Simultaneous Wireless Information and Power Transfer ... - IEEE Xplore

Simultaneous Wireless Information and Power Transfer for MIMO Amplify-and-Forward Relay Systems Fatma Benkhelifa 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,slim.alouini}@kaust.edu.sa

Abstract—In this paper, we investigate two-hop Multiple-Input Multiple-Output (MIMO) Amplify-and-Forward (AF) relay communication systems with simultaneous wireless information and power transfer (SWIPT) at the multi-antenna energy harvesting relay. We derive the optimal source and relay covariance matrices to characterize the achievable region between the sourcedestination rate and the harvested energy at the relay, namely Rate-Energy (R-E) region. In this context, we consider the ideal scenario where the energy harvester (EH) receiver and the information decoder (ID) receiver at the relay can simultaneously decode the information and harvest the energy at the relay. This scheme provides an outer bound for the achievable R-E region since practical energy harvesting circuits are not yet able to harvest the energy and decode the information simultaneously. Then, we consider more practical schemes which are the power splitting (PS) and the time switching (TS) proposed in [1] and which separate the EH and ID transfer over the power domain and the time domain, respectively. In our study, we derive the boundary of the achievable R-E region and we show the effect of the source transmit power, the relay transmit power and the position of the relay between the source and the destination on the achievable R-E region for the ideal scenario and the two practical schemes. Index Terms—Energy harvesting, Simultaneous Wireless Information and Power Transfer (SWIPT), MIMO relay systems, Amplify-and-Forward (AF), Rate-Energy (R-E) region, Power Splitting (PS), Time Switching (TS).

I. Introduction Harvesting energy from the environment is a promising technique to prolong the lifetime of the battery powered wireless networks and make the wireless communication systems self-sustaining. Solar, wind, and vibration are known to be the classical sources of harvesting energy at the transmitter. However, the radio frequency (RF) signals from ambient transmitters have recently been considered as another possible source of wireless power transfer (WPT). On the other hand, it is well known that RF signals are used for wireless information transfer (WIT). As a result, simultaneous wireless information and power transfer (SWIPT) has been widely investigated where RF signals are simultaneously used to transmit information from the transmitter and scavenge energy at the receiver. The optimal tradeoff between information rate and energy transfer was investigated in a single-input single-output (SISO) flat-fading channels and frequency selective channels in [2], [3]. These works assumed that the information detection (ID) and energy harvesting (EH) can be done with the same circuit technologies, namely co-located receiver. However, practical This paper was funded by a grant from the office of competitive research funding (OCRF) at KAUST, Saudi Arabia.

receiver designs are not yet able to decode information and harvest energy simultaneously. Thus, [1] extended the work done in [2], [3] and considered two practical schemes where 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). More specifically, in [1], multi-antenna broadcasting system was considered, and the rate-energy (R-E) region was characterized for the co-located and separated receivers. In [4], a SISO case was considered where a general receiver employing a dynamic power splitting (DPS) receiver was proposed which dynamically split the received signals over two time slots with different power ratio. In [5], a two-hop SISO amplifyand-forward (AF) relay system was considered where the relay harvests energy from the transmitted signal from the source and uses the harvested energy 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 [6], 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 tradeoffs between the harvested power at the EH receiver and the information rate at the destination node. In [6], the harvested power from the signal transmitted by the source and the relay was gathered in an independent node that coexists with the two-hop Multiple-Input Multiple-Output (MIMO) relay system. In this paper, we propose to study the two-hop MIMO AF relay systems where the relay is itself an EH multi-antenna node. We assume that CSI is available at the three nodes. We derive the optimal source covariance matrix and the optimal relay amplification matrix to characterize the R-E region. In this context, we consider the ideal scenario where the EH and ID receivers at the relay can simultaneously decode the information and harvest the energy at the relay. This scheme will be an outer bound for the achievable R-E region since the practical energy harvesting circuits are not yet able to decode the information simultaneously. Then, we consider two practical schemes which are the PS and TS schemes. II. System Model and Problem Formulation We consider a two-hop MIMO AF relay communication system where the source, the relay, and the destination are

