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Robust Cognitive Beamforming With Bounded Channel Uncertainties Gan Zheng, Member, IEEE, Kai-Kit Wong, Senior Member, IEEE, and Björn Ottersten, Fellow, IEEE

Abstract—This paper studies the robust beamforming design for a multi-antenna cognitive radio (CR) network, which transmits to multiple secondary users (SUs) and coexists with a primary network of multiple users. We aim to maximize the minimum of the received signal-to-interference-plus-noise ratios (SINRs) of the SUs, subject to the constraints of the total SU transmit power and the received interference power at the primary users (PUs) by optimizing the beamforming vectors at the SU transmitter based on imperfect channel state information (CSI). To model the uncertainty in CSI, we consider a bounded region for both cases of channel matrices and channel covariance matrices. As such, the optimization is done while satisfying the interference constraints for all possible CSI error realizations. We shall first derive equivalent conditions for the interference constraints and then convert the problems into the form of semi-definite programming (SDP) with the aid of rank relaxation, which leads to iterative algorithms for obtaining the robust optimal beamforming solution. Results demonstrate the achieved robustness and the performance gain over conventional approaches and that the proposed algorithms can obtain the exact robust optimal solution with high probability. Index Terms—Cognitive radio, interference control, MIMO, robust beamforming.

I. INTRODUCTION

R

ADIO spectrum is a very precious resource for wireless communications. According to federal communications commission (FCC) [1], spectrum utilization depends very much upon place and time and yet most spectrum is underutilized. Cognitive radio (CR), first proposed by Mitola and Maguire in 1999 [2], is a new paradigm for exploiting the spectrum resources in a dynamic way [3], [4] and has been adopted in IEEE 802.22 Wireless Regional Area Networks (WRANs) for license-exempt devices to use the spectrum on a non-interfering basis. In CR, the spectrum usage is administered in a hierarchical manner. The primary users (PUs) who are the licensed users

Manuscript received March 18, 2009; accepted June 10, 2009. First published July 14, 2009; current version published November 18, 2009. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Xiang-Gen Xia. This work was supported by EPSRC under Grants EP/D058716/1 and EP/E022308/1, and the British Council Research Exchange Programme. G. Zheng and K.-K. Wong are with the Department of Electrical and Electronic Engineering, University College London, London, WC1E 7JE, U.K. (e-mail: [email protected]). B. Ottersten is with the School of Electrical Engineering, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden. He is also with securityandtrust.lu, University of Luxembourg. 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.2009.2027462

have the priority rights to access the spectrum, while the secondary users (SUs) (also known as cognitive users) can occupy the spectrum only if they do not interrupt the communication of PUs.1 In practice, this will require the SUs to sense the spectrum and detect any PU’s activities by using, for instance, the likelihood ratio test [5], energy detection [6], matched-filtering-based technique [7] and cyclostationary detection [8], etc., so that the decision for reusing the spectrum by the SUs can be made. Spectrum holes are the most obvious opportunities to be exploited by CR [9], but higher spectrum utilization is anticipated if coexistence between PUs and SUs is permitted. The latter scenario is indeed possible if the interference caused by the SUs can be properly controlled and managed [10]. In this respect, multiple-input multiple-output (MIMO) antenna system has been recognized as an effective means to mitigate co-channel interference and has been widely used in traditional fixed-spectrum-allocation based communications systems. The use of beamforming for interference control in the context of CR is challenging, as the interference seen by the licensed PUs from the license-exempt SUs must be kept below a certain tolerable limit, if the SUs are granted access to the spectrum of the PUs. In light of this, a number of beamforming techniques have been proposed for the uplink CR network to control interference as well as achieve various objectives, such as capacity maximization [11], signal-to-interference-plus-noise ratio (SINR) balancing [11], and the minimization of transmit power with SUs’ quality-of-service (QoS) constraints [12]. Most recently, [13] used the uplink–downlink duality to study the optimal beamforming for downlink CR networks. Thus far, however, previous works on CR beamforming were based by and large on the exploitation of perfect channel state information (CSI). Although the CSI may be estimated from the uplink of the primary network if the SU transmitter continuously senses the PU’s transmission and the uplink and downlink for the PUs are reciprocal, in practice, the CSI will never be perfect, due to estimation and quantization errors as well as Doppler spread. In CR, more CSI errors are anticipated due to limited cooperation between PUs and SUs. Robust optimization, which tackles optimization with uncertainties, is usually addressed by either a stochastic or a worstcase approach and has been actively researched for MIMO systems in recent years. In particular, the CSI errors are often modeled as Gaussian random variables [14], [15], and the robustness can be provided in the statistical sense. Alternatively, another way to achieve robustness is by worst-case optimization, which designs the system to operate under the worst-case channel condition [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.

