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IEEE COMMUNICATIONS LETTERS, VOL. 20, NO. 8, AUGUST 2016 1563
Wireless Information and Power Transfer in Two-Way
OFDM Amplify-and-Forward Relay Networks
Gaofei Huang, Member, IEEE, and Dong Tang, Member, IEEE
Abstract— Energy harvesting (EH) is promising to solve
energy scarcity problem in energy-constrained wireless
networks. In this letter, we investigate simultaneous power
and information transfer in two-way amplify-and-forward
orthogonal-frequency-division-multiplexing relay networks,
where an energy-constrained relay node equipped EH device
harvests energy from two terminals and uses the energy to
forward information in a time-switching (TS) manner. By jointly
designing TS ratios of EH and information processing at the
relay, as well as power allocation over all the subcarriers at two
terminals and relay, our objective is to maximize end-to-end
achievable rate of two-way relay networks subject to transmit
power constraint at two terminals and EH constraint at
relay. The formulated problem is hard to tackle because the
EH constraint is non-convex and the objective rate function
has a highly non-convex structure. We propose to solve this
non-convex problem by transforming it into a nonlinear
fractional problem, which can be solved by successive convex
approximation and Dinkelbach’s procedure, and thus propose an
efficient iterative algorithm that achieves superior performance
with fast convergence speed.
Index Terms—Power allocation, orthogonal frequency division
multiplexing (OFDM), energy harvesting (EH), relay.
I. INTRODUCTION
SIMULTANEOUS wireless information and power trans-
fer (SWIPT) is a promising energy harvesting (EH) tech-
nique. In wireless relay networks, a relay node equipped with
EH devices is able to harvest energy from radio frequency (RF)
signals transmitted from a source node [3]–[5]. This may pro-
long lifetime of an energy-constrained relay. To perform EH
while relaying, the relay may operate in a time-switching (TS)
or power-splitting (PS) manner. The TS-based relay switches
over time between EH and information processing, while
the PS-based relay splits the received signals into two
streams of different powers for EH and information processing
separately [3].
As orthogonal frequency division multiplexing (OFDM)
is a spectrally efficient technique in broadband wireless
communications, OFDM relay networks with SWIPT have
been investigated recently in [6]–[8]. In [6] and [7],
Manuscript received March 28, 2016; revised May 4, 2016;
accepted May 16, 2016. Date of publication May 19, 2016; date of
current version August 10, 2016. This work was supported in part by the
National Natural Science Foundation of China under Grant 61472458 and
Grant 61173148, in part by Guangdong Natural Science Foundation under
Grant 2014A030310349, in part by Guangdong Science & Technology
Project under Grant 2016A010101032 and 2013B010402018, in part by
Guangzhou Science & Technology Project under Grant 2014J4100142,
Grant 2014J4100233 and Grant 2014A030310349, and in part by Guangzhou
Colleges and Universities Project under Grant 1201421329. The associate
editor coordinating the review of this letter and approving it for publication
was C.-H. Lee.
The authors are with the School of Mechanical and Electrical
Engineering, Guangzhou University, Guangzhou 510006, China (e-mail:
huanggaofei@gzhu.edu.cn; tangdong@gzhu.edu.cn).
Digital Object Identifier 10.1109/LCOMM.2016.2570751
resource allocations in PS-based and TS-based decode-and-
forward (DF) OFDM relay networks with SWIPT have been
studied respectively. In [8], resource allocations in a PS-based
or TS-based amplify-and-forward (AF) OFDM network with
multiple antennas have been studied.
Meanwhile, since two-way relaying may improve the
spectrum efficiency of one-way relaying, two-way relay
networks (TWRNs) with SWIPT have been investigated
in [9]–[13] under narrow-band channels. In [9]–[11], the per-
formance of TS-based or PS-based two-way AF or DF relay
networks has been studied. In [12] and [13], resource alloca-
tions in two-way AF or DF relay networks have been studied
respectively.
In this letter, we investigate resource allocations in a broad-
band OFDM AF TWRN with SWIPT. We focus on TS-based
relaying, because TS-based EH receivers have low imple-
mentation complexity since current commercial circuits are
usually designed to process information and harvest energy
separately, operating with very different power sensitivities [1].
