BS. Therefore, necessary channelinformation should be acquired at the macro BSs to addressthis issue. Considering limited backhaul between the macro BSand the small-cell BS in many scenarios, we assume that eachmacro BS has the knowledge of the channel variance matricesof the channels from the BS to its overlaid small-cell users.Since the channel variance change slowly, they can be estimatedby the small-cell users and then convey to the macro BS via thesmall-cell BS, or they can be directly estimated by the macroBS based on the uplink-downlink reciprocity.1 To complete theformulation, let Gk,s ∈ C

Mk×Mk denote the average channelvariance matrix from BS-k to the s-th small-cell users in macro-cell k, ∀ s ∈ S = {1, . . . , T} and ∀ k ∈ K.

B. Problem Formulation

Now we focus on the coordinated beamforming design atthe macro BSs to consider both the inter-layer and cross-layerinterference. Our objective is the maximization of the systemEE with specific rate constraints from the users. Different formthe conventional approaches [20], [21] we consider per-cell EEmeasure so as to satisfy possible heterogeneous or priority-based requirements, which is defined as the ratio of the per-cellweighted SR to the per-cell total power consumption, given by

fk(W ) =

∑m∈Uk

θk,mRk,m

�k

∑m∈Uk

Tr(W k,m) +MkPc + P0, ∀ k,

(2)

where W k,m = wk,mwHk,m, W denotes the collection of all

beamforming vectors, Pc is the constant circuit power con-sumption per antenna including power dissipations in the trans-mit filter, the mixer, the frequency synthesizer, and thedigital-to-analog (D/A) converter, which is independent of theactual transmitted power, P0 is the basic power consumed atthe BS which is independent of the number of transmit antennas[10], �k ≥ 1 is a constant accounting for the inefficiency of thepower amplifier of BS-k, Rk,m denote the instantaneous rate of

1As we all know, the coverage radius of a small-cell BS is very smallcompared with the coverage radius of the macro-cell BS, due to the factthat the small-cell BS transmit power is often lower than the macro-cell BStransmit power. It also means that the small-cell users and the small-cell BShave similar distance from macro-cell BS and all are surrounded by a similarscatter environment, i.e., they have similar large scale fading coefficients andhave similar channel variance matrices. Based on these observations, in thispaper, we assume that each macro-BS has the knowledge of the average channelvariance matrix of the channel coefficient to each of the small-cell users whichlocate in the same small cell [32]–[34].


user-(k,m), θk,m is the weighting factor associated with therate of user user-(k,m), which is typically included to achievea certain fairness index among users. For example, to obtainthe proportional fairness, we can set θk,m = 1/Rk,m, whereRk,m is the average data rate of user-(k,m) in the previoustime slots [35]. The user rate Rk,m is calculated as Rk,m =ln(1 + SINRk,m) and is unit of Nat/s/Hz,2 with SINRk,m beingthe SINR of user-(k,m) and given by

SINRk,m =1

Ik,m

∥∥hHk,k,mwk,m

∥∥2 , (3)

where Ik,m denotes the interference-plus-noise which includesthe inter-user interference and the additive Gaussian whitenoise, given by

Ik,m =∑n∈Uk,

n �=m

∥∥hHk,k,mwk,n

∥∥2+

∑j∈K,j �=k

∑l∈Uj

∥∥hHj,k,mwj,n

∥∥2 + σ2k,m. (4)

To balance the system SR and the total power consump-tion, in this paper, a new EE optimization problem named asWSPEEMax problem is formulated as follows.3

maxW

∑k∈K

αkfk(W )

s.t.∑m∈Uk

Tr(Gk,sW k,m) ≤ �k, ∀ k ∈ K, s ∈ S,

Rk,m ≥ γk,m, ∀ k ∈ K, m ∈ Uk∑m∈Uk

Tr(W k,m) ≤ Pk, ∀ k ∈ K, (5)

where Pk is power constraint of BS-k, αk is the weightingfactor accounting for different EE priorities, giving a largervalue to the user with a higher EE requirement in heterogeneousnetworks, γk,m denotes the minimum user rate demand of user-(k,m), also in unit of Nat/s/Hz, �k denotes the maximuminstantaneous interference that the BS-m allows to generateto the overlaid small-cell users. In order to further investi-gate the relationship between the SE maximization and theEE maximization, the conventional WSRMax problem is alsoconsidered, given as

maxW

∑k∈K

∑m∈Uk

θk,mRk,m

s.t.∑m∈Uk

Tr(Gk,sW k,m) ≤ �k, ∀ k ∈ K, s ∈ S,

2To simplify the description of the following derivations, we adopt thelogarithm with e as the base throughout this paper, which can be easilyconverted to bit/s/Hz and has no effect on our algorithm derivation.

3In this section, we focus only on the maximization of the weighted sum ofmacro-cell EE without taking the small-cell EE into account due to the limitedbackhaul link between the small-cell BS and the macro-cell BS. To furtherimprove the network EE, later, we develop a leakage-ware based on energy-efficient transmission algorithm for each small-cell network.

Rk,m ≥ γk,m, ∀ k ∈ K, m ∈ Uk∑m∈Uk

Tr(W k,m) ≤ Pk, ∀ k ∈ K. (6)

It is well known that the user rate achieved in interferencechannels is a challenging non-convex function due to the cou-pling among the optimization variables. Therefore, problem(5) and (6) are both non-convex. Furthermore, the non-convexper-user minimum user rate constraints make problem (5) and(6) more intractable and more difficult to solve directly thanthe conventional weighted SR maximization problem. Besides,the sum-of-ratio form of the objective function in problem (5)brings more obstacles to solving directly problem (5). In whatfollows, we focus on finding solutions to these optimizationproblems in (5) and (6) by introducing some problem transfor-mation methods. Note that there also exists the feasibility issuein the above two optimization problems, which is similar to thepower minimization problem subject to a given minimum targetrate demand set in [36]. Throughout this paper, we assume thatthe set of per-user target rate requirement is feasible and onlyfocus on finding the solution to the problem.

III. PROBLEM TRANSFORMATION

In this section, we investigate some useful problem trans-formations to derive algorithms for finding the solutions tothe WSPEEMax problem (5) and the WSRMax problem (6).More specifically, problem (5) and problem (6) are firstlytransformed into a tractable form by utilizing the relationshipbetween the user rate and the MMSE. Then, the fractional formobjective function of problem (5) is further reformulated into aparametric subtractive quadratic form by using the Lagrangianduality theorem.

