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Energy Efficiency in Energy Harvesting Cooperative
Networks with Self-Energy Recycling
Shiyang Hu1, Zhiguo Ding1, Qiang Ni1, Wenjuan Yu1and Zhengyu Song2
1School of Computing and Communications, Lancaster University, LA1 4WA, UK
2School of Information and Electronics, Beijing Institute of Technology, 100081, Beijing, China
Abstract—Cooperative communication has been identified as
an important component in the 5G system. This paper considers a
decode-and-forward (DF) relaying wireless cooperative network,
in which the self-energy recycling relay is powered by radio-
frequency (RF) signal from the source and its transmitted power
from the loop-back channel. The harvested energy is used to
support the relay transmissions. Based on a self-energy recycling
relaying protocol, we study the optimization of energy efficiency
in wireless cooperative networks. Although the formulated op-
timization problem is not convex, it can be re-constructed to a
parametric problem in the convex form by using the non-linear
fractional programming, to which closed form solutions can be
found by using the Lagrange multiplier method. The simulation
results are presented to verify the effectiveness of this solution
proposed in this paper.
I. INTRODUCTION
Wireless energy harvesting is a promising approach to
charge batteries in the future 5G wireless communication
network [1]–[3]. For many extreme conditions, the traditional
power supplies are impossible to recharge or to replace [1].
For example, more and more medical devices are implanted
under the patient’s skin for recording body data. These devices
are difficult and costly to replace the battery. Therefore,
wirelessly recharging the battery from the external source
has a high demand, which motivates the energy harvesting
technology emerging. Some works pointed out that the power
of solar, wind and thermoelectric phenomena can be used as
the external sources to wirelessly charge [4]–[7]. Besides, the
radio-frequency (RF) signals as a potential way to achieve
the wireless energy harvesting has drawn the considerable
attention [8].
However, the practical circuit limits the development of
simultaneously receiving information and harvesting energy at
the receiver node. Hence a more practical design was proposed
in [3], where an energy harvesting node performs information
decoding and energy harvesting separately according to the
time switching protocol or the power splitting protocol. Some
works studied the new energy harvesting protocols for point-
to-point communication networks [1], [8]–[10]. Besides, the
authors in [11] researched the time switching protocol and
the power splitting protocol in wireless cooperative networks.
Specifically, in [11], the authors employed energy harvesting
protocol in an amplify-to-forward (AF) cooperative network
and studied the outage probability and the throughput in the
system. The authors in [12] studied power allocation strategies
in a cooperative network with one energy harvesting relay and
multiple pairs of sources and destinations.
The half-duplex cooperative network is extensively studied
in the wireless energy harvesting study. There are some
researchers focusing on the full-duplex structure study in the
energy harvesting. The full-duplex wireless powered networks
based on time-switching protocol were studied in [13], [14].
In the full-duplex wireless powered network, the node is
capable of transmitting energy and receiving information si-
multaneously. In [15], a new self-energy recycling protocol
was proposed. The self-energy recycling relaying protocol is
based on a two-phase transmission protocol. The full-duplex
structure relay is equipped with two antennas. It receives
the information from the source node in the first phase.
In the second phase, the energy harvesting relay uses its
receiving antenna to collect power from the source and uses
its transmission antenna to relay the decoded information to
the destination node. The advantage of this protocol is that
the energy harvesting relay not only harvests power form the
source node, but also reuses the energy from its transmitted
power by its loop-back channel. This protocol was set up in
a MISO relaying channel in [15].
Motivated by this, we study energy-efficiency maximization
in decode-and-forward (DF) wireless cooperative network with
the self-energy recycling protocol. The concept of energy-
efficient study in wireless communication has drawn much
attention recently [16]–[18]. Solutions of the energy efficiency
(bit-per-Joule) optimization problem can ensure that wireless
communication systems utilize energy in a more environment-
friendly way. The energy-efficient maximization problem is
defined as a ratio of the channel capacity and overall power
consumption in the block time. Particularly the energy-efficient
maximization problems are formulated based on the self-
energy recycling protocol. The formulated optimization prob-
lem is not in standard convex form, but can be transformed to a
parametric problem in the convex form by applying non-linear
fractional programming. The re-formulated problems can be
solved by applying the Lagrange multiplier method and the
gradient method. Simulation results are provided to verify that
the proposed algorithm improves energy utilization compared
to the case without applying energy-efficient optimization. The
trade-off between the energy efficiency and system parameters
is also analysed by using the provided numerical results.
The rest of this paper is organized as follows. Section II
presents the system model of an energy harvesting wireless
cooperative network. Section III presents the details for the
transmission model based on the self-energy recycling protocol
and formulates the corresponding energy-efficient problem.
Section IV presents the solutions to transform original prob-
lems into solvable convex optimization form. The numerical
results are provided in V and section VI concludes this paper.