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equipped with N s , Nr , and Nd antennas, respectively. The direct link between the source and the destination is not considered. The relay operates in a half-duplex 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 ∈ CNr ×Ns and G ∈ CNd ×Nr , respectively, and are assumed to be quasi-static block-fading channels. Indeed, the symbol duration is assumed to be sufficiently small compared to the coherence time of the channel. We assume that the CSI is perfectly known at the three nodes. While both the source and the destination are battery powered, the relay is an energy harvesting node. It harvests the energy from the received RF signal from the source and uses the harvested energy to forward the received signal to the destination. In a realistic scenario, this might happen when the direct link is weak and is suffering from the path attenuation and the shadowing and we rely on the intermediate relay to opportunistically assist the transmission from the source to the destination. The relay is equipped with EH and ID receivers. We assume that the EH and ID receivers are operating at the same frequency. The interference at the relay from other signals is not considered in this paper and can be a further extension of this work. 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 the additive white Gaussian noise (AWGN) vector nr . The Nr × 1 received vector at the relay yr is given by (1) yr = Hx s + nr ,

the relay amplifies the received signal yr by the Nr × Nr relay precoder matrix F. The baseband transmitted signal from the relay, denoted as xr , is then given by (3) xr = Fyr = FHx s + Fnr ,

where x s is the transmitted signal  from the source whose covariance matrix is S = E x s xHs , nr is the Nr × 1 AWGN vector whose entries are independent identically distributed (i.i.d.) and drawn from the Gaussian distribution with zero mean and variance equal to σ2r . Furthermore, we assume that  the source has an average transmit power constraint, i.e. E xs 2 = tr (S) ≤ Ps , where tr(.) denotes the trace operator. The EH receiver at the relay converts the RF received signal yr to a direct current (DC) by the rectifier without need for the conversion from RF band to baseband [4]. Thanks to the law of energy conservation, the harvested energy at the relay (energy normalized by the symbol duration in unit of energy), denoted as Qr , is given by    1  1 Qr = ζ E Hx s 2 = ζ tr HSH H , (2) 2 2 where ζ ∈ [0, 1] is the conversion efficiency and the factor 1/2 is because the EH mode only occurs during the first half of the time slot. Note that the relay is operating in a harvest-save-use mode, rather than a harvest-use mode. The harvest-save-use mode allows the relay to use the harvested energy only from the previous time slot and save the current harvested energy for the next time slot. The harvest-use mode enables the relay to use the harvested energy immediately during the first half of the time slot. The harvest-save mode was only chosen for the sake of simplicity. The harvest-use mode will be studied in the extension of this work. The harvest-save-mode can also be optimized in future works. During the second half of the time slot, the ID receiver of

III. Ideal Scenario: Optimal Source and Relay Design Let us consider the ideal scenario when the ID and EH receivers at the relay can operate simultaneously. In this section, we derive the optimal source covariance matrix S and the optimal relay amplification matrix F to achieve the Rate-Energy (R-E) region. The R-E region characterizes the achievable region between the source-destination rate and the harvested energy at the relay for a given transmit power constraint:      1 (P ) (7) CR−E s = R, Qr ; R ≤ log2  I + GFHSH H FH GH  2  −1     1 × σ2r GFFH GH + σ2d I Nd , Qr ≤ ζ tr HSH H ,  2      H 2 H tr F HSH + σr INr F ≤ Pr , tr (S) ≤ P s , S  0 .

where the relay  transmit  power  is given by   E xr xrH = tr F HSHH + σ2r INr FH ,

(4)

which should be constrained by the average transmit power constraint at the relay Pr . As said previously, we assume that the relay is allowed to use only the harvested energy from the previous time slot, i.e. Pr = 2 Qr,−1 , where Qr,−1 is the harvested energy during the previous time slot and the factor 2 is due to the fact that Qr,−1 is used during half-time the symbol duration. The fact that we use the current harvested energy Qr , which is the harvest-use mode, will be studied in the extension of this work. Afterwards, the Nd × 1 received vector at the destination, yd , is expressed as (5) yd = Gxr + nd = GFHx s + GFnr + 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 . The source-destination achievable rate of the MIMO AF relay system (normalized by the symbol duration), in bits/s/Hz, is known to be given by [7]    −1   1 H H H 2 H H 2 R (S, F) = log2  I + GFHSH F G σr GFF G + σd I Nd .   2 (6) In the following section, we find the optimal source covariance matrix S and the optimal relay precoder matrix F that achieve the tradeoff between the information rate at the destination and the harvested energy at the relay.

