# Performance Analysis and Optimization of Bidirectional Overlay Cognitive
Radio Networks with Hybrid-SWIPT

Addanki Prathima, Devendra S. Gurjar, , Ha H. Nguyen, , and Ajay Bhardwaj
Copyright (c) 2015 IEEE. Personal use of this material is permitted. However,
permission to use this material for any other purposes must be obtained from
the IEEE by sending a request to<EMAIL_ADDRESS>author: Devendra S. Gurjar (devendra.gurjar@ieee.org).Addanki Prathima and
Devendra Singh Gurjar are with the Department of Electronics and Communication
Engineering, National Institute of Technology, Silchar, Cachar, Assam, 788010,
India (e-mails<EMAIL_ADDRESS>devendra.gurjar@ieee.org)Ha H.
Nguyen is with the Department of Electrical and Computer Engineering,
University of Saskatchewan, Saskatoon, SK S7N 5C5, Canada (e-mail:
ha.nguyen@usask.ca).Ajay Bhardwaj is with School of Computing and Electrical
Engineering, Indian Institute of Technology Mandi, 175005, India (e-mail:
ajaybhardwaj@ieee.org).A preliminary version of some part of this paper has
been presented at the IEEE VTC 2018-Fall, Chicago, USA, August 27-30, 2018.

###### Abstract

This paper considers a cooperative cognitive radio network with two primary
users (PUs) and two secondary users (SUs) that enables two-way communications
of primary and secondary systems in conjunction with non-linear energy
harvesting based simultaneous wireless information and power transfer (SWIPT).
With the considered network, SUs are able to realize their communications over
the licensed spectrum while extending relay assistance to the PUs. The overall
bidirectional end-to-end transmission takes place in four phases, which
include both energy harvesting (EH) and information transfer. A non-linear
energy harvester with a hybrid SWIPT scheme is adopted in which both power-
splitting and time-switching EH techniques are used. The SUs aid in relay
cooperation by performing an amplify-and-forward operation, whereas selection
combining technique is adopted at the PUs to extract the intended signal from
multiple received signals broadcasted by the SUs. Accurate outage probability
expressions for the primary and secondary links are derived under the
Nakagami-$m$ fading environment. Further, the system behavior is analyzed with
respect to achievable system throughput and energy efficiency. Since the
performance of the considered system is strongly affected by the spectrum
sharing factor and hybrid SWIPT parameters, particle swarm optimization is
implemented to optimize the system parameters so as to maximize the system
throughput and energy efficiency. Simulation results are provided to
corroborate the performance analysis and give useful insights into the system
behavior concerning various system/channel parameters.

###### Index Terms:

Cognitive radio network, amplify-and-forward, Nakagami-$m$ fading, outage
probability, non-linear energy harvester, particle swarm optimization.

##  I Introduction

Spectrum scarcity is one of the key challenges in current wireless
communications, therefore, improving spectrum efficiency has become a crucial
design objective in fifth-generation (5G) wireless communication networks [1].
The applicability of spectrum sharing techniques has been identified for
large-scale wireless networks to improve the spectrum utilization and to
accommodate a large number of devices [2]. In particular, cognitive radio
networks (CRNs) can resolve the problem of spectrum scarcity by adopting
different spectrum sharing techniques such as interweave, overlay, and
underlay [3]. In underlay spectrum sharing, the secondary users (SUs) can
utilize the licensed spectrum if the interference caused by their
transmissions to the primary system remains below a certain threshold.
Moreover, in interweave spectrum sharing, the SUs opportunistically harness
free spectrum (spectrum holes) for conducting their communications without
causing any interference to the primary communications. Whereas, in overlay
spectrum sharing, both PUs and SUs can avail the same licensed band for the
transmission of their signals provided the SUs facilitate relay cooperation to
the PUs. These techniques help autonomous users to make use of the licensed
spectrum for carrying out their communications without affecting the
performance of legitimate users.

Another critical design aspect of next-generation wireless communication
systems is the network lifetime. As most systems being energy-constraint,
there is a need to prolong the network lifetime and making them energy self-
sufficient [4]. In many cases, for battery-operated systems, recharging or
restoring batteries can become inconvenient and undesirable. Also,
conventional energy harvesting (EH) sources, e.g., solar, thermoelectric,
etc., are highly intermittent as they rely on the surroundings and
environmental conditions [5]. Therefore, a promising solution takes advantage
of the fact that energy can be effectively harvested from ambient radio-
frequency (RF) signals [6]. As a consequence, simultaneous wireless
information and power transfer (SWIPT) is gaining strong interests in both
research and industry and is considered as the future of self-sustainable
wireless networks.

In RF-based EH, the antenna receives the transmitted signal (RF radiation) and
harvests energy using appropriate circuitry [7], which can then be converted
into direct current and stored in a battery. There are two popular approaches,
namely power splitting (PS) and time switching (TS), for realizing the SWIPT
technology in wireless networks [8]. In a PS approach, the total power of the
received RF signal is divided for EH and information processing (IP), whereas
in a TS scheme, the receiving node switches in time for enabling EH and IP
operations. It is pointed out, however, that the conventional linear EH model
is not practically feasible since an EH circuit comprises of non-linear
elements such as diodes, inductors, capacitors etc. Therefore, non-linear
energy harvester is a more realistic approach towards energy harvesting [9].

###  I-A Prior Works

In recent years, SWIPT-based cooperative relaying networks have been studied
from the perspectives of link reliability and network lifetime [11]-[18].
Specifically, one-way cooperative relay system is considered in [11] and [12]
whereby energy is harvested at the relay node from the RF signals. Focusing on
solving the spectral inefficiency of one-way relay networks, an amplify-and-
forward (AF) based two-way relaying system has been considered with SWIPT in
[13]-[15]. Further, in [16] and [17], the authors have employed the decode-
and-forward (DF) based relaying strategy for two-way SWIPT-enabled relay
systems. Recently, the authors in [18] have analyzed the impact of transceiver
hardware impairments on the performance of a DF-based two-way relay network
that supports RF energy harvesting.

Apart from cooperative relay networks, research works in [19]-[32] have
exploited the advantages of SWIPT in CRNs and cellular networks. Specifically,
in [19], the authors have proposed a SWIPT-based CRN, whereby the secondary
node provides relay cooperation or transmits its own signal in separate phases
by extracting power from the RF signals. Different from [19], the SU can
utilize an overlay mode in [20] to simultaneously transmit both primary and
secondary information signals. Herein, the authors have derived the
expressions of outage probability (OP) for both systems under the Rayleigh
fading environment. Furthermore, the authors in [21] and [22] have considered
an underlay spectrum sharing scheme along with EH and investigated outage
performance of the system. As an extension to the system model of [21] and
[22], the authors in [23] have assumed multiple primary interferences and
quantified the outage and ergodic capacity for the secondary system. The
authors in [24] have utilized energy harvesting in DF-based one-way
cooperative CRN (CCRN) and analyzed outage probability and throughput
performance for both primary and secondary systems. A similar model has been
adopted in [25], where the authors have introduced a dynamic SWIPT protocol
with opportunistic relaying. In [26], the authors have studied OP performance
for a cooperative CRN enabled with energy harvesting under the Nakagami-$m$
fading. Recently, the authors in [27] have considered bidirectional
transmission in a SWIPT-CCRN system under the Rayleigh fading with the DF
relaying strategy and analyzed its outage performance. In this work, only PS-
based SWIPT has been considered for energy harvesting.

The authors in [28] have studied energy efficiency (EE) optimization for SWIPT
by adopting the PS-based scheme in multiple-input multiple-output (MIMO) two-
way AF networks. In their work, the objective was to maximize the EE of the
network. Further, in [29], the authors aimed to optimize the EE in coordinated
multi-point SWIPT heterogeneous networks. Focusing on potential applications
of spectrum sharing and RF energy harvesting in IoT, the authors in [30] have
introduced a DF-based SWIPT-enabled CCRN by considering a linear model for EH
and analyzed system performance in terms of OP, system throughput, and end-to-
end transmission delay. Different from the above-mentioned works, the authors
in [31] have considered a CCRN system with an underlay scheme in which
transmission of data from a secondary transmitter to a secondary receiver
occurs with the help of multiple relays. The authors then analyzed outage
performance for various relaying schemes. Further, an opportunistic ambient
backscatter transmission model considering opportunistic spectrum sensing has
been proposed in [32] for an RF-powered CRN, where the authors have formulated
an optimization problem to maximize the energy efficiency of the considered
network. Very recently, research works in [33] and [34] have addressed the
security issue of SWIPT-enabled cognitive radio networks by employing physical
layer security techniques. Focusing on relay selection, the authors in [35]
have utilized a neural network to select the best relay in a two-way SWIPT-
enabled CRN. Most of the previous works discussed above on one-way/two-way
CRNs have adopted either a linear EH model with Nakagami-$m$ fading channels,
or a non-linear EH model with Rayleigh fading channels and considered a single
SWIPT receiver design. To the authors’ best knowledge, no work has yet
investigated performance of a bidirectional PS-TS SWIPT-based CRN with a non-
linear EH circuit over Nakagami-$m$ fading channels.

###  I-B Main Contributions

Motivated by the above observations, this paper introduces an AF-based111While
AF and DF are two popular relaying techniques for cooperative communication
systems, we focus on AF relaying due to its lower complexity as compared to DF
relaying. bidirectional cognitive radio network. Different from [30], a hybrid
PS-TS based SWIPT is adopted in this paper. Thus, the obtained analysis can be
used for both popular EH receiver designs. Furthermore, unlike [30], a non-
linear energy harvesting model222We consider a simplified piecewise linear EH
model with two linear segments to make the analysis tractable. It is
interesting and left as a future work to carry out the analysis for the more
general piecewise linear EH model (with $N$ segments) as introduced in [36,
37] so that the non-linearity of a practical EH circuit can be captured more
accurately. is considered so that more practical insights can be gathered from
the performance analysis. The harvested energy is employed at secondary users
(SUs) for facilitating relay assistance to the primary users (PUs) and their
own information exchange. Moreover, this paper explores cooperative
communications to ensure the quality of service (QoS) of the PUs and enhances
the overall spectrum efficiency by enabling bidirectional transmissions of
primary and secondary users. Another objective of this paper is to exploit
soft computing techniques to maximize the achievable system throughput and
energy efficiency of the considered system while maintaining the required QoS
of the primary system.

The major contributions are summarized below:

  * •

The paper introduces a non-linear EH-based hybrid SWIPT cognitive radio
network in which primary and secondary users can realize their bidirectional
transmissions.

  * •

With the introduced scheme, accurate expressions of OP are derived for both
primary and secondary users under Nakagami-$m$ fading. Subsequently,
expressions of the overall system throughput and energy efficiency are
obtained for the delay-limited scenario.

  * •

The critical value of the spectrum sharing factor is determined to obtain the
feasible range of the spectrum sharing factor for which the proposed scheme
has lower OP than the direct transmission (i.e., without relay cooperation) to
maintain the QoS of the primary system.

