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Unification of RF Energy Harvesting Schemes under Mixed Rayleigh-Rician Fading Channels-Paper without Code

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  • The University of Oklahoma Tulsa OK USA

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In this work, a generalized approach is proposed to review the performance of relaying designs with energy harvesting capability. The unified modeling of generalized energy harvesting relaying (GEHR) design covers the non-energy harvesting designs and the well-known energy harvesting designs, i.e., time-based relaying (TR) and power-based relaying (PR). Moreover, the hybrid of both TR and PR designs is also catered. We find the mathematical representations for the outage probability, ergodic capacity and average throughput in Rayleigh fading channels for amplify-and-forward (AF) and decode-and-forward (DF) relaying modes. The closed-form expressions are derived for the outage probability. To validate that GEHR design is a generalization of TR and PR designs, we study the individual cases of GEHR design from the perspectives of schematic diagram, signal analysis and the performance evaluation parameters comparison. Furthermore, the GEHR design is studied for the mixed Rayleigh-Rician fading channels. We considered two sub-cases of mixed fading, i.e., in case 01, source to relay (SR) link is considered as Rayleigh channel and relay to destination (RD) link is considered as a Rician channel. Conversely, in case 02, SR link is taken as Rician channel and RD link is a Rayleigh channel. We find the mathematical expressions for the ergodic capacity, outage probability and average throughput for DF and AF relaying for both cases of mixed fading channels. The analytical results in both channel configurations are presented for throughput and verified using extensive Monte-Carlo simulations. The results show that the proposed GEHR design can be set to work as not only for the conventional TR and 2 PR designs but also for hybrid of them. Furthermore, with some slight modifications in the proposed design, it can work as a conventional non-energy harvesting cooperative relaying model. Index Terms Ergodic capacity, outage probability, average throughput, delay tolerant and delay limited communication , mixed Rayleigh-Rician fading channels.
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1
Unification of RF Energy Harvesting Schemes
under Mixed Rayleigh-Rician Fading
Channels-Paper without Code
Waseem Raza1, Hina Nasir2,3, Nadeem Javaid4,
1The University of Lahore, Lahore 54000, Pakistan
2International Islamic University, Islamabad 44000, Pakistan
3Air University, Islamabad 44000, Pakistan
4COMSATS University Islamabad, Islamabad 44000, Pakistan
Correspondence: nadeemjavaidqau@gmail.com; https://www.njavaid.com
Abstract
In this work, a generalized approach is proposed to review the performance of relaying designs with
energy harvesting capability. The unified modeling of generalized energy harvesting relaying (GEHR)
design covers the non-energy harvesting designs and the well-known energy harvesting designs, i.e.,
time-based relaying (TR) and power-based relaying (PR). Moreover, the hybrid of both TR and PR
designs is also catered. We find the mathematical representations for the outage probability, ergodic
capacity and average throughput in Rayleigh fading channels for amplify-and-forward (AF) and decode-
and-forward (DF) relaying modes. The closed-form expressions are derived for the outage probability.
To validate that GEHR design is a generalization of TR and PR designs, we study the individual cases
of GEHR design from the perspectives of schematic diagram, signal analysis and the performance
evaluation parameters comparison. Furthermore, the GEHR design is studied for the mixed Rayleigh-
Rician fading channels. We considered two sub-cases of mixed fading, i.e., in case 01, source to relay
(SR) link is considered as Rayleigh channel and relay to destination (RD) link is considered as a Rician
channel. Conversely, in case 02, SR link is taken as Rician channel and RD link is a Rayleigh channel.
We find the mathematical expressions for the ergodic capacity, outage probability and average throughput
for DF and AF relaying for both cases of mixed fading channels. The analytical results in both channel
configurations are presented for throughput and verified using extensive Monte-Carlo simulations. The
results show that the proposed GEHR design can be set to work as not only for the conventional TR and
2
PR designs but also for hybrid of them. Furthermore, with some slight modifications in the proposed
design, it can work as a conventional non-energy harvesting cooperative relaying model.
Index Terms
Ergodic capacity, outage probability, average throughput, delay tolerant and delay limited commu-
nication, mixed Rayleigh-Rician fading channels.
I. INTRODUCTION
Energy harvesting in wireless communication is becoming an efficient approach to lengthen the
lifetime of battery-operated network devices. It is also being proposed to include it in the vision
of upcoming 5th generation technology [1] and [2]. Radio frequency (RF) energy harvesting
aims to extract energy from the received radio signals to operate the network devices. It has
gained notable recognition from the research community in academia and industry as it promises
to fulfill the energy requirements of battery-constraint devices without causing any significant
decrease in the data rates [3]. Simultaneous wireless information and power transfer (SWIPT)
facilitates processing information and harvesting energy at the same time [4]. The advancement
in circuit technology has enabled to scavenge the significant amount of energy from the acquired
signals while maintaining the original purpose of information transfer simultaneously.
On the other hand, cooperative relaying is an attractive solution to enhance the performance
of wireless communication system. It involves a relay node which helps in transmitting data
from source to intended destination. SWIPT combined with cooperative relaying offers ben-
efits in terms of increased diversity gain as well as increased lifetime of battery constrained
devices by recharging node’s battery [59]. Some of the earliest works on cooperative energy
harvested SWIPT relaying include [10] and [11]. In [10], the authors propose and analyze two
energy harvesting designs named as time-based relaying (TR) and power-based relaying (PR) for
amplify-and-forward (AF) relaying conditions. The authors consider the delay-tolerant (DT) and
delay-limited (DL) cases and derive the mathematical representations for the ergodic capacity and
outage probability in closed-form for these cases, respectively. This work is further extended for
the case of the decode-and-forward (DF) model in [11], where the mathematical representations
are derived for the ergodic capacity for TR and PR designs in closed-form. SWIPT for randomly
distributed relays is studied for DF relaying in [12]. Also, the outage probability analysis of
the PR using DF is given in [13]. Apart from the mathematical representations of the outage
3
probability, the theoretical expressions of the optimum harvested energy and most suitable relay
location are also determined, which are utilized to minimize the outage probability. In [14],
the authors propose the optimal TR protocols for DF relay network, where the relay nodes
are equipped with rechargeable batteries. The proposed system aims to optimally adapt the
system power and energy harvesting time according to the channel conditions and battery status.
In [15], a TR energy harvesting for mixed Rayleigh-additive white Gaussian noise cooperative
relaying channels is proposed. The authors find the mathematical representations for the outage
probability, average throughput and the optimal energy harvesting time in a closed-form. The
work in [16] focuses on adaptively adjust the time switching factor according to the channel
conditions, the energy stored at the relay and the threshold signal to noise ratio (SNR) for DF and
AF relaying. The authors also propose a framework to reduce the complexity of obtaining channel
state information. In [17], the outage performance of a PR design is studied in the presence of
a direct connection linking source and destination. The authors also study the diversity gain
of the proposed scheme. RF energy harvesting in AF relaying systems over log-normal fading
channels is studied in [18] and [19]. Specifically, in [18], the authors present the exact analytical
representations for the ergodic outage probability and capacity in half-duplex relaying. Whereas,
in [19], the authors extend their previous work and include the case of full-duplex relaying with
RF energy harvesting capability. The authors in [20] study the outage performance analysis of an
energy harvesting multi-relay cooperative communication system for Rayleigh fading channels.
In [21], the authors studied the impact of power allocation factor on the outage probability
performance of transmit antenna selection scheme and found the optimal value using Golden-
Section search method.
RF energy harvesting in the presence of an interferer is studied in [2227]. Specifically, in [22],
the authors propose to rethink the role of interference for the benefits of the wireless battery-
constraint system. In this proposal, instead of avoiding the interfering signals, the authors use
it for beneficial purposes in three different ways. The first concept considers the interference at
modulation levels resulting in the energy-saving multiuser precoding approaches, whereas, in the
second case, the interfering signal is used as the source of green energy. Moreover, in the third
case, they use interfering signals for security and jamming purposes. In [23], the authors study the
ergodic capacity and average throughput of energy harvesting system considering the presence
of multiple interfering signals at the relay node. The authors also derive the ergodic and outage
capacities for RF-based energy harvesting in an interference-limited cooperative relaying system
4
in [28]. In [25], the authors consider the presence of an interfering node at the relay and study
the outage probability and throughput for TR, PR and hybrid relaying designs. The works in [24]
and [26] study the resource and power allocation for interference-based RF energy harvesting in
DF relaying systems. The interference-based SWIPT in cognitive radios with various transceivers
is also studied in [27].
Most of the aforementioned work is studied for Rayleigh fading channels, however, RF energy
harvesting is also studied for some other fading channels, like for Rician in [2933], Nakagami-m
in [3443] and αµin [45] and [46], etc. Specifically, in [29], the authors derived mathematical
representations for the average throughput in closed-form for energy harvesting DF relaying
scheme in Rician fading channels. The authors in [30] and [31] show that energy collected
is linearly proportional to the amount of incident energy and derive the exact mathematical
representations of the statistical characteristics of energy harvested over Nblocks in closed-
form. The authors in [32] carry out the analysis of bidirectional energy harvesting relaying
scheme with hardware impairments under Rician fading channels. The outage probability, average
throughput and symbol error rate are considered as the performance evaluation metrics. In [33],
the authors derive the mathematical representations for outage probability and upper bound of
channel capacity in closed-form for DF mode.
