Cognitive and Cellular Networks-2014

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 2, JANUARY 15, 2014 295

Multi-Cell Beamforming With DecentralizedCoordination in Cognitive and Cellular Networks

Harri Pennanen, Student Member, IEEE, Antti Tölli, Member, IEEE, and Matti Latva-aho, Senior Member, IEEE

Abstract—This paper considers a downlink beamformingproblem in a cognitive radio network where multiple primaryand secondary cells coexist. Each multiantenna primary andsecondary transmitter serves its own set of single antenna users.The optimization objective is to minimize the sum transmissionpower over secondary transmitters while guaranteeing the min-imum SINR for each secondary user and satisfying the maximumaggregate interference power constraint for each primary user.We propose a decentralized beamforming algorithm where theoriginal centralized problem is decomposed via primal decompo-sition method into two levels, i.e., transmitter-level subproblemsmanaged by a network-level master problem. The master problemis solved independently at each secondary transmitter using aprojected subgradient method requiring limited backhaul sig-naling among secondary transmitters. To solve the independenttransmitter-level subproblems, we propose three alternative ap-proaches which are based on second order cone programming,semidefinite programming and uplink–downlink duality. Specialemphasis is put on the last approach, which is also considered ina multi-cell MISO cellular network. Numerical results show thatthe proposed algorithm achieves close to optimal solution evenafter a few iterations in quasi-static channel conditions. Moreover,near centralized performance is demonstrated in time-correlatedchannels.

Index Terms—Decentralized coordinated beamforming, powerminimization, primal decomposition, uplink-downlink duality.

I. INTRODUCTION

C OGNITIVE RADIO (CR) is a promising approach toeffectively utilize the radio spectrum by allowing cog-

nitive secondary users (SUs) to access the bandwidth of thelicensed primary users (PUs) [1], [2]. Many of the traditionalCR approaches are designed to exploit the spectrum holes ofthe licensed band [3]. However, higher spectrum utilization

Manuscript received March 29, 2013; revised August 27, 2013; accepted Oc-tober 04, 2013. Date of publication October 31, 2013; date of current versionDecember 24, 2013. The associate editor coordinating the review of this man-uscript and approving it for publication was Prof. Xiqi Gao. This work hasbeen supported by the Finnish Funding Agency for Technology and Innova-tion (TEKES). During this work the first author was supported in part by theGraduate School in Electronics, Telecommunications and Automation (GETA),the Riitta and Jorma J. Takanen foundation, the Tauno Tönning foundation andthe Walter Ahlström foundation. Parts of this paper have been presented at theTwenty-Second Annual IEEE International Symposium on Personal Indoor andMobile Radio Communications, Toronto, ON, Canada, September 2011, and atthe Forty-Fifth Annual Asilomar Conference on Signals, Systems, and Com-puters, Pacific Grove, California, USA, November 2011.The authors are with the Centre for Wireless Communications, University of

Oulu, 90014 Oulu, Finland (e-mail: [email protected]; [email protected], [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2013.2287681

is achieved in underlay spectrum sharing systems where theprimary and secondary networks are allowed to share the sameradio resources, requiring that the caused interference on eachPU is kept below a tolerable level [4], [5]. In this respect,multiantenna beamforming is seen as a promising approachto provide efficient spectrum usage while satisfying the PUspecific interference power constraints since the generatedinterference can be spatially controlled [6]. Beamforming hasbeen widely studied for traditional wireless communication sys-tems [7], [8], such as cellular networks. Recent achievements incellular beamforming, especially in coordinated beamforming,have evoked the interest to extend these solutions to spectrumsharing CR networks.Coordinated beamforming has been recognized as a powerful

approach for improving the performance of cellular systems,especially at the cell-edge area, by controlling inter-cell in-terference [8]. Inter-cell interference occurs when the sameradio resources are re-used in neighboring cells without propercoordination. Inter-cell interference is a major limiting factorin modern and future cellular systems, such as LTE [9] andLTE-Advanced [10]. In coordinated beamforming, each datastream is transmitted from a single base station (BS) and thetransmissions are dynamically coordinated between multipleBSs to control the generated inter-cell interference. Coordi-nated beamforming is more practical than joint transmission[10] (also known as network MIMO [8]), which is another formof coordinated multipoint transmission [8], [10], since backhaulsignaling load is reduced and carrier phase synchronism is notrequired. Coordinated beamforming can either be centralizedor decentralized. In the centralized case, coordination is per-formed via a central controlling unit which requires access toglobal channel state information (CSI). In the decentralizedcase, each BS acquires only local CSI and the coordinationis performed directly between BSs via backhaul links and/orover-the-air signaling. In general, decentralized approachesare more practical due to their potentially reduced backhaulsignaling loads, lower computational requirements and simplernetwork architectures.Coordinated beamforming approaches with various opti-

mization objectives and quality of service (QoS) constraintshave gained a lot of attention in the wireless communicationresearch community; see for example [8], and the refer-ences therein. The most common optimization criteria areweighted sum rate maximization [11]–[13], SINR/rate bal-ancing [14]–[17] and transmission power minimization withper user QoS constraints [18]–[24]. The first approach is alwaysfeasible and some fairness is included via priority weights. Onthe other hand, the weighted SINR/rate balancing approach isfair between users. However, in general, either one of them

