# On the Secrecy Rate of Downlink NOMA in Underlay Spectrum Sharing with
Imperfect CSI

Vaibhav Kumar1, Mark F. Flanagan1, Daniel Benevides da Costa2, and Le-Nam
Tran1  1School of Electrical and Electronic Engineering, University College
Dublin, Belfield, Dublin 4, Ireland
2Future Technology Research Center, National Yunlin University of Science and
Technology, Douliu, Yunlin 64002, Taiwan
Email: {vaibhav.kumar, mark.flanagan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we present the ergodic sum secrecy rate (ESSR) analysis of an
underlay spectrum sharing non-orthogonal multiple access (NOMA) system. We
consider the scenario where the power transmitted by the secondary transmitter
(ST) is constrained by the peak tolerable interference at multiple primary
receivers (PRs) as well as the maximum transmit power of the ST. The effect of
channel estimation error is also taken into account in our analysis. We derive
exact and asymptotic closed-form expressions for the ESSR of the downlink NOMA
system, and show that the performance can be classified into two distinct
regimes, i.e., it is dictated either by the interference constraint or by the
power constraint. Our results confirm the superiority of the NOMA-based system
over its orthogonal multiple access (OMA) based counterpart. More
interestingly, our results show that NOMA helps in maintaining the secrecy
rate of the strong user while significantly enhancing the secrecy performance
of the weak user as compared to OMA. The correctness of the proposed
investigation is corroborated through Monte Carlo simulation.

††footnotetext: This publication has emanated from research conducted with the
financial support of Science Foundation Ireland (SFI) and is co-funded under
the European Regional Development Fund under Grant Number 17/CDA/4786.

##  I Introduction

Non-orthogonal multiple access (NOMA) has gained tremendous attention as a
potential multiple access technology for the next-generation wireless systems.
It is proven to be capable of providing massive connectivity, low latency and
higher achievable rate as compared to the traditional orthogonal multiple
access (OMA) system [1]. In order to serve multiple users using a given
resource block (time slot, frequency band, and/or spreading code), NOMA uses
power-division multiplexing at the transmitter’s side and successive
interference cancellation (SIC) at the receivers’ side. Underlay spectrum
sharing is another potential technology to mitigate the problem of spectrum
scarcity, where an unlicensed/secondary network simultaneously uses the
spectrum owned by a licensed/primary network in such a manner that the
interference inflicted by the secondary network on the primary one remains
below a certain threshold [2]. The benefits of underlay spectrum sharing NOMA
systems over their OMA-based counterparts were discussed in [3, 4, 5].

In today’s privacy-concerned society, the broadcast nature of radio waves
poses a significant risk that wireless communications may be intercepted by an
eavesdropper. Therefore, physical-layer security (PLS), which provides an
additional layer of security from an information-theoretic perspective, has
gained significant attention in the past couple of decades [6]. The secrecy
outage probability (SOP) analysis of a multiple-relay-assisted two-user
downlink NOMA system was presented in [7], where the authors proposed
different relay selection schemes to enhance the secrecy performance of the
system. The ergodic sum secrecy rate (ESSR) analysis of a two-user downlink
and uplink cooperative NOMA system with untrusted relaying was performed in
[8], while the authors in [9] presented the ergodic secrecy rate (ESR) and SOP
analysis of a full-duplex relay-assisted two-user downlink NOMA system with
energy harvesting.

On the other hand, some recent contributions on the secrecy analysis of
underlay spectrum sharing systems include [10, 11, 12]. Specifically, the
secrecy throughput and energy efficiency analysis of a spectrum sharing system
consisting of one primary source-destination pair, multiple secondary source-
destination pairs and an eavesdropper was presented in [10], where primary and
secondary networks either interfere or cooperate with each other in order to
improve the secrecy performance. In [11], the authors derived closed-form
expressions for the non-zero secrecy rate, ESR and SOP of an outage-
constrained spectrum sharing system with transmitter selection and unreliable
backhaul. The analysis of the achievable secrecy rate for an underlay spectrum
sharing multi-user massive multiple-input multiple-output (mMIMO) system was
presented in [12]. However, to the best of the authors’ knowledge, the secrecy
rate analysis of an underlay spectrum sharing system where the secondary
transmitter communicates with the secondary receivers using NOMA has not yet
been investigated in the literature. Therefore, in this paper, we present the
ESSR analysis of a two-user downlink NOMA system in an underlay spectrum
sharing scenario. The main contributions in this paper are listed below:

  * •

We derive an exact closed-form expression for the ESSR of a downlink NOMA
system in an underlay spectrum sharing scenario with multiple primary
receivers (PRs) considering the effect of imperfect channel state information
(CSI). We show the effect of the peak tolerable interference power at the PRs,
the maximum power budget at the secondary transmitter (ST), channel estimation
error and the number of PRs on the ESSR of the NOMA system.

  * •

We derive an asymptotic expression for the ESSR of the downlink NOMA system
for large values of peak tolerable interference at the PRs. The asymptotic
analysis confirms that the slope of the ESSR tends to zero (w.r.t. the peak
tolerable interference at the PRs), and the ESSR becomes independent of the
number of PRs as well as the quality of the channel between the ST and PRs.

  * •

Using the exact closed-form expressions for the ESR, we perform a one-
dimensional numerical search to allocate power between the NOMA users in such
a manner that the ESR of the strong user in the NOMA system and the
corresponding OMA system becomes equal, and then the remaining power is
allocated to the weak NOMA user. Such a power allocation policy ensures that
the secrecy performance of the strong user in the NOMA system is the same as
that of the strong user in the corresponding OMA system while the secrecy
performance of the weak user in the NOMA system is enhanced significantly as
compared to that in the OMA system, resulting in a performance superiority of
the NOMA system over its OMA-based counterpart in terms of ESSR.

##  II System Model

Figure 1: System model for underlay spectrum sharing NOMA with multiple
primary receivers.

Consider an underlay spectrum sharing system as shown in Fig. 1, consisting of
$M$ PRs $\left(\mathrm{PR_{1},PR_{2},\ldots,PR}_{M}\right)$, one ST, two111As
suggested by the 3GPP-LTE, one resource block should be allocated for two NOMA
users. Therefore, in this paper, we consider only two downlink NOMA users.
Similar assumptions were considered in [7, 8, 9]. However, our analysis can be
extended to multiple downlink NOMA users by adopting the user pairing scheme
proposed in [9]. secondary (NOMA) receivers, denoted by $\mathrm{U_{n}}$ (near
user) and $\mathrm{U_{f}}$ (far user), and one eavesdropper $\mathrm{E}$. It
is assumed that all nodes are single-antenna devices. The distances of
$\mathrm{PR}_{m}\left(m\in\left\\{1,2,\ldots,M\right\\}\right)$,
$\mathrm{U_{n}}$, $\mathrm{U_{f}}$ and $\mathrm{E}$ from the ST are denoted by
$d_{m},d_{\mathrm{n}},d_{\mathrm{f}}$ and $d_{\mathrm{e}}$, respectively. The
channel fading coefficient for ST-$\mathrm{PR}_{m}$, ST-$\mathrm{U_{n}}$,
ST-$\mathrm{U_{f}}$ and ST-$\mathrm{E}$ links are, respectively, given by
$h_{m}\sim\mathcal{CN}\left(0,\Omega_{m}=d_{m}^{-\alpha}\right)$,
$h_{\mathrm{n}}\sim\mathcal{CN}\left(0,\Omega_{\mathrm{n}}=d_{\mathrm{n}}^{-\alpha}\right)$,
$h_{\mathrm{f}}\sim\mathcal{CN}\left(0,\Omega_{\mathrm{f}}=d_{\mathrm{f}}^{-\alpha}\right)$
and
$h_{\mathrm{e}}\sim\mathcal{CN}\left(0,\Omega_{\mathrm{e}}=d_{\mathrm{e}}^{-\alpha}\right)$,
with $\alpha$ denoting the path loss exponent. The corresponding channel gain
is given by
$g_{i}\triangleq\left|h_{i}\right|^{2},i\in\left\\{1,2,\ldots,M,\mathrm{n,f,e}\right\\}$.
We assume that the ST has imperfect instantaneous CSI of the
ST-$\mathrm{PR}_{m}$, ST-$\mathrm{U_{n}}$, ST-$\mathrm{U_{f}}$ and
ST-$\mathrm{E}$ links. Let the estimate of $h_{i}$ be denoted by
$\tilde{h}_{i}$, and $h_{i}=\tilde{h}_{i}+\epsilon_{i}$, where
$\epsilon_{i}\sim\mathcal{CN}\left(0,\sigma_{\epsilon}^{2}\right)$ is the
channel estimation error. Therefore,
$\tilde{h}_{i}\sim\mathcal{CN}\left(0,\tilde{\Omega}_{i}\right)$, where
$\tilde{\Omega}_{i}=\Omega_{i}-\sigma_{\epsilon}^{2}$ and the corresponding
channel gain is denoted by $\tilde{g}_{i}$. The peak (instantaneous)
interference that the PRs can tolerate from the secondary network is denoted
by $I_{\mathrm{p}}$, and the maximum transmit power at the ST is denoted by
$P_{\max}$. Therefore, the instantaneous transmit power of the ST is given by
$P=\min\left\\{P_{\max},\tfrac{I_{\mathrm{p}}}{\tilde{g}_{1}},\tfrac{I_{\mathrm{p}}}{\tilde{g}_{2}},\ldots,\tfrac{I_{\mathrm{p}}}{\tilde{g}_{M}}\right\\}=\min\left\\{P_{\max},\tfrac{I_{\mathrm{p}}}{\tilde{g}_{\mathrm{p}}}\right\\}$,
where $\tilde{g}_{\mathrm{p}}\triangleq\max_{m}\tilde{g}_{m}$. We denote the
probability density function (PDF), cumulative distribution function (CDF) and
complementary CDF (CCDF) of a random variable $\mathcal{X}$ by
$f_{\mathcal{X}}\left(\cdot\right)$, $F_{\mathcal{X}}\left(\cdot\right)$ and
$\mathcal{F_{X}}\left(\cdot\right)$, respectively, and we define
$\mathscr{F}_{\mathcal{X}}\left(c_{1},c_{2}\right)\triangleq\int_{c_{1}}^{c_{2}}f_{\mathcal{X}}\left(t\right)\mathrm{d}t$.