In [34], energy harvesting in the Nakagami-m fading channel is viewed bearing interference at
the relay node. The authors obtain representations for the outage probability and average through-
put for TR and PR designs in AF relaying. The non-linear energy harvesting under Nakagami-m
fading channels is examined in [35]. The authors derive the mathematical representations of the
complementary cumulative distribution function (CDF) of end-to-end SNR and outage capacity
in closed-form. Furthermore, the work in [36] considers an interference-limited DF relaying
based energy harvesting system for Nakagami-m fading channels. The work in [37] studies a
dual-hop DF cooperative communication system that can store and harvest energy at randomly
located relay nodes. The authors in [38] study the outage probability analysis for Nakagami-m
fading channels in energy harvesting two-way DF relaying system. The work in [39] also studies
Nakagami-m fading for energy harvesting for underlay cognitive networks for DF relaying. The
authors place a limit on the maximum power of secondary origin and relay and the latter is set
to collect energy from the signal received from the former node and primary transmitter node. F.
Nawaz et al., also study the outage probability under Nakagami-m fading channels for PR based
DF relaying in the presence of a direct connection linking source and destination in [40]. The
5
work in [41] presents the performance analysis of energy harvesting in the cluster-based multi-
hop network for Nakagami-m fading channels. For this system, the authors assume that all nodes
harvest energy from the signals of power beacon and propose three relay selection schemes based
on harvested energy, channel quality, maximum energy and data rate. The outage probability of
the three schemes for multi-hop cluster-based systems is evaluated. For Nakagami-m fading, bit
error rate (BER) analysis of energy harvesting relaying is discussed in [42]. In this work, the
authors derive the mathematical representations of bit error probability for TR, PR and ideal
relaying in AF and DF modes in closed-form. Furthermore, the optimization is performed to
maximize system performance. In [43], the TR in hybrid decode-amplify-and-forward with two-
way relay network bearing a direct connection under Nakagami-m fading channels is studied.
The authors adopt the TR based approach where the time is divided into four phases; one
phase is practised for energy collection and the rest three phases are utilized for data transfer.
The outage probability and average achievable throughput expressions are derived using the
selection combining techniques at the destination. Authors study the throughput performance of
full duplex DF SWIPT system for Nakagami-m faded channels in [44]. Closed-form expressions
for the outage probability are derived. In [45], G Nauryzbayev et al., present the ergodic capacity
analysis of RF energy harvesting over αµfading channels whereas in [46], the ergodic outage
probability is evaluated along with the ergodic capacity for AF relaying in αµfading channels.
Wireless energy harvesting is also studied for mixed fading channels, for example in [4750].
The authors consider a Rayleigh-Rician fading channel. They obtain the mathematical represen-
tations for the outage probability, average throughput and average symbol error probability in
closed-form for TR design in AF relaying in [47]. The work in [48] considers a TR based energy
harvesting relaying design under mixed Nakagami-m Rayleigh distributions, where relay node
employs maximal ratio combining on the received signals from the source node and transmit
antenna selection for the signal sent to the destination node. In [49], the authors discuss the effect
of a line of sight path on energy harvesting in AF relaying system under mixed Rician-Rayleigh
fading environment. O. S. Badarneh et al., study the outage analysis of energy harvesting DF
relaying system for generalized αµand ηµfading channels in [50]. The authors also
present the close relations of BER for different coherent modulations schemes. In [51], the
authors implement the optimum time or power fraction for energy harvesting in cooperative
relaying system using TR or PR protocols. The authors analysed both TR and PR protocols for
the outage probability. In [52], the authors study the two and three slot power time splitting-
6
based SWIPT protocols for bi-directional AF channels. The closed-form expressions for delay
limited and delay tolerant transmission modes are derived for throughput performances. In [53],
the authors explored incremental relaying cooperative communication using SWIPT to derive
closed form expression for the outage probability.
Majority of the works as mentioned earlier generally consider two important energy harvesting
designs, i.e., TR and PR as their primary work. Although, the operation of TR and PR designs is
entirely different and related to the time of signal transmission and power of the received signal,
respectively. However, the mathematical relations involving TR and PR designs are remarkably
the same. Hence, it provides an opportunity to formulate a unified framework to describe both
of these designs. To realize this, we introduce two binary variables; αband ρbrelated to TR and
PR, respectively. We aim to propose a mathematical formulation of a generalized design which
would be able to work; as TR when αb= 1 & ρb= 0 and as PR when αb= 0 & ρb= 1. We
also aim that this generalized design works as a hybrid of both TR and PR designs when we
set αb= 1 & ρb= 1.
It is to note here that there are some other works related to the unification and hybridization
of TR and PR designs in [5458]. However, our proposed work is different from these works in
various aspects as discussed below. The work in [54] proposes time-power switching protocol for
AF relaying. Although this work manages to provide a hybrid sense of time and power domains,
it skipped the generalization aspects being proposed in our paper which retains the working of
TR and PR as the independent designs. Similarly, the authors in [55] also propose a hybrid TR-
PR design for AF relaying only in Rayleigh fading channels. The work in [56] also considers
a hybrid TR-PR design in Nakagami-m fading channels for AF relaying with non-linear energy
harvesting. The work in [57] proposes a hybrid TR-PR design over Nakagami-m fading channels
for a bidirectional DF relaying system. While as discussed earlier, our work is focused on the
generalization of relaying design, which is also able to provide hybrid relaying as its particular
case. Finally, another effort to propose a unified design over Nakagami-m fading channels
is discussed in [58] for various cases of the multiple-input-multiple-output (MIMO) relaying
system. This work is different from our design because it does not focus on the unification and
hybridization of the working of TR and PR designs despite being inclined towards the unified
performance analysis of these designs for the MIMO relaying system. In the following, we
summarize the key points of our proposed work.
7
A. Contributions
In this work, we propose a generalization framework for the energy harvesting relaying designs.
This framework not only unifies the two existing designs; TR and PR, but also accommodates
the non-energy harvesting case and the hybrid of both TR and PR designs, i.e., hybrid energy
harvested relaying (HEHR) design. The generalized design is termed as the generalized energy
harvested relaying (GEHR). Key points of our contribution are summarized below.
To realize the proposed GEHR design, we introduce two binary variables. The four combi-
nations of these two variables correspond to the conventional non-energy harvested relaying,
TR, PR and HEHR designs as given in Table I.
We find the mathematical representations for the outage probability, ergodic capacity and
average throughput in Rayleigh fading channels for AF and DF relaying modes. The closed-
form expressions are derived for the outage probability. To validate that GEHR design is
a generalization of TR and PR designs, we study the individual cases of GEHR design
from the perspectives of schematic diagram, signal analysis and the performance evaluation
parameters comparison.
As an extended contribution, we further study the GEHR design for the mixed Rayleigh-
Rician fading channels. We consider two sub-cases of mixed fading, i.e., in case 01, source
to relay (SR) link is considered as Rayleigh channel and relay to destination (RD) link is
considered as a Rician channel. Conversely, in case 02, SR link is taken as Rician channel
and RD link is a Rayleigh channel.
We find the mathematical expressions for the ergodic capacity, outage probability and
average throughput for DF and AF relaying for both cases of mixed fading channels.
The rest of the paper is arranged as follows. In Section II, the system model, assumptions
and basics of fading models under consideration are discussed. The proposed GEHR design is
presented in Section III which also includes an overview of TR and PR designs. In the first
two subsections of Section IV, signal and SNR analysis of the proposed GEHR design for AF
and DF relaying modes are given in detail. Whereas, in the last subsection of Section IV, it is
discussed from different perspectives that TR and PR designs are special cases of GEHR design.
Furthermore, the analysis of proposed GEHR design in mixed Rayleigh-Rician fading is carried
out in Section V. Performance evaluation and results of the proposed GEHR design in Rayleigh
and mixed fading channels are presented in Section VI. Finally, in Section VII, we conclude the
8
R
DS
Fig. 1: A two hop cooperative relaying system with an EH relay.
work with some possible future aspects to explore.
II. SY ST EM DESIGN
We study a cooperative system of [10] that has two hops and comprises 1source, 1destination
and 1relay denoted as S,Dand R, respectively, as shown in Fig. 1. The distances between SR
and RD are represented as d1and d2, respectively. Similarly, the channel gains of SR and RD
are respectively denoted as h1and h2. It is believed that direct communication between Sand
Dis infeasible as it is in deep fade because of path loss and shadowing effects. Therefore, the
source communicates with the destination with the help of a relay. The relay is hybrid and fitted
with harvesting energy equipment. All nodes are half-duplex with a single antenna resource.