Our objective is to maximize end-to-end achievable rate by
jointly designing TS ratios of EH and information processing
at relay as well as power allocation (PA) over all subcarriers
at two terminals and relay, subject to transmit power con-
straints at two terminals and a non-convex EH constraint at
relay. Compared with the optimization problems for TS-based
DF [7] and AF [8] OFDM one-way relay networks (OWRNs),
the formulated problem in our work is more challenging.
This is because the non-convex objective rate function of the
OFDM AF TWRN has more complex structure than those of
the OWRNs in [7] and [8]. In [7], due to the simple structure
of the rate function in the DF relay network, the optimization
problem can be decomposed into a convex problem of PA
at source and a quasi-convex problem of PA at relay. In [8],
the rate function in the AF relay network is concave as PA
at source or PA at relay is fixed, or can be approximated as
a concave function at high SNR (signal-to-noise ratio) region.
However, for the OFDM AF TWRN in our work, due to
the complex structure of the rate function, the optimization
problem cannot be decomposed. Also, the rate function is not
concave at high SNR region or as any one of PAs (i.e. PAs
at relay and two terminals) is fixed. Thus, the optimization
problem in our work is more difficult to tackle. In this
letter, we propose to transform the non-convex problem into
a nonlinear fractional programming problem, and then solve it
with an iterative successive convex approximation algorithm
embedded with a Dinkelbach’s procedure.
II. SYSTEM MODEL AND PROBLEM FORMULATION
Consider an AF TWRN with Nsubcarriers, where two
terminals TAand TBexchange information with each other
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1564 IEEE COMMUNICATIONS LETTERS, VOL. 20, NO. 8, AUGUST 2016
by virtue of an EH relay R(Fig.1). TAcannot communicate
with TBdirectly, e.g., due to physical obstacles, which is
valid in many real-world communication scenarios [3]–[15].
Both TAand TBhave fixed energy supply while Ris battery-
powered and thus energy-constrained [3]–[12]. To address the
energy scarce problem, Rharvests energy from the RF signals
transmitted from TAand TBand uses the harvested energy
to forward signals to TAand TB. The circuitries’ processing
power at relay is assumed to be negligible [3]–[12].
The information exchange between TAand TBis on a time-
frame basis. Each time frame with equal duration, denoted
as Tf, is divided into three time slots with one fraction of ρTf
and two successive fractions of (1−ρ)Tf/2. At the first
time slot, Rharvests energy transferred by TAand TB.At
the second time slot, both TAand TBtransmit information
to R. At the third time slot, by reallocating the harvested
energy over all subcarriers, Ramplifies the received signals
and broadcasts them to TAand TB. It is worth noting that the
assumption of symmetric information transmission (IT) at the
last two time slots is necessary since an AF relay amplifies
the information signals received from an incoming subcarrier
and directly forwards it to the next node over an outgoing
subcarrier. However, for DF relaying, dynamic time allocation
for IT may further improve the system performance [16].
We will study such a scenario in our future work.
The channel in the OFDM TWRN is assumed to be recip-
rocal [14]. Over subcarrier n, the responses of the two-hop
links are denoted as h1(n)and h2(n)respectively. Then,
the harvested energy at relay is E=ηρ TfPEwith PE=
N
n=1pA,E
n|h1(n)|2+pB,E
n|h2(n)|2[1], where pA,E
nand
pB,E
ndenote the power allocated over subcarrier nin the
duration of energy transfer and 0 <η<1isenergy
conversion efficiency. Define Hkmaxn|hk(n)|2,k∈
{1,2}. Obviously, PE≤PAH1+PBH2,wherePAand PB
denote the maximum allowable transmit power of TAand TB
respectively. This indicates that as TAand TBallocate all
available power over the subcarrier with largest channel gain,
the relay can harvest the maximum energy E=ρGT
fwith
G=PAH1+PBH2. Moreover, the end-to-end achievable rate
of the OFDM AF TWRN can be expressed as [14]
R=(1−ρ)
2(RA+RB)(1)
where RA=N
n=1log 1+ϒA
n(pn)and RB=
N
n=1log 1+ϒB
n(pn)in which
ϒA
n(pn)=pA
nγ1(n)pR
nγ2(n)
pA
nγ1(n)+pR
nγ2(n)+pB
nγ2(n)+1(2)
and
ϒB
n(pn)=pB
nγ2(n)pR
nγ1(n)
pA
nγ1(n)+pR
nγ1(n)+pB
nγ2(n)+1(3)
with pn=pA
n,pR
n,pB
nT,pA
n,pB
nand pR
ndenote the transmit
power of TA,TBand Rover subcarrier nfor IT respectively,
γ1(n)=|h1(n)|2
σ2and γ2(n)=|h2(n)|2
σ2. Here, σ2denotes
the variance of additive Gaussian noise over each subcarrier.