  * •

An optimization algorithm based on particle swarm optimization (PSO) has been
implemented to obtain optimal system design parameters (i.e. spectrum sharing
factor, TS factor, and PS factor) such that the overall system throughput and
energy efficiency are maximized.

  * •

The paper discusses the effect of various system/channel parameters on the
system performance by means of comprehensive numerical and simulation results.

###  I-C Organization

The remainder of the paper is organized as follows. In Section II, the system
model and the proposed spectrum sharing scheme are described. Section III
evaluates system performance in terms of OP for both PUs and SUs, system
throughput and energy efficiency. The optimization framework is implemented in
Section III-F. Numerical results are presented in Section IV. Section V
concludes the paper.

Notations: $\textmd{Pr}[\cdot]$, $f_{X}(\cdot)$, and $F_{X}(\cdot)$ denote
respectively the probability, probability density function (PDF) and the
cumulative distribution function (CDF) of a random variable $X$. The upper
incomplete, the lower incomplete, and the complete gamma functions are
represented as $\Gamma[\cdot,\cdot]$, $\Upsilon[\cdot,\cdot]$, and
$\Gamma[\cdot]$, respectively [38, eq. (8.350)]. $\mathcal{K}_{v}(\cdot)$
represents $v^{\textmd{th}}$ order modified Bessel function of second kind
[38, eq. (8.432.1)] and $\mathcal{W}_{u,v}[\cdot]$ denotes Whittaker function
[38, eq. (9.222)].

##  II System Model and Scheme

Figure 1: System model for hybrid TS/PS SWIPT in cognitive radio networks.
Figure 2: Transmission block of hybrid-SWIPT in a cooperative CRN.

As illustrated in Fig. 1, a cooperative CRN enhanced with the SWIPT technology
is considered wherein two PUs, denoted as ${\sf PU}_{a}$ and ${\sf PU}_{b}$,
wish to exchange information to each other. However, the direct transmission
link between the two PUs is not capable enough to provide the required target
rates because of heavy shadowing. Therefore, two SUs in their vicinity,
denoted as ${\sf SU}_{1}$ and ${\sf SU}_{2}$, provide cooperative relaying of
primary users’ signals and also carry out their own communications over the
same spectrum allocated to the primary system. The PUs and SUs are assumed to
be single-antenna devices and operate in a half-duplex mode. The overall EH
and bidirectional information exchange between the primary and secondary
systems are executed in four phases, as depicted in Fig. 2. Specifically, in
the first two phases, the SUs harvest energy using hybrid TS and PS SWIPT.
This harvested energy is then stored and used for their mutual information
exchange and cooperative relaying to the PUs.

In the first phase, SUs adopt the TS technique to harvest energy from the
signals received from the PUs, whereas, in the second phase, they use the PS
technique for the same purpose. The remaining fraction of power of the signal
received in the second phase is used for information processing and
broadcasting. In the third phase, the SU applies the AF technique to the
relaying signal and combines it with the signal that is to be transmitted to
the other SU. This phase is referred to as broadcast channel (BC) phase. In
the fourth phase, which is the second BC phase, the other SU performs the same
task as that of the secondary user in the first BC phase (i.e., the third
phase). The PUs therefore receive two signal copies from the two SUs, and they
apply a selection combination (SC) technique to extract the desired signal.

The system is assumed to operate in a block fading environment wherein the
channel coefficients stay constant for the entire duration of the block
(comprising of four phases). The channel coefficients are represented as
${h}_{j,i}$ and ${h}_{i,\hat{j}}$, respectively, for the communication links
between ${\sf PU}_{j}$ to ${\sf SU}_{i}$ and ${\sf SU}_{i}$ to ${\sf
PU}_{\hat{j}}$, for $i\in\\{1,2\\}$ and $j,\hat{j}\in\\{a,b\\}$, where
$j\neq\hat{j}$. Likewise, ${h}_{i,\hat{i}}$ denotes the channel gain for the
communication links between ${\sf SU}_{i}$ to ${\sf SU}_{\hat{i}}$, where
$i\neq\hat{i}$. Further, it is assumed that the channel gains of individual
hops are reciprocal, e.g., ${h}_{i,\hat{j}}={h}_{\hat{j},i}$ and
${h}_{i,\hat{i}}={h}_{\hat{i},i}$. All the signals received at PUs and SUs are
affected by additive white Gaussian noise (AWGN). Moreover, the channel
coefficient $h_{j,i}$ is considered to follow the Nakagami-$m$ distribution
with average power $\Omega_{ij}$ and severity parameter of fading $m_{ij}$.

###  II-A Energy Harvesting

The structure of one complete block duration of the considered cooperative CRN
is shown in Fig. 2. The entire duration $T$ of one block is initially split
into two time slots, each of duration $\alpha T$ and $(1-\alpha)T$,
respectively, where $0\leq\alpha<1$. The first slot of duration $\alpha T$ is
allocated for harvesting energy by the TS approach, whereas the second slot of
duration $(1-\alpha)T$ is used for both EH with the PS technique and carrying
out the end-to-end bidirectional communications. The energy harvested in the
first slot at ${\sf SU}_{i}$ is given as [39]

$\displaystyle\mathcal{E}^{(\textmd{TS})}_{i}=\eta_{i}\alpha
T(P_{a}|h_{a,i}|^{2}+P_{b}|h_{b,i}|^{2}),\quad i\in\\{1,2\\}$ (1)

where $P_{a}\,\&\,P_{b}$ are the transmit powers at ${\sf PU}_{a}\,\&\,{\sf
PU}_{b}$, respectively, and $\eta_{i}$ denotes the efficiency of energy
conversion of the EH circuit (in the linear region) at ${\sf SU}_{i}$, where
$0<\eta_{i}<1$. The second slot $(1-\alpha)T$ is again split into three
phases, each of duration $(1-\alpha)T/3$. In the first slot, both the PUs
transmit their signals to the SUs, and is referred to as multiple access
channel (MAC) phase. With $x_{a}$ and $x_{b}$ being unit-energy symbols
transmitted, respectively, by ${\sf PU}_{a}$ and ${\sf PU}_{b}$ in the MAC
phase, the signal received at ${\sf SU}_{i}$ is given as
$y_{i}=\sqrt{P_{a}}h_{a,i}x_{a}+\sqrt{P_{b}}h_{b,i}x_{b}+n_{i}$, where
$n_{i}\sim\mathcal{CN}(0,\sigma^{2}_{i})$ is the AWGN component at ${\sf
SU}_{i}$. The received power at each SU is split into two parts using a power
splitting circuit. Specifically, $(\sqrt{\beta}y_{i})^{2}$ amount of power is
utilized for harvesting energy and the remaining
$\left(\sqrt{(1-\beta)}y_{i}\right)^{2}$ power is utilized for completing the
information transmission. Thus, the harvested energy at ${\sf SU}_{i}$ in the
MAC phase can be expressed as

$\displaystyle\mathcal{E}^{(\textmd{PS})}_{i}=\frac{\beta(1-\alpha)T\eta_{i}}{3}(P_{a}|h_{a,i}|^{2}+P_{b}|h_{b,i}|^{2}).$
(2)

Considering a non-linear EH model as in [9], from (1) and (2), the total power
transmitted at ${\sf SU}_{i}$ is given as

$\displaystyle
P_{i}\\!\\!=\\!\\!\begin{cases}\Delta_{i}(P_{a}|h_{a,i}|^{2}\\!+\\!P_{b}|h_{b,i}|^{2}),\\!\\!\\!&P_{a}|h_{a,i}|^{2}\\!+\\!P_{b}|h_{b,i}|^{2}\leq
P_{\text{th}}\\\
\Delta_{i}P_{\text{th}},&P_{a}|h_{a,i}|^{2}\\!+\\!P_{b}|h_{b,i}|^{2}>P_{\text{th}}\end{cases}$
(3)

where $\Delta_{i}=\left(\frac{3\eta_{i}\alpha}{1-\alpha}+\beta\eta_{i}\right)$
and $P_{\text{th}}$ is the saturation threshold power of the EH circuit. It is
further assumed that the amounts of energy used for signal processing at the
relaying SUs are negligible compared to that used for broadcasting the
combined signals. Therefore, all the energy harvested at SUs can be utilized
for broadcasting operation [5].

###  II-B Instantaneous Signal-to-Noise Ratios (SNRs)

The signal received at ${\sf SU}_{i}$ in the MAC phase for information
processing and broadcasting, can be written as

$\displaystyle
y^{(\textmd{rem})}_{i}=\sqrt{1-\beta}\left(\sqrt{P_{a}}h_{a,i}x_{a}+\sqrt{P_{b}}h_{b,i}x_{b}+n_{i}\right)+n_{c_{i}}$
(4)

where $n_{c_{i}}\sim\mathcal{CN}(0,\sigma^{2}_{c_{i}})$ represents RF-to-
baseband conversion noise. The received signal at ${\sf SU}_{i}$ in MAC phase
then undergoes AF operation and is combined with the signal to be transmitted
to the other SU. The ${\sf SU}_{i}$ then broadcasts the combined signal. Thus,
the broadcast signal in the first BC phase is given as

$\displaystyle
x^{(\textmd{BC})}_{i}=\sqrt{\frac{\mu_{i}P_{i}}{\mathcal{G}}}y^{(\textmd{rem})}_{i}+\sqrt{(1-\mu_{i})P_{i}}x_{i}$
(5)

where $\mu_{i}$ represents the power splitting factor for spectrum sharing and
$\mathcal{G}=(1-\beta)(P_{a}|h_{a,i}|^{2}+P_{b}|h_{b,i}|^{2}+\sigma^{2}_{i})$
is the AF gain. Further, the signal received at ${\sf PU}_{j}$, for
$j\in\\{a,b\\}$, can be given as

$\displaystyle
y_{i,j}=\left(\sqrt{\frac{\mu_{i}P_{i}}{\mathcal{G}}}y^{(\textmd{rem})}_{i}+\sqrt{(1-\mu_{i})P_{i}}x_{i}\right)h_{i,j}+n_{j}$
(6)

where $n_{j}\sim\mathcal{CN}(0,\sigma^{2}_{j})$ represents AWGN variable at
${\sf PU}_{j}$. The PUs are aware of their transmitted signals and hence they
can eliminate the self-interference caused from the received signal to extract
the desired signal. Approximating
$\mathcal{G}\approx{(1-\beta)(P_{a}|h_{a,i}|^{2}+P_{b}|h_{b,i}|^{2})}$ in (6),
the instantaneous SNR at ${\sf PU}_{j}$ is expressed as

$\displaystyle\gamma_{i,j}=\begin{cases}\gamma_{i,j}^{(\text{lin})},&P_{a}|h_{a,i}|^{2}+P_{b}|h_{b,i}|^{2}\leq
P_{\text{th}}\\\
\gamma_{i,j}^{(\text{sat})},&P_{a}|h_{a,i}|^{2}+P_{b}|h_{b,i}|^{2}>P_{\text{th}}\end{cases}$
(7)

Further, $\gamma_{i,j}^{(\text{lin})}$ and $\gamma_{i,j}^{(\text{sat})}$ can
be expressed as follows.

