IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009 4871

Robust Cognitive Beamforming With BoundedChannel Uncertainties

Gan Zheng, Member, IEEE, Kai-Kit Wong, Senior Member, IEEE, and Björn Ottersten, Fellow, IEEE

Abstract—This paper studies the robust beamforming design fora multi-antenna cognitive radio (CR) network, which transmits tomultiple secondary users (SUs) and coexists with a primary net-work of multiple users. We aim to maximize the minimum of the re-ceived signal-to-interference-plus-noise ratios (SINRs) of the SUs,subject to the constraints of the total SU transmit power and thereceived interference power at the primary users (PUs) by opti-mizing the beamforming vectors at the SU transmitter based on im-perfect channel state information (CSI). To model the uncertaintyin CSI, we consider a bounded region for both cases of channelmatrices and channel covariance matrices. As such, the optimiza-tion is done while satisfying the interference constraints for all pos-sible CSI error realizations. We shall first derive equivalent condi-tions for the interference constraints and then convert the problemsinto the form of semi-definite programming (SDP) with the aid ofrank relaxation, which leads to iterative algorithms for obtainingthe robust optimal beamforming solution. Results demonstrate theachieved robustness and the performance gain over conventionalapproaches and that the proposed algorithms can obtain the exactrobust optimal solution with high probability.

Index Terms—Cognitive radio, interference control, MIMO, ro-bust beamforming.

I. INTRODUCTION

R ADIO spectrum is a very precious resource for wirelesscommunications. According to federal communications

commission (FCC) [1], spectrum utilization depends very muchupon place and time and yet most spectrum is underutilized.Cognitive radio (CR), first proposed by Mitola and Maguire in1999 [2], is a new paradigm for exploiting the spectrum re-sources in a dynamic way [3], [4] and has been adopted inIEEE 802.22 Wireless Regional Area Networks (WRANs) forlicense-exempt devices to use the spectrum on a non-interferingbasis.

In CR, the spectrum usage is administered in a hierarchicalmanner. The primary users (PUs) who are the licensed users

Manuscript received March 18, 2009; accepted June 10, 2009. First publishedJuly 14, 2009; current version published November 18, 2009. The associate ed-itor coordinating the review of this manuscript and approving it for publicationwas Prof. Xiang-Gen Xia. This work was supported by EPSRC under GrantsEP/D058716/1 and EP/E022308/1, and the British Council Research ExchangeProgramme.

G. Zheng and K.-K. Wong are with the Department of Electrical andElectronic Engineering, University College London, London, WC1E 7JE, U.K.(e-mail: g.zheng@ee.ucl.ac.uk).

B. Ottersten is with the School of Electrical Engineering, Royal Instituteof Technology (KTH), SE-100 44 Stockholm, Sweden. He is also with secu-rityandtrust.lu, University of Luxembourg.

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

Digital Object Identifier 10.1109/TSP.2009.2027462

have the priority rights to access the spectrum, while the sec-ondary users (SUs) (also known as cognitive users) can occupythe spectrum only if they do not interrupt the communication ofPUs.1 In practice, this will require the SUs to sense the spectrumand detect any PU’s activities by using, for instance, the likeli-hood ratio test [5], energy detection [6], matched-filtering-basedtechnique [7] and cyclostationary detection [8], etc., so that thedecision for reusing the spectrum by the SUs can be made.

Spectrum holes are the most obvious opportunities to be ex-ploited by CR [9], but higher spectrum utilization is anticipatedif coexistence between PUs and SUs is permitted. The latter sce-nario is indeed possible if the interference caused by the SUs canbe properly controlled and managed [10]. In this respect, mul-tiple-input multiple-output (MIMO) antenna system has beenrecognized as an effective means to mitigate co-channel interfer-ence and has been widely used in traditional fixed-spectrum-al-location based communications systems.

The use of beamforming for interference control in the con-text of CR is challenging, as the interference seen by the licensedPUs from the license-exempt SUs must be kept below a certaintolerable limit, if the SUs are granted access to the spectrum ofthe PUs. In light of this, a number of beamforming techniqueshave been proposed for the uplink CR network to controlinterference as well as achieve various objectives, such ascapacity maximization [11], signal-to-interference-plus-noiseratio (SINR) balancing [11], and the minimization of transmitpower with SUs’ quality-of-service (QoS) constraints [12].Most recently, [13] used the uplink–downlink duality to studythe optimal beamforming for downlink CR networks.