Furthermore, the consumed energy for IT at relay should be
less than or equal to the harvested energy [3]–[5],
(1−ρ)
2
N
n=1
pR
n≤ρG.(4)
Thus, joint TS and PA optimization problem is formulated as
max
p≥0,ρ∈[0,1]Rs.t.(4),
N
n=1
pA
n≤PA,
N
n=1
pB
n≤PB(5)
where p=vec {p1,p2,··· ,pN}.
III. JOINT TS AND PA OPTIMIZATION
In problem (5), both the objective function provided in (1)
and the EH constraint in (4) are non-convex.Thus, it is difficult
to solve this problem directly. To make it tractable, we rewrite
the EH constraint (4) as ρ≥N
n=1pR
n
ξwith ξ=N
n=1pR
n+2G.
Since the objective function is a non-increasing function of ρ,
the optimal ρsatisfies ρ=N
n=1pR
n
ξ. Then, we can rewrite
problem (5) as
max
p≥0(RA+RB)s.t.
N
n=1
pA
n≤PA,
N
n=1
pB
n≤PB(6)
where =G
ξ.
Problem (6) is still hard to tackle due the highly non-convex
objective function. Nevertheless, we observe that for any z≥0
and z≥0, the following lower bound holds [17]
αlog z+β≤log(1+z)(7)
where αand βare defined as
α=z
1+z,β=log(1+z)−z
1+zlog z(8)
and the bound is tight for z=z. Thus, the lower
bounds to RAand RBcan be expressed as
RA=N
n=1αA
nlog ϒA
n(pn)+βA
nand and
RB=
N
n=1αB
nlog ϒB
n(pn)+βB
nrespectively, in which all
αA
n,βA
n,αB
nand βB
nare approximation constants computed
as (8) for zA
n=ϒA
n(pn)and zB
n=ϒB
n(pn)with some
given pnto be specified later.
Furthermore, consider the transformation pA
n=exp qA
n,
pB
n=exp qB
nand pR
n=exp qR
n.Letqn=qA
n,qR
n,qB
nT
and q=vec {q1,q2,··· ,qN}. Then using the lower bounds
RAand
RB, problem (6) can be rewritten as
max
qQ
RA+
RB(9a)
s.t.
N
n=1
exp qA
n≤PA,
N
n=1
exp qB
n≤PB(9b)
where Q=G
ζ,
RA=N
n=1αA
nqA
n+qR
n+log ψ−log ζA+
βA
n]and
RB=N
n=1αB
nqB
n+qR
n+log ψ−log ζB+βB
nin
which ζ=N
n=1exp qR
n+2G,ψ=γ1(n)γ2(n),ζA=
exp qA
nγ1(n)+exp qR
nγ2(n)+exp qB
nγ2(n)+1andζB=
exp qA
nγ1(n)+exp qR
nγ1(n)+exp qB
nγ2(n)+1. It can
be verified that each constraint in problem (9) is convex and
HUANG AND TANG: WIRELESS INFORMATION AND POWER TRANSFER IN TWO-WAY OFDM AF RELAY NETWORKS 1565
Algorithm 1 Dinkelbach’s Procedure to Solve Problem (9)
1: Initialization: Set tolerance >0, μ=0andFLAG=0.
2: Repeat:
Update qby solving problem (10).
if h(q,μ
)<,FLAG=1;
else μ=G(
RA+
RB)
ζ;end if.
3: Until: FLAG =1.
