Simultaneous Wireless Information and Power Transfer for Decode ...

Simultaneous Wireless Information and Power Transfer for Decode-and-Forward MIMO Relay Communication Systems Fatma Benkhelifa, Ahmed Kamal Sultan Salem and Mohamed-Slim Alouini Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division King Abdullah University of Science and Technology (KAUST) Thuwal, Makkah Province, Saudi Arabia {fatma.benkhelifa,ahmed.salem,slim.alouini}@kaust.edu.sa

Abstract—In this paper, we investigate the simultaneous wireless information and power transfer (SWIPT) for a decodeand-forward (DF) multiple-input multiple-output (MIMO) relay system where the relay is an energy harvesting node. We consider the ideal scenario where both the energy harvesting (EH) receiver and information decoding (ID) receiver at the relay have access to the whole received signal and its energy. The relay harvests the energy while receiving the signal from the source and uses the harvested power to forward the signal to the destination. We obtain the optimal precoders at the source and the relay to maximize the achievable throughput rate of the overall link. In the numerical results, the effect of the transmit power at the source and the position of the relay between the source and the destination on the maximum achievable rate are investigated. Index Terms—Energy harvesting, simultaneous wireless information and power transfer (SWIPT), MIMO relay systems, decode-and-forward, maximum achievable rate.

I. Introduction Harvesting energy from the environment is a promising technique for future green communication systems. It allows, at least theoretically, the construction and the operation of perpetually powered communication networks. Solar, wind, and vibration are commonly suggested sources for energy harvesting at the transmit nodes. Radio frequency (RF) signals from ambient transmitters have recently been considered as a possible source of wireless power transfer (WPT). Passive radio-frequency identification (RFID) systems [1], and body sensor networks for medical implants (BSNs) [2] are, for instance, flagship applications that have successfully implemented RF energy harvesting. Moreover, simultaneous wireless information and power transfer (SWIPT) has recently gained a lot of research interest in order to study wireless communication systems when RF signals are simultaneously used to transmit information from the transmitter and scavenge energy at the receiver. In [3] and [4], a SWIPT single-input single-output (SISO) system was considered in flat-fading and frequency selective channels where the optimal tradeoff between information rate and energy transfer was investigated. In these two works, the information decoding (ID) receiver and energy harvesting (EH) receiver were assumed to be co-located. An extension of the work done in [3] and [4] was presented in [5] where two practical schemes were considered. The two schemes assume that the receiver separates the ID and EH transfer over the power domain, known as the power splitting (PS) scheme, or the time domain, known as the time switching (TS) scheme. In [5], a multi-antenna broadcasting system was considered and the rate-energy (R-E) region was characterized for co-located This paper was funded by a grant from the office of competitive research funding (OCRF) at KAUST, Saudi Arabia.

and separated receivers. In [6], a two-hop SISO amplify-andforward (AF) relay system was considered where the relay harvests energy from the transmitted signal from the source and uses the harvested power to forward the signal to the destination. This work analyzed the outage probability and the ergodic capacity for delay-limited and delay-tolerant transmission modes. However, the channel state information (CSI) is assumed to be known only at the destination. In [7], the authors extended the work done in [6] proposing a continuous and discrete adaptive time-switching protocol where both AF and decode-and forward (DF) relay networks were considered. Analytic expressions of the achievable throughput for both cases were derived. In [8], a two-hop multi-antenna AF relay system was investigated in the presence of a multi-antenna EH receiver where the source and the relay nodes employ orthogonal space-time block codes (STBC) for data transmission. The optimal source and relay precoders were jointly optimized to achieve the rate-energy tradeoff between the harvested power at the EH receiver and the information rate at the destination node. In [9], a two-hop SISO orthogonal frequency division multiplexing (OFDM) DF relay system was investigated where the relay harvests the energy from the signals transmitted from the source. The power splitting scheme was considered at the relay and the resource allocation was studied in order to maximize the total achievable throughput rate. In this paper, we study a two-hop MIMO DF relay system where the relay is an energy harvesting node. We consider the ideal scenario where each of the EH and ID receivers, at the relay use all the available received energy. Our design goal is to maximize the achievable rate between the source and the destination via optimizing the precoding matrices employed at the source and the relay. The optimization problem is convex, and we provide a detailed study of its solution. To the best of our knowledge, this paper is the first to consider the achievable rate in a two-hop MIMO DF relay system where the relay is a multi-antenna EH node. II. System Model and Problem Formulation We consider a two-hop MIMO DF relay communication system where the source S , relay R, and destination D are equipped with N s , Nr , and Nd antennas, respectively. We assume that the direct link between S and D suffers from severe path attenuation and shadowing, thereby rendering it unusable for reliable communications. The relay operates in a halfduplex mode, i.e. the signal from the source to the relay and the signal from the relay to the destination are transmitted over two separate time slots. The channel between the source and the relay and the channel between the relay and the destination are denoted H ∈ Nr ×Ns and G ∈ Nd ×Nr , respectively, and are assumed to be quasi-static block-fading channels. The slot