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channel is optimized to achieve a given QoS for every possible CSI error if the problem is feasible, so that absolute robustness can indeed be achieved. Interestingly, it was also shown in [17] that the worst-case model can be viewed as the deterministic analog of the stochastic CSI model in [18]. A bounded worstcase model is also believed to be able to cope with quantization errors in CSI [19]. In this paper, we shall employ the worst-case approach to address the beamforming design in downlink CR with bounded CSI errors. Since the worst-case based techniques in the MIMO literature are highly related to our study, some are reviewed as follows. In [20], a robust maximin approach was developed for a single-user MIMO system based on convex optimization. Later in [17], the robust transmission schemes to maximize the compound capacity for single and multiuser rank-one Ricean MIMO channels were addressed, based on the uncertainty set in [21]. Robust beamforming for multiuser multiple-input single-output (MISO) downlink channels with individual QoS constraints under an imperfect channel covariance matrix was studied in [16], [22]. Recently in [23], the optimal power allocation over fixed beamforming vectors was obtained in the presence of errors in CSI matrices. Most recently, efficient numerical solutions to find conservative robust beamforming for multiuser MISO systems with mean-square-error (MSE) and SINR constraints and different bounded CSI errors have been developed in [24]. In contrast, robust designs in CR are much less understood. One such attempt appeared in [25] where the optimization of a CR network exploiting perfect CSI from the SU transmitter and receiver but imperfect CSI from the SU transmitter and the PU receiver, was considered. In particular, it was illustrated that the beamforming to maximize the rate of the SU while keeping the interference to the PU under a given level in high probability can be obtained by solving a semi-infinite programming (SIP) problem. The limitation of [25] is, however, that only one SU receiver and one PU receiver are considered. With MIMO, multiple users sharing the same spectrum are anticipated, but the results in [25] do not reveal the optimal robust beamforming solution for CR systems with multiple PUs and SUs with CSI uncertainty. This paper studies a downlink CR network where there are multiple PUs and SUs occupying the same spectrum and a multi-antenna SU transmitter is communicating with multiple single-antenna SUs. We aim to obtain the optimal robust beamforming solutions for maximizing the minimum of the SUs’ SINRs, which is subjected to the constraints of the overall SU transmit power and the received interference power at the PUs, by exploiting CSI at the SU transmitter with its errors modeled by an ellipsoidal region in either channel matrices or channel covariance matrices. In particular, we have made the following contributions. • Unlike previous approaches that often restrict the feasible region when handling SINR constraints, using S-Procedure as a tool, we derive equivalent constraints to the SINR and interference power constraints in the case of ellipsoidal channel matrices errors. • We also derive equivalent constraints for the case of ellipsoidal channel covariance matrices uncertainties, with con-

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

sideration on the positive semi-definiteness of the covariance matrices, which is usually ignored in the literature. • Moreover, we propose bisection search based iterative algorithms, which use rank relaxation and the equivalent SDP constraints, to obtain the robust beamforming solutions for both cases of channel matrices and covariance uncertainties.2 In addition, numerical results show that the optimal robust solutions can indeed be found (which appears when the rank relaxation becomes exact) with very high probability. The remainder of this paper is structured as follows. Section II introduces the system model for the multiuser MISO CR system and formulates the SINR balancing problem. In Section III, we define the uncertainty region for the channel matrices uncertainties and then derive the equivalent robust constraints, while the channel covariance matrices uncertainties will be addressed in Section IV. Section V presents the algorithms to find the solutions to the robust problems using convex optimization and rank relaxation. Simulation results will be given in Section VI and finally, we conclude this paper in Section VII. Throughout this paper, complex scalar is represented by a denotes its modulus. denotes the lowercase letter and expectation of an input random entity. Vectors and matrices are represented by bold lowercase and uppercase letters, respeccomputes the Frobenius norm. The th entry tively, and or . The superscripts , of a matrix is denoted as and are used to denote the transpose, conjugate and Hermitian transpose of a vector or matrix, respectively. and mean that is positive semi-definite and strictly gives the positive definite, respectively. In addition, deminimum eigenvalue of a square matrix while notes the trace of , is a vector formed by stacking the returns an square matrix columns of , and of the elements of an -vector . gives the Kronecker denotes a vector product of and . Finally, of complex Gaussian entries with a mean vector of and a covariance matrix of . II. SYSTEM MODEL We consider a CR downlink network with PUs and SUs, where the SU common transmitter has antennas and each SU receiver has only one antenna. The received signal at the th SU, 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 beamdenotes the channel between the SU transforming vector, is the mitter and the th SU receiver, and additive noise. The total transmit power from the SUs is given and required to be limited by . by 2It is noted that the proposed approach can also handle the total power minimization of the SUs and more convex constraints, such as per-antenna power constraints, can also be easily incorporated.