We treat interference as noise and consider linear equal-izer strategy so that the estimated signal is given by xk,m =μ∗k,myk,m, ∀ k ∈ K, ∀m ∈ Uk, where μk,m denotes the re-

ceiver filter at User-(k,m). Then, the MSE of user-(k,m) iscalculated as

ek,m =Ex,z {(xk,m − xk,m)(xk,m − xk,m)∗}=μ∗

k,mJk,mμk,m − μk,mwHk,mhk,k,m

− μ∗k,mhH

k,k,mwk,m + 1, (7)

where Jk,m =∑

j∈K∑

n∈Uj‖hH

j,k,mwj,n‖2 + σ2k,m. Fixing

all the transmit beamformers and minimizing the MSE(weighted sum-MSE, WSMSE) lead to the well-known MMSEreceiver, i.e.,

μoptk,m = J−1

k,mhHk,k,mwk,m, (8)

and the corresponding MMSE is calculated as

emmsek,m = 1− J−1

k,m

∥∥hHk,k,mwk,m

∥∥2 . (9)

Proposition 1: The WSPEEMax optimization problem (5)is equivalent to the quasi-WSMSE optimization problem


defined by

maxW ,U ,S

∑k∈K

αkhk(W ,U ,S)

gk(W )

s.t.∑m∈Uk

Tr(Gk,sW k,m) ≤ �k, ∀ k ∈ K, s ∈ S,

hk,m(W ,U ,S) ≥ γk,m, ∀ k ∈ K, m ∈ Uk∑m∈Uk

Tr(W k,m) ≤ Pk, ∀ k ∈ K, (10)

where hk,m = −sk,mek,m + ln(sk,m) + 1, and⎧⎨⎩hk(W ,U ,S) =

∑m∈Uk

θk,mhk,m, (11a)

gk(W ) = �k

∑m∈Uk

Tr(W k,m) +MkPc + P0, (11b)

where sk,m is an auxiliary variable of user-(k,m), ∀ k,m.4

Proof: Note that function hk(W ,U ,S) andhk,m(W ,U ,S) are concave in each of the optimizationvariables W , U , S, and function gk(W ) is convex withrespect to the optimization variables W . However, they areall not joint concave or convex in all variables. By exploitingthe characteristics that each linear equalizer μk,m and eachauxiliary variable sk,m possess separable nature for each user,it is easy to know that the optimum solution of μk,m is givenby μopt

k,m in (8), since μoptk,m minimizes the MSE of User-(k,m),

∀ k,m, i.e., maximizes the objective function of problem (10)[11], [12]. Also, it is easily verified that the optimum solutionof sk,m is e−1

k,m by using the Lagrangian duality theorem. Bysubstituting the optimal μk,m and sk,m, ∀ k,m, in (10), we havethe following optimization problem:

maxW

∑k∈K

αk

∑m∈Uk

θk,m ln(

1emmsek,m

)gk(W )

s.t.∑m∈Uk

Tr(Gk,sW k,m) ≤ �k, ∀ k ∈ K, s ∈ S,

−ln(emmsek,m

)≥ γk,m, ∀ k ∈ K, m ∈ Uk∑

m∈Uk

Tr(W k,m) ≤ Pk, ∀ k ∈ K. (12)

According to (9), we have ln(1/emmsek,m ) = Rk,m. Therefore, it

is easily seen that problem (12) is equivalent to problem (5). �Proposition 2: The WSRMax problem (6) is equivalent to

the WSMSE optimization problem defined by

maxW

∑k∈K

hk(W ,U ,S)

s.t.∑m∈Uk

Tr(Gk,sW k,m) ≤ �k, ∀ k ∈ K, s ∈ S,


Tr(W k,m) ≤ Pk, ∀ k ∈ K, (13)

4For notional convenience, let U and S denote respectively the collection ofall linear equalizers and the collection of all auxiliary variables.

where hk(W ,U ,S) and hk,m(W ,U ,S) are given inProposition 1, respectively.

According to Proposition 1, problem (5) can be transformedinto problem (10) which has some pleasant properties for thenew form of rate constraint. However, it is still challengingdue to the sum-of-ratio form in the objective [37]–[40]. Toovercome it, the objective function of problem (10) is furthertransformed into an equivalent parametric subtractive form byintroducing firstly auxiliary variable β = {β1, . . . , βK} andrewriting as.

maxW ,S,U ,β

∑k∈K

αkβk

s.t.hk(W ,U ,S)

gk(W )≥ βk, ∀ k ∈ K,∑

m∈Uk

Tr(Gk,sW k,m) ≤ �k, ∀ k ∈ K, s ∈ S,


Tr(W k,m) ≤ Pk, ∀ k ∈ K. (14)

We should note that though in general the new form problem(14) is still not joint convex in all variables W , U , S, and β,the functions hk(W ,U ,S), hk,m(W ,U ,S), and gk(W ) areconcave or convex in each individual variable, by which theproblem can be solved through handing a parametric subtractivequadratic optimization problem. The following theorem estab-lishes the equivalence between the weighted sum maximizationproblem (14) with fractional constraints and a parametric sub-tractive quadratic optimization problem.

Theorem 1: If (W ,U ,S,β) is the solution of problem (14),then there exist λ, such that (W ,U ,S) satisfies the Karush-Kuhn-Tucker (KKT) of the following problem for λ = λ andβ = β.

maxW ,U ,S

∑k∈K

λk (hk(W ,U ,S)− βkgk(W ))

s.t.∑m∈Uk

Tr(Gk,sW k,m) ≤ �k, ∀ k ∈ K, s ∈ S,

hk,m (W ,U ,S) ≥ γk,m, ∀ k ∈ K, m ∈ Uk∑m∈Uk

Tr (W k,m) ≤ Pk, ∀ k ∈ K. (15)

And (W ,U ,S) also satisfies the following system equationsfor λ = λ and β = β.{λk = αk

gk(W ) , ∀ k ∈ K, (16a)

βk = hk(W ,U ,S)gk(W ) , ∀ k ∈ K. (16b)

On the contrary, if (W ,U ,S) is a solution of problem (15)and satisfies system (16) for λ = λ and β = β, (W ,U ,S,β)also satisfies the KKT conditions of problem (14) for Lagrangemultipliers λ = λ associated with the EE constraints.

