IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 5, MAY 2014
789
Antenna Selection for Simultaneous Wireless Information and Power Transfer in MIMO Systems Sai Zhao, Quanzhong Li, Qi Zhang, and Jiayin Qin
Abstract—For simultaneous wireless information and power transfer in multiple-input multiple-output broadcast systems, we propose to investigate the antenna selection (AS) design problem. The problem is formulated as joint AS and transmit covariance matrix design optimization problem which maximizes the achievable rate from the transmitter to the informationdecoding receiver subject to the energy-harvesting constraint and the transmit power constraint. To solve the problem, we relax the binary constraints on the AS matrices and restrict the transmit covariance matrix to be diagonal. The AS matrices and the transmit covariance matrix are optimized iteratively by our proposed iterative AS algorithm. We also propose a lowcomplexity non-iterative norm-based algorithm which optimizes the AS matrices and the transmit covariance matrix sequentially. It is shown from simulation results that the achievable rates of proposed algorithms approach that of the AS scheme which is optimized by exhaustive search. Index Terms—Antenna selection (AS), energy harvesting (EH), multiple-input multiple-output (MIMO), simultaneous wireless information and power transfer (SWIPT).
I. I NTRODUCTION
T
HE simultaneous wireless information and power transfer (SWIPT), which belongs to energy harvesting (EH) techniques, is promising to solve the energy scarcity problem in wireless communications [1]. The SWIPT schemes for multiple-input-multiple-output (MIMO) and multiple-inputsingle-output (MISO) broadcast systems have been investigated in [1]–[3]. The SWIPT schemes for MIMO relay networks were studied in [4]. For MIMO systems, one drawback is that the multiple antennas should be associated with the multiple radio frequency (RF) chains, which are costly in terms of size, power, and hardware [5]–[7]. One feasible solution to overcome this drawback is the antenna selection (AS) scheme [8]–[11], which provides a good tradeoff between cost, complexity and performance. The key idea of AS scheme is to allocate the limited available RF chains to the transmit and receive antennas, between which the wireless links have the highest signal-to-noise ratio (SNR). Considering the SWIPT scheme, the AS design problem in the MIMO broadcast system is
Manuscript received January 20, 2014. The associate editor coordinating the review of this letter and approving it for publication was H. Suraweera. This work was supported in part by the National Natural Science Foundation of China under Grant 61173148 and Grant 61202498, and in part by the Scientific and Technological Project of Guangzhou City under Grant 12C42051578. The authors are with the School of Information Science and Technology, Sun Yat-Sen University, Guangzhou 510006, Guangdong, China (email:
[email protected],
[email protected], {zhqi26, issqjy}@mail.sysu.edu.cn). The authors are also with the SYSU-CMU Shunde International Joint Research Institute, Shunde, China. Digital Object Identifier 10.1109/LCOMM.2014.031514.140136
1 H Transmitter
M 1 ID Receiver N G
Antenna Selection
1 L EH Receiver
Fig. 1. The system model of antenna selection for SWIPT in a MIMO broadcast system.
important. To the best of our knowledge, the research on above-mentioned problem is missing. In this letter, we study the antenna selection design problem for SWIPT in a three-node MIMO broadcast system which consists of a transmitter, an information-decoding (ID) receiver and a EH receiver. We formulate the joint AS and transmit covariance matrix design as the optimization problem which maximizes the achievable rate from transmitter to ID receiver subject to the EH constraint at EH receiver and the transmit power constraint at transmitter. The formulated optimization problem is a mixed integer programming problem which is non-convex because of the binary constraints on the AS matrices and the non-convexity of the objective function. To solve the problem, we relax the binary constraints and restrict the transmit covariance matrix to be diagonal. The AS matrices and the transmit covariance matrix are optimized iteratively by our proposed iterative AS algorithm. Because of the computational complexity of proposed iterative algorithm and the unnecessary diagonal restriction on the transmit covariance matrix, we also propose a low-complexity non-iterative normbased algorithm which optimizes the AS matrices and the transmit covariance matrix sequentially. Notations: Boldface lowercase and uppercase letters denote vectors and matrices, respectively. The A† , det(A), Tr(A) denote the conjugate transpose, determinant, and trace of the matrix A, respectively. The A(i, j) denotes the entry in the ith row and j th column of A. By A 0, we mean that A is positive semidefinite. The CN (0, I) denotes the distribution of a circularly symmetric complex Gaussian vector with mean vector 0 and covariance matrix I. II. S YSTEM M ODEL Consider a three-node SWIPT MIMO broadcast system which consists of a transmitter, an information-decoding (ID)
c 2014 IEEE 1089-7798/14$31.00
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IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 5, MAY 2014
receiver and an energy-harvesting (EH) receiver, as shown in Fig. 1. The transmitter, ID receiver and EH receiver are equipped with N , M , and L antennas, respectively. To reduce the hardware cost, the transmitter has only n (n ≤ N ) radio frequency (RF) chains and the ID receiver has only m (m ≤ M ) RF chains. Thus, the transmitter employs n antennas to transmit signals and the ID receiver employs m antennas to receiver signals. When the transmitter transmits symbol s ∈ CN ×1 to the ID receiver, the received signal at the ID receiver, denoted as y, is given by
to the modified optimization problem is obtained when the constraints Tr(P) ≤ n and Tr(Q) ≤ m are active, i.e., Tr(P) = n and Tr(Q) = m. According to [1], if the EH constraint, τ , is chosen such that 0 ≤ τ ≤ τmax where
y = QHPs + Qz
III. T HE P ROPOSED I TERATIVE AS A LGORITHM
(1)
where H ∈ CM×N is the channel response matrix from transmitter to ID receiver; z ∼ CN (0, σ 2 I) is the additive Gaussian noise vector at ID receiver; P ∈ RN ×N and Q ∈ RM×M are the antenna selection (AS) matrices at the transmitter and ID receiver, respectively. The AS matrices, P and Q, are diagonal matrices where the diagonal entries are either 1 or 0. If the (i, i)th , i ∈ N = {1, 2, · · · , N }, entry of P is 1, the ith antenna is selected for transmission; otherwise, the ith antenna is not selected. If the (j, j)th , j ∈ M = {1, 2, · · · , M }, entry of Q is 1, the j th antenna is selected for receiving signals; otherwise, the j th antenna is not selected. The achievable rate between transmitter and ID receiver is given by [10] (2) R(S, P, Q) = log2 det I + σ −2 QHPSP† H† Q† where S denotes the transmit covariance matrix, S = E ss† .
(3)
When the transmitter transmits signals to ID receiver, it also transfers energy to EH receiver simultaneously. The received power at EH receiver is expressed as [1] E(S, P) = Tr(GPSP† G† )
(4)
where G ∈ CL×N is the channel response matrix from transmitter to EH receiver. The entries of channel response matrices, H and G, are zero-mean circularly symmetric complex Gaussian distributed random variables. We assume that the perfect channel state information (CSI) is known at the transmitter and ID receiver. Our objective is to design the transmit covariance matrix S and the AS matrices, P and Q, which maximize the achievable rate subject to the EH constraint and the transmit power constraint. The optimization problem is formulated as follows, R(S, P, Q)
max
P(diag.),Q(diag.),S0
s.t.
(5)
Tr(S) ≤ Γ, E(S, P) ≥ τ, P(i, i) ∈ {0, 1}, ∀ i ∈ N , Q(j, j) ∈ {0, 1}, ∀ j ∈ M, Tr(P) = n, Tr(Q) = m
where Γ is the transmit power constraint at transmitter and τ is the EH constraint at EH receiver. It is noted that the constraints Tr(P) = n and Tr(Q) = m in the problem (5) mean that all the RF chains are always activated. This is reasonable because if we modify the aforementioned constraints as Tr(P) ≤ n and Tr(Q) ≤ m, it can be proved that the optimal solution
τmax = αΓ
(6)
in which α is the largest singular value of G, the problem (5) is feasible.
The optimization problem (5) is a mixed integer programming which is non-convex because of the binary constraints and the non-convexity of the objective function. To solve the problem, we propose the iterative AS algorithm. In the proposed iterative AS algorithm, the joint optimization of S, P and Q is decoupled into two phases where in the first phase, we obtain the optimal P and Q and in the second phase, we obtain the optimal S. It is noted that in the first phase, to obtain the optimal P and Q is to select most suitable transmit and receive antennas for transmission, i.e., to select the transmit and receive antenna pairs which have the highest channel gains. Although the optimization of P and Q is also related with the optimization of S, the function of S is only for proper power allocation over the selected antennas. Thus, in the first phase of proposed iterative AS algorithm, we restrict S to be diagonal. It is found that in the problem (5), if any two among P, Q, and S are known, the objective function is concave. We first assume that Q and S are known. Because of the restriction that S are diagonal, we have PSP† = S1/2 PS1/2 .
(7)
When Q and S are known, the problem (5) is rewritten as follows max
log2 det(I + σ −2 QHS1/2 PS1/2 H† Q† )
s.t.
P(i, i) ∈ {0, 1}, ∀ i ∈ N , Tr(P) = n.
P(diag.)
(8)
The problem (8) is still non-convex because of the binary constraints. To solve the problem, we relax the binary constraints as 0 ≤ P(i, i) ≤ 1, ∀ i ∈ N .
(9)
Thus, the problem (8) is transformed into max
log2 det(I + σ −2 QHS1/2 PS1/2 H† Q† )
s.t.
0 ≤ P(i, i) ≤ 1, ∀ i ∈ N , Tr(P) = n.
P(diag.)
(10)
The problem (10) is convex and can be solved effectively using the interior point method [12]. When P and S are known, according to the determinant property, det(I + AB) = det(I + BA) (11) where AB and BA are square matrices, we have det(I + σ −2 QHPSP† H† Q† ) = det(I + σ −2 P† H† Q† QHPS).
(12)
ZHAO et al.: ANTENNA SELECTION FOR SIMULTANEOUS WIRELESS INFORMATION AND POWER TRANSFER IN MIMO SYSTEMS
791
Algorithm 1 The Proposed Iterative AS Algorithm
Since Q† Q = Q,
(13)
1: 2:
we have det(I + σ −2 QHPSP† H† Q† ) = det(I + σ
−2 1/2
S
†
†
P H QHPS
(14) 1/2
).
(15)
By relaxing the binary constraints Q(j, j) ∈ {0, 1}, ∀ j ∈ M in (5) as 0 ≤ Q(j, j) ≤ 1, ∀ j ∈ M,
(16)
the problem (5) is transformed into max
log2 det(I + σ −2 S1/2 P† H† QHPS1/2 )
s.t.
0 ≤ Q(j, j) ≤ 1, ∀ j ∈ M, Tr(Q) = m,
Q(diag.)
(17)
given that P and S are known. The problem (17) is convex. When P and Q are known, the problem (5) is simplified as follows max
S(diag.)0
log2 det(I + σ −2 QHPSP† H† Q† ) Tr(S) ≤ Γ, E(S, P) ≥ τ.
s.t.
(18)
The problem (18) is convex. By solving the problems (10), (17) and (18), we iteratively update P, Q, and S. Because the problems (10), (17) and (18) are convex, iteratively updating P, Q, and S will monotonically increase the value of objective function. Since the value of objective function is upper bounded by the transmit power constraint, the iterative algorithm of iteratively updating P, Q, and S will converge to the optimal solution to the following problem, max
R(S, P, Q)
s.t.
Tr(S) ≤ Γ, E(S, P) ≥ τ,
P(diag.),Q(diag.),S(diag.)0
3: 4: 5:
(0) (0) (0) Initialization: (0) k =(0)0, P(0) , Q , S ; ; Obtain R S , P , Q Repeat: k = k + 1; Solve (10) with Q(k−1) and S(k−1) . Output P(k) ; Solve (17) with P(k) and S(k−1) . Output Q(k) ; (k) (k) Solve (18) with P(k) and Q . Output S ; (k) (k) (k) ; Obtain R S , P , Q Until: (k) (k) (k) R S , P , Q −R S(k−1) , P(k−1) , Q(k−1) ≤ ; Recover the binary constraints and obtain Po and Qo ; Obtain So by solving (5) with Po and Qo .
IV. T HE P ROPOSED N ON -I TERATIVE N ORM -BASED AS A LGORITHM The proposed iterative AS algorithm has high computational complexity. Furthermore, in the proposed algorithm, we introduce the diagonal restriction on S. To reduce computational complexity and remove the diagonal restriction on S, we propose a low-complexity non-iterative norm-based AS algorithm in this section. In [9], the norm-based AS algorithm was proposed for the point-to-point MIMO communication system. The norm-based AS algorithm selects the transmit and receive antennas which corresponding to the columns and rows of channel matrix with the largest column and row norms, respectively. The normbased algorithm is also employed in [10]. Let F = diag[f1 , f2 , · · · , fN ]
(20)
fj = G(:, j)
(21)
where
(19)
0 ≤ P(i, i) ≤ 1, ∀ i ∈ N , 0 ≤ Q(j, j) ≤ 1, ∀ j ∈ M, Tr(P) = n, Tr(Q) = m. After obtaining the optimal solution to problem (19), we recover the binary constraints over the AS matrices by assigning the largest n and m diagonal entries of P and Q to be 1, respectively. The other diagonal entries of P and Q are assigned to be 0. The aforementioned procedure is the first phase of the proposed iterative AS algorithm. In the second phase, denoting the AS matrices which satisfy the binary constraints as Po and Qo , we solve the problem (5) by substituting the Po and Qo into (5), where the diagonal restriction on S is removed. We summarize the proposed iterative algorithm in Algorithm1 where is a small positive value. Remark 1: In Section II, we assume that all the receive antennas at the EH receiver participate the EH. Our proposed iterative AS algorithm can be straightforwardly generated to the scenario where only the partial receive antennas at the EH receiver participate the EH by adding an iterative step in Step 2 of Algorithm 1.
for j ∈ N , in which G(:, j) denotes the j th column of G. At the transmitter, we solve the following problem max HFP
P(diag.)
s.t.
(22)
P(i, i) ∈ {0, 1}, ∀ i ∈ N , Tr(P) = n.
The solution of problem (22), denoted as Pb , is that the diagonal entries of Pb corresponding to the n columns of HF which have the largest n column norms are 1 and the other entries to be 0 [9]. When the AS matrix at the transmitter, Pb , is obtained, at the ID receiver, we solve the following problem max QHPb
Q(diag.)
s.t.
(23)
Q(j, j) ∈ {0, 1}, ∀ j ∈ M, Tr(Q) = m.
The solution of problem (23), denoted as Qb , is that the diagonal entries of Qb corresponding to the m rows of HPb which have the largest m row norms are 1 and the other entries to be 0 [9].
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IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 5, MAY 2014
When the AS matrices, Pb and Qb , are obtained, we solve the following problem S0
s.t.
AS−ES AS−Iter AS−Non−Iter AS−Diff
20
(24)
Tr(S) ≤ Γ, E(S, Pb ) ≥ τ.
The problem (24) is convex and can be solved effectively using the interior point method [12]. Remark 2: Our proposed non-iterative norm-based AS algorithm can be straightforwardly generated to the scenario where only the partial receive antennas at the EH receiver participate the EH.
18 Achievable Rate (bps/Hz)
max R(S, Pb , Qb )
22 L=M=N=6, m=n=3
16 14 12 10 8 L=M=N=10, m=n=2
6 4
V. S IMULATION R ESULTS In this section, we evaluate the performance of the proposed algorithms via computer simulations. In the MIMO broadcast system, we assume that the channel matrices H and G have independent complex Gaussian entries with zero mean and variances 1 and 1.1, respectively. The reason that the variances of the entries in G are larger than those in H is that the EH receiver is generally nearer to the transmitter than the ID receiver as in [1]. The EH constraint, τ , is set to be τ = 0.5τmax . In the legends of all our plot, “AS-ES” represents the AS scheme which is optimized by exhaustive search of all possible combinations of antennas where the optimization of S given the selected antennas is a convex problem. Therefore, the AS-ES scheme provides the optimal AS solution. The “ASIter” and “AS-Non-Iter” represent the proposed iterative and non-iterative norm-based AS algorithms, respectively. We also present the performance of the difference AS scheme proposed in [8], denoted as “AS-Diff” in the legend. From [10] and [13], the worst-case complexities of AS-ES, AS-Iter, AS-Non-Iter, and AS-Diff schemes are M N (25) O · N 3.5 log −1 , m n O η · max{M, N }3.5 log −1 , (26) −1 3.5 , and (27) O N log M N O + N 3.5 log −1 , (28) m n respectively, where denotes the accuracy requirement and η denotes the average number of iteration times for the ASIter scheme. From (25)-(28), the AS-Iter scheme has much lower complexity than the AS-ES scheme, especially when M and N are large. The AS-Non-Iter scheme has the lowest complexity among all the aforementioned schemes. In Fig. 2, we present the achievable rate comparison of AS-ES, AS-Iter, AS-Non-Iter and AS-Diff schemes. In the simulations, the signal-to-noise ratio (SNR), Γ/σ 2 , sweeps from 0 dB to 20 dB. From Fig. 2, it is observed that the performance gap between the AS-ES and AS-Iter schemes is negligible. The AS-Iter scheme has the larger achievable rate than the AS-Non-Iter and AS-Diff schemes. The AS-Non-Iter scheme outperforms the AS-Diff scheme. VI. C ONCLUSION In this letter, we have proposed an iterative AS algorithm and a non-iterative norm-based algorithm to optimize the AS
2
0
5
10
15
20
2
Γ/σ (dB)
Fig. 2. Achievable rate versus Γ/σ2 ; comparison of the AS-ES, AS-Iter, AS-Non-Iter and AS-Diff schemes.
matrices and the transmit covariance matrix for SWIPT in MIMO broadcast systems. It is shown from simulation results that the achievable rates of proposed algorithms approach that of the AS scheme which is optimized by exhaustive search. R EFERENCES [1] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” IEEE Trans. Wireless Commun., vol. 12, no. 5, pp. 1989–2001, May 2013. [2] C. Xing, N. Wang, J. Ni, Z. Fei, and J. Kuang, “MIMO beamforming designs with partial CSI under energy harvesting constraints,” IEEE Signal Process. Lett., vol. 20, no. 4, pp. 363–366, Apr. 2013. [3] Z. Xiang and M. Tao, “Robust beamforming for wireless information and power transmission,” IEEE Wireless Commun. Lett., vol. 1, no. 4, pp. 372–375, Aug. 2012. [4] B. K. Chalise, W. K. Ma, Y. D. Zhang, H. A. Suraweera, and M. G. Amin, “Optimum performance boundaries of OSTBC based AF-MIMO relay system with energy harvesting receiver,” IEEE Trans. Signal Process., vol. 61, no. 17, pp. 4199–4213, Sept. 2013. [5] S. Sanayei and A. Nosratinia, “Antenna selection in MIMO systems,” IEEE Commun. Mag., vol. 42, no. 10, pp. 68–73, Oct. 2004. [6] A. F. Molish and M. Z. Win, “MIMO systems with antenna selection,” IEEE Microw. Mag., vol. 5, no. 1, pp. 46–56, Mar. 2004. [7] X. Zeng, Q. Li, Q. Zhang, and J. Qin, “Joint beamforming and antenna subarray formation for MIMO cognitive radios,” IEEE Signal Process. Lett., vol. 20, no. 5, pp. 479–482, May 2013. [8] Y. Wang and J. P. Coon, “Difference antenna selection and power allocation for wireless cognitive systems,” IEEE Trans. Commun., vol. 59, no. 12, pp. 3494–3503, Sept. 2011. 2010. [9] P. J. Smith, T. W. King, L. M. Garth, and M. Dohler, “A power scaling analysis of norm-based antenna selection techniques,” IEEE Trans. Wireless Commun., vol. 7, no. 8, pp. 3140–3149, Aug. 2008. [10] 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. [11] J. Li, Q. Zhang, Q. Li, L. Luo, and J. Qin, “Joint single transmit and receive antenna selection for MIMO cognitive radios without channel state information,” Electron. Lett., vol. 49, no. 13, pp. 479–482, May 2013. [12] S. Boyd and L. Vandenberghe, Convex Optimization.. Cambridge University Press, 2004. [13] I. Polik and T. Terlaky, “Interior point methods for nonlinear optimization,” in Nonlinear Optimization, 1st ed., G. Di Pillo and F. Schoen, Ed. Springer, 2010, ch. 4.
