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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 wire-
less information and power transfer (SWIPT) for a decode-
and-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 in-
formation 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 imple-
mented RF energy harvesting. Moreover, simultaneous wire-
less 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 tradeoffbetween 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-and-
forward (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 trans-
mission 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 orthogo-
nal space-time block codes (STBC) for data transmission. The
optimal source and relay precoders were jointly optimized to
achieve the rate-energy tradeoffbetween 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 Dare
equipped with Ns,Nr, and Ndantennas, 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 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∈Nr×Nsand G∈Nd×Nr, respectively, and
are assumed to be quasi-static block-fading channels. The slot
978-1-4799-8088-8/15/$31.00 ©2015 IEEE
durationisassumedtobesufficiently 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 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 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 Ns×1 precoded vector xsto the relay through the Nr×Ns
block-fading channel Hwhich is corrupted by additive white
Gaussian noise (AWGN) vector nr.TheNr×1 received vector
at the relay, yr, is given by
yr=Hxs+nr,(1)
where xsis the transmitted signal from the source whose
covariance matrix is Rs=xsxH
s,nris 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 σ2
r. Furthermore, we assume that
the source has an average power constraint, i.e. xs2=
tr (Rs)≤Ps, where tr(·) denotes the trace operator.
The EH receiver at the relay converts the RF received signal
yrto 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,isgiven
by
Qr=ζHxs2=ζtr HRsHH,(2)
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 yrand forwards the
baseband transmitted signal from the relay xrwhich has a
covariance matrix given by the Nr×Nrrelay precoding matrix
Rr. The relay is constrained to an average transmit power
constraint given by xrxH
r=tr (Rr)≤Qr. Afterwards, the
Nd×1 received vector at the destination, yd, is expressed as
yd=Gxr+nd,(3)
where ndis the Nd×1 AWGN vector whose entries are i.i.d.
and drawn from the Gaussian distribution with zero mean and
variance equal to σ2
d.
Using Gaussian codebooks, the source-destination achiev-
able rate of the MIMO DF relay system, in bits/s/Hz, is known
to be given by [11]
R(Rs,Rr)=min RS−R(Rs),RR−D(Rr)(4)
=1
2min log2
I+HRsHH
,log2
I+GRrGH,
(5)
where RS−Rand RS−Rare 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 Rsand Rrare the solutions to the
optimization problem (P):
(P) : max
Rs,Rr
R(Rs,Rr)(6a)
s.t. tr (Rs)≤Ps(6b)
tr (Rr)≤ζtr HRsHH(6c)
Rs0,Rr0,(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 optimiza-
tion 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 Rsand 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 Hand G
are H=UHD1/2
H(VH)Hand G=UGD1/2
G(VG)H, respectively,
where UH,VH,UG, and VGare unitary matrices with di-
mensions Nr×r1,r1×Ns,Nd×r2, and r2×Nr, respectively,
DH=diag λH,1,...,λ
H,r1and DG=diag λG,1,...,λ
G,r2are
the diagonal matrices containing the eigenvalues arranged in a
decreasing order of HHHand GGH, respectively, and r1and
r2are the rank of Hand G, respectively.
Let R∗
sbe the optimal solution to (P). That is, R∗
sis the
optimal source precoding matrix that maximizes the end-to-
end rate. Knowing R∗
s, the relay can maximize the achievable
rate of the R-D link by solving the optimization problem
(P1) : max
Rr
1
2log2
I+GRrGH
(7)
s.t. tr (Rr)≤Pr(8)
Rr0,(9)
where Pr=ζtr HR∗
sHH. The optimal solution R∗
rof (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
(P2) : max
Rs
1
2log2
I+HRsHH
(10)
s.t. tr (Rs)≤Ps(11)
ζtr HRsHH≥Qr(12)
Rs0,(13)
where Qris 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 tradeoffbetween the achievable throughput rate and the
achievable harvested power of a one-hop MIMO system [5].
Hence, the achievable throughput rate of the S-R link RS−Ris
nonincreasing with respect to Qr.
Let us discuss the relevance of the constraint Qrin (P2).
•If the value of Qris zero, the energy harvesting con-
straint in (12) is trivially satisfied. Problem (P2) be-
comes equivalent to maximizing the rate of a one-hop
MIMO system and its optimal solution ˆ
Rsis presented
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
relay. Let Qmin =ζtr Hˆ
RsHHbe the corresponding
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.
Let ˆ
Rrbe the solution to (P1) when Pr=Qmin .
•If the value of Qris very large, (P2) becomes infeasible.
In order to obtain the maximum possible value for Qrfor
which (P2) is feasible, we solve the following optimiza-
tion problem
(P3) : max
Rs
Qr=ζtr HRsHH(14)
s.t. tr (Rs)≤Ps,Rs0.(15)
The optimal solution Rsof (P3) is presented in Appendix
B-2. The corresponding quantity Qmax =ζtr H RsHH
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 Rrbe 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 Rsfor the
precoding matrix at the source and Rrfor 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 Qma x.
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
optimal solution to (P) is ˆ
Rsfor the precoding matrix at
the source and ˆ
Rrfor 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,maxand
RR−D,min,RR−D,ma xis 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 R-
D link are equal. This is due to the fact that the achievable
Rmax
RS−R,max
RR−D,min
RS−R,min
RR−D,max
Rate
Qr
S−R link
R−D link
QID
Qmin Qmax
Figure 1. Illustration of the possible solution of (P) in Case III.
Algorithm 1 Proposed Solution of (P)
Inputs: H,G,σ2
r,σ2
d,Ps,ζ.
Outputs: R∗
s,R∗
r.
Given Ps,H,ζ, and σ2
r, compute RS−R,max ,Qmin,Qma x, and
RS−R,min as in (20), (21), (26), and (27), respectively.