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296 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 2, JANUARY 15, 2014

cannot guarantee the per user QoS targets. In this respect, wefocus on the third optimization criterion which satisfies thepredefined QoS constraints. Furthermore, the overall interfer-ence in the network is decreased since the transmission poweris minimized. Optimal centralized solutions were proposed in[18], [19] and [20] based on uplink-downlink duality, semidef-inite programming (SDP) and second order cone programming(SOCP), respectively. Optimal decentralized solutions were ob-tained via uplink-downlink duality, dual decomposition, primaldecomposition and alternating direction method of multipliers(ADMM) methods in [21]–[23] and [24], respectively.Extending cellular beamforming approaches to CR networks

requires introducing additional constraints on the maximum al-lowed interference levels experienced by the PUs [6]. Recently,cognitive beamforming approaches have been widely studiedwith different secondary network optimization objectives, e.g.,sum rate maximization [25], SINR/rate balancing [26]–[28]and power minimization with QoS constraints [29]–[37].Cognitive multicast and robust MISO beamforming strategieswere studied in [31], [36] and [34], [35], respectively. In[37], non-robust and robust MIMO beamforming strategieswere proposed for CR networks. In this paper, our particularinterest is in cognitive (non-robust) MISO beamforming forpower minimization. In this respect, convex optimization anduplink-downlink duality based beamforming solutions wereproposed for a CR network with a single secondary and primarytransmitter in [30], [32] and [29], [30], [32], respectively. In[33], [32] was extended to the cognitive MISO interferencechannel (IC), i.e., a CR network with multiple secondary/pri-mary transmitter-receiver pairs. The aforementioned cognitivebeamforming approaches are inherently centralized. Hence,they would require a central controlling unit with global CSIfor the secondary network coordination in a general multi-cellmultiuser CR network setting.In [27], minimum power beamformers were solved as an

intermediate result of the original rate balancing problem in thecognitive MISO IC. It was shown that the centralized problemcan be cast as an SOCP and solved efficiently. In addition, theproblem was solved in a decentralized manner via a two-levelalgorithm, where the outer and inner optimizations weresolved using a subgradient method and an uplink-downlinkduality based approach, similar to that in [21], respectively.The outer optimization requires limited backhaul signalingbetween secondary transmitters, whereas the inner optimiza-tion requires real physical transmissions and receptions alongwith some over-the-air signaling. In general, decentralizeduplink-downlink duality based approaches, such as [21] and[27], usually need to converge before they can satisfy the userspecific SINR constraints. This may cause long delays and highsignaling/computational load to the system. It is challengingto apply this kind of decentralized algorithms to realistic timevarying channel conditions, where the channel changes fasterthan the algorithm converges. Moreover, the algorithm in [27]is designed for the cognitive IC instead of a general cognitiveinterference broadcast channel (IBC), where each transmittercan serve multiple simultaneous users. Thus, there is seeminglya lack of (practical) decentralized beamforming algorithms,especially ones that are designed for the cognitive IBC.In this paper, we address this challenge and propose a novel

decentralized beamforming algorithm for the spectrum sharing

multi-cell multiuser MISO CR networks (i.e., the cognitiveIBC). We aim to minimize the total transmission power ofsecondary transmitters while providing the minimum SINRfor each SU and keeping the maximum aggregate interferencepower at each PU below a predefined level. The optimizationproblem is coupled between secondary transmitters, and thus,it is inherently centralized. In order to obtain a decentralizedimplementation, we first equivalently reformulate the problemby introducing two sets of new auxiliary variables, i.e., SUand PU specific inter-cell interference terms. Now, we pro-pose a primal decomposition method to turn the reformulatedoptimization problem into two-levels: a network-level masterproblem which controls transmitter-level subproblems. Thenetwork-level optimization of the SU and PU specific inter-ference terms can be solved independently and in parallel ateach secondary transmitter via a projected subgradient methodwith the aid of limited backhaul signaling. Herein, we referthe backhaul signaling to exchanging a small amount of in-formation, i.e., transmitter-level subgradients, between thecoupled secondary transmitters via backhaul links. We proposeSOCP, SDP and uplink-downlink duality based approachesto solve the independent and signaling-free transmitter-levelsubproblems for the beamformers and subgradients. The firsttwo approaches can be solved optimally using standard convexoptimization software packages. Special emphasis is put on thelast approach which obtains an optimal solution without theneed of explicit convex optimization tools. The last approachis also considered briefly in a conventional multi-cell multiuserMISO cellular system.The proposed decentralized algorithm converges to the op-

timal solution in static channel conditions. Unlike the previousdecentralized approach in [27], the proposed algorithm canprovide feasible beamformers, which satisfy the per SU SINRtargets and per PU interference constraints, at intermediate iter-ations. Therefore, increased delay and signaling/computationalload can be avoided in practice by stopping the algorithm aftera limited number of iterations. This comes at a possible cost ofsub-optimal performance. However, simulation results demon-strate close to optimal performance even after a few iterationsin quasi-static channels. Furthermore, the proposed algorithmobtains near to centralized performance in time-correlatedchannel conditions, where the backhaul signaling is outdateddue to channel variations.The rest of the paper is organized as follows. In Section II, we

introduce the generic multi-cell multiuser MISO system modelfor both CR and cellular networks. The optimization problemsare formulated in Section III. In Section IV, we propose a cog-nitive decentralized beamforming algorithm and provide a de-tailed derivation of it. In Section V, decentralized beamformingis briefly considered in cellular networks and a novel approachis introduced. The performance of the proposed algorithms areevaluated through numerical examples in Section VI. Finally,conclusion is drawn in Section VII.Throughout the paper we use the following notations. We de-

note matrices by bold face upper-case letters and vectors by boldface lower-case letters. Notations and denotereal, non-negative real, positive real and complex spaces, re-spectively. Statistical expectation is . The transpose, Hermi-tian transpose and inverse of a matrix are denoted byand , respectively. The identity matrix is denoted by . No-

PENNANEN et al.: MULTI-CELL BEAMFORMING WITH DECENTRALIZED COORDINATION IN COGNITIVE AND CELLULAR NETWORKS 297

tation indicates matrix inequality with respect to the cone ofsemidefinite matrices. The cardinality of the set is denoted by. Selecting th element of a vector and an element of a matrix

on row and column is denoted by and , respectively.