##  III Analysis of the ESSR

Based on the estimated CSI at the ST, the users $\mathrm{U_{n}}$ and
$\mathrm{U_{f}}$ are further categorized as $\mathrm{U_{s}}$ (strong user) and
$\mathrm{U_{w}}$ (weak user), where
$\mathrm{s}\triangleq\mathrm{argmax}_{j\in\left\\{\mathrm{n,f}\right\\}}\
\tilde{g}_{j}$ and
$\mathrm{w}\triangleq\mathrm{argmin}_{j\in\left\\{\mathrm{n,f}\right\\}}\
\tilde{g}_{j}$. The signal received at
$\mathrm{U}_{k},k\in\left\\{\mathrm{s},\mathrm{w}\right\\}$ is given by

$y_{k}=\left(\tilde{h}_{k}+\epsilon_{k}\right)\left(\sqrt{a_{\mathrm{s}}P}x_{\mathrm{s}}+\sqrt{a_{\mathrm{w}}P}x_{\mathrm{w}}\right)+n_{k},$

where $x_{k}$ is the unit-energy information-bearing complex constellation
symbol and $n_{k}\sim\mathcal{CN}\left(0,1\right)$ is the additive white
Gaussian noise (AWGN) at node $\mathrm{U}_{k}$. Following the NOMA principle,
it is assumed that $a_{\mathrm{s}}<a_{\mathrm{w}}$ and
$a_{\mathrm{s}}+a_{\mathrm{w}}\mathrm{=1}$. The weak user decodes
$x_{\mathrm{w}}$ by treating the interference due to $x_{\mathrm{s}}$ as
additional noise, whereas the strong user first decodes $x_{\mathrm{w}}$ by
treating the interference due to $x_{\mathrm{s}}$ as additional noise and then
applies successive interference cancellation (SIC) to decode $x_{\mathrm{s}}$.
Therefore, the signal-to-interference-plus-noise ratio (SINR) at
$\mathrm{U_{s}}$ and $\mathrm{U_{w}}$ are, respectively, given by