Algorithm 2 The Proposed Joint TS and PA Optimization
Algorithm for the OFDM AF TWRN
1: Initialization: Set l=0, tolerance >0andp=p(0);
2: Repeat:
Calculate zA
n(l)=ϒA
n(pn(l))and zB
n(l)=ϒB
n(pn(l));
Set αA
n(l)and βA
n(l)with zA
n(l)and set αB
n(l)and βB
n(l)
with zB
n(l)as in (8);
Solve problem (9) using Algorithm 1 to obtain p(l+1);
l=l+1;
3: Until: p(l+1)−p(l)<.
the function
RA+
RBis concave [18]. Then, problem (9) is a
concave-convex fractional problem [19], which can be solved
by solving the convex problem as follows
max
qh(q,μ
)=G
RA+
RB−μζ
s.t.
N
n=1
exp qA
n≤PA,
N
n=1
exp qB
n≤PB.(10)
Let F(μ) =max
qh(q,μ) and f(μ) =argmaxqh(q,μ).
It has been proven in [19] that problem (9) and (10) are
equivalent to each other if and only if F(μ∗)=0and
f(μ∗)=q∗. Then, problem (9) can be solved by using
Dinkelbach’s procedure [19] as illustrated in Algorithm 1.
As the optimal solution to problem (9) is obtained, we pro-
pose to solve (6) by iteratively optimizing PA and improving
the bound as illustrated in Algorithm 2.
As to Algorithm 2, we have the following proposition.
Proposition 1: Algorithm 2 monotonically improves the
value of the objective in problem (6) at each iteration and
will always converge. At convergence, the obtained PA p∗
satisfies the Karush-Kuhn-Tucker (KKT) optimality conditions
of problem (6).
Proof : Denote the objective function in problem (6) and
problem (9) as R(p)and
R(q), respectively. Let p(l)=
exp(q(l)) be the optimized values after the lth iteration and
let x(l)be the vector stacked by {αA
n(l)}N
n=1,{βA
n(l)}N
n=1,
{αB
n(l)}N
n=1and {βB
n(l)}N
n=1, which are the constants calculated
with p(l)by (8). Then, as l≥1,we have
R(p(l−1))(a)
=
R(q(l−1);x(l−1))(b)
≤
R(q(l);x(l−1))
(c)
≤R(p(l))(a)
=
R(q(l);x(l))(b)
≤
R(q(l+1);x(l))
(11)
where (a) holds because the lower bounds
RAand
RBare tight
at the current values of PA, (b) holds because the Dinkelbach’s
procedure in Algorithm 1 computes the globally optimal solu-
tion to problem (9) and (c) holds because
R(q(l);x(l−1))
is the lower bound of R(p(l)). (11) indicates that the value
of the objective in problem (6) is improved at each iteration.
Since R(p)is bounded above, the procedure must converge.
In addition, after reverting back to the p-space, the first
order necessary conditions of problem (9) can be written as
αA
n1−pA
nγ1(n)
φA−αB
npA
nγ1(n)
φB−λApA
n=0,
αB
n1−pB
nγ2(n)
φB−αA
npB
nγ2(n)
φA−λBpB
n=0,
αA
n1−pR
nγ2(n)
φA+αB
n1−pR
nγ1(n)
φB=G
(12)
where φA=pA
nγ1(n)+pR
nγ2(n)+pB
nγ2(n)+1andφB=
pA
nγ1(n)+pR
nγ1(n)+pB
nγ2(n)+1. Meanwhile, the first order
necessary conditions of problem (6) can be expressed as
A1−pA
nγ1(n)
φA−αB
npA
nγ1(n)
φB−λApA
n=0,
B1−pB
nγ2(n)
φB−αA
npB
nγ2(n)
φA−λBpB
n=0,
A1−pR
nγ2(n)
φA+B1−pR
nγ1(n)
φB=G
(13)
where A=ϒA
n(pn)
1+ϒA
n(pn)and B=ϒB
n(pn)
1+ϒB
n(pn). Denote the opti-
mal solution to problem (9) as p∗. Then, there exists λ∗
Aand
λ∗
Bsuch that the triplet p∗,λ
∗
A,λ
∗
Bsatisfies (12). Calculating
αA∗
nand αB∗
nin (12) by (8) with p∗and substituting them
into (12), (12) has exactly the same form as (13). Similarly,
the complementary slackness conditions of problem (9) have
the same forms as those of problem (6). Thus, the triplet
p∗,λ
∗
A,λ
∗
Bsatisfies KKT conditions (13).