II. SYSTEM MODEL

Consider a spectrum sharing-based CR network consisting ofprimary and secondary transmitters with and

transmit antennas, respectively. For notational convenience, wedenote . There are PUs and SUs in thesystem. Each of them is equipped with a single receive antenna.The total number of cells and users in the system are denotedby and , respectively. Eachuser is served by a single transmitter. The serving transmitterfor the user is denoted by . User association is assumed tobe predefined and fixed. We denote the sets of all the primaryand secondary transmitters and all the PUs and SUs by

and , respectively. The received signal at the user canbe expressed as

(1)

where and denote thetransmit beamforming vector, the data symbol withand the additive white Gaussian noise sample for the user .The channel vector from the th transmitter to the th user isdenoted by .The total transmission powers across pri-

mary and secondary transmitters are expressed asand

, respectively.

The subsets and with sizesand include all the PUs and SUs served by theirrespective transmitters and , respectively. The receivedSINR of the SU can be written as

(2)

Interference from the primary transmissions to the SU is de-noted by . We assume that is knownat the serving secondary transmitter in order to guarantee theSU specific SINR constraints. Hence, we obtain performanceupper bounds for more practical cases where only partial oraverage interference knowledge is available at the secondarytransmitters. The SINR for the th PU is given by

(3)

Note that a conventional one-tier cellular network can be seenas a special case of a two-tier CR network. Hence, the receivedsignal model in (1) is also valid for a multi-cell multiuser MISOcellular system with BSs and single antenna users.

III. PROBLEM STATEMENT

The optimization target is to minimize the sum transmis-sion power of secondary transmitters while satisfying the SU

specific minimum SINR targets and the PU specificmaximum aggregate interference power constraints .Mathematically, the optimization problem can be written as

(4)

For completeness of the paper, we show in the following propo-sition that (4) can be reformulated and solved optimally usingstandard convex optimization packages, assuming that there ex-ists a central controlling unit with access to global CSI.Proposition 1: Problem (4) can be cast as an SOCP.Proof: See Appendix I.

In (4), the sum power and the SINR constraints are alwaystight at the optimal solution. However, there may be occasionswhen the PU specific aggregate interference constraints are in-active, i.e., the interference is below a predefined maximumlevel at the optimal solution. In that case, removing the con-straints would not have any impact for solving the problem.Herein, we assume that the aggregate interference constraintsare always active at the optimal solution. The problem (4) canbe infeasible in some channel conditions and network scenarios,e.g., the number of users or the SINR targets are too high. Since(4) can be solved as an SOCP, infeasibility can be recognized. Inthis case, admission control needs to adjust the number of usersor the SINR constraints accordingly.For the rest of this paper, we assume that (4) is strictly fea-

sible, and an optimal solution exists. For strict feasibility, weneed to have , since interference power cannotbe negative. A thorough study of the feasibility conditions wasprovided in [27] for a MISO CR system with multiple primaryand secondary transmitter-receiver pairs. As an example, a suf-ficient condition for (4) to be always feasible is when ,even if , provided that the user channels arei.i.d. We also assume that each secondary transmitter has onlylocal CSI, i.e., the knowledge of its own channels to each SUand PU in the CR system. Local CSI can be acquired, e.g., byutilizing reciprocity between uplink and downlink channels intime division duplexing-based systems.In a cellular system, minimum power beamforming problem

reduces to the following:

(5)

where is the sum power. We denote the sets of all the BSs,users and users served by the BS by and , respectively.The problem (5) can be efficiently solved using the algorithms


proposed in [18]–[24]. In Section V, we propose a novel decen-tralized beamforming approach which combines the ideas intro-duced in [23] and [38].

IV. DECENTRALIZED BEAMFORMING IN CR NETWORKS

In this section, we propose a novel decentralized cognitivebeamforming approach which is based on primal decomposi-tion. In general, primal decomposition method turns the originalone-level optimization problem into two optimization levels: ahigher level master problem which is in charge of lower levelsub-problems [39]. This problem structure can be exploited tosolve the original problem in a decentralized way. In the fol-lowing subsections, we will show how the primal decomposi-tion method is utilized in our cognitive beamforming problemleading to an optimal decentralized algorithm. In particular, wepropose a projected subgradient method to solve the masterproblem. Moreover, three alternative approaches are proposedto solve the subproblems: SOCP, SDP and uplink-downlink du-ality-based methods.

A. Equivalent Problem Reformulation

Primal decomposition method is applicable for the problemswhich include coupling variables such that by fixing them theproblem decouples [39]. In this respect, we introduce two setsof auxiliary variables which are coupled between the secondarytransmitters. The auxiliary variables are inter-cell interferencepower from the secondary transmitter to the SU and to thePU , and they are denoted by and

respectively. The resulting problem is expressedas

(6a)

(6b)

(6c)

(6d)

(6e)

(6f)

where the vectors and are composed of the elements of thesets and , respectively. Wedenote the sets of all the SUs except for those served by thetransmitter by and all the transmitters exceptby . The reformulated SINR for the SU isgiven by

(7)

If a centralized implementation is assumed, the last set ofconstraints in (6f) can be replaced by

. For a decentralized case, however, the redundantterm is needed. The rationale behind this isgiven in Section IV.C. The optimal solution of (4) is equivalentto that of (6) since all the inequality constraints in (6) hold withequality at the optimal point. For strict feasibility assumption,we need to have and

. The following proposition shows that (6)can be reformulated as a convex problem.Proposition 2: Problem (6) can be cast as an SOCP.Proof: See Appendix II.