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

The instantaneous received SINR at the pressed as

th SU can be ex-

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and are the estimated CSI known at the SU where and are the CSI errors. We transmitter while assume that the CSI errors are bounded by ellipsoids, i.e.,

(2) (9) which is required to be at least where are the preset SINR balancing parameters providing the SINR pattern of the SUs and is an auxiliary variable for scaling the SINRs. To control the interferences, at the th PU, the received interference power is limited by (3) where denotes the channel between the SU transmitter and the th PU receiver. In particular, our problem of interest is to maximize the minimum of the received SINRs of the SUs given the parameters and by optimizing the transmit beamforming vectors with the aid of CSI. Mathematically, it is expressed as (4a)

(10) determine the quality of where the matrices CSI and are assumed known. Moreover, the CSI becomes perand approach infinity while the CSI is the fect when worst if they are zeros. The above model also embraces the Frobenius norm bounded error. B. Equivalent Worst-Case Constraints To take the CSI errors into account, the SINR balancing problem in (6) based on worst-case criterion can be formulated as (11a)

(11b) (4b) Before we proceed, we shall first investigate on rewriting the constraints in (11b). In particular, the SU’s SINR constraint [i.e., the first constraint in (11b)] can be rewritten as 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 will be spent on solving (6) with CSI errors. III. BEAMFORMING WITH CHANNEL MATRIX UNCERTAINTY

(12) Note that in (12), both sides contain a common ellipsoidal uncertainty region and it is not known to be tractable [26]. One possible 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 proposed to deal with this problem [19], [24] but the resulting solutions appear to degrade the performance and reduce the feasible region considerably. Our approach here is different in that we do not impose any restriction on the optimizing variables but derive the exact equivalent condition directly from the robust constraint (12). The main result is summarized in the following Theorem 1. Theorem 1: The robust SU’s SINR constraint (12) is equivalent to

A. Ellipsoidal Uncertainty Model It is assumed that the SU transmitter has neither perfect CSI nor statistical knowledge of the uncertainty. In particular, the channels are modeled as (7) (8)

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

(14)

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Though is not convex as is not positive semidefinite, according to S-Procedure [28], (14) is an equivalent statement of (13), which completes the proof.3 Similarly, we next derive the equivalent interference constraint [i.e., the second constraint in (11b)] and the result is summarized in Theorem 2. Theorem 2: The robust interference constraint (11b) is equivalent to (15)

Similar as before, the matrices and are given 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 written as

Proof: It follows similarly as that of Theorem 1 and is omitted. C. Reformulation and SDP Relaxation With Theorems 1 and 2, the original robust problem (11) can be equivalently expressed as

th SU can be

(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)

(16a) (24b)

(16b)

The above reformulation is straightforward and equivalent to the original problem (11), which permits a possible exact optimal solution to be found. The proposed algorithm we develop to solve (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 covariances. In this case, the uncertainty in the channel covariance matrices has to be properly modeled and considered in (11).

It is worth noting that there have been a number of researches attempting to address uncertainties in the covariance matrices of channel or signal, e.g., [16], [22], and [29]–[32]. However, an important property that the covariance matrices must be positive semi-definite (PSD), is often assumed [29] or ignored [30] and thus the solution will be over pessimistic because some unrealistic errors are considered as well. This constraint was considered in [31] by modeling the error of the “square root” of the signal covariance matrix but not the covariance matrix itself. 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

A. Ellipsoidal Uncertainty Model As in Section III, we use an ellipsoid to model the channel covariance uncertainty such that (17) (18) where and are the estimated channel covariance matrices from the SU transmitter to the th SU and from the SU transmitter to the th PU, respectively, and and denote the corresponding uncertainties that are considered to be bounded in the sets

(19)

(25) As such, it suffices to derive the maximum interference, which we shall address in Theorem 3. Before we present this theorem, we find the following lemma useful. Hermitian matrices, and , such Lemma 1: For and , we have the following gradient that results: (26) Proof: See Appendix A. Theorem 3: Let be an Then, we have

complex Hermitian matrix.