Let (Qmax , REH ) and (QID , Rmax ) be the boundary points of the R-E region corresponding to the maximum achievable harvested energy at the relay and the maximum achievable end-to-end rate, respectively. These boundary points will be defined below. The remaining boundary of the R-E region over REH < R < Rmax is defined for QID ≤ Qr ≤ Qmax . Consequently, we consider the following optimization problem (8a) (P) : max R (S, F) , S,F

s.t. tr (S) ≤ P s ,     tr F HSH H + σ2r INr FH ≤ Pr ,

(8b) (8c)

  1 ζ tr HSH H ≥ Qr , S  0, (8d) 2 where (8b) is the average power constraint at the source, (8d) is the energy harvesting constraint at the relay, and (8c) is the average transmit power constraint at the relay. Note that Pr is within the interval 2QID,−1 ≤ Pr ≤ 2Qmax,−1 , where QID,−1 and Qmax,−1 are the boundary points of the R-E region at the previous time slot. Obviously, the problem (P) is not a convex optimization problem [8] since the objective function is not concave with respect to F, and the constraints are not over a convex set. Global optimal solutions of S and F are not guaranteed. First, let us investigate the optimal structure of the source covariance matrix S and the relay amplification matrix F. The equivalent optimization problem is then modified to obtain an equivalent convex optimization problem where the global convergence is guaranteed. Let us denote the singular value decomposition (SVD) of H and G as H (9) H = U H D1/2 H (V H ) , G = UG DG1/2 (V G )H ,

where DS and DF are Nb × Nb diagonal matrices, and V H,1 , V G,1 , and U H,1 contain the first Nb columns of V H , V G , and U H , respectively. Proof: The details of the proof are GIVEN in the Appendix A. Proposition 1 states that the optimal source and relay covariance matrices jointly diagonalize the source-relay-destination channel which becomes equivalent to a set of parallel singleinput single-output (SISO) channels. This result is alike to the MIMO AF relay system with a non-energy harvesting relay. Given (11) and (12), all we need to optimize now is DS and DF , which verifies the following optimization problem

Nb 1 ai bi xi yi (P1) : max log2 1 + , (13a) xi ,yi 2 i=1 bi yi + ai xi + 1 s.t.

xi ≤ P s ,

Nb 

yi ≤ Pr ,

i

i=1

1 ˆ ζ 2

xi ≥ 0, yi ≥ 0, ∀i = 1, . . . , Nb , where ζˆ = σ2r ζ, ai =

1 σ2r

DH (i, i), bi =

1 σ2d

s.t.

(10)

respectively, where U H , V H , UG , and V G are unitary matrices with dimensions Nr × r1 , r1 × N s , Nd × r2 , and r2 × Nr , respectively, DH , and DG 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. Given the fact that the objective function is Schurconcave (equivalently, (−R (S, F)) is Schur-convex [9]), we are able to investigate the structure of the optimal S and F of (P). Proposition 1: Assuming the fact that rank (S) = rank (F) = Nb ≤ min (r1 , r2 ), the optimal source covariance matrix S and the optimal relay amplification matrix F have the following structure

(11) S = V H,1 DS V H,1 H ,

H F = V G,1 DF U H,1 , (12)

Nb 

and y are symmetric. However, the iterative algorithm does not guarantee a global convergence and is very sensitive to the initial starting point. Thus, we will present a more efficient and robust method to solve (P1) that guarantees global convergence. As stated previously, the objective function of (P1) is not a concave function, even though it is the sum of quasiconcave functions [10]. However, for moderate to high SNR values of the links between the source and the relay and between the relay and the destination which occurs when ai xi + bi yi >> 1, the objective function can be shown to be equivalent to the sum of concave functions and is hence concave [11]. In this case, (P1) becomes equivalent to the convex optimization problem (P2) given by

Nb 1 ai bi xi yi log2 1 + , (14a) (P2) : max xi ,yi 2 i=1 bi yi + ai xi

Nb 

ai xi ≥ Q¯ r ,

i=1

(13b) (13c)

DG (i, i), xi = DS (i, i),

yi = σ2r fi (ai xi + 1), and fi = D2F (i, i). The objective function of the problem (P1) is non convex and a global-optimal solution to (P1) is intractable. Local optimal solution of (P1) can be obtained by an iterative algorithm since the roles of x

Nb 

Nb 1 ˆ  ai xi ≥ Q¯ r , ζ 2 i i=1 i=1 (14b) (14c) xi ≥ 0, yi ≥ 0, ∀i = 1, . . . , Nb .

xi ≤ P s ,

Nb 

yi ≤ P r ,

Now, let us introduce a slack variable τi , for i = 1, . . . , Nb , as follows Nb 1 (15a) log2 (1 + τi ) , (P3) : max τi ,xi ,yi 2 i=1 s.t.