$\gamma_{\mathrm{s}}=\dfrac{a_{\mathrm{s}}\tilde{g}_{\mathrm{s}}P}{1+\sigma_{\epsilon}^{2}P},\qquad\gamma_{\mathrm{w}}=\dfrac{a_{\mathrm{w}}\tilde{g}_{\mathrm{w}}P}{1+a_{\mathrm{s}}\tilde{g}_{\mathrm{w}}P+\sigma_{\epsilon}^{2}P}.$

Following [13, Ch. 15], we consider the case that $\mathrm{E}$ follows the
same decoding order as that of the legitimate users. Therefore, the SINR at
$\mathrm{E}$ for decoding $x_{\mathrm{w}}$ and $x_{\mathrm{s}}$ are
respectively given by

$\gamma_{\mathrm{e,w}}=\dfrac{a_{\mathrm{w}}\tilde{g}_{\mathrm{e}}P}{1+a_{\mathrm{s}}\tilde{g}_{\mathrm{e}}P+\sigma_{\epsilon}^{2}P},\qquad\gamma_{\mathrm{e,s}}=\dfrac{a_{\mathrm{s}}\tilde{g}_{\mathrm{e}}P}{1+\sigma_{\epsilon}^{2}P}.$

Let
$\hat{g}_{\mathrm{s}}\triangleq\sigma_{\epsilon}^{2}+a_{\mathrm{s}}\tilde{g}_{\mathrm{s}}$,
$\hat{g}_{\mathrm{w}}\triangleq\sigma_{\epsilon}^{2}+a_{\mathrm{s}}\tilde{g}_{\mathrm{w}}$,
$\check{g}_{\mathrm{w}}\triangleq\sigma_{\epsilon}^{2}+\tilde{g}_{\mathrm{w}}$
and
$\hat{g}_{\mathrm{e}}\triangleq\sigma_{\epsilon}^{2}+a_{\mathrm{s}}\tilde{g}_{\mathrm{e}}$.
Then using the standard statistical procedure of transformation of random
variables, it follows that

$f_{\hat{g}_{\mathrm{s}}}\\!\left(x\right)\\!=\\!\\!\\!\sum_{\ell\in\left\\{1,2,3\right\\}}\\!\\!\\!\\!\\!\\!\hat{A}_{\ell}\exp\\!\left(\\!\dfrac{-x}{a_{\mathrm{s}}\Xi_{\ell}}\\!\right),\mathcal{F}_{\hat{g}_{\mathrm{s}}}\\!\left(x\right)\\!=\\!\\!\\!\sum_{\ell\in\left\\{1,2,3\right\\}}\\!\\!\\!\\!\\!\\!\hat{\mathcal{A}}_{\ell}\exp\\!\left(\\!\dfrac{-x}{a_{\mathrm{s}}\Xi_{\ell}}\\!\right),$

$f_{\hat{g}_{\mathrm{w}}}\left(x\right)=-\hat{A}_{2}\exp\left(\dfrac{-x}{a_{\mathrm{s}}\Xi_{2}}\right),\mathcal{F}_{\hat{g}_{\mathrm{w}}}\left(x\right)=-\hat{\mathcal{A}}_{2}\exp\left(\dfrac{-x}{a_{\mathrm{s}}\Xi_{2}}\right),$
$f_{\check{g}_{\mathrm{w}}}\left(x\right)=\check{A}_{2}\exp\left(\dfrac{-x}{\Xi_{2}}\right),\mathcal{F}_{\check{g}_{\mathrm{w}}}\left(x\right)=\check{\mathcal{A}}_{2}\exp\left(\dfrac{-x}{\Xi_{2}}\right),$