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duration is assumed to be sufficiently small compared to the coherence time of the channel. We assume that channel state information (CSI) is perfectly known at all the nodes. While both the source and the destination are batterypowered, the relay is an energy harvesting node. It harvests the energy from the received RF signal from the source, and uses the harvested power to forward the received signal to the destination. The relay is equipped with EH and ID receivers. We assume that the EH and ID receivers are operating over the same frequency. We also assume that the processing power used by the receive and transmit circuits at the relay is negligible compared to the transmit power of the relay. During the first half of the time slot, the source transmits the N s × 1 precoded vector x s to the relay through the Nr × N s block-fading channel H which is corrupted by additive white Gaussian noise (AWGN) vector nr . The Nr × 1 received vector at the relay, yr , is given by (1) yr = Hx s + nr , where x s is the transmitted signal  from the source whose x s xHs , nr is the Nr × 1 AWGN covariance matrix is R s = vector whose entries are independent identically distributed (i.i.d.) and drawn from the Gaussian distribution with zero that mean and variance equal to σ2r . Furthermore, we assume   the source has an average power constraint, i.e. x s 2 = tr (R s ) ≤ P s , where tr(·) denotes the trace operator. The EH receiver at the relay converts the RF received signal yr to a direct current (DC) by a rectifier without the need to convert from RF band to baseband [10]. Thanks to the law of energy conservation, the harvested power at the relay (energy normalized by the symbol duration), denoted as Qr , is given by     (2) Qr = ζ Hx s 2 = ζ tr HR s H H , where ζ ∈ [0, 1] is the conversion efficiency. During the second half of the time slot, the ID receiver of the relay decodes the received signal yr and forwards the baseband transmitted signal from the relay xr which has a covariance matrix given by the Nr × Nr relay precoding matrix Rr . The relay is constrained   to an average transmit power constraint given by xr xrH = tr (Rr ) ≤ Qr . Afterwards, the Nd × 1 received vector at the destination, yd , is expressed as (3) yd = Gxr + nd , where nd is the Nd × 1 AWGN vector whose entries are i.i.d. and drawn from the Gaussian distribution with zero mean and variance equal to σ2d . Using Gaussian codebooks, the source-destination achievable rate of the MIMO DF relay system, in bits/s/Hz, is known to be given by [11]  R (R s , Rr ) = min RS −R (R s ) , RR−D (Rr ) (4)          1 = min log2  I + HR s H H , log2  I + GRr GH  ,     2 (5) where RS −R and RS −R are the rate of the first hop (namely the S-R link) and the rate of the second hop (namely the R-D link), respectively. For a matrix A, | A| denotes its determinant. III. Optimal Source and Relay Design Let us consider the ideal scenario when the ID and EH receivers at the relay are able to operate simultaneously– each making use of the full received signal. Our main design goal is to maximize the source-to-destination achievable rate. The

optimal covariance matrices R s and Rr are the solutions to the optimization problem (P): (6a) (P) : max R (R s , Rr ) R s ,Rr

s.t. tr (R s ) ≤ P s



tr (Rr ) ≤ ζ tr HR s H R s  0, Rr  0,

H



(6b) (6c) (6d)