Nb 

xi ≤ P s ,

Nb  i

i=1

τi ≤

Nb 1 ˆ  yi ≤ Pr , ζ ai xi ≥ Q¯ r , 2 i=1 (15b)

ai bi xi yi , xi ≥ 0, yi ≥ 0, ∀i = 1, . . . , Nb . bi yi + ai xi (15c)

i xi yi Given the fact that bai yi bi +a can be written as i xi ai bi xi yi 1 = ai xi + bi yi − ai xi (ai xi + bi yi )−1 ai xi bi yi + ai xi 2

− bi yi (ai xi + bi yi )−1 bi yi , (16)

and using the Schur-complement theorem [8], we can show that (P3) is equivalent to the semidefinite programming (SDP) formulation given by Nb 1 (P4) : max (17a) log2 (1 + τi ) , τi ,xi ,yi 2 i=1 s.t.

Nb 

xi ≤ P s ,

i=1

Nb 

yi ≤ Pr ,

i

Nb 1 ˆ  ai xi ≥ Q¯ r , ζ 2 i=1

(17b) ⎤ bi y i ⎥⎥⎥ ⎥⎥⎥  0, ai xi ⎥⎦ ai xi + bi yi − 2τi (17c) (17d) xi ≥ 0, yi ≥ 0, ∀i = 1, . . . , Nb , ⎡ ⎢⎢⎢ai xi + bi yi ⎢⎢⎢ 0 ⎢⎣ bi yi

0 ai xi + bi yi ai xi

or equivalently, in matrix notation, as

1 tr log2 (1 + Dτ ) , (P5) : max (18a) Dτ , D x , D y 2   1 s.t. tr (D x ) ≤ P s , tr Dy ≤ Pr , ζˆ tr (Da D x ) ≥ Q¯ r , (18b) 2

   ⎡ ⎢⎢⎢ Da D x + Db Dy 0 Db Dy ii ii   ⎢⎢⎢⎢ ( Da D x )ii 0 Da D x + Db Dy ⎢⎢⎢   ii  ⎢⎣ ( Da D x )ii D b Dy Da D x + Db Dy − 2 Dτ ii

D x  0, Dy  0, Dτ  0,

⎤ ⎥⎥⎥ or, equivalently, ⎥⎥⎥

Nb ⎥⎥⎥  0, 1 ai bi xi yi ⎥⎥⎦ log2 1 + max , (25a) xi ,yi 2 i=1 bi yi + ai xi + 1 ii (18c) Nb Nb   xi ≤ P s , yi ≤ Pr , xi ≥ 0, yi ≥ 0, ∀i = 1, . . . , Nb , s.t. (18d)

where D x , Dy , Dτ , Da , and Db are diagonal matrices that contain the elements of x, y, τ, a, and b, respectively. Note that (P4) or (P5) can be easily solved by the CVX software [12] in Matlab. Next, let us define the boundary points of the R-E region (Qmax , REH ) and (QID , Rmax ). The boundary point (Qmax , REH ) corresponds to the case when we are interested in maximizing the achievable harvested energy, regardless of the achievable rate. The boundary point (QID , Rmax ) corresponds to the case when we are interested in maximizing the achievable rate, regardless of the achievable harvested energy at the relay. A. Maximum Achievable Harvested Energy at the Relay When Qr is very large, (P) becomes infeasible. Hence, we propose to investigate the maximum achievable harvested energy at the EH receiver of the relay and we solve the following optimization problem where the objective function to be maximized is the harvested energy at the relay   1 (19a) (P6) : max Qr (S) = ζ tr HSH H S 2 s.t. tr (S) ≤ P s , S  0. (19b) This optimization problem (P6) is only with respect to S and independent of F, and hence it is equivalent to the one-hop MIMO systems which were previously studied in [1]. In [1], it has been shown that the optimal source covariance matrix S is given by (20) SEH = P s vH,1 vH H,1 , where v1 is the eigenvector of H H H which corresponds to the maximum eigenvalue a1 of H H H. The maximum harvested energy at the relay, given SEH , is expressed as 1 Qmax = ζ a1 P s . (21) 2 From (20), we can see that the optimal SEH is ranked one, and the maximum harvested energy is obtained by energy beamforming along the strongest eigenmode of H H H. Consequently, the optimal FEH should be ranked one and is given by Pr (22) vG,1 uH FEH = 2 H,1 , σr (a1 P s + 1) where vG,1 , and uH,1 is the first column and the first row of V G and U H . The end-to-end rate corresponding to the maximum harvested energy at the relay is given by

