# Massive MIMO Relay Systems with Multipair Wireless Information and Power Transfer

Authors: Hongwu Liu; Kyung Hi Chang; Kyung Sup Kwak
Journal: Mobile Information Systems
 (2017)

---

## Abstract

This paper investigates destination-aided simultaneous wireless information and power transfer (SWIPT) for a decode-and-forward relay network, in which massive multiple-input multiple-output antennas are deployed at relay to assist communications among multiple source-destination pairs. During relaying, energy signals are emitted from multiple destinations when multiple sources are sending their information signals to relay. With power splitting and unlimited antennas at relay, asymptotic expression of harvested energy is derived. The analysis reveals that asymptotic harvested energy is independent of fast fading effect of wireless channels; meanwhile transmission powers of each source and destination can be scaled down inversely proportional to the number of relay antennas. To significantly reduce energy leakage interference and multipair interference, zero-forcing processing and maximum-ratio combing/maximum-ratio transmission are employed at relay. Fundamental trade-off between harvested energy and achievable sum rate is quantified. It is shown that asymptotic sum rate is neither convex nor concave with respect to power splitting and destination transmission power. Thus, a one-dimensional embedded bisection algorithm is proposed to jointly determine the optimal power splitting and destination transmission power. It shows that destination-aided SWIPT are beneficial for harvesting energy and increasing sum rate. The significant sum rate improvements of the proposed schemes are verified by numerical results.


---

## Body

## 1. Introduction

Recently, simultaneous wireless information and power transfer (SWIPT) have been envisioned for wireless relay networks [1–3]. The focuses on SWIPT for relay networks mainly include two relaying protocols, that is, time switching (TS) relaying and power splitting (PS) relaying [1]. In TS relaying, a relay harvests energy from a source-emitted RF signal and forwards source signal in a time-division manner. In PS relaying, a relay extracts energy for forwarding from its received source signal with PS operation. Generally, PS relaying can increase its spectrum efficiency by reducing the consumed time slots compared to that of TS relaying. In attempting to effectively utilize available time slots, SWIPT schemes with destination-aided energy flow (EF) have emerged [2–4]. In [2], the destination-aided EF was proposed for TS relaying protocol. In [3], an autonomous multiple-input multiple-output (MIMO) relay employing PS-based energy receiver was investigated with destination-aided EF. For cognitive relay systems, the authors in [4] proposed sending EFs from destination to energy-constrained relay.In the above mentioned destination-aided EF transmission schemes [2, 3], only a single source-destination pair was considered. However, the demands for relay-assisted SWIPT with multiple source-destination pairs have emerged. For instance, authors in [5] proposed several power allocation schemes for a relay network, where multiple source-destination pairs communicate with each other via a common energy harvesting relay. In [6], the outage performances of SWIPT in large-scale networks with/without relay were analyzed, where multiple receivers harvest energy using PS technique. By employing multiple energy harvesting relays to assist multipair communications, the authors in [7] proposed distributed PS for SWIPT by using game theory. In [8], multiple relays were applied to assist communications from multiple base stations to multiple cell-edge users and PS-based energy receivers were deployed at multiple relays. For wireless power transfer enabled collaborative mobile clouds, cellular data distributing among multiple mobile users was investigated in [9]. It has shown that the energy-constrained relays need to harvest energy from ambient radio-frequency (RF) signals to support multipair communications in future dense heterogeneous networks [6–10].One challenge in relay-assisted SWIPT is a limited amount of harvested energy. Due to path-loss and inefficient RF-to-DC conversion, only a small fraction of energy emitted by a source can be harvested at relay. In order to harvest energy efficiently, smart antennas were employed in SWIPT systems (see [11] and references therein). In [12], TS and PS receivers were proposed for SWIPT in MIMO broadcasting channels. A low-complexity antenna switching for MIMO relay networks with SWIPT was investigated in [13]. Moreover, multiple antennas were deployed at relay to harvest energy from EF emitted by a single destination [3]. Although MIMO techniques have improved SWIPT performance to a certain extent, harvesting enough energy cannot be guaranteed by a limited number of antennas.Recently, massive MIMO techniques were applied to improve wireless transmission capacity by exploiting its large array gain [14, 15]. A review of massive MIMO can be found in [16]. By employing linear signal processing [17], massive MIMO techniques have obtained increased signal-to-interference-plus-noise ratio (SINR) and power efficiency, which makes massive MIMO suitable to be deployed in practical SWIPT prototypes [18]. In [19], massive MIMO antennas were employed at a hybrid data-and-energy access point such that the minimum data rate among user nodes can be maximized whenever the number of antennas at the access point grows without bound. In [20], the authors proposed a low-complexity antenna partitioning algorithm for SWIPT systems with massive MIMO. Subjecting to a minimum harvested energy constraint, interference can be mitigated and throughput can be maximized [20]. Note that all the above mentioned works on SWIPT with massive MIMO were constrained in single-hop communications systems. To the best of our knowledge, only a few works have been conducted for multipair massive MIMO relay networks with SWIPT; for example, the authors in [21] proposed TS and PS relaying protocols for multiway amplify-and-forward (AF) relay networks, where information sharing among multiple users was aided by a massive MIMO relay.In this paper, we propose a two-phase relaying protocol with destination-aided EFs for a decode-and-forward (DF) relay network, where multiple source-destination pairs communicate via a massive MIMO relay [10]. Different from the work of [7], where synchronization among multiple relays is required, we consider only a single relay with massive MIMO antennas and destination-aided EFs. This model has many interests, for example, in environmental sensor networks, Internet of things, and future dense heterogeneous networks, where interuser communications can be realized with the aid of an energy-constrained relay [3–10]. The main contributions of this paper can be summarized as follows:(i)
The important performance metrics including asymptotic harvested energy and asymptotic achievable rate are derived for the case in which the number of relay antennas grows without bound.(ii)
We show that destination-aided EFs are beneficial for boosting energy harvesting and, hence, the achievable sum rate can be improved significantly with the aid of destination-aided EFs. The optimal destination transmission power that maximizes the achievable sum rate is derived in closed-form. The optimal PS factor that maximizes the achievable sum rate is derived in closed-form.(iii)
We also show that by using zero-forcing (ZF) processing and maximum-ratio combing/maximum-ratio transmission (MRC/MRT) at relay, energy leakage interference (ELI) and multipair interference (MI) can be cancelled completely as the number of relay antennas grows without bound. When the number of relay antennas is large, transmission powers of each source and each destination can be scaled inversely proportional to the number of relay antennas.(iv)
We show that asymptotic sum rate is neither convex nor concave with respect to PS and destination transmission power, such that conventional convex optimization methods cannot be applied. We propose a one-dimensional embedded bisection (EB) algorithm to jointly determine the optimal PS and destination transmission power, so that the sum rate can be improved significantly.The rest of this paper is organized as follows. Section2 describes the system model and presents ZF processing and MRC/MRT processing. Section 3 presents the asymptotic analysis of system performance. The one-dimensional EB algorithm is also presented in Section 3. Section 4 presents numerical results and discusses the system performance of the proposed schemes. Finally, Section 5 summarizes the contributions of this study.Notation. The superscripts ·T, ·∗, and ·H represent the transpose, conjugate, and conjugate-transpose, respectively. E· and Var· stand for the expectation and variance operations, respectively. CM×N denotes the M×N complex space and CN0,Σ stands for the circularly symmetric complex Gaussian distribution with zero mean and variance matrix Σ. Unless otherwise stated, vectors and matrices are, respectively, represented by bold lowercase and uppercase letters. IK is the K×K identity matrix, 0M×K is the M×K zero matrix, and 0K is the K×K zero matrix. TrΣ denotes trace operation of a matrix Σ.