(6b) is the transmit average power constraint at the source, and (6c) is the transmit average power constraint at the relay. Problem (P) is a convex optimization problem since the objective function is concave and the constraints are affine [12]. This problem can be solved using the convex optimization tools available in Matlab such as the CVX software [13]. Nevertheless, a mere numerical answer does not provide insight into the structure of the solution. In the sequel, we provide a detailed study of the solution to (P) and an explicit characterization of the optimal precoders at the source and the relay. Note that if the transmit power at the relay in (6c) was independent of the covariance matrix of the source, the joint optimization of (P) can be split into two independent optimization problems where the rate of the S-R link and the rate of the R-D link are maximized independently and the overall rate corresponds to the minimum of the two maximums [11]. However, the joint optimization of the problem (P) cannot immediately split into two independent sub-problems since (6c) depends on both R s and Rr . Let us denote by Rmax the maximum achievable rate solution to (P) and QID the corresponding harvested power. The singular value decompositions (SVDs) of H and G 1/2 H H are H = U H D1/2 H (V H ) and G = UG DG (V G ) , respectively, where U H , V H , UG , and V G are unitary matrices with dimensions N r × r1 , r1 × N s , Nd × r2 , and r2 × Nr , respectively,

DH = diag λH,1 , . . . , λH,r1 and DG = diag λG,1 , . . . , λG,r2 are the diagonal matrices containing the eigenvalues arranged in a decreasing order of HH H and GGH , respectively, and r1 and r2 are the rank of H and G, respectively. Let R∗s be the optimal solution to (P). That is, R∗s is the optimal source precoding matrix that maximizes the end-toend rate. Knowing R∗s , the relay can maximize the achievable rate of the R-D link by solving the optimization  problem   1 log2  I + GRr GH  (P1) : max (7)   Rr 2 (8) s.t. tr (Rr ) ≤ Pr (9) Rr  0,   where Pr = ζ tr HR∗s H H . The optimal solution R∗r of (P1) is presented in Appendix A. The achievable throughput rate over the R-D link is increasing with respect to Pr , or equivalently Qr . Now, let us consider the following optimization problem at the source    1 H  log2  I + HR s H  (P2) : max (10)   Rs 2 (11) s.t. tr (R s ) ≤ P s   H (12) ζ tr HR s H ≥ Qr (13) R s  0, where Qr is a lower bound on the energy harvested at the relay. The relevance of the constraint is explained below. The optimal solution to (P2) is presented in Appendix B. There is a tradeoff between the achievable throughput rate and the achievable harvested power of a one-hop MIMO system [5].

s.t. tr (R s ) ≤ P s , R s  0.

(15)

The optimal solution R s of (P3) is presented in Appendix B-2. The corresponding quantity Qmax = ζ tr HR s H H is the maximum power that can be harvested at the relay. Let RS −R,min be the achievable rate over the S-R link. Given Pr = Qmax , the achievable rate over the R-D link corresponds to its maximum value denoted by RR−D,max . Let Rr be the corresponding precoding matrix at the relay solution to (P1) for Pr = Qmax . A. Case I: When RS −R,min ≥ RR−D,max In Case I, we consider the case when the harvested power at the relay is maximized and, still, the maximum achievable rate over the R-D link, RR−D,max , is below the achievable rate over the S-R link. Then, the optimal solution to (P) is R s for the precoding matrix at the source and Rr for the precoding matrix at the relay. The maximum achievable source-destination rate Rmax is the maximum achievable rate over the R-D link RR−D,max . The corresponding achievable harvested power at the relay QID is equal to Qmax . B. Case II: When RS −R,max ≤ RR−D,min In Case II, we consider the case when the harvested power at the relay is equal or above its minimum value Qmin and, still, the maximum achievable rate over the S-R link RS −R,max is below the achievable rate over the R-D link. Then, the ˆ s for the precoding matrix at optimal solution to (P) is R ˆ the source and Rr for the precoding matrix at the relay. The maximum achievable source-destination rate Rmax is the maximum achievable rate over the S-R link RS −R,max . The corresponding achievable harvested power at the relay QID is equal to Qmin .  C.  Case III: When the intersection of RS −R,min , RS −R,max and RR−D,min , RR−D,max is nonempty In Case III, the maximum achievable throughput rate of the overall link occurs when the achievable throughput rate of the S-R link and the achievable throughput rate of the RD link are equal. This is due to the fact that the achievable