1 a1 b1 P s Pr REH = log2 1 + . (23) 2 a1 P s + b1 Pr + 1 B. Maximum Achievable End-to-End Rate When Qr is zero, the constraint (8d) is trivially satisfied. Problem (P) becomes equivalent to maximizing the rate of a two-hop MIMO AF relay system regardless of the harvested energy at the relay. As a result, Problem (P) simplifies to (24a) (P7) : max R (S, F) , S,F

(24b) s.t. tr (S) ≤ P s ,     H 2 H tr F HSH + σr INr F ≤ Pr , S  0, (24c)

i=1

i

(25b) which can be solved in a similar way as above by assuming the moderate to high SNR regime and using the slack variable τi , for i = 1, ..., Nb . This problem is similar to the nonenergy harvesting relay node which was previously solved in [9]. Let us denote its solutions by SID and FID . Hence, the corresponding achievable rate is denoted by Rmax and the corresponding achievable harvested energy is given by   1 QID = ζ tr HSID H H . (26) 2 IV. Practical Schemes The ideal scenario where the EH receiver and the ID receiver can operate simultaneously at the relay remains beyond the existing technology of energy harvesting circuits. In this section, we present practical schemes which are the PS and TS schemes. A. Power Splitting (PS) The PS scheme splits the power between the EH receiver and the ID receiver at the relay. We consider the case when we have uniform power splitting among the receiving antennas and let ρ be the proportion of the power split to the antennas of the EH receiver. The achievable R-E region, in this case, is given by    1 PS CR−E (P s ) = ∪ (R, Qr ) ; R ≤ log2  I + (1 − ρ)GFHSH H  0≤ρ≤1 2   −1  H H 2 H H 2 × F G σr GFF G + σd I Nd ,      H 2 H tr F (1 − ρ) HSH + σr INr F ≤ Pr ,    1 H Qr ≤ ζ ρ tr HSH , tr (S) ≤ P s , S  0 . (27) 2 It is obvious that the two boundary points (0, Qmax ) and (Rmax , 0) can be achieved when ρ = 1 and ρ = 0, respectively. B. Time Switching (TS) The TS scheme operates over two orthogonal time slots, one for energy harvesting and the other for information transmission. Let α denotes the time proportion of the time transmission allocated for the EH mode, with 0 ≤ α ≤ 1. In this case, the achievable R-E region is given by   1−α TS (P s ) = ∪ (R, Qr ) ; R ≤ log2  I + GFHS1 H H CR−E  0≤α≤1 2   −1  H H 2 H H 2 × F G σr GFF G + σd I Nd ,      H 2 H (1 − α) tr F HS1 H + σr INr F ≤ Pr ,   Qr ≤ αζ tr HS2 H H ,  (28) tr (S1 ) ≤ P s , tr (S2 ) ≤ P s , S1  0, S2  0 .

Alike the PS scheme, the two boundary points (0, Qmax ) and (Rmax , 0) can be achieved when α = 1 and α = 0, respectively.

region for harvest-use mode which allows the relay to use the harvested energy immediately and forward the signal to the destination.