A. Channel Model
We consider a mixed fading model where two different combinations of fading characteristics
are considered for SR and RD links. The channel fading coefficient his assumed to be either
Rayleigh or Rician. The probability density function (PDF) of Rayleigh distributed ith channel
with channel coefficient hiis given in the following equation,
fhi(xi) = xi
σ2
i
exp x2
i
2σ2
i,(1a)
where, σis the scaling parameter of Rayleigh distribution. The CDF of Rayleigh distribution is
given as,
Fhi(xi) = 1 exp x2
i
2σ2
i.(1b)
The channel gain is represented by an exponentially distributed random variable Xi=|hi|2with
PDF,
fXi(xi) = 1
¯
Xi
exp xi
¯
Xi,(1c)
9
whereas, the CDF of exponentially distributed random variable X1is given below as,
FXi(xi) = 1 exp xi
¯
Xi.(1d)
When signal amplitude hiis a random variable with Rician distribution, its PDF is given as,
fhi(xi) = 2xi(K+ 1)eK
¯
Xi
e(K+1)x2
i
¯
XiI0
2sK(K+ 1)x2
i
¯
Xi
,(2a)
where, Kis shape, ¯
Xiis the scale parameter of Rician distribution and I0(.)is a zero-order
modified Bessel function. The CDF of Rician distributed is given in terms of first order Marcum
Q-function as,
Fhi(xi) = 1 Q1
2K, s2(K+ 1)x2
i
¯
Xi
(2b)
For Rician distributed hi, the signal gain Xi=|hi|2is non-central chi-square distributed with
PDF given in the following equation.
fXi(xi) = (K+ 1)eK
¯
Xi
exp (K+ 1)xi
¯
XiIo
2sK(K+ 1)xi
¯
Xi
,(2c)
Using the relation Io(x) = P
l=0
(x/2)2l
(l!)2from [59, 8.447.1], the aforementioned PDF is rewritten
as,
fXi(xi) = βeiexp (βixi)
X
l=0
(Kβixi)l
(l!)2,(2d)
where, βi=K+1
¯
Xi, βei=βi
eK. The CDF of non-central chi-square distributed random variable is
given in terms of Marcum-Q function,
FXi(xi) = 1 Q12K, p2βixi.(2e)
The aforementioned equation can be written in the following form as,
FXi(xi) = 1
X
l=0
l
X
m=0
βlmβm
ixm
iexp(βixi),(2f)
where, βlm =Kl
l!m!eK.
III. THE PROPOSED SCHEME
To explain the proposed generalized and hybrid designs, we first give a brief overview of the
TR and PR models given in [10].
10
(a) Energy harvesting with TR.
(b) Energy harvesting with PR.
Fig. 2: Cooperative energy harvesting relaying protocols
A. TR and PR Overview
The TR design is shown in Fig. 2a.Tis the entire time duration needed for the transmission
of a packet from Sto D. It has two intervals. Specifically, the relay is permitted to harvest
energy for a fraction of the time αT , where αis the time switching factor. Whereas, the left
time, (1 α)T, is shared among two phases. The SR transmission is happened in the first phase
and RD transmission in the next phase.
In the PR model, the fraction of power ρP , where ρis power splitting factor, of the signal is
used for energy harvesting and the same signal with the remaining power (1 ρ)Pis practised
for SR transmissions in the first phase as depicted in Fig. 2b. While in the second phase, the
relay transmits the received signal to the destination using the harvested energy. The energies
harvested by the TR and PR designs are calculated in [10] and given in the following equations.
ET R
hr =ηPS|h1|2
d
1
αT, (3a)
EP R
hr =ηρPS|h1|2
d
1
T
2,(3b)
11
where, PSis the transmitting power of the source, ηis the harvesting capability, h1is the channel
coefficient, d1is the distance of SR and is path loss exponent. After harvesting the energy,
the relay sends to the destination in both designs, with the transmit powers calculated using the
following equations.
PT R
hr =2ηPS|h1|2αT
(1 α)d
1
,(4a)
PP R
hr =ηρPS|h1|2
d
1
.(4b)
In the following section, we discuss the proposed generalization approach to include both TR
and PR designs in a single description.
B. Generalization: GEHR
The generalized energy harvesting relaying aims to unify the working of both TR and PR
designs by a single mathematical formulation. For this purpose, we consider two binary variables
αband ρb. The four possible combinations of these variables not only include the TR and PR
designs as the special cases of GEHR; however, they also accommodate the cases of non-energy
harvesting relaying and hybrid of TR and PR relaying designs as shown in Table I.
The working model of the GEHR design, is similar to the TR and PR designs and their
adaptive versions, as given in Fig. 3. In this design, the relay harvests energy from the received
source’s signal for the time ´ααbT, where ´αis the time switching factor of the proposed GEHR
design similar to the αof the conventional TR design. Then, for the half of the remaining
time, i.e., (1 ´ααb)T/2, the relay collects energy from the received signal using power splitting
approach with power splitting factor ´ρρb. The remaining power (1 ´ρρb)Pis used for the SR
transmission. In the remaining time, i.e., (1´ααb)T/2, RD transmission takes place. The binary
variables αband ρbcorresponds to TR and PR designs and setting the value of αband ρbfrom
Table Iinto the Fig. 3, this diagram reduces to TR and PR cases given in Fig. 2a and Fig. 2b,
respectively. Like (αand ρ) of TR and PR designs, respectively in [11], we have defined new
variables (´αand ´ρ), for the proposed GEHR design. The relations between the new variables
of GEHR and old variables of TR and PR will be discussed later. The energy harvested in the
GEHR design is determined by adding the energy harvested as,
EGER
hr =ηPS|h1|2
d
1
´ααbT+η´ρρbPS|h1|2
d
1
(1 ´ααb)T
2.(5a)
12
Energy
Harvesting at
Relay
Relay to Destination
Transmission
´ bP
T
Source to Relay
Transmission
(1- ´ b)P
´ bT(1-
´
b) T/2 (1-
´
b) T/2
Fig. 3: Proposed model for generalized energy harvesting relaying.
TABLE I: Working modes of the GEHR design with respect to different values of the binary
variables.
αbρbScheme
0 0 No Energy Harvesting (NEH)
0 1 PR
1 0 TR
1 1 HEHR
Simplifying the aforementioned equation, we get,
EGER
hr = (2´ααb+ ´ρρb´ααb´ρρb)
| {z }
gc
ηPS|h1|2T
2d
1
.(5b)
Let the under-braced factor in the aforementioned equation be gcfor the convenience of descrip-
tion in the further discourse.
In the following subsection, we separately discuss the signal analysis for the proposed GEHR
design for AF and DF relaying modes.
IV. SIGNA L AN D SNR ANALYSIS
Let Stransmits xswith the power PSto R. The signal at Rin the GEHR design is given as,
yr=s(1 ´ρρb)PS
d
1
h1xs+p(1 ´ρρb)na
r+nc
r,(6)
13
where, na
ris the additive white Gaussian noise and nc
ris additive noise due to RF to baseband
conversion. The energy harvested from this signal is given in (5b). The instantaneous relay SNR
is given as,
γR=γsr =(1 ρ´ρb)PS|h1|2
d
1σ2
nr
.(7)
where, nr=p(1 ´ρρb)na
r+nc
r. The received signal at the relay is either amplified or regenerated
before it is sent to the destination. The relay transmit power Pris given as,
Pr=ηPS|h1|2gc
(1 ´ααb)d
1
(8)
Similar to gc, we substitute (1 ´ααb)as gaand (1 ´ρρb)as grin the rest of the discussion.
A. AF Relaying
The relay transmits the following signal expressed as,
xr=
h1qη Psgc
d
1gaqgrPs
d
1h1xs+grna
r+nc
r
qgrPs|h1|2
d
1+grσ2
na
r+σ2
nc
r
,(9)
where, the amplification factor Gis given by the following,
G=1
qgrPs|h1|2
d
1+grσ2
na
r+σ2
nc
r
(10)
The received signal at D is expressed as,
yAF
d=s1
d
2
h2xr+grna
d+nc
d.(11)
Equating the values of xrfrom equation (9), we get the relation of ydgiven in (12). Rearranging
and collecting the coefficients of xs, we get ydin (13). To find the SNR, we find the ratio of
the average power of the signal part and noise part of (13). The SNR of AF relaying case is
given in the (14), where nr=grna
r+na
rand nd=grna
d+nc
dand σ2
nr=grσ2
na
r+σ2
nc
rand
σ2
nd=grσ2
na
d+σ2
nc
d.
B. DF Relaying
The signal received at destination in DF relaying is given in the following equation,
yDF
d=sPr
d
2
h2ˆxr+grna
d+nc
d,(15)
14
yAF
d=
ηPSgch1h2qgrPS
d
1h1xs+grna
r+nc
r
rgad
1d
2grPS|h1|2
d
1+grσ2
na
r+σ2
nc
r+grna
d+nc
d(12)
yAF
d=qη grgc
d
1h2
1h2PSxs
rgad
1d
2grPS|h1|2
d
1+σ2
nr+η PSgch1h2nr
rgad
1d
2grPS|h1|2
d
1+σ2
nr+nd(13)
γAF
d=η Ps2gcgr|h1|4|h2|2
η Psgcd
1|h1|2|h2|2σ2
nr+Psgagrd
1d
2|h1|2σ2
nd+gad2
1d
2σ2
ndσ2
nr
(14)
where, ˆxris the regenerated signal at the relay. Equating the value of transmission power of
relay node from (8), we get,
yDF
d=sηPsgc
gad
1d
2
h1h2ˆxr+grna
d+nc
d.(16)
The SNR of at the destination for DF relaying is given by the following equation,
γDF
d=γrd =η PSgc|h1|2|h2|2
gad
1d
2σ2
nd
(17)
Now, we discuss GEHR with respect to its special cases.