$\displaystyle\gamma_{i,j}^{(\text{lin})}=\frac{\mu_{i}\Delta_{i}P_{\hat{j}}|h_{\hat{j},i}|^{2}}{\omega_{ij}|h_{i,j}|^{2}+\omega_{i\hat{j}}|h_{\hat{j},i}|^{2}+\varepsilon_{ij}}$
(8)

and

$\displaystyle\gamma_{i,j}^{(\text{sat})}=\frac{\mu_{i}\Delta_{i}P_{\text{th}}P_{\hat{j}}|h_{i,j}|^{2}|h_{\hat{j},i}|^{2}}{\phi_{ij}|h_{i,j}|^{2}+\Phi_{ij}|h_{i,j}|^{4}+\Phi_{i\hat{j}}|h_{i,j}|^{2}|h_{\hat{j},i}|^{2}+\varphi_{\hat{j}j}|h_{\hat{j},i}|^{2}}$
(9)

where
$\varepsilon_{ij}=\left(\mu_{i}\Delta_{i}\sigma^{2}_{i}+\frac{\mu_{i}\Delta_{i}\sigma^{2}_{c_{i}}}{1-\beta}\right)$,
$\omega_{ij}=(1-\mu_{i})\Delta_{i}P_{j}$,
$\omega_{i\hat{j}}=(1-\mu_{i})\Delta_{i}P_{\hat{j}}$,
$\phi_{ij}=\left(\mu_{i}\Delta_{i}P_{\text{th}}\sigma^{2}_{i}+\frac{\mu_{i}\Delta_{i}P_{\text{th}}\sigma^{2}_{c_{i}}}{1-\beta}+P_{j}\sigma^{2}_{j}\right)$,
$\Phi_{ij}=\left(1-\mu_{i}\right)\Delta_{i}P_{\text{th}}P_{j}$,
$\Phi_{i\hat{j}}=\left(1-\mu_{i}\right)\Delta_{i}P_{\text{th}}P_{\hat{j}}$ and
$\varphi_{\hat{j}j}=P_{\hat{j}}\sigma^{2}_{j}$ for $i\in\\{1,2\\}$ and
$\\{j,\hat{j}\\}\in\\{a,b\\}$, with $j\neq\hat{j}$. To obtain the expression
of $\gamma^{\textmd{(lin)}}_{i,j}$, the effect of noise terms in the
corresponding BC phase is considered negligible as compared to RF-to-baseband
conversion noise [12] and interference of secondary transmission. Meanwhile,
the signal received in the first BC phase at the other SU can be expressed as

$\displaystyle
y_{i,\hat{i}}=\left(\sqrt{\frac{\mu_{i}P_{i}}{\mathcal{G}}}y^{(\textmd{rem})}_{i}+\sqrt{(1-\mu_{i})P_{i}}x_{i}\right)h_{i,\hat{i}}+n_{\hat{i}}$
(10)

where $n_{\hat{i}}\sim\mathcal{CN}(0,\sigma^{2}_{\hat{i}})$ is the AWGN
component at ${\sf SU}_{\hat{i}}$.

The signals transmitted by PUs in the MAC phase are received by both the SUs.
Assuming multiuser decoding scenario as adopted in [42] and [43], the
interference due to primary signals at ${\sf SU}_{\hat{i}}$ in (10) is thus
eliminated. Therefore, the instantaneous SNR at ${\sf SU}_{\hat{i}}$ can be
expressed as

$\displaystyle\gamma_{i,\hat{i}}=\begin{cases}\gamma_{i,\hat{i}}^{(\text{lin})},&P_{a}|h_{a,i}|^{2}+P_{b}|h_{b,i}|^{2}\leq
P_{\text{th}}\\\
\gamma_{i,\hat{i}}^{(\text{sat})},&P_{a}|h_{a,i}|^{2}+P_{b}|h_{b,i}|^{2}>P_{\text{th}}\end{cases}$
(11)

where $\gamma_{i,\hat{i}}^{(\text{lin})}$ and
$\gamma_{i,\hat{i}}^{(\text{sat})}$ are given as follows.

$\displaystyle\gamma_{i,\hat{i}}^{(\text{lin})}=\frac{\zeta_{i}\left(P_{a}|h_{a,i}|^{2}+P_{b}|h_{b,i}|^{2}\right)|h_{i,\hat{i}}|^{2}}{\xi_{i}|h_{i,\hat{i}}|^{2}+\sigma^{2}_{\hat{i}}}$
(12)

and

$\displaystyle\gamma_{i,\hat{i}}^{(\text{sat})}=\frac{\Psi_{i}\left(P_{a}|h_{a,i}|^{2}+P_{b}|h_{b,i}|^{2}\right)|h_{i,\hat{i}}|^{2}}{\mathcal{C}_{i}|h_{i,\hat{i}}|^{2}+\mathcal{D}_{\hat{i}}\left(P_{a}|h_{a,i}|^{2}+P_{b}|h_{b,i}|^{2}\right)}$
(13)

where $\zeta_{i}=(1-\mu_{i})\Delta_{i}$,
$\xi_{i}=\mu_{i}\Delta_{i}\left(\sigma^{2}_{i}+\frac{\sigma^{2}_{c_{i}}}{1-\beta}\right)$,
$\Psi_{i}=(1-\mu_{i})(1-\beta)\Delta_{i}P_{\text{th}}$,
$\mathcal{C}_{i}=\mu_{i}\Delta_{i}P_{\text{th}}\left((1-\beta)\sigma^{2}_{i}+\sigma^{2}_{c_{i}}\right)$
and $\mathcal{D}_{\hat{i}}=(1-\beta)\sigma^{2}_{\hat{i}}$.

Likewise in the second BC phase, ${\sf SU}_{\hat{i}}$ broadcasts its own
signal $x_{\hat{i}}$ meant for ${\sf SU}_{i}$ along with the primary signals.
The end-to-end instantaneous SNR at each PU for this signal transfer is
obtained by replacing $i$ with $\hat{i}$ in (7). Similarly, the instantaneous
SNR at ${\sf SU}_{i}$ can be formulated by replacing $i$ with $\hat{i}$ and
vice versa in (11).

##  III Performance Analysis

###  III-A Outage Probability of the Primary System

Outage probability is a crucial performance measure that can assess
reliability of communication link of a wireless system operating in a fading
channel. In the considered network, the relaying SUs broadcast two signal
copies in two BC phases to the PUs. Thus, an outage event at any PU, after
considering both the intended signal copies, occurs when the instantaneous
data rate is lower than a pre-defined target data rate. Accordingly, the OP
for the PUs can be defined as

$\displaystyle\mathcal{P}_{\textmd{out},j}$
$\displaystyle=\textmd{Pr}\left[\mathcal{R}_{\textmd{sc},j}<r_{j}\right]=\textmd{Pr}\left[\max\left(\gamma_{i,j},\gamma_{\hat{i},j}\right)<\gamma_{j}\right]$
(14)

where the instantaneous rate
$\mathcal{R}_{\text{sc},j}=\frac{1-\alpha}{3}\log_{2}\left(1+\max\left(\gamma_{i,j},\gamma_{\hat{i},j}\right)\right)$
and $\gamma_{j}=2^{\frac{3r_{j}}{1-\alpha}}-1$, for $j\in\\{a,b\\}$,
$i,\hat{i}\in\\{1,2\\}$ and $i\neq\hat{i}$. Further, (14) can be expressed as

$\displaystyle\mathcal{P}_{\textmd{out},j}=\prod^{2}_{i=1}F_{\gamma_{i,j}}(\gamma_{j})$
(15)

where

$\displaystyle
F_{\gamma_{i,j}}(\gamma_{j})=F_{\gamma_{i,j}^{(\text{lin})}}(\gamma_{j})+F_{\gamma_{i,j}^{(\text{sat})}}(\gamma_{j}).$
(16)

Accurate expressions of the CDF of
$F_{\gamma_{i,j}^{(\text{lin})}}(\gamma_{j})$ and
$F_{\gamma_{i,j}^{(\text{sat})}}(\gamma_{j})$ are provided in the following
lemma.

###### Lemma 1

The expression of $F_{\gamma^{(\text{lin})}_{i,j}}(\gamma_{j})$, for
$j\in\\{a,b\\}$ can be presented as

$\displaystyle\\!\\!\\!F_{\gamma^{(\text{lin})}_{i,j}}(\gamma_{j})$
$\displaystyle\\!=\\!\frac{\Gamma[m_{ij}]-\Gamma\left[m_{ij},\frac{m_{ij}P_{\text{th}}}{\Omega_{ij}P_{j}}\right]}{\Gamma[m_{ij}]}-\frac{\left(\frac{m_{ij}}{\Omega_{ij}}\right)^{m_{ij}}}{\Gamma[m_{ij}]}\mathrm{e}^{\left(-\frac{m_{i\hat{j}}P_{\text{th}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)}$
$\displaystyle\\!\times\\!\\!\sum_{k=0}^{m_{i\hat{j}-1}}\frac{\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)^{k}}{k!}\sum_{q=0}^{k}P_{\text{th}}^{q}\binom{k}{q}(-P_{j})^{k-q}$
$\displaystyle\\!\times\\!\\!\sum_{t=0}^{k+m_{ij}-q-1}\frac{(-1)^{t}t!\binom{k+m_{ij}-q-1}{t}}{\left(\frac{m_{i\hat{j}}P_{j}}{\Omega_{i\hat{j}}P_{\hat{j}}}-\frac{m_{ij}}{\Omega_{ij}}\right)^{t+1}}$
$\displaystyle\\!\times\\!\\!\left(\left(\frac{P_{\text{th}}}{P_{j}}\right)^{k+m_{ij}-q-t-1}\mathrm{e}^{\frac{P_{\text{th}}}{P_{j}}\left(\frac{m_{i\hat{j}}P_{j}}{\Omega_{i\hat{j}}P_{\hat{j}}}-\frac{m_{ij}}{\Omega_{ij}}\right)}\right.$
$\displaystyle-\left.\iota_{1}^{k+m_{ij}-q-t-1}\mathrm{e}^{\iota_{1}\left(\frac{m_{i\hat{j}}P_{j}}{\Omega_{i\hat{j}}P_{\hat{j}}}-\frac{m_{ij}}{\Omega_{ij}}\right)}\right)$
$\displaystyle-\frac{\left(\frac{m_{ij}}{\Omega_{ij}}\right)^{m_{ij}}}{\Gamma[m_{ij}]}\mathrm{e}^{\left(-\frac{m_{i\hat{j}}\varepsilon_{ij}\gamma_{j}}{\Omega_{i\hat{j}}\Xi}\right)}\sum_{p=0}^{m_{i\hat{j}}-1}\frac{\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}\Xi}\right)^{p}}{p!}$
$\displaystyle\\!\times\\!\\!\sum_{n=0}^{p}\varepsilon_{ij}^{n}\gamma_{j}^{p}\binom{p}{n}\omega_{ij}^{p-n}\left(\frac{\omega_{ij}\gamma_{j}m_{i\hat{j}}}{\Omega_{i\hat{j}}\Xi}\\!+\\!\frac{m_{ij}}{\Omega_{ij}}\right)^{\\!-\\!(m_{ij}\\!-\\!n\\!+\\!p)}$
$\displaystyle\\!\times\\!\\!\Upsilon\left[m_{ij}-n+p,\left(\frac{m_{ij}}{\Omega_{ij}}+\frac{\omega_{ij}\gamma_{j}m_{i\hat{j}}}{\Omega_{i\hat{j}}\Xi}\right)\iota_{1}\right]$
(17)

where
$\iota_{1}=\frac{P_{\text{th}}\Xi-\varepsilon_{ij}\gamma_{j}P_{\hat{j}}}{\omega_{ij}\gamma_{j}P_{\hat{j}}+P_{j}\Xi}$
and $\Xi=\mu_{i}\Delta_{i}P_{\hat{j}}-\omega_{i\hat{j}}\gamma_{j}$. On the
other hand, the expression for $F_{\gamma^{(\text{sat})}_{i,j}}(\gamma_{j})$
is as follows:

$\displaystyle\\!\\!\\!F_{\gamma^{(\text{sat})}_{i,j}}(\gamma_{j})$
$\displaystyle=\sum_{p=0}^{m_{ij}-1}\frac{\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)^{p}}{p!}\sum_{n=0}^{p}\binom{p}{n}P_{\text{th}}^{n}(-P_{\hat{j}})^{p-n}\frac{\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}}\right)^{m_{i\hat{j}}}}{\Gamma[m_{i\hat{j}}]}$
$\displaystyle\\!\times\\!\\!\mathrm{e}^{-\frac{m_{ij}P_{\text{th}}}{\Omega_{ij}P_{j}}}\sum_{s=0}^{\infty}\frac{\left(\frac{m_{ij}P_{\hat{j}}}{\Omega_{ij}P_{j}}\right)^{s}}{s!}\Upsilon\left[m_{i\hat{j}}-n+p+s,\frac{m_{i\hat{j}}\iota_{2}}{\Omega_{i\hat{j}}}\right]$
$\displaystyle\\!\times\\!\\!\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}}\right)^{-(m_{i\hat{j}}-n+p+s)}$
$\displaystyle+\mathrm{e}^{\left(\frac{\mathcal{T}_{1}m_{ij}}{\mathcal{T}_{2}\Omega_{ij}}\right)}\frac{\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}}\right)^{m_{i\hat{j}}}}{\Gamma[m_{i\hat{j}}]}\sum_{k=0}^{m_{ij}-1}\frac{\left(\frac{m_{ij}}{\mathcal{T}_{2}\Omega_{ij}}\right)^{k}}{k!}\sum_{q=0}^{k}\binom{k}{q}$
$\displaystyle\\!\times\\!\\!(-\mathcal{T}_{1})^{k-q}\left(\frac{m_{ij}}{\mathcal{T}_{2}\Omega_{ij}}+\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}}\right)^{-(m_{i\hat{j}}+q)}$
$\displaystyle\\!\times\\!\Gamma\left[m_{i\hat{j}}+q,\iota_{2}\left(\frac{m_{ij}}{\mathcal{T}_{2}\Omega_{ij}}+\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}}\right)\right]$
(18)

where
$\iota_{2}=\frac{\mathcal{T}_{2}P_{\text{th}}+\mathcal{T}_{1}P_{j}}{P_{j}+\mathcal{T}_{2}P_{\hat{j}}}$.

###### Proof:

Please see Appendix A. ∎

Inserting (1) and (1) into (15), we can get the required expression of OP for
the primary system.

###  III-B Outage Probability of the Secondary System

Outage occurs at the SU if the instantaneous data rate achieved at that node
falls below a pre-defined target data rate. Therefore, the OP at the SU is
given as

$\displaystyle\mathcal{P}_{\textmd{out},\hat{i}}=\textmd{Pr}[\mathcal{R}_{i,\hat{i}}<r_{\hat{i}}]$
(19)

for $i,\hat{i}\in\\{1,2\\}$ and $i\neq\hat{i}$, where $r_{\hat{i}}$ is the
target rate required at ${\sf SU}_{\hat{i}}$. Moreover,
$\mathcal{R}_{i,\hat{i}}=((1-\alpha)/3)\log_{2}(1+\gamma_{i,\hat{i}})$ is the
instantaneous rate achieved at ${\sf SU}_{\hat{i}}$.
$\textmd{Pr}[\mathcal{R}_{i,\hat{i}}<r_{\hat{i}}]$ in (19) can be formulated
as,

$\displaystyle\textmd{Pr}[\mathcal{R}_{i,\hat{i}}<r_{\hat{i}}]=F_{\gamma_{i,\hat{i}}}(\gamma_{\hat{i}})$
(20)

and

$\displaystyle
F_{\gamma_{i,\hat{i}}}(\gamma_{\hat{i}})=F_{\gamma_{i,\hat{i}}^{(\text{lin})}}(\gamma_{\hat{i}})+F_{\gamma_{i,\hat{i}}^{(\text{sat})}}(\gamma_{\hat{i}})$
(21)

where $\gamma_{\hat{i}}=2^{\frac{3r_{\hat{i}}}{1-\alpha}}-1$. The required CDF
expression $F_{\gamma_{i,\hat{i}}}(\gamma_{\hat{i}})$ is provided in the
following Lemma.

###### Lemma 2

The CDF $F_{\gamma_{i,\hat{i}}^{\text{lin}}}(\gamma_{\hat{i}})$ can be
expressed as

$\displaystyle\\!\\!\\!F_{\gamma^{(\text{lin})}_{i,\hat{i}}}(\gamma_{\hat{i}})$
$\displaystyle\\!=\\!\lambda\frac{\Upsilon\left[m_{i\hat{i}},\frac{m_{i\hat{i}}\iota_{3}}{\Omega_{i\hat{i}}}\right]}{\Gamma[m_{i\hat{i}}]}\left(\Lambda\left(k,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)-\\!\\!\\!\\!\\!\\!\sum_{t=0}^{-k+m_{ij}+m_{i\hat{j}}+s-2}\right.$
$\displaystyle\times\left.\frac{\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)^{t}}{t!}\Lambda\left(k+t,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}+\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)\right)\right)$
$\displaystyle+\lambda\left(\rho\left(k,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)-\sum_{t=0}^{-k+m_{ij}+m_{i\hat{j}}+s-2}\right.$
$\displaystyle\times\left.\frac{\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)^{t}}{t!}\rho\left(k+t,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}+\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)\right)\right)$

where $\lambda$,
$\Lambda\left(k,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)$, and
$\rho\left(k,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)$ are defined
on the next page in (2), (24), and (2), respectively and
$\iota_{3}=\frac{\gamma_{\hat{i}}\sigma_{\hat{i}}^{2}}{\zeta_{i}P_{\text{th}}-\xi_{i}\gamma_{\hat{i}}}$.

$\displaystyle\lambda$
$\displaystyle=\frac{\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)^{m_{ij}}\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)^{m_{i\hat{j}}}}{\Gamma[m_{ij}]\Gamma[m_{i\hat{j}}]}\sum_{k=0}^{m_{ij}-1}(-1)^{-k+m_{ij}-1}\binom{m_{ij}-1}{k}\sum_{s=0}^{\infty}\frac{\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)^{s}}{s!}\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)^{-(-k+m_{ij}+m_{i\hat{j}}+s-1)}$
$\displaystyle\times\Gamma[-k+m_{ij}+m_{i\hat{j}}+s-1]$ (23)
$\displaystyle\Lambda\left(k,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)=\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)^{-(k+1)}\Upsilon\left[k+1,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)P_{\text{th}}\right]$
(24)
$\displaystyle\rho\left(k,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)$
$\displaystyle\\!=\\!\left(\Gamma[k+1]\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)^{-(k+1)}\frac{\Gamma\left[m_{i\hat{i}},\frac{m_{i\hat{i}}\iota_{3}}{\Omega_{i\hat{i}}}\right]}{\Gamma[m_{i\hat{i}}]}-\Gamma[k+1]\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)^{-(k+1)}\mathrm{e}^{\left(-\frac{\xi_{i}\gamma_{\hat{i}}}{\zeta_{i}}\right)\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)}\sum_{q=0}^{k}\frac{\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)^{q}}{q!}\right.$
$\displaystyle\times\frac{\left(\frac{m_{i\hat{i}}}{\Omega_{i\hat{i}}}\right)^{m_{i\hat{i}}}}{\Gamma[m_{i\hat{i}}]}\frac{1}{\zeta_{i}^{q}}\sum_{p=0}^{q}\binom{q}{p}(\xi_{i}\gamma_{\hat{i}})^{p}\left(\gamma_{\hat{i}}\sigma_{\hat{i}}^{2}\right)^{q-p}\left(2\left(\frac{m_{i\hat{i}}}{\Omega_{i\hat{i}}}\right)^{-\frac{1}{2}(m_{i\hat{i}}+p-q)}\left(\left(\frac{\gamma_{\hat{i}}\sigma_{\hat{i}}^{2}}{\zeta_{i}}\right)\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)^{\frac{1}{2}(m_{i\hat{i}}+p-q)}\right.$
$\displaystyle\\!\times\\!\\!\mathcal{K}_{m_{i\hat{i}}+p-q}\left(2\sqrt{\frac{m_{i\hat{i}}\left(\gamma_{\hat{i}}\sigma_{\hat{i}}^{2}\right)\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)}{\zeta_{i}\Omega_{i\hat{i}}}}\right)-\sum_{u=0}^{\infty}\frac{\left(-\frac{m_{i\hat{i}}}{\Omega_{i\hat{i}}}\right)^{u}}{u!}\left(\left(\frac{\gamma_{\hat{i}}\sigma_{\hat{i}}^{2}}{\zeta_{i}}\right)\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)^{(m_{i\hat{i}}+p+u-q)}$
$\displaystyle\\!\times\\!\left.\left(\left(\frac{\gamma_{\hat{i}}\sigma_{\hat{i}}^{2}}{\zeta_{i}}\right)\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\frac{1}{\iota_{3}}\right)^{-\frac{1}{2}(m_{i\hat{i}}+p+u-q+1)}\mathrm{e}^{-\left(\frac{\gamma_{\hat{i}}\sigma_{\hat{i}}^{2}}{\zeta_{i}}\right)\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\frac{1}{2\iota_{3}}}\right.$
$\displaystyle\\!\times\\!\\!\left.\left.\mathcal{W}_{-\frac{1}{2}(m_{i\hat{i}}+p+u-q+1),\frac{1-(m_{i\hat{i}}+p+u-q+1)}{2}}\left[\left(\frac{\gamma_{\hat{i}}\sigma_{\hat{i}}^{2}}{\zeta_{i}}\right)\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\frac{1}{2\iota_{3}}\right]\right)\right)$

The CDF $F_{\gamma_{i,\hat{i}}^{(\text{sat})}}(\gamma_{\hat{i}})$ can be
expressed as,

$\displaystyle\\!\\!\\!F_{\gamma_{i,\hat{i}}^{(\text{sat})}}(\gamma_{\hat{i}})$
$\displaystyle\\!=\\!\lambda\left(\varrho\left(k,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)-\sum_{t=0}^{-k+m_{ij}+m_{i\hat{j}}+s-2}\right.$
$\displaystyle\times\left.\frac{\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)^{t}}{t!}\varrho\left(k+t,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}+\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)\right)\right.$
$\displaystyle+\left.\epsilon\left(k,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)-\sum_{t=0}^{-k+m_{ij}+m_{i\hat{j}}+s-2}\frac{\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)^{t}}{t!}\right.$
$\displaystyle\times\left.\epsilon\left(k+t,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}+\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)\right)\right)$
(26)

where
$\iota_{4}=\frac{\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}P_{\text{th}}}{\Psi_{i}P_{\text{th}}-\mathcal{C}_{i}\gamma{\hat{i}}}$.
$\varrho\left(k,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)$ and
$\epsilon\left(k,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)$ are
defined in (III-B) and (28), respectively.