(20) 3Similar

result has been obtained in [27].

(27)

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

and the maximum is attained when

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In [22], [32], the error model in (19) was considered with for some , and it has been proposed to optimize the system based on the lower bound of the SINR (28)

(38)

Proof: Now, we consider the problem (29) The Lagrangian can be written as (30)

However, as this lower bound is generally loose, severe performance 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

is the dual variable. Setting the derivative to be in which zero and using Lemma 1 in the Appendix, we obtain (39)

(31) and (37) becomes

Using Kronecker operation, it yields (32)

(40)

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 optimal implies objective value equals . Further, , which enables us to solve for by that

(34) As a result, we have proved the desired result (27) and the op, is given by (28), which completes the timal , denoted by proof. Using Theorem 3, the optimal to (25), denoted by , is given by

This solution is simple but problematic as it may not satisfy the positive semi-definiteness property of the covariance matrix, i.e., because is in general not positive semi-definite, and this condition will likely be violated if the error bound is large. Although (40) simplifies the problem and is automatically satisfied, it is sometimes emphasized that in general the solution using (39) only serves as a lower bound for the actual achievable performance. We now overcome 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

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

(42) with the worst-case

given by

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

Proof: First, we consider the problem (44a) (44b)

(37)

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

(45a)

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

(45b)

Since (44) is a convex problem, it has the same optimal objective value as its dual (45). In addition, statement i) is equivalent to 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 has completed the proof. C. Reformulation and SDP Relaxation With (36) and Theorem 4, we can then rewrite (24) in an equivalent form as

(46a)

V. PROPOSED ALGORITHM In this section, we present an algorithm to solve the optimization problems (16) and (46). Although our previous efforts have made to rewrite the constraints into SDP, the problems are still not convex due to the rank-1 constraints on the beamforming matrices and also the dependence of with other variables. To overcome this, we note that if we relax the rank-1 constraints, then both (16) and (46) will become quasi-convex problems and the optimal solutions can be found by checking the feasibility of an SDP problem. In particular, both (16) and (46) can be solved using 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 iterate and the “problem” can be either (16) or (46). Algorithm 1: Robust Cognitive Beamforming using Bisection Search 1: Input: and

,

,

,

,

,

,

,

.

2: begin Initialize the iteration index the upper bound .

4:

Initialize

5:

For a given , check the feasibility of the problem without the rank-1 constraints.

6:

if the problem is feasible, then

(46b)

7:

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

8:

Update

9:

Update the sets and

10: 11:

, then

.

end else

12:

Update

13:

end

14:

Set

15:

Find matrices

.

and go to line 5 if

greater than a given threshold. and obtain the beamforming .

16: end (47)

and

.

if it also outputs rank-1 solutions for all

D. Numerical Example To conclude this section, we here provide a numerical example to illustrate that the robust constraints (46b) are necessary and if we replace them with (40), a valid worst-case channel covariance realization is not met, and as a result, the performance is unnecessarily degraded. In this example, we consider the case where , , , 5 dB, 20 dB, 0 dB, and

, the lower bound

3:

17: Extract the beamforming solution by matrix decomposition. 18: Output:

and

.

from

is

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

In line 5 of Algorithm 1, it requires detecting the feasibility of a convex SDP problem, which can be achieved by checking if there exist any feasible solution of the SDP. Rank relaxation has been widely used in communications optimization such as maximum-likelihood (ML) detection (see [35], [36] and the references therein). In that particular application, some randomization methods were proposed to recover the required rank-1 solution based on the higher-rank solution. This is also required in line 17 of Algorithm 1, if any of the obtained appear to be of higher-rank. beamforming matrices Nevertheless, it is noted that the known randomization methods are not applicable here because our problems have complex robust constraints required by the PUs and the SUs, and it is not clear even how to construct a feasible rank-1 solution based on some higher-rank solutions. To tackle the rank issue, we store all those iterations that return rank-1 solutions. After the algorithm converges, if the final beamforming solution is of rank-1, then it is guaranteed to be optimal; otherwise, we shall use the rank-1 solution that achieves the largest , but in this case, its optimality is not guaranteed. As a final remark, it is worth mentioning that it is possible to , such as per incorporate more convex constraints about transmit antenna power, into the proposed algorithm. In addition, the proposed techniques can also be used to address the power minimization problem subject to the robust SINR and interference constraints as the equivalent tractable robust constraints have been derived in SDP. VI. SIMULATION RESULTS A. Setup and Assumptions Simulations are conducted to assess the performance of the proposed algorithms in Rayleigh flat-fading channels, so that . For the case the entries of the channel matrices follow of channel covariance in Section IV, we considered the channel model in [32]. In particular, it is assumed that the th SU is surrounded by a large number of local scatterers corresponding to . For a uniform linear array with elements a spread angle of spaced half a wavelength apart, we have the normalized spatial covariance matrix as