## 2. System Model

A block diagram of the considered multipair massive MIMO DF relay network is depicted in Figure1, where K single-antenna equipped sources (S1,S2,…,SK) transmit their information signals to K single-antenna equipped destinations (D1,D2,…,DK) via an M-antenna equipped wireless-powered massive MIMO relay (R). Without loss of generality, we assume that M is larger than K in this paper. We also assume that a direct link between the source and destination of each pair does not exist due to large path-loss or obstacles. All nodes work in half-duplex mode. Moreover, noises at relay’s information detecting (ID) receiver and destinations’ receivers are modeled as complex zero-mean additive noises, while noise power at relay’s energy harvesting (EH) receiver is assumed to be small, so that it is neglected.Figure 1
Multipair massive MIMO relay network.The relaying protocol consists of two time phases. In phase I, all source nodes transmit their information signals R for forwarding. Meanwhile, all destination nodes transmit their EFs to R. Denote the PS factor byθ∈0,1; the received signal power at R is split into a ratio of θ:1-θ for EH and ID processing, respectively. Denote the transmitted signals from the K sources and K destinations by xS≜xS,1xS,2⋯xS,KT and xD≜xD,1xD,2⋯xD,KT, respectively, where xS,k is the information signal transmitted by Sk and xD,k is the EF signal transmitted by Dk. We assume ExSxSH=IK and ExDxDH=IK. The received signal at the input of the ID receiver can be written as(1)yR,ID=1-θ1/2PSGSRxS+PDGDRxD+nR,where GSR∈CM×K and GDR∈CM×K are the channel matrices from the K sources and K destinations to R, respectively, PS and PD are the average transmission powers of each source and each destination, respectively, and nR is the effective additive noise vector at R satisfying EnRnRH=σR2IM. In practice, the effective noise can be defined as nR≜1-θnR′+nR′′, where nR′ and nR′′ are additive noise before and after the passive PS, respectively. The channel matrices GSR and GDR are, respectively, modeled as(2)GSR=HSRDSR1/2,GDR=HDRDDR1/2,where HSR~CNM×K0M×K,IM⊗IK and HDR~CNM×K0M×K,IM⊗IK are the small-scale fading matrices modeling independent fast fading and DSR and DDR are the K×K diagonal matrices modeling the distance-related path-loss. The kth diagonal elements of DSR and DDR are denoted by βSR,k and βDR,k, respectively. Moreover, βSR,k and βDR,k are assumed to be constant over many channel coherence intervals as they change very slowly with time. Next, by exploiting reciprocity property of wireless channels, the channel matrix from R to the K destinations in phase II can be defined as(3)GRD=GDRT=DRD1/2HRD,where HRD=HDRT and DRD=DDRT.AtR, the harvested energy at the EH receiver can be expressed as(4)Eh=τuηθPSTrGSRHGSR+PDTrGDRHGDR,where τu is the time duration of each phase and η is the RF-to-DC conversion efficiency. Thanks to the harvested energy in phase I, the relay transmission power PR=Eh/τu can be expressed as(5)PR=ηθPSTrGSRHGSR+PDTrGDRHGDR.In phase II, the relay forwards the decoded and processed signal, xR∈CM×1, to the K destinations by using the harvested energy during phase I. The received signals at the K destinations can be written in a vector form as(6)yD=PRGRDxR+nD,where nD is the additive noises at the K destinations satisfying EnDnDT=σD2IK.


### 2.1. Channel Estimation

We adopt the minimum mean-squared error (MMSE) method to obtain the channel state information (CSI) estimation. A part of coherence time is used for channel estimation. All the sources and destinations simultaneously send their pilot sequences ofτp symbols to R. The received signal at R is given by(7)Yp=τpPpGSRΦS+τpPpGDRΦD+Np,where Pp is the transmission power of each symbol, Np is the additive noise matrix whose elements have zero mean and variance σR,p2. In addition, the kth rows of ΦS∈CK×τp and ΦD∈CK×τp are the pilot sequences transmitted from Sk and Dk, respectively. It is required that τp≥2K, so that the pairwise orthogonality of the pilot sequences, that is, ΦSΦSH=IK, ΦDΦDH=IK, and ΦSΦDH=0K, can be guaranteed.The MMSE channel estimates ofGSR and GRD are, respectively, given by [17](8)G^SR=GSRD~SR+1τpPpNpΦSHD~SR,G^DR=GDRD~DR+1τpPpNpΦDHD~DR,where D~SR≜σR,p2DSR-1/τpPp+IK-1 and D~DR≜σR,p2DDR-1/τpPp+IK-1. Denote the estimation error matrices of GSR and GDR by ESR and ESR, respectively. Then, we have(9)GSR=G^SR+ESR,GDR=G^DR+EDR.Since G^SR, ESR, G^DR, and EDR are independent, we have G^SR~CN0M×K,D^SR, ESR~CN0M×K,DSR-D^SR, G^DR~CN0M×K,D^DR, and EDR~CN0M×K,DDR-D^DR, where D^SR and D^DR are diagonal matrices whose kth elements are given by σSR,k2≜τpPpβSR,k2/τpPpβSR,k+σR,p2 and σDR,k2≜τpPpβDR,k2/τpPpβDR,k+σR,p2, respectively. Due to wireless channel reciprocity, G^RD=G^DRT is used as the MMSE channel estimate for GRD.


### 2.2. Linear Processing at Relay

The relay employs linear processing for DF relaying. In phase I, R uses a linear receiver matrixWH to separate the received signal into K streams. With the linear receiver, the received signal becomes(10)r=WHyR,ID=1-θPSWHGSRxS+1-θPDWHGDRxD+WHnR.Then, the kth element of r is used for detecting the signal transmitted from Sk, which can be expressed as(11)
r
k
=
1
-
θ
P
S
w
k
H
g
S
R
,
k
x
S
,
k
︸
desired  signal
+
1
-
θ
P
S
∑
j
≠
k
K
w
k
H
g
S
R
,
j
x
S
,
j
︸
multipair  interference
+
1
-
θ
P
D
w
k
H
G
D
R
x
D
︸
energy  leakage  interference
+
w
k
H
n
R
︸
noise
,where wk and gSR,k are the kth columns of W and GSR, respectively.After detecting the received signal, the relay node applies linear precoding to the detected signal before broadcasting. With DF relaying, the relay first decodes the original source signal and then regenerates the signal [22]. Thus, the transmitted signal at R can be expressed as(12)xR=PRAxS,where A∈CM×K is a linear precoding matrix. From (6) and (12), the received signal at Dk can be written as(13)yD,k=PRgDR,kTakxS,k+PR∑j≠kKgDR,kTajxS,j+nD,k,where gDR,k and ak are the kth columns of GDR and A, respectively, and nD,k is the kth element of nD.


#### 2.2.1. ZF Processing

Since both theK sources and K destinations transmit their signals to R in phase I and R broadcasts the processed streams to the K destinations in phase II, MI will affect both the received signals at R and K destinations. For the received signal at R, MI is the term (1-θ)PS∑j≠kKwkHgSR,jxS,j, whereas, for the received signal at Dk, MI is PR∑j≠kKgDR,kTajxS,j. Furthermore, the received signal at R also suffers from ELI resulting from EFs transmitted from the K destinations to R in phase I, which can be represented by 1-θPDwkHGDRxD. In such a case, the relay applies the ZF receiver and ZF precoding to process the signal. By ZF processing, both MI and ELI are removed via projecting each intended information stream onto the orthogonal complement of MI and ELI. To this end, perfect CSI is required at R. However, when only the estimate of CSI is available at R, MI and ELI still exist.The ZF processing matrices atR are given by(14)WH=WZFH≜G^SRHG^SR-1G^SRH,(15)A=AZF≜αZFG^RDHG^RDG^RDH-1,respectively. In (15), the normalization factor αZF is chosen to satisfy xR2=PR. Therefore, we have [23](16)αZF≜1TrG^RDHG^RD-1.