Proof: Introducing Lagrange multipliers λ = {λ1, . . . ,λK} associated with the EE constraints, ζ = {ζ1, . . . , ζK}associated with interference limitation constraints whereζk = {ζk,1, . . . , ζk,T }, ν = {ν1, . . . ,νK} associated with the


minimum user rate constraints where νi = {νi,1, . . . ,νi,|Ui|},∀ i ∈ K, and ς = {ς1, . . . , ςK} associated with the transmitpower constraints, respectively. Thus, the Lagrange function ofproblem (14) is given by

L(W ,U ,S,β,λ, ζ,ν, ς)

=∑k∈K

αkβk +∑k∈K

λk (hk(W ,U ,S)− βkgk(W ,U ,S))

−∑k∈K

∑s∈S

ζk,s

( ∑m∈Uk

Tr(Gk,sW k,m)− �k

)+

∑k∈K

∑m∈Uk

νk,m (hk,m(W ,U ,S)− γk,m)

−∑k∈K

ςk

( ∑m∈Uk

Tr(W k,m)− Pk

). (17)

As (W ,U ,S,β) is the solution of problem (14), there existλ, ζ, ν, and ς such that the corresponding KKT conditions ofproblem (14) are as follows [41]

∂L∂W

=∑k∈K

λk

(∇hk(W ,U ,S)− βk∇gk(W )

)−

∑k∈K

ςk∑m∈Uk

∇(Tr(W k,m)− Pk

)−

∑k∈K

∑s∈S

ζk,s∇( ∑

m∈Uk

Tr(Gk,sW k,m)− �k

)+∑k∈K

∑m∈Uk

νk,m∇(hk,m(W ,U ,S)−γk,m

)=0, (18)

∂L∂U

=∑k∈K

λk


)+

∑k∈K

∑m∈Uk

νk,m∇ (hk,m(W ,U ,S)−γk,m)=0, (19)

∂L∂S

=∑k∈K

λk


)+

∑k∈K

∑m∈Uk

νk,m∇ (hk,m(W ,U ,S)−γk,m)=0, (20)

∂L∂βk

= αk − λkgk(W ) = 0, ∀ k ∈ K, (21)

λk∂L∂λk

=λk

(hk(W ,U ,S)−βkgk(W )

)=0, ∀ k∈K, (22)

ζk,s∂L∂ζk,s

= ζk,s

( ∑m∈Uk

Tr (Gk,sW k,m)− �k

)= 0,

∀ k ∈ K, s ∈ S, (23)

νk,m∂L

∂νk,m= νk,m (hk,m(W ,U ,S)− γk,m) = 0,

∀ k ∈ K, m ∈ Uk, (24)

ςk∂L∂ςk

= ςk

( ∑m∈Uk

Tr(W k,m)−Pk

)=0, ∀ k∈K, (25)⎧⎪⎪⎨⎪⎪⎩

∑m∈Uk

Tr(Gk,sW k,m)≤�k, ζk,s≥0, ∀ k∈K, s∈S, (26a)

hk,m(W ,U ,S)≥γk,m, ∀νk,m≥0, k∈K,m∈Uk, (26b)∑m∈Uk

Tr(W k,m)≤Pk, λk≥0, ςk≥0, ∀∈K. (26c)

Since gk(W ) > 0, ∀ k ∈ K, for arbitrary W , (21) is equiv-alent to

λk =αk

gk(W ), (27)

and (22) is equivalent to

βk =hk(W ,U ,S)

gk(W ). (28)

Moreover, the system (18), (23)–(26) are just the KKT con-ditions of the following problem for parameters λ = λ andβ = β.

maxW ,U ,S

∑k∈K

λk (hk(W ,U ,S)− βkgk(W ))

s.t.∑m∈Uk

Tr(Gk,sW k,m) ≤ �k, ∀ k ∈ K, s ∈ S


Tr(W k,m) ≤ Pk, ∀ k ∈ K. (29)

Therefore, the first conclusion in Theorem 1 holds. Followinga similar procedure, it is easy to prove that the contrary conclu-sion also holds. �

According to Theorem 1, we can easily know that thesolution of the problem (14) can be obtained by finding theparticular one that satisfies system (16) among the solutions ofproblem (15). Furthermore, if the achieved result is unique, thenthe solution is just the global solution of problem (14). It is alsoworth noting that the parameterized non-fractional objectiveof problem (15) facilitates the solution development with aneffective approach, which is addressed in the next section.

IV. SOLUTIONS OF OPTIMIZATION PROBLEMS

A. Algorithms of WSPEEMax Problem

As shown in the previous section, the function hk(W ,U ,S)and hk,m(W ,U ,S) are concave in each of the optimizationvariables W , U , S and the function gk(W ) is convex in theoptimization variables W . This means that the cost function ofproblem (15) is convex in each of the optimization variables W ,U , and S for given auxiliary variables λ and β. In what follows,we adopt the block coordinate optimization method [41] tosolve problem (15) and focus our energy on the solution of thetransmit precoding matrices W . First, for a given fixed transmitbeamformers, it is easily known that the optimal solution ofμk,m and the optimal solution of sk,m are given by (8) and(emmse

k,m )−1, respectively. Then we resort to solve problem (15)with respect to the optimization variables W . By plugging (7)and (11) in (15), we arrive at the equivalent form of problem(15), given by (30), shown at the bottom of the next page, fora given linear equalizers U and auxiliary variables S, whereGk,s = G

(1/2)k,s , ∀ k ∈ K,m ∈ Uk. It is well known that the

above problem (30) can be solved by using the classical secondorder conic programming (SOCP) method [42].

Based on the aforementioned analysis, here, we adoptthe block coordinate descent method to solve problem (15).


Specifically, we solve the parametric subtractive quadratic formoptimization problem by sequentially fixing two of the threevariables W , U , S, and updating the third. The detailed steps 2of the proposed energy-efficient algorithm is summarized asAlgorithm 1, where⎧⎨⎩

ψk(λk) = λkgk(W )− αk, ∀ k ∈ K, (31a)ϕk(βk) = βkgk(W )− hk(W ,U ,Σ), ∀ k ∈ K, (31b)χk = 1

gk(W ) , ∀ k ∈ K, (31c)

and ρ denotes the objective value of problem (15), and η is apredefined threshold.

Algorithm 1 Macro-cell Energy-Efficient Transmission

1: Let n = 0, choose ∀ξ ∈ (0, 1), ∀ε ∈ (0, 1) and choosearbitrarily W (n) such that the constraints are satisfied,and compute U (n) and S(n). Let ρ(0) = 0 and⎧⎪⎨⎪⎩

λ(n)k = αk

gk(W (n)), ∀ k, (32a)

β(n)k =

hk(W (n),U(n),S(n))gk(W (n))

, ∀ k. (32b)

2: Solve problem (30) with U (n) and S(n), then obtainW (∗); update U with (8) and W (∗), then obtain U (∗);update S with (9), W (∗) and U (∗), then obtain S(∗).