(48) where is the nominal direction to the source, and the covariance matrices to the PUs as

(49) and denote, respectively, the nominal direcin which tion to the source and the angular spread. Furthermore, we conand sidered a system with three SUs, where , and . The parameter will be varied to generate various channel environments. Moreover, the CSI uncertainties are assumed to be truncated Gaussian distributed within the bounded uncertainty sets. In the simulations, unless explicitly stated, we considered that

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all users have the same target SINR, CSI error bounds and indB , terference power limits, i.e., and , while is the parameter to adjust the CSI quality and set to 0.1 unless specified otherwise. We also as, is sumed that the average total transmit SNR, defined as to denote 10 dB. For convenience, we use the triple transmit antennas and PUs. The folan -SU system with lowing benchmarks are compared with the proposed algorithm. • Non-robust design—This optimizes the users’ beamforming vectors based on the estimated CSI matrices. , , , , The CSI uncertainty regions, are ignored, so this method is expected to have a high probability of the constraints being violated. • Conservative channel-based robust solution—This was proposed in [24] to address the imperfect channel matrices. This solution is based on the equivalent MSE constraints. • Conservative covariance-based robust solution—This was proposed in [32] when CSI is in the form of channel covariance matrices and is imperfect. Before presenting the results, we first comment on the complexity per iteration of the proposed algorithm and that of the conservative approaches in [24], [32]. It is known that the number of variables, the number of SDP constraints and the size of SDP dominate the overall complexity [24], [37]. For the channel-based problem, the virtual MSE-based algorithm in real-valued design variables and slack [24] has SDP constraints of size variables. Moreover, there are also . By contrast, our proposed algorithm has real-valued design variables and slack variables, with SDP constraints of size and another SDP constraints . There are also some computations required of size to perform matrix decomposition in the proposed method. Due to the total power and interference constraints, both will have . one scalar constraint and SDP constraints of size On the other hand, as far as the covariance-based problem is real-valued design concerned, the method in [32] has scalar constraints, while our algorithm has variables and real-valued design variables but slack variSDP constraints of size and ables are needed, with scalar constraints. Similarly, both will have scalar constraints. To summarize, in general, the complexities of the proposed algorithms are slightly higher than those of the conservative approaches, but it will be apparent that substantial performance gain can be achieved because more degree of freedom is exploited. B. Results In Fig. 2, the cumulative density functions (CDFs) of the received interference at the first PU receiver are provided for a 10 dB for the case of imperfect (3,3,2) system with channel 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 while the non-robust scheme can rarely satisfy the interference constraints due to the ignorance of imperfect CSI. In Fig. 3, the CDFs of the received SINR at the first SU receiver are provided for the same systems as before. It is observed

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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 highest SINR but at the price of severe interference to the PUs, which has been seen in Fig. 2. On the other hand, results illustrate that the proposed algorithm achieves about 5- and 1-dB gains in the received SINR over the conservative method, respectively, for the cases of imperfect channel matrices and channel covariance matrices.