V. Numerical Results In this section, we present some selected numerical simulations of the boundary of the R-E region. We consider the case when H and G represent Rayleigh fading channels with path loss. The Rayleigh fading is assumed to have a unit variance. The path loss exponent is taken equal to m = 2.7 which corresponds to an urban cellular network environment. All the simulations are obtained by averaging over 100 independent realizations of H and G. We denote by d sr , drd , and d sd the distance between the source and the relay, the distance between the relay and the destination, and the distance between the source and the destination, respectively. We assume that σ2r = σ2d = 1, and ζ = 1. In Fig. 1, we have plotted the R-E region between the achievable end-to-end rate and the achievable harvested energy for different positions of the relay between the source and the destination. The transmit power at the source and the relay are both chosen equal to 20 dB. The number of transmit antennas at the source, the relay, and the destination is all equal to 2. In Fig. 1, we can see that the closest the relay to the source is, the largest the maximum achievable harvested energy is. We can see also that the maximum achievable throughput rate is achieved when the relay is 0.5 away from the destination. When the relay is at a distance closer to the destination or the source, the maximum achievable throughput rate decreases. This can be explained by the fact that the channel between the source and the relay (respectively the channel between the relay and the destination) diminishes as the relay gets closer to the destination (respectively, closer to the source). In Fig. 2, we have plotted the R-E region between the achievable end-to-end rate and the achievable harvested energy for different values of the transmit power at the source and the relay. The number of transmit antennas at the source, the relay, and the destination is all equal to 2. The relay is equidistant from the source and the destination. In Fig 2, we can see that as we increase the transmit power at the source, the maximum achievable harvested energy increases. Compared to Fig. 1 (a) where Pr = 20 dB, we can see that the maximum achievable throughput rate increases when we increase the relay transmit power, or equivalently the harvested energy at the relay during the previous time slot, as in Fig 2 (a). Same comment is valid for (b) and (c) in Fig. 2. We can see also in Figs. 1 and 2 that TS PS (P s ) ⊆ CR−E (P s ). we have CR−E

Appendix A Proof of Proposition 1 In this section, we prove the result given in Proposition 1. Let Nb = rank (S) = rank (F), and let B a N s × Nb matrix defined such as S = BBH . Let H = GFHS1/2 , and C = σ2r GFFH GH + σ2d I Nd . Hence, the achievable rate in (6) is equivalent to   H R (S, F) = log2  I Ns + H C−1 H . (29)

VI. Conclusion In this paper, we considered a two-hop MIMO AF relay communication system and we investigated the R-E region when the relay is itself an EH node. We have considered the ideal scenario where the EH and ID receivers at the relay are operating simultaneously. Although it is not practical, this scheme offers an outer bound for the achievable R-E region. Subsequently, more practical schemes were considered which are the PS and TS schemes. We derived the achievable R-E region for the ideal scenario and the two practical schemes. We also showed its dependence on the transmit power at the source, the transmit power at the relay, and the position of the relay between the source and the destination. An immediate extension of this work is to investigate the boundary of R-E

It has been shown that the achievable rate in (6) is fully characterized by the diagonal elements of the minimum mean square error (MMSE) [9]. In fact, we have (30) R (S, F) = − log2 |MMSE| , where the MMSE is given by [13] −1 H MMSE = I Ns + H C−1 H .

(31)

As such, minimizing the MMSE is equivalent to maximizing the achievable rate. Let us denote ˆ = 1 HB = U A D1/2 V H (32) H A A σr ˆ = σr GF = X I Nr + A −1/2 , F (33) σd where U A and V A are unitary matrices with dimensions Nr ×Nb and Nb ×Nb , respectively, and DA is the Nb ×Nb diagonal matrix where the diagonal elements are the eigenvalues sorted in an increasing order of A, where A and X are defined as follows ˆH ˆ H = 1 HSH H = U A DA U H (34) A=H A σ2r

σr X= GF I Nr + A 1/2 = U X DX V H (35) X, σd where U X , and V X are unitary matrices with dimensions Nd × Nb , Nr × Nb , respectively, and DX is Nb × Nb diagonal matrix containing the singular values of X in an increasing order. In [9], it has been shown that (31) can be written as (36) MMSE = I Ns − V A D1 Q2H D2 Q2 D1 V H A,   −1/2 −1 , D2 = I Nb + D−2 , and where D1 = I Nb + D−1 A X H Q2 = V X U A . Using Lemma 1 and 2 in [9], we have that the column vector containing the diagonal elements of V A D1 Q2H D2 Q2 D1 V H A is weakly submajorized [14] when V A = ΦNb and Q2 = ΦNb , where ΦNb is a Nb × Nb matrix where the diagonal elements have unit norm and the non-diagonal elements are zero. Given the fact that the achievable rate is Schur-concave and increasing with respect to the column vector containing the diagonal elements of V A D1 Q2H D2 Q2 D1 V H A and using Lemma 3 in [9], the achievable rate is maximized when V A = ΦNb and Q2 = ΦNb . In particular, we choose V X = U A , without loss of generality. Now let us investigate the optimal structure of S and F. First, we assume that the optimal S is given and we look for the structure of the optimal F. Recall that it is well known that the maximum of a multivariable objective function can be computed as max f (x, y) = max max f (x, y) [8]. The problem x,y

x

y

(P) with respect to F, for a given S, is given by (P8) : max R (S, F) F     s.t. tr F HSH H + σ2r INr FH ≤ Pr .