A key aspect of designing our decentralized beamforming ap-proach is that (6) can be solved via its Lagrange dual problem.This is addressed in the following.Proposition 3: Strong duality holds for problem (6).Proof: See Appendix III.

B. Two-Level Problem Formulation via Primal Decomposition

The problem (6) is coupled between the secondary transmit-ters by the variables and . Precisely, each el-ement of couples two secondary transmitters, whereas eachelement of couples all the secondary transmitters. The objec-tive of (6) is inherently separable between secondary transmit-ters, i.e., . If and are fixed, (6) decou-ples between secondary transmitters. Hence, primal decompo-sition is an adequate method to decompose (6) into a higher levelmaster problem and lower level subproblems, one for eachsecondary transmitter. Let us now introduce secondary trans-mitter-specific interference vectors and ,which consist of all and that couple the transmitterwith the other transmitters. Precisely, the elements of

are taken from the sets and . The

vector is defined by since the transmitter is cou-pled with all the other transmitters by all .For the fixed and , the lower level subproblem at the

secondary transmitter can be written as

(8a)

(8b)

(8c)

(8d)

(8e)

(8f)

In the rest of the paper, we assume that (8) is strictly feasible,and there exists an optimal solution. Propositions 2 and 3 holdtrue for (8) since it is a reduced case of (6), i.e., (8) can be castas an SOCP and strong duality holds for it.


The network-level master problem controls the subproblemsby updating and . The master problem isexpressed as

(9)

where denotes the optimal objective valueof (8) for a given and . Sets and are denotedby and

. The problem (9) isconvex since its objective and constraint functions are convex.

C. Network-Level Optimization

The problem (9) can be solved via a projected subgradientmethod with the following updates:

(10)

(11)

where and are the projections onto the sets and ,respectively. The step-sizes at iteration are denoted byand . The scalars and are any (network-level)subgradients of (9) evaluated at the points and ,respectively.In the literature, there are many results on convergence of the

(projected) subgradient method with different step-size rules,see [40], [41]. For example, the projected subgradient methodconverges to the optimal value for a convex problem when thenon-negative step-size is nonsummable and diminishingwith , i.e., and . In thisrespect, a valid step-size is , where gets a fixed andpositive value. Monotonic convergence is not guaranteed for thesubgradient method, and thus, one must keep track of the bestsolution among the previous iterations. Next, we introduce validsubgradients for (9).Proposition 4: Valid subgradients for (9) at points andare given by

(12)

(13)

where and are the optimal Lagrange multiplierscorresponding to the SU specific SINR and inter-cell interfer-ence constraints (8c) and (8d) in th and th subproblems, re-spectively. Similarly, and are the optimal La-grange multipliers corresponding to the PU specific interferenceconstraints (8e) and (8f) in th and th subproblems, respec-tively.

Proof: See Appendix IV.It is worth mentioning that the update process of would

be different in a decentralized case if the constraint (6f) wasequivalently replaced by

. In this case, the constraintvanishes in the corresponding subproblem leading to a sub-gradient of at the point . The subgradientdepends only on the subproblem resulting in an alwaysincreasing (before the projection ) at each subgradientiteration, i.e., . Byhaving the term in (6f), always increasing

can be avoided. Instead, the subgradient in (13) dependson all the subproblems leading to a proper update process of

.Since each is projected such that , the

constraints (8e) and (8f) must be the same, and thus, their La-grange multipliers are also the same, i.e.,

. Hence, the term in (13) can be

equivalently rewritten as . Since the subgradient

update is now obtained by solving only , the con-straint (8f) can be removed. Thus, (8) can be reformulated ac-cordingly. In the rest of the paper, all the related problems willbe formulated accordingly.If the exchange of the subgradients is allowed between the

coupled secondary transmitters via backhaul, and canbe solved independently at the secondary transmitter , for all

in parallel. The proposed network-level optimization issummarized in Algorithm 1.

Algorithm 1: Network-level optimization

1: Set . Initialize and .2: repeat3: Compute and . Communicate

each and to the coupled secondary trans-mitters via backhaul. Optional: compute .

4: Update using (10).5: Update using (11).6: Set .7: until Network-level stopping criterion is satisfied8: Compute .

D. Transmitter-Level Optimization

There are several alternative methods to compute the La-grange multipliers (and beamformers) at step 3 in Algorithm1. In the following subsections, we propose three approacheswhich are based on SOCP, SDP and uplink-downlink duality.The first two approaches can be solved optimally using standardconvex optimization packages. The last approach finds an op-timal solution via a projected subgradient method and a simplefixed-point iteration, and thus, any external convex optimiza-tion package is not required. In general, fixed-point iteration haslower computational complexity compared to SOCP and SDP.For simplicity of notation, we drop the iteration index from thevectors and in the following subsections.1) SOCP Approach: In general, by solving a convex opti-

mization problem using any standard convex optimization soft-ware package, e.g., cvx [42], the optimal Lagrange multipliersare provided as a certificate for optimality. In this respect, (8)can be cast as a convex SOCP by following the principles in


Appendix I. The resulting SOCP problem is expressed as

(14)

where . The optimal Lagrange multipliers

and can now be obtained as a sideinformation by solving (14) for and using anystandard SOCP solver.2) SDP Approach: Since strong duality holds for (8), the op-

timal Lagrange multipliers and can besolved via the Lagrange dual problem of (8), which is expressedas

(15)

where and

. Since the objective function is linear and theinequality constraints are linear matrix inequalities, (15) can becast as a standard form SDP by turning the maximization intominimization and changing the sign of the objective function.Thus, it can be efficiently solved via standard SDP optimizationpackages. In order to find the optimal beamformers ,we can utilize the uplink-downlink duality results proposed inthe next section. Precisely, (44) and (25) need to be solved.3) Uplink-Downlink Duality Based Approach: We

start by equivalently splitting the dual problem (15) intoan outer maximization of and and an innermaximization of . The vectors and con-sist of the Lagrange multipliers corresponding to theSU specific inter-cell interference and SINR constraints,respectively, i.e., and