#### 2.2.2. MRC/MRT Processing

Since ZF processing neglects the noise effect, the corresponding detection performance is poor when the signal-to-noise ratio (SNR) is low. By contrast, MRC/MRT processing neglects MI to maximize the received SNR. As a result, MRC/MRT processing achieves a better detection performance than that of ZF processing in the low SNR region at the cost of achieving a worse detection performance than that of ZF processing in the high SNR region. Similarly to ZF processing, MRC/MRT processes the received signal and amplifies the processed signal according to (10) and (12). The MRC receiver and MRT precoding matrices can be, respectively, expressed as [23](17)WH=WMRCH≜G^SRH,A=AMRT≜αMRTG^RDH.Furthermore, the normalization factor for the MRT precoding is given by [23](18)αMRT≜TrG^RDHG^RD-1/2.


## 2.1. Channel Estimation

We adopt the minimum mean-squared error (MMSE) method to obtain the channel state information (CSI) estimation. A part of coherence time is used for channel estimation. All the sources and destinations simultaneously send their pilot sequences ofτp symbols to R. The received signal at R is given by(7)Yp=τpPpGSRΦS+τpPpGDRΦD+Np,where Pp is the transmission power of each symbol, Np is the additive noise matrix whose elements have zero mean and variance σR,p2. In addition, the kth rows of ΦS∈CK×τp and ΦD∈CK×τp are the pilot sequences transmitted from Sk and Dk, respectively. It is required that τp≥2K, so that the pairwise orthogonality of the pilot sequences, that is, ΦSΦSH=IK, ΦDΦDH=IK, and ΦSΦDH=0K, can be guaranteed.The MMSE channel estimates ofGSR and GRD are, respectively, given by [17](8)G^SR=GSRD~SR+1τpPpNpΦSHD~SR,G^DR=GDRD~DR+1τpPpNpΦDHD~DR,where D~SR≜σR,p2DSR-1/τpPp+IK-1 and D~DR≜σR,p2DDR-1/τpPp+IK-1. Denote the estimation error matrices of GSR and GDR by ESR and ESR, respectively. Then, we have(9)GSR=G^SR+ESR,GDR=G^DR+EDR.Since G^SR, ESR, G^DR, and EDR are independent, we have G^SR~CN0M×K,D^SR, ESR~CN0M×K,DSR-D^SR, G^DR~CN0M×K,D^DR, and EDR~CN0M×K,DDR-D^DR, where D^SR and D^DR are diagonal matrices whose kth elements are given by σSR,k2≜τpPpβSR,k2/τpPpβSR,k+σR,p2 and σDR,k2≜τpPpβDR,k2/τpPpβDR,k+σR,p2, respectively. Due to wireless channel reciprocity, G^RD=G^DRT is used as the MMSE channel estimate for GRD.


## 2.2. Linear Processing at Relay

The relay employs linear processing for DF relaying. In phase I, R uses a linear receiver matrixWH to separate the received signal into K streams. With the linear receiver, the received signal becomes(10)r=WHyR,ID=1-θPSWHGSRxS+1-θPDWHGDRxD+WHnR.Then, the kth element of r is used for detecting the signal transmitted from Sk, which can be expressed as(11)
r
k
=
1
-
θ
P
S
w
k
H
g
S
R
,
k
x
S
,
k
︸
desired  signal
+
1
-
θ
P
S
∑
j
≠
k
K
w
k
H
g
S
R
,
j
x
S
,
j
︸
multipair  interference
+
1
-
θ
P
D
w
k
H
G
D
R
x
D
︸
energy  leakage  interference
+
w
k
H
n
R
︸
noise
,where wk and gSR,k are the kth columns of W and GSR, respectively.After detecting the received signal, the relay node applies linear precoding to the detected signal before broadcasting. With DF relaying, the relay first decodes the original source signal and then regenerates the signal [22]. Thus, the transmitted signal at R can be expressed as(12)xR=PRAxS,where A∈CM×K is a linear precoding matrix. From (6) and (12), the received signal at Dk can be written as(13)yD,k=PRgDR,kTakxS,k+PR∑j≠kKgDR,kTajxS,j+nD,k,where gDR,k and ak are the kth columns of GDR and A, respectively, and nD,k is the kth element of nD.


### 2.2.1. ZF Processing

Since both theK sources and K destinations transmit their signals to R in phase I and R broadcasts the processed streams to the K destinations in phase II, MI will affect both the received signals at R and K destinations. For the received signal at R, MI is the term (1-θ)PS∑j≠kKwkHgSR,jxS,j, whereas, for the received signal at Dk, MI is PR∑j≠kKgDR,kTajxS,j. Furthermore, the received signal at R also suffers from ELI resulting from EFs transmitted from the K destinations to R in phase I, which can be represented by 1-θPDwkHGDRxD. In such a case, the relay applies the ZF receiver and ZF precoding to process the signal. By ZF processing, both MI and ELI are removed via projecting each intended information stream onto the orthogonal complement of MI and ELI. To this end, perfect CSI is required at R. However, when only the estimate of CSI is available at R, MI and ELI still exist.The ZF processing matrices atR are given by(14)WH=WZFH≜G^SRHG^SR-1G^SRH,(15)A=AZF≜αZFG^RDHG^RDG^RDH-1,respectively. In (15), the normalization factor αZF is chosen to satisfy xR2=PR. Therefore, we have [23](16)αZF≜1TrG^RDHG^RD-1.


### 2.2.2. MRC/MRT Processing

Since ZF processing neglects the noise effect, the corresponding detection performance is poor when the signal-to-noise ratio (SNR) is low. By contrast, MRC/MRT processing neglects MI to maximize the received SNR. As a result, MRC/MRT processing achieves a better detection performance than that of ZF processing in the low SNR region at the cost of achieving a worse detection performance than that of ZF processing in the high SNR region. Similarly to ZF processing, MRC/MRT processes the received signal and amplifies the processed signal according to (10) and (12). The MRC receiver and MRT precoding matrices can be, respectively, expressed as [23](17)WH=WMRCH≜G^SRH,A=AMRT≜αMRTG^RDH.Furthermore, the normalization factor for the MRT precoding is given by [23](18)αMRT≜TrG^RDHG^RD-1/2.


## 2.2.1. ZF Processing

Since both theK sources and K destinations transmit their signals to R in phase I and R broadcasts the processed streams to the K destinations in phase II, MI will affect both the received signals at R and K destinations. For the received signal at R, MI is the term (1-θ)PS∑j≠kKwkHgSR,jxS,j, whereas, for the received signal at Dk, MI is PR∑j≠kKgDR,kTajxS,j. Furthermore, the received signal at R also suffers from ELI resulting from EFs transmitted from the K destinations to R in phase I, which can be represented by 1-θPDwkHGDRxD. In such a case, the relay applies the ZF receiver and ZF precoding to process the signal. By ZF processing, both MI and ELI are removed via projecting each intended information stream onto the orthogonal complement of MI and ELI. To this end, perfect CSI is required at R. However, when only the estimate of CSI is available at R, MI and ELI still exist.The ZF processing matrices atR are given by(14)WH=WZFH≜G^SRHG^SR-1G^SRH,(15)A=AZF≜αZFG^RDHG^RDG^RDH-1,respectively. In (15), the normalization factor αZF is chosen to satisfy xR2=PR. Therefore, we have [23](16)αZF≜1TrG^RDHG^RD-1.