Thus far, however, previous works on CR beamforming werebased by and large on the exploitation of perfect channel stateinformation (CSI). Although the CSI may be estimated from theuplink of the primary network if the SU transmitter continuouslysenses the PU’s transmission and the uplink and downlink forthe PUs are reciprocal, in practice, the CSI will never be perfect,due to estimation and quantization errors as well as Dopplerspread. In CR, more CSI errors are anticipated due to limitedcooperation between PUs and SUs.

Robust optimization, which tackles optimization with uncer-tainties, is usually addressed by either a stochastic or a worst-case approach and has been actively researched for MIMO sys-tems in recent years. In particular, the CSI errors are often mod-eled as Gaussian random variables [14], [15], and the robustnesscan be provided in the statistical sense. Alternatively, anotherway to achieve robustness is by worst-case optimization, whichdesigns the system to operate under the worst-case channel con-dition [16] if the CSI uncertainty is bounded. In this case, the

1By SU(s) and PU(s), we mean the SU receiver(s) and the PU receiver(s),respectively; otherwise, we shall clearly specify that they are transmitters.

1053-587X/$26.00 © 2009 IEEE

4872 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009

channel is optimized to achieve a given QoS for every possibleCSI error if the problem is feasible, so that absolute robustnesscan indeed be achieved. Interestingly, it was also shown in [17]that the worst-case model can be viewed as the deterministicanalog of the stochastic CSI model in [18]. A bounded worst-case model is also believed to be able to cope with quantizationerrors in CSI [19]. In this paper, we shall employ the worst-caseapproach to address the beamforming design in downlink CRwith bounded CSI errors.

Since the worst-case based techniques in the MIMO liter-ature are highly related to our study, some are reviewed asfollows. In [20], a robust maximin approach was developedfor a single-user MIMO system based on convex optimization.Later in [17], the robust transmission schemes to maximize thecompound capacity for single and multiuser rank-one RiceanMIMO channels were addressed, based on the uncertaintyset in [21]. Robust beamforming for multiuser multiple-inputsingle-output (MISO) downlink channels with individual QoSconstraints under an imperfect channel covariance matrix wasstudied in [16], [22]. Recently in [23], the optimal powerallocation over fixed beamforming vectors was obtained inthe presence of errors in CSI matrices. Most recently, efficientnumerical solutions to find conservative robust beamformingfor multiuser MISO systems with mean-square-error (MSE)and SINR constraints and different bounded CSI errors havebeen developed in [24].

In contrast, robust designs in CR are much less understood.One such attempt appeared in [25] where the optimization ofa CR network exploiting perfect CSI from the SU transmitterand receiver but imperfect CSI from the SU transmitter and thePU receiver, was considered. In particular, it was illustrated thatthe beamforming to maximize the rate of the SU while keepingthe interference to the PU under a given level in high proba-bility can be obtained by solving a semi-infinite programming(SIP) problem. The limitation of [25] is, however, that only oneSU receiver and one PU receiver are considered. With MIMO,multiple users sharing the same spectrum are anticipated, butthe results in [25] do not reveal the optimal robust beamformingsolution for CR systems with multiple PUs and SUs with CSIuncertainty.

This paper studies a downlink CR network where there aremultiple PUs and SUs occupying the same spectrum and amulti-antenna SU transmitter is communicating with multiplesingle-antenna SUs. We aim to obtain the optimal robust beam-forming solutions for maximizing the minimum of the SUs’SINRs, which is subjected to the constraints of the overall SUtransmit power and the received interference power at the PUs,by exploiting CSI at the SU transmitter with its errors modeledby an ellipsoidal region in either channel matrices or channelcovariance matrices. In particular, we have made the followingcontributions.

• Unlike previous approaches that often restrict the feasibleregion when handling SINR constraints, using S-Proce-dure as a tool, we derive equivalent constraints to the SINRand interference power constraints in the case of ellipsoidalchannel matrices errors.

• We also derive equivalent constraints for the case of ellip-soidal channel covariance matrices uncertainties, with con-

sideration on the positive semi-definiteness of the covari-ance matrices, which is usually ignored in the literature.