C. TR and PR Schemes as the Special Cases of GEHR
In this part, we examine the GEHR design from the perspective of its cases, i.e., GEHR as TR
design and GEHR as PR design. We compare the schematic diagrams, mathematical relations of
signal analysis and performance evaluation parameters. From these discussions, we prove that
the proposed GEHR design works as TR and PR designs by setting corresponding values of
binary variables given in Table I.
For this purpose, we first compare the schematic diagrams. It is straight forward that the
diagram of GEHR in Fig. 3becomes similar to that of TR shown in Fig. 2a, when αb=
1and ρb= 0. Same is the case for PR given in Fig. 2b when αb= 0 and ρb= 1. Also, when
we set αb= 1 and ρb= 1, we get the schematic diagram of HEHR design which considers
both TR and PR designs simultaneously. Likewise, by setting the value of αb= 1 and ρb= 0
in (5b) and (8), we get (3a) and (4a) which correspond to the energy harvested and power of
the relay node for TR design. Using the same approach, it can easily be verified that by setting
15
αb= 0 and ρb= 1, the energy and power relations of GEHR from (5b) and (8) are same as
those of PR design given in (3b) and (4b).
Comparing the signal analysis of GEHR with previous designs, we find the values of ga, grand gc
for different combinations of αband ρbas given in Table II. Setting the specific combination of
values enables the equations to work for a particular case. For instance, when αb= 0 and ρb= 1,
we get ga= 1, gr= 1 ´ρand gc= ´ρ. Using these values in (6) and (12), we get the signal
received at relay and destination for PR design in AF relaying. The equations of signal analysis
can be verified for TR and HEHR cases as well. SNR equations of GEHR for AF and DF relaying,
given in (14) and (17) are also generic and include the TR and PR designs as their special cases
which can be verified by the simple substitution. Now, we compare the performance evaluating
parameters of the GEHR design to those of its special cases and verify that the generality of
GEHR is also evident while finding the performance evaluation parameters. In the following
subsection, we study the performance parameters of GEHR and their relations concerning those
of TR and PR designs. For this purpose, we derive the ergodic capacity, outage probability and
the average throughput of GEHR design in AF and DF relaying in Rayleigh fading channels.
TABLE II: Working modes of the GEHR design with respect to different values of the binary
variables.
Scheme αbρbgagrgcCoefficients
NEH 0 0 1 1 0
a=Psd
1d
2σ2
ndγth
b=d2
1d
2σ2
nrσ2
ndγth
c= 0
d= 0
PR 0 1 1 1- ´ρ´ρ
a=Psd
1d
2σ2
ndγth (1 ´ρ)
b=d2
1d
2σ2
nrσ2
ndγth
c=ηP 2
S´ρ(1 ´ρ)
d=ηPSd
1σ2
nrγth ´ρ
TR 1 0 1-´α1 2 ´α
a=Psd
1d
2σ2
ndγth (1 ´α)
b=d2
1d
2σ2
nrσ2
ndγth (1 ´α)
c=ηP 2
S2´α
d=ηPSd
1σ2
nrγth2 ´α
HEHR 1 1 1-´ρ1- ´α2 ´α+ ´ρ´ρ´α
a=Psd
1d
2σ2
ndγth (1 ´α) (1 ´ρ)
b=d2
1d
2σ2
nrσ2
ndγth (1 ´α)
c=ηP 2
S(1 ´ρ) (2 ´α+ ´ρ´α´ρ)
d=ηPSd
1σ2
nrγth (2´α+ ´ρ´α´ρ)
16
1) Outage Probability in AF Relaying: The relation of the outage probability is derived by
substituting the γAF
dfrom (14) in the following equation,
PAF
out =pr γAF
d< γth(18)
Rearranging the aforementioned equation for X2, we have,
pr X2PSgagrd
1d
2σ2
ndγthX1+gad2
1d
2σ2
ndσ2
nrγth
η Ps2gcgrX2
1η Psgcd
1σ2
nrγthX1.(19)
Simplifying the above equation by rewriting it in terms of the coefficients ´a,´
b,´cand ´
das given
in the following equations below.
pr X2´aX1+´
b
´cX2
1´
dX1!,(20)
where,
´a=Psgagrd
1d
2σ2
ndγth,(21a)
´
b=gad2
1d
2σ2
nrσ2
ndγth,(21b)
´c=η Ps2gcgr,(21c)
´
d=η Psgcd
1σ2
nrγth.(21d)
Following the methodology adopted in [10], we cater for the negative and positive values of the
denominator ´cX2
1´
dX1in (20), the conditions for the outage probability of the GEHR design
are given as,
PAF
out =pr ´cX 2
1´
dX1X2<´aX1+´
b
=
pr X2<´aX1+´
b
´cX2
1´
dX1, X1>´
d/´c
pr X2>´aX1+´
b
´cX2
1´
dX1= 1, X1<´
d/´c
(22)
The inequality X2>´aX1+´
b
´cX2
1´
dX1follows the fact that if X1<´
d/´c, the denominator ´cX 2
1´
dX1is
a negative value, therefore, the probability is always 1.
PAF
out =Z´
d
´c
0
fX1(x1)pr X2>´aX1+´
b
´cX2
1´
dX1!dx1+Z
´
d
´c
FX2(Xth)fX1(x1)dx1
=Z´
d
´c
0
fX1(x1)dx1+Z
´
d
´c
FX2(Xth)fX1(x1)dx1.(23)
17
The above relation is simplified by doing necessary manipulations as,
PAF
out = 1 1
¯
X1Z
´
d
´c
exp z
¯
X1
+´az +´
b
cz2´
dz)¯
X2!dz. (24)
Where, ¯
X1and ¯
X2represent the mean of random variables X1and X2, respectively. The
integration in (24) cannot be solved further. Therefore, we apply high SNR approximation.
At high SNR, ´
b0. Thus, outage probability is approximated as,
PAF
out 11
¯
X1Z
´
d
´c
exp z
¯
X1
+´a
cz ´
d)¯
X2!dz. (25)
Redefining the variable of integration, let y= ´cz ´
d, the approximated outage probability is
written as,
PAF
out 1exp( ´
d
´cX1)
´c¯
X1Z
0
exp y
´c¯
X1
+´a
y¯
X2dy. (26)
The above integration takes the form R
0exp(β
4yαy)dy =qβ
αK1(βα),
PAF
out 1exp ´
d
´c¯
X1! sa
´c¯
X1¯
X2!K1 sa
´c¯
X1¯
X2!.(27)
where, K1(.)is the modified Bessel function of second kind and first order. Now, we focus on
comparing the individual cases of the coefficients of the GEHR design given in (21) with those
of TR and PR designs. It can be easily verified that these coefficients are reduced to TR and PR
designs when the corresponding values of ga, gr,and gcare substituted as given in Table II.
0 0.2 0.4 0.6 0.8 1
Time Switching Factor
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Throughput
TSR
GEHR with b=1 and b=0
0 0.2 0.4 0.6 0.8 1
Power Splitting Factor
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Throughput
PSR
GEHR with b=0 and b=1
Fig. 4: Throughput of GEHR design in terms of its special cases.
18
2) Outage Probability in DF Relaying: The outage probability of DF relaying system is given
as,
PDF
out =pr(γsr < γth) + pr(γrd < γth, γsr > γth )(28)
The first part, i.e., pr(γsr < γth)is given as,
pr(γsr < γth) = pr X1<γthd
1σ2
nr
grPs=FX1γth
kr,(29)
where, kr=grPs
d
1σ2
nr
. The second part of equation is expressed as,
pr(γrd < γth , γsr > γth) = Z
γth
kr
FX2γth
kdx1fx1(x1)dx1,(30)
where, kd=η Psgc
gad
1d
2σ2
nd
. Using these equations, the final relation of outage probability in DF
relaying is given by the following equation.
PDF
out = 1 1
¯
X1Z
γth
kr
exp γth
kd¯
X2x1exp(x1
¯
X1
)dx1.(31)
The exact expression in closed-form for this integral is unknown to us, however, we follow an
approximation of exponential function; exp (a/x) = P
k=0
1
k!a
xkand solve the integral by
the following approach.
PDF
out = 1 1
¯
X1
X
k=0
1
k!γth
kd¯
X2kZ
γth
kr
exp x1
¯
X1
xk1+1
1
| {z }
Z1
dx1.(32)
Z1is solved using [3.351.4 59] and the final result is given by the following equation.
PDF
out = 1 1
¯
X1
X
k=0
1
k!γth
kd¯
X2k
(1)k(¯
X1)1kEi γth
kr¯
X1
(k1)! +
exp γth
kr¯
X1
γth
kr(k1)
k2
X
l=0
(1)l(¯
X1)lγth
krl
(k1)(k2)...(k1l)
(33)
In the following subsections, we find the ergodic capacity for AF and DF relaying.