## 2.2.2. MRC/MRT Processing

Since ZF processing neglects the noise effect, the corresponding detection performance is poor when the signal-to-noise ratio (SNR) is low. By contrast, MRC/MRT processing neglects MI to maximize the received SNR. As a result, MRC/MRT processing achieves a better detection performance than that of ZF processing in the low SNR region at the cost of achieving a worse detection performance than that of ZF processing in the high SNR region. Similarly to ZF processing, MRC/MRT processes the received signal and amplifies the processed signal according to (10) and (12). The MRC receiver and MRT precoding matrices can be, respectively, expressed as [23](17)WH=WMRCH≜G^SRH,A=AMRT≜αMRTG^RDH.Furthermore, the normalization factor for the MRT precoding is given by [23](18)αMRT≜TrG^RDHG^RD-1/2.


## 3. Asymptotic Analysis

In this section, the asymptotic harvested energy is derived for the case in whichM→∞ and K is fixed. The asymptotic symmetric sum rate over i.i.d. fading is derived for the case in which M→∞, K→∞, and K/M is a constant.


### 3.1. Asymptotic Harvested Energy Analysis

To begin with, the transmission powers at each source and each destination are, respectively, scaled inversely proportional to the number of antennas atR, namely, PS=ES/M and PD=ED/M, while keeping ES and ED fixed.Proposition 1.
Assume that the number of the source-destination pairs is fixed, the harvested energy at the relay asM→∞ is given by(19)Eh→a.s.τuηθESTrDSR+EDTrDDR,where →a.s. denotes almost sure convergence.Proof.
See AppendixA.To obtain further insights, by lettingDSR=βSRIK and DDR=βDRIK, the harvested energy expression in (19) over i.i.d. fading can be asymptotically expressed as(20)Eh∞=τuηθKβSRES+βDRED.Remark 2.
The asymptotic harvested energy in (20) is independent of fast fading components of wireless channels. Furthermore, the asymptotic harvested energy increases with respect to the source and destination transmission powers, respectively. Therefore, for an expected value of the harvested energy at the relay, the source transmission power can be tuned down by tuning up the destination transmission power and vice versa. Moreover, since the array gain increases monotonically with respect to M, both the source and destination transmission powers can be scaled down inversely proportional to M without any asymptotic performance degradation.


### 3.2. Using a Large Receive Antenna Array

To cancel ELI, orthogonal projection can be applied in detecting the desired signal. However, projecting ELI into its orthogonal space may contaminate the desired signal. Note that if the subspace spanned by ELI is orthogonal to the desired signal’s subspace whenM is large, the orthogonal projection scheme can work well. Thanks to the massive array implementation, the channel vectors of the desired signal and ELI become nearly orthogonal when M grows large. Therefore, we adopt the linear receiver as an orthogonal projection of the ELI. Consequently, ELI can be cancelled significantly by using the linear receiver with a large value of M. The main result is summarized in the following proposition.Proposition 3.
Assume that the number of source-destination pairs is fixed; the received signal at the relay for decoding the signal of thekth source as M→∞ is given by(21)rk→a.s.1-θPSxS,k,forZFreceiver,(22)rkMσSR,k2→a.s.1-θPSxS,k,forMRCreceiver.Proof.
See AppendixB.Remark 4.
The result in Proposition3 shows that ELI and MI can be cancelled completely when M grows to infinity. The received signal at the relay after using linear processing includes only the desired signal. Therefore, the capacity of Sk→R link grows without bound as M approaches infinity.


### 3.3. Using a Large Transmit Antenna Array

Proposition 5.
Assume that the number of source-destination pairs is fixed and the transmission power at the relay is supplied by the harvested energy, the received signal atDk for decoding the relayed information signal as M→∞ is given by(23)yD,kMM-K→a.s.ηθσDR,k4TrD^RD-1ESTrDSR+EDTrDDRxS,kfor ZF processing and(24)yD,kM→a.s.ηθσDR,k4ESTrDSR+EDTrDDRTrD^DRxS,kfor MRC/MRT processing.Proof.
See AppendixC.To obtain further insights, the expressions in (23) and (24) of the received signal for decoding over i.i.d. fading can be, respectively, expressed as(25)yD,kMM-K→a.s.ηθK2σDR2βSRES+βDREDxS,k,yD,kM→a.s.ηθσDR2βSRES+βDREDxS,k,where σDR2≜τpPpβDR2/τpPpβDR+σR,p2.Remark 6.
By using linear processing atR, the received signal at the destination Dk includes only the desired signal. Therefore, the capacity of R→Dk link grows without bound, which is similar to that of Sk→R link. Furthermore, we can see that the noise and MI disappeared when M grows to infinity. It can be clearly seen from (25) that the EFs emitted by the destinations provide gain for decoding the information signal. Moreover, the received desired signals in (25) are independent of small-scale fading of wireless channels.