• Moreover, we propose bisection search based iterative al-gorithms, which use rank relaxation and the equivalentSDP constraints, to obtain the robust beamforming solu-tions for both cases of channel matrices and covarianceuncertainties.2 In addition, numerical results show that theoptimal robust solutions can indeed be found (which ap-pears when the rank relaxation becomes exact) with veryhigh probability.

The remainder of this paper is structured as follows. Section IIintroduces the system model for the multiuser MISO CR systemand formulates the SINR balancing problem. In Section III, wedefine the uncertainty region for the channel matrices uncertain-ties and then derive the equivalent robust constraints, while thechannel covariance matrices uncertainties will be addressed inSection IV. Section V presents the algorithms to find the solu-tions to the robust problems using convex optimization and rankrelaxation. Simulation results will be given in Section VI and fi-nally, we conclude this paper in Section VII.

Throughout this paper, complex scalar is represented by alowercase letter and denotes its modulus. denotes theexpectation of an input random entity. Vectors and matrices arerepresented by bold lowercase and uppercase letters, respec-tively, and computes the Frobenius norm. The th entryof a matrix is denoted as or . The superscripts ,

and are used to denote the transpose, conjugate and Her-mitian transpose of a vector or matrix, respectively.and mean that is positive semi-definite and strictlypositive definite, respectively. In addition, gives theminimum eigenvalue of a square matrix while de-notes the trace of , is a vector formed by stacking thecolumns of , and returns an square matrixof the elements of an -vector . gives the Kroneckerproduct of and . Finally, denotes a vectorof complex Gaussian entries with a mean vector of and a co-variance matrix of .

II. SYSTEM MODEL

We consider a CR downlink network with PUs and SUs,where the SU common transmitter has antennas and each SUreceiver has only one antenna. The received signal at the thSU, denoted by , can be written in vector form as

(1)

in which denotes the symbol intended to the th SU with, denotes the corresponding transmit beam-

forming vector, denotes the channel between the SU trans-mitter and the th SU receiver, and is theadditive noise. The total transmit power from the SUs is givenby and required to be limited by .

2It is noted that the proposed approach can also handle the total power min-imization of the SUs and more convex constraints, such as per-antenna powerconstraints, can also be easily incorporated.

ZHENG et al.: ROBUST COGNITIVE BEAMFORMING WITH BOUNDED CHANNEL UNCERTAINTIES 4873

The instantaneous received SINR at the th SU can be ex-pressed as

(2)

which is required to be at least where are the presetSINR balancing parameters providing the SINR pattern of theSUs and is an auxiliary variable for scaling the SINRs. Tocontrol the interferences, at the th PU, the received interferencepower is limited by

(3)

where denotes the channel between the SU transmitter andthe th PU receiver.

In particular, our problem of interest is to maximize the min-imum of the received SINRs of the SUs given the parameters

and by optimizing the transmit beamforming vectorswith the aid of CSI. Mathematically, it is expressed

as(4a)

(4b)

For ease of exposition, we define ,and

(5)

As a result, (4) can be compactly rewritten as(6a)

(6b)

With perfect CSI at the SU transmitter, the optimal solution for(6) has recently been found in [13]. In the sequel, our effort willbe spent on solving (6) with CSI errors.

III. BEAMFORMING WITH CHANNEL MATRIX UNCERTAINTY

A. Ellipsoidal Uncertainty Model

It is assumed that the SU transmitter has neither perfect CSInor statistical knowledge of the uncertainty. In particular, thechannels are modeled as

(7)

(8)

where and are the estimated CSI known at the SUtransmitter while and are the CSI errors. Weassume that the CSI errors are bounded by ellipsoids, i.e.,

(9)

(10)

where the matrices determine the quality ofCSI and are assumed known. Moreover, the CSI becomes per-fect when and approach infinity while the CSI is theworst if they are zeros. The above model also embraces theFrobenius norm bounded error.