. Since (15) is concave, both theouter and inner problems are also concave.The outer maximization can be expressed as

(16)

where is the optimal objective value of the innermaximization on for given and . Sets and are

denoted by and

. We can use a projected subgradient method to optimally

solve (16) for and . The projected subgradient updatesare given by

(17)

(18)

where and are the projections onto the sets and ,respectively. At iteration , the step-sizes are denoted byand . Based on Proposition 4, the subgradients and

at the points and can be expressedas

(19)

(20)

In order to solve (19) and (20), the optimal beamformersneed to be found at each iteration . For ease of

presentation, we omit the iteration index with respect toand in the rest of this subsection.Let us write the inner optimization problem on as

(21)

Note that the term is

omitted from the objective since it is fixed, and thus, does nothave any impact on finding the optimal . Inspired by [20],[21], [38], we will next show how to find the optimal and

with the aid of uplink-downlink duality.Theorem 1: The problem (21) is equivalent to the following

problem:

(22)

where is interpreted as a virtual uplink beamformerfor the SU . The problem (22) can be interpreted as a virtual


dual uplink problem for a CR system where the SU specificSINR constraints remain the same as in the downlink.

Proof: See Appendix V.Next, we show how to optimally solve (22) for .Proposition 5: The problem (22) is solved optimally for

via the following fixed point iteration:

(23)

where

(24)

Proof: See Appendix VI.For the fixed (optimal) , the optimal virtual uplink beam-

formers can be computed using the linear MMSE re-ceiver, which is presented in (44) in Appendix V. The followingproposition shows how the optimal downlink beamformers canbe acquired with the aid of the optimal uplink beamformers.Proposition 6: The optimal downlink beamformers

are solved via the optimal virtual uplink beam-formers by scaling, i.e.,

(25)

The scaling factors are solved via thematrix equation:

(26)

where the -th and th elements of the matrix and thevector are given by

(27)

and , respectively.Proof: See Appendix VII.

Following from the previous findings and Proposition 3, theoptimal value of the downlink problem (8) is the same as the op-timal value obtained from solving the outer maximization (16)via the projected subgradient method (17)–(18) and the innerminimization via the virtual uplink problem (22). These opti-mization steps can be solved independently at each secondarytransmitter , for all in parallel. The proposed trans-mitter-level optimization is summarized in Algorithm 2. Algo-rithm 2 is guaranteed to converge to the globally optimal solu-tion due to the convergence of the projected subgradient methodand fixed-point iteration, and the uplink-downlink duality re-sults obtained in this subsection.

TABLE ITOTAL SIGNALING LOAD PER NETWORK-LEVEL ITERATION

Algorithm 2: Transmitter-level optimization

1: Set and . Initialize and.

2: repeat3: repeat4: Compute using (23).5: Set .6: until stopping criterion is satisfied7: Compute uplink beamformers using (44).8: Compute downlink beamformers using (25).

9: Update using (17).10: Update using (18).11: Set .12: until transmitter-level stopping criterion is satisfied

E. Backhaul Signaling and Practical Considerations

Table I presents the amount of the required backhaul sig-naling at each network-level iteration in Algorithm 1. In orderto achieve optimal performance, Algorithm 1 needs to be rununtil convergence. However, aiming to the optimal solutionis somewhat impractical since the more iterations are run, thehigher the signaling/computational load and the longer thecaused delay. In this respect, unlike the existing decentralizedalgorithm in [27], Algorithm 1 naturally lends itself to a morepractical case, where a fixed number of iterations can be usedas a stopping criterion. This is due to an inherent property ofthe primal decomposition method that feasible beamformerscan be provided at intermediate iterations, i.e., beamformerswhich satisfy all the SINR and interference constraints. Hence,Algorithm 1 can be stopped at any feasible iteration leading toa reduced delay and signaling/computational load. This comesat a possible cost of sub-optimal performance.If the secondary transmitters do not have any prior informa-

tion on the “optimal” interference levels, it is fair to use equalelements in the initialization of and at step 1 in Algo-rithm 1, i.e., and

. In practice, there can be a mechanism that stopsafter a fixed number of initialization tries if feasible initializa-tion is not found, and declares the problem “infeasible”. Then,it is up to admission control to loosen the system requirements,e.g., to lower the SINR targets or to drop some users.

F. Alternative Beamforming Approaches

The problem (9) can be also solved using a hierarchical two-level primal-primal decomposition approach where one set ofcoupling variables, e.g., , is decomposed at a higher level andthe other set, i.e., , at a lower level. Primal-primal decompo-sition approach converges if the lower level master problem issolved in a faster timescale than the higher level master problem.See [39] for more details on solving problems with variables op-timized in different timescales. There are also other options toreformulate (4) and achieve a decentralized beamforming de-sign, i.e., using dual decomposition [39] or alternating direction


method of multipliers (ADMM) [43]. Both approaches lead tothe optimal solution. Thorough study of these approaches is notin the scope of this paper, and thus, we omit the derivation of thealgorithms. Note that (4) can also be solved in a decentralizedway via its dual problem and with the aid of the uplink-down-link duality using the similar principles as in [27]. However, theresulting approach might be somewhat impractical as alreadydiscussed in Section I.

G. Special Beamforming Cases

Algorithm 1 allows some special cases where the number ofoptimization variables are reduced leading to lower computa-tional and backhaul signaling loads. This comes at the cost ofsomewhat decreased performance. Some possible special casesare presented below:• Group-dependent SU specific interference constraints:

, where theset consists of any arbitrary group of SUs, i.e., thecardinality of the set can range from 1 to .