### 3.4. Sum Rate and Harvested Energy Trade-Off

The end-to-end (e2e) achievable rate of the transmission linkSk→R→Dk is given by(26)Rk=min⁡RSR,k,RRD,k,where RSR,k and RRD,k are the achievable rates of the transmission links Sk→R and R→Dk, respectively. From (11), the received signal used for detecting xS,k at the relay can be expressed as(27)rk=1-θPSEg^SR,kHgSR,kxS,k︸desired  signal+n~R,k︸effective  noise,where the effective noise n~R,k is given by(28)n~R,k=1-θPSg^SR,kHgSR,k-Eg^SR,kHgSR,kxS,k+1-θPS∑j≠kKg^SR,kHgSR,jxS,j+1-θPDg^SR,kHGDRxD+g^SR,kHnR.It can be seen that the “desired signal” and “effective noise” in (27) are uncorrelated. Based on the fact that the worst-case uncorrelated additive noise is independent Gaussian noise of the same variance (see [17] and references therein), the achievable rate can be obtained as(29)RSR,k=log2⁡1+1-θPSEwkHgSR,k21-θPSVarwkHgSR,k+MIk+ELIk+ANk,where MIk, ELIk, and ANk represent the MI, ELI, and additive noise effect, respectively, and are, respectively, given by(30)MIk≜1-θPS∑j≠kKEwkHgSR,j2,(31)ELIk≜1-θPDEwkHGDR2,(32)ANk≜EwkHnR2.Similarly, considering (13), the achievable rate of the R→Dk link can be expressed as(33)RRD,k=log21+EgDR,kTak2VargDR,kTak+∑j≠kKEgDR,kTaj2+σD2/pR.Although (33) implies that only statistical knowledge of the channel gains has been applied at Dk in decoding the information signal, the performance degradation compared to genie receiver is negligible for a large value of M [17].Theorem 7.
For a finite number of relay antennas, the achievable rate of the system with ZF processing over i.i.d. fading is given by(34)Rsum=μKlog2⁡1+min⁡M-K1-θPSσSR2K1-θPSβSR-σSR2+PDβDR+σR2,ηθMM-KPSβSR+PDβDRηθKMσDR-2PSβSR+PDβDRβDR-σDR2+σD2σDR-2,where σSR2≜τpPpβSR2/τpPpβSR+σR,p2, σDR2≜τpPpβDR2/τpPpβDR+σR,p2, and the pre-log factor μ≜τu/2τu+τp resulted from the pilot transmission and half-duplex relaying.Proof.
See AppendixD.Remark 8.
We can see from (34) that the achievable rates of the dual-hop links grow without bound as M→∞. Therefore, the e2e performance can be always improved by using a more large antenna array when ZF processing is employed. When M→∞, K→∞, and K/M is a constant, the asymptotic sum rate over i.i.d. fading becomes(35)Rsum∞=μKlog2⁡1+min⁡γSR,γRD,where(36)γSR≜1-θESσSR21/M/K-11-θESβSR-σSR2+EDβDR+1/1-K/MσR2,(37)γRD≜ηθM-KESβSR+EDβDRηθKσDR-2ESβSR+EDβDRβDR-σDR2+σD2σDR-2denote the asymptotic SINRs of the first-hop and second-hop for each source-destination pair, respectively. This expression shows that, although the SINR of the Sk→R link is decreased by ELI, the SINR of the R→Dk link can be improved by employing destination-aided EFs. Therefore, the e2e performance can be improved when it is limited by the R→Dk link. Furthermore, we can reduce the transmission power of each source and each destination proportional to 1/M, meanwhile maintaining a given quality-of-service.Proposition 9.
With ZF processing, the asymptotic sum rate is neither convex nor concave with respect to destination transmission power and the optimal destination transmission power that achieves the allowable maximum asymptotic sum rate is given by(38)EDZF=0,ES≥EZFβSREZF-ESβSRβDR,ES<EZFβSR,where EZF≜-b1+c1+b1-c12+4a1d1/2a1 with a1≜K/(M-K)ESσSR2, b1≜MσR2/M-K1-θESσSR2-K/M-K, c1≜KσDR-2βDR-σDR2/M-K, and d1≜σD2σDR-2/ηθM-K.Proof.
See AppendixE.Proposition 10.
With ZF processing, the asymptotic sum rate is concave with respect toθ and the optimal θ that achieves the allowable maximum asymptotic sum rate is given by(39)θZF=a2+b2-c2+d2-a2+b2-c22-2a2-b2-c2d2+d222a2-c2,where a2≜KESβSR-σSR2+EDβDR/M-KESσSR2, b2≜MσR2/M-KESσSR2, c2≜KσDR-2βDR-σDR2/M-K, and d2≜σD2σDR-2/ηM-KESβSR+EDβDR.Proof.
See AppendixF.Theorem 11.
With ZF processing, the asymptotic sum rate is neither convex nor concave with respect toθ,ED and there is a unique optimal θ∗,ED∗ that achieves the allowable maximum asymptotic sum rate.Proof.
See AppendixG.Since the asymptotic sum rate is neither convex nor concave with respect toθ,ED, the optimal θ∗,ED∗ cannot be obtained by applying conventional convex optimization methods [24]. However, Theorem 11 shows that the optimal θ∗,ED∗ is unique and Proposition 10 shows that the asymptotic sum rate is concave with respect to θ, whereas Proposition 9 shows that for any given θ the corresponding optimal EDZF is uniquely determined. Thus, the two-dimensional searching for the optimal θ∗,ED∗ can proceed with a one-dimensional concave optimizing for θ∗ with the corresponding ED∗ explicitly determined by (38). Since bisection is effective in searching optimal solution of a concave function [24], this paper proposes a one-dimensional EB algorithm to search the optimal (θ∗,ED∗) with noting that the optimal ED∗ is embedded in one-dimensional bisection. The proposed EB algorithm is presented in Algorithm 1.Algorithm 1:One-dimensional EB algorithm.
(1) Initialize(θa,θb)⊂(0,1) and ε. (2) Ifθb-θa>ε, go to step (3), otherwise go to step (6). (3) Setθc=θa+θb/2, θa′=θa+θc/2, and θb′=θb+θc/2. ComputeEDZFθa′, EDZFθb′, and EDZF(θc′) according to (38); ComputeRsum∞θa′,EDZFθa′, Rsum∞θb′,EDZFθb′, and Rsum∞θc,EDZFθc according to (35). (4) IfRsum∞θa′,EDZFθa′≤Rsum∞θc,EDZFθc≤Rsum∞θb′,EDZFθb′, set θa=θa′; IfRsum∞θa′,EDZFθa′≥Rsum∞θc,EDZFθc≥Rsum∞θb′,EDZFθb′, set θb=θb′; IfRsum∞θc,EDZFθc≥max⁡Rsum∞θa′,EDZFθa′,Rsum∞θb′,EDZFθb′, set θa=θa′ and θb=θb′. (5) Go to step(2). (6) Returnθc, EDZFθc, and Rsum∞=Rsum∞θc,EDZFθc.To quantify the trade-off between the harvested energy and achievable sum rate, we solve forθ in (35) and substitute it into (20). The trade-off between the asymptotic harvested energy and sum rate with ZF processing can be derived as(40)Eh∞=τuηKβSRES+βRDED×d2Λc2Λ+1,θ<θZF1+b2Λa2Λ-1,θ≥θZF,where a2, b2, c2, and d2 are the same as those of Proposition 10 and(41)Λ≜2Rsum∞/μK-1.Theorem 12.
For a finite number of relay antennas, the achievable rate of the system with MRC/MRT processing over i.i.d. fading is given by(42)Rsum=μKlog2⁡1+min⁡M1-θPSσSR2K1-θPSβSR+PDβDR+σR2,M2ηθσDR2PSβSR+PDβDRKMηθβDRPSβSR+PDβDR+σD2.Proof.
See AppendixH.Remark 13.
We can see from (42) that, as M→∞, the achievable rates of the dual-hop links grow without bound. Thus, the e2e performance can be always improved by using a more large antenna array when MRC/MRT processing is employed. When M→∞, K→∞, and K/M is constant, the asymptotic symmetric sum rate over i.i.d. fading is given by(43)Rsum∞=μKlog2⁡1+min⁡1-θESσSR21-θESβSR+EDβDRK/M+σR2,MηθσDR2ESβSR+EDβDRKηθβDRESβSR+EDβDR+σD2.Similarly to the case of ZF processing, the above expression implies that the e2e performance with MRC/MRT processing can be improved by destination-aided EFs when it is limited by the R→Dk link.Proposition 14.
With MRC/MRT processing, the asymptotic sum rate is neither convex nor concave with respect to destination transmission power and the optimal destination transmission power that achieves the allowable maximum asymptotic sum rate is given by(44)EDMRC/MRT=0,ES≥EMRC/MRTβSREMRC/MRT-ESβSRβDR,ES<EMRC/MRTβSR,where EMRC/MRT≜-b3+c3+b3-c32+4a3d3/2a3, a3≜K/MESσSR2, b3≜σR2/1-θESσSR2, c3≜KβDR/MσDR2, and d3≜σD2/MηθσDR2.Proof.
By following similar procedure as the proof of Proposition9 in Appendix E, Proposition 14 can be reached.Proposition 15.
With MRC/MRT processing, the asymptotic sum rate is concave with respect toθ and the optimal θ that achieves the allowable maximum asymptotic sum rate is given by(45)θMRC/MRT=a4+b4-c4+d4-a4+b4-c42-2a4-b4-c4d4+d422a4-c4,where a4≜KESβSR+EDβDR/MESσSR2, b4≜σR2/ESσSR2, c4≜KβDR/MσDR2, and d4≜σD2/MησDR2ESβSR+EDβDR.Proof.
By following similar procedure as the proof of Proposition10 in Appendix F, Proposition 15 can be reached.Theorem 16.
With MRC/MRT processing, the asymptotic sum rate is neither convex nor concave with respect toθ,ED and there is a unique optimal θ∗,ED∗ that achieves the allowable maximum asymptotic sum rate.Proof.
By following similar procedure as the proof of Theorem11 in Appendix G, Theorem 16 can be reached.Since Propositions14 and 15, and Theorem 16, have the similar expressions as those of Propositions 9 and 10, and Theorem 11, respectively, the one-dimensional EB algorithm can be applied to the case of MRC/MRT processing straightforwardly. Similarly to (40), the asymptotic harvested energy versus sum rate trade-off with MRC/MRT processing can be derived as(46)Eh∞=τuηKβSRES+βRDED×d4Λc4Λ+1,θ<θMRC/MRT1+b4Λa4Λ-1,θ≥θMRC/MRT,where a4, b4, c4, and d4 are the same as those of Proposition 15.