B. Equivalent Worst-Case Constraints

To take the CSI errors into account, the SINR balancingproblem in (6) based on worst-case criterion can be formulatedas

(11a)

(11b)

Before we proceed, we shall first investigate on rewriting theconstraints in (11b). In particular, the SU’s SINR constraint [i.e.,the first constraint in (11b)] can be rewritten as

(12)Note that in (12), both sides contain a common ellipsoidal un-certainty region and it is not known to be tractable [26]. Onepossible remedy is to derive a bound for the constraint for any

and then perform optimization based on the bound[27]. Similar conservative methods have recently been proposedto deal with this problem [19], [24] but the resulting solutionsappear to degrade the performance and reduce the feasible re-gion considerably.

Our approach here is different in that we do not impose any re-striction on the optimizing variables but derive the exact equiva-lent condition directly from the robust constraint (12). The mainresult is summarized in the following Theorem 1.

Theorem 1: The robust SU’s SINR constraint (12) is equiv-alent to

(13)Proof: It is easy to see that (12) can be equivalently written

as

(14)

4874 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009

Though is not convex as is not positive semi-definite, according to S-Procedure [28], (14) is an equivalentstatement of (13), which completes the proof.3

Similarly, we next derive the equivalent interference con-straint [i.e., the second constraint in (11b)] and the result issummarized in Theorem 2.

Theorem 2: The robust interference constraint (11b) is equiv-alent to

(15)

Proof: It follows similarly as that of Theorem 1 and isomitted.

C. Reformulation and SDP Relaxation

With Theorems 1 and 2, the original robust problem (11) canbe equivalently expressed as

(16a)

(16b)

The above reformulation is straightforward and equivalent to theoriginal problem (11), which permits a possible exact optimalsolution to be found. The proposed algorithm we develop tosolve (16) will be presented later in Section V.

IV. BEAMFORMING WITH CHANNEL COVARIANCE

UNCERTAINTY

In practice, the CSI may be estimated in the form of covari-ances. In this case, the uncertainty in the channel covariancematrices has to be properly modeled and considered in (11).

A. Ellipsoidal Uncertainty Model

As in Section III, we use an ellipsoid to model the channelcovariance uncertainty such that

(17)

(18)

where and are the estimated channel covariance ma-trices from the SU transmitter to the th SU and from the SUtransmitter to the th PU, respectively, and and de-note the corresponding uncertainties that are considered to bebounded in the sets

(19)

(20)

3Similar result has been obtained in [27].

Similar as before, the matrices and aregiven and define the quality of CSI in covariances. In addition,we assume that they can be decomposed into

(21)

(22)

with .Under this model, the received SINR at the th SU can be

written as

(23)

and the resulting interference at the th PU is given by. As a result, the beamforming optimization in (6)

can be rewritten as(24a)

(24b)

It is worth noting that there have been a number of researchesattempting to address uncertainties in the covariance matricesof channel or signal, e.g., [16], [22], and [29]–[32]. However,an important property that the covariance matrices must be pos-itive semi-definite (PSD), is often assumed [29] or ignored [30]and thus the solution will be over pessimistic because some un-realistic errors are considered as well. This constraint was con-sidered in [31] by modeling the error of the “square root” ofthe signal covariance matrix but not the covariance matrix it-self. By contrast, our approach addresses the uncertainties of

directly.

B. Equivalent Worst-Case Constraints

To proceed, we first consider the interference constraint [i.e.,the second constraint in (24b)], which can be reexpressed as

(25)As such, it suffices to derive the maximum interference, whichwe shall address in Theorem 3. Before we present this theorem,we find the following lemma useful.

Lemma 1: For Hermitian matrices, and , suchthat and , we have the following gradientresults:

(26)

Proof: See Appendix A.Theorem 3: Let be an complex Hermitian matrix.

Then, we have

(27)

ZHENG et al.: ROBUST COGNITIVE BEAMFORMING WITH BOUNDED CHANNEL UNCERTAINTIES 4875

and the maximum is attained when

(28)

Proof: Now, we consider the problem

(29)

The Lagrangian can be written as

(30)

in which is the dual variable. Setting the derivative to bezero and using Lemma 1 in the Appendix, we obtain

(31)

Using Kronecker operation, it yields

(32)

Therefore, the dual objective is given by

(33)

where we have used (32) and the fact that at the optimum,. Because (29) is a convex problem and has

a nonempty interior, strong duality holds [33] and its optimalobjective value equals . Further, impliesthat , which enables us to solve for by

(34)

As a result, we have proved the desired result (27) and the op-timal , denoted by , is given by (28), which completes theproof.