• Fixed SU specific interference constraints:, where is a predefined constant. An

interference nulling approach is also possible, i.e.,.

• Fixed PU specific per BS interference constraints:where is a predefined con-

stant and the following must hold:. One such case is when all are equal for

a given PU , i.e., .The last case can be combined with both of the former ones,and thus, leading to even further reduced signaling load andcomplexity.

V. DECENTRALIZED BEAMFORMING IN CELLULAR NETWORKS

In this section, we propose a decentralized beamforming al-gorithm for a one-tier multi-cell multiuserMISO cellular systemwhich is a special case of a two-tier CR system. At a high level,the proposed approach combines the ideas in [23] and [38],i.e., primal decomposition and uplink-downlink duality. Moreprecisely, primal decomposition method is first used to decom-pose the original problem into network-level and BS-level opti-mizations. The network-level optimization is solved via a pro-jected subgradient method. In the BS-level optimization, up-link-downlink duality is utilized. As a result, the main benefitsof [38] and [23] are achieved, i.e., explicit convex optimiza-tion solvers are not required to find an optimal solution andfeasible beamformers that satisfy the SINR targets can be ob-tained at intermediate iterations of the algorithm, respectively.In the practical point of view, the resulting approach is more ap-pealing than the convex optimization based algorithm in [23].Moreover, unlike in [21], long delays and high signaling/com-putational load can be avoided by stopping the algorithm aftera limited number of iterations. The proposed approach is sim-ilar to that introduced in Section IV.D.3 with the difference thatnow the optimization on is omitted. Mathematical analysis inSection IV is valid herein. Hence, most of the details are omittedto avoid redundancy. The cellular beamforming design is sum-marized in Algorithm 3 and Algorithm 4. Both algorithms areperformed independently and in parallel at each BS.

Algorithm 3: Network-level optimization

1: Set . Initialize .2: repeat3: Solve Algorithm 4 and communicate each to the

coupled BSs via backhaul.4: Update using projected subgradient method:

(28)5: Set .6: until Network-level stopping criterion is satisfied

Algorithm 4: BS-level optimization

1: Set and . Initialize .2: repeat3: repeat4: Compute using fixed point algorithm:

(29)where

(30)

5: Set .6: until Stopping criterion is satisfied7: Compute uplink beamformers

(31)8: Compute downlink beamformers

(32)

9: Update using projected subgradient method:

(33)

10: Set .11: until BS-level stopping criterion is satisfied


Fig. 1. Average normalized sub-optimality ofAlgorithm 1 versus network-leveliteration .

VI. NUMERICAL RESULTS

Let us consider a two tier cognitive radio network withprimary transmitter and multiple secondary transmitters. Eachtransmitter serves a predefined set of two single antenna users,i.e., . The pathloss be-tween a transmitter and each user is 0 dB, i.e., the path gain tonoise ratio is 1. This can be interpreted as a worst case scenario,where all the PUs and SUs are at the same cell-edge area. Weassume that the primary transmitter employs the optimal min-imum power beamforming for serving its PUs without beingconcerned on the caused interference to the SUs. For ease ofpresentation, we set and

. In the simulations, we use the following initializationof the interference terms: and

, where is chosen empirically.

A. Quasi-Static Fading Scenario

We consider a quasi-static flat Rayleigh fading scenario withuncorrelated channels between antennas, i.e., each element ofeachchannelrealizationmatrix is i.i.d.complexGaussianrandomvariable with zero mean and unit variance. In all the quasi-staticsimulations, we have used nonsummable and diminishing stepsizes, i.e., , for theprojectedsubgradientmethods.First, we examine the average convergence behavior of Algo-

rithm 1 with different simulation parameters. Fig. 1 presents theaveraged normalized sub-optimality of Algorithm 1 as a func-tion of the network-level iteration number . The normalizedsub-optimality isgivenby ,where

is the sum power at iteration and is the optimal sumpower. InFig. 1, eachpoint is obtainedby averagingoverquasi-static channel realizations. Simulation results imply thatdecreasing or increasing convergence speed becomesslower. Notice that the convergence can be non-monotonic,which is an inherent feature of the projected subgradient method[41].Fig. 2 shows the average converged sum power versus the

SINR target. We compare Algorithm 1 and an interferencenulling beamforming strategy, where . Simula-tion results are achieved by averaging over channelrealizations. We have also set for Fig. 3. Results

Fig. 2. Average sum power versus SINR target forAlgorithm 1 and interferencenulling beamforming.

Fig. 3. Average sum powers of secondary and primary networks versus PUspecific aggregate interference constraint.

demonstrate that Algorithm 1 significantly outperforms theinterference nulling, especially at low and medium SINRs.However, the performance gain decreases when the SINRtarget increases.Fig. 3 illustrates the average converged transmit powers of

both the secondary and primary networks as a function of themaximum aggregate interference power level . Results showthat the secondary network’s sum power decreases about 1 dBwhen increases from dB to dB. The transmit powerof primary network is increased the same amount within thesame change of . Clearly, there is a trade-off between the pri-mary and secondary network performance. It is worth notingthat this trade-off is an interesting techno-economical researchtopic on itself. One can see that the performance of both net-works degrades rapidly when dB. The reason forthis behavior is that being large, the secondary transmissionscause severe interference to the PUs, which leads the primarytransmitter to increase its power to satisfy the PUs’ SINR tar-gets. This again causes severe interference to the SUs, and thus,the secondary transmitters have to raise their powers to satisfythe SINR targets of the SUs.


