IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 5, MAY 2013
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Joint Beamforming and Antenna Subarray Formation for MIMO Cognitive Radios Xinpeng Zeng, Quanzhong Li, Qi Zhang, and Jiayin Qin
Abstract—The antenna subarray formation (ASF) is a promising technique for multiple-input multiple-output (MIMO) receiver. For MIMO cognitive radio systems, we propose a joint beamforming and ASF scheme in this letter which maximizes the cognitive achievable capacity subject to the peak transmit power constraint at the secondary transmitter, peak interference power constraint at the primary receiver, and the limited number of nonzero elements in the ASF matrix. To solve the joint optimization problem, we propose a relax-and-recover scheme. Simulation results have shown that the proposed scheme outperforms the conventional antenna selection scheme. Index Terms—Antenna selection (AS), antenna subarray formation (ASF), beamforming, cognitive radio, multiple-input multipleoutput (MIMO).
I. INTRODUCTION
F
OR cognitive radios (CR), the multiple-input multiple-output (MIMO) CR system is capable to improve the overall end-to-end throughput [1]. However, MIMO technique significantly increases the hardware cost because each antenna requires a radio frequency (RF) chain which is expensive. One way to reduce the number of required RF chains in MIMO system is to employ antenna selection (AS) scheme [2], [3]. The key idea of AS is to select the most advantageous antennas to transmit or receive signals given the number of available RF chains. The AS scheme exploits only partial array gain while the MIMO system with full complexity exploits the full array gain. To exploit more array gain at the MIMO receiver, the AS scheme is further developed into an antenna subarray formation (ASF) scheme [4], [5]. Unlike the AS scheme where each RF chain is allocated to a single antenna, the ASF scheme allocates each RF chain to a subarray of antennas. For the ASF scheme, each antenna output in a subarray is complexly weighted using a variable gain low noise amplifier (vg-LNA) and a programmable phase shifter. The complex weighted antenna outputs in a subarray are constructively combined to exploit the array gain. Compared to the MIMO system with full complexity, the ASF scheme achieves the decreased
Manuscript received December 20, 2012; revised February 28, 2013; accepted March 08, 2013. Date of publication March 14, 2013; date of current version March 21, 2013. This work was supported by the National Natural Science Foundation of China under Grants 61173148 and 61202498, and by the Scientific and Technological Project of Guangzhou City under Grants 12C42051578 and 11A11060133. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Min Dong. The authors are with the Department of Electronics and Communications Engineering, Sun Yat-Sen University, Guangzhou 510006, China (e-mail:
[email protected];
[email protected];
[email protected];
[email protected]). Digital Object Identifier 10.1109/LSP.2013.2252897
Fig. 1. The MIMO CR system model.
receiver hardware complexity, since less downconverters and analog-to-digital converters (ADCs) are needed [4], [5]. In this letter, we propose a joint beamforming and ASF scheme at the MIMO CR secondary receiver (SU-RX). We adopt a relaxed-structured ASF (RS-ASF) technique as in [6]. Furthermore, we propose to jointly design the beamforming at the secondary transmitter (SU-TX) to satisfy the peak transmit power (PTP) constraint at SU-TX and peak interference power (PIP) constraint at the primary receiver (PU-RX). The interference from primary transmitter (PU-TX) to SU-RX is also considered. To the best of our knowledge, such problem has not been studied in the literature. means The following notations are used in this letter. that is a positive definite matrix and means that is a denotes the determinant. positive semi-definite matrix. denotes its conjugate transpose For any general matrix , denotes its rank. denotes statistical expectaand rank tion. denotes the space of matrices with complex endenotes the distritries. denotes the identity matrix. bution of a circularly symmetric complex Gaussian vector with the mean vector and the covariance matrix . II. SYSTEM MODEL The MIMO CR system model, shown in Fig. 1, involves a primary link and a secondary link. The primary link consists of a primary transmitter (PU-TX) with antennas and a primary receiver (PU-RX) with antennas. The secondary link conantennas and sists of a secondary transmitter (SU-TX) with a secondary receiver (SU-RX) with antennas. To reduce the hardware cost, the SU-RX has only RF chains. The to SU-RX. The signal reSU-TX transmits symbol ceived at the SU-RX is given by (1) is the channel response matrix from where the SU-TX to the SU-RX, is the received interference from PU-TX, and is the additive white Gaussian noise (AWGN) vector. Thus, the interference-plus-noise covariance
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matrix is
where is the channel response matrix from the PU-TX to the SU-RX [3]. We adopt the relaxed-structured ASF (RS-ASF) as in [6] at the SU-RX to reduce hardware complexity. The received signal after RS-ASF is given by
It is found that (6) is convex and the optimal solution can be obtained by semi-definite programming (SDP) [9]. Then, we fix and obtain by solving (7)
(2) is the ASF matrix in which the subscript where denotes the number of nonzero elements in . In the ASF matrix , the elements in each row correspond to the complex weights for multiple antennas in a subarray. In (2), is column vector where the element in each row correan sponds to a RF chain input where the RF chain includes a downconverter and an ADC. Thus, the multiple antenna outputs in a subarray are complexly weighted and combined. The weighted and combined output, after downconversion and analog-to-digital conversion, constitutes an element in . The subarrays determines correspond to elements in . The parameter the number of variable gain linear amplifiers and phase shifters available at the SU-RX [5]. The achievable capacity of secondary link is [7] (3) is the transmit covariance matrix. The mawhere trix determines the beamforming employed at the SU-TX. Our objective is to maximize the achievable capacity by designing the optimal ASF matrix and the transmit covariance matrix , i.e.,
. It is noted that the right hand side where of (7) can be viewed as the matrix-variable form of the generalized Rayleigh quotient. After tedious derivation as shown in the Appendix A, we obtain the optimal solution of (7), which is (8) consists of the first columns of and where is obtained by the eigenvalue decomposition (EVD), . By (6) and (8), we update and iteratively. Because (6) is convex and (8) is the closed-form optimal solution, updating and iteratively will only increase and or maintain the objective value of (5). By updating iteratively, we obtain a monotonically increasing sequence of the objective values of (5) which has the upper bound due to power constraint. Therefore, the algorithm for updating and iteratively converges to a stationary solution for the problem (5). B. The Recover Step In this step, we propose to zero elements in to recover the constraint . For simplicity, we use and to denote the final solutions obtained in the relax step. An approach is to search all the
(4) is the channel response matrix from the where is the transmit power constraint at SU-TX to the PU-RX, the SU-TX and is the interference power constraint from the SU-TX to the PU-RX. III. JOINT BEAMFORMING AND ASF DESIGN Because of the constraint of only nonzero elements in , the optimization problem (4) is the mixed integer nonlinear programming [8]. In this letter, we propose a relax-and-recover scheme to solve the optimization problem (4), which consists of two steps, the relax step and the recover step. A. The Relax Step In this step, we solve the following relaxed problem of (4) (5) where the constraint of only nonzero elements in is relaxed to be that all the elements on can be nonzero. To solve (5), we propose an iterative algorithm where the matrices and are optimized iteratively. Specifically, in the th iteration, we fix and solve the following optimization problem (6)
possible combinations for and choose the one that has the largest capacity as the final solution. However, such exhaustive search needs to compute two determinants, and , for
times, which causes high computation complexity. To reduce the complexity, we employ the greedy search method whose main idea is to zero one element which minimizes the capacity loss each time. We assume that after -times zeroing, the ASF and the capacity is . At the matrix is -times zeroing, we derive the capacity loss with respect to zeroing the th element in , denoted by . The ASF th element in is expressed matrix after zeroing the as (9) denotes a matrix with the where and the others being 0. Let
th element being 1 (10)
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and denote as a column vector with the th element being 1 and the others being 0. We have (11) where (12) and (13) Applying the matrix inverse lemma and the matrix determinant lemma [10], we obtain (14) where (15) (16) , we fiPerforming some similar manipulations for nally obtain the data rate loss with respect to zeroing the th element in as follows
(17) where (18) (19) (20) (21) From (17), we obtain the capacity loss with respect to zeroing each of all nonzero elements in . Thus, we zero . It is the element with the smallest capacity loss to obtain noted that obtaining only needs two matrix inversions, and . To zero elements in , we are for required to evaluate
times. Since the complexities to compute determinant and matrix inversion are comparable, our proposed greedy search method has much lower complexity than the exhaustive search. IV. SIMULATION RESULTS In this section, we present the computer simulation results of our proposed joint beamforming and ASF scheme for the MIMO CR system. We consider a MIMO CR system with
Fig. 2. Average capacity versus SNR; comparison of the proposed joint beamin [2] and the conforming and ASF scheme, the AS scheme . ventional MIMO system with full-complexity (
, , , , and . As in [1], we assume that all the channel matrices have independent complex Gaussian entries with zero mean and variances 1, 0.1, 0.1 for , , and , respectively. The interference power constraint at PU is set to be . In the legends of all our plots, “Full system” represents the conventional MIMO system with full-complexity where the number of RF chains is equal to the number of receive antennas at the SU-RX, , and the full array gain are exploited; “ASF” represents the proposed joint beamforming and ASF scheme; “AS” represents the AS scheme in [2]; “ES” represents that the AS scheme or the ASF scheme is optimized by the exhaustive search. In Fig. 2, we present the average capacity comparison of the proposed joint beamforming and ASF scheme, the AS scheme in [2] and the conventional MIMO system with . The SU transmit power constraint full-complexity ( sweeps from 1 to 100, i.e., the SNR goes from 0 dB to 20 dB. From Fig. 2, it is observed that with the same number of RF chains, the proposed scheme with or outperforms the AS scheme in the whole SNR regime. This is because that the AS scheme can be viewed as a special case of the proposed scheme with the elements of the ASF matrix being only 0 or 1. From Fig. 2, it is also observed that our proposed greedy search method achieves almost the same average capacity as the exhaustive search method. In Fig. 3, we present the capacity cumulative distribution functions (CDFs) of the proposed joint beamforming and ASF scheme, the AS scheme in [2] and the conventional MIMO system with full-complexity when the SNR is 10 dB. From Fig. 3, it is observed that with the same number of RF chains, the proposed scheme with or outperforms the AS scheme. In Fig. 4, we show the effect of parameter on the average capacity for the proposed joint beamforming and ASF scheme when the SNR is 10 dB. From Fig. 4, it is found that with the increase of , the performance of the proposed scheme approaches that of the conventional MIMO system with full-complexity.
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Let (23) and (24) We have (25) where Fig. 3. The capacity CDFs when SNR is 10 dB; comparison of the proposed in [2] and joint beamforming and ASF scheme, the AS scheme . the conventional MIMO system with full-complexity
(26) The equality occurs when (27) where consists of the first columns of eigenvalue decomposition (EVD), In fact, when ,
which is from the .
(28) Hence, (29) Fig. 4. Average capacity versus when SNR is 10 dB; comparison of the proposed joint beamforming and ASF scheme, the AS scheme in [2] and the conventional MIMO system with full-complexity .
V. CONCLUSION In this letter, we propose a relax-and-recover scheme for joint design of beamforming and ASF in MIMO CR systems. Simulation results have shown that the proposed joint beamforming and ASF scheme outperforms the AS with exhaustive search. The proposed scheme approaches the full complexity MIMO system with the increase of , which determines the number of variable gain linear amplifiers and phase shifters available at the SU-RX. APPENDIX DERIVATION OF (8) In this appendix, we will use the following theorem [11] to derive (8). Theorem 1: Given two Hermite matrices and , , , is a matrix, . Let denote the th eigenvalue of matrix in descending order. We have (22)
Equation (8) is proved. REFERENCES [1] R. Zhang and Y. C. Liang, “Exploiting multi-antennas for opportunistic spectrum sharing in cognitive radio networks,” IEEE J. Sel. Topics Signal Process, vol. 2, no. 1, pp. 88–102, Feb. 2008. [2] S. Sanayei and A. Nosratinia, “Antenna selection in MIMO systems,” IEEE Commun. Mag., vol. 42, no. 10, pp. 68–73, Oct. 2004. [3] M. F. Hanif, P. J. Smith, D. P. Taylor, and P. A. Martin, “MIMO cognitive radios with antenna selection,” IEEE Trans. Wireless Commun., vol. 10, no. 11, pp. 3688–3699, Nov. 2011. [4] P. D. Karamalis, N. D. Skentos, and A. G. Kanatas, “Adaptive antenna subarray formation for MIMO systems,” IEEE Trans. Wireless Commun., vol. 5, no. 11, pp. 2977–2982, Nov. 2006. [5] A. G. Kanatas, “A receive antenna subarray formation algorithm for MIMO systems,” IEEE Commun. Lett., vol. 11, no. 5, pp. 396–398, May 2007. [6] P. Theofilakos and A. G. Kanatas, “Capacity performance of adaptive receive antenna subarray formation for MIMO systems,” EURASIP J.Wirel. Commun. Netw., vol. 2007, pp. 1–12, Dec. 2007. [7] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Europ Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov. 1999. [8] I. Nowak, Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming. Basel, Switzerland: Birkhäuser, 2005. [9] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [10] D. A. Harville, Matrix Algebra from a Statistician’s Perspective. New York, NY, USA: Springer-Verlag, 1997. [11] A. J. Scott and G. P. H. Styan, “On a separation theorem for generalized eigenvalues and a problem in the analysis of sample surveys,” Lin. Alg. Applicat., vol. 70, pp. 209–224, Oct. 1985.