Using Theorem 3, the optimal to (25), denoted by, is given by

(35)

Since , [34, Ch. 13] and clearly itsatisfies and is therefore a valid solution.Now, substituting (35) into (25), the th interference constraintcan be equivalently written as

(36)

Next, we proceed to study the robust SINR constraint [i.e.,the first constraint in (24b)], which can be rewritten as follows:

(37)

In [22], [32], the error model in (19) was considered withfor some , and it has been proposed to optimize the

system based on the lower bound of the SINR

(38)

However, as this lower bound is generally loose, severe perfor-mance degradation is expected.

Alternatively, our approach is to tackle the constraint (37)directly. Following the same method as in the derivation of

, the optimal can be found as

(39)

and (37) becomes

(40)

This solution is simple but problematic as it may not satisfythe positive semi-definiteness property of the covariance matrix,i.e., because is in general not positivesemi-definite, and this condition will likely be violated if theerror bound is large. Although (40) simplifies the problem andsometimes is automatically satisfied, it isemphasized that in general the solution using (39) only serves asa lower bound for the actual achievable performance. We nowovercome this by deriving an exact equivalent condition of (37)and the main result is summarized in Theorem 4 as follows.

Theorem 4: The following two statements are equivalent:i)

(41)

ii) There exist and such that

(42)

with the worst-case given by

(43)Proof: First, we consider the problem

(44a)

(44b)

4876 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009

In Appendix B, the dual problem of (44) is derived as

(45a)

(45b)

Since (44) is a convex problem, it has the same optimal objec-tive value as its dual (45). In addition, statement i) is equivalentto say that the optimal objective value of (44) is non-negative,which in turn implies that the optimal objective value of (45)is also non-negative and this is exactly statement ii), which hascompleted the proof.

C. Reformulation and SDP Relaxation

With (36) and Theorem 4, we can then rewrite (24) in anequivalent form as

(46a)

(46b)

The algorithm that can solve (46) will be presented in detail inSection V.

D. Numerical Example

To conclude this section, we here provide a numerical ex-ample to illustrate that the robust constraints (46b) are necessaryand if we replace them with (40), a valid worst-case channel co-variance realization is not met, and as a result, the performanceis unnecessarily degraded.

In this example, we consider the case where , ,, 5 dB, 20 dB,

0 dB, and

(47)

Under this setting, the output solutions based on the constraints(40) and the robust constraints (46b) are given, respectively, asin Fig. 1. As we can see, if (46b) is not considered, then thepositive-definiteness constraint on the channel covariance willbe violated, and in particular, the achieved based on (40) issmaller than that achieved based on (46b).

V. PROPOSED ALGORITHM

In this section, we present an algorithm to solve the optimiza-tion problems (16) and (46). Although our previous efforts havemade to rewrite the constraints into SDP, the problems are stillnot convex due to the rank-1 constraints on the beamformingmatrices and also the dependence of with other variables. Toovercome this, we note that if we relax the rank-1 constraints,then both (16) and (46) will become quasi-convex problems andthe optimal solutions can be found by checking the feasibility ofan SDP problem. In particular, both (16) and (46) can be solvedusing a bisection search with the aid of SDP and rank relaxation,which is described by Algorithm 1 as follows. In the algorithm,we have the superscript to denote the variables at the th it-erate and the “problem” can be either (16) or (46).

Algorithm 1: Robust Cognitive Beamforming using BisectionSearch

1: Input: , , , , , , ,and .

2: begin

3: Initialize the iteration index , the lower bound andthe upper bound .

4: Initialize

5: For a given , check the feasibility of the problemwithout the rank-1 constraints.

6: if the problem is feasible, then

7: Update .

8: if it also outputs rank-1 solutions for all , then

9: Update the setsand .

10: end

11: else

12: Update .

13: end

14: Set and go to line 5 if is

greater than a given threshold.

15: Find and obtain the beamformingmatrices .

16: end

17: Extract the beamforming solution fromby matrix decomposition.

18: Output: and .

ZHENG et al.: ROBUST COGNITIVE BEAMFORMING WITH BOUNDED CHANNEL UNCERTAINTIES 4877

In line 5 of Algorithm 1, it requires detecting the feasibilityof a convex SDP problem, which can be achieved by checkingif there exist any feasible solution of the SDP.