3: Compute ρ(∗) with W (∗), U (∗), S(∗), λ(n), and β(n).If |ρ(∗) − ρ(n)| ≤ δ where δ is an arbitrary small posi-tive number, then let W (n+1) = W (∗), U (n+1) = U (∗),S(n+1) = S(∗), ρ(n+1) = ρ(∗), and go to step 4, oth-erwise, let W (n) = W (∗), U (n) = U (∗), S(n) = S(∗),ρ(n) = ρ(∗), and go to step 2.

4: If the following conditions are satisfied,⎧⎪⎪⎪⎨⎪⎪⎪⎩∣∣∣λ(n)

k gk

(W (n+1)

)− αk

∣∣∣ ≤ δ, ∀ k, (33a)∣∣∣∣∣∣β(n)k gk

(W (n+1)

)−hk

(W (n+1),U (n+1),S(n+1)

)∣∣∣∣∣∣ ≤ δ, ∀ k. (33b)

then output the optimal solutions W (n+1), U (n+1)

and S(n+1), and stop the algorithm. Otherwise, let

n = n+ 1 and⎧⎨⎩λ(n)j = λ

(n−1)j − ξi

(n)χjψj

(λ(n−1)j

), ∀ j (34a)

β(n)j = β

(n−1)j − ξi

(n)χjϕj

(β(n−1)j

), ∀ j (34b)

with W (n), U (n), Σ(n), and go to step 2, where i(n)

denotes the smallest integer among i ∈ {0, 1, 2, . . . . . .}satisfying∑

j

∣∣∣ψj

(λ(n)j −ξiχjψj

(λ(n)j

))∣∣∣2+∣∣∣ϕj

(β(n)j −ξiχjϕj

(β(n)j

))∣∣∣2≤

(1−εξi

)2 ∑j

(∣∣∣ψj

(λ(n)j

)∣∣∣2+∣∣∣ϕj

(β(n)j

)∣∣∣2) (35)

for W (n+1), U (n+1), and Σ(n+1).

B. Algorithms of WSRMax Problem

According to the conclusion of Proposition 2 and (11), weknow that for a fixed W , the optimal solutions of μk,m and sk,min problem (13) are given with (8) and (emmse

k,m )−1, respectively.Furthermore, the above problem can be reformulated into thefollowing equivalent problem which can be solved with SOCPfor given U and S.

minW

∑k∈K

∑m∈Uk

∑(j,n) =(k,m)

∥∥∥√θk,msk,mμ∗k,mhH

j,k,mwj,n

∥∥∥2+∑k∈K

∑m∈Uk

∥∥∥√θk,msk,m(1−μ∗

khHk,k,mwk,m

)∥∥∥2s.t.

∑(j,n) =(k,m)

∥∥√sk,mμ∗k,mhH

j,k,mwj,n

∥∥2+∥∥√sk,m

(1− μ∗

khHk,k,mwk,m

)∥∥2+ sk,mσ2

k,m|μk,m|2 ≤ ln(sk,m) + 1− γk,m,

∀ k ∈ K,m ∈ Uk∑m∈Uk

∥∥∥Gk,swk,m

∥∥∥2 ≤ �k, ∀ k ∈ K, s ∈ S,∑m∈Uk

‖wk,m‖2 ≤ Pk, ∀ k ∈ K. (36)

minW

∑k∈K

∑m∈Uk

∑(j,n) =(k,m)

∥∥∥√λkθk,msk,mμ∗k,mhH

j,k,mwj,n

∥∥∥2+

∑k∈K

∑m∈Uk

∥∥∥√λkθk,msk,m(1− μ∗

khHk,k,mwk,m

)∥∥∥2 +∑k∈K

∑m∈Uk

∥∥∥√λkβk�kwk,m

∥∥∥2 ,s.t.

∑(j,n) =(k,m)

∥∥√sk,mμ∗k,mhH

j,k,mwj,n

∥∥2 + ∥∥√sk,m(1− μ∗

k,mhHk,k,mwk,m

)∥∥2+ sk,mσ2

k,m ‖μk,m‖2 ≤ ln (sk,m) + 1− γk,m, ∀ k ∈ K,m ∈ Uk,∑m∈Uk

∥∥∥Gk,swk,m

∥∥∥2 ≤ �k, ∀ k ∈ K, s ∈ S,∑m∈Uk

‖wk,m‖2 ≤ Pk, ∀ k ∈ K (30)


As a consequence, problem (13) can be solved with the fol-lowing Algorithm 2 where R denotes the SR of the consideredproblem.

Algorithm 2 WSRMax Optimization

1: Let n = 0, choose arbitrarily W (n) such that the con-straints are satisfied, and compute U (n) and S(n), com-pute R(n) with W (n), U (n), S(n).

2: Solve problem (36) with U (n), and S(n), and obtainW (n+1); update U with (8) and W (n+1), and obtainU (n+1), update S with (9), W (n+1), and U (n+1), andobtain S(n+1).

3: Compute R(n+1) with W (n+1), U (n+1), and S(n+1). If|R(n+1) −R(n)| ≤ η, then stop, otherwise, let n = n+1, and then go to step 2.

Remark 1: Based on the equivalent problem transformationbased on Proposition 1 or 2 and Theorem 1, we propose a uni-fied approach to solve the energy-efficient beamforming designproblem and also the WSRMax problems for heterogeneousdownlink networks. Especially, we can address the possiblerate constraints and the interference leakage constraints inconjunction with the energy-efficient transmission, thus ourproposed approach can also apply to other scenarios such asthe homogeneous network and the cognitive radio. Moreover,it is worth noting that when the considered problem (15) issimplified by removing the rate and the interference leakageconstraints, for given ss,k and μk,m, the optimal beamformercan be analytically derived with a similar procedure used in[23, Section III-B], given by

woptk,m = λkθk,msk,mμk,m

(Ξk + ςoptk I

)−1hk,k,m, (37)

where Ξk =∑

j∈K∑

n∈Ujλjθj,nsj,n|μj,n|2hk,j,nh

Hk,j,n +

λkβk�kI whose eigen-value decomposition (EVD) is givenas ΨkΛkΨ

Hk , ςoptk denotes the optimal Lagrange multiplier

associated with the transmit power constraint and satisfies thefollowing condition

Mk∑v=1

[Φk]v,v([Λk]v,v + ςk

)2 = Pk, (38)

where Φk=λ2kΨ

Hk

∑m∈Uk

‖θk,msk,mμk,m‖2hk,k,mhHk,k,mΨk.

In practical use ςoptk can be efficiently obtained using the bi-section method, in this case the complex SOCP in (30) can beavoided in calculating the beamforming vector, and thereforesignificantly reduces the computational complexity, especiallywhen the number of antennas Mk is large.