3) Ergodic Capacity in AF Relaying: Similar to the previous discussion, we need to find the
relations of ergodic capacity of GEHR and compare it with those of TR and PR designs. Ergodic
capacity of GEHR design is given by the following equation,
CAF
erg =Z
0Z
´
d
´c
´az +´
b´cz2
cz2´
dz)2¯
X1¯
X2γ
exp z
¯
X1
+´az +´
b
cz2´
dz)¯
X2!log2(1+γ)dz(34)
where the coefficients of this equation are obtained by substituting γth with γin (21).
19
4) Ergodic Capacity in DF Relaying: In this part, we derive the ergodic capacity relations
of GEHR design for DF relaying. Following the same approach as in [11], the ergodic capacity
of SR link in GEHR design for Rayleigh fading environment is given as,
Csr =
¯
X1exp ´e
¯
X1E1´e
¯
X1
´elog(2) ,(35)
where, ´e=d
1σ2
nr
grPsand E1(x) = R
t=x
et
tdt is the exponential integral. Similarly, the ergodic
capacity of RD link in GEHR design is given by the following equation,
Crd =2´
f
¯
X1¯
X2log(4)G3 1
1 3 [0,]
[1,0,0,]
´
f
¯
X1¯
X2!(36)
where, ´
f=d
1d
2σ2
ndga
η Psgcand Gis MeijerG function defined in [59, 3.471.9]. The ergodic capacity
in DF relaying is given by the following relation.
CDF
erg = min (Csr, Crd )(37)
The coefficients ´eand ´
fcan be reduced to those of TR and PR designs given in [11].
The results of the GEHR design in Rayleigh fading is compared with those of TR and PR
designs in Fig. 4. From this figure, it is evident that the working of TR and PR designs can
be realized from its unified version, i.e., GEHR design. Hence, from various aspects, we have
shown that the proposed GEHR design is a valid generalization of TR and PR designs. In the
following section, we extend the analysis of proposed GEHR design for two cases of mixed
fading environments.
V. GEHR ANA LYSIS IN MIXED FADING ENVIRONMENT
In this section, we formulate the relations of the ergodic capacity, outage probability and the
average throughput of the proposed GEHR design in mixed Rayleigh-Rician fading environment.
First, we rewrite (7) as γsr =krX1, and (17) as γrd =kdX1X2. It is to be noted that the relation
of γAF
din (14) can also be derived by considering the relations of γsr and γrd from (7) and (17)
in the following formula,
γAF
d= min (γsr , γrd) = γsrγrd
γsr +γrd + 1 krkdX1X2
kr+kdX2
(38)
20
A. Ergodic Capacity
To compute the ergodic capacity for both the AF and DF relaying cases, we use the following
relation,
C(γ) = 1
ln(2) Z
0
(1 + γ)1¯
Fγ(γ)dγ. (39)
where, ¯
Fγis the complementary CDF (CCDF) defined as ¯
Fγ(γ) = 1 Fγ(γ).
1) DF Relaying: To find the ergodic capacity in DF relaying, we need to find CCDFs
separately for SR and RD links.
We first find the CCDF of SR link for the case of Rician and Rayleigh fading in the following
equations respectively as,
¯
Fγsr (γ) =
X
l=0
l
X
m=0
βlm β1
krm
γmexp(β1
kr
γ)(40a)
¯
Fγsr (γ) = exp γ
kr¯
X1(40b)
Now, using these relations, the ergodic capacity is derived for the following cases.
Case 01: When h1is Rayleigh and h2is Rician Faded:
For this case, the CCDF for Rician fading defined above is utilized in the following equation,
C(1)
sr (γ) = 1
ln(2) Z
0
(1 + γ)1exp γ
kr¯
X1dγ. (41)
Solving the above integral using [59, 3.382.4],
C(1)
sr (γ) = 1
ln(2) exp 1
kr¯
X1Γ0,1
kr¯
X1.(42)
Case 02: When h1is Rician and h2is Rayleigh Faded:
The CCDF for Rayleigh fading defined earlier is used in the following equation,
C(2)
sr (γ) = 1
ln(2)
X
l=0
l
X
m=0
βlm β1
krmZ
0
γm
1 + γexp β1
kr
γdγ. (43)
The integral involved in this equation can easily be solved using mathematical tool such as
Maple or Mathematica. The solved equation is written as,
C(2)
sr (γ) = 1
ln(2)
X
l=0
l
X
m=0
βlm β1
krm
exp β1
krΓm, β1
krΓ (1 + m).(44)
Now, we find the ergodic capacity of RD link for both cases.
The generic relation for the CDF of RD link is given as,
Fγrd (γ) = Z
0
fX1(x1)FX2γ
kdx1dx1.(45)
21
This is employed for the two cases of mixed fading channels in the following.
Case 01: When h1is Rayleigh and h2is Rician Faded:
For this case, CCDF of γrd is shown as,
¯
Fγrd (γ) = 1
¯
X1
X
l=0
l
X
m=0
βlm β2γ
kdmZ
0
xm
1exp β2γ
kdx1x1
¯
X1dx1(46)
The above integral is solved using [3.471.9 59].
¯
Fγrd (γ) = 2
X
l=0
l
X
m=0
βlm (µ1)ν1γν1K1m(2µ1γ),(47)
where, µ1=β2
kd¯
X1and ν1=m+1
2which results in 1m= 2 2ν1. Using the CCDF of this
equation, the ergodic capacity of RD link in case 01 is given as,
C(1)
rd (γ) = 2
ln(2)
X
l=0
l
X
m=0
βlmµν1
1Z
0
γν1
1 + γK22ν1(2µ1γ)dγ. (48)
Solving the aforementioned integral in Maple, results in the following equation.
C(1)
rd (γ) = 1
ln(2)
X
l=0
l
X
m=0
βlmµν1
1G3 1
1 3 ν1
1ν111,ν1µ1,(49)
where, Gis MeijerG function defined in [9.301 59]. In the following discussion, we follow the
same approach to find the ergodic capacity of DF link in case 02.
Case 02: When h1is Rician and h2is Rayleigh Faded:
For this case CCDF of γrd is given in the following equation,
¯
Fγrd (γ) = 2βe1
X
l=0
(Kβl
1)
(l!)2Z
0
xl
1exp γ
kd¯
X2x1β1x1dx1,(50)
solving the involved integral using [3.471.9 59] we have,
¯
Fγrd (γ) = 2βe1
X
l=0
(Kβl
1)
(l!)2µ2
β2
1ν2
γν2Kl+1 (2µ2γ).(51)
where, µ2=β1
kd¯
X2and ν2=l+1
2. Using this relation of CCDF the ergodic capacity of RD link
in case 02 is given in following equation,
C(2)
rd (γ) = 2βe1
ln(2)
X
l=0
(Kβ1)lµν2
2
(l!)2β2ν2
1Z
0
γν2
1 + γK2ν2(2µ2γ)(52)
Similar to the case 01, integral involved in this relation is also solved using Maple. Final equation
of ergodic capacity for case 02 is given as,
C(2)
rd (γ) = βe1
ln(2)
X
l=0
(Kβ1)l
(l!)2µ2
β2
1ν2
G3 1
1 3 ν2
ν2,ν22µ2(53)
22
As mentioned earlier, overall ergodic capacity in DF relaying is obtained by finding of capacities
of SR and RD links for each respective case. In the following subsection, we discuss the ergodic
capacity of AF relaying design for two cases of mixed fading channels.
2) AF Relaying: For capacity evaluation in this AF scenario, we need to find the CDF and
CCDF of γAF
dusing the following general formula.
FAF
γd(γ) = Zγ
kr
0
fX1(x1)dx1+Z
γ
kr
FX2krγ
krkdx1kdγfX1(x1)dx1(54)
This generic relation utilizes the respective PDF and CDF relations to find the CCDF and
eventually ergodic capacity for two cases of mixed fading channels in the following description.
Case 01: When h1is Rayleigh and h2is Rician Faded:
For this case, the following relation of CCDF is obtained,
¯
FAF
γd(γ) =
X
l=0
l
X
m=0
βlm
(β2krγ)m
krkd¯
X1exp γ
krZ
0
ym
1exp β2γ kr
y1y1
krkddy1(55)
where, y1=krkdx1kdγ. Solving the involved integral using [3.471.9 59], CCDF is given as,
¯
FAFc1
γd(γ) = 2
X
l=0
l
X
m=0
βlmµν1
1γν1exp γ
kr¯
X1K22ν1(2µ1γ)(56)
Using this relation of CCDF ergodic capacity is given as,
CAFc1
γd(γ) = 2
ln(2)
X
l=0
l
X
m=0
βlmµν1
1Z
0
γν1
1 + γexp γ
kr¯
X1K22ν1(2µ1γ)(57)
In the following discussion, we study the ergodic capacity of AF relaying in case 02 of mixed
fading channels.
Case 02: When h1is Rician and h2is Rayleigh Faded:
For this case, CCDF is given in the following equation.