## 3.1. Asymptotic Harvested Energy Analysis

To begin with, the transmission powers at each source and each destination are, respectively, scaled inversely proportional to the number of antennas atR, namely, PS=ES/M and PD=ED/M, while keeping ES and ED fixed.Proposition 1.
Assume that the number of the source-destination pairs is fixed, the harvested energy at the relay asM→∞ is given by(19)Eh→a.s.τuηθESTrDSR+EDTrDDR,where →a.s. denotes almost sure convergence.Proof.
See AppendixA.To obtain further insights, by lettingDSR=βSRIK and DDR=βDRIK, the harvested energy expression in (19) over i.i.d. fading can be asymptotically expressed as(20)Eh∞=τuηθKβSRES+βDRED.Remark 2.
The asymptotic harvested energy in (20) is independent of fast fading components of wireless channels. Furthermore, the asymptotic harvested energy increases with respect to the source and destination transmission powers, respectively. Therefore, for an expected value of the harvested energy at the relay, the source transmission power can be tuned down by tuning up the destination transmission power and vice versa. Moreover, since the array gain increases monotonically with respect to M, both the source and destination transmission powers can be scaled down inversely proportional to M without any asymptotic performance degradation.


## 3.2. Using a Large Receive Antenna Array

To cancel ELI, orthogonal projection can be applied in detecting the desired signal. However, projecting ELI into its orthogonal space may contaminate the desired signal. Note that if the subspace spanned by ELI is orthogonal to the desired signal’s subspace whenM is large, the orthogonal projection scheme can work well. Thanks to the massive array implementation, the channel vectors of the desired signal and ELI become nearly orthogonal when M grows large. Therefore, we adopt the linear receiver as an orthogonal projection of the ELI. Consequently, ELI can be cancelled significantly by using the linear receiver with a large value of M. The main result is summarized in the following proposition.Proposition 3.
Assume that the number of source-destination pairs is fixed; the received signal at the relay for decoding the signal of thekth source as M→∞ is given by(21)rk→a.s.1-θPSxS,k,forZFreceiver,(22)rkMσSR,k2→a.s.1-θPSxS,k,forMRCreceiver.Proof.
See AppendixB.Remark 4.
The result in Proposition3 shows that ELI and MI can be cancelled completely when M grows to infinity. The received signal at the relay after using linear processing includes only the desired signal. Therefore, the capacity of Sk→R link grows without bound as M approaches infinity.


## 3.3. Using a Large Transmit Antenna Array

Proposition 5.
Assume that the number of source-destination pairs is fixed and the transmission power at the relay is supplied by the harvested energy, the received signal atDk for decoding the relayed information signal as M→∞ is given by(23)yD,kMM-K→a.s.ηθσDR,k4TrD^RD-1ESTrDSR+EDTrDDRxS,kfor ZF processing and(24)yD,kM→a.s.ηθσDR,k4ESTrDSR+EDTrDDRTrD^DRxS,kfor MRC/MRT processing.Proof.
See AppendixC.To obtain further insights, the expressions in (23) and (24) of the received signal for decoding over i.i.d. fading can be, respectively, expressed as(25)yD,kMM-K→a.s.ηθK2σDR2βSRES+βDREDxS,k,yD,kM→a.s.ηθσDR2βSRES+βDREDxS,k,where σDR2≜τpPpβDR2/τpPpβDR+σR,p2.Remark 6.
By using linear processing atR, the received signal at the destination Dk includes only the desired signal. Therefore, the capacity of R→Dk link grows without bound, which is similar to that of Sk→R link. Furthermore, we can see that the noise and MI disappeared when M grows to infinity. It can be clearly seen from (25) that the EFs emitted by the destinations provide gain for decoding the information signal. Moreover, the received desired signals in (25) are independent of small-scale fading of wireless channels.