###### Proof:

Please see Appendix B. ∎

$\displaystyle\varrho\left(k,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)$
$\displaystyle\\!=\\!\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)^{-(k+1)}\left(k!\frac{\Upsilon\left[m_{i\hat{i}},\frac{m_{i\hat{i}}}{\Omega_{i\hat{i}}}\iota_{4}\right]-\Upsilon\left[m_{i\hat{i}},\frac{m_{i\hat{i}}}{\Omega_{i\hat{i}}}\frac{\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}{\Psi_{i}}\right]}{\Gamma[m_{i\hat{i}}]}-k!\frac{\left(\frac{m_{i\hat{i}}}{\Omega_{i\hat{i}}}\right)^{m_{i\hat{i}}}}{\Gamma[m_{i\hat{i}}]}\sum_{p=0}^{k}\frac{\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)^{p}}{p!}(\mathcal{C}_{i}\gamma_{\hat{i}})^{p}\frac{1}{\Psi_{i}^{m_{i\hat{i}}+p}}\right.$
$\displaystyle\\!\times\\!\left.\mathrm{e}^{-\left(\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\frac{\mathcal{C}_{i}\gamma_{\hat{i}}}{\Psi_{i}}+\frac{m_{i\hat{i}}}{\Omega_{i\hat{i}}}\frac{\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}{\Psi_{i}}\right)}\\!\\!\sum_{q=0}^{m_{i\hat{i}}+p-1}\binom{m_{i\hat{i}}+p-1}{q}(\mathcal{D}_{\hat{i}}\gamma_{\hat{i}})^{q}\sum_{u=0}^{\infty}\frac{\left(-\frac{m_{i\hat{i}}}{\Psi_{i}\Omega_{i\hat{i}}}\right)^{u}}{u!}\left(\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\frac{\mathcal{C}_{i}\gamma_{\hat{i}}}{\Psi_{i}}\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}\right)^{m_{i\hat{i}}-q+u}\right.$
$\displaystyle\times\left.\left(\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\frac{\mathcal{C}_{i}\gamma_{\hat{i}}}{\Psi_{i}}\frac{\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}{\Psi_{i}\iota_{4}-\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}\right)^{-\frac{1}{2}(m_{i\hat{i}}-q+u+1)}\mathrm{e}^{-\frac{1}{2}\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\frac{\mathcal{C}_{i}\gamma_{\hat{i}}}{\Psi_{i}}\frac{\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}{\Psi_{i}\iota_{4}-\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}}\right.$
$\displaystyle\times\left.\mathcal{W}_{-\frac{1}{2}(m_{i\hat{i}}-q+u+1),\frac{1}{2}(1-(m_{i\hat{i}}-q+u+1))}\left[\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\frac{\mathcal{C}_{i}\gamma_{\hat{i}}}{\Psi_{i}}\frac{\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}{\Psi_{i}\iota_{4}-\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}\right]\right.$
$\displaystyle-\left.\Upsilon\left[k+1,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)P_{\text{th}}\right]\frac{\Upsilon\left[m_{i\hat{i}},\frac{m_{i\hat{i}}\iota_{4}}{\Omega_{i\hat{i}}}\right]-\Upsilon\left[m_{i\hat{i}},\frac{m_{i\hat{i}}}{\Omega_{i\hat{i}}}\frac{\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}{\Psi_{i}}\right]}{\Gamma[m_{i\hat{i}}]}\right)$
(27)
$\displaystyle\epsilon\left(k,\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)\right)$
$\displaystyle=\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)^{-(k+1)}\Gamma\left[k+1,\frac{m_{ij}}{\Omega_{ij}P_{j}}P_{\text{th}}\right]\frac{\Upsilon\left[m_{i\hat{i}},\frac{m_{i\hat{i}}}{\Omega_{i\hat{i}}}\frac{\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}{\Psi_{i}}\right]}{\Gamma[m_{i\hat{i}}]}$
(28)

###  III-C Spectrum Sharing

The secondary system employed for relay cooperation should assist in providing
better OP at the primary nodes compared to direct end-to-end transmission.
Hence, it is useful to analyze the range of spectrum sharing factor such that
the system shows better OP than that of the direct link. When the PUs are
communicating directly without the help of relaying SUs, the rate achieved at
the PUs can be given as

$\displaystyle\mathcal{R}_{j,\hat{j}}^{(\textmd{D})}=\frac{1}{2}\log_{2}\left(1+\frac{P_{j}\left|h_{j,\hat{j}}\right|^{\textmd{2}}}{\sigma^{2}}\right).$
(29)

The overall end-to-end communication in direct transmission occurs in two
phases and hence there is a pre-log factor $1/2$ in the above expression.
Thus, the OP for the direct transmission between two PUs is

$\displaystyle\mathcal{P}_{\textmd{out},\hat{j}}^{(\textmd{D})}=\Pr\left[\frac{P_{j}\left|h_{j,\hat{j}}\right|^{\textmd{2}}}{\sigma^{2}}<\widetilde{\gamma_{\hat{j}}}\right]=\frac{\Upsilon\left[m_{j\hat{j}},\frac{m_{j\hat{j}}\widetilde{\gamma_{\hat{j}}}\sigma^{2}}{\Omega_{j\hat{j}}P_{j}}\right]}{\Gamma\left[m_{j\hat{j}}\right]}$
(30)

where $\widetilde{\gamma_{\hat{j}}}=2^{2r_{\hat{j}}}-1$. For effective
spectrum sharing, the SWIPT-enabled CCRN should lead to lower or equal OP than
that of direct transmission (without relaying) under the same required target
rate [30], i.e.,

$\displaystyle\mathcal{P}_{\textmd{out},a}\leq\mathcal{P}_{\textmd{out},a}^{(\textmd{D})}\text{
and
}\mathcal{P}_{\textmd{out},b}\leq\mathcal{P}_{\textmd{out},b}^{(\textmd{D})}.$
(31)

On the other hand, maintaining QoS of the primary system is also essential
while enabling spectrum sharing for the secondary transmission. For
maintaining QoS of the primary system, the term corresponding to OP of the
direct transmission ($\mathcal{P}_{\textmd{out},b}^{(\textmd{D})}$) can be
replaced by a predefined target OP to support the required target rate. Using
the above expressions, the critical value of spectrum sharing factor
$\mu^{\star}$, beyond which the proposed system achieves lower OP than direct
transmission link, can be obtained using numerical methods. Therefore, the
network can assure effective spectrum sharing for $\mu_{i}\geq\mu^{\star}$.
Note that, the value of $\mu^{\star}$ is crucial to define the range of
constraints for solving the optimization problem in (37). Thus, it has been
ensured that the system under consideration satisfies the QoS constraint.

###  III-D System Throughput

Under the delay-limited transmission mode of the considered scheme, throughput
can be calculated as the sum of average target rates of both the primary and
secondary systems that can be attained successfully over the fading channels
[39, 30]. Thus, system throughput can be defined based on OPs of primary and
secondary systems, each at a fixed transmission rate. Here, all the individual
data rates are considered to be equal to the target transmission data rate
$r_{\textmd{th}}$ in bits/sec/Hz. Therefore, the system throughput can be
expressed as

$\displaystyle\mathcal{S}_{\textmd{T}}=\mathcal{S}_{\textmd{PU}}+\mathcal{S}_{\textmd{SU}}$
(32)

where $\mathcal{S}_{\textmd{PU}}$ and $\mathcal{S}_{\textmd{SU}}$ are the
throughputs of primary and secondary users, respectively. Furthermore,
$\mathcal{S}_{\textmd{PU}}$ and $\mathcal{S}_{\textmd{SU}}$ can be expressed
in terms of their respective individual outage probabilities as

$\displaystyle\mathcal{S}_{\textmd{PU}}=\frac{\left(1-\alpha\right)}{3}\bigg{(}\left(1-\mathcal{P}_{\textmd{out},a}\right)r_{a}+\left(1-\mathcal{P}_{\textmd{out},b}\right)r_{b}\bigg{)}$
(33)
$\displaystyle\mathcal{S}_{\textmd{SU}}=\frac{\left(1-\alpha\right)}{3}\bigg{(}\left(1-\mathcal{P}_{\textmd{out},1}\right)r_{1}+\left(1-\mathcal{P}_{\textmd{out},2}\right)r_{2}\bigg{)}$
(34)

where the primary outage probabilities $\mathcal{P}_{\textmd{out},a}$ $\&$
$\mathcal{P}_{\textmd{out},b}$ are defined in (15), and secondary outage
probabilities $\mathcal{P}_{\textmd{out},1}$ $\&$
$\mathcal{P}_{\textmd{out},2}$ are given in (19).

###  III-E Energy Efficiency

For any communication system, it is desirable to support an increased system
throughput while minimizing energy consumption, and hence contributing to
environment-friendly transmission. Thus, energy efficiency has become an
important parameter in designing and analyzing the system performance. Energy
efficiency is defined as [39]

$\displaystyle\textmd{Energy efficiency}=\frac{\textmd{Total amount of data
delivered}}{\textmd{Total amount of energy consumed}}$ (35)

It follows that the energy efficiency for the considered system in the delay-
limited scenario is given by

$\displaystyle\eta_{\rm
EE}=\frac{\mathcal{S}_{\textmd{T}}}{\left(\alpha+\frac{\beta\left(1-\alpha\right)}{3}\right)\left(P_{a}+P_{b}\right)}.$
(36)

###  III-F Optimization

Given the above system throughput and energy efficiency expressions and
variables, we now formulate an optimization problem with an objective of
maximizing the system throughput and energy efficiency while satisfying the
system parameters constraints. Here, we strive to find the optimal values of
TS, PS and spectrum sharing factors, i.e., $\alpha$, $\beta$ and $\mu$ that
maximize the system throughput. The formulated optimization problem is as
follows:

$\displaystyle\mathbf{P1:}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\underset{\alpha,\beta,\mu}{\textrm{maximize\leavevmode\nobreak\
}}\mathbb{U}(\alpha,\beta,\mu)$ (37) s.t. (Time switching constraint):
$\displaystyle\text{C}_{1}:\;0\leq\alpha<1$ (Power splitting constraint):
$\displaystyle\text{C}_{2}:\;0\leq\beta<1$ (Spectrum sharing constraint):
$\displaystyle\text{C}_{3}:\;\mu^{\star}\leq\mu<1,$

where $\mathbb{U}\in\\{\mathcal{S}_{\rm T},\eta_{\rm EE}\\}$ is the objective
function and the expressions of $\mathcal{S}_{\rm T}$ and $\eta_{\rm EE}$ are
provided in (32) and (36), respectively.