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

In Fig. 4, the output SINR results for the case with imperfect CSI covariance matrices are plotted against the angular separation . The dashed lines are for the systems without the interference power constraints. As we can see, the performance for all schemes improves as the angular separation increases. However, the solid lines which correspond to the systems with the interference constraints, clearly show the impact of the positions of the PUs, i.e., if the SUs are not separated far enough and close to the PUs, the performance may degrade as the angular separation increases. In particular, the worst performance appears 10 . when In Fig. 5, the output SINR results for the both cases are provided for various error bound, . The setting is the same as that 16 . Results demonstrate that the proused in 2 except that posed algorithms significantly outperforms the conservative approaches, especially for the case of imperfect channel matrices. Finally, we investigate the optimality of the proposed algorithm by studying the ranks of the beamforming solutions of the SDP for the case with imperfect channel matrices. For the case with imperfect channel covariance matrices, numerical results indicate that the optimal rank-1 solutions can always be obtained, which means that the proposed algorithm always provides the optimal solutions. In other words, we can measure the number of times that the proposed SDPs return rank-1 beamforming matrices and declare global optimality in these channels. In Table I, we show such probability results that the proposed algorithm is guaranteed to achieve the robust optimum for various CSI error bound . Results reveal that the proposed

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received interference power at the PUs. For CSI of channel matrices and channel covariance matrices, we derived the equivalent robust constraints in the form of SDP using S-Procedure. A bisection based algorithm has been devised to obtain the robust beamforming solutions with rank relaxation. Simulation results have shown the achieved robustness against the CSI uncertainties and the effectiveness to meet the required output SINRs at the SUs. APPENDIX A THE PROOF OF LEMMA 1 Proof: First, note that for a general matrix complex vari, the gradient of a real-value function is defined as , and that and are treated as two independent as variables. Now, we expand

able

(50) which has the gradient

(51) Note that for a general complex matrix without any special structure, the above gradient should be zero. Similarly, we can as follows: find the gradient for

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

and then TABLE I THE PROBABILITY RESULTS THAT THE PROPOSED ALGORITHM GUARANTEES TO ACHIEVE OPTIMALITY

(53) As a result, we have the desired results, which completes the proof. APPENDIX B DERIVATION OF THE DUAL PROBLEM OF (44)

algorithm can achieve the global optimal solutions at very high probability.

Here, we derive the dual of (44). First, we have the Lagrangian of (44) given by

VII. CONCLUSION This paper has addressed the worst-case robust beamforming design of a downlink multiuser multi-antenna CR system coexisted with a number of PUs, in which only imperfect CSI is exploited at the SU transmitter. With ellipsoidal uncertainty sets on the CSI, our objective was to balance the SU SINRs with the constraints of the total transmit power of the SUs and the

(54) where

, are the dual variables. Setting and using Lemma 1, we then have (55)

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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 at the optimum, we have (59) Combining (55) with or

and , we

then have

(60)

(61) As a result, we arrive at (43) and (45). REFERENCES [1] Spectrum Policy Task Force, Federal Communications Commission, Rep. ET Docket No. 02-135, Nov. 2002. [2] J. Mitola and G. Q. Maguire, “Cognitive radio: Making software radios more personal,” IEEE Pers. Commun., vol. 6, no. 6, pp. 13–18, Aug. 1999. [3] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [4] S. M. Mishra, A. Sahai, and R. W. Brodensen, “Cooperative sensing among cognitive radios,” in Proc. IEEE Int. Conf. Commun., Istanbul, Turkey, Jun. 11–15, 2006, pp. 1658–1663. [5] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory. Englewood Cliffs, NJ: Prentice-Hall, 1998, vol. 2. [6] R. Tandra and A. Sahai, “Fundamental limits on detection in low SNR under noise uncertainty,” presented at the WirelessCom 2005, Maui, HI, Jun. 13–16, 2005. [7] D. Cabric, A. Tkachenko, and R. W. Brodersen, “Spectrum sensing measurements of pilot, energy, and collaborative detection,” in Proc. Military Comm. Conf., Washington, DC, Oct. 23–25, 2006, pp. 1–7. [8] W. A. Gardner, “Exploitation of spectral redundancy in cyclostationary signals,” IEEE Signal Process. Mag., vol. 8, pp. 14–36, 1991. [9] Y. C. Liang, Y. Zeng, E. C. Y. Peh, and A. T. Hoang, “Sensingthroughput tradeoff for cognitive radio networks,” IEEE Trans. Wireless Commun., vol. 7, no. 4, pp. 1326–1337, Apr. 2008. [10] A. Ghasemi and E. S. Sousa, “Fundamental limits of spectrum-sharing in fading environments,” IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 649–658, Feb. 2007. [11] L. Zhang, Y. C. Liang, and Y. Xin, “Joint beamforming and power control for multiple access channels in cognitive radio networks,” IEEE J. Sel. Areas Commun., vol. 26, no. 1, pp. 38–51, Jan. 2008. [12] M. H. Islam, Y. C. Liang, and A. T. Hoang, “Joint power control and beamforming for cognitive radio networks,” IEEE Trans. Wireless Commun., vol. 7, no. 7, pp. 2415–2419, Jul. 2008.