Rate

lin k D R−

RS−R,max

k lin

Rs

RR−D,max

R S−

Hence, the achievable throughput rate of the S-R link RS −R is nonincreasing with respect to Qr . Let us discuss the relevance of the constraint Qr in (P2). • If the value of Qr is zero, the energy harvesting constraint in (12) is trivially satisfied. Problem (P2) becomes equivalent to maximizing the rate of a one-hop ˆ s is presented MIMO system and its optimal solution R in Appendix B-1. Let us denote the resulting rate by RS −R,max . This is the maximum achievable rate over the S-R link regardless of the energy harvested at the  ˆ s H H be the corresponding relay. Let Qmin = ζ tr H R harvested power. Note that for any Qr ∈ [0, Qmin ] the solution to (P2) yields the same rate-maximizing solution. Given Pr = Qmin , the achievable rate over the R-D link corresponds to its minimum value denoted by RR−D,min . ˆ r be the solution to (P1) when Pr = Qmin . Let R • If the value of Qr is very large, (P2) becomes infeasible. In order to obtain the maximum possible value for Qr for which (P2) is feasible, we solve the following optimization problem   (14) (P3) : max Qr = ζ tr HR s H H

Rmax RR−D,min RS−R,min

Qmin

Q

ID

Qmax

Qr

Figure 1. Illustration of the possible solution of (P) in Case III.

Algorithm 1 Proposed Solution of (P) Inputs: H, G, σ2r , σ2d , P s , ζ. Outputs: R∗s , R∗r . Given P s , H, ζ, and σ2r , compute RS −R,max , Qmin , Qmax , and RS −R,min as in (20), (21), (26), and (27), respectively. Given Qmin , Qmax , G, and σ2d , compute RR−D,min and RR−D,max as in Appendix A for Pr equal to Qmin and Qmax , respectively. if RS −R,min ≥ RR−D,max then - QID = Qmax . - R∗s = R s in (25) presented in Appendix B-2. - R∗r = Rr in (16) for Pr = Qmax . else if RS −R,max ≤ RR−D,min then - QID = Qmin . ˆ s in (19) presented in Appendix B-1. - R∗s = R ∗ ˆ r in (16) for Pr = Qmin . - Rr = R else - R∗s and R∗r are given in (18) and (16), respectively. of the equality between - QID is

the argument

RS −R R∗s = RR−D R∗r . The value of QID is efficiently obtained by using the bisection method for a precision of the order 10−5 . end if end if throughput rate of the S-R link is nonincreasing [5] while the achievable throughput rate of the R-D link is increasing with respect to Qr . Consequently, the maximum of the minimum of two rates Rmax happens when they are equal as illustrated in Fig. 1. In order to find this maximum, we need to find the optimum value QID ∈ (Qmin , Qmax ) which satisfies the equality between the rates of the two hops. This optimum value of QID can be efficiently obtained using the bisection method [13]. The procedure for solving (P) is summarized in Algorithm 1 where we present a pseudo-code describing all steps to find the optimal precoding matrices at the source and the relay. IV. Numerical Results In this section, we present some selected simulations of the achievable rate of the source-destination link. We consider the case where the elements of H and G are Rayleigh fading channels with path loss. The Rayleigh fading is assumed to have unit variance. The path loss exponent is taken equal to m = 2.7. 1 1 0.5 All the simulations are obtained for H = m 0.5 1 and d sd 1 1 0.5 G = m 0.5 1 . We denote by d sr , drd , and d sd the distance drd

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Figure 2. The maximum achievable rate Rmax and the corresponding harvested energy QID versus the transmit power at the source P s for d sr /d sd = 0.5, ζ = 1, and N s = Nr = Nd = 2.