Given Qmin,Qmax ,G, and σ2
d, compute RR−D,min and RR−D,max
as in Appendix A for Prequal to Qmin and Qma x, respec-
tively.
if RS−R,min ≥RR−D,max then
-QID =Qmax .
-R∗
s=Rsin (25) presented in Appendix B-2.
-R∗
r=Rrin (16) for Pr=Qmax .
else
if RS−R,max ≤RR−D,min then
-QID =Qmin.
-R∗
s=ˆ
Rsin (19) presented in Appendix B-1.
-R∗
r=ˆ
Rrin (16) for Pr=Qmin.
else
-R∗
sand R∗
rare given in (18) and (16), respectively.
-QID is the argument of the equality between
RS−RR∗
s=RR−DR∗
r.ThevalueofQID 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,Qma x)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 Hand Gare Rayleigh fading chan-
nels with path loss. The Rayleigh fading is assumed to have
unit variance. The path loss exponent is taken equal to m=2.7.
All the simulations are obtained for H=1
dm
sd 10.5
0.51
and
G=1
dm
rd 10.5
0.51
. We denote by dsr ,drd, and dsd the distance
−20 −10 0 10 20 30 40
0
2
4
6
8
10
12
14
16
18
20
Transmit Power at the Source P
s (dB)
Maximum Achievable End−to−End Rate (bits/s/Hz)
(a)
RateRD,max
RateRD,min
RateSR,max
Rmax (cvx)
Rmax (semi opt)
RateSR,min
−20 −10 0 10 20 30 40
10−2
10−1
100
101
102
103
104
105
Transmit Power at the Source P
s (dB)
Harvested Energy at thre Relay (Unit of energy)
(b)
Qmax
Optimal scheme (CVX)
Optimal scheme (semi opt)
Qmin
Figure 2. The maximum achievable rate Rmax and the corresponding harvested
energy QID versus the transmit power at the source Psfor dsr/dsd =0.5, ζ=1,
and Ns=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 σ2
r=σ2
d=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 Ps. 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,
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
4
6
8
10
12
14
16
18
20
Ratio Between the Distances dsr/dsd
Maximum Achievable End−to−End Rate (bits/s/Hz)
(a)
RateRD,max
RateRD,min
RateSR,max
Rmax (cvx)
Rmax (semi opt)
RateSR,min
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
101
102
103
104
105
106
Ratio Between the Distances dsr/dsd
Harvested Energy at thre Relay (Unit of energy)
(b)
Qmax
Optimal scheme (CVX)
Optimal scheme (semi opt)
Qmin
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 dsr/dsd for Ps=20 dB, ζ=1, and Ns=Nr=Nd=2.
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 dsr/dsd , 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 dsr/dsd . 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. C onclusion
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 end-
to-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∗
rof (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
R∗
r=VGD∗
rVH
G,(16)
where the elements of the diagonal matrix D∗
r=
diag λr,1,...,λ
r,r2are given by the water-filling solution
as [14] λr,i=1
β−1
λG,i+
,for i=1,...,r2, where βis the
Lagrange multiplier satisfying the constraint (8) with equality.
Hence, the corresponding rate of the R-D link is given by
RR−D=1
2
r2
i=1
log21+λG,iλr,i.(17)
Appendix B
Optimal Solution R∗
sof (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∗
shas been shown to be
R∗
s=W−1/2ˆ
VHD∗
sˆ
VH
HW−1/2,(18)
where W=βINs−θHHH,βand θare the Lagrange multipliers
satisfying the constraints (11) and (12), respectively, ˆ
VHis an
r1×Nsunitary matrix corresponding to the SVD of HW−1/2=
ˆ
UHˆ
D1/2
Hˆ
VHHwith ˆ
DH=diag ˆ
λH,1,...,ˆ
λH,r1, and D∗
s=
diag λs,1,...,λ
s,r1with λs,i=1−1
ˆ
λH,i+
,fori=1,...,r1.
1) If Qr=0:
Note that when the value of Qris zero, the energy harvesting
constraint in (12) is trivially satisfied. The problem (P2)
becomes equivalent to the throughput maximization of one-
hop MIMO system given only an average transmit power
constraint at the transmitter (11) and the optimal solution to
(P2) simplifies to be ˆ
Rs=VHˆ
DsVH
H,(19)
where ˆ
Ds=diag λs,1,...,λ
s,r1are given by the water-filling
solution as [14] λs,i=1
θ−1
λH,i+
,for i=1,...,r1, where θ
is the Lagrange multiplier satisfying the constraint (11) with
equality. Hence, the corresponding achievable rate is given by
RS−R,max =1
2
r1
i=1
log21+λH,iλs,i,(20)
and the corresponding harvested power is given by
Qmin =ζ
r1
i=1
λH,iλs,i.(21)
2) If Qr=Qmax :
On the other hand, if the value of Qris equal to its maximum
value, the solution to (P2) is equivalent to
(P3) : max
Rs
Qr=ζtr HRsHH(22)
s.t. tr (Rs)≤Ps(23)
Rs0.(24)
Optimization problem (P3) is with respect to Rsonly 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 Rsis given by
Rs=PsvH,1vH
H,1,(25)
where v1is the eigenvector of HHHwhich corresponds to
the maximum eigenvalue a1of HHH. Then, the maximum
harvested power at the relay, given Rs, is expressed as
Qmax =ζa1Ps.(26)
From (25), we can see that the optimal Rsis ranked one,
and the maximum harvested power is obtained by energy
beamforming along the strongest eigenmode of HHH.The
corresponding rate of the S-R link is given by
RS−R,min =1
2log2(1 +Psa2
1).(27)
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