[13] L. Zhang, R. Zhang, Y. C. Liang, Y. Xin, and H. V. Poor, “On Gaussian MIMO BC-MAC duality with multiple transmit covariance constraints,” [Online]. Available: http://www.citebase.org/abstract?id=oai:arXiv.org:0809.4101 [14] X. Zhang, D. P. Palomar, and B. Ottersten, “Statistically robust design of linear MIMO transceivers,” IEEE Trans. Signal Process., vol. 56, no. 8, pp. 3678–3689, Aug. 2008. [15] B. K. Chalise, S. Shahbazpanahi, A. Czylwik, and A. B. Gershman, “Robust downlink beamforming based on outage probability specifications,” IEEE Trans. Wireless Commun., vol. 6, no. 10, Oct. 2007. [16] M. Bengtsson and B. Ottersten, “Optimal and suboptimal transmit beamforming,” in Handbook of Antennas in Wireless Communications, L. C. Godara, Ed. Boca Raton, FL: CRC Press, Aug. 2001. [17] A. Wiesel, Y. C. Eldar, and S. Shamai, “Optimization of the MIMO compound capacity,” IEEE Trans. Wireless Commun., vol. 6, no. 3, pp. 1094–1101, Mar. 2007. [18] N. Jindal, “MIMO broadcast channels with finite rate feedback,” IEEE Trans. Inf. Theory, vol. 52, no. 11, pp. 5045–5059, Nov. 2006. [19] M. Botros and T. N. Davidson, “Convex conic formulations of robust downlink precoder designs with quality of service constraints,” IEEE J. Sel. Topics Signal Process., vol. 1, no. 4, pp. 714–724, Dec. 2007. [20] A. Pascual-Iserte, D. P. Palomar, A. I. Perez-Neira, and M. A. Lagunas, “A robust maximin approach for MIMO communications with partial channel state information based on convex optimization,” IEEE Trans. Signal Process., vol. 54, no. 1, pp. 346–360, Jan. 2006. [21] J. Wolfowitz, Coding Theorems of Information Theory, 3rd ed. Berlin, Germany: Springer-Verlag, 1978. [22] M. Biguesh, S. Shahbazpanahi, and A. B. Gershman, “Robust downlink power control in wireless cellular systems,” EURASIP J. Wireless Commun. Netw., vol. 2004, no. 2, pp. 261–272, Dec. 2004. [23] M. Payaró, A. Pascual, and M. Lagunas, “Robust power allocation designs for multiuser and multiantenna downlink communication systems through convex optimization,” IEEE J. Sel. Areas Commun., vol. 25, no. 7, pp. 1390–1401, Sep. 2007. [24] N. Vucic and H. Boche, “Robust QoS-constrained optimization of downlink multiuser MISO systems,” IEEE Trans. Signal Process., vol. 57, no. 2, pp. 714–725, Feb. 2009. [25] L. Zhang, Y. C. Liang, and Y. Xin, “Robust cognitive beamforming with partial channel state information,” in Proc. 42nd Conf. Info. Sciences Systems, Princeton, NJ, Mar. 19–21, 2008, pp. 890–895. [26] D. Bertsimas and M. Sim, “Tractable approximations to robust conic optimization problems,” Math. Program., vol. 107, no. 1, 2006. [27] G. Zheng, K. K. Wong, and T. S. Ng, “Robust MIMO in the downlink: A worst-case optimization with ellipsoidal uncertainty regions,” EURASIP J. Adv. Signal Process., vol. 2008, 2008, Article ID 609028, 15 pp. [28] A. Beck and Y. Eldar, “Strong duality in nonconvex quadratic optimization with two quadratic constraints,” SIAM J. Optimiz., vol. 17, no. 3, pp. 844–860, 2006. [29] Y. C. Eldar and N. Merhav, “A competitive minimax approach to robust estimation of random parameters,” IEEE Trans. Signal Process., vol. 52, no. 7, pp. 1931–1946, Jul. 2004. [30] S. Shahbazpanahi, A. B. Gershman, Z.-Q. Luo, and K. M. Wong, “Robust adaptive beamforming for general-rank signal models,” IEEE Trans. Signal Process., vol. 51, no. 9, pp. 2257–2269, Sep. 2003. [31] H. Chen and A. B. Gershman, “Robust adaptive beamforming for general-rank signal models using positive semi-definite covariance constraint,” in Proc. Int. Conf. Acoustics, Speech, Signal Process., Las Vegas, NV, Apr. 2008, pp. 2341–2344. [32] M. Bengtsson and B. Ottersten, “Optimal downlink beamforming using semidefinite optimization,” in Proc. 37th Annu. Allerton Conf. Commun., Control, Comput., Monticello, IL, Sep. 22–24, 1999, pp. 987–996. [33] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [34] A. J. Laub, Matrix Analysis for Scientists and Engineers. Philadelphia, PA: SIAM, 2005. [35] W. K. Ma, P. C. Ching, and Z. Ding, “Semidefinite relaxation based multiuser detection for -ary PSK multiuser systems,” IEEE Trans. Signal Process., vol. 52, no. 10, pp. 2862–2872, Oct. 2004. [36] W. K. Ma, C. C. Su, J. Jalden, T. H. Chang, and C. Y. Chi, “The equivalence of semidefinite relaxation MIMO detectors for higher-order QAM,” IEEE J. Sel. Topics Signal Process. Aug. 2008 [Online]. Available: http://arxiv.org/abs/0809.4529, submitted for publication [37] A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, ser. MPSSIAM Series on Optimization. Philadelphia, PA: SIAM, 2001.