Rank relaxation has been widely used in communicationsoptimization such as maximum-likelihood (ML) detection (see[35], [36] and the references therein). In that particular applica-tion, some randomization methods were proposed to recover therequired rank-1 solution based on the higher-rank solution. Thisis also required in line 17 of Algorithm 1, if any of the obtainedbeamforming matrices appear to be of higher-rank.Nevertheless, it is noted that the known randomization methodsare not applicable here because our problems have complex ro-bust constraints required by the PUs and the SUs, and it is notclear even how to construct a feasible rank-1 solution based onsome higher-rank solutions.

To tackle the rank issue, we store all those iterations that re-turn rank-1 solutions. After the algorithm converges, if the finalbeamforming solution is of rank-1, then it is guaranteed to be op-timal; otherwise, we shall use the rank-1 solution that achievesthe largest , but in this case, its optimality is not guaranteed.As a final remark, it is worth mentioning that it is possible toincorporate more convex constraints about , such as pertransmit antenna power, into the proposed algorithm. In addi-tion, the proposed techniques can also be used to address thepower minimization problem subject to the robust SINR andinterference constraints as the equivalent tractable robust con-straints have been derived in SDP.

VI. SIMULATION RESULTS

A. Setup and Assumptions

Simulations are conducted to assess the performance of theproposed algorithms in Rayleigh flat-fading channels, so thatthe entries of the channel matrices follow . For the caseof channel covariance in Section IV, we considered the channelmodel in [32]. In particular, it is assumed that the th SU is sur-rounded by a large number of local scatterers corresponding toa spread angle of . For a uniform linear array with elementsspaced half a wavelength apart, we have the normalized spatialcovariance matrix as

(48)

where is the nominal direction to the source, and the co-variance matrices to the PUs as

(49)in which and denote, respectively, the nominal direc-tion to the source and the angular spread. Furthermore, we con-sidered a system with three SUs, where and

, and . Theparameter will be varied to generate various channel environ-ments. Moreover, the CSI uncertainties are assumed to be trun-cated Gaussian distributed within the bounded uncertainty sets.In the simulations, unless explicitly stated, we considered that

all users have the same target SINR, CSI error bounds and in-terference power limits, i.e., dB ,

and , while is the parameter to adjust theCSI quality and set to 0.1 unless specified otherwise. We also as-sumed that the average total transmit SNR, defined as , is10 dB. For convenience, we use the triple to denotean -SU system with transmit antennas and PUs. The fol-lowing benchmarks are compared with the proposed algorithm.

• Non-robust design—This optimizes the users’ beam-forming vectors based on the estimated CSI matrices.The CSI uncertainty regions, , , , ,are ignored, so this method is expected to have a highprobability of the constraints being violated.

• Conservative channel-based robust solution—This wasproposed in [24] to address the imperfect channel matrices.This solution is based on the equivalent MSE constraints.

• Conservative covariance-based robust solution—This wasproposed in [32] when CSI is in the form of channel covari-ance matrices and is imperfect.

Before presenting the results, we first comment on thecomplexity per iteration of the proposed algorithm and that ofthe conservative approaches in [24], [32]. It is known that thenumber of variables, the number of SDP constraints and thesize of SDP dominate the overall complexity [24], [37]. For thechannel-based problem, the virtual MSE-based algorithm in[24] has real-valued design variables and slackvariables. Moreover, there are also SDP constraints of size

. By contrast, our proposed algorithm hasreal-valued design variables and slack variables, withSDP constraints of size and another SDP constraintsof size . There are also some computations requiredto perform matrix decomposition in the proposed method. Dueto the total power and interference constraints, both will haveone scalar constraint and SDP constraints of size .On the other hand, as far as the covariance-based problem isconcerned, the method in [32] has real-valued designvariables and scalar constraints, while our algorithm has

real-valued design variables but slack vari-ables are needed, with SDP constraints of size and

scalar constraints. Similarly, both will have scalarconstraints. To summarize, in general, the complexities ofthe proposed algorithms are slightly higher than those of theconservative approaches, but it will be apparent that substantialperformance gain can be achieved because more degree offreedom is exploited.