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II. SYSTEM MODEL
The wireless cooperative network considered here includes
one source-destination pair and one energy harvesting relay.
The source node and the destination node are equipped one
antenna respectively. The relay node has two antennas, one
antenna is used for information transmission while the other
one is used for receiving. We assume that there is no direct
link between the source node and the destination node, i.e.
the source node intends to transmit message to the destination
node with the assist of the relay node. Channels are modelled
as quasi-static block fading channels. The perfect channel state
information is available at the destination node.
The energy harvesting relay is solely powered by the source.
It can harvest energy from the source node and utilize the
energy to relay the source information. This assumption is used
in [11]. The battery capacity of the energy harvesting relay is
assumed as infinite. The decode-and-forward (DF) scheme is
employed in the cooperative network. A new energy harvesting
protocol with self-energy recycling relay is considered in this
paper. The detailed analysis based on this protocol is given in
the following sections. We also assume that at the relay, the
power consumed to process the harvested energy is negligible,
when compared to the power used in transmitting information
to the destination.
III. ENERGY EFFICIENCY WITH SELF-ENERGY
RECYCLING RELAY
In this section, we study the energy-efficient optimization
problem in self-energy recycling relay networks. The trans-
mission model is given as follows.
A. Relay Protocol and Transmission Model
The whole transmission process is operated in the block
time, denoted by T. Without loss of generality, Tis normalized
to be unity. The information transmission process is split into
two phases. In the first phase, the source transmits information
to the relay for T/2time and relay uses its receiving antenna to
receive information. In the second phase, the source transmits
RF signals to power the energy harvesting relay and the relay
sends decoded information to the destination. Recall that the
energy harvesting relay is equipped with one transmission
antenna and one receiving antenna, therefore it is capable of
relaying the information and collecting energy simultaneously.
The energy harvesting relay first receives the information
from the source with its receiving antenna. The received signal
at the relay can be expressed as
y1
r=pPshxs+nr,(1)
where Psis the transmitted power, his the channel gain from
the source to the receiving antenna of the relay, and xsis
the normalized transmitted signal with unit power and nris
the baseband additive white Gaussian noise (AWGN) from the
receiving antenna.
In the second phase, we assume that the energy harvesting
relay can decode the source message successfully. The relay
uses its transmission antenna to relay the information to the
destination node. The received signal at the destination is given
by
yd=pPrgxs+nd,(2)
where Pris the power transmitted from the relay, gis the
channel gain between the transmission antenna of the relay
and the destination node, xsis the decoded signal for the
destination and ndis the AWGN from the receiving antenna.
Concurrently, the energy harvesting relay is wireless powered
by the source node with dedicated energy-bearing signal. The
received signal at the relay is
y2
r=pPshxe+pPrfxs+nr+nc,(3)
where xeis normalized signal with unit power transmitted by
the source node, fis the channel gain of the loop channel at
the relay node, and ncis the sampled AWGN from the RF
band to baseband conversion [11]. The relay node not only
collects energy from the source node, but also recycles part of
its transmitted power due to its two antenna being activated at
the same time. In the other full-duplex relaying structure, the
relay employs interference cancellation techniques to eliminate
the loop-back interference signal. In the energy harvesting
cooperative network, the loop-back signal can be reused at the
relay as the transmitted power. The harvest energy is given by
E=⌘(Ps|h|2+Pr|f|2)T
2,(4)
where ⌘is the energy conversion efficiency coefficient. Then
the transmitted power at the relay node is E
T/2which can be
expressed as
Pr=⌘Ps|h|2
1⌘|f|2,(5)
In the case of the cooperative network using DF protocols
without the direct link between the source and the destination,
the channel capacity can be calculated as
C=min(Csr,C
rd),(6)
where Csr and Crd are the channel capacity from the source
to the relay and from the relay to the destination respectively.
Csr and Crd can be express as
Csr =1
2log ✓1+Ps|h|2
2
r◆,(7)
and
Crd =1
2log ✓1+ ⌘Ps|h|2|g|2
(1 ⌘|f|2)2
d◆,(8)
where 2
rand 2
dare the variances of the nrand nd.
B. Problem Formulation
In this section, we consider the energy efficiency problem
for the considered network. The energy efficiency is a ratio
between the system channel capacity and the overall power
consumption in one transmission block time. It is assumed
that each node has a constant circuit power consumption for
signal processing, which is independent of the power used for
transmitting signal or the harvested power. We denote P1,P2
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and P3by the circuit consumed power in the source, the relay
and the destination, respectively.
According to the above relaying protocol, nodes are acti-
vated in the corresponding transmission phase. Therefore the
total consumed power in this wireless cooperative network is
given by
Pc=P1+P2+1
2P3+Ps,(9)
This work aims to maximize the energy efficiency in the
considered model. The optimization problem is given by
max
Ps
C
Pc
s.t. C1:PsPs,
C2:CC,
C3:Ps0.(10)
where the predefined variable Psis the maximum transmit
power limitation at the source. Cis the minimum required
channel capacity in order to meet the QoS criterion.