Similar to [44], it can be seen that problem $\mathbf{P}_{1}$ is non-convex in
nature, and finding an optimal solution is computationally challenging.
Therefore, in this paper, the particle swarm optimization technique is
implemented to maximize the system throughput, energy efficiency and thereby
obtain near optimal values of TS, PS and spectrum sharing factors (i.e.,
$\alpha$, $\beta$ and $\mu$, respectively). PSO is a population-based
stochastic optimization technique known for its ease of implementation,
accuracy and robustness. The algorithm is initialized with $N$ number of
random solutions called as population. Here, a solution includes the values of
$\alpha$, $\beta$ and $\mu$. The system throughput and energy efficiency
corresponding to a solution set are evaluated using (32) and (36). The
algorithm updates the population for every iteration based on the individual
solutions’ best performance, called $p^{(\rm best)}$ or local best
performance, and the overall best performance of the population set, called
$g^{(\rm best)}$ or global best performance. Therefore, $p^{(\rm best)}$
implies the solutions’ individual best performance which gives maximum system
throughput and energy efficiency, whereas $g^{(\rm best)}$ implies the best
solution in the entire population set which yields the maximum system
throughput and energy efficiency. In this algorithm, the solutions are updated
for every iteration using two parameters called _velocity_ and _position_
[40]. The velocity of a solution implies the values by which $\alpha$, $\beta$
and $\mu$ have to be changed in order to update the solution. The position of
a solution is the updated set of $[\alpha,\beta,\mu]$. Here, velocity has been
evaluated separately for $\alpha$, $\beta$ and $\mu$. The velocity $v$ and
position $\psi$ of the solution are updated using the following equations:

$\displaystyle
v^{(\textmd{next})}_{i}=w*v^{\textmd{(current)}}_{i}+c_{1}*\textmd{rand()}*(p^{(\rm
best)}_{i}-\psi_{i})$ $\displaystyle+c_{2}*\textmd{rand()}*(g^{(\rm
best)}-\psi_{i})$ (38)
$\displaystyle\psi^{(\textmd{next})}_{i}=\psi^{\textmd{(current)}}_{i}+v^{(\textmd{next})}_{i}.$
(39)

In the above expressions, $v_{i}$ is the velocity of the $i^{\textmd{th}}$
solution, $w$ is the inertia weight factor, $c_{1}$ and $c_{2}$ are learning
factors, $\mathrm{rand}()$ is a uniformly distributed random number between 0
and 1, $p^{(\rm best)}_{i}$ is the individual best performance of the
$i^{\textmd{th}}$ solution, $g^{(\rm best)}$ is the best solution of the
population set and $\psi_{i}$ denotes the position of $i^{\textmd{th}}$
solution. If the updated position is found to be out of the search space, then
its value is set to be within the bounds and the corresponding velocity of the
solution is equated to zero. The termination criterion of the algorithm is
given by the number of iterations.

The proposed PSO Algorithm 1 process in three steps: (i) initialization, (ii)
update, and (iii) termination. Line 1 and 2 are initializations and contribute
one operation for $N$ times each. Line 4 performs $N$ iterations to find
system throughput and energy efficiency. The calculation of internal
expressions in system throughput and energy efficiency is ignored as they can
be saved in a lookup table. In line 5, using bubble sort, we find
$p^{\textrm{(best)}}$ which requires $N\log(N)$ operations, and finding
$g^{\textrm{(best)}}$ in $N$ solutions requires $\log(N)$ operations. In line
6, velocity update for N particles and three parameters require 15N
multiplications and 12N additions. To update its values, one more addition is
required. All the above operations are performed $t$ times to check the
termination condition. Therefore, the worst-case complexity is $O(N\cdot
t\cdot\log(N))$.

Table I: Number of terms required in infinite series for achieving required accuracy. Index $s$ |  $P_{\text{out},a}$ in (14) |  $P_{\text{out},1}$ in (19)
---|---|---
$\mathrm{SNR}=15$dB | 20dB | 20dB | 25dB
1 | 0.111445 | 0.0133768 | 0.330062 | 0.20458
2 | 0.111445 | 0.0133768 | 0.4865 | 0.292226
3 | 0.111445 | 0.0133768 | 0.608107 | 0.356532
4 | 0.111445 | 0.0133768 | 0.696492 | 0.400999
5 | 0.111445 | 0.0133768 | 0.758037 | 0.430666
6 | 0.111445 | 0.0133768 | 0.799646 | 0.450000
7 | 0.111445 | 0.0133768 | 0.827179 | 0.462397
8 | 0.111445 | 0.0133768 | 0.845107 | 0.470251
9 | 0.111445 | 0.0133768 | 0.856636 | 0.475184
10 | 0.111445 | 0.0133768 | 0.863977 | 0.478261
11 | 0.111445 | 0.0133768 | 0.868615 | 0.48017
12 | 0.111445 | 0.0133768 | 0.871526 | 0.481349
13 | 0.111445 | 0.0133768 | 0.873344 | 0.482075
14 | 0.111445 | 0.0133768 | 0.874473 | 0.48252
15 | 0.111445 | 0.0133768 | 0.875173 | 0.482793
16 | 0.111445 | 0.0133768 | 0.875605 | 0.482959

Input : ($\alpha$, $\beta$ and $\mu$)

Output : Maximal system throughput, energy efficiency and optimal solution
$[\alpha^{*},\beta^{*},\mu^{*}]$

1 Initialize all the required parameters and the range (search space) of
values of $\alpha$, $\beta$ and $\mu$.

2Randomly initialize a population set of solutions ($[\alpha,\beta,\mu]$),
with an initial random position and velocity values.

3while  _termination criterion not met_

4 Evaluate system throughput and energy efficiency of every solution using
(32) and (36).

5 Find $p^{(\rm best)}$ of every solution and $g^{(\rm best)}$ solution of the
population set.

6 Update the position and velocity of all the solutions using (III-F) and (39)

7 if  _position of the solution is out of bounds_ then

8 Set the position equal to the bound.

9 Set the velocity of the corresponding solution equal to zero.

10 end if

11

12 Update the population set.

13 Repeat the procedure.

14 end while

15

16

Algorithm 1 Algorithm for maximizing system throughput and energy efficiency

##  IV Numerical and Simulation Results

###  IV-A Outage Probability

This section discusses the outage performance at both the primary and
secondary users with respect to various system parameters. All the numerical
and simulation results are obtained under the assumption that $P_{a}=P_{b}=P$,
$\sigma^{2}_{i}=\sigma^{2}_{j}=\sigma^{2}_{c_{i}}=\sigma^{2}$ for
$i\in\\{1,2\\}$ and $j\in\\{a,b\\}$, and the SNR at the transmitting end is
defined as ${P}/{\sigma^{2}}$ [20].

Furthermore, a 2-D network topology is considered where $\sf PU_{a}$ and $\sf
PU_{b}$ are placed at coordinates (0, 0), (4, 0), and $\sf SU_{1}$ and $\sf
SU_{2}$ are at $(d,0)$, $(d,2)$, respectively, where $d=2$. Regarding the
path-loss model, the path-loss exponent is selected as $\nu=3$, while the
channel coefficient is modeled in terms of the distance between the end nodes
and the path-loss exponent. For links ${\sf PU}_{a}\rightarrow{\sf SU}_{1}$
and ${\sf PU}_{a}\rightarrow{\sf SU}_{2}$, variances of the channel gains are
represented as $\Omega_{1a}=d_{1a}^{-\nu}$ and $\Omega_{2a}=d_{2a}^{-\nu}$,
respectively. Likewise, for ${\sf PU}_{b}\rightarrow{\sf SU}_{1}$, ${\sf
PU}_{b}\rightarrow{\sf SU}_{2}$, and ${\sf SU}_{2}\rightarrow{\sf SU}_{1}$
links, variances of the channel gains are $\Omega_{1b}=d_{1b}^{-\nu}$,
$\Omega_{2b}=d_{2b}^{-\nu}$, and $\Omega_{12}=d_{12}^{-\nu}$, respectively
[18], [39]. Also, $P_{\text{th}}$ is set as $-10$dBm and the noise power at
both primary and secondary users is assumed to be $-60$dBm, unless specified
otherwise. The fading severity parameters are assumed as $m_{1a}=m_{2a}=m_{a}$
and $m_{1b}=m_{2b}=m_{b}$. The other parameters vary with each figure and are
defined therein. Table I shows the required number of terms in infinite series
to achieve a fair accuracy in evaluating (14) and (19) for the considered set
of parameters.

Figure 3: OP versus SNR for ${\sf PU}_{b}\rightarrow{\sf PU}_{a}$ link. Figure
4: OP versus SNR for ${\sf SU}_{2}\rightarrow{\sf SU}_{1}$ link.

Fig. 3 shows the primary link outage probability-versus-SNR curves for
different target data rates. In plotting this figure, the parameters are set
as $\alpha=\beta=0.2$, $\mu=0.8$, $m_{a}=3$ and $m_{b}=2$. From this figure,
one can notice that the OP performance improves with the increase in SNR
values up to a certain value and remains constant afterward in the high SNR
range. This is due to the fact that, since a non-linear EH model is used, the
total transmit power becomes constant after it exceeds the threshold value
$P_{\text{th}}$. Also, the OP performance of the primary link is better for
lower target rates because lower target rates result in relatively lower
target SNRs. Hence, the probability of outage occurrence becomes comparatively
lower with lower values of target SNRs. Furthermore, Fig. 4 plots the
secondary link OP-versus-SNR curves for different target data rates. With the
increase in SNR values, the behavior of OP performance at the SU is similar to
that of the OP performance at PU. Here also, the OP curves become saturated at
the high SNR range, which is attributed to the non-linearity of the considered
energy harvester.

Figure 5: OP versus $\mu$ for ${\sf PU}_{b}\rightarrow{\sf PU}_{a}$ link.
Figure 6: OP versus $\mu$ for ${\sf SU}_{2}\rightarrow{\sf SU}_{1}$ link.
Figure 7: OP versus $\alpha$ and $\beta$ for ${\sf PU}_{b}\rightarrow{\sf
PU}_{a}$ link.

###  IV-B OP for PUs and SUs with Spectrum Sharing Factor

For Fig. 5, the values of different parameters are considered as
$\eta_{1}=\eta_{2}=0.7$, $\alpha=0.2$, $\beta=0.2$, $\mu_{1}=\mu_{2}=\mu$ and
$\textmd{SNR}=20$dB. This figure shows the OP of the primary link ${\sf
PU}_{b}\rightarrow{\sf PU}_{a}$ versus spectrum sharing factor $\mu$, for
various target rates. On analyzing the OP performance of the primary system
for varying fading severity parameters, it is observed that the OP performance
improves with increasing value of $m_{b}$. This is in accordance with the fact
that, when the system operates in a less severe fading environment, it has
better link reliability. Note that for a specific target data rate, the value
of $\mu_{i}$ should be greater than a certain value, in order to enable
spectrum sharing. For $r_{j}=1/3$ bps/Hz and $\alpha=0.2$, the acceptable
spectrum sharing factor range is $0.62<\mu_{i}<1$, while for $\mu_{i}<0.62$
outage occurs at the primary system (i.e., the OP equals one). Along with the
target rate required at the node, the TS factor $\alpha$ also determines the
minimum possible value of $\mu_{i}$. Further, with the increase in the value
of $\mu_{i}$, the primary link OP also improves due to the fact that a larger
portion of power is being allotted for primary transmissions.