Figure 3. The maximum achievable rate Rmax and the corresponding harvested energy QID versus the position of the relay between the source and the destination d sr /d sd for P s = 20 dB, ζ = 1, and N s = Nr = Nd = 2.

between the source and the relay, the distance between the relay and the destination, and the distance between the source and the destination, respectively. In our simulation, we assume that σ2r = σ2d = 1. In all figures, we have plotted the solution obtained by the CVX software as well as our proposed solution in Section III. The bisection method, that is used in Case III, is implemented with a precision of the order 10−5 [14]. In Fig. 2, we have plotted the maximum achievable throughput rate Rmax and the corresponding harvested energy QID versus the transmit power at the source P s . The number of transmit antennas at the source, the relay, and the destination are all equal to 2. We have considered that the relay is equidistant from the source and the destination. We can see that our proposed solution to (P) gives exactly the same result as the CVX software. In Fig. 2(a), we can see that as we increase the transmit power at the source, the maximum achievable throughput rate Rmax increases. Similarly, in Fig. 2(b), we can also see that the achievable harvested power QID increases. In addition, we have QID is always equal to Qmin which corresponds to Case II described in section III. Hence,

the maximum achievable rate is equal to RS −R,max which is in agreement with Fig. 2(a). In Fig. 3, we have plotted the maximum achievable rate Rmax and the corresponding harvested energy QID versus the position of the relay between the source and the destination. The transmit power at the source is equal to 20 dB. The number of transmit antennas at the source, the relay, and the destination are all equal to 2. We can see that our proposed solution to (P) gives exactly the same result as the CVX software. In Fig. 3(a), we can see that as we increase the distance between the source and the relay, the achievable rate over the S-R link decreases as we increase d sr /d sd , while the achievable rate over the R-D link is maximized when the relay is equidistant from the source and the destination. In addition, the maximum achievable rate Rmax decreases as we increase d sr /d sd . Similarly, in Fig. 3(b), we can also see that the achievable harvested power QID decreases. Moreover, we have QID = Qmin , and the maximum achievable rate is equal to RS −R,max . This case corresponds to Case II described in Section III.

V. Conclusion In this paper, we have considered a two-hop MIMO DF relay communication system and focused on the ideal scenario where the EH receiver and the ID receiver at the relay are operating simultaneously. We have obtained the optimal source and relay precoding matrices that maximize the endto-end rate. Although this scheme is not practical, it offers an outer bound for the system’s achievable rate. More practical scenarios like the PS and TS schemes will be the object of our future work. Appendix A Optimal Solution R∗r of (P1) Problem (P1) is equivalent to the throughput maximization of one-hop MIMO system given an average transmit power constraint at the transmitter in (8) and its optimal solution R∗r has the following form (16) R∗r = V G D∗r V GH , where the elements of the diagonal matrix D∗r =

diag λr,1 , . . . , λr,r2 are given by the water-filling solution + 1 1 − , for i = 1, . . . , r2 , where β is the as [14] λr,i = β λG,i Lagrange multiplier satisfying the constraint (8) with equality. Hence, the corresponding rate of the R-D link is given by r2

1 log2 1 + λG,i λr,i . (17) RR−D = 2 i=1 Appendix B Optimal Solution R∗s of (P2) Problem (P2) maximizes the achievable rate of one-hop MIMO system given an average transmit power constraint at the transmitter (11) and an energy harvesting constraint at the receiver (12). This problem was studied in [5] and its optimal solution R∗s has been shown to be H (18) R∗s = W −1/2 Vˆ H D∗s Vˆ H W −1/2 , where W = βI Ns −θH H H, β and θ are the Lagrange multipliers satisfying the constraints (11) and (12), respectively, Vˆ H is an −1/2 = r1 × N s unitary  to the SVD of HW H matrix corresponding 1/2  ∗ ˆ ˆ ˆ ˆ ˆ ˆ with DH = diag λH,1 , . . . , λH,r1 , and D s = U H DH V H  +

diag λ s,1 , . . . , λ s,r1 with λ s,i = 1 − λˆ 1 , for i = 1, . . . , r1 . H,i

1) If Qr = 0: Note that when the value of Qr is zero, the energy harvesting constraint in (12) is trivially satisfied. The problem (P2) becomes equivalent to the throughput maximization of onehop MIMO system given only an average transmit power constraint at the transmitter (11) and the optimal solution to (P2) simplifies to be ˆ sV H ˆ s = VH D (19) R H,