One can easily verify that the proposed problem above
is not a convex optimization problem, because the objective
function is a ratio of the channel capacity and total consumed
power. The solution to this formulated problem is given in the
following section.
IV. ENERGY-EFFICI EN T POWER ALLOCATION
The proposed problem is not in the standard convex form.
However the problem can be transformed to a parametric
optimization problem by exploring the properties of the non-
linear fractional programming [19]. According to the para-
metric method, we introduce a new variable q⇤to denote the
optimum energy efficient for the proposed problem. q⇤can be
expressed as
q⇤=C⇤
Pc⇤,
=max
Ps
C
Pc
.(11)
Therefore, the transformed parametric problem is given by
max
Ps
Cq⇤Pc
s.t. C1:PsPs,
C2:CC,
C3:Ps0.(12)
Here we have the theorem to show the relationship between
the transformed problem and the original problem.
Theorem 1. The optimum energy efficient q⇤=C⇤
Pc
⇤=
max
Ps
C
Pc
if and only if q⇤satisfies
max
Ps
Cq⇤Pc,
=C⇤q⇤Pc⇤,
=0.(13)
Proof. Please refer to [19]. ⌅
By introducing the above theorem, the objective function of
the original problem can be be transformed into a subtractive
form, which is equivalent to its original one. Since the both
problems have the same optimum solution, we solve the
transformed problem as follows.
It can be confirmed that @2C
@2Ps<0. Therefore, the trans-
formed problem is convex and can be solved by its Largrange
dual problem due to the strong duality existing in it. The
Lagrangian function of the transformed problem is given by
L(,,Ps)=Cq⇤Pc(PsPs)(CC),(14)
where 0and 0are the Lagrange multipliers
corresponding the constraints C1and C2in the problem. The
dual problem can by expressed by
min
,0max
Ps
L(,,Ps),(15)
The dual problem can be solved by updating the optimum
solution of the transmitted power with Lagrange multipliers.
According to KKT conditions with @L(,,Ps)
@Ps=0, the
optimum value of Psin the cooperative network with the self-
energy recycling relaying protocol can be obtained as
Ps=[ 1+
2ln2(q⇤+)2
r
|h|2]+,(16)
or
Ps=[ 1+
2ln2(q⇤+)2
d⌘|f|22
d
⌘|h|2|g|2]+,(17)
There are two close forms of the transmitted power because the
different channel state information of the considered coopera-
tive network leads to different formulas to calculate the chan-
nel capacity in the DF protocol. Specifically, if Csr <C
rd, the
optimum value of transmitted power is calculated by (16). If
not, the optimum value of transmitted power is calculated by
(17). Unlike other cooperative networks, the channel capacity
of the source-relay link and that of the relay-destination link
both are the function of the transmitted power. Therefore we
know the formulation to calculate the channel capacity if the
channel state information is available.
So far, the optimum Pscan be obtained with given Lagrange
multipliers and we can use the gradient method in the iteration
algorithm to update and , which are given by
(n+ 1) = [(n) (PsPs)]+,(18)
(n+ 1) = [(n) (CC)]+,(19)
where and are positive step sizes. nis the iteration
index. With these updated Lagrange multipliers, the
optimum energy efficiency can be obtained. Inspired by the
Dinkelbach method [19], the algorithm is proposed as follows
Algorithm: Energy-Efficient optimization Iteration Algorithm
1: Initialize the q⇤, Lagrange multipliers and , the
maximum number of iteration nmax and the maximum
tolerance ✏.
2: Obtain Psin one of (16) and (17).
3: Use (18) and (19) to update the and in the iteration
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procedure.
4: If C⇤q⇤Pc⇤>✏, then set q⇤=C⇤
Pc
⇤and repeat the step
2 and 3. If C⇤q⇤Pc⇤<✏, the optimum Ps⇤and q⇤are
obtained.
From the proposed algorithm, we can see that the trans-
formed parametric problem can be solved by the Lagrange
multiplier method with the given q⇤. The convergence of the
optimum q⇤in its parametric problem is guaranteed.
Proof. Please refer to Appendix. ⌅
V. NUMERICAL RESULTS
In this section, simulation results are provided to evaluate
the performance of the proposed energy-efficient optimization
algorithm. The distances from the source to the relay and
the relay to the destination are 10m with the path loss
exponent m=3. The antenna noise variance and conversion
noise variance are equal to 50dbm. The energy harvesting
coefficient is set as 0.5. The circuit-consumed power at the
source, the relay and the destination is all set as 20dbm. The
maximum transmitted power is 30dbm. the minimum required
channel capacity is 1bits/sec/Hz. The channel between the
transmit-receive antenna pair is simulated as h=(d)m
2ejw.