Fig. 6 shows the OP curves for the link ${\sf SU}_{2}\rightarrow{\sf SU}_{1}$
versus $\mu$, under different fading scenarios and target rates. All the
parameters considered here are the same as in Fig. 5. In contrast to Fig. 5,
as the value of $\mu_{i}$ increases, the secondary link OP performance
deteriorates. This is because the transmitted power at the secondary system is
scaled by a factor of $(1-\mu_{i})$. Here also, it can observed that when the
fading severity parameter $m_{12}$ increases, the secondary system shows
better performance due to the fact that link reliability is better in a less
severe fading environment. Unlike the primary link, the secondary nodes can
have reliable communication for the complete range of $\mu_{i}$. Likewise, the
OP performance of the secondary system also degrades with increasing target
rate. All the simulation results and analytical plots of Figs. 5 and 6 are in
good consonance, which confirms the accuracy of the theoretical analysis.

###  IV-C OP for PUs with TS and PS Factors

In the considered CCRN, hybrid SWIPT is employed and therefore it is of
particular interest to investigate the joint effect of both TS and PS factors
on the OP performance. Here, the noise power at PUs is set as $-10$dBm. The OP
curves are plotted against TS factor $\alpha$ and PS factor $\beta$ for the
primary system as shown in Fig. 7. For the considered parameters, the primary
link achieves the minimum OP when $\alpha=0.08$ and $\beta=0.75$. In contrast,
when $\alpha=0$, the primary link attains the minimum OP at $\beta=0.8$. For a
certain $\alpha$ value and increasing $\beta$ value, the OP of the primary
link decreases up to a certain point and then starts increasing. Also, it can
be observed that for certain $\alpha$ and $\beta$ values, the OP is very low
compared to that at other values.

###  IV-D Effective Spectrum Sharing

Figure 8: OP versus $\mu$ for ${\sf PU}_{b}\rightarrow{\sf PU}_{a}$ link with
proposed scheme and direct transmissions.

Fig. 8 plots OP curves for the primary link ${\sf PU}_{b}\rightarrow{\sf
PU}_{a}$ versus the spectrum sharing factor $\mu$. This figure demonstrates
the outage performance of both the proposed scheme and direct transmission of
the primary link and offers some key insights into the system design
constraints for effective spectrum sharing with respect to $\mu$. Here, the
parameters are considered as $\alpha=0.2$, $\beta=0.2$ and
$\textmd{SNR}=20$dB. It can be observed from the figure that the critical
value of $\mu_{i}$, denoted as $\mu^{\star}$, is the point at which the OPs of
the proposed system and that of the direct transmission are equal. For
$\mu_{i}>\mu^{\star}$, the OP of the primary system shows better performance
than the direct transmission link. Therefore, $\mu^{\star}<\mu_{i}<1$ can be
considered as the feasible range of $\mu_{i}$ for effective spectrum sharing.
Below the critical value $\mu^{\star}$, the system shows poor outage
performance compared to direct primary transmission link. It is also noted
that, with increasing data rates, the value of $\mu^{\star}$ also increases.
This is in agreement with the fact that, as higher data rates result in higher
target SNRs, the fraction of power allocated for primary signal transmission
in BC phases should be greater for higher data rates. Consequently, the
critical value of $\mu_{i}$, which is the power splitting factor for spectrum
sharing, also becomes larger.

###  IV-E System Throughput

Figure 9: System throughput versus SNR for different target rates.

Fig. 9 shows the system throughput in bps/Hz versus SNR in dB. System
throughput curves are plotted for different values of target rate
$r_{\textmd{th}}$. The parameters are considered as $\alpha=\beta=0.2$,
$\mu=0.8$, $\eta_{1}=\eta_{2}=0.7$, $m_{a}=3$, $m_{b}=2$, $m_{12}=m_{21}=1$
and $r_{a}=r_{b}=r_{1}=r_{2}=r_{\textmd{th}}$. The target rates of primary and
secondary users are assumed to be equal. It can be observed that, at the low
SNR range, the system throughput corresponding to higher target rates is very
small compared to that at lower target rates. This is attributed to the fact
that, as the target rates increase, the target SNR also increases, but because
the considered SNR is low, the outage probabilities at both PUs and SUs
increase, which results in decreased system throughput values. When the SNR is
in the medium to high range, the outage probabilities improve with increasing
target rates and hence the system throughput also increases with the target
rates. Also, it is observed that after a certain SNR value (which varies with
target rates), the system throughput becomes saturated and serves as the
maximum achievable throughput for the defined parameter set.

###  IV-F Energy Efficiency

Figure 10: Energy efficiency versus SNR for different target rates.

In Fig. 10, the energy efficiency of the system versus SNR is studied under
different target rates. The system parameters considered are the same as that
considered for Fig. 9. It is evident from the figure that energy efficiency of
the system corresponding to a certain target rate is high only for a specific
range of SNR values, and this range of SNR varies for different target rates.
For lower target rates, maximum energy efficiency is achieved at lower SNR
values and for higher target rates, maximum energy efficiency is attained at
higher values of SNR. Also, as the target rates increase, the maximum
attainable energy efficiency decreases. Thus, it can be deduced that the
maximum energy efficiency for a certain target rate can be obtained only for a
specific range of SNR values.

###  IV-G Results with Optimized System Parameters

Figs. 11 and 12 plot the system throughput and energy efficiency,
respectively, for varying SNR. In Fig. 11 the curves are plotted for two sets
of $\alpha$, $\beta$ and $\mu$ and for different target data rates. One set
comprises of typical values as considered in previous figures whereas the
other is the optimized set obtained from the optimization algorithm described
in Section III-F. It can be observed that the system throughput corresponding
to the optimized set of parameters shows a much superior performance compared
to the curves corresponding to typical parameters. The difference in between
the optimized system throughput and the typical system throughput is much
higher in the medium to high SNR range as compared to the low SNR range
because the optimal parameters are obtained for SNR $=20$ dB. Since the system
throughput depends on the TS factor $\alpha$ and OP, and the optimization
algorithm results in minimized OP and low $\alpha$ value, the resultant
optimized system throughput is comparatively high in the high SNR range.
Whereas in the low SNR region, outage occurs due to the higher target SNRs and
thus the system throughput is zero for the low SNR range. After a saturation
point, the system throughput remains constant at the maximum achievable
throughput.

Table II: Optimal system parameters Target rate | Optimal system parameters
---|---
$\alpha$ | $\beta$ | $\mu$
1/2 | 0.01 | 0.6918 | 0.95
1/3 | 0.01 | 0.7125 | 0.95
1/4 | 0.01 | 0.6847 | 0.8165
1/5 | 0.01 | 0.7101 | 0.7544
1/6 | 0.01 | 0.7140 | 0.7732

The optimal values of $\alpha,\beta$, and $\mu$, to plot curves in Fig. 11,
are given in Table II for SNR $=20$ dB. From this table, one can see that the
values of $\alpha$ are close to zero. This is because of a well known fact
that PS-based SWIPT shows better system throughput than TS-based SWIPT in the
high SNR region [45].

Similarly, energy efficiency curves are plotted in Fig. 12 for specific target
data rates and for typical and optimized sets of $\alpha$, $\beta$ and $\mu$.
There is a significant difference between the optimal and typical energy
efficiencies in the medium SNR range. This is again due to the fact that the
optimized values of $\alpha$ and $\beta$ are very low compared to the typical
values. Therefore in the mid SNR range the energy efficiency is very high
compared to its typical counterpart. In the high SNR range, i.e., after the
system attains the maximum throughput, the energy efficiency degrades with
increasing SNR because as the SNR becomes high, the transmitted power, which
is assumed to be proportional to the system SNR up to a certain threshold
value, also increases and hence the energy efficiency decreases. Furthermore,
with increasing SNR, the energy efficiency remains constant due to the non-
linearity of the EH receiver. Thus, the system can achieve maximum throughput
and energy efficiency for optimized values of $\alpha$, $\beta$ and $\mu$ at
particular SNR value and target data rate.

Figure 11: System throughput versus SNR for optimized and typical input
parameters with different target rates. Figure 12: Energy efficiency versus
SNR for optimized and typical input parameters with different target rates.

##  V Conclusions

This paper has investigated a new hybrid SWIPT-based spectrum sharing system
to enable two-way communications of PUs and SUs over the same licensed
spectrum. A pair of AF-based SWIPT-enabled SUs provides cooperative relaying
for primary signal transmissions. The primary signals are relayed by the SUs
in two successive phases, thereby improving the reliability of the
communication links. Outage performance of both primary and secondary systems
with the considered scheme operating under the Nakagami-$m$ fading environment
is analyzed. Further, the feasible range of spectrum sharing factor $\mu$ is
deduced, for which the OP performance of the proposed scheme is superior to
that of the direct transmission. The system throughput and energy efficiency
expressions are formulated and they are maximized by means of particle swarm
optimization. Numerical and simulation results confirmed the accuracy of the
derived analytical expressions. The impact of crucial design parameters, e.g.,
spectrum sharing factor, TS and PS factors is revealed on performance of the
considered system.