Fig. 4. Sum power comparison of Algorithm 1, interference nulling and cen-tralized beamforming strategies in quasi-static fading scenario.

In Fig. 4, the sum power is presented for 20 quasi-staticchannel realizations. We compare Algorithm 1 with differentnumber of iterations to the interference nulling and centralizedbeamforming strategies. As can be observed, Algorithm 1obtains near optimal performance even after a few iterations.Moreover, the interference nulling beamforming scheme isoutperformed significantly.

B. Time-Correlated Fading Scenario

We consider a time-correlated flat fading channel with Jakes’Doppler spectrum. The parameter that determines how fast thechannel changes is , where and denote the backhaulsignaling period and the maximum Doppler shift, respectively.We compare the proposed decentralized algorithm to the inter-ference nulling and centralized beamforming strategies. A fixedstep size is used in the projected subgradient method for all thetime-correlated fading simulations.In Fig. 5, we present the sum power for 100 time-correlated

channel realizations. We set , which can be inter-preted, for example, as 2 ms reporting rate and 2.7 km/h uservelocity at 2 GHz carrier frequency. One can see that near (op-timal) centralized performance is achieved using Algorithm 1.Furthermore, Algorithm 1 has significantly better performancethan that of the interference nulling.Let us now consider a cellular network with two BSs each

serving two single antenna users. Each BS is equipped with fourantennas. Fig. 6 illustrates the performance of Algorithm 3 incellular time-correlated fading scenario. Now, we set, which can be interpreted as 2 ms reporting rate and 27 km/h

user speed with 2 GHz carrier frequency. One can observe thatAlgorithm 3 has close to the centralized performance and it sig-nificantly outperforms the interference nulling strategy.

VII. CONCLUSION

This paper proposed a decentralized MISO beamformingalgorithm for multi-cell CR networks. The system optimizationobjective is to minimize the sum power among secondary

Fig. 5. Sum power comparison of Algorithm 1, interference nulling and cen-tralized beamforming strategies in time-correlated fading scenario.

Fig. 6. Sum power comparison of Algorithm 3, interference nulling and cen-tralized beamforming strategies in cellular time-correlated fading scenario.

transmitters while guaranteeing the minimum SINRs for theSUs and satisfying the maximum aggregate interference con-straints for the PUs. Decentralized implementation is achievedthrough a primal decomposition method which decomposes theproblem into a network-level and transmitter-level optimiza-tions. The network-level optimization is solved via projectedsubgradient method relying on the limited backhaul signalingbetween secondary transmitters. The transmitter-level opti-mizations can be solved using three alternative approaches:SOCP, SDP or uplink-downlink duality. The last method is alsoconsidered in multi-cell multiuser MISO cellular systems. Theproposed algorithm converges to the globally optimal solutionin static channel conditions. Numerical results showed signif-icant gains over the interference nulling strategy and close tooptimal performance even after a few iterations in quasi-staticchannel conditions. In addition, near centralized performanceis demonstrated in time-correlated channel conditions, wherethe backhaul signaling is outdated due to time varying channel.An interesting future work is to extend the proposed algorithmto solve a robust cognitive beamforming problem where onlyimperfect local CSI is available at each transmitter.


APPENDIX IPROOF OF PROPOSITION 1

In this Appendix, we show that (4) can be expressed as a stan-dard form SOCP. The proof is an extension of a result describedin [20]. We begin with an observation that an arbitrary phaserotation of beamforming vectors does not affect the values ofthe objective and constraint functions in (4). In other words,if beamformers are optimal, so are ,where is an arbitrary phase. Hence, we can consider such

which results in being real. This allows us to turnthe constraints of (4) into SOC constraints by rearranging themand taking the square roots of the both sides of the resulting in-equalities. After denoting , we can recast (4) as

(34)

where

and

. The problem (34) is a standardform SOCP [44] since the objective function is linear and theconstraints are SOC constraints.

APPENDIX IIPROOF OF PROPOSITION 2

Following from the previous Appendix, (6) can be reformu-lated as an SOCP. First, we denote and .The resulting SOCP is obtained

(35)where

and .

APPENDIX IIIPROOF OF PROPOSITION 3

It will be proved in this Appendix that strong duality holdsfor (6). Strong duality implies that the duality gap between a

primal problem and its Lagrangian dual problem is zero, i.e.,both problems have the same solution [44]. Therefore, primalproblem can be solved via its dual problem. Strong duality holdsfor convex problems which are strictly feasible, i.e., Slater’sconditions hold [44]. Since we assume strict feasibility for (6),strong duality holds for the reformulated convex SOCP problem(35). Given the fact that equivalent optimization problems, suchas (6) and (35), may have different Lagrange dual problems [44],we need to show that the Lagrangians of (6) and (35) are thesame. Therefore, the dual problems also must be the same. Thisimplies that strong duality holds for (6).First, we formulate the Lagrangians of (6) and (35) as

(36)

(37)

Next, we use a similar procedure as in [38] to reformulate (37).Let us write

(38)

(39)


and so forth for the rest of the terms in (37). By substitutingthese terms into (37), we get

(40)

Since and are all strictly positive, we can per-form a change of the optimization variables for (40), i.e.,

and .Now, the Lagrangians of the original problem and the SOCPproblem, i.e., (36) and (40), are exactly the same, leading to thesame dual problem.

APPENDIX IVPROOF OF PROPOSITION 4

It is shown in this Appendix that (12) and (13) are valid sub-gradients of (9) at the points and , respectively. Theproof is inspired by a result in [38]. Let us start by equivalentlyrewriting the objective value of (9) by , where is theoptimal objective value of (6) at the point , and .Since strong duality holds for (6), can be achieved bysolving the Lagrange dual problem of (6) at the point . Thiscan be expressed as

(41)

where the vectors and consist of the elements of thesets and ,respectively.