The loop channel path loss is 15db [20]. Simulation results
were averaged over 1000 independent trials.
1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Number of Iterations
System energy efficiency (bit/Joule)
Proposed algorithm, η=1
Proposed algorithm, η=0.5
EE without optimization, η=1
EE without optimization, η=0.5
Fig. 1. The performance of proposed algorithm with the different levels
of energy conversion coefficient ⌘and system performance without energy
efficiency optimization
In Fig. 1, the convergence of the proposed algorithm for
energy-efficient maximization in cooperative networks with
the self-energy recycling relaying protocol is shown, where
the energy efficiency for the case without optimization is also
provided as a benchmark. As can be seen from the figure,
the proposed algorithm converges within 5iterations. The
concavity of problems can be guaranteed. Since the proposed
algorithm always converges, the optimum energy efficiency
can be obtained by using the proposed algorithm. The Fig. 1
also presents the performance with different levels of the en-
ergy conversion coefficient ⌘. With the bigger ⌘, the relay node
can harvest more power from the source node and its loop-back
channel. Therefore the system performance achieves better. To
facilitate a better performance evaluation, the energy efficiency
without optimization with different ⌘is also provided. The
value of energy efficiency without the proposed algorithm
is obtained with the maximum allowed transmitted power.
It is obvious that the energy efficiency can not outperform
the proposed algorithm. The reason for this performance gain
is following. Recall that the energy efficiency is calculated
from the ratio of the channel capacity and the overall power
consumption. Increasing the power can achieve larger channel
capacity, but it consumes more power. The proposed algorithm
guarantees that better energy efficiency is achieved.
1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Number of Iterations
System energy efficiency (bit/Joule)
Proposed algorithm, original parameters
Proposed algorithm, d=15
Proposed algorithm, noise variance=−40dbm
Fig. 2. Energy efficiency optimization for the self-energy recycling relaying
protocol with different parameters
Fig. 2 illustrates the performance of system energy effi-
ciency maximization with different noise variance and distance
based on the self-energy recycling relaying protocol. As can be
seen from the picture, the energy efficiency is deteriorated by
the increasing noise variance and distances. The reason is that
when noise variance and distance increase, channel capacity
will reduce and then lead to the decrease of energy efficiency.
VI. CONCLUSIONS
This paper proposed feasible solutions for the maximization
of energy efficiency in wireless cooperative networks based on
a self-energy recycling relaying protocol. The original problem
was first transformed to a parametric problem by using non-
linear fractional programming and then solved by applying the
Lagrange multiplier method. Simulation results confirm the
energy efficiency of the proposed algorithm in the considered
model. The trade-off between energy efficiency and system
parameters is also analysed by using simulation results.
APPENDIX
A. Convergence analysis
We prove that the convergence of the transformed paramet-
ric problem can be guaranteed. First of all, we introduce two
lemmas.
Lemma 1. The object function Cq⇤Pcis a monotonic
decreasing function of q⇤.
Proof. Given two power allocation results Ps1and Ps2with
their corresponding energy efficiency q⇤
1and q⇤
2. We define
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q⇤
1>q
⇤
2, then we have
max
Ps
Cq⇤
2Pc,
=C(Ps2)q⇤
2Pc(Ps2),
>C(Ps1)q⇤
2Pc(Ps1),
C(Ps1)q⇤
1Pc(Ps1),
=max
Ps
Cq⇤
1Pc.(20)
⌅
Lemma 2. There is an energy efficient solution q⇤to ensure
Cq⇤Pc=0.
Proof. It can be proved that the objective function is contin-
uous in q⇤. If q⇤is plus infinity, the value of the objective
function is minus infinity and vice-versa. Hence there is a q⇤
to let Cq⇤Pc=0.⌅
With the above lemmas, we can prove the convergence of
the transformed parametric problem. We denote Psnand q⇤
n
as the energy efficiency policy in the n-th iteration. Recall that
in the algorithm, q⇤
n+1 =C(Psn)
Pc(Psn). Then we have
C(Psn)q⇤
nPc(Psn),
=q⇤
n+1Pc(Psn)q⇤
nPc(Psn),
=(q⇤
n+1 q⇤
n)Pc(Psn).(21)
where Pc(Psn)is greater than 0, therefore q⇤
n+1 q⇤
n0.
The q⇤
nis non-decreasing in the iterative process. In lemma
1, the objective function is monotonic decreasing in q⇤
nand
q⇤
nis non-decreasing in the iteration, therefore the objective
function is non-increasing in the iteration. In lemma 2, the
optimum energy efficiency converges when Cq⇤Pc=0. If
the iteration index is big enough, Cq⇤Pcwill equal to 0
and the energy efficiency will be obtained.
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