¯
FAFc2
γd(γ) = βe1
X
l=0
l
X
m=0 l
mβl
1(kdγ)ml
(krkd)l+1 expβ1γ
krZ
0
yl
1exp krγ
y1β1y1
krkddy1,(58)
the integral involved in this relation is solved using [3.471.9 59],
¯
FAFc2
γd(γ) = 2βe1
X
l=0
l
X
m=0 l
mKlβ
2l1m
2
1kml
r
(l!)2kd¯
X2ν1γlm
2+1
2exp β1γ
krK(m+1) (2µ2γ).(59)
Using this relation of CCDF, ergodic capacity is given as,
CAFc2
γd(γ) = 2βe1
ln(2)
X
l=0
l
X
m=0 l
mKlβ
2l1m
2
1kml
r
(l!)2kd¯
X2ν1
Z
0
γlm
2+1
2
1 + γexp β1γ
krK(m+1) (2µ2γ)(60)
23
Authors are unaware of the closed-form expression of the integral involved in this relation;
however, it can be numerically solved using MATLAB. Since the results of this function match
with the Monte-Carlo simulations, we have relied on it. Also, it is worth mentioning that the
approximations of functions involved in this integral may simplify it and make it easy to solve.
B. Outage Probability
The outage probability of the relaying system is defined as the probability that SNR at the
destination is less than a specified threshold value. We first derive the outage probability of the
proposed design for DF relaying mode followed by the derivation in AF relaying mode.
1) Decode and Forward Relaying: To calculate the outage probability of GEHR design for
DF model, we follow the methodology discussed in [19]. The following equation yields the
outage probability as,
PDF
out =pr(γsr > γth)pr(γrd < γth|γsr > γth ) + pr(γsr < γth)
=pr(γrd < γth , γsr > γth)
| {z }
c12
+pr(γsr < γth)
| {z }
c11
(61)
Using the relations of γsr and γrd,c12 is written in the following form as,
c12 =pr(X2< γth/kdx1, γsr > γth/kr)(62)
=Z
γth
kr
FX2(γth
kdx1
)fX1(x1)dx1.(63)
We evaluate c12 separately for case 01 and case 02 in the following discussion.
Case 01: When h1is Rayleigh and h2is Rician Faded:
In this case, we consider X1is a square of Rayleigh distributed random variable and X2as a
square of Rician distributed variable. Using the respective CDF and PDF of X1and X2, we
have,
c12 =Z
γth
kr"1
X
l=0
l
X
m=0
βlmβm
2γth
kdx1m
×exp β2γth
kdx1fX1(x1)dx1(64)
Using the PDF of exponentially distributed X1and substituting the result of c12 in (61), we get
the following result for the outage probability for DF relaying mode as,
PDF
out = 1 1
¯
X1
X
l=0
l
X
m=0
βlmβm
2γth
kdmZ
γth
kr
xm
1exp β2γth
kdx1x1
¯
X1dx1.(65)
24
As mentioned earlier, the exact closed-form expression of the integral involved in this equation
is unknown to us, however, it can be solved by using the approximation of exponential function
discussed in (32).
PDF
out = 11
¯
X1
X
l=0
l
X
m=0
X
n=0
(1)nβlmβm+n
2
k!γth
kdm+nZ
γth
kr
xmn
1exp x1
¯
X1dx1.(66)
PDF
out = 1 1
¯
X1
X
l=0
l
X
m=0
X
n=0
(1)nβlmβm+n
2
k!γth
kdm+n
(1)mn(1
¯
X1)m+n1Ei γth
krX1
(mn1)! +
exp γth
krX1
γth
kr(m+n1)
m+n2
X
p=0
(1)p(¯
X)pγth
krp
(m+n1)(m+n2)...(m+np)
(67)
Case 02: When h1is Rician and h2is Rayleigh Faded:
Evaluating (62), for this case as,
c12 =Z
γth
kr1exp γth
kdx1¯
X2fX1(x1)dx1(68)
Using the PDF of Rician distributed X1,c12 for this case is obtained, which is then used in (61)
to get the following relation of outage probability,
PDF
out = 1 βe1
X
l=0
(Kβ1)l
(l!)2Z
γth
kr
xl
1exp γth
kd¯
X2x1β1x1dx1.(69)
Similar to the previous cases, the integral involved in this equation is solved by using the
power series expansion of exponential function resulting in the following relation of the outage
probability,
PDF
out = 1 βe1
X
l=0
X
m=0
(Kβ1)l
(l!)2m!γth
kd¯
X2mZ
γth
kr
xlm
1exp (β1x1)dx1.(70)
Approximating the exponential function, we get the outage probability in DF relaying as,
PDF
out = 1 βe1
X
l=0
X
m=0
(Kβ1)l
(l!)2m!γth
kd¯
X2m
(1)ml(β1)ml1Ei β1γth
kr
(ml)!
+
exp β1γth
kr
γth
kr(ml1)
ml2
X
n=0
(1)n(β1)nγth
krn
(ml1)(ml2)...(mln)
.(71)
25
2) Amplify and Forward Relaying: The outage probability for both of the considered cases
in AF relaying is given by using the CCDF relations derived earlier.
Case 01: When h1is Rayleigh and h2is Rician Faded:
In this case, the outage probability is derived by substituting γ=γth in the CCDF derived
in (56).
PAF
out = 1 2
X
l=0
l
X
m=0
βlmµν1
1γν1
th exp γth
kr¯
X1K22ν1(2µ1γth)(72)
Case 02: When h1is Rician and h2is Rayleigh Faded:
Similar to the previous case, the outage probability in this case is derived by using γ=γth
in (59).
PAF
out = 1 2βe1
X
l=0
l
X
m=0 l
mKlβ
2l1m
2
1kml
r
(l!)2kd¯
X2ν1γlm
2+1
2
th exp β1γth
krK(m+1) (2µ2γth)(73)
C. Average Throughput
The average throughput of the proposed GEHR design is evaluated separately for the case
of delay tolerant and delay constrained cases using the relations of the outage probability and
average capacity derived in the previous section.
1) Delay Tolerant Case: In this case, the average throughput is dependent on the average
ergodic capacity given by the following equation,
τdt =(1 ´ααb)C
2,(74)
where the generic variable Crepresents the ergodic capacity of underlaying relaying channel.
2) Delay Limited Case: The average throughput for the DL case is calculated using the rela-
tions of the outage probability and effective communication time. If the effective communication
time of the GEHR design is (1 ´ααb)T /2and the transmission rate is Rbits/sec/Hz, the average
throughput for the DL case is given by the following relation.
τdl =(1 Pout) (1 ´ααb)R
2(75)
VI. PERFORMANCE EVALUATION
In this section, we evaluate the performance of the proposed GEHR design for different
parameters in Rayleigh fading and mixed Rayleigh-Rician fading environments separately. For
26
performance evaluation, we consider the average throughput because this parameter is dependent
on the outage probability in DL case and on the ergodic capacity in DT case as discussed in
the previous section. To validate the analytical results, we perform the Monte-Carlo simulations
with 1,000,000 iterations. The naming convention of the legends is explained as follows. For
example, in GEHR 01 dt df the;01 refers to the PR design [10], other options are
10 referring to TR [10] and 11 to HEHR, dt is related to DT case while another one is dl for
DL case. df is related to decode-and-forward relaying and its other counterpart is AF relaying
referred as af. Finally, the last keyword the refers to theoretical results while sim refers to
Monte-Carlo simulation results. All the legends are following a similar pattern. The first part
of the results is to prove the generic nature of the GEHR design as shown in Fig. 5. While in
the second subsection, we present the analytical and Monte-Carlo simulation results of GEHR
design for mixed Rayleigh-Rician fading channels. In order to map the plots with the analytical
expressions, the details are as follows. Figs. 5 and 6 are plotted using (74) and (75). The outage
relations required in (75) are picked from (33) and the capacity equation required for (74) is
picked from (34). Similarly, in Figs. 8 and 9, (74) and (75) are used to plot the throughput of
mixed fading channel conditions using Ergodic Capacity and Outage Probability. The outage
relations required in (75) for AF relations are picked from (72) and (73) for case 01 and case
02, respectively. For DF relaying, (67) and (71) equations are used for case 01 and case 02
respectively. The ergodic capacity relations required in (74) for AF relaying is picked from (57)
and (60) for case 01 and case 02 respectively. For the DF relaying, (42) and (49) equations for
SR and RD links are used for case 01. Similarly (44) and (53) are used for SR and RD links in
case 02.
27
1 3 5 7 9 11 13
SNR [dB]
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Average Throughput
GEHR-10-dt-df-the
GEHR-10-dt-df-sim
GEHR-10-dl-df-the
GEHR-10-dl-df-sim
GEHR-01-dt-df-the
GEHR-01-dt-df-sim
GEHR-01-dl-df-the
GEHR-01-dl-df-sim
(a) GEHR working as TR (α= 0.3) and PR
(ρ= 0.3) in DF relaying.
1 3 5 7 9 11 13
SNR [dB]
0.5
1
1.5
2
2.5
3
3.5
4
Average Throughput
GEHR-10-dt-af-the
GEHR-10-dt-af-sim
GEHR-10-dl-af-the
GEHR-10-dl-af-sim
GEHR-01-dt-af-the
GEHR-01-dt-af-sim
GEHR-01-dl-af-the
GEHR-01-dl-af-sim
(b) GEHR working as TR (α= 0.3) and PR
(ρ= 0.3) in AF relaying.