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

Gan Zheng (S’05–M’09) received the B.Eng. and M.Eng. degrees from Tianjin University, China, in 2002 and 2004, respectively, both in electronic and information engineering, and the Ph.D. degree in electrical and electronic engineering from The University of Hong Kong, Hong Kong, in 2008. He has been working as a Research Associate at University College London since December 2007. His research interests are in the general area of signal processing for wireless communications, with particular emphasis on the design, optimization and analysis for multiuser multiple-input multiple-output (MIMO) antenna systems, cooperative network, and cognitive radio.

Kai-Kit Wong (S’98–M’01–SM’08) received the B.Eng., M.Phil., and Ph.D. degrees, all in electrical and electronic engineering, from the Hong Kong University of Science and Technology, Hong Kong, in 1996, 1998, and 2001, respectively. After graduation, he joined the Department of Electrical and Electronic Engineering, the University of Hong Kong as a Research Assistant Professor. From July 2003 to December 2003, he visited the Wireless Communications Research Department of Lucent Technologies, Bell-Labs, Holmdel, NJ, as a Visiting Research Scholar. After that, he then joined the Smart Antennas Research Group of Stanford University, Stanford, CA, as a Visiting Assistant Professor. From 2005 to August 2006, he was a Lecturer with the Department of Engineering, the University of Hull, U.K. Since August 2006, he has been with 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 Japan Chapter Award of the International IEEE Vehicular Technology Conference—Spring 2000 and was also a corecipient of the First Prize Paper Award in the IEEE Signal Processing Society Postgraduate Forum Hong Kong Chapter in 2004. In 2002 and 2003, he received, respectively, the SY King Fellowships and the WS Leung Fellowships from the University of Hong Kong. Also, he was awarded the Competitive Earmarked Research Grant Merit and Incentive Awards in 2003–2004.

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Björn Ottersten (F’04) was born in Stockholm, Sweden, in 1961. He received the M.S. degree in electrical engineering and applied physics from Linköping University, Linköping, Sweden, in 1986 and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 1989. He has held research positions at the Department of Electrical Engineering, Linköping University; the Information Systems Laboratory, Stanford University; and the Katholieke Universiteit Leuven, Leuven. During 1996–1997, he was Director of Research at ArrayComm Inc, San Jose, CA, a startup company based on Ottersten’s patented technology. In 1991, he was appointed Professor of Signal Processing at the Royal Institute of Technology (KTH), Stockholm. From 2004 to 2008, he was Dean of the School of Electrical Engineering at KTH, and from 1992 to 2004 he was head of the department for Signals, Sensors, and Systems at KTH. He is also Director of securityandtrust.lu at the University of 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 Society Best Paper Award in 1993, 2001, and 2006. He has served as Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and on the editorial board of the IEEE Signal Processing Magazine. He is currently Editor-in-Chief of the EURASIP Signal Processing Journal and a member of the Editorial Board of the EURASIP Journal of Applied Signal Processing. Dr. Ottersten is a Fellow of EURASIP. He is a first recipient of the European Research Council advanced research grant.