Since the dual problem (41) is concave, we can define a sub-gradient of at the point to be any vector , which satisfiesthe following [41]: , for all . We canexpress using the objective of the dual problem(41). First, let us assume that and are the optimal Lagrangemultiplier vectors at the points and , respectively. Replacingwith in the dual objective expression of , we can write

the following:

(42)

Based on the subgradient definition, one can see that

and are the subgradients of at the pointsand , respectively.

APPENDIX VPROOF OF THEOREM 1

The proof is a modification of the uplink-downlink duality re-sults proposed in [38]. Let us begin by formulating a virtual dualuplink problem in CR system, where the SU specific SINR con-straints are the same as in the downlink case. This is expressedas

(43)

where and denote the virtual uplink power and its (con-stant) scaler for the SU , respectively. One can observe that(43) is exactly the same as (22) if we denote

and

. The vectors andcan be interpreted as the virtual dual uplink powers for the

SUs at the serving cell, SUs at the other cells and PUs, respec-tively. Moreover, the virtual uplink power of the SU is scaledin the objective by the constant that is a sum of the noise andfixed interference powers experienced by the SU in downlink.When is fixed, an explicit optimal solution to (22) is

found by using the MMSE receiver which maximizes the SINR.The MMSE receiver is given by

(44)


By substituting into (22) and using Lemma 1 from[38], the SINR constraints in (22) can be expressed as

(45)

These constraints are the same as in (21) except they are re-versed. If we change the minimization into the maximizationand reverse the SINR constraints in (22), the resulting problemis exactly (21). This reversing does not change the optimal valueof the problem since the SINR constraints are met with equalityat the optimal solution. Thus, (21) and (22) are equivalent prob-lems having identical solutions.

APPENDIX VIPROOF OF PROPOSITION 5

We will prove that is optimally solved using (23). Thisproof is inspired by the previous works in [20], [21], [45]. First,we set the gradient of the Lagrangian of (8) with respect to

to zero. This is expressed as

(46)

After rearranging, similar to that in [21], the function (23)is obtained. We can rewrite (23) as follows:

. The function is a standard func-tion if it satisfies the properties of 1) positivity, 2) monotonicityand 3) scalability, i.e.,1) If , then .

2) If , then.

3) For .We omit the proof which shows that these properties are satisfiedby (23) since it follows the principles in [20]. Since standardfunction always converges to a unique fixed point for any initialvalue [45], the obtained is optimal.

APPENDIX VIIPROOF OF PROPOSITION 6

In this Appendix, we show that the optimal downlink beam-formers can be solved via the optimal uplink beamformers byscaling. The proof is a modification of the results presented in[38]. After manipulation of (46), similar to that in [38], we can

achieve the following expression for

(47)

If this is compared to the MMSE expression of in (44),it can be observed that is a scaled version of , i.e.,

, where .

Since still depends on , we need to find an expressionwhere are expressed only via . In thisrespect, we can use the fact that the SINR constraints (8c) aresatisfied with equality at the optimal point of (8). By substi-tuting into the SINR constraints (8c), we canwrite

(48)

Now, (48) can be solved for via (26).

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Harri Pennanen (S’07) received his M.Sc. (Tech.)degree in electrical engineering from the Universityof Oulu, Oulu, Finland in December 2007. He is cur-rently working towards the D.Sc. (Tech.) degree atthe Centre for Wireless Communications (CWC) atthe University of Oulu. His research interests are inradio resource management for wireless communica-tions systems with special focus on interference coor-dination and decentralized algorithms in cellular andcognitive networks. He has authored or coauthored20 international journal and conference papers on the

topics in wireless communications.

Antti Tölli (M’08) received his D.Sc. (Tech.) degreein electrical engineering from the University ofOulu, Oulu, Finland in June 2008. Before joiningthe Department of Communication Engineeringand Centre for Wireless Communications (CWC)at the University of Oulu, he worked five years forNokia Networks, IP Mobility Networks division asa Research Engineer and Project Manager both inFinland and Spain. Currently he works as a SeniorResearch Fellow and Adjunct Professor at the DCEof the University of Oulu. He served as a General

co-chair of IEEE WDN in 2010 and 2011, as well as Finance chair in IEEECTW 2011. His main research interests are in radio resource managementand transceiver design for broadband wireless communications with specialemphasis on distributed interference management in heterogeneous wirelessnetworks. A. Tölli has published about 90 papers in peer-reviewed internationaljournals and conferences, as well as, several patents all in the area of signalprocessing and wireless communications.

Matti Latva-aho (SM’06) was born in Kuivaniemi,Finland in 1968. He received the M.Sc. (Tech.),Lic.Sc. (Tech.) and D.Sc. (Tech.) degrees in elec-trical engineering from the University of Oulu, Oulu,Finland in 1992, 1996 and 1998, respectively. From1992 to 1993, he was a Research Engineer at NokiaMobile Phones, Oulu, Finland. During the years1994–1998 he was a Research Scientist at Telecom-munication Laboratory and Centre for WirelessCommunications (CWC) at the University of Oulu.Currently he is the Department Chair Professor of

Digital Transmission Techniques and Head of Department at the Universityof Oulu, Department for Communications Engineering. Prof. Latva-aho wasDirector of Centre for Wireless Communications at the University of Ouluduring the years 1998-2006. His research interests are related to mobile broad-band wireless communication systems. Prof. Latva-aho has published over 200conference or journal papers in the field of wireless communications. He hasbeen TPC Chairman for PIMRC’06, TPC Co-Chairman for ChinaCom’07 andGeneral Chairman for WPMC’08. He acted as the Chairman and vice-chairmanof IEEE Communications Finland Chapter in 2000–2003.

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