Remark 1: Proposition 1 indicates that the iterations in
Algorithm 2 guarantees at least a local optimal solution
to problem (6). We will show that the proposed algorithm
achieves superior performance by simulations in section IV.
IV. SIMULATION RESULTS
We evaluate the performance of our proposed scheme over
the OFDM AF TWRN by Monte-Carlo simulations. In simula-
tions, the fading statistics of different subcarriers are assumed
to be independent and identically Rayleigh distributed and the
variances are assigned by adopting a path loss model of the
form ∝d−2.5
pq ,wheredpq denotes the distance between
nodes pand q,p,q∈{TA,TB,R}. The distance of TAand
TBis set as dAB =20m. The location of relay is indicated
by κ=dAR
dAB ,wheredAR is the distance between TAand R.
The maximum allowable transmit power at TAand TBis set
as PA=PB=Pmax =20mW respectively. The number of
subcarriers is set as N=32 and the noise power over each
subcarrier is set as σ2=10−8
NW.
In Fig.1, we present the average rate comparison of our
proposed scheme (denoted as “Rate by Proposed” in the
legend) and the fixed-TS scheme (denoted as “Rate by Fixed
TS”) with ρ=0.1andρ=0.5 for different values of κ.
The fixed-TS scheme is obtained by solving problem (5) with
the algorithm proposed in [15]. For our proposed scheme,
1566 IEEE COMMUNICATIONS LETTERS, VOL. 20, NO. 8, AUGUST 2016
Fig. 1. Average achievable rates and the obtained EH TS ratio ρversus κ.
Fig. 2. Convergence of the achievable end-to-end rate (with κ=0.5).
p(0)in Algorithm 2 is set as Pmax/N. Also, we illustrate
the average rate of the OFDM DF OWRN achieved in [7]
(denoted as “Rate of OWRN”) in Fig.1 by letting TAbe the
source and TBbe the destination where the source power is
set as PS=PA+PB. Moreover, in Fig.1, we compare our
proposed scheme with the optimal solution of (6) obtained
by exhaustive search (ES) (denoted as “Rate by ES” in the
legend) when N=2 since the exhaustive search for the
case when N=32 is too complex. Furthermore, we plot
the EH TS ratio ρobtained by our proposed scheme (denoted
as “EH TS Ratio ρby Proposed”) in Fig.1. From Fig.1, it
is observed that our proposed scheme achieves significant rate
gains over the fixed-TSscheme with ρ=0.5 for all values of κ
and the fixed-TS scheme with ρ=0.1asκapproaches 0.5.
Specially, when κapproaches 0.1 or 0.9, the performance of
our proposed scheme is similar to that of the fixed-TS scheme
with ρ=0.1. This is because, as κapproaches 0.1 or 0.9,
the optimal EH TS ratio ρis approximately equal to 0.1, which
is just illustrated by the dotted curve. Moreover, by Fig.1,
two-way relaying achieves much larger rates than one-way
relaying, and the average transmission rate of our proposed
scheme is very close to that of optimal solution.
Fig.2 shows the convergence of the rate achieved by
Algorithm 2 as κ=0.5 for different initial p(0).ByFig.2,
Algorithm 2 converges to the optimal solution from differ-
ent initial points with fast convergence speed. In addition,
the complexity of a single iteration in Algorithm 2 is mainly
tied to the optimization of PA by Dinkelbach’s procedure in
Algorithm 1. Since Dinkelbach’s procedure has a super-linear
convergence rate [19] and only requires solving a convex
problem in each iteration with polynomial time [18], our
proposed algorithm has low computational complexity and
thus is efficient.
V. CONCLUSIONS
In this letter, we have proposed an efficient joint TS
and PA optimization algorithm for OFDM AF TWRNs with
SWIPT. Simulation results demonstrate that our proposed
algorithm achieves superior performance with fast convergence
speed.
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