Robust Cognitive Beamforming With Partial Channel State Information{}^{\star}

# Robust Cognitive Beamforming With Partial Channel State Information⋆

Lan Zhang, Ying-Chang Liang, Yan Xin, and H. Vincent Poor The work is supported by the National University of Singapore (NUS) under Grants R-263-000-314-101, R-263-000-314-112, and R-263-000-478-112, by a NUS Research Scholarship, and by the U.S. National Science Foundation under Grants ANI-03-38807 and CNS-06-25637. This work was done while Y. Xin was visiting Princeton University.L. Zhang and Y. Xin are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 118622. (email: zhanglan@nus.edu.sg; elexy@nus.edu.sg)Y.-C. Liang is with Institute of Infocomm Research, A*STAR, 21 Heng Mui Keng Terrace, Singapore 119613. (email: ycliang@i2r.a-star.edu.sg)H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA, (email: poor@princeton.edu).
###### Abstract

This paper considers a spectrum sharing based cognitive radio (CR) communication system, which consists of a secondary user (SU) having multiple transmit antennas and a single receive antenna and a primary user (PU) having a single receive antenna. The channel state information (CSI) on the link of the SU is assumed to be perfectly known at the SU transmitter (SU-Tx). However, due to loose cooperation between the SU and the PU, only partial CSI of the link between the SU-Tx and the PU is available at the SU-Tx. With the partial CSI and a prescribed transmit power constraint, our design objective is to determine the transmit signal covariance matrix that maximizes the rate of the SU while keeping the interference power to the PU below a threshold for all the possible channel realization within an uncertainty set. This problem, termed the robust cognitive beamforming problem, can be naturally formulated as a semi-infinite programming (SIP) problem with infinitely many constraints. This problem is first transformed into the second order cone programming (SOCP) problem and then solved via a standard interior point algorithm. Then, an analytical solution with much reduced complexity is developed from a geometric perspective. It is shown that both algorithms obtain the same optimal solution. Simulation examples are presented to validate the effectiveness of the proposed algorithms.
Keywords: Cognitive radio, interference constraint, multiple-input single-output (MISO), partial channel state information, power allocation, rate maximization.

Suggested Editorial Areas:

[0.5cm] Cognitive radio, Multiple-input single-output (MISO), Partial channel state information, Power allocation, Wireless networks.

## I Introduction

One of the fundamental challenges faced by the wireless communication industry is how to meet rapidly growing demands for wireless services and applications with limited radio spectrum. Cognitive radio (CR) technology has been proposed as a promising solution to tackle such a challenge [1, 2, 3, 4, 5, 6, 7, 8]. In a spectrum sharing based CR network, the secondary users (SUs) are allowed to coexist with the primary user (PU), subject to the constraint, namely the interference constraint, that the interference power from the SU to the PU is less than an acceptable value. Evidently, the purpose of the imposed interference constraint is to ensure that the quality of service (QoS) of the PU is not degraded due to the SUs. To be aware of whether the interference constraint is satisfied, the SUs needs obtain knowledge of the radio environment cognitively.

In this paper, we consider a spectrum sharing based CR communication scenario, in which the SU uses a multiple-input single-output (MISO) channel and the primary user (PU) has one receive antenna. We assume that the channel state information (CSI) about the SU link is perfectly known at the SU transmitter (SU-Tx). However, owing to loose cooperation between the SU and the PU, only the mean and covariance of the channel between the SU-Tx and the PU is available at the SU-Tx. With this CSI, our design objective is, for a given transmit power constraint, to determine the transmit signal covariance matrix that maximizes the rate of the SU while keeping the interference power to the PU below a threshold for all the possible channel realizations within an uncertainty set. We term this design problem the robust cognitive beamforming design problem.

In non-CR settings, the study of multiple antenna systems with partial CSI has received considerable attention in the past [9, 10]. Specifically, the paper [10] considers the case in which the receiver has perfect CSI but the transmitter has only partial CSI (mean feedback or covariance feedback). It was proved in [10] that the optimal transmission directions are the same as those of the eigenvectors of the channel covariance matrix. However, the optimal power allocation solution was not given in an analytical form. A universal optimality condition for beamforming was explored in [11], and quantized feedback was studied in [12].