(37a) (37b)

(a) dsr/dsd= 0.1

10

8

6

4

2

0 0

1

(c) dsr/dsd= 0.9

(b) dsr/dsd= 0.5 1400

Optimal scheme (CVX) PS scheme TS scheme

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x 10

2 3 4 5 Achievable throughput rate (bits/s/Hz)

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7

Optimal scheme (CVX) PS scheme TS scheme

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50

0 0

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5

6

Figure 1. Rate-Energy region for N s = Nr = Nd = 2, P s = 20 dB, Pr = 20 dB, and different values of d sr /d sd . (a) Ps= Pr= 30 dB

(a) Ps=20 dB, Pr= 2*Qmax 14000

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Optimal scheme (CVX) PS scheme TS scheme

Optimal scheme (CVX) PS scheme TS scheme

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Achievable harvested energy (Unit of energy)

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Optimal scheme (CVX) PS scheme TS scheme

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Figure 2. Rate-Energy region for N s = Nr = Nd = 2, d sr /d sd = 0.5, and different values of P s and Pr .

Hence, the problem is exactly like the conventional relay-only design formulation which was previously studied [9], [15]– [17] and it has been shown that if the objective function is Schur-concave [14], the structure of the optimal relay amplification matrix is given by (12) and corresponds to U X = UGH , V X = U A = U H , and ΦNb = I Nb . Now, let us investigate the structure of the optimal S. Recall that the problem (P) is different from the optimization problem in [9] since the constraint (8d) reveals from the energy harvesting constraint. However, the MMSE in (36) (or equivalently the objective function) and the constraints (8d) and (8c) depend only on DA , independently of the structure of S. The structure of S is only significant in (8b). This implies that the structure of the optimal S is exactly the same as the one when we have only an average power constraint at the source [9], which is equal to (11). This concludes our proof. References [1] 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. [2] L. Varshney, “Transporting information and energy simultaneously,” in IEEE International Symposium on Information Theory (ISIT’2008), July 2008, pp. 1612–1616. [3] 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. [4] 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, November 2013.

[5] 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. [6] B. Chalise, Y. Zhang, and M. Amin, “Energy harvesting in an OSTBC based amplify-and-forward MIMO relay system,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP’2012), Kyoto, Japan, March 2012, pp. 3201–3204. [7] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York, NY, USA: Wiley-Interscience, 1991. [8] S. Boyd and L. Vandenberghe, Convex Optimization. New York, NY, USA: Cambridge University Press, 2004. [9] Y. Rong, X. Tang, and Y. Hua, “A unified framework for optimizing linear nonregenerative multicarrier MIMO relay communication systems,” IEEE Transactions on Signal Processing, vol. 57, no. 12, pp. 4837–4851, December 2009. [10] K. J. Arrow and A. C. Enthoven, “Quasi-concave programming,” Econometrica, vol. 29, no. 4, pp. pp. 779–800. [11] B. Chalise, Y. Zhang, and M. Amin, “Simultaneous transfer of energy and information for MIMO-OFDM relay system,” in 1st IEEE International Conference on Communications in China (ICCC’2012), Beijing, China, August 2012, pp. 481–486. [12] M. Grant and S. Boyd, “CVX: Matlab Software for Disciplined Convex Programming.” [Online]. Available: http://stanford.edu/ boyd/cvx [13] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Upper Saddle River, NJ, USA: Prentice-Hall, Inc., 1993. [14] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications. New York: Academic Press, 1979. [15] X. Tang and Y. Hua, “Optimal design of non-regenerative MIMO wireless relays,” IEEE Transactions on Wireless Communications, vol. 6, no. 4, pp. 1398–1407, April 2007. [16] Z. Fang, Y. Hua, and J. Koshy, “Joint source and relay optimization for a non-regenerative MIMO relay,” in Fourth IEEE Workshop on Sensor Array and Multichannel Processing (SAM’2006), Waltham, MA, USA, July 2006, pp. 239–243. [17] L. Sanguinetti, A. D’Amico, and Y. Rong, “A tutorial on the optimization of amplify-and-forward MIMO relay systems,” IEEE Journal on Selected Areas in Communications, vol. 30, no. 8, pp. 1331–1346, September 2012.

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