1 3 5 7 9 11 13
SNR [dB]
0.5
1
1.5
2
2.5
3
Average Throughput
GEHR-10-dt-df-the
GEHR-10-dt-df-sim
GEHR-10-dl-df-the
GEHR-10-dl-df-sim
GEHR-10-dt-af-the
GEHR-10-dt-af-sim
GEHR-10-dl-af-the
GEHR-10-dl-af-sim
(c) HEHR: (TR,PR) = (α= 0.3,ρ= 0.3) in
AF and DF relaying.
Fig. 5: Average throughput of GEHR with respect to SNR of the transmitted signal in Rayleigh
fading environment.
28
A. GEHR Scheme Evaluated in Rayleigh Fading
In Fig. 5, the average throughput for different cases of GEHR design is evaluated against
SNR in DT and DL scenarios. The average throughput of GEHR with GEHR-10 and GEHR-01
cases for α=ρ= 0.3in DF relaying is given in Fig. 5a and in AF relaying in Fig. 5b. In both
of these cases, the simulation results match the corresponding analytical curves which validate
the correctness of the analytical findings. Furthermore, it is to be noted that; with the increasing
SNR, the average throughput in each case is generally increased. However, at high SNR, the
throughput gets saturated. Also, from these figures, it is observed that the average throughput
of DL cases is significantly less than that of DT cases for each value of SNR. Furthermore,
comparing the results between the individual cases GEHR-10 and GEHR-01, when α=ρ= 0.3,
it is depicted that GEHR-01, i.e., PR design outperforms the GEHR-10 with a significant margin
in both AF and DF modes. Finally, comparing these results for DF and AF relaying in Figs. 5a
and 5b, respectively, it is evident that the average throughput of DF mode outperforms its AF
counterpart in the DT case, whereas, in the DL case, these results are matching with each other.
The throughput in DF is more. The reason for better throughput in DF is as follows. AF has the
problem of noise propagation. The signal is neither decoded nor corrected at relay, rather, it is
amplified along with noise and transmitted to the destination. Whereas, in DF, it is decoded and
corrected and fresh copy is transmitted to the destination. The error correction at relay helps DF
to achieve better throughput. The average throughput of HEHR design for AF and DF relaying
modes for DL and DT cases is given in Fig. 5c. Similar to the previous designs, the average
throughput with respect to SNR increases for all cases. Also, the average throughput of the DT
case is larger than that of the DL case. Similarly, the throughput in DF relaying is more than
that of AF relaying.
Fig. 6shows the average throughput of GEHR design with respect to αand ρ. Similar to the
trends in [10], the average throughput against the corresponding factor gradually increases to
a maximum value and then gradually decreases to the minimum. The maximum value for TR
design is at 0.3, and the maximum value for PR is at 0.66. To understand this trend, we need
to focus on the role of time switching or power splitting parameters on the average throughput,
as given in (75). Specifically, this parameter has a two-sided impact on the average throughput.
On one side, it has a role to play in the calculation of the outage probability, i.e., Pout in (75) is
a function of αb, and on the other side, it relates to the transmission rate R. This process makes
29
0 0.1667 0.3333 0.5 0.6667 0.8333 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Average Throughput
GEHR-10-dl-af-the
GEHR-10-dl-af-sim
GEHR-10-dt-af-the
GEHR-10-dt-af-sim
GEHR-10-dl-df-the
GEHR-10-dl-df-sim
GEHR-10-dt-df-the
GEHR-10-dt-df-sim
(a) GEHR working as TR.
0 0.1667 0.3333 0.5 0.6667 0.8333 1
0
0.5
1
1.5
2
Average Throughput
GEHR-01-dl-af-the
GEHR-01-dl-af-sim
GEHR-01-dt-af-the
GEHR-01-dt-af-sim
GEHR-01-dl-df-the
GEHR-01-dl-df-sim
GEHR-01-dt-df-the
GEHR-01-dt-df-sim
(b) GEHR working as PR.
0 0.1667 0.3333 0.5 0.6667 0.8333 1
and
0
0.25
0.5
0.75
1
1.25
1.5
Average Throughput
GEHR-11-dl-af-the
GEHR-11-dl-af-sim
GEHR-11-dt-af-the
GEHR-11-dt-af-sim
GEHR-11-dl-df-the
GEHR-11-dl-df-sim
GEHR-11-dt-df-the
GEHR-11-dt-df-sim
(c) GEHR working as HEHR.
Fig. 6: Average throughput of GEHR with respect to αand ρin Rayleigh fading environment
working as TR, PR and HEHR.
30
the overall expression in (75) quadratic in nature and gives parabolic curves. This initial rise is
due to an increase in time (in TR) or power (in PS) for energy harvesting. When more time or
power is allocated for energy harvesting, relays can harvest more energy; however, it brings a
toll on the time or power required to transmit from relay to the destination. Hence, the TR or
PS factor cannot be increased beyond a specific limit as it disturbs the communication process
to such an extent that the purpose of energy harvesting becomes purposeless.
Especially, in Fig. 6a, the throughput of GEHR-10 is compared with respect to time switching
factor for DL and DT in AF and DF relaying. For each combination, the simulation results match
with the analytical curves which validate the correctness of the GEHR design. As discussed
earlier, the average throughput of the DL case is significantly greater than the DT counterparts
in both DF and AF relaying modes.
Also, the average throughput of DF relaying outperforms its AF counterparts in both DT and
DL cases as shown in Fig. 6a. Further, in Fig. 6b, the average throughput of GEHR-01 is given
for DT and DL scenarios in AF and DF relaying conditions. In this figure, the average throughput
of DL case with DF relaying shows anomalous behavior which is explained as follows. In PR
design, which is given by GEHR-01 in this paper, received power is split for energy harvesting
and signal processing. Moreover, in DF relaying, a regenerated signal is transmitted from relay
to destination with the transmit power directly related to the harvested energy. Hence, when
the power splitting factor is increased, it leads to an increased transmit power of RD signal
transmission. It should be noted that in DF relaying; we assume that the regenerated signal
from the relay is the correct form of the received signal and we generally do not dive into the
relationship between the received signal at the relay and the regenerated signal at the relay.
Hence, a more significant value of power splitting (or even time switching) factor will although
increase energy harvesting, however, it also causes a decrease in time and power for correctly
decoding the received signal at the relay. Eventually, it makes the SR signal un-decodable at the
relay. Thus, there should be an absolute limit to the time switching or power splitting factor.
Finally, we discuss the results of HEHR design given in Fig. 6c with respect to joint time
switching and power splitting factor which is the average of αand ρ. The trend of these results is
similar to the previous designs. HEHR causes an increase in the harvested energy and transmit
power of the relay node, which in return decreases the outage probability of the RD link.
Hence, the average throughput of DL-DF case in HEHR is significantly more than the previous
counterparts as shown in Fig. 6c.
31
1 3 5 7 9 11 13
SNR [dB]
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Average Throughput
GEHR-10-dt-df-c1
GEHR-10-dt-df-c2
GEHR-10-dt-af-c1
GEHR-10-dt-af-c2
GEHR-10-dt-sim
GEHR-10-dl-df-c1
GEHR-10-dl-df-c2
GEHR-10-dl-af-c1
GEHR-10-dl-af-c2
GEHR-10-dl-sim
(a) GEHR as TR with α= 0.4in AF and DF
Relaying for case 01 and case 02.
1 3 5 7 9 11 13
SNR [dB]
0
0.5
1
1.5
2
2.5
Average Throughput
GEHR-01-dt-df-c1
GEHR-01-dt-df-c2
GEHR-01-dt-af-c1
GEHR-01-dt-af-c2
GEHR-01-dt-sim
GEHR-01-dl-df-c1
GEHR-01-dl-df-c2
GEHR-01-dl-af-c1
GEHR-01-dl-af-c2
GEHR-01-dl-sim
(b) GEHR as PR (ρ= 0.4) in AF and DF
Relaying.
1 3 5 7 9 11 13
SNR [dB]
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Average Throughput
GEHR-11-dt-df-c1
GEHR-11-dt-df-c2
GEHR-11-dt-af-c1
GEHR-11-dt-af-c2
GEHR-11-dt-sim
GEHR-11-dl-df-c1
GEHR-11-dl-df-c2
GEHR-11-dl-af-c1
GEHR-11-dl-af-c2
GEHR-11-dl-sim
(c) GEHR as TR and PR in AF and DF
Relaying.
Fig. 7: Average throughput of GEHR with respect to SNR in mixed fading environment: K=
2, d1= 3m, d2= 1m, Ps= 1W.
32
B. GEHR Scheme Evaluated in Mixed Fading Channels
This work on GEHR design considers two subcases of mixed fading channels as explained in
Section V. Now, we show the theoretical and simulation results of the GEHR design in mixed
fading conditions. To diversify the analysis, we consider that in the mixed fading analysis, the
distances between SR and RD are not necessarily the same. Similar to non-mixed scenario, we
consider the average throughput as the performance evaluation metric for two subcases of mixed
fading and find the average throughput of DT and DL cases with respect to various parameters
like SNR, time switching and power splitting factors, relay position, energy conversion efficiency
and Rician K factor as shown in Figs. 7,8and 9, respectively.