ˆ s = diag λ s,1 , . . . , λ s,r1 are given by the water-filling where D  + 1 1 − , for i = 1, . . . , r1 , where θ solution as [14] λ s,i = θ λH,i is the Lagrange multiplier satisfying the constraint (11) with equality. Hence, the corresponding achievable rate is given by r1

1 log2 1 + λH,i λ s,i , (20) RS −R,max = 2 i=1 and the corresponding harvested power is given by r1  λH,i λ s,i . Qmin = ζ i=1

(21)

2) If Qr = Qmax : On the other hand, if the value of Qr is equal to its maximum value, the solution to (P2) is equivalent  to  (22) (P3) : max Qr = ζ tr HR s H H Rs

s.t. tr (R s ) ≤ P s R s  0.

(23) (24)

Optimization problem (P3) is with respect to R s only and is independent of Rr . Hence, it is equivalent to the maximization problem of the harvested energy for one-hop MIMO system which was previously studied in [5]. In [5], it has been shown that the optimal source covariance matrix R s is given by R s = P s vH,1 vH (25) H,1 , where v1 is the eigenvector of H H H which corresponds to the maximum eigenvalue a1 of H H H. Then, the maximum harvested power at the relay, given R s , is expressed as (26) Qmax = ζa1 P s . From (25), we can see that the optimal R s is ranked one, and the maximum harvested power is obtained by energy beamforming along the strongest eigenmode of H H H. The corresponding rate of the S-R link is given by 1 RS −R,min = log2 (1 + P s a21 ). (27) 2 References [1] R. Want, “Enabling ubiquitous sensing with RFID,” Computer, vol. 37, no. 4, pp. 84–86, Apr. 2004. [2] F. Zhang, S. Hackworth, X. Liu, H. Chen, R. Sclabassi, and M. Sun, “Wireless energy transfer platform for medical sensors and implantable devices,” in Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC’2009), Sept. 2009, pp. 1045– 1048. [3] L. Varshney, “Transporting information and energy simultaneously,” in IEEE International Symposium on Information Theory (ISIT’2008), July 2008, pp. 1612–1616. [4] P. Grover and A. Sahai, “Shannon meets tesla: Wireless information and power transfer,” in IEEE International Symposium on Information Theory Proceedings (ISIT’2010), June 2010, pp. 2363–2367. [5] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” IEEE Transactions on Wireless Communications, vol. 12, no. 5, pp. 1989–2001, May 2013. [6] A. Nasir, X. Zhou, S. Durrani, and R. Kennedy, “Relaying protocols for wireless energy harvesting and information processing,” IEEE Transactions on Wireless Communications, vol. 12, no. 7, pp. 3622–3636, July 2013. [7] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Wireless energy harvesting and information relaying: Adaptive time-switching protocols and throughput analysis,” CoRR, vol. abs/1310.7648, 2013. [8] B. Chalise, W.-K. Ma, Y. Zhang, H. Suraweera, and M. Amin, “Optimum performance boundaries of OSTBC based AF-MIMO relay system with energy harvesting receiver,” IEEE Transactions on Signal Processing, vol. 61, no. 17, pp. 4199–4213, Sept. 2013. [9] X. Di, K. Xiong, and Z. Qiu, “Simultaneous wireless information and power transfer for two-hop OFDM relay system,” CoRR, vol. abs/1407.0166, 2014. [Online]. Available: http://arxiv.org/abs/1407.0166 [10] X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and power transfer: Architecture design and rate-energy tradeoff,” IEEE Transactions on Communications, vol. 61, no. 11, pp. 4754–4767, Nov. 2013. [11] S. Simoens, O. Munoz-Medina, J. Vidal, and A. del Coso, “On the gaussian MIMO relay channel with full channel state information,” IEEE Transactions on Signal Processing, vol. 57, no. 9, pp. 3588–3599, Sept. 2009. [12] S. Boyd and L. Vandenberghe, Convex Optimization. New York, NY, USA: Cambridge University Press, 2004. [13] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming.” [Online]. Available: http://stanford.edu/ boyd/cvx [14] T. M. Cover and J. A. Thomas, Elements of information theory. New York, NY, USA: Wiley-Interscience, 1991.