B. Results

In Fig. 2, the cumulative density functions (CDFs) of the re-ceived interference at the first PU receiver are provided for a(3,3,2) system with 10 dB for the case of imperfectchannel matrices and a (6,3,2) system with 0 dB and

10 for the case of imperfect channel covariance matrices.Results show that for both cases, using the proposed algorithm,the interference power is always below the required limit whilethe non-robust scheme can rarely satisfy the interference con-straints due to the ignorance of imperfect CSI.

In Fig. 3, the CDFs of the received SINR at the first SU re-ceiver are provided for the same systems as before. It is observed

4878 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009

Fig. 1. Numerical example to show the importance of the constraints (46b).

Fig. 2. CDFs of the received interference power at the first PU.

Fig. 3. CDFs of the received SINR at the first SU.

that as expected, the non-robust scheme results in the highestSINR but at the price of severe interference to the PUs, whichhas been seen in Fig. 2. On the other hand, results illustrate thatthe proposed algorithm achieves about 5- and 1-dB gains in thereceived SINR over the conservative method, respectively, forthe cases of imperfect channel matrices and channel covariancematrices.

Fig. 4. Output worst-case SINR results for various angular separation �.

In Fig. 4, the output SINR results for the case with imperfectCSI covariance matrices are plotted against the angular separa-tion . The dashed lines are for the systems without the interfer-ence power constraints. As we can see, the performance for allschemes improves as the angular separation increases. However,the solid lines which correspond to the systems with the inter-ference constraints, clearly show the impact of the positions ofthe PUs, i.e., if the SUs are not separated far enough and closeto the PUs, the performance may degrade as the angular sep-aration increases. In particular, the worst performance appearswhen 10 .

In Fig. 5, the output SINR results for the both cases are pro-vided for various error bound, . The setting is the same as thatused in 2 except that 16 . Results demonstrate that the pro-posed algorithms significantly outperforms the conservative ap-proaches, especially for the case of imperfect channel matrices.

Finally, we investigate the optimality of the proposed algo-rithm by studying the ranks of the beamforming solutions ofthe SDP for the case with imperfect channel matrices. For thecase with imperfect channel covariance matrices, numerical re-sults indicate that the optimal rank-1 solutions can always beobtained, which means that the proposed algorithm always pro-vides the optimal solutions. In other words, we can measure thenumber of times that the proposed SDPs return rank-1 beam-forming matrices and declare global optimality in these chan-nels. In Table I, we show such probability results that the pro-posed algorithm is guaranteed to achieve the robust optimumfor various CSI error bound . Results reveal that the proposed

ZHENG et al.: ROBUST COGNITIVE BEAMFORMING WITH BOUNDED CHANNEL UNCERTAINTIES 4879

Fig. 5. Output worst-case SINR results for various error bound �. (a) Channel-based; (b) covariance-based.

TABLE ITHE PROBABILITY RESULTS THAT THE PROPOSED ALGORITHM

GUARANTEES TO ACHIEVE OPTIMALITY

algorithm can achieve the global optimal solutions at very highprobability.

VII. CONCLUSION

This paper has addressed the worst-case robust beamformingdesign of a downlink multiuser multi-antenna CR system coex-isted with a number of PUs, in which only imperfect CSI is ex-ploited at the SU transmitter. With ellipsoidal uncertainty setson the CSI, our objective was to balance the SU SINRs withthe constraints of the total transmit power of the SUs and the

received interference power at the PUs. For CSI of channel ma-trices and channel covariance matrices, we derived the equiva-lent robust constraints in the form of SDP using S-Procedure. Abisection based algorithm has been devised to obtain the robustbeamforming solutions with rank relaxation. Simulation resultshave shown the achieved robustness against the CSI uncertain-ties and the effectiveness to meet the required output SINRs atthe SUs.

APPENDIX ATHE PROOF OF LEMMA 1

Proof: First, note that for a general matrix complex vari-able , the gradient of a real-value function is defined as

, and that and are treated as two independentvariables. Now, we expand as

(50)

which has the gradient

(51)

Note that for a general complex matrix without any specialstructure, the above gradient should be zero. Similarly, we canfind the gradient for as follows:

(52)

and then

(53)

As a result, we have the desired results, which completes theproof.