In Fig. 7, the average throughput is evaluated against SNR. The keywords c1 and c2 in labels
refer to the case 01 and case 02 of mixed fading channels. For all these cases, we assume that
relay is located far from the source and nearer to the destination with settings d1= 3 and d2= 1.
Other values of relevant parameters are given in the caption. Simulations results match with the
analytical curves which validate the correctness of the GEHR design. From these figures, the
general trend of increase in the average throughput with respect to SNR is observed for GEHR-
10, 01 and 11 designs. Also, for the given settings of relay location, the average throughput of
case 02 is significantly greater than that of case 01 for each combination of relaying (AF and
DF) and transmission modes (DL and DT). This is because EH is directly related to the channel
quality of the SR link. Also, in case 02, SR link is considered as Rician fading channel, which
increases the energy harvesting capability of the relay node because of the dominant line of
sight path. Contrary to this, when both SR and RD distances are equal, the average throughput
for both c1 and c2 cases is similar.
In the following discussion of plots, we take the selective approach where instead of discussing
each and every case we choose some candidate cases and assume that the other cases follow the
same trend. For instance in Fig. 8, we present the GEHR-10 and GEHR-01 designs assuming their
DT with DF conditions and DL with AF conditions for case 01 and case 02. For this comparison,
GEHR-01 and GEHR-10 designs are evaluated to their corresponding time switching or power
splitting factors. The general trend of GEHR designs with respect to corresponding factors is
similar to the previous one shown in Fig. 6. However, the average throughput of case 02 is
greater than that of case 01 especially for the larger value of corresponding factors as given
in Fig. 8. Also for both of these cases, simulation results match the analytical curves for each
33
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1.2
Average Throughput
GEHR-10-dt-df-c1
GEHR-10-dl-af-c1
sim-c1
GEHR-10-dt-df-c2
GEHR-10-dl-af-c2
sim-c2
(a) GEHR as TR (α= 0.3) in AF and DF
Relaying.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.25
0.5
0.75
1
1.25
1.5
Average Throughput
GEHR-01-dt-df-c1
GEHR-01-dl-af-c1
sim-c1
GEHR-01-dt-df-c2
GEHR-01-dl-af-c2
sim-c2
(b) GEHR as PR (ρ= 0.3) in AF and DF
Relaying.
Fig. 8: Average throughput of GEHR with respect to the corresponding factor in mixed fading
environment.
combination of relaying and transmission modes.
In Fig. 9, the GEHR designs is evaluated for average throughput with respect to different
factors. The average throughput of GEHR-10 and GEHR-01 designs is plotted with respect
to SR and RD distances (d1=d2) in Fig. 9a. Increasing the SR and RD distances decreases
the average throughput in all the combinations of GEHR design. However, it is clear that the
average throughput of case 02 is greater than case 01 at lower distances. While for the larger
distances, there is not a significant difference between the average throughputs of case 01 and
case 02. Furthermore, in Fig. 9b, the average throughput is evaluated with respect to the energy
conversion efficiency factor η. In this figure, it is observed that with the increase of conversion
efficiency, the average throughput rapidly increases and gradually gets saturated. Also, in these
results, it is evident that the average throughput of case 02 is significantly greater than that of
case 01. Finally, in Fig. 9c, the average throughput of GEHR designs is evaluated with respect
to the Rician K factor. There occurs a slight increase in the average throughput for Rician K
factor in case 01, whereas, in case 02, the increase in the average throughput is more significant.
This trend is observed because the increase in Rician K factor corresponds to the SR channel
in case 02, which improves the energy harvested and power required for RD transmission.
34
1 1.5 2 2.5 3
Distance (m)
0
0.5
1
1.5
2
2.5
Average Throughput
GEHR-10-dt-df-c1
GEHR-10-dt-df-c2
GEHR-01-dl-af-c1
GEHR-01-dl-af-c2
GEHR-10-dt-af-c1
GEHR-10-dt-af-c2
GEHR-01-dl-af-c1
GEHR-01-dl-af-c2
(a) GEHR as TR (α= 0.3) in AF and DF
Relaying.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Average Throughput
GEHR-10-dt-df-c1
GEHR-10-dt-df-c2
GEHR-01-dl-af-c1
GEHR-01-dl-af-c2
GEHR-10-dt-af-c1
GEHR-10-dt-af-c2
GEHR-01-dl-af-c1
GEHR-01-dl-af-c2
(b) GEHR as PR (ρ= 0.3) in AF and DF
Relaying.
123456
K
0.6
0.8
1
1.2
1.4
1.6
1.8
Average Throughput
GEHR-10-dt-df-c1
GEHR-10-dt-df-c2
GEHR-01-dl-af-c1
GEHR-01-dl-af-c2
GEHR-10-dt-af-c1
GEHR-10-dt-af-c2
GEHR-01-dl-af-c1
GEHR-01-dl-af-c2
(c) GEHR as TR and PR in AF and DF
Relaying.
Fig. 9: Average throughput of GEHR in mixed fading environment working as TR and PR for
AF and DF relaying modes.
35
VII. CONCLUSION AND FUTURE WOR K
In this work, a framework has been proposed to generalize the two conventional energy
harvesting relaying designs named as TR and PR. The motivation of this generalization has
been to exploit the similar nature of the signal and performance analysis related to these designs
and propose a hybrid version of both designs, named as HEHR. Hence, both the TR and PR along
with the HEHR have been posed merely as the individual cases of GEHR design. Signal and
SNR analysis of GEHR design have been carried out and relations of the outage probability and
ergodic capacity are derived for the cases of AF and DF relaying in Rayleigh fading channels.
Furthermore, considering the two sub-cases of cooperative relaying in mixed Rayleigh-Rician
fading channels, above mentioned performance evaluation parameters have been evaluated for
DF and AF relaying separately for each case. The average throughput has been presented as a
single performance evaluation parameter for the case of delay limited and delay tolerant cases.
It has been concluded that the proposed GEHR design can be set to work as not only for the
conventional TR and PR designs but also for hybrid TR and PR design. Furthermore, with
some slight modifications in the proposed design, it can also be used to work as a conventional
non-energy harvesting cooperative relaying design.
This work is limited to half-duplex energy harvesting relaying with a simple three-node
network with no interference at the relay and without the appearance of a direct connection.
In the future, we aim to modify this work for full-duplex and two way relaying systems with
the possibility of interference at relay or destination. Another aspect of extending this work is to
consider the secrecy performance analysis in the cognitive radio paradigm. The notion of energy
harvesting in hybrid decode-amplify-and-forward relaying also provides another dimension to
explore with the appearance of a direct connection in conventional and incremental relaying
conditions.
36
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CD-ROM (Windows, Macintosh and UNIX).
Waseem Raza is currently doing PhD in Electrical and Comupter Engineering and working as Graduate
Research Assistant in AI4Networks lab at Oklahoma University USA. Previously he worked as a Lecturer
in Department of Technology, The University of Lahore, Lahore, Pakistan. He has been working as a
Research Associate in Communications over Sensors (ComSens) Lab, in COMSATS Institute of Informa-
tion Technology, Islamabad, Pakistan from June 2016 to December 2017. He did his Bachelor and Master
in Telecommunication Engineering from UET Taxila Pakistan both under the supervision of Dr. Nadeem
Javaid.
42
Hina Nasir received her B.S. degree in information and communication systems engineering from NUST-
SEECS, Islamabad in 2008. She completed both her M.S. and Ph.D theses under the supervision of Dr.
Nadeem Javaid in computer science from International Islamic University, Islamabad, in 2015. Currently,
she is working in the Department of Computer Science, Air University Islamabad. She was awarded the
university Gold Medal during her M.S. studies. Her research interests include cooperative communication,
energy harvesting, IoT, and WSNs.
Nadeem Javaid received the bachelor degree in computer science from Gomal University, Dera Ismail
Khan, Pakistan, in 1995, the master degree in electronics from Quaid-i-Azam University, Islamabad,
Pakistan, in 1999, and the Ph.D. degree from the University of Paris-Est, France, in 2010. He is currently
an Associate Professor and the Founding Director of the Communications Over Sensors (ComSens)
Research Laboratory, Department of Computer Science, COMSATS University Islamabad, Islamabad.
He has supervised 120 master and 16 Ph.D. theses. He has authored over 900 articles in technical journals
and international conferences. His research interests include energy optimization in smart/micro grids, wireless sensor networks,
big data analytics in smart grids, and blockchain in WSNs, smart grids, etc. He was recipient of the Best University Teacher
Award from the Higher Education Commission of Pakistan, in 2016, and the Research Productivity Award from the Pakistan
Council for Science and Technology, in 2017. He is also Associate Editor of IEEE Access, Editor of the International Journal
of Space-Based and Situated Computing and editor of Sustainable Cities and Society.
ResearchGate has not been able to resolve any citations for this publication.
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