$f_{\hat{g}_{\mathrm{e}}}\\!\left(x\right)\\!=\\!\hat{C}_{\mathrm{e}}\exp\left(\\!\dfrac{-x}{a_{\mathrm{s}}\tilde{\Omega}_{\mathrm{e}}}\\!\right),\mathscr{F}_{\hat{g}_{\mathrm{e}}}\\!\left(\sigma_{\epsilon}^{2},x\right)\\!=1-\hat{\mathscr{C}}_{\mathrm{e}}\exp\left(\\!\dfrac{-x}{a_{\mathrm{s}}\tilde{\Omega}_{\mathrm{e}}}\\!\right),$
$\mathscr{F}_{\check{g}_{\mathrm{e}}}\left(\sigma_{\epsilon}^{2},x\right)=1-\check{\mathscr{C}}_{\mathrm{e}}\exp\left(\dfrac{-x}{\tilde{\Omega}_{\mathrm{e}}}\right),$

where $\Xi_{1}=\tilde{\Omega}_{\mathrm{n}}$,
$\Xi_{2}=\tilde{\Omega}_{\mathrm{n,f}}$,
$\Xi_{3}=\tilde{\Omega}_{\mathrm{f}}$,
$\tilde{\Omega}_{\mathrm{n,f}}=\left(\tilde{\Omega}_{\mathrm{n}}^{-1}+\tilde{\Omega}_{\mathrm{f}}^{-1}\right)^{-1}$,
$\hat{A}_{\ell}=\tfrac{(-1)^{\ell+1}}{a_{\mathrm{s}}\Xi_{\ell}}\exp\left(\tfrac{\sigma_{\epsilon}^{2}}{a_{\mathrm{s}}\Xi_{\ell}}\right)$,
$\hat{\mathcal{A}}_{\ell}=a_{\mathrm{s}}\Xi_{\ell}\hat{A}_{\ell}$,
$\check{A}_{2}=\tfrac{1}{\Xi_{2}}\exp\left(\tfrac{\sigma_{\epsilon}^{2}}{\Xi_{2}}\right)$,
$\check{\mathcal{A}}_{2}=\Xi_{2}\check{A}_{2}$,
$\hat{C}_{\mathrm{e}}=\tfrac{1}{a_{\mathrm{s}}\tilde{\Omega}_{\mathrm{e}}}\exp\left(\tfrac{\sigma_{\epsilon}^{2}}{a_{\mathrm{s}}\tilde{\Omega}_{\mathrm{e}}}\right)$
and
$\hat{\mathscr{C}}_{\mathrm{e}}=a_{\mathrm{s}}\tilde{\Omega}_{\mathrm{e}}\hat{C}_{\mathrm{e}}$.
From [14], the PDF of $\tilde{g}_{\mathrm{p}}$ is given by

$f_{\tilde{g}_{\mathrm{p}}}\left(x\right)=-\sum_{\boldsymbol{\eta}\in\Phi}\kappa_{\boldsymbol{\eta}}B_{\boldsymbol{\eta}}\exp\left(-B_{\boldsymbol{\eta}}x\right),$

where
$\Phi=\left\\{\boldsymbol{\eta}=[\eta_{1}\eta_{2}\cdots\eta_{M}]\in\mathbb{Z}_{2}^{M}\left|\sum\nolimits_{m=1}^{M}\eta_{m}>0\right.\right\\}$
represents the set of all nonzero binary vectors of length $M$,
$\mathbb{Z}_{2}=\left\\{0,1\right\\}$,
$\kappa_{\boldsymbol{\eta}}=\prod_{m=1}^{M}(-1)^{\eta_{m}}$ and
$B_{\boldsymbol{\eta}}=\sum_{m=1}^{M}\left(\eta_{m}/\tilde{\Omega}_{m}\right)$.

Finally, the ESR of the strong user can be written as

$\displaystyle\bar{R}_{\mathrm{s}}$
$\displaystyle=\mathbb{E}\left\\{\max\left\\{\log_{2}\left(1+\gamma_{\mathrm{s}}\right)-\log_{2}\left(1+\gamma_{\mathrm{e,s}}\right),0\right\\}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\max\left\\{\log_{2}\left(1+\hat{g}_{\mathrm{s}}P\right)-\log_{2}\left(1+\hat{g}_{\mathrm{e}}P\right),0\right\\}\right\\}.$
(1)