APPENDIX BDERIVATION OF THE DUAL PROBLEM OF (44)

Here, we derive the dual of (44). First, we have the La-grangian of (44) given by

(54)

where , are the dual variables. Settingand using Lemma 1, we then have

(55)

4880 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009

We can solve

(56)The dual objective can also be found as

(57)

(58)

where (57) is due to (55) and (58) comes from the fact that atthe optimum, we have

(59)

Combining (55) with andor , we

then have

(60)

(61)As a result, we arrive at (43) and (45).

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ZHENG et al.: ROBUST COGNITIVE BEAMFORMING WITH BOUNDED CHANNEL UNCERTAINTIES 4881

Gan Zheng (S’05–M’09) received the B.Eng. andM.Eng. degrees from Tianjin University, China,in 2002 and 2004, respectively, both in electronicand information engineering, and the Ph.D. degreein electrical and electronic engineering from TheUniversity of Hong Kong, Hong Kong, in 2008.

He has been working as a Research Associate atUniversity College London since December 2007.His research interests are in the general area ofsignal processing for wireless communications, withparticular emphasis on the design, optimization

and analysis for multiuser multiple-input multiple-output (MIMO) antennasystems, cooperative network, and cognitive radio.

Kai-Kit Wong (S’98–M’01–SM’08) received theB.Eng., M.Phil., and Ph.D. degrees, all in electricaland electronic engineering, from the Hong KongUniversity of Science and Technology, Hong Kong,in 1996, 1998, and 2001, respectively.

After graduation, he joined the Department ofElectrical and Electronic Engineering, the Universityof Hong Kong as a Research Assistant Professor.From July 2003 to December 2003, he visited theWireless Communications Research Department ofLucent Technologies, Bell-Labs, Holmdel, NJ, as

a Visiting Research Scholar. After that, he then joined the Smart AntennasResearch Group of Stanford University, Stanford, CA, as a Visiting AssistantProfessor. From 2005 to August 2006, he was a Lecturer with the Departmentof Engineering, the University of Hull, U.K. Since August 2006, he has beenwith University College London, where he is a Senior Lecturer.

Dr. Wong is on the editorial board of the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS. He won the IEEE Vehicular Technology Society JapanChapter Award of the International IEEE Vehicular Technology Confer-ence—Spring 2000 and was also a corecipient of the First Prize Paper Award inthe IEEE Signal Processing Society Postgraduate Forum Hong Kong Chapterin 2004. In 2002 and 2003, he received, respectively, the SY King Fellowshipsand the WS Leung Fellowships from the University of Hong Kong. Also, hewas awarded the Competitive Earmarked Research Grant Merit and IncentiveAwards in 2003–2004.

Björn Ottersten (F’04) was born in Stockholm,Sweden, in 1961. He received the M.S. degree inelectrical engineering and applied physics fromLinköping University, Linköping, Sweden, in 1986and the Ph.D. degree in electrical engineering fromStanford University, Stanford, CA, in 1989.

He has held research positions at the Departmentof Electrical Engineering, Linköping University;the Information Systems Laboratory, StanfordUniversity; and the Katholieke Universiteit Leuven,Leuven. During 1996–1997, he was Director of

Research at ArrayComm Inc, San Jose, CA, a startup company based onOttersten’s patented technology. In 1991, he was appointed Professor of SignalProcessing at the Royal Institute of Technology (KTH), Stockholm. From 2004to 2008, he was Dean of the School of Electrical Engineering at KTH, andfrom 1992 to 2004 he was head of the department for Signals, Sensors, andSystems at KTH. He is also Director of securityandtrust.lu at the Universityof Luxembourg. His research interests include wireless communications,stochastic signal processing, sensor array processing, and time-series analysis.

Dr. Ottersten coauthored papers that received an IEEE Signal Processing So-ciety Best Paper Award in 1993, 2001, and 2006. He has served as AssociateEditor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and on the editorialboard of the IEEE Signal Processing Magazine. He is currently Editor-in-Chiefof the EURASIP Signal Processing Journal and a member of the Editorial Boardof the EURASIP Journal of Applied Signal Processing. Dr. Ottersten is a Fellowof EURASIP. He is a first recipient of the European Research Council advancedresearch grant.