Remark 2: Note that the implementation of the proposed twoalgorithms both involve the initialization of W , which can beaddressed by solving the following sum power minimizationproblem subject to the same constraints.

minW

K∑k=1

Nk∑m=1

‖wk,m‖2

s.t. SINRk,m ≥ eγk,m − 1, ∀ k ∈ K, m ∈ Uk∑m∈Uk

‖Gk,swk,m‖2 ≤ �k, ∀ k ∈ K, s ∈ S

∑m∈Uk

‖wk,m‖2 ≤ Pk, ∀ k ∈ K. (39)

Assume that W(∗)

is the optimal solution of problem (39),

then let wk,m = W(∗)k,m, ∀ k ∈ K,m ∈ Uk, respectively. The

feasiblity of the target user rates can be checked by solvingproblem (39). In other words, the user target rates are feasibleonce problem (39) is solvable.

C. Algorithm Analysis

It is easy to find that the steps 2∼3 in Algorithm 1 areessentially the same with the steps 2∼3 in Algorithm 2.Therefore, these steps have the same convergence property andsimilar computational complexity. In what follows, we focuson analyzing the convergence property and the computationalcomplexity of Algorithm 2.

Lemma 1: The sequence generated by Algorithm 2 guaran-tees to converge.

Proof: It is easy to know that the updates of steps 2 all aimto improving the SR at each iteration such that an increasingsequence is generated while running Algorithm 2. Furthermore,the SR is bounded by the limited transmit power constraint.Therefore, the convergence of Algorithm 2 can be guaranteedby the monotonic boundary sequence theorem [43]. �

Based on the conclusion of Theorem 1, the convergence ofthe steps 2–3 in Algorithm 1 is also guaranteed by the mono-tonic boundary sequence theorem. In addition, the convergenceof the outer iteration loop in Algorithm 1 can be guaranteed bythe gradient methods [41].

The computational complexity of the proposed two algo-rithms mainly involves some complex multiplication oper-ations and the SOCP algorithm [42]. Specifically, problem(30) has M = 2

∑Kj=1 MjNj real optimization variables, N =∑K

j=1 Nj target user rate second-order-cone (SOC) constraints

where each of them consists of M real dimensions, K leakage-aware SOC constraints where the kth leakage-aware SOCconsists of MkNk real dimensions, ∀ k and K transmit powerSOC constraints where the kth transmit power SOC con-straint consists of MkNk real dimensions, ∀ k. According to[42], the computational complexity of the power minimizationproblem in terms of number of iterations is upper bounded

by O(√N + 2K) where the complexity of each iteration is

within the order of O(2M3(N + 2)). Thus, the total worst-case computational complexity of the SOCP algorithm is given

by O(2√

N + 2KM3(N + 2)). Similarly, the computationalcomplexity of problem (36) can also be analyzed and is hereomitted.

V. SMALL-CELL ENERGY-EFFICIENT TRANSMISSION

Note that in problem (5) we only consider the weighted sumEE of the macro-cells in the objective, it is straightforward to


extend into considering both the macro-cell and small-cells.However, it requires global CSI and is usually hard to realizein practice due to limited backhaul capacity. Alternatively, herewe focus on the energy-efficient beamforming design for smallcell only based on local CSI, so as to release the backhaulburden. The considered small-cell energy-efficient problem isformulated as5

max{ws,l}

Ns∑l=1

θs,lRs,l

ωs

Ns∑l=1

‖ws,l‖2 +MsPc + P0

,

s.t.Ns∑l=1

‖ws,l‖2 ≤ Ps, Rs,l ≥ γs,l, ∀ l, (40)

where Ns and Ms represent respectively the number of usersin the s-th small-cell and the number of the transmit anten-nas of the s-th small-cell BS, θs,l is the weighting factor,ws,l ∈ C

Ms×1 denotes the beamforming vector for user-(s, l)in the s-th small-cell BS, Ps denotes the maximum transmitpower constraint of the s-th small-cell BS, γs,l and Rs,l denoterespectively the target user rate and the user rate of the user-(s, l) in the s-th small-cell, Rs,l is calculated as

Rs,l = ln

⎛⎜⎜⎜⎝1 +

∥∥(hs,l)Hws,l

∥∥2Ns∑

n=1,n=l

‖(hs,n)Hws,n‖2 + �s,l

⎞⎟⎟⎟⎠ , (41)

where hs,l denotes the channel coefficient between the macro-cell BS and the user-(s, l) in the s-th small-cell BS in macro-cell k, �s,l = �+ σ2

s,l can be regarded as the noise varianceof user-(s, l) in the s-th small-cell and � denotes the leakageinterference threshold.6 This problem is hard to tackle directly.To address it, we reformulate it into the following form basedon the result of Proposition 1,

max{ws,l,μs,l,φs,l}

Ns∑l=1

θs,l (−φs,les,l + ln(φs,l) + 1)

ωs

Ns∑l=1


,

s.t.Ns∑l=1

‖ws,l‖2≤Ps,−φs,les,l

+ ln(φs,l) + 1 ≥ γs,l, ∀ l, (42)

where φs,l is an auxiliary variable for user-(s, l) in the s-thsmall-cell, es,l is the MMSE of user-(s, l) in the s-th small-cellBS and is given by

es,l = ‖μs,l‖2Js,l − μs,l(ws,l)Hhs,l − (μs,l)

∗(hs,l)Hws,l+1,

(43)

5The index of macro-cell k is neglected for notational simplicity due to thefact that all small-cell BSs can be adopted by a similar method to optimize theenergy-efficient transmission beamforming.