######  Theorem 1.

A closed-form expression for the ESR of $\mathrm{U_{s}}$ can be derived as

$\bar{R}_{\mathrm{s}}=\mathscr{I}_{\mathrm{s,1}}+\mathscr{I}_{\mathrm{s,2}}-\mathscr{I}_{\mathrm{s,3}}-\mathscr{I}_{\mathrm{s,4}},$
(2)

where the expressions for
$\mathscr{I}_{\mathrm{s,1}},\mathscr{I}_{\mathrm{s,2}},\mathscr{I}_{\mathrm{s,3}}$
and $\mathscr{I}_{\mathrm{s,4}}$ are given by (3) – (6), shown at the top of
the next page, $\Theta\triangleq I_{\mathrm{p}}/P_{\max}$,
$\Xi_{\ell,e}\triangleq\left(\Xi_{\ell}^{-1}+\tilde{\Omega}^{-1}\right)^{-1}$,
and $\mathrm{Ei}\left(\cdot\right)$ and $\mathrm{E_{1}}\left(\cdot\right)$ are
the exponential integrals.

$\mathscr{I}_{\mathrm{s,1}}=-\log_{2}\left(e\right)a_{\mathrm{s}}\text{$\left[\sum_{\ell\in\left\\{1,2,3\right\\}}\hat{A}_{\ell}\left\\{\Xi_{\ell}\mathbb{J}_{1}\left(a_{\mathrm{s}}\Xi_{\ell}\right)-\hat{\mathscr{C}}_{\mathrm{e}}\Xi_{\ell,\mathrm{e}}\mathbb{J}_{1}\left(a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}\right)\right\\}\right]$}\left[\sum_{\boldsymbol{\eta}\in\Phi}\kappa_{\boldsymbol{\eta}}\left\\{1-\exp\left(-B_{\boldsymbol{\eta}}\Theta\right)\right\\}\right],$
(3)

$\mathscr{I}_{\mathrm{s,2}}=-\log_{2}\left(e\right)a_{\mathrm{s}}\sum_{\ell\in\left\\{1,2,3\right\\}}\sum_{\boldsymbol{\eta}\in\Phi}\hat{A}_{\ell}\kappa_{\eta}\left[\Xi_{\ell}\mathbb{J}_{2}\left(a_{\mathrm{s}}\Xi_{\ell}\right)-\hat{\mathscr{C}}_{\mathrm{e}}\Xi_{\ell,\mathrm{e}}\mathbb{J}_{2}\left(a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}\right)\right],$
(4)

$\mathscr{I}_{\mathrm{s,3}}=-\log_{2}\left(e\right)a_{\mathrm{s}}\hat{C}_{\mathrm{e}}\left[\sum_{\ell\in\left\\{1,2,3\right\\}}\hat{\mathcal{A}}_{\ell}\Xi_{\ell,\mathrm{e}}\mathbb{J}_{1}\left(a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}\right)\right]\left[\sum_{\boldsymbol{\eta}\in\Phi}\kappa_{\boldsymbol{\eta}}\left\\{1-\exp\left(-B_{\boldsymbol{\eta}}\Theta\right)\right\\}\right],$
(5)

$\mathscr{I}_{\mathrm{s,4}}=-\log_{2}\left(e\right)a_{\mathrm{s}}\hat{C}_{\mathrm{e}}\sum_{\ell\in\left\\{1,2,3\right\\}}\sum_{\boldsymbol{\eta}\in\Phi}\hat{\mathcal{A}}_{\ell}\kappa_{\boldsymbol{\eta}}\Xi_{\ell,2}\mathbb{J}_{2}\left(a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}\right),$
(6)

$\mathbb{J}_{1}\left(\Omega\right)=\exp\left(\dfrac{-\sigma_{\epsilon}^{2}}{\Omega}\right)\ln\left(1+\sigma_{\epsilon}^{2}P_{\max}\right)+\exp\left(\dfrac{1}{\Omega
P_{\max}}\right)\mathrm{E}_{1}\left[\dfrac{1}{\Omega}\left(\dfrac{1}{P_{\max}}+\sigma_{\epsilon}^{2}\right)\right],$
(7)

$\displaystyle\mathbb{J}_{2}\left(\Omega\right)\triangleq$
$\displaystyle\exp\left(\dfrac{-\sigma_{\epsilon}^{2}}{\Omega}\right)\mathrm{Ei}\left(-B_{\boldsymbol{\eta}}\Theta\right)-\dfrac{B_{\boldsymbol{\eta}}^{-1}}{B_{\boldsymbol{\eta}}^{-1}-I_{\mathrm{p}}\Omega}\left\\{\exp\left(I_{\mathrm{p}}\sigma_{\epsilon}^{2}B_{\boldsymbol{\eta}}-\dfrac{\sigma_{\epsilon}^{2}}{\Omega}\right)\mathrm{Ei}\left(-B_{\boldsymbol{\eta}}\left(\Theta+I_{\mathrm{p}}\sigma_{\epsilon}^{2}\right)\right)-\exp\left(\dfrac{\Theta}{I_{\mathrm{p}}\Omega}-B_{\boldsymbol{\eta}}\Theta\right)\right.$
$\displaystyle\left.\times\mathrm{Ei}\left(\dfrac{-\Theta-
I_{\mathrm{p}}\sigma_{\epsilon}^{2}}{I_{\mathrm{p}}\Omega}\right)\right\\}+\exp\left(-B_{\boldsymbol{\eta}}\Theta-\dfrac{\sigma_{\epsilon}^{2}}{\Omega}\right)\left\\{\ln\left(1+\dfrac{I_{\mathrm{p}}\sigma_{\epsilon}^{2}}{\Theta}\right)+\exp\left(\dfrac{\Theta+I_{\mathrm{p}}\sigma_{\epsilon}^{2}}{I_{\mathrm{p}}\Omega}\right)\mathrm{E}_{1}\left(\dfrac{\Theta+I_{\mathrm{p}}\sigma_{\epsilon}^{2}}{I_{\mathrm{p}}\Omega}\right)\right\\}.$
(8)

###### Proof:

See Appendix A. ∎

On the other hand, the ESR for the weak user $\mathrm{U_{w}}$ can be written
as

$\displaystyle\bar{R}_{\mathrm{w}}=\mathbb{E}\left\\{\max\left\\{\log_{2}\left(1+\gamma_{\mathrm{w}}\right)-\log_{2}\left(1+\gamma_{\mathrm{e,w}}\right),0\right\\}\right\\}$
$\displaystyle=\mathbb{E}\left\\{\max\left\\{\log_{2}\left(\dfrac{1+\left(\tilde{g}_{{}_{\mathrm{w}}}+\sigma_{\epsilon}^{2}\right)P}{1+\left(a_{\mathrm{s}}\tilde{g}_{\mathrm{{}_{\mathrm{w}}}}+\sigma_{\epsilon}^{2}\right)P}\right)\right.\right.$
$\displaystyle\qquad\qquad\qquad-\left.\left.\log_{2}\left(\dfrac{1+\left(\tilde{g}_{{}_{\mathrm{e}}}+\sigma_{\epsilon}^{2}\right)P}{1+\left(a_{\mathrm{s}}\tilde{g}_{{}_{\mathrm{e}}}+\sigma_{\epsilon}^{2}\right)P}\right),0\right\\}\right\\}.$
(9)

######  Theorem 2.

A closed-form expression for the ESR of $\mathrm{U_{w}}$ can be derived as

$\bar{R}_{\mathrm{w}}=\mathscr{I}_{\mathrm{w,1}}+\mathscr{I}_{\mathrm{w,2}}-\mathscr{I}_{\mathrm{w,3}}-\mathscr{I}_{\mathrm{w,4}},$
(10)

where the expressions for
$\mathscr{I}_{\mathrm{w,1}},\mathscr{I}_{\mathrm{w,2}},\mathscr{I}_{\mathrm{w,3}}$
and $\mathscr{I}_{\mathrm{w,4}}$ are given by (11) – (14), shown on the next
page.

$\mathscr{I}_{\mathrm{w,1}}\\!=\\!-\log_{2}\left(e\right)\\!\sum_{\boldsymbol{\eta}\in\Phi}\\!\kappa_{\boldsymbol{\eta}}\left\\{1\\!-\\!\exp\left(\\!-B_{\boldsymbol{\eta}}\Theta\right)\right\\}\\!\\!\left[\\!\check{A}_{2}\Xi_{2}\mathbb{J}_{1}\\!\left(\Xi_{2}\right)\\!-\\!\check{A}_{2}\check{\mathscr{C}}_{\mathrm{e}}\Xi_{2,\mathrm{e}}\mathbb{J}_{1}\\!\left(\Xi_{2,\mathrm{e}}\right)\\!+\\!\hat{A}_{2}a_{\mathrm{s}}\Xi_{2}\mathbb{J}_{1}\\!\left(a_{\mathrm{s}}\Xi_{2}\right)\\!-\\!\hat{A}_{2}\hat{\mathscr{C}}_{\mathrm{e}}a_{\mathrm{s}}\Xi_{2,\mathrm{e}}\mathbb{J}_{1}\\!\left(a_{\mathrm{s}}\Xi_{2,\mathrm{e}}\right)\right],$
(11)

$\mathscr{I}_{\mathrm{w,2}}=\\!-\log_{2}\left(e\right)\sum_{\boldsymbol{\eta}\in\Phi}\kappa_{\boldsymbol{\eta}}\left[\check{A}_{2}\Xi_{2}\mathbb{J}_{2}\left(\Xi_{2}\right)\\!-\\!\check{A}_{2}\check{\mathscr{C}}_{\mathrm{e}}\Xi_{2,\mathrm{e}}\mathbb{J}_{2}\left(\Xi_{2,\mathrm{e}}\right)+\hat{A}_{2}a_{\mathrm{s}}\Xi_{2}\mathbb{J}_{2}\left(a_{\mathrm{s}}\Xi_{2}\right)-\hat{A}_{2}\hat{\mathscr{C}}_{\mathrm{e}}a_{\mathrm{s}}\Xi_{2,\mathrm{e}}\mathbb{J}_{2}\left(a_{\mathrm{s}}\Xi_{2,\mathrm{e}}\right)\right],$
(12)

$\mathscr{I}_{\mathrm{w,3}}=-\log_{2}\left(e\right)\sum_{\boldsymbol{\eta}\in\Phi}\kappa_{\boldsymbol{\eta}}\left\\{1-\exp\left(-B_{\boldsymbol{\eta}}\Theta\right)\right\\}\left[\check{\mathcal{A}}_{2}\check{C}_{\mathrm{e}}\Xi_{2,\mathrm{e}}\mathbb{J}_{1}\left(\Xi_{2,\mathrm{e}}\right)+\hat{\mathcal{A}}_{2}\hat{C}_{\mathrm{e}}a_{\mathrm{s}}\Xi_{2,\mathrm{e}}\mathbb{J}_{1}\left(a_{\mathrm{s}}\Xi_{2,\mathrm{e}}\right)\right],$
(13)

$\mathscr{I}_{\mathrm{w,4}}=-\log_{2}\left(e\right)\sum_{\boldsymbol{\eta}\in\Phi}\kappa_{\boldsymbol{\eta}}\left[\check{\mathcal{A}}_{2}\check{C}_{\mathrm{e}}\Xi_{2,\mathrm{e}}\mathbb{J}_{2}\left(\Xi_{2,\mathrm{e}}\right)+a_{\mathrm{s}}\hat{\mathcal{A}}_{2}\hat{C}_{\mathrm{e}}\Xi_{2,\mathrm{e}}\mathbb{J}_{2}\left(a_{\mathrm{s}}\Xi_{2,\mathrm{e}}\right)\right].$
(14)

###### Proof:

See Appendix B. ∎

Using (2) and (10), a closed-form expression for the ESSR of the NOMA system
is given by
$\bar{R}_{\mathrm{sum}}=\bar{R}_{\mathrm{s}}+\bar{R}_{\mathrm{w}}$.

### ESSR for OMA

For the case of an underlay spectrum sharing OMA system, the ST transmits
$\sqrt{P}x_{\mathrm{s}}$ to $\mathrm{U_{s}}$ in the first time slot, and
$\sqrt{P}x_{\mathrm{w}}$ to $\mathrm{U_{w}}$ in the second time slot. For a
fair comparison between the NOMA and OMA systems, we consider the same value
of $I_{\mathrm{p}}$ for both systems. Therefore, the ESSR for the OMA system
is given by

$\displaystyle\bar{R}_{\mathrm{sum