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WIRELESS COMMUNICATIONS AND MOBILE COMPUTING Wirel. Commun. Mob. Comput. (2013) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/wcm.2401

RESEARCH ARTICLE

Optimal precoder design for non-regenerative multiple-input multiple-output cognitive relay systems with perfect and imperfect channel state information Quanzhong Li1* , Liping Luo2 and Jiayin Qin1 1 2

Sun Yat-Sen University, School of Information Science and Technology Guangxi University for Nationalities, College of Information Science and Engineering

ABSTRACT This paper studies optimal precoder design for non-regenerative multiple-input multiple-output (MIMO) cognitive relay systems, where the secondary user (SU) and relay station (RS) share the same spectrum with the primary user (PU). We aim to maximize the system capacity subject to the transmit power constraints at the SU transmitter (SU-Tx) and RS, and the interference power constraint at the PU. We jointly optimize precoders for the SU-Tx and RS with perfect and imperfect channel state information (CSI) between the SU-Tx/RS and PU, where our design approach is based on the alternate optimization method. With perfect CSI, we derive the optimal structures of the RS and SU-Tx precoding matrices and develop the gradient projection algorithm to find numerical solution of the RS precoder. Under imperfect CSI, we derive equivalent conditions for the interference power constraints and convert the robust SU-Tx precoder optimization into the form of semi-definite programming. For the robust RS precoder optimization, we relax the interference power constraint related with the RS precoder to be convex by using an upper bound and apply the gradient projection algorithm to deal with it. Simulation results demonstrate the effectiveness of the proposed schemes. Copyright © 2013 John Wiley & Sons, Ltd. KEYWORDS cognitive radio; non-regenerative relay; precoder; MIMO; perfect and imperfect CSI *Correspondence Quanzhong Li, School of Information Science and Technology, Sun Yat-sen University. E-mail: [email protected]

1. INTRODUCTION Cognitive radio (CR) [1,2] is regarded as a promising technology to alleviate the spectrum shortage problem and to improve the spectrum utilization. In CR networks, the secondary user (SU) is allowed to share the spectrum in two most common models [3,4]: overlay and underlay. In spectrum overlay, SUs are only allowed to access spectrum resources owned by the primary network provider if these resources are not being used by primary users (PU). Whereas in spectrum underlay, the PUs and SUs share the same spectrum and can transmit simultaneously, as long as the SU causes a tolerable level of interference to the primary receiver. With underlay spectrum sharing, higher spectrum utilization is anticipated if the interference to the PUs caused by the SUs can be properly controlled and managed [5]. Copyright © 2013 John Wiley & Sons, Ltd.

Recently, introducing cooperative relay into CR networks has received great attentions because of its capability to improve the overall end-to-end throughput of the SU significantly. For spectrum underlay, many works [6–8] focus on the regenerative relays for which a decode-andforward scheme is needed. Compared with regenerative relaying, non-regenerative relaying offers lower complexity and smaller processing delay because the relays just amplify and forward the received signals. The authors in [9–11] investigate non-regenerative cognitive relay systems with underlay spectrum sharing, assuming that all the nodes involved are equipped with a single antenna. However, in underlay spectrum sharing model, [12] shows that exploiting multi-antennas at the SU can effectively enhance the system capacity as well as control the interference at the PU under a certain level. In order to further increase the spectrum efficiency, our previous work

Precoder design for non-regenerative multiple-input multiple-output cognitive relay systems

[13] studied the non-regenerative multiple-input multipleoutput (MIMO) cognitive relay system, where the SU and relay station (RS) share the same spectrum with the PU, and all of them are equipped with multiple antennas. However, [13] only presented the optimal structure of the relay precoder with the assumption that the perfect channel state information (CSI) between the SU transmitter (SU-Tx)/RS and PU is available. In this paper, we consider the same system model as [13] and focus on jointly designing optimal precoders for both the SU-Tx and the RS to maximize the achievable rate subject to the transmit power constraints at the SUTx and RS, and the interference power constraint at the PU. We not only consider the case of the perfect CSI, but also the case of the imperfect CSI between the SU-Tx/RS and PU where the interference power constraints at the PU may be broken [14]. However, this violation cannot be tolerated in the CR systems; and thus, we need to consider the CSI imperfectness between the SU-Tx/RS and PU. Our approach to jointly design precoders for both SU-Tx and RS with perfect and imperfect CSI is based on the alternate optimization method, which has been widely used to design precoders for non-regenerative MIMO relay systems [15,16]. We summarize the main contributions of this paper as follows: With perfect CSI between the SU-Tx/RS and PU

available, we derive the optimal structure of the RS precoding matrix, which reduces the number of complex-valued design variables in the RS precoding optimization. We develop the gradient projection (GP) algorithm to find the numerical solution of the optimal RS precoder because its closed-form solution does not exist. When the RS precoder is fixed, we derive the optimal structure of the SU-Tx precoding matrix, which only involves four real-valued optimization variables and can be solved by the interiorpoint method with a fast computation speed. When perfect CSI between the SU-Tx/RS and PU is not available, we adopt the worst-case approach to model the imperfect CSI. Using S-Procedure as a tool, we derive equivalent constraints to the interference power constraints in the case of ellipsoidal channel matrices errors. These equivalent constraints are convex with respect to (w.r.t.) the SU-Tx precoder, thus the robust SU-Tx precoder optimization can be converted into the convex semi-definite programming (SDP). However, the equivalent constraints are non-convex w.r.t. the RS precoder. To make the robust RS precoder optimization tractable, we adopt an upper bound to relax the interference power constraint related with the RS precoder to be convex. Thus, the GP algorithm we have developed can be used to compute the robust RS precoder effectively. The rest of this paper is structured as follows. Section 2 introduces the system model for the non-regenerative MIMO cognitive relay systems and formulates the rate

Q. Li, L. Luo and J. Qin

Figure 1. The schematic model of a non-regenerative multipleinput multiple-output cognitive relay system.

maximization problem. In Section 3, we provide efficient algorithms to jointly design the optimal precoders for SU-Tx and RS with the perfect CSI. Section 4 presents robust precoder optimization when the CSI between SU-Tx/RS and PU is imperfect. Simulation results will be given in Section 5 and finally, we conclude this paper in Section 6. Notation: Boldface lowercase letters denote vectors, whereas boldface uppercase letters denote matrices. The notations ZT , Z , Z , and Z1 denote the transpose, Hermitian, conjugate and inverse of the matrix Z, and Tr.Z/, det.Z/, and kZk represent the trace, determinant, and Frobenius norm of the matrix Z, respectively. The notations A B and A B denote A B is a positive definite/semidefinite matrix. The operation vec.Z/ stacks the columns of the matrix Z into a single vector. The symbol ˝ represents Kronecker product. The notation A 2 C M 0 denotes A is an empty matrix.

2. SYSTEM MODEL The non-regenerative MIMO cognitive relay system is shown in Figure 1, where the SU-Tx, SU receiver (SU-Rx), RS, and PU are equipped with Nt , Nd , M , and L antennas, respectively. With underlay spectrum sharing, we assume that all the channels are flat fading over a common narrow band. We also assume that there is no direct link between the SU-Tx and the SU-Rx, and the use of relay is to establish the reliable communication link. The system operates in the time division duplex mode. During the first time slot, the received signal at the RS is† yR D H1 Wx C nR

(1)

where H1 2 C M Nt represents the channel matrix from the SU-Tx to the RS. The matrix W 2 C Nt Nt denotes the †

We adopt the assumption that the interference from the PU to SU is neglected as [9–13]. In IEEE 802:22 standard, the secondary wireless regional area network is located far away from the primary transmitter and hence the interference from the primary transmitter can be neglected at the secondary receiver. Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm



precoding matrix at the SU-Tx, and x 2 C Nt 1 is the transmitted symbol with E.xx / D I. The symbol nR 2 C M 1 is the noise assumed to be complex white Gaussian, i.e., nR CN .0; I/. The interference power at the PU is given by i D Tr G1 WW G1

(2)

where G1 2 C LNt is the channel from the SU-Tx to the PU. During the second time slot, the RS transmits yO R D FyR to the SU-Rx where F 2 C M M is the RS precoding matrix. The transmit power at the RS and the interference power at the PU are pR D Tr FH1 WW H1 F C FF

(3)

to solve Equation (7), which is based on the alternate optimization method. That is, in the following subsections, we will optimize the RS precoder F with fixed W and the SU precoding matrix W with fixed F, alternately until the convergence reaches. 3.1. Optimal relay station precoder with fixed secondary user transmitter precoding In this subsection, we focus on designing optimal precoding matrix F for the RS with fixed W. We assume that the number of antennas at the RS is no less than that at the SU-Tx. This assumption is reasonable because when spatial multiplexing is applied at the SU-Tx, it requires M Nt such that the compound channel can support Nt independent substreams [17]. Define the singular value decomposition (SVD):

and I D Tr G2 FH1 WW H1 F G2 C G2 FF G2

(4)

where G2 2 C LM denotes the channel matrix from the RS to the PU. The signal received at the SU-Rx is yD D H2 FH1 Wx C H2 FnR C nD

(5)

where H2 2 C Nd M is the channel matrix from the RS to the SU-Rx, and the noise nD is assumed to be CN .0; I/. The achievable capacity at the SU-Rx is 12 log2 det.X/ where [15] 1 (6) X D I C H2 FH1 WW H1 F H2 H2 FF H2 C I In this paper, our goal is to jointly design optimal precoding matrices for the SU and RS to maximize the achievable rate of the system subject to the transmit power constraints at the SU-Tx and RS and the interference power constraints at the PU. This optimization problem can be formulated as follows: max F;W

s:t:

1 log2 det X 2

(7a)

pR PR ; pS PS

(7b)

I ; i

(7c)

WW

. PS and PR are the transmit where pS D Tr power constraints at the SU-Tx and RS, respectively. is the interference power constraint at the PU.

3. OPTIMAL PRECODER DESIGN WITH RESPECT TO PERFECT CHANNEL STATE INFORMATION The problem (Equation 7) is quite complex and no global optimal solution exists because it is a highly non-convex problem. In this paper, we develop an efficient algorithm Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

H1 W D U1 †1 V1

H2 D U2 †2 V2

(8)

(9)

where U1 2 C M Nt , V1 2 C Nt Nt , U2 2 C Nd d , V2 2 C M d with d D min.M ; Nd /, and †1 ; †2 are Nt Nt , d d diagonal matrix, respectively. Then we have the following theorem. Theorem 1. The optimal RS precoding matrix F has the following structure: i h (10) F D V2 V? 2 PU1 M .M d / is such that where P i2 C M Nt and V? 2 2 C h V2 V? 2 is unitary.

The proof of Theorem 1 can be found in [13], which is omitted here for brevity. Under some special conditions, we can obtain a simpler relay precoder in the following two cases [13]: Case 1. If the RS operates in the low signal-to-noise ratio (SNR) regime, the optimal relay precoder F is the singular vector matching (SVM) precoder given by

F D V2 ƒU1

(11)

where ƒ is an Nt d diagonal matrix. Case 2. If D 0 and L < M , the optimal relay precoder F is the projected-channel SVD (P-SVD) precoder given by

O 2 ƒU O FDV 1

(12)

O 2 2 C M K comes from the SVDs: H2 I where V O 2† O and G2 D UG †G V , and O 2V VG2 VG D U 2 2 G2 2 2 O D ƒKK 0K.N K/ with K D min.d ; M L/. ƒ


Although the optimal RS precoder in Theorem 1 still contains matrix optimization variable P, the design dimensions have reduced from M 2 to MNt . Let us define F D VPU

(13)

i h where V D V2 V? 2 and U D U1 . Because there is no closed-form solution of the optimization variable P, we need to develop an effective algorithm to find its numerical solution. By replacing F with VPU , we can rewrite the problem (Equation 7) with fixed W as 1 R.P/ D log2 det X 2 s:t: pR PR ; I

min P

(14)

In the previous problem, we can prove that the constraint set P D fP j pR PR ; I g is convex (please refer to Appendix A.1). Therefore, Equation (14) can be viewed as non-linear optimization over a convex set and can be solved by exploiting the GP method [18]. To make the GP method work for our particular problem, we need to compute the gradient rP R.P/. We first derive the general derivative w.r.t. P as 1 T @ ln det YPZP YCI D ZP Y YPZP YCI Y @P (15) where the detailed derivations can be found in Appendix A.2. From (15), the gradient rP R.P/ is given by i h N PP H N 2 .H N 2 C I/1 H N rP R.P/ D log2 e P H 2 2 i h N PWP O H N 2 .H N 2 C I/1 H N O H log2 e WP 2

2

(16)

N 2 D V H . O D U H1 WW H U C I and H where W 1 2 O In addition, we need to project P D P rP R.P/ on the constraint set P, that is to find PN 2 P to minimize O PN P/ O /. Obviously, this projected operation is Tr..PN P/. a simple convex optimization and can be solved using the interior-point method [19]. Now, we summarize the GP algorithm as follows: (1) Initialize P0 by a feasible matrix and set k D 0; (2) Compute the gradient rPk R.Pk / according to (16); (3) Find the projection PN k using the method previously described; (4) Update PkC1 D Pk C ˛k .PN k Pk /; (5) If jjrPk R.Pk /jj or kPkC1 Pk k < , stop; otherwise, set k D k C 1 and go to step 2. In the previously discussed algorithm, we adopt Armijo rule along the feasible direction [18] to determine the step size ˛k , which provides provable convergence. Let


˛k D ˇ mk , where ˇ mk is the first non-negative integer that satisfies R.Pk / R.PkC1 / ˇ m Tr rPk R.Pk / PN k Pk (17) Here, 2 .0; 1/, ˇ 2 .0; 1/ are fixed scalars and we choose D 0:1, ˇ D 0:5 in the simulation settings. Remark (complexity of GP). The number of iterations required in the gradient-related algorithm is known to be at most O. 2 / when the iterations are set to stop under the condition that the magnitude of the gradient is less or equal to [20]. So the complexity of the proposed GP algorithm is conjectured to be O. 2 / if the stopping criterion is set to be jjrPk R.Pk /jj . 3.2. Optimal secondary user transmitter precoder with fixed relay station precoding In this subsection, we consider the optimal precoder design for the SU-Tx with the fixed RS precoder F. We first rewrite the problem (Equation 7) w.r.t. the optimization variable W as 1 min R.W/ D log2 det I C W ˆW 2 W s:t: Tr.W ‰i W/ PNi ; i D 1; 2; 3; 4

(18)

1 H2 FH1 , ‰1 D where ˆ D H1 F H2 H2 FF H2 C I

I, ‰2 D H1 F FH1 , ‰3 D G1 G1 , ‰4 D H1 F G2 G2 FH1 , PN1 D PS , PN2 D PR Tr FF , PN3 D , and PN4 D Tr G2 FF G2 . Then the following Theorem establishes the structure of the optimal SU-Tx precoder W. Theorem 2. The optimal SU-Tx precoding matrix W in Equation (18) has the following structure: 1 W D Tw V1 D

(19)

P where Tw Tw D 4iD1 i ‰i , the eigenvalue decomposi 1 tion (EVD): T1 D Vw ƒw Vw , D is an N1 N1 w ˆ.Tw / diagonal matrix, V1 contains N1 columns of Vw associated with eigenvalues being greater than one, and fi 0g4iD1 are the solution to the following Lagrange dual problem of Equation (18): max

fi 0g4 i D1

ln det I C .ƒw I/C 4 X C C Tr .I ƒ1 / i PNi w

(20)

iD1

The proof of Theorem 2 can be found in Appendix A.3. Because the Lagrange dual problem is convex [19], the problem (Equation 20), which only has four real valued Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm



optimization variables that can be solved by the interiorpoint method with a fast computation speed. The number of iterations required in the interior-point method is proportional to log.0 =/, which is the log of the ratio of the initial duality gap 0 to the final duality gap [19]. Until now, we have proved that in order to jointly maximize the system capacity, both the optimal F and W have a general precoding structure as given in Equations (10) and (19), respectively. Exploiting these optimal structures, the problem (Equation 7) can be solved by an alternate iterative algorithm. This algorithm is first initialized at W0 , which is a feasible solution to Equation (18). Then F is updated according to Equation (10) with a fixed W, and W is updated according to Equation (19) with a given F. The updating of F and W is operated in an alternating fashion. The monotonic convergence of F and W will be obtained because the updates of F and W may either increase or maintain but cannot decrease the objective value of Equation (7).

4. ROBUST PRECODER OPTIMIZATION As described in the preceding text, with perfect CSI between the SU-Tx/RS and PU, we can always control the interference power at the PU under a certain level. In practice, however, the CSI between the SU-Tx/RS and PU is hard to be known perfectly because of the loose or usually no cooperation between the SU-Tx/RS and PU. Consequently, although designated to satisfy the interference power constraints, the SU-Tx/RS may still break these limitations because of imperfect CSI between the SU-Tx/RS and PU. This violation of the interference power constraints cannot be tolerated in the CR systems; and therefore, the CSI imperfectness between the SU-Tx/RS and PU has to be taken into account in the system design. 4.1. The equivalent interference constraint set To characterize the CSI imperfectness between the SUTx/RS and PU, we adopt the following common imperfect CSI model [14,21]. That is, the actual channel is assumed to be within the neighborhood of a nominal channel, whereas the nominal channel could be the estimated or feedback channel. Specifically, the SU-Tx/RS is assumed Q i of the actual channel Gi , to know a noisy version G i D 1; 2, given by Q i D Gi Gi G

(21)

where Gi represents the CSI uncertainty, which is bounded by the elliptical region o n (22) Gi D Gi jTr Gi Ti Gi i2 where Ti 0 is a given matrix that determines the orientation of the region, and the parameter i2 controls the size of Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

the region. As the interference limits must be strictly met for any channel (error) in the uncertainty region, the SUTx/RS should satisfy the robust interference constraints in the following, instead of the constraints in (7c), 8 ˆ Q 1 C G1 /Q.G Q 1 C G1 / ; Tr . G ˆ ˆ < Q 2 C G2 /‚.G Q 2 C G2 / ; (23) Tr .G ˆ ˆ ˆ : 8Gi 2 Gi ; i D 1; 2 where Q D WW , which is defined as the transmit covari ance matrix at the SU-Tx, and ‚ D FH1 QH1 F C FF . We note that ‚ is a linear function of Q for a given F, while it is a quadratic function of F for a given Q. Using S-procedure [22] as a tool, the constraint set in Equation (23) can be rewritten in a more convenient form as given in the following theorem. Theorem 3. Given fTi 0g2iD1 and fi2 > 0g2iD1 , the constraint set in Equation (23) is equivalent to 9 fi 0g2iD1 ; 2 . T Q/T ˝ I 6 1 1 4 Q 1Q vec G 2 6 4

.2 T2 ‚/T ˝ I Q 2‚ vec G

3 Q 1Q vec G 7 5 0; 2 Q Q Tr G1 QG1 1 1

(24a) 3

Q 2‚ vec G 7 5 0: 2 Q 2 Q 2 ‚G Tr G 2

2

(24b) The proof of Theorem 3 can be found in Appendix A.4. We note that the constraint set (Equation 24) is convex w.r.t Q when F is fixed because the constraints (Equation 24a) and (Equation 24b) are linear matrix inequalities [19] in Q. However, when Q is fixed, Equation (24) is non-convex w.r.t F, because the left hand size of the constraint (24b) is a quadratic matrix function of F. 4.2. Robust precoder design Using the robust interference constraint set in Equation (24), the robust precoder optimization problem can be expressed as max

F;Q;1 ;2

s:t:

1 1 log2 det I C H2 FH1 QH1 F H2 H2 FF H2 C I 2 Tr FH1 QH1 F C FF PR ; Tr.Q/ PS ; Q 0; " # Q 1 Q/ .1 T1 Q/T ˝ I vec. G Q 1 QG Q 1 Q/ Q 1 2 Tr G vec .G 1

0; 91 0 " .2 T2 ‚/T ˝ I Q 2 ‚/ vec .G 0; 92 0

1

Q 2 ‚/ vec. G Q 2 ‚G Q 2 2 Tr G 2

#

2

(25)


The problem (Equation 25) is also a non-convex problem and no global optimal solution exists. Our approach to solve Equation (25) is still based on the alternate optimization method as described in Section 3. When the RS precoder F is fixed, from Theorem 3, we can easily see that Equation (25) is a convex problem w.r.t. Q and fi g2iD1 , and can be solved using semi-definite programming [23], where the associated complexity is O Nt6:5 per accuracy digit [23]. However, when Q and fi g2iD1 are given, the GP algorithm can not be applied to find the RS precoder F directly because the interference constraint (Equation 24b) is non-convex w.r.t. F. In order to make the robust RS precoder optimization tractable, we need to do some relaxation. We rewrite the second interference constraint in Equation (23) as

Q Q Tr G2 C G2 ‚ G2 C G2 Q G Q D Tr .G 2 2 C E/‚ ; 8G2 2 G2

Q Q G2 C G G kEk D kG 2 2 2 C G2 G2 k

Q G2 k C kG G Q kG 2 2 2 k C kG2 G2 k Q 2 kkG2 k C kG2 k2 2kG

(27)

Because Tr.G2 T2 G2 / 22 , we have kG2 k p 2 max

(31) Comparing Equation (31) with Equation (14), we can see that the structure of the optimal RS precoder proposed in Equation (10) is also optimal to Equation (31), and the GP algorithm developed in Section 3.1 can be also used to solve Equation (31) with just slight modification about the projected operation.

In this section, we investigate the performances of the proposed schemes. In the simulations, we assume that the entries of the channel matrices fHi g2iD1 and fGi g2iD1 are independent circularly symmetric complex Gaussian random variables with zero mean and unit variance. For the sake of simplicity, we assume that the distances between the SU-Tx/RS and the PU are equal and denoted by l1 . We also assume that the RS is located between the SUTx and SU-Rx with the same distance denoted by l2 . The same path loss model is used to describe all the channels and the path loss exponent is chosen to be 3. We define SNRS D PS =Nt ; SNRR D PR =M , and normalize l2 D 1.

where max is the maximum eigenvalue of T2 .

Thus, using Equation (27), it is possible to choose D 2 Q 2 k C 2 . Then, we can rewrite Equation (26) 2 p2 kG as

1 log2 det I C H2 FH1 QH1 F H2 2 F 1 H2 FF H2 C I s:t: Tr FH1 QH1 F C FF PR ; Q Q G FH1 QH1 F C FF Tr G 2 2 C I

min

5. SIMULATION RESULTS (26)

Q G2 C G G Q where E D G 2 2 2 C G2 G2 is a normbounded matrix kEk and has the following relation:


max

max

max Tr

kEk

Q Q G G 2 2CE ‚

(28)

We adopt an upper bound proposed in [24], that is max Tr

kEk

5.1. Convergence performance of the proposed alternate optimization method

Q 2 C E ‚ Tr G Q 2 C I ‚ Q G Q G G 2

2

(29) Now, we obtain a relaxed version of the interference constraint (Equation 26) as Tr

Q Q G G 2 2 C I ‚

(30)

Following the steps in Appendix A.1, it is not hard to verify that the interference constraint (Equation 30) is convex w.r.t F for a given Q. Using Equation (30), the robust RS precoder optimization with a fixed Q can be formulated as follows:

In this subsection, we investigate the convergence performance of the proposed alternate optimization method under perfect and imperfect CSI. In Figure 2, we plot the average rate as the function of the alternate iteration number for perfect CSI ( D 0) and imperfect CSI ( D 0:05 and D 0:1). We observe in Figure 2 that the average rate converges to a limit after no more than 15 alternate iterations both for perfect and imperfect CSI. We also see that the convergence rate for perfect CSI is slightly faster than that for imperfect CSI. This can be explained by the fact that the average rate grows faster when the CSI has higher quality. 5.2. Performance of the proposed precoding scheme with perfect channel state information We first investigate the performances of average rate by different precoding schemes. The scheme of “No precoding" means that the SU-Tx and RS precoders are W D ˛I and F D ˇI, respectively, where ˛ and ˇ satisfy the transmit Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm



Average capacity (bps/Hz)

5

4.5

4

3.5

3

2.5 0

Perfect CSI ξ=0 Imperfect CSI ξ=0.05 Imperfect CSI ξ=0.1

5

10

15

20

25

Number of alternate iterations

and P-SVD (Equation 12) RS precoders in the whole SNR regime. In the low SNR regime, the rate by the OPT and SVM are close and better than that by the P-SVD, because the rate is mainly limited by the transmit power constraint. As SNR increases, the gap between the rate by the OPT and that by the SVM becomes larger. In the high SNR regime, the P-SVD outperforms the SVM, because the rate is mainly limited by the inter-process communication and the P-SVD can eliminate the interference more efficiently than the SVM. In Figure 3, we also observe some performance floors when using the schemes “No precoding” “SU-Tx only” and “RS only by (11)”. This is because these schemes cannot eliminate the interference from the RS to the PU. When the SNR goes to high values, the average rate by these schemes remains unchanged, because it is limited by the interference constraints. Similar results are also

Figure 2. Average rate versus number of alternate iterations. M D 6, Nt D Nd D 4, L D 2, SNR D SNRS D SNRR D 10dB, D 10dB, l1 D 2.

5 4.5

Average Rate(bit/s/Hz)

5 4.5

Joint precoding RS only by (10) RS only by (12) RS only by (11) SU−Tx only No precoding


5.5

4 3.5 3 2.5

3.5 3 2.5 2 1.5 1

2

0.5 −5

1.5 1 −5

4

Joint precoding RS only by (10) RS only by (12) RS only by (11) SU−Tx only No precoding

0

5

10

15

20

SNR(dB) 0

5

10

15

20

SNR(dB) Figure 3. Average rate by different precoding schemes versus SNR. M D 6, Nt D Nd D 4, L D 2, SNR D SNRS D SNRR , D 10dB, l1 D 2.

Figure 4. Average rate by different precoding schemes versus SNR. M D 6, Nt D 3, Nd D 4, L D 2, SNR D SNRS D SNRR , D 10dB, l1 D 2.

power constraints at the SU-Tx and RS and the interference power constraints at the PU. The scheme of “SU-Tx only" denotes that the SU-Tx precoder is optimized using Equation (19), but the RS precoder is chosen as in “No precoding". Similarly, the scheme of “RS only by (10)" denotes that the RS precoder is optimized using Equation (10), but the SU-Tx precoder is chosen as in “No precoding". The scheme of “Joint precoding" denotes that the SUTx and RS precoders are optimized jointly using Equations (19) and (10). Figure 3 shows the average rate by various precoding schemes versus SNR with M D 6, Nt D Nd D 4, and L D 2. From Figure 3, we can see that the scheme of “Joint precoding" has obtained substantial rate improvements over other precoding schemes from low to high SNR regime. Figure 3 also shows that the optimal (OPT) RS precoder (Equation 10) outperforms the SVM (Equation 11) Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm


3

2.5

Joint precoding RS only by (10) RS only by (11) SU−Tx only No precoding

2

1.5

1

0.5 −5

0

5

10

15

20

SNR(dB) Figure 5. Average rate by different precoding schemes versus SNR. M D 4, Nt D 3, Nd D 4, L D 8, SNR D SNRS D SNRR , D 5dB, l1 D 2.


shown in Figure 4 with M D 6, Nt D 3, Nd D 4, and L D 2. In Figure 5, we plot the average rate with M D 4, Nt D 3, Nd D 4, and L D 8. We can see that the average rate by all the schemes will not increase when the SNR reaches a certain value, which is mainly because all the schemes can not eliminate the interference to the PU when L is larger than Nt , Nd , and M . In Figure 6, we depict the average rate by the scheme of “Joint precoding” versus the distance l1 under different transmit power constraints. An increase of the distance l1 leads to a decrease of the interference power at the PU. As shown in Figure 6, with an increase of l1 , the average rate increases due to the lower interference power. Until the distance l1 reaches a certain value, the average rate remains unchanged, because the transmit power constraint dominates the average rate, and the interference power constraint becomes inactive. In Figure 7, we investigate the effect of the number of antennas at the RS and SU-Tx on the average rate by

the scheme of “Joint precoding”. We define k D M =N and plot the average rate versus k. We see that both N and M have a significant effect on the average rate. The effect of M becomes more significant as N becomes larger. This property makes Theorem 1 important in decreasing the number of optimization variables for reducing search complexity, especially when M is much large than N . 5.3. Comparison of the robust and non-robust precoding Following the worst-case robustness philosophy, we compare the robust and the non-robust precoding schemes through their worst-case performance, i.e., the worst-case interference power generated at the PU by the worst

5.5 5


9 8

Average Rate(bps/Hz)

10 SNR=10dB SNR=5dB SNR=0dB

7 6 5

non−robust robust

4.5 4 3.5 3

4 2.5 3 2 −5

2

0

0 1

1.5

2

2.5

3

3.5

4

4.5

5

9 N=4 N=3 N=2

7 6 5 4 3 2 1

1

10

15

20

1.5

Figure 8. Average rate by the robust and non-robust precoding schemes. SNR D SNRS D SNRR , M D 6, Nt D Nd D 4, L D 2, D 10dB, l1 D 2.

Average Worst−case Interference Power(dB)

Figure 6. Average rate versus l1 . M D 6, Nt D Nd D 4, L D 2, SNR D SNRS D SNRR , D 10dB.

8

5

SNR(dB)

1



2

2.5

3

3.5

4

4.5

5

Figure 7. Average rate versus k D M =N . N D Nt D Nd ; L D 2, SNRS D SNRR D 10dB, D 10dB, l1 D 2.

−4 −5

non−robust robust

−6 −7 −8 −9 −10

Interference constraint

−11 −12 −13 −5

0

5

10

15

20

SNR(dB) Figure 9. Average worst-case interference power by the robust and non-robust schemes. SNR D SNRS D SNRR , M D 6, Nt D Nd D 4, L D 2, D 10dB, l1 D 2.

Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm



channel error. Here, the non-robust precoding denotes the Q i g2 scheme of “Joint precoding” using the estimated fG iD1 as the perfect CSI. For a specific precoding scheme, the worst channel error is approximately obtained by choosing the one raising the maximum interference power among 1000 randomly generated errors on the boundary of the uncertainty region. In Figures 8 and 9, we show the average rate and the average worst-case interference power by different precoding schemes, respectively. The average Q i g2 and fHi g2 , whose elements are is taken over fG iD1 iD1 Gaussian random variables with zero mean and unit variance. For the sake of simplicity, we consider the same spherical channel uncertainty regions (Ti D I; i D ; 8i ). From Figure 8, we see that the robust precoding scheme has rate loss as compared with the non-robust precoding. As the channel uncertainty region (i.e., ) enlarges, the rate loss becomes larger. Whereas in Figure 9, we can observe that the non-robust precoding scheme may generate the interference power at the PU dramatically higher than the given threshold even for a small amount of uncertainty. However, the PU’s communications can be efficaciously protected by using the robust precoding scheme that always satisfies the interference power constraints at the PU.

6. CONCLUSIONS In this paper, we have studied optimal precoder design for non-regenerative MIMO cognitive relay systems with the objective of maximizing the capacity of the secondary user. Both perfect and imperfect CSI between the SU-Tx/RS and PU are considered in our design. We have transformed the design of the SU-Tx precoder to convex programming and have developed the GP algorithm to compute the RS precoder. We alternately compute the SU-Tx precoder and RS precoder until convergence. Monte Carlo simulations have confirmed the effectiveness of our proposed schemes. Before closing this paper, we should highlight the contributions on the basic of [13] for four points: (i) under perfect CSI, we develop the GP algorithm to compute the numerical value of the RS precoder based on the structure given in [13]; (ii) under perfect CSI, we derive the structure of the SU-Tx precoder and compute its numerical value effectively using convex programming; (iii) under imperfect CSI, we transform the design of the SU-Tx precoder to convex semi-definite programming through S-Procedure; and (iv) under imperfect CSI, we transform the design of the RS precoder to the problem with convex constraints and use the GP algorithm to compute its numerical value.

N G ; Tr G2 F2 WF N G Tr G2 F1 WF 1 2 2 2 (A.1) Define FN D ˇ1 F1 C ˇ2 F2 where ˇ1 ; ˇ2 0 and ˇ1 C ˇ2 D 1. Then N .ˇ1 F1 C ˇ2 F2 / G Tr G2 .ˇ1 F1 C ˇ2 F2 / W 2 N G C ˇ 2 G2 F2 WF N G D Tr ˇ12 G2 F1 WF 2 1 2 2 2 N G C G2 F2 WF N G C ˇ1 ˇ2 Tr G2 F1 WF 2 2 1 2 (A.2) N 12 and B , G2 F2 W N 12 , using the fact that LetA , G2 F1 W Tr AB C BA Tr AA C BB [25], we obtain N G C G2 F2 WF N G Tr G2 F1 WF 2 2 1 2 N G C G2 F2 WF N G Tr G2 F1 WF 1

2

2

2

From (A.1) to (A.3), we have N FN G ˇ 2 C ˇ 2 C 2ˇ1 ˇ2 D (A.4) Tr G2 FN W 1 2 2 N FN / PR . Therefore the Similarly, we have Tr.FN W constraint set w.r.t. F is convex, and it is also convex w.r.t P because F and P is linear relation. A.2. Derivation of Equation (15) To derive the derivative w.r.t. P, we need the properties: d ln det.X/ D Tr.X1 dX/, d.XY/ D dXY C XdY [26]. Then we have d ln det YPZP Y C I

1 d YPZP Y C I DTr YPZP Y C I

1 YdPZP Y C YPZdP Y DTr YPZP Y C I

1 YdP DTr ZP Y YPZP Y C I

1 (A.5) YPZdP C Tr Y YPZP Y C I

APPENDIX A A.1. Proof of the convexity of the constraint set in Equation (14) N and N D H1 WW H C I, then pR D Tr FWF Let W 1 N G . We assume F1 ; F2 2 C M M I D Tr G2 FWF 2 such that Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

(A.3)

Using the relations [26]: df .X/ D Tr.AdX/ , AT

and

@f .X/ @X

@f .X/ @X

D

D 0, we have

1 T @ ln det YPZP YCI D ZP Y YPZP Y CI Y @P (A.6)


A.3. Proof of Theorem 2


From Equation (A.11), Equation (A.10) is equivalent to

The Lagrangian function associated with the problem (Equation 18) is given by

1 1 1 1 2 2 D ƒ1 I C ƒ1 DD ƒ1 ƒ12 D D 0

1 2

1

4 X L D ln det I C W ˆW C i Tr W ‰i W PNi iD1

(A.7) where, w.l.o.g, log2 has been replaced by ln. Let the derivative of L w.r.t. W be zero, we can obtain 1 2

1 @L D I C W ˆW W ˆ C W Tw Tw D 0 @W (A.8) P where Tw Tw D 4iD1 i ‰i . We claim that the matrix T! is non-singular, which is based on the following facts: first, i cannot be all zero. We note that ˆ is positivedefinite. If i are all zero, then from Equation (A.8), we must have W D 0. However, W D 0 is not the optimal solution, by contradiction, i cannot be all zero. Second, if Nt L, then ‰i are all positive-definite matrices, which leads to that the matrix T! is non-singular. If Nt > L, then the SU-Tx must use the maximum power to launch because it can eliminate the interference to the PU using its nullspace. That is, the power constraint Tr.W ‰1 W/ PN1 is and the Lagrange multiplier 1 > 0. Therefore, Pactive 4 iD1 i ‰i is positive-definite and T! is non-singular. Because Tw is non-singular, Equation(A.8) can be equivalently written as 1

1 1 ˆ T T W W Tw T1 ICW Tw T1 w w w w ˆ Tw C W Tw D 0

(A.9)

1 D Vw ƒw Vw D Let the EVD: T1 w ˆ.Tw / ŒV1 V2 diag.ƒ1 ; ƒ2 /ŒV1 V2 where all the diagonal elements of ƒ1 are larger than one, and all the diagonal elements of are ƒ2 less than or equal to one. Applying the EVD into Equation (A.9) and assuming Tw W D V1 D where D is non-singular, we have

.I C D ƒ1 D/1 D ƒ1 V1 C D V1 D 0

(A.10)

Applying the matrix identity .I C XX /1 X D X.I C X X/1 [25], we obtain

1 1 1 1 1 1 I C D ƒ1 D D ƒ12 D D ƒ12 I C ƒ12 DD ƒ12 (A.11)

, ƒ12

1

1

I C ƒ12 DD ƒ12

1

1

ƒ12 I D 0

1 1 1 D ƒ1 , I C ƒ12 DD ƒ12 1 , DD D I ƒ1 1

(A.12)

where the first equivalence relation comes from that the matrix D is non-singular. From (A.12), we have D D 12 X where X satisfies XX D I. Then we I ƒ1 1 1 12 V1 I ƒ1 X. Note that can obtain W D Tw 1 1 2 C 12 V1 D Vw I ƒ1 Vw where V1 I ƒ1 w 1 .x/C D max.x; 0/. Applying W back to (A.7) and employing the identity det.I C AB/ D det.I C BA/ and Tr.AB/ D Tr.BA/, we have the Lagrange dual problem of Equation (18) as max ln det I C .ƒw I/C fi 0g4 i D1

C Tr

I ƒ1 w

C

4 X

i PNi

(A.13)

iD1

We observe that the problem (Equation A.13) is independent of the matrix X. Hence, without loss of generality, we 12 . Finally, we obtain can take X D I, i.e., D D I ƒ1 1 1 12 1 V1 I ƒ1 . W D Tw A.4. Proof of Theorem 3 To prove Theorem 3, we need the S-procedure for complex case [22] as follows: S-procedure [22]: Given Hermitian matrices Ai 2 C nn , vectors bi 2 C n1 , and numbers ci 2 R for i D 1; 2. Define the functions fi .x/ D x Ai xC2Refbi xgCci . n1 If there exists a vector x 2 C such that f1 .x/ > 0, then the following two conditions are equivalent: (1) f1 .x/ > 0 for every x 2 C n1 such that f2 .x/ 0; (2) there exists 0 such that # " # " A2 b2 A1 b1 0 (A.14) b1 c1 b2 c2 Proof . The first constraint in Equation (23) can be rewritten as 8 < Tr G1 AG C G1 B C BG C C 1 1 : 8G1 W Tr G1 T1 G1 12 (A.15) Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm



Q 1 Q, C D G Q 1 QG Q . Using the where A D Q, B D G 1 T T T identity Tr.WXYZ/ D vec .Z /.Y ˝ W/vec.X/ [25], Equation (A.15) is equivalent to n o 8 < g .AT ˝ I/g 2Re vec .B/g Tr.C/ C 0 : 2 8g W g TT 1 ˝ I g C 1 0 (A.16) where g D vec.G1 /. Applying the S-Procedure in the preceding text, one can see that Equation (A.16) holds if and only if there exists 1 0 such that "

.1 T1 A/T ˝ I

vec.B/

vec .B/

N 1 12

7.

8.

# 0

(A.17) 9.

where N D Tr.C/ and the condition f1 .x/ > 0 in the S-Procedure is readily satisfied becauase 12 > 0. Similarly, the second constraint in Equation (23) is equal to "

6.

.2 T2 ‚/T ˝ I

Q 2 ‚/ vec.G

Q 2 ‚/ vec .G

O 2 22

10.

# 0

(A.18) 11.

Q 2 ‚G Q /. Combining Equations (A.17) where O D Tr.G 2 and (A.18), we last obtain (24). 12.

ACKNOWLEDGEMENTS This work was supported by the Scientific and Technological Project of Guangzhou City (No. 12C42051578 and 11A11060133), the National Natural Science Foundation of China (No. 61173148 and 61202498), and Guangxi Natural Science Foundation (No. 2012GXNSFBA053162).

13.

14.

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Liping Luo was born in Hubei province, China in 1980. She received her PhD degree in Sun Yat-sen University, China in 2011. She is now a lecturer in Guangxi University for Nationalities, China. Her research interests focus on the signal processing in wireless communication systems and the performance analysis of wireless communication systems.

Jiayin Qin received his MS degree in Radio Physics from Huazhong Normal University, China, in 1992 and PhD degree in Electronics from Sun Yat-Sen University, Guangzhou, China, in 1997. Since 1999, he has been a professor in the Department of Electronic and Communication Engineering. His research areas include wireless communication and submillimeter wave technology.

AUTHORS’ BIOGRAPHIES Quanzhong Li received his BS degree from Sun Yat-sen University, Guangzhou, China, in 2009, and from then on, he is working toward a PhD degree. His research interests are in wireless digital communications and wireless networks with emphasis on MIMO communications, cooperation network, and cognitive radio.

Wirel. Commun. Mob. Comput. (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/wcm