6We would like to point out that the interference leakage threshold � can bereplaced with the other values, such as the received average interference signalstrength from other BSs.

where Js,l =∑Ns

m=1 ‖(hs,l)Hws,m‖2 + �s,l and μs,l denotes

the receiver filter at user-(s, l) in the s-th small-cell. It has beenrevealed in [37]–[39] that problem (40) is equivalent to lookingup the optimal value of ηs that the optimal objective value ofthe following problem equal to zero.

max{ws,l,μs,l,φs,l}

Ns∑l=1

θs,l (−φs,les,l + ln(φs,l) + 1)

− ηs

(ωs

Ns∑l=1


),

s.t.Ns∑l=1

‖ws,l‖2 ≤ Ps,−φs,les,l

+ ln(φs,l) + 1 ≥ γs,l, ∀ l. (44)

According to the conclusion obtained in proposition 1, for fixed{ws,l}, the optimal solution of μs,l and ss,l of problem (44) aregiven as

μ(opt)s,l =

1

Js,l(hs,l)

Hws,l, (45)

φ(opt)s,l =

(1− 1

Js,l

∥∥(hs,l)Hws,l

∥∥2)−1

. (46)

Fixed {ss,l, us,l} and ηs, problem (44) can be reformulated as

max{ws,l}

Ns∑l=1

Ns∑m=1,m =l

∥∥∥√θs,lφs,l(μs,m)∗(hs,m)Hws,m

∥∥∥2

+

Ns∑l=1

∥∥∥√θs,lφs,l

(1− (μs,l)

∗(hs,l)Hws,l

)∥∥∥2

+ ηsωs

Ns∑l=1

‖ws,l‖2,

s.t.Ns∑

m=1,m =l

∥∥∥√φs,l(μs,m)∗ (hs,m)H ws,m

∥∥∥2+∥∥∥√φs,l

(1− (μs,l)

∗(hs,l)Hws,l

)∥∥∥2+ φs,l�s,l|μs,l|2 ≥ ln(φs,l) + 1− γs,l, ∀ l,

Ns∑l=1

‖ws,l‖2 ≤ Ps. (47)

The above problem is a classical SOCP problem and canbe solved with the standard convex package. Based on theabove derivations, it is easy to develop an alternating algorithmto solve problem (42), summarized as Algorithm 3 wherefs(ws,l, μs,l, ss,l, ηs) denotes the objective value of problem(44). We also point out that the initialization of W can berealized via solving power minimization problem subject to thesame constraints. Furthermore, the convergence of Algorithm 3can be easily proven with a similar method used in [20], [23].


Algorithm 3 Small-cell Energy-Efficient Transmission

1: Choose arbitrarily {w(0)s,l } such that the constraints are

satisfied, and then compute respectively {μ(0)s,l , phi

(0)s,l }

with (45) and (46), let η(0)s =

∑Ns

l=1 θs,l(−φs,les,l +

ln(φs,l) + 1)/ωs

∑Ns

l=1 ‖ws,l‖2 +MsPc + P0 with

{w(0)s,l }, {μ(0)

s,l }, and {φ(0)s,l }.

2: Solve problem (47) with {μ(0)s,l , φ

(0)s,l } and η

(0)s , then obtain

{w(Temp)s,l }.

3: Update {μs,l} with (45) and (46) and {w(Temp)s,l }, then

obtain {μ(Temp)s,l }, update {φs,l} with (46), {w(Temp)

s,l }and {μ(Temp)

s,l }, then obtain {φ(Temp)s,l }.

4: If

∣∣∣∣ fs(w(Temp)

s,l,μ

(Temp)

s,l,φ

(Temp)

s,l,η

(0)s

)−fs

(w

(0)

s,l,μ

(0)

s,l,φ

(0)

s,l,η

(0)s

) ∣∣∣∣ ≤ δ, where δ is an

arbitrary small positive number, then go stop step 5,otherwise, let w(0)

s,l = w(Temp)s,l , μ(0)

s,l = μ(Temp)s,l , φ(0)

s,l =

φ(Temp)s,l , then go to step 2.

5: If |fs(w(Temp)s,l , μ

(Temp)s,l , φ

(Temp)s,l , η

(0)s )| ≤ δ, then stop,

otherwise let η(0)s =∑Ns

l=1 θs,l(−φs,les,l + ln(φs,l) + 1)/

ωs

∑Ns

l=1 ‖ws,l‖2 +MsPc + P0 with {w(Temp)s,l ,

μ(Temp)s,l , φ

(Temp)s,l } and then go to step 2.

VI. NUMERATION RESULTS

In this section, numerical simulations are performed to evalu-ate the performance of the proposed scheme in a heterogeneousmulticell multiuser MISO downlink network. Consider a clusterof K = 3 hexagonal adjacent macro-cells each overlaid withT small-cells. Assume that the kth macro-cell BS is equippedwith Mk transmit antennas and serves Nk single antenna users.Each small-cell consists of a multi-antenna low power BS anda plurality of single-antenna users. The radius of the macro-celland the small-cell are set to be 500 m and 40 m, respectively.The distance between the macro-cell BS and the small-cell BSis 350 m or 380 m. In the considered propagation environment,the macro-cell path loss and the small-cell path loss are given as34.5 + 38 log10(d[m]) and 37 + 32 log10(d[m]), respectively.Each component of the small scale fading coefficients is mod-eled as independent and identically distributed (i.i.d.) complexGaussian random variable with zero mean and unit variance.The shadow fading is modeled as log-normal distribution withzero mean and standard deviation 8 dB. The noise figure ateach user terminal is 9 dB. The circuit power per antenna isPc = 30 dBm, and the basic power consumed at the BS is P0 =40 dBm [20]. The values of θk,m, �k, and αk are set to be unit,∀ k ∈ K,m ∈ Uk, respectively. The leakage power constraint�k is set to be 27 dBm, ∀ k. Assume that all BSs have thesame maximum transmit power constraints, maximum leakagepower thresholds and minimum per-cell target rate thresholds.The maximum transmit power and the number of transmitantenna of small-cell BS are 15 dBm and 2, respectively [32].The energy-efficient transmission beamformers of the macro-

cell BS and the small-cell BS are designed respectively by theAlgorithm 1 and Algorithm 3.

In our simulation, the user target rates are set to be the userrates achieved by the normalized maximum ratio transmission(MRT) beamforming with the power allocation optimized tosatisfy the leakage interference constraint and the transmitpower constraint, i.e., the power allocation factor is given by

χk = min

⎛⎜⎝mins∈S

�k∑m∈Uk

Tr(Gk,sW k,m),

Pk∑m∈Uk

Tr(W k,m)

⎞⎟⎠(48)

and the corresponding transmit beamforming is wk,m =χk(hk,k,m/‖hk,k,m‖), ∀ k ∈ K,m ∈ Uk. For the case wherethe leakage interference mitigation is not considered, the usertarget rates are set to be the user rates achieved by normalizedMRT beamforming and equal power allocation at each BS andthe value of �k is set to be the actual received interferencestrength generated by its serving BS.

For comparison, the performance of the power minimizationalgorithm (PM Algorithm) that aims to minimize the transmitpower subject to the per-user rate demands and the per-BStransmit power constraints and of the MRT beamforming trans-mission with equal power allocation (MRT Algorithm) subjectto the per-BS transmit power constraints are also evaluatedin our numerical simulations. In the legend of the figures,(w/LA) denotes the algorithm design considering leakage in-terference constraints, while (w/o LA) denotes that withoutleakage interference controlling and (w/o LA&R) denotes thatwithout leakage interference controlling and minimum user raterequirement constraint.