## 3.4. Sum Rate and Harvested Energy Trade-Off

The end-to-end (e2e) achievable rate of the transmission linkSk→R→Dk is given by(26)Rk=min⁡RSR,k,RRD,k,where RSR,k and RRD,k are the achievable rates of the transmission links Sk→R and R→Dk, respectively. From (11), the received signal used for detecting xS,k at the relay can be expressed as(27)rk=1-θPSEg^SR,kHgSR,kxS,k︸desired  signal+n~R,k︸effective  noise,where the effective noise n~R,k is given by(28)n~R,k=1-θPSg^SR,kHgSR,k-Eg^SR,kHgSR,kxS,k+1-θPS∑j≠kKg^SR,kHgSR,jxS,j+1-θPDg^SR,kHGDRxD+g^SR,kHnR.It can be seen that the “desired signal” and “effective noise” in (27) are uncorrelated. Based on the fact that the worst-case uncorrelated additive noise is independent Gaussian noise of the same variance (see [17] and references therein), the achievable rate can be obtained as(29)RSR,k=log2⁡1+1-θPSEwkHgSR,k21-θPSVarwkHgSR,k+MIk+ELIk+ANk,where MIk, ELIk, and ANk represent the MI, ELI, and additive noise effect, respectively, and are, respectively, given by(30)MIk≜1-θPS∑j≠kKEwkHgSR,j2,(31)ELIk≜1-θPDEwkHGDR2,(32)ANk≜EwkHnR2.Similarly, considering (13), the achievable rate of the R→Dk link can be expressed as(33)RRD,k=log21+EgDR,kTak2VargDR,kTak+∑j≠kKEgDR,kTaj2+σD2/pR.Although (33) implies that only statistical knowledge of the channel gains has been applied at Dk in decoding the information signal, the performance degradation compared to genie receiver is negligible for a large value of M [17].Theorem 7.
For a finite number of relay antennas, the achievable rate of the system with ZF processing over i.i.d. fading is given by(34)Rsum=μKlog2⁡1+min⁡M-K1-θPSσSR2K1-θPSβSR-σSR2+PDβDR+σR2,ηθMM-KPSβSR+PDβDRηθKMσDR-2PSβSR+PDβDRβDR-σDR2+σD2σDR-2,where σSR2≜τpPpβSR2/τpPpβSR+σR,p2, σDR2≜τpPpβDR2/τpPpβDR+σR,p2, and the pre-log factor μ≜τu/2τu+τp resulted from the pilot transmission and half-duplex relaying.Proof.
See AppendixD.Remark 8.
We can see from (34) that the achievable rates of the dual-hop links grow without bound as M→∞. Therefore, the e2e performance can be always improved by using a more large antenna array when ZF processing is employed. When M→∞, K→∞, and K/M is a constant, the asymptotic sum rate over i.i.d. fading becomes(35)Rsum∞=μKlog2⁡1+min⁡γSR,γRD,where(36)γSR≜1-θESσSR21/M/K-11-θESβSR-σSR2+EDβDR+1/1-K/MσR2,(37)γRD≜ηθM-KESβSR+EDβDRηθKσDR-2ESβSR+EDβDRβDR-σDR2+σD2σDR-2denote the asymptotic SINRs of the first-hop and second-hop for each source-destination pair, respectively. This expression shows that, although the SINR of the Sk→R link is decreased by ELI, the SINR of the R→Dk link can be improved by employing destination-aided EFs. Therefore, the e2e performance can be improved when it is limited by the R→Dk link. Furthermore, we can reduce the transmission power of each source and each destination proportional to 1/M, meanwhile maintaining a given quality-of-service.Proposition 9.
With ZF processing, the asymptotic sum rate is neither convex nor concave with respect to destination transmission power and the optimal destination transmission power that achieves the allowable maximum asymptotic sum rate is given by(38)EDZF=0,ES≥EZFβSREZF-ESβSRβDR,ES<EZFβSR,where EZF≜-b1+c1+b1-c12+4a1d1/2a1 with a1≜K/(M-K)ESσSR2, b1≜MσR2/M-K1-θESσSR2-K/M-K, c1≜KσDR-2βDR-σDR2/M-K, and d1≜σD2σDR-2/ηθM-K.Proof.
See AppendixE.Proposition 10.
With ZF processing, the asymptotic sum rate is concave with respect toθ and the optimal θ that achieves the allowable maximum asymptotic sum rate is given by(39)θZF=a2+b2-c2+d2-a2+b2-c22-2a2-b2-c2d2+d222a2-c2,where a2≜KESβSR-σSR2+EDβDR/M-KESσSR2, b2≜MσR2/M-KESσSR2, c2≜KσDR-2βDR-σDR2/M-K, and d2≜σD2σDR-2/ηM-KESβSR+EDβDR.Proof.
See AppendixF.Theorem 11.
With ZF processing, the asymptotic sum rate is neither convex nor concave with respect toθ,ED and there is a unique optimal θ∗,ED∗ that achieves the allowable maximum asymptotic sum rate.Proof.
See AppendixG.Since the asymptotic sum rate is neither convex nor concave with respect toθ,ED, the optimal θ∗,ED∗ cannot be obtained by applying conventional convex optimization methods [24]. However, Theorem 11 shows that the optimal θ∗,ED∗ is unique and Proposition 10 shows that the asymptotic sum rate is concave with respect to θ, whereas Proposition 9 shows that for any given θ the corresponding optimal EDZF is uniquely determined. Thus, the two-dimensional searching for the optimal θ∗,ED∗ can proceed with a one-dimensional concave optimizing for θ∗ with the corresponding ED∗ explicitly determined by (38). Since bisection is effective in searching optimal solution of a concave function [24], this paper proposes a one-dimensional EB algorithm to search the optimal (θ∗,ED∗) with noting that the optimal ED∗ is embedded in one-dimensional bisection. The proposed EB algorithm is presented in Algorithm 1.Algorithm 1:One-dimensional EB algorithm.
(1) Initialize(θa,θb)⊂(0,1) and ε. (2) Ifθb-θa>ε, go to step (3), otherwise go to step (6). (3) Setθc=θa+θb/2, θa′=θa+θc/2, and θb′=θb+θc/2. ComputeEDZFθa′, EDZFθb′, and EDZF(θc′) according to (38); ComputeRsum∞θa′,EDZFθa′, Rsum∞θb′,EDZFθb′, and Rsum∞θc,EDZFθc according to (35). (4) IfRsum∞θa′,EDZFθa′≤Rsum∞θc,EDZFθc≤Rsum∞θb′,EDZFθb′, set θa=θa′; IfRsum∞θa′,EDZFθa′≥Rsum∞θc,EDZFθc≥Rsum∞θb′,EDZFθb′, set θb=θb′; IfRsum∞θc,EDZFθc≥max⁡Rsum∞θa′,EDZFθa′,Rsum∞θb′,EDZFθb′, set θa=θa′ and θb=θb′. (5) Go to step(2). (6) Returnθc, EDZFθc, and Rsum∞=Rsum∞θc,EDZFθc.To quantify the trade-off between the harvested energy and achievable sum rate, we solve forθ in (35) and substitute it into (20). The trade-off between the asymptotic harvested energy and sum rate with ZF processing can be derived as(40)Eh∞=τuηKβSRES+βRDED×d2Λc2Λ+1,θ<θZF1+b2Λa2Λ-1,θ≥θZF,where a2, b2, c2, and d2 are the same as those of Proposition 10 and(41)Λ≜2Rsum∞/μK-1.Theorem 12.
For a finite number of relay antennas, the achievable rate of the system with MRC/MRT processing over i.i.d. fading is given by(42)Rsum=μKlog2⁡1+min⁡M1-θPSσSR2K1-θPSβSR+PDβDR+σR2,M2ηθσDR2PSβSR+PDβDRKMηθβDRPSβSR+PDβDR+σD2.Proof.
See AppendixH.Remark 13.
We can see from (42) that, as M→∞, the achievable rates of the dual-hop links grow without bound. Thus, the e2e performance can be always improved by using a more large antenna array when MRC/MRT processing is employed. When M→∞, K→∞, and K/M is constant, the asymptotic symmetric sum rate over i.i.d. fading is given by(43)Rsum∞=μKlog2⁡1+min⁡1-θESσSR21-θESβSR+EDβDRK/M+σR2,MηθσDR2ESβSR+EDβDRKηθβDRESβSR+EDβDR+σD2.Similarly to the case of ZF processing, the above expression implies that the e2e performance with MRC/MRT processing can be improved by destination-aided EFs when it is limited by the R→Dk link.Proposition 14.
With MRC/MRT processing, the asymptotic sum rate is neither convex nor concave with respect to destination transmission power and the optimal destination transmission power that achieves the allowable maximum asymptotic sum rate is given by(44)EDMRC/MRT=0,ES≥EMRC/MRTβSREMRC/MRT-ESβSRβDR,ES<EMRC/MRTβSR,where EMRC/MRT≜-b3+c3+b3-c32+4a3d3/2a3, a3≜K/MESσSR2, b3≜σR2/1-θESσSR2, c3≜KβDR/MσDR2, and d3≜σD2/MηθσDR2.Proof.
By following similar procedure as the proof of Proposition9 in Appendix E, Proposition 14 can be reached.Proposition 15.
With MRC/MRT processing, the asymptotic sum rate is concave with respect toθ and the optimal θ that achieves the allowable maximum asymptotic sum rate is given by(45)θMRC/MRT=a4+b4-c4+d4-a4+b4-c42-2a4-b4-c4d4+d422a4-c4,where a4≜KESβSR+EDβDR/MESσSR2, b4≜σR2/ESσSR2, c4≜KβDR/MσDR2, and d4≜σD2/MησDR2ESβSR+EDβDR.Proof.
By following similar procedure as the proof of Proposition10 in Appendix F, Proposition 15 can be reached.Theorem 16.
With MRC/MRT processing, the asymptotic sum rate is neither convex nor concave with respect toθ,ED and there is a unique optimal θ∗,ED∗ that achieves the allowable maximum asymptotic sum rate.Proof.
By following similar procedure as the proof of Theorem11 in Appendix G, Theorem 16 can be reached.Since Propositions14 and 15, and Theorem 16, have the similar expressions as those of Propositions 9 and 10, and Theorem 11, respectively, the one-dimensional EB algorithm can be applied to the case of MRC/MRT processing straightforwardly. Similarly to (40), the asymptotic harvested energy versus sum rate trade-off with MRC/MRT processing can be derived as(46)Eh∞=τuηKβSRES+βRDED×d4Λc4Λ+1,θ<θMRC/MRT1+b4Λa4Λ-1,θ≥θMRC/MRT,where a4, b4, c4, and d4 are the same as those of Proposition 15.