## Appendix A

Let $X_{i}\triangleq|h_{j,i}|^{2}$ and $Y_{i}\triangleq|h_{\hat{j},i}|^{2}$
for $i\in\\{1,2\\}$, $j,\hat{j}\in\\{a,b\\}$ with $j\neq\hat{j}$, be Gamma
distributed random variables. Their PDFs can be given as
$f_{X_{i}}(x_{i})=\left(\frac{m_{ij}}{\Omega_{ij}}\right)^{m_{ij}}\frac{x^{m_{ij}-1}_{i}}{\Gamma[m_{ij}]}\mathrm{e}^{-\frac{m_{ij}x_{i}}{\Omega_{ij}}},\,x_{i}\geq
0$, and
$f_{Y_{i}}(y_{i})=\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}}\right)^{m_{i\hat{j}}}\frac{y^{m_{i\hat{j}}-1}_{i}}{\Gamma[m_{i\hat{j}}]}\mathrm{e}^{-\frac{m_{i\hat{j}}y_{i}}{\Omega_{i\hat{j}}}},\,y_{i}\geq
0$. The CDF $F_{\gamma_{i,j}^{(\text{lin})}}(\gamma_{j})$ can be formulated
using (8), as

$\displaystyle F_{\gamma_{i,j}^{(\text{lin})}}(\gamma_{j})$
$\displaystyle=\textmd{Pr}\Bigg{[}\frac{\mu_{i}\Delta_{i}P_{\hat{j}}X_{i}Y_{i}}{\varepsilon_{ij}X_{i}+\omega_{ij}X_{i}^{2}+\omega_{i\hat{j}}X_{i}Y_{i}+\sigma^{2}_{j}}<\gamma_{j},$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,\,P_{j}X_{i}+P_{\hat{j}}Y_{i}\leq
P_{\text{th}}\Bigg{]}$
$\displaystyle=\textmd{Pr}\left[Y_{i}<\frac{\varepsilon_{ij}\gamma_{j}+\omega_{ij}\gamma_{j}X_{i}}{\Xi},Y_{i}\leq\frac{P_{\text{th}}-P_{j}X_{i}}{P_{\hat{j}}}\right]$
(40)

where $\Xi=\mu_{i}\Delta_{i}P_{\hat{j}}-\omega_{i\hat{j}}\gamma_{j}$. From
(A), one can observe that when the term $\Xi\leq 0$, i.e.,
$\left(\mu_{i}-(1-\mu_{i})\gamma_{j}\right)\leq 0$, the CDF
$F_{\gamma_{i,j}^{(\text{lin})}}(\gamma_{j})=1$. On the other hand, when
$\left(\mu_{i}-(1-\mu_{i})\gamma_{j}\right)>0$, the expression of
$F_{\gamma_{i,j}^{\text{lin}}}(\gamma_{j})$ can be given as

$\displaystyle F_{\gamma_{i,j}^{\text{lin}}}(\gamma_{j})$
$\displaystyle=\int^{\iota_{1}}_{x_{i}=0}\\!\\!\\!f_{X_{i}}(x_{i})\\!\\!\int^{\frac{\varepsilon_{ij}\gamma_{j}+\omega_{ij}\gamma_{j}x_{i}}{\Xi}}_{y_{i}=0}\\!\\!f_{Y_{i}}\\!(\\!y_{i}\\!)dy_{i}\,dx_{i}$
$\displaystyle+\int^{\frac{P_{\text{th}}}{P_{j}}}_{x_{i}=\iota_{1}}\\!\\!\\!f_{X_{i}}(x_{i})\\!\\!\int^{\frac{P_{\text{th}}-P_{j}x_{i}}{P_{\hat{j}}}}_{y_{i}=0}\\!\\!f_{Y_{i}}\\!(\\!y_{i}\\!)dy_{i}\,dx_{i}$
(41)

The CDF $F_{\gamma_{i,j}^{(\text{sat})}}(\gamma_{j})$ can be formulated using
(9), as,

$\displaystyle F_{\gamma_{i,j}^{(\text{sat})}}(\gamma_{j})$
$\displaystyle=\textmd{Pr}\left[\frac{\mu_{i}\Delta_{i}P_{\text{th}}P_{\hat{j}}X_{i}Y_{i}}{\phi_{ij}X_{i}+\Phi_{ij}X_{i}^{2}+\Phi_{i\hat{j}}X_{i}Y_{i}^{2}+\varphi_{\hat{j}j}Y_{i}}<\gamma_{j},\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad\qquad
P_{j}X_{i}+P_{\hat{j}}Y_{i}>P_{\text{th}}\right]$
$\displaystyle=\textmd{Pr}\left[X_{i}>\frac{Y_{i}-\mathcal{T}_{1}}{\mathcal{T}_{2}},X_{i}>\frac{P_{\text{th}}-P_{\hat{j}}Y_{i}}{P_{j}}\right].$
(42)

where
$\mathcal{T}_{1}=\frac{\phi_{ij}\gamma_{j}}{\mu_{i}\Delta_{i}P_{\text{th}}P_{\hat{j}}-\Phi_{i\hat{j}}\gamma_{j}}$
and
$\mathcal{T}_{2}=\frac{\Phi_{ij}\gamma_{j}}{\mu_{i}\Delta_{i}P_{\text{th}}P_{\hat{j}}-\Phi_{i\hat{j}}\gamma_{j}}$.

$\displaystyle F_{\gamma_{i,j}^{\text{sat}}}(\gamma_{j})$
$\displaystyle=\int^{\iota_{2}}_{y_{i}=0}\\!\\!\\!f_{Y_{i}}(y_{i})\\!\\!\int^{\infty}_{x_{i}=\frac{P_{\text{th}}-P_{\hat{j}}y_{i}}{P_{j}}}\\!\\!f_{X_{i}}\\!(\\!x_{i}\\!)dx_{i}\,dy_{i}$
$\displaystyle+\int^{\infty}_{y_{i}=\iota_{2}}\\!\\!\\!f_{Y_{i}}(y_{i})\\!\\!\int^{\infty}_{x_{i}=\frac{y_{i}-\mathcal{T}_{1}}{\mathcal{T}_{2}}}\\!\\!f_{X_{i}}\\!(\\!x_{i}\\!)dx_{i}\,dy_{i}.$
(43)

After substituting the PDFs in (A) and (A) and applying [38, eqs. 3.381.1,
3.471.9], the required solutions can be given as in Lemma 1.

## Appendix B

Let $Z\triangleq|h_{i,\hat{i}}|^{2}$ be a random variable with PDF
$f_{Z}(z)=\left(\frac{m_{i\hat{i}}}{\Omega_{i\hat{i}}}\right)^{m_{i\hat{i}}}\frac{z^{m_{i\hat{i}}-1}}{\Gamma[m_{i\hat{i}}]}\mathrm{e}^{-\frac{{m_{i\hat{i}}}z}{\Omega_{i\hat{i}}}},\,z\geq
0$, and $W\triangleq P_{j}X_{i}+P_{\hat{j}}Y_{i}$ be a random variable whose
PDF can be expressed as,

$\displaystyle\\!\\!\\!f_{W}(w)$
$\displaystyle\\!\\!=\\!\\!\frac{\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)^{m_{ij}}\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)^{m_{i\hat{j}}}}{\Gamma[m_{ij}]\Gamma[m_{i\hat{j}}]}\\!\\!\sum_{k=0}^{m_{ij}\\!-\\!1}(\\!-\\!1)^{-k\\!+\\!m_{ij}\\!-\\!1}\binom{m_{ij}\\!-\\!1}{k}$
$\displaystyle\times\sum_{s=0}^{\infty}\frac{\left(\frac{m_{ij}}{\Omega_{ij}P_{j}}\right)^{s}}{s!}\left(\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}\right)^{-(-k+m_{ij}+m_{i\hat{j}}+s-1)}w^{k}$
$\displaystyle\times\mathrm{e}^{-\frac{m_{ij}}{\Omega_{ij}P_{j}}w}\Upsilon\left[-k+m_{ij}+m_{i\hat{j}}+s-1,\frac{m_{i\hat{j}}}{\Omega_{i\hat{j}}P_{\hat{j}}}w\right].$
(44)

From (12), the CDF $F_{\gamma_{i,\hat{i}}^{\text{lin}}}(\gamma_{\hat{i}})$ can
be formulated as

$\displaystyle F_{\gamma_{i,\hat{i}}^{\text{lin}}}(\gamma_{\hat{i}})$
$\displaystyle=\textmd{Pr}\left[\frac{\zeta_{i}WZ}{\xi_{i}Z+\sigma^{2}_{\hat{i}}}<\gamma_{\hat{i}},W\leq
P_{\text{th}}\right]$
$\displaystyle=\textmd{Pr}\left[W<\frac{\xi_{i}\gamma_{\hat{i}}Z+\gamma_{\hat{i}}\sigma_{\hat{i}}^{2}}{\zeta_{i}Z},W\leq
P_{\text{th}}\right].$ (45)

Following (B), the CDF
$F_{\gamma_{i,\hat{i}}^{\text{lin}}}({\gamma_{\hat{i}}})$ can be expressed in
integration form as

$\displaystyle F_{\gamma_{i,\hat{i}}^{\text{lin}}}(\gamma_{\hat{i}})$
$\displaystyle=\int^{\iota_{3}}_{z=0}\\!\\!\\!f_{Z}(z)\\!\\!\int^{P_{\text{th}}}_{w=0}\\!\\!f_{W}\\!(\\!w\\!)dw\,dz$
$\displaystyle+\int^{\infty}_{z=\iota_{3}}\\!\\!\\!f_{Z}(z)\\!\\!\int^{\frac{\xi_{i}\gamma_{\hat{i}}z+\gamma_{\hat{i}}\sigma_{\hat{i}}^{2}}{\zeta_{i}z}}_{w=0}\\!\\!f_{W}\\!(\\!w\\!)dw\,dz.$
(46)

The CDF of $F_{\gamma_{i,\hat{i}}^{\text{sat}}}(\gamma_{\hat{i}})$ can be
formulated using (13) as

$\displaystyle F_{\gamma_{i,\hat{i}}^{\text{sat}}}(\gamma_{\hat{i}})$
$\displaystyle=\textmd{Pr}\left[\frac{\Psi_{i}WZ}{\mathcal{C}_{i}|Z+\mathcal{D}_{\hat{i}}W}<\gamma_{\hat{i}},W>P_{\text{th}}\right]$
$\displaystyle=\textmd{Pr}\left[W<\frac{\mathcal{C}_{i}\gamma_{\hat{i}}Z}{\Psi_{i}Z-\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}},W>P_{\text{th}}\right]$
$\displaystyle=\textmd{Pr}\left[W<\frac{\mathcal{C}_{i}\gamma_{\hat{i}}Z}{\Psi_{i}Z-\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}},W>P_{\text{th}},Z>\frac{\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}{\Psi_{i}}\right]$
$\displaystyle+\textmd{Pr}\left[W>\frac{\mathcal{C}_{i}\gamma_{\hat{i}}Z}{\Psi_{i}Z-\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}},W>P_{\text{th}},Z<\frac{\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}{\Psi_{i}}\right].$
(47)

Following (B), the CDF
$F_{\gamma_{i,\hat{i}}^{\text{sat}}}({\gamma_{\hat{i}}})$ can be expressed in
integration form as

$\displaystyle F_{\gamma_{i,\hat{i}}^{\text{sat}}}(\gamma_{\hat{i}})$
$\displaystyle=\int^{\iota_{4}}_{z=\frac{\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}{\Psi_{i}}}\\!\\!\\!f_{Z}(z)\\!\\!\int^{\frac{\mathcal{C}_{i}\gamma_{\hat{i}}z}{\Psi_{i}z-\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}}_{w=P_{\text{th}}}\\!\\!f_{W}\\!(\\!w\\!)dw\,dz$
$\displaystyle+\int^{\frac{\mathcal{D}_{\hat{i}}\gamma_{\hat{i}}}{\Psi_{i}}}_{z=0}\\!\\!\\!f_{Z}(z)\\!\\!\int^{\infty}_{w=P_{\text{th}}}\\!\\!f_{W}\\!(\\!w\\!)dw\,dz.$
(48)

On substituting the respective PDFs and applying the required mathematical
formulations [38, eqs. 3.381.1, 3.471.9], (B) and (B) can be expressed as in
Lemma 2.

## Acknowledgment

The authors would like to thank the editor and anonymous reviewers for their
valuable comments and suggestions, which helped to improve the clarity and
quality of the paper.

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