Fig. 2 illustrates the average network EE comparison of var-ious coordinated transmission algorithms under 1000 randomchannel realizations, where the distance between the macro-cell BS and the small-cell BS is 350 m.7 It can be seenthat in the lower transmit power region, such as 26–38 dBm,Algorithm 1 (w/LA) and Algorithm 2 (w/LA) achieve thesame network EE. However, in the middle-high transmit powerregion, Algorithm 1 (w/LA) obtains a better EE performancethan Algorithm 2 (w/LA). This is because that in Algorithm 2the SRMax algorithm always suggests all BSs transmitting withfull power even when the increase of transmit power has lowefficiency in improving the rates, leading to the EE performancedegradation. It is also observed that both Algorithm 1 (w/o LA)and Algorithm 2 (w/o LA) outperform the MRT algorithm interms of the EE metrics, due to the fact that the MRT algorithmonly tries to maximize the signal power of its serving userwithout considering any intra-cell and inter-cell interferencemitigation.

Figs. 3 and 4 shows the average small-cell EE and SRcomparison of various coordinated transmission algorithms for

7The channel coefficient between macro-cell BS-k and the l-th user in small-cell s in macro-cell k is described as εs,l(Gk

s )(1/2)g

k(w)s,l

where εs,l denotes

the large scale fading coefficient, Gks denotes the channel variance matrix, and

gk(w)s,l

denotes the small scale fading coefficient whose elements are modeledas i.i.d. complex Gaussian random variables with zero mean and unit variance.


Fig. 2. Networks EE comparison, Mk = 8, Ms = 4, Nk = 2, Ns = 2,∀ k, s, η = δ = 10−3. (a) T = 2. (b) T = 3.

Fig. 3. Small-cell EE comparison, Mk = 8, Ms = 4, Nk = 2, Ns = 2,∀ k, s, η = δ = 10−3. (a) T = 2. (b) T = 3.

small-cell users under 1000 random channel realizations, wherethe distance between the macro-cell BS and the small-cell BSis 350 m. It is seen from the performance of Algorithm 1(w/LA) and Algorithm 2 (w/LA) that the small-cell EE can beimproved through controlling the interference power generatedby the macro-cell BS. It is seen from the performances ofAlgorithm 1 (w/o LA) and Algorithm 2 (w/o LA), if the leakagepower of macro-cell BS is not controlled, the EE performanceof the small-cell users gradually reduces with the transmitpower of macro-cell BS due to the increased interference. Sim-ilarly, the SR of the small-cell users also gradually deteriorates

Fig. 4. Small-cell SR comparison, Mk = 8, Ms = 4, Nk = 2, Ns = 2,∀ k, s, η = δ = 10−3. (a) T = 2. (b) T = 3.

with the transmit power of macro-cell BS increasing. By con-trast, if leakage interference is considered in the macro-cell BSbeamforming design using Algorithm 1 (w/LA) or Algorithm 2(w/LA), it is illustrated that the small-cell performance in termsof both the EE and the SR remain unchanged with the transmitpower of macro-cell BS increasing. These results suggest thatit is particulary important to control the interference fromthe macro-cell BS to the small-cell users in a heterogeneousnetwork. Compared with Algorithm 1 (w/o LA), Algorithm 2(w/o LA), and the MRT algorithm (w/o LA), PM algorithm(w/o LA) can achieve a better performance in terms of the SRand EE. The reason is that the interference generated by themacro-cell BSs with the beamforming obtained by Algorithm 1(w/o LA), Algorithm 2 (w/o LA), and MRT algorithm (w/o LA)is greater than that generated by the macro-cell BSs with the PMbeamforming (w/o LA).

Fig. 5 illustrates the average network EE comparison andthe average small-cell EE of various coordinated transmissionalgorithms under 1000 random channel realizations, where thedistance between the macro-cell BS and the small-cell BS is380 m. Compared with Figs. 2 and 3, it is easy to seen that thenetwork EE performance improves slightly with the increaseof the distance between the macro-cell BS and the small-cellBS. This is because the interference strength generated by themacro-cell BS to the small-cell users reduces with the distanceincreasing, and the macro-cell BS can allocate more transmitpower to improve its corresponding SE or EE while remainingthe leakage interference level to the small cells.

Fig. 6 shows the average network EE performance of theabove algorithms varying with the number of transmit antennasat each macro-cell BS, where the distance between the macro-cell BS and the small-cell BS is 350 m and the transmit powerconstraint of each macro-cell BS is 46 dBm. It can be seenthat for a fixed number of the macro-cell users, such as 2,the network EE performance of all algorithms increases with


Fig. 5. T = 2, Mk = 8, Ms = 4, Nk = 2, Ns = 2, ∀ k, s, η = δ = 10−3.(a) Networks EE comparison. (b) Small-cell EE comparison.

Fig. 6. Network EE comparison, Ms = 4, Ns = 2, ∀ s, η = δ = 10−3.(a) The number of severed users of macro-cell is 2. (b) The ratio of the numberof transmit antennas to the number of served users of macro-cell BS is setto 8 : 1.

the number of transmit antennas Mk. However, when Mk >16, ∀ k, the network EE performance of all algorithms decreaseswhen the number of transmit antennas Mk increases. This isbecause the circuit power consumption MkPc also increasesrapidly when the number of transmit antenna Mk increases.It means that there is an optimal number of transmit antennaswhen the number of served users, receiver antennas and datastreams are fixed, for example the optimal number of per-BStransmit antennas in the considered configuration is 16. Whilethe ratio of the number of transmit antennas to the number of

Fig. 7. Macro-cell EE comparison, η = δ = 10−3.

the served users at each macro-cell BS is set to 8 : 1, numericalresults corroborate that the network EE of Algorithm 1 (w/oLA&R) and Algorithm 2 (w/o LA&R) both increase with thenumber of transmit antennas, while Algorithm 1 (w/o LA&R)exhibits obvious advantage over Algorithm 2 (w/o LA&R)especially when the number of transmit antennas is not solarge. In this case, the network EE performance of the MRTAlgorithm slightly increases. It also means that multiplexingto many users rather than beamforming to a single user andincreasing the number of service antennas can benefit thenetwork EE performance [21]. It is also observed that the gainof Algorithm 1 (w/o LA&R) over Algorithm 2 (w/o LA&R)tends to shrink with the number of transmit antennas, implyingthat the SE and EE can be simultaneously improved in themassive MIMO system.