## 4. Numerical Results

This section presents some numerical results to verify the performance of the proposed protocol. For simplicity of illustration, the circuit power consumption and EH receiver sensitivity at the relay are ignored [21]. In all the illustrative examples, we choose K=10, τp=2K, τu=98, η=0.7, pp=30 dBm, and σR,p2=σR2=σD2=σ2=-90 dBm. Furthermore, we set DSR=DDR=βIK to model i.i.d. fading, where β=-65 dB corresponds to a path-loss related to a distance of about 10 m for the carrier frequency of 915 MHz.In Figure2, the trade-off between the harvested energy and achievable sum rate is investigated for ZF processing. Five levels of destination-aided EFs, that is, ED=35 dBm, ED=30 dBm, ED=20 dBm, ED=0 mW, and EDZF, are considered, respectively, whereas ES is fixed at 25 dBm. Note that ED=0 mW corresponds to the scenario where the destinations do not transmit. The asymptotic trade-off curves are plotted by using (40). Furthermore, a set of trade-off curves are plotted by using Monte Carlo simulations with gradually increasing M from 40 to 10000. In Figure 2, the points of the trade-off curves corresponding to a higher harvested energy refer to a larger PS ratios and, consequently, this scenario leads to a higher sum rate. Nevertheless, as θ→1, the achievable sum rates of all the fixed ED drop down dramatically and approach zero. It can be seen from Figure 2 that the trade-off between the harvested energy and sum rate can be improved significantly by using a very large antenna array. For example, for ED equal to 20 dBm and 30 dBm, destination-aided EFs can effectively improve the trade-off between harvested energy and sum rate. However, for ED=35 dBm, destination-aided EFs deteriorate the trade-off. Furthermore, since a fixed ED can only achieve a part of the sum rate compared to that of EDZF, Figure 2 shows that the curves are overlapped partially, whereas EDZF achieves the highest trade-off between harvested energy and sum rate.Figure 2
Trade-off betweenEh and sum rate with ZF processing.In Figure3, we investigate the trade-off between the harvested energy and achievable sum rate for MRC/MRT processing. Four levels of destination-aided EFs, that is, ED=30 dBm, ED=20 dBm, ED=0 mW, and EDZF, are considered, respectively, whereas ES is fixed at 25 dBm. The asymptotic trade-off curves are plotted by using (46). The trade-off curves for M→∞ are also presented in Figure 3. As can be seen from Figure 3, the trade-off curves for M→∞ serve as the trade-off bounds. When M is a limited number, similar to the case of ZF processing, a larger PS ratio leads to a higher harvested energy as well as a higher achievable sum rate. Also, the achievable sum rate drops down dramatically as θ→1.Figure 3
Trade-off betweenEh and sum rate with MRC/MRT processing.In Figures4 and 5, we investigate sum rate versus ED for ZF processing and MRC/MRT processing, respectively. The curves in Figures 4 and 5 verify the correctness of Propositions 9, 10, 14, and 15. Compared with fixed θ and fixed ED, the corresponding EDZF (EDMRC/MRT) and θZF (θMRC/MRT) can, respectively, improve the sum rate. Furthermore, Figures 4 and 5 show that the proposed one-dimensional EB algorithm achieves the allowable maximum sum rate. Figures 4 and 5 also show that the sum rate increases as M increases from 40 to 10000.Figure 4
Sum rate versusED with ZF processing.Figure 5
Sum rate versusED with MRC/MRT processing.In Figures6 and 7, we investigate sum rate versus PS factor for ZF processing and MRC/MRT processing, respectively. The results in Figures 6 and 7 verify the correctness of the analytical θZF and θMRC/MRT, respectively. As can be seen from Figures 6 and 7, the optimal PS factor almost keeps unchanged with the variable M for the considered same power budget. Furthermore, the optimal PS factor for ZF processing is almost the same as that for MRC/MRT processing. Moreover, Figures 6 and 7 verify that the sum rate is concave with respect to θ. Figures 6 and 7 also verify that the one-dimensional EB algorithm achieves the allowable maximum sum rate.Figure 6
Sum rate versusθ with ZF processing.Figure 7
Sum rate versusθ with MRC/MRT processing.In Figures8 and 9, average sum rates are plotted against source transmission power for ZF processing and MRC/MRT processing, respectively. The analytical curves are plotted by using (34) and (42) for ZF processing and MRC/MRT processing, respectively, whereas the exact sum rate curves are plotted by using Monte Carlo simulations. Figures 8 and 9 clearly show that the closed-form achievable rates in (34) and (42) are accurate in computing the sum rate based on channel statistics-aided decoding with a finite number of relay antennas. Furthermore, it can be clearly seen from Figures 8 and 9 that destination-aided EFs can improve the sum rate significantly. This verifies that ELI can be effectively cancelled out by massive MIMO processing. Moreover, the highest sum rate is achieved by the proposed one-dimensional EB algorithm. Note that EDZF and EDZF achieve the second highest sum rate in Figures 8 and 9, respectively, the correctness of Propositions 9 and 14 is verified, respectively. Also, the piecewise sum rate curves verify the correctness of Propositions 10 and 15. By comparing the curves of M=40 and M=10000, we observe that the gains of sum rate can be obtained by increasing the number of relay antennas while keeping the source transmission power in useful regions. Notably, Figures 8 and 9 show that the sum rate achieved by ZF processing is higher than that of MRC/MRT processing in the high ES region, whereas both ZF processing and MRC/MRT processing achieve almost the same sum rate in the low ES region.Figure 8
Sum rate versusES with ZF processing.Figure 9
Sum rate versusES with MRC/MRT processing.The sum rate versus the number of the relay antennas is investigated in Figure10 with ES=25 dBm. As M increases, the corresponding sum rate increases accordingly. The curves in Figure 10 show that ZF processing achieves a higher sum rate than that of MRC/MRT processing with the considered power budget. Furthermore, in the whole region of the value of M, the highest sum rate is achieved by the one-dimensional EB algorithm and the second highest sum rate is achieved by EDZF (and EDMRC/MRT). Compared to ED=0 mW, the proposed destination-aided EFs can significantly improve the sum rate in the whole region of the value of M.Figure 10
Sum rate versus the number of the relay antennas.


## 5. Conclusion

A destination-aided SWIPT relaying protocol has been proposed for a multipair massive MIMO relay network, in which a PS relay employs linear processing to cancel MI and ELI. The expressions of asymptotic harvested energy and symmetric sum rate with massive MIMO relay have been derived in closed-form. The trade-off between asymptotic harvested energy and achievable sum rate has been quantified. The effect of destination-aided EFs on the sum rate has been investigated and our results reveal that destination-aided EFs can boost the energy harvesting, so that achievable sum rate can be significantly improved by destination-aided EFs. Meanwhile, the detrimental impact of ELI on sum rate performance can be cancelled out by using a massive MIMO relay with linear processing. It has shown that asymptotic sum rate is neither convex nor concave with respect to PS and destination transmission power and a one-dimensional EB algorithm has been proposed to obtain the optimal PS and destination transmission power. The significant sum rate improvement of the proposed scheme has been verified by numerical results.



---

*Source: 1760187-2017-02-26.xml*