In CR settings, power allocation strategies have been developed for multiple access channels (MAC) [13] and for point-to-point multiple-input multiple-output (MIMO) channels [14]. Particularly, the solution developed in [14] can be viewed as cognitive beamforming since the SU-Tx forms its main beam direction with awareness of its interference to the PU. A closed-form method has been present in [14]. A water-filling based algorithm is proposed in [13] to obtain the suboptimal power allocation strategy. However, the papers [13] and [14] assume that perfect CSI about the link from the SU-Tx to the PU is available at the SU-Tx. Due to loose cooperation between the SU and the PU, it could be difficult or even infeasible for the SU-Tx to acquire accurate CSI between the SU-Tx to the PU.

In this paper, we formulate the robust cognitive beamforming design problem as a semi-infinite programming (SIP) problem, which is difficult to solve directly. The contribution of this paper can be summarized as follows.

1. Several important properties of the optimal solution of the SIP problem, the rank-1 property, and the sufficient and necessary conditions of the optimal solution, are presented. These properties would transform the SIP problem into a finite constraint optimization problem.

2. Based on these properties, we show that the SIP problem can be transformed into a second order cone programming (SOCP) problem, which can be solved via a standard interior point algorithm.

3. By exploiting the geometric properties of the optimal solution, a closed-form solution for the SIP problem is also provided.

The rest of this paper is organized as follows. Section II describes the SU MISO communication system model, and the problem formulation of the robust cognitive beamforming design. Section III presents several important lemmas that are used to develop the algorithms. Two different algorithms, the SOCP based solution and the analytical solution, are developed in Section V and Section IV, respectively. Section VI presents simulation examples, and finally, Section VII concludes the paper.

The following notation is used in this paper. Boldface upper and lower case letters are used to denote matrices and vectors, respectively, and denote the conjugate transpose and transpose, respectively, denotes an identity matrix, denotes the trace operation, and denotes the rank of the matrix .

## Ii Signal Model and Problem Formulation

With reference to Fig. 1, we consider a point-to-point SU MISO communication system, where the SU has transmit antennas and a single receive antenna. The signal model of the SU can be represented as , where and are the received and transmitted signals respectively, denotes the channel response from the SU-Tx to the SU-Rx, and is independent and identically distributed (i.i.d.) Gaussian noise with zero mean and unit variance111Since the SU receiver cannot differentiate the interference from the PU from the background noise, the term can be viewed as the summation of the interference and the noise. The variance of does not influence the algorithms discussed later. Moreover, the variance of can be measured at the SU receiver [13]. . Suppose that the PU has one receive antenna. The channel response from the SU-Tx to the PU is denoted by an vector . Further, assume that the SU-Tx has perfect CSI for its own link, i.e., is perfectly known at the SU-Tx. However, due to the loose cooperation between the SU and the PU, only partial CSI about is assumed to be available at the SU-Tx. We assume that and are the mean and covariance of , respectively222Due to the cognitive property, we assume that the SU can obtain the pilot signal from the PU, and thus can detect the channel information from the PU to the SU. Moreover, since the SU shares the same spectrum with the PU, based on the channel from the PU to the SU, the statistics of the channel from the SU to the PU can be obtained [15]. Therefore, we can assume that and are known to the SU.. In previous work [10, 16, 15, 17], partial CSI has been considered in two extreme cases in a non-CR setting. One is the mean feedback case, , where can be viewed as the variance of the estimation error; and the other is the covariance feedback case, where is a zero vector. In this paper, we study the case where the SU-Tx knows both the mean and covariance of in a CR setting.

The objective of this paper is to determine the optimal transmit signal covariance matrix such that the information rate of the SU link is maximized while the QoS of the PU is guaranteed under a robust design scenario, i.e., the instantaneous interference power for the PU should remain below a given threshold for all the in the uncertain region. Mathematically, the problem is formulated as follows:

 Robust design problem (P1): max\small\boldmathS≥0 log(1+\boldmathhHs\boldmathS\boldmathhs)subject to:tr(\boldmathS)≤¯P, and \boldmathhH\boldmathS\boldmathh≤Pt for (\boldmathh−% \boldmathh0)H\boldmathR−1(\boldmathh−%\boldmath$h$0)≤ϵ, (1)

where is the transmit signal covariance matrix, is the transmit power budget, is the interference threshold of the PU, and is a positive constant. The parameter characterizes the uncertainty of at the SU. According to the definition of the uncertainty in [18], belongs to a type of ellipsoid uncertainty problem, i.e., the uncertain parameter is confined in a range of an ellipsoid , where . Thus, the optimal solution of problem can guarantee the interference power constraint of the PU for all the , and thus the robustness of is in the worst case sense [19], i.e., in the worst case channel realization, the interference constraint should also be satisfied. If the primary transmission does not exist, then the interference constraint is excluded, and thus the problem reduces to a trivial beamforming problem. Hence, we only focus on the case where the both PU and SU transmission exist.

###### Remark 1

An important observation is that the objective function in problem remains invariant when undergoes an arbitrary phase rotation. Without loss of generality, we assume, in the sequel, that and have the same phase, i.e., .

Since problem has a finite number of decision variable , and is subjected to an infinite number of constraints with respect to the compact set , problem is an SIP problem [20]. One obvious approach for an SIP problem is to transform it into a finite constraint problem. However, there is no universal algorithm to determine the equivalent finite constraints such that the transformed problem has the same solution as the original SIP problem. In the following section, we first study several important properties of problem , which would be used to transform the SIP problem into its equivalent finite constraint counterpart.

## Iii Properties of The Optimal Solution

The maximization problem is a convex optimization problem, and thus has a unique optimal solution. The following lemma presents a key property of the optimal solution of problem (see Appendix A for the proof).

###### Lemma 1

The optimal covariance matrix for problem is a rank-1 matrix.

###### Remark 2

Lemma 1 indicates that beamforming is the optimal transmission strategy for problem , and the optimal transmit covariance matrix can be expressed as , where is the optimal transmit power and is the optimal beamforming vector with . Therefore, the ultimate objective of problem is to determine and .

According to Lemma 1, a necessary and sufficient condition for the optimal solution of problem is presented as follows (refer to Appendix B for the proof).

###### Lemma 2

A necessary and sufficient condition for to be the globally optimal solution of problem is that there exists an such that

 \boldmathSopt=argmax\small% \boldmathS,plog(1+\boldmathhHs\boldmathS\boldmathhs), subject to: tr(\boldmathS)≤p, 0≤p≤¯P, \boldmathhHopt\boldmathS\boldmathhopt≤Pt, (2)

where

 \boldmathhopt=argmax\small% \boldmathh\boldmathhH\boldmathSopt\boldmathh, for (\boldmathh−\boldmathh0)H\boldmathR−1(\boldmathh−\boldmathh0)≤ϵ. (3)
###### Remark 3

The vector is a key element for all , in the sense that, for the optimal solution, the constraint dominates the whole interference constraints, i.e., all the other interference constraints are inactive. Thus, if we can determine , the SIP problem is transformed into a finite constraint problem (2). It is worth noting that the problem (2) has the same form as the problem discuss in [14], in which the CSI on the link of the SU and the link between SU-Tx and PU are perfectly known at the SU-Tx. However, unlike the problem in [14], in (2) is an unknown parameter.

In the following lemma (see Appendix C for the proof), the optimal beamforming vector is shown to lie in a two-dimensional (2-D) space spanned by and the projection of into the null space of . Define and , where . Hence, we have with .

###### Lemma 3

The optimal beamforming vector is of the form with .

###### Remark 4

According to Lemma 3, we can search for the optimal beamforming vector on the 2-D space spanned by and , which simplifies the search process significantly. The optimal found in this 2-D space, is also the globally optimal solution of the original problem . As depicted in Fig. 2, problem is transformed into the problem of determining the beamforming vector in the 2-D space and the corresponding power . Combining Lemma 2 and Lemma 3, it is easy to conclude that lies in the space spanned by and .

## Iv Second Order Cone Programming Solution

In this section, we solve problem via a standard interior point algorithm [19, 21, 22]. We first transform the SIP problem into a finite constraint problem, and further transform it into a standard SOCP form, which can be solved by using a standard software package such as SeDuMi [23]. One key observation is that if , i.e., the worst case interference constraint of is satisfied, then the interference constraint of holds. Combining this observation with Lemma 1, problem can be transformed as:

 {Equivalent problem (P2):}maxp≥0,∥\small\boldmathv∥=1log(1+p\boldmathhHs\boldmathv\boldmathv% H\boldmathhs)subject to:p≤¯P, max% \small\boldmathh∈H(ϵ) p\boldmathhH\boldmathv\boldmathvH\boldmathh≤Pt, (4)

where . It is clear that maximizing is equivalent to maximizing . By defining , the objective function can be rewritten as . Similarly, the interference power can be expressed as . Thus, problem can be further transformed to

 max\small\boldmathw|\boldmathhHs\boldmathw|subject to:∥\boldmathw∥≤√¯P, max\small\boldmathh∈H(ϵ)|\boldmathhH\boldmathw|≤√Pt. (5)

According to the definition of , we can rewrite the worst-case constraint in (5) as

 max\small\boldmathh∈H(ϵ)|%\boldmath$h$H\boldmathw|=max\small\boldmathh1∈H1(ϵ)|(\boldmathh0+\boldmathh1)H\boldmathw|≤√Pt, (6)

where . By applying the triangle inequality and the fact that for (refer to Appendix D for details), the interference power can be transformed as follows:

 |(\boldmathh0+\boldmathh1)H% \boldmathw| ≤|\boldmathhH0\boldmathw|+|\boldmathhH1\boldmathw|≤|\boldmathhH0\boldmathw|+√ϵ∥\boldmathQ% \boldmathw∥, (7)

where with and being obtained by the eigenvalue decomposition of as . Moreover, since the arbitrary phase rotation of does not change the value of the objective function or the constraints, according to Remark 1 and Lemma 3, we can assume that , , and have the same phase, i.e.,

 Re{\boldmathwH\boldmathhs}≥0, Im{\boldmathwH\boldmathh0}=0, and Im{\boldmathwH\boldmathhs}=0. (8)

Hence, the interference constraint can be transformed into two second order cone inequalities as follows

 √ϵ∥\boldmathQ\boldmathw∥+\boldmathhH0\boldmathw≤√Pt, and √ϵ∥\boldmathQ% \boldmathw∥−\boldmathhH0\boldmathw≤√Pt. (9)

By combining (5), (9), with (8), problem is transformed into the standard SOCP problem as follows

 max\small\boldmathw \boldmathhHs\boldmathwsubject to:∥\boldmathw∥≤√¯P, Im{\boldmathwH\boldmathh0}=0, √ϵ∥\boldmathQ\boldmathw∥+\boldmathhH0\boldmathw≤√Pt, √ϵ∥\boldmathQ\boldmathw∥−\boldmath% hH0\boldmathw≤√Pt. (10)

Since the parameters and , and the variable in (10) have complex values, we first convert them to its corresponding real-valued form in order to simplify the solution. Define , , , , and

We then can rewrite the standard SOCP problem (10) as

 max~\small\boldmathw%  ~\boldmathhHs~\boldmathwsubject to: ∥~\boldmathw∥≤√¯P, ˇ\boldmathhH0~% \boldmathw=0,√ϵ∥~\boldmathQ~\boldmathw∥+~\boldmathhH0~\boldmathw≤√Pt, √ϵ∥~% \boldmathQ~\boldmathw∥−~\boldmathhH0~\boldmathw≤√Pt. (11)

Problem (11) can be solved by a standard interior point program SeDuMi [23], which has a polynomial complexity. In the next section, we develop an analytical algorithm to solve problem , which reduces the complexity of the interior point based algorithm substantially.

## V An Analytical Solution

In this section, we present a geometric approach to problem . We begin by studying a special case, the mean feedback case, i.e., . Due to its special geometric structure, the mean feedback case problem can be solved via a closed-form algorithm. We next show that problem can be transformed into an optimization problem similar to the mean feedback case. Based on the closed-form solution derived for the mean feedback case, the analytical solution to problem with a general form of a covariance matrix is presented in Subsection V-B.

### V-a Mean Feedback Case

Based on the observation in Lemma 1 and the definition of the mean feedback, the special case of problem with mean feedback can be written as follows.

 Mean feedback problem (P3): maxp≥0,∥\small\boldmathv∥=1log(1+p\boldmathhHs\boldmathv% \boldmathvH\boldmathhs)subject to:p≤¯P, p% \boldmathhH\boldmathv\boldmathvH% \boldmathh≤Pt, for ∥\boldmathh−% \boldmathh0∥2≤ϵσ2. (12)

Problem has two constraints, i.e., the transmit power constraint and the interference constraint. Similar to the idea in [13], the two-constraint problem is decoupled into two single-constraint subproblems:

 Subproblem 1 (SP1): maxp≥0,∥\small\boldmathv∥=1 log(1+p\boldmathhHs\boldmathv\boldmathvH\boldmathhs) (13) subject to: p≤¯P. (14) Subproblem 2 (SP2): maxp≥0,∥\small\boldmathv∥=1 log(1+p\boldmathhHs\boldmathv\boldmathvH\boldmathhs) (15) subject to: p\boldmathhH\boldmathv\boldmathvH\boldmathh≤Pt, for ∥\boldmathh−\boldmathh0∥2≤ϵσ2. (16)

In the sequel, we present the algorithm to obtain the optimal power and the optimal beamforming vector for both subproblems in subsection V-A1, and describe the relationship between the subproblems and problem in subsection V-A2.

#### V-A1 Solution to subproblems

For , the optimal power is constrained by the transmit power constraint, and thus . Moreover, since there does not exist any constraints on the beamforming direction, it is obvious that the optimal beamforming direction is equal to , i.e., . Thus, the optimal covariance matrix for is . In the following, we focus on the solution to .

has infinitely many interference constraints, and thus is an SIP problem too. By following a similar line of thinking as in Lemma 2, can be transformed into an equivalent problem that has finite constraints (refer to Appendix E for the proof) as follows.

###### Lemma 4

and the following optimization problem:

 maxp≥0,∥\small\boldmathv∥=1log(1+p% \boldmathhHs\boldmathv\boldmathvH% \boldmathhs), subject to: p\boldmathhHopt\boldmathv\boldmathvH\boldmathhopt≤Pt, (17)

where , have the same optimal solution.

According to Lemma 4, problem (17) has the same optimal solution as . Moreover, according to Lemma 3, the optimal solution of problem (17) lies in the plane spanned by and . We next apply a geometric approach to search the optimal solution, i.e., by restricting our search space to a 2-D space. As shown in Fig. 3, assume that the angle between and is , and the angle between and is . It is easy to observe that 333This follows because if , we can always replace by without affecting the final result, and the angle between and is less than .. Since lies in a 2-D space, can be uniquely identified by the angle . Hence, we need only to search for the optimal angle . By exploiting the relationship between , , and , the two-variable optimization problem (17) can be further transformed into an optimization problem with a single variable , which can be readily solved.

By observing Fig. 3, the angle between and is , and hence the objective function of (17) can be expressed as

 max∥\boldmathv∥=1log(1+p\boldmathhHs\boldmathv\boldmathvH\boldmathhs)=maxβlog(1+p∥\boldmathhs∥2cos2(β−α)). (18)

Clearly, the maximum rate is achieved if the following function

 f(β):=p∥\boldmathhs∥2cos2(β−α) (19)

is maximized.

Moreover, it can be proved by contradiction that the interference constraint is satisfied with equality, i.e., . Thus, we have

 p\boldmathhHopt\boldmathv\boldmathvH\boldmathhopt=p(\boldmathh0+√ϵσ\boldmathv)H\boldmathv\boldmathvH(\boldmathh0+√ϵσ%\boldmath$v$)=p(∥\boldmathh0∥cosβ+√ϵσ)2=Pt. (20)

Hence, the interference constraint is transformed into

 p=Pt(∥\boldmathh0∥cosβ+√ϵσ)2. (21)

By substituting (21) into (19), we have

 f(β)=p∥\boldmathhs∥2cos2(β−α)=∥\boldmathhs∥2Ptcos2(β−α)(∥% \boldmathh0∥cos(β)+√ϵσ)2. (22)

Thus, the optimal can be expressed as

 βopt=argmaxf(β)=argmax∥\boldmathhs∥2Ptcos2(β−α)(∥\boldmathh0∥cos(β)+√ϵσ)2. (23)

The problem of (23) is a single variable optimization problem. It is easy to observe that the feasible region for is . According to the sufficient and necessary condition for the optimal solution of an optimization problem, lies either on the border of the region ( or ) or on the point which satisfies . Since

 ∂f(β)∂β=2∥\boldmathhs∥2Ptcos(β−α)(sinα−sin(β−α)√ϵσ/∥\boldmathh0∥)∥\boldmathh0∥2(cosβ+√ϵσ/∥\boldmathh0∥)3, (24)

we can obtain a locally optimal solution by solving the equation . In the case when , is a non-decreasing function. Hence, the optimal is , and we define for this case. Therefore, the globally optimal solution is

 βopt=argmax(f(α),f(π/2),f(β1)). (25)

The optimal power can be further obtained by substituting into (21). According to the definition of and Lemma 3, we have

 \boldmathvopt=av^\boldmathh+bv^\boldmathh⊥, (26)

where and . In summary, can be solved by Algorithm 1 as described in Table I.

#### V-A2 Optimal solution to problem P3

In the preceding subsection, we presented the optimal solutions for the two subproblems. We now turn our attention to the relationship between problem and the subproblems, and present the complete algorithm to solve problem . Since the convex optimization problem has two constraints, the optimal solution can be classified into three cases depending on the activeness of the constraints: 1) only the transmit power constraint is active; 2) only the interference constraint is active; and 3) both constraints are active. Relying on this classification, the relationship between the solutions of problem and the two subproblems is described as follows (refer to Appendix F for the proof).

###### Lemma 5

If the optimal solution of satisfies the constraint of , then is the optimal solution of problem . If the optimal solution of satisfies the constraint of , then is the optimal solution of problem . Otherwise, the optimal solution of problem simultaneously satisfies the transmit power constraint and with equality.

###### Remark 5

To apply Lemma 5, we need to test whether and satisfy both constraints. The condition that satisfies the interference constraint is

 Pint≤Pt,where Pint=max\small% \boldmathh\boldmathhH\boldmathS1% \boldmathh, for ∥\boldmathh−\boldmathh0∥2≤ϵσ2, (27)

where can be obtained by the method discussed in Appendix D. The condition that satisfies the transmit power constraint is .

We next discuss the method for finding the solution in the case where neither nor is the optimal solution of problem . Similarly to the method in the preceding subsection, we solve this case from a geometric perspective. According to Lemma 5, in the case in which neither nor is the feasible solution, the optimal covariance must satisfy both constraints with equality, i.e.,

 popt=¯P, and popt% \boldmathhHopt\boldmathvopt% \boldmathvHopt\boldmathhopt=Pt. (28)

Combining these two equalities, we have

 ¯P(∥\boldmathh0∥cos(β)+√ϵσ)2=Pt. (29)

Thus,

 βopt=arccos(√Pt/¯P−√ϵσ∥\boldmathh0∥). (30)

Based on , we can obtain from (26). We summarize the procedure called Algorithm 2, which solves the case where both constraints are active for problem , in Table II. Furthermore, we are now ready to present the complete algorithm, namely Algorithm 3, to solve problem in Table III.

In Algorithm 3, we obtain the optimal solutions to and and the optimal solution to the case where both constraints are active separately. According to Lemma 5, the final solution obtained in Algorithm 3 is thus the optimal solution of problem .

###### Proposition 1

Algorithm 3 obtains the optimal solution of problem .

### V-B The Analytical Method for Problem P1

In the preceding subsection, the mean feedback problem is solved via a closed-form algorithm. Unlike problem , problem has a non-identity-matrix covariance feedback. To exploit the closed-form algorithm, we first transform problem into a problem with the mean feedback form as follows.

 Equivalent problem (P4): maxp,¯\small\boldmathvlog(1+p¯\boldmathhHs¯\boldmathv¯\boldmathvH¯\boldmathhs)subject to:p∥Δ1/2¯\boldmathv∥2≤¯P, p¯\boldmathhH¯\boldmathv¯\boldmathvH¯\boldmathh≤Pt, % for ∥¯\boldmathh−¯\boldmathh0∥2≤ϵ, (31)

where obtained by eigen-decomposing , , , , and . By substituting these definitions into (31), it can be observed that the achieved rates and constraints of both problem and are equivalent. Thus, the optimal solution of can be obtained by solving its equivalent problem . Moreover, the optimal beamforming vector of problem can be easily transformed into the optimal solution for problem by letting . Note that it is not necessary that in (31).

In the preceding subsection, decoupling the multiple constraint problem into several single constraint subproblems facilitates the analysis and simplifies the process of solving the problem. For problem , it can also be decoupled into two subproblems as follows.

 Subproblem 3 (SP3): maxp,¯\small\boldmathvlog(1+p¯\boldmathhHs¯\boldmathv¯\boldmath% vH¯\boldmathhs) (32) subject to: p∥Δ1/2¯% \boldmathv∥2≤¯P. (33) Subproblem 4 (SP4): maxp,¯\small\boldmathvlog(1+p¯\boldmathhHs¯\boldmathv¯\boldmath% vH¯\boldmathhs) (34) subject to: p¯\boldmathhH¯% \boldmathv¯\boldmathvH¯\boldmathh≤Pt for ∥¯\boldmathh−¯\boldmathh0∥2≤ϵ. (35)

It is easy to observe that is equivalent to , and the optimal transmit covariance matrix of can be obtained in the same way as that for . Moreover, is the same as , and thus it can be solved by Algorithm 1 discussed in Subsection V-A1.

The relationship between problem and subproblems and is similar to the one between and corresponding subproblems as depicted in Lemma 5, i.e., if either optimal solution of or satisfies both constraints, then it is the globally optimal solution; otherwise, the optimal solution satisfies both constraints with equalities. We hereafter need to consider only the case in which the solutions of both subproblems are not feasible for problem . For this case, the two equality constraints can be written as follows.

 ∥Δ1/2¯\boldmathv∥=1, % and max(¯\boldmathhH¯\boldmathv¯\boldmathvH¯\boldmathh)=Pt¯P, for ∥¯\boldmathh−¯% \boldmathh0∥2≤ϵ. (36)

Assume that the angle between and is , and that . Similar to Lemma 3, the optimal lies in a plane spanned by and , where , , and . Thus, if we can determine and from (36), then the optimal can be identified by

 ¯\boldmathv=¯p(cos(¯β)^¯% \boldmathh+sin(¯β)^¯\boldmathh⊥). (37)

Based on the new variables and , the constraints (36) can be transformed as follows.

 ¯p∥∥Δ1/2(cos(¯β)^¯\boldmathh+sin(¯β)^¯\boldmathh⊥)∥∥ =1, (38) and, ¯p(cos(¯β)∥¯% \boldmathh0∥+√ϵ) =√Pt¯P. (39)

According to (38), we have

 ¯p=1∥∥Δ1/2(cos(¯β)^¯\boldmathh+sin(¯β)^¯% \boldmathh⊥)∥∥. (40)

Substituting (40) into (39), we have

 (41)

Hence, the optimal can be obtained by solving (41), and can be obtained by substituting into (37). In summary, the procedure to solve the case in which both constraints are active is listed as Algorithm 4 in Table IV. Moreover, we are now ready to present the complete algorithm, namely Algorithm 5, for solving problem in Table V.

In Algorithm 5, we obtain the optimal solutions to and and the optimal solution to the case where both constraints are active separately. According to Lemma 5, the final result obtained in Algorithm 5 is thus the optimal solution of problem .

###### Proposition 2

Algorithm 5 achieves the optimal solution of problem .

###### Remark 6

The complexity of the interior point algorithm for the SOCP problem (11) is , where denotes the error tolerance. For Algorithm 5, a maximum of operations is needed to solve (41), and the complexity for each operation is . Hence, the computation complexity required for Algorithm 5 is , which is much less than that of the interior point algorithm.

## Vi Simulations

Computer simulations are provided in this section to evaluate the performance of the proposed algorithms. In the simulations, it is assumed that the entries of the channel vectors and are modeled as independent circularly symmetric complex Gaussian random variables with zero mean and unit variance. Moreover, we denote by the distance between the SU-Tx and the SU-Rx, and by the distance between the SU-Tx and the PU. It is assumed that the same path loss model is used to describe the transmissions from the SU-Tx to the SU-Rx and to the PU, and the path loss exponent is chosen to be . The noise power is chosen to be , and the transmit power and interference power are defined in dB relative to the noise power. For all cases, we choose dB.

### Vi-a Comparison of the Analytical Solution and the Solution Obtained by the SOCP Algorithm

In this simulation, we compare the two results obtained by a standard SOCP algorithm (SeDuMi) and Algorithm 3. We consider the system with , , and ranging from 3 dB to 10 dB. In Fig. 4, we can see that the results obtained by different algorithms coincide. This is because both algorithms determine the optimal solution. Compared with the SOCP algorithm solution, Algorithm 3 obtains the solution directly, and thus it has lower complexity. In Fig. 5, we compare the two results obtained by SeDuMi and Algorithm 5. We consider the system with , = 5 dB, and ranging from 1 to 10. The covariance matrix is generated by , where each element of follows Gaussian distribution with zero mean and unit variance. From Fig. 5, we can see that the results obtained by the two algorithms coincide again. Moreover, we note that the achievable rate with is always greater than or equal to the rate with , since a larger corresponds to the stricter constraints.

### Vi-B Effectiveness of the Interference Constraint

In this simulation, we apply Algorithm 3 to solve problem . In Fig. 6, we depict the achievable rate versus the ratio under different transmit power constraints. The increase of the ratio corresponds the decrease of the interference power constraint. As shown in Fig. 6, with an increase of , the achievable rate increases due to the lower interference constraint. Until the ratio reaches a certain value, the achievable rate remains unchanged, since the transmit power constraint dominates the result, and the interference constraint becomes inactive.

### Vi-C The Activeness of the Constraints

In this simulation, we compare the achieved rates of problem with a single transmit power constraint, a single interference constraint and both constraints. Here, we choose , , and generate in the same way as in the first simulation example. Fig. 7 plots three achievable rates for different constraints, respectively. It can be observed from Fig. 7 that the rate under two constraints is always less than or equal to the rate under a single constraint. Obviously, this is due to the fact that extra constraints reduce the degree of freedom of the transmitter.

## Vii Conclusions

In this paper, the robust cognitive beamforming design problem has been investigated, for the SU MISO communication system in which only partial CSI of the link from the SU-Tx to the PU is available at the SU-Tx. The problem can be formulated as an SIP optimization problem. Two approaches have been proposed to obtain the optimal solution of the problem; one approach is based on a standard interior point algorithm, while the other approach solves the problem analytically. Simulation examples have been used to present a comparison of the two approaches as well as to study the effectiveness and activeness of imposed constraints.

This work initiates research in robust design of cognitive radios. We are currently extending these methods to the more general case with multiple receive antennas and multiple PUs. Other interesting extensions include more practical scenarios, such as the case in which the SU channel information is also partially known at the SU-Tx.

### A Proof of Lemma 1

Problem involves infinitely many constraints. Denote the set of active constraints by , the cardinality of the set by , and the channel response related to the th element of the set by . According to the Karush-Kuhn-Tucker (KKT) conditions for , we have:

 \boldmathhs(1+\boldmathhHs% \boldmathS\boldmathhs)−1