Fig. 7 shows the average EE of Algorithm 1, Algorithm 2and the PM Algorithm without taking the leakage interferenceconstraint into account, under 1000 random channel realiza-tions, where the number of the served users at each macro-cell BS is configured to increase with the number of transmitantennas according to a fixed ratio that is set to 1 : 2. It canbe seen that the average EE of all algorithms increase withthe number of the transmit antennas. In particular, the EEof Algorithm 1 (w/o LA), Algorithm 2 (w/o LA), and PMAlgorithm improves by up to 72.75%, 86.88% and 53.88%,respectively, when the number of transmit antenna at the BSincreases from 4 to 8. Also, it is observed that in both antennaconfigurations Algorithm 1 (w/o LA) and Algorithm 2 (w/oLA) achieve the same EE performance at lower transmit powerregion. While at high transmit power region, Algorithm 1 (w/oLA) outperforms Algorithm 2 (w/o LA) in terms of the EEmeasure. In addition, the EE performances of both Algorithm 1(w/o LA) and Algorithm 2 (w/o LA) decrease at high transmitpower region, this is due to the fact that the target user rate setin our simulation increases with the transmit power constraintat each BS and the increased rate can not make up the increasedpower consumption leads to lower EE.

Fig. 8 illustrates the average EE performances of theabove three algorithms for different values of circuit power


Fig. 8. Macro-cell EE comparison, Mk = 4, Nk = 2, ∀ k, η = δ = 10−3.

consumption. It can be seen that the performances of all algo-rithms increase with the circuit power consumption per antennadecreasing. In particular, when the value of Pc is reduced from40 dBm to 30 dBm, the EEs of Algorithm 1 (w/o LA) and thePM algorithm both improve by over 100%. Moreover, it is alsoobserved that the upper bound of the transmit power regionwhere Algorithm 1 (w/o LA) and Algorithm 2 (w/o LA) achievethe same EE performance moves from 42 dBm to 38 dBm,implying that Algorithm 1 (w/o LA) achieves more EE gainover Algorithm 2 (w/o LA) if the circuit power is reduced.

VII. CONCLUSION

In this paper, coordinated energy-efficient transmissiondesign for the heterogeneous multicell multiuser downlinksystems was investigated. Contrary to the conventional EEcriterion, we consider a new objective based on per-cell EEmeasure which was defined as the ratio of per-cell weightedSR and the per-cell total power consumption. The optimizationproblem of interest was formulated as maximizing the WSPEEsubject to multiple constraints in terms of per-cell BS power,per-cell leakage interference power, and per-user minimumrate demands. Then, an alternating optimization algorithm wasproposed to solve the given optimization problem by trans-forming the original problem into an equivalent parametricsubtractive quadratic form optimization problem. The proposedalgorithm was further modified to solve the WSRMax problem.To improve the network EE, an energy-efficient transmissionalgorithm based on the leakage-control was also developed.Finally, the effectiveness of our developed schemes was finallyverified via extensive numerical results.

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Shiwen He received the M.S. degree from ChengduUniversity of Technology, Chengdu, China, in 2009.In 2013, he received the Ph.D. degree in informa-tion and communication engineering from SoutheastUniversity, Nanjing, China.

He is currently a Postdoctoral Researcher with theState Key Laboratory of Millimeter Waves, Depart-ment of Radio Engineering, Southeast University.His main research interests include multiuser MIMOwireless communications, cooperative wireless com-munications, energy-efficient wireless communica-

tions, millimeter-wave communications, and optimization theory.

Yongming Huang (M’10) received the B.S. andM.S. degrees from Nanjing University, Nanjing,China, in 2000 and 2003, respectively, and the Ph.D.degree in electrical engineering from Southeast Uni-versity, Nanjing, in 2007.

Since March 2007, he has been a faculty memberwith the School of Information Science and Engi-neering, Southeast University, China. In 2008–2009,he visited the Signal Processing Laboratory, Schoolof Electrical Engineering, Royal Institute of Tech-nology (KTH), Stockholm, Sweden. His current re-

search interests include space–time wireless communications, cooperativewireless communications, energy-efficient wireless communications, and op-timization theory. He serves as an Associate Editor for the IEEE TRANS-ACTIONS ON SIGNAL PROCESSING, the EURASIP Journal on Advances inSignal Processing, and the EURASIP Journal on Wireless Communications andNetworking.

Haiming Wang (M’08) received the M.S. andPh.D. degrees in electrical engineering from South-east University, Nanjing, China, in 2002 and 2009,respectively.

Since April 2002, he has been with the Schoolof Information Science and Engineering, South-east University, where he is currently an Asso-ciate Professor. His current research interests includesignal processing for MIMO wireless communica-tions and millimeter-wave wireless communications.Dr. Wang received the first-class Science and Tech-

nology Progress Award of Jiangsu Province of China in 2009.

Shi Jin (S’06–M’07) received the B.S. degree incommunications engineering from Guilin Universityof Electronic Technology, Guilin, China, in 1996; theM.S. degree from Nanjing University of Posts andTelecommunications, Nanjing, China, in 2003; andthe Ph.D. degree in communications and informa-tion systems from the Southeast University, Nanjing,in 2007.

From June 2007 to October 2009, he was a Re-search Fellow with the Adastral Park Research Cam-pus, University College London, London, U.K. He is

currently with the faculty of the National Mobile Communications ResearchLaboratory, Southeast University. His research interests include space–timewireless communications, random matrix theory, and information theory.Dr. Jin serves as an Associate Editor for the IEEE COMMUNICATIONS

LETTERS and IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. Heand his coauthors have been awarded the 2011 IEEE Communications SocietyStephen O. Rice Prize Paper Award in the field of communication theory and a2010 Young Author Best Paper Award by the IEEE Signal Processing Society.

Luxi Yang (M’96) received the M.S. and Ph.D.degrees in electrical engineering from SoutheastUniversity, Nanjing, China, in 1990 and 1993,respectively.

Since 1993, he has been with the Department ofRadio Engineering, Southeast University, where heis currently a Professor of information systems andcommunications, and the Director of Digital SignalProcessing Division. His current research interestsinclude signal processing for wireless communica-tions, MIMO communications, cooperative relaying

systems, and statistical signal processing. He is the author or coauthor of twopublished books and more than 100 journal papers. He is a holder of tenpatents. Prof. Yang received the first- and second-class prizes of Science andTechnology Progress Awards of the State Education Ministry of China in 1998and 2002. He is currently a member of the Signal Processing Committee ofChinese Institute of Electronics.

Documents

Leakage-Aware Energy-Efficient Beamforming for Heterogeneous Multicell Multiuser Systems