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Relay Beamforming for Amplify-and-Forward Multi-Antenna Relay Networks with Energy Harvesting Constraint Jianli Huang, Quanzhong Li, Qi Zhang, Guangchi Zhang, and Jiayin Qin
Abstract—For amplify-and-forward multi-antenna relay networks with energy harvesting (EH) constraint, we study the optimal relay beamforming problem which maximizes the achievable rate from source to information-decoding receiver subject to the transmit power constraint at relay and the EH constraint at EH receiver. Because of the EH constraint, the beamforming problem is not convex. We propose the optimal beamforming scheme by converting the beamforming problem into a convex semidefinite programming with the rank-one relaxation and Charnes-Cooper transformation. We also propose a suboptimal closed-form beamforming scheme. It is shown from simulations that when the maximum allowable relay transmit power to noise power ratio is high, the performance of proposed suboptimal scheme approaches that of the optimal scheme. Index Terms—Amplify-and-forward (AF), beamforming, energy harvesting (EH), relay networks, semidefinite programming (SDP).
I. INTRODUCTION
E
NERGY HARVESTING (EH) techniques allow battery operated wireless communication systems to harvest energy from ambient radio signals. In [1]–[3], the wireless communications in multiple-input-multiple-output (MIMO) channel and multiple-input-single-output (MISO) broadcast channel with EH constraints were investigated. In [4], with the help of orthogonal space-time block codes (OSTBC), Chalise et al. proposed a novel joint optimal source and relay precoding scheme where the MIMO channels are decoupled into parallel single-input-single-output (SISO) channels. The proposed scheme in [4] is actually a space-time beamforming (STB)
Manuscript received December 29, 2013; revised February 08, 2014; accepted February 08, 2014. Date of publication February 11, 2014; date of current version February 19, 2014. This work was supported by the National Natural Science Foundation of China under Grants 61173148 and 61102070, the Scientific and Technological Project of Guangzhou City under Grants 12C42051578 and 2013J2200071, and by the Natural Science Foundation of Guangdong Province under Grant S2011040004135. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Rodrigo C. de Lamare. J. Huang, Q. Li, Q. Zhang, and J. Qin are with the School of Information Science and Technology, Sun Yat-Sen University, Guangzhou 510006, Guangdong, China (e-mail:
[email protected]; liquanzhong2009@gmail. com;
[email protected];
[email protected]). G. Zhang is with the School of Information Engineering, Guangdong University of Technology, Guangzhou 510006, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2014.2305737
scheme. Compared with the conventional space beamforming scheme [5], the STB scheme introduces the dimension of time for beamforming design. For the half duplex relay network, the STB scheme achieves better performance than the space beamforming while the system complexity of the former remains the same as that of the latter. For the full duplex relay network where the relay is able to transmit and receive signals in the same frequency band and in the same time slot, the space beamforming scheme may still perform worst than the STB scheme. However, the space beamforming scheme may have less system complexity. This is because for the space beamforming, the relay is not required to have the whole radio-frequency chains where the output signals are the weighted sums of the input signals. For the STB, the relay requires additional time-delay elements. If the input signals are wideband, the time-delay elements have high system complexity. Without OSTBC, the conventional space relay beamforming for amplify-and-forward (AF) multi-antenna relay networks without EH constraint was well-studied in the literature [5]. Recently, Michalopoulos et al. proposed the relay selection scheme for decode-and-forward (DF) relay networks with EH constraint [6]. However, to our best knowledge, the research on optimal space relay beamforming for AF multi-antenna relay networks with EH constraint is missing. In this letter, considering the EH constraint, we investigate the optimal beamforming problem for AF multi-antenna relay networks which maximizes the achievable rate from transmitter to information-decoding (ID) receiver subject to the transmit power constraint at relay and the EH constraint at EH receiver. Because of the EH constraint, the beamforming problem is not convex. We propose the optimal beamforming scheme by converting the beamforming problem into a convex semidefinite programming (SDP) with the rank-one relaxation and Charnes-Cooper transformation [7]. To reduce computational complexity, we also propose a suboptimal closed-form beamforming scheme. Although the half duplex relay network is considered in this letter, our investigation may provide insight on the full duplex relay network. Notations: Boldface lowercase and uppercase letters denote vectors and matrices, respectively. The , , , , and denote the transpose, conjugate, conjugate transpose, Frobenius norm and trace of the matrix , respectively. The denotes to stack the columns of a matrix into a single vector . The denotes the Kronecker product. By ,
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we mean that the matrix is positive semidefinite. denotes the distribution of a circularly symmetric complex Gaussian vector with mean and covariance .
harvested power is the sum of the signal power and the noise power. The harvested power at EH receiver should satisfy the EH constraint, , where is the predefined threshold.
II. SYSTEM MODEL
III. RELAY BEAMFORMING WITH EH CONSTRAINT
Consider a two-hop AF relay network, which consists of a source, a multi-antenna relay, an ID receiver and an EH receiver [1]. The relay is equipped with antennas. Each of other nodes is equipped with a single antenna. The scenario is typical for device-to-device communications where two mobile phones (the source and the ID receiver) communicate directly with the help of an AF relay. Simultaneously, the relay is responsible to charge a senor (the EH receiver) which harvestes energy to sense the environment. We assume that the direct link between source and ID receiver is sufficiently weak to be ignored. This occurs when the direct link is blocked due to long-distance path loss or obstacles. The network operates in a half-duplex mode. During the first time slot, the source transmits signals to the relay. During the second time slot, the relay multiplies the received signals by a beamforming matrix and forwards them to the ID receiver. When the relay forwards signals to the ID receiver, it also transfers energy to the EH receiver simultaneously. We assume that the channel state information (CSI) of whole network is available at the relay such that optimal beamforming can be performed. When the source transmits signal , the received signal at the ID receiver, denoted as , is given by
The objective of relay beamforming design problem is to maximize the achievable rate from source to ID receiver subject to the transmit power constraint at relay and the EH constraint at EH receiver, which is formulated as
(1) where and denote the channel response vectors from source to relay and from relay to ID receiver, respectively; denotes the additive Gaussian noise vector received at the relay; denotes the additive Gaussian noise received at the ID receiver; denotes the linear beamforming matrix employed at the relay. In (1), the transmit power of source is . The transmit power of the relay is
(5) where the factor 1/2 is included because the signals are transmitted in two time-slots. The optimization problem (5) is a nonconvex optimization problem which is difficult to be solved. By using the equality [8] (6) where and are arbitrary matrices with compatible dimensions, we equivalently express the problem (5) as
(7) where
, , and
,
,
,
.
A. The Optimal Relay Beamforming with EH Constraint In (7), the multiple quadratical constraints cause the optimization problem to be a fractional quadratically constrained quadratic problem (QCQP) which is intractable [8]. In this letter, we propose a semidefinite relaxation based solution to the problem (7). We transform the problem (7) to the following fractional semidefinite programming (SDP)
(2) which is constrained by the maximum allowable relay transmit power, denoted as . Because of the broadcast nature of wireless transmission, the relay forwards signals to the ID receiver and transfers energy to the EH receiver simultaneously. According to (1), the received signal-to-noise ratio (SNR) at the ID receiver, denoted as , is expressed as (3)
(8) where
,
,
,
, and . It is noted that in (8), the rank-one constraint has been removed and the problem (8) is a quasi-convex optimization problem. Let (9)
The harvested power at EH receiver during the second time slot is
By using the Charnes-Cooper transformation [7], we transform the quasi-convex problem (8) into a convex SDP as follows
(4) denotes the channel response vector from where relay to EH receiver. It is noted that the signal and the amplified and forwarded noise are independent which causes that the
(10)
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We have the following proposition. Proposition 1: Suppose that is the optimal solution to the problem (10). The optimal solution to the problem (8) is . Proof: Suppose that and are the optimal values of the problems (8) and (10), respectively. If , Proposition 1 is proved. In the following, we first prove that . It is noted that . If , which leads to since is positive definite. If is the optimal solution of problem (10), it can be verified that is feasible for the problem (8). Furthermore, the objective value in the problem (8) at the point is (11) Therefore, . Similarly, we can prove that . Combining with , we have . The objective function of problem (10) is a linear function of semidefinite cone and the constraints of problem (10) specify a convex set of . Thus, the problem (10) is a convex SDP which can be solved efficiently using the interior-point method [10]. From Proposition 1, if the optimal solution to problem (10) is rank-one, the optimal solution to the problem (7) is , where . If , from the rank-one decomposition theorem [9], we can find a rank-one matrix in polynomial-time such that (12) The optimal solution to (7) is
From the definition of achievable rate-energy region (13), regardless of information rate, we obtain by solving (14) , is The optimal solution to problem (14), denoted as [8], where denotes the generalized eigenvector corresponding to the largest generalized eigenvalue of the matrix pair and (15) Substituting into (13), we obtain of objective function of (14) is
. The optimal value (16)
where denotes the largest eigenvalue of matrix . Regardless of harvested power, we obtain by solving
The optimal solution to problem (17), denoted as [8], where . Thus, we obtain
(17) , is and
(18) The optimal value of objective function of (17) is (19)
.
B. Achievable Rate-Energy Region For wireless communications with EH constraints, the achievable rate-energy region is important to characterize all the achievable rate and energy pairs under a given transmit power constraint [1]. In this letter, the achievable rate-energy region is defined as
Remark 1: In this letter, we consider that the source is equipped with a single antenna. If the source has multiple antennas, the derivations in Sections III-A and III-B are also applicable, where the closed-form expressions of , , and cannot be obtained. This is because that the obtained may not be rank-one if the source has multiple antennas. However, from the rank-one decomposition theorem [9], we can find the optimal solution in polynomial-time to problem (17). If either the ID receiver or the EH receiver has multiple antennas, the derivations in Sections III-A and III-B are not applicable. C. Suboptimal Relay Beamforming Scheme
(13) is the optimal solution to (7). According to [1], to where obtain the achievable rate-energy region requires to identify two special boundary points, denoted by and , where the former corresponds to the maximum harvested power at the EH receiver regardless of the information rate and the latter corresponds to the maximum achievable rate at the ID receiver regardless of the harvested power.
To reduce computational complexity, we also propose a suboptimal closed-form solution to the problem (7). It is noted that the transmit power constraint is active at the optimum, i.e., (20) Substituting (20) into the problem (7), we have
(21)
HUANG et al.: RELAY BEAMFORMING FOR AMPLIFY-AND-FORWARD MULTI-ANTENNA RELAY NETWORKS
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Define as a matrix which consists of the eigenvectors of matrix whose corresponding eigenvalues are no greater than zero. We have , where denotes the suboptimal solution to problem (7) and is an arbitrary vector such that (22) Substituting
into the problem (21), we have (23)
It is noted that the problem (23) is the generalized Rayleigh quotient whose optimal solution is [8] (24) where
is a scalar satisfying the transmit power constraint,
Fig. 1. Achievable rate versus the maximum transmit power of the relay to ; performance comparison of the proposed optimal noise power ratio, and suboptimal relay beamforming schemes.
(25) Remark 2: It is noted that both optimal and suboptimal relay beamforming schemes requires perfect CSIs of whole relay network. From [11], the complexity of solving an SDP problem is where denotes the number of semidefinite cone constraints, denotes the dimension of the semidefinite cone, and is the accuracy of solving the SDP. Compare the SDP (10) with the standard form in [11], we have and . Thus, the computational complexity of our proposed optimal relay beamforming scheme is . Since the computational complexities of matrix multiplication, matrix inversion, eigen decomposition and Cholesky decomposition are not greater than where denotes the dimension of the matrix, the computational complexity of our proposed suboptimal scheme is where . IV. SIMULATION RESULTS In this section, we present the computer simulation results of our proposed relay beamforming schemes. We assume that all the entries in channel response vectors are independent complex Gaussian random variables with zero mean and unit variance. The variances of noises are . The transmit power at source to noise power ratio is dB. In all simulations, the performance of proposed relay beamforming schemes is the result averaging over 1000 randomly generated channel realizations. In Fig. 1, we present the achievable rate comparison of proposed optimal and suboptimal relay beamforming schemes (denoted as “Optimal” and “Suboptimal” in the legend, respectively) for different number of antennas at the relay and different EH constraints. From Fig. 1, it is observed that when is high, the performance of proposed suboptimal scheme approaches that of the optimal scheme. We also compare the proposed scheme with the conventional relay beamforming scheme without the EH constraint in [5] (denoted as “No-EH” in the
Fig. 2. Achievable rate-energy regions for the AF relay network.
legend). From Fig. 1, it is observed that when dB, the proposed optimal scheme performs almost the same as the conventional relay beamforming scheme without the EH constraint. This is because that when is high, the EH constraint is always satisfied. In Fig. 2, we present the achievable rate-energy regions for the AF relay network. From Fig. 2, it is found that the achievable rate-energy regions improve with the increase of the transmit power at relay, , and the number of antennas at relay, . With more antennas, the relay exploits the array gain to achieve the larger rate-energy region. V. CONCLUSION In this letter, we have proposed the optimal and suboptimal relay beamforming schemes for the AF multi-antenna relay network with EH constraint. It is shown from simulations that when the maximum allowable relay transmit power to noise power ratio, i.e., , is high, the performance of proposed suboptimal scheme approaches that of the optimal scheme.
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[6] D. S. Michalopoulos, H. A. Suraweera, and R. Schober, “Relay selection for simultaneous information transmission and wireless energy transfer: A tradeoff perspective,” [Online]. Available: http://arxiv.org/ abs/1303.1647. [7] A. Charnes and W. W. Cooper, “Programming with linear fractional functionals,” Naval Res. Logist. Quart., vol. 9, pp. 181–186, 1962. [8] R. A. Horn and C. R. Johnson, Matrix Analysis. : Cambridge University Press, 1985. [9] W. Ai, Y. Huang, and S. Zhang, “New results on hermitian matrix rank-one decomposition,” Math. Progr., vol. 128, no. 1-2, pp. 253–283, 2011. [10] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [11] I. Polik and T. Terlaky, “Interior point methods for nonlinear optimization,” in Nonlinear Optimization, G. Di Pillo and F. Schoen, Eds., 1st Ed. ed. Berlin, Germany: Springer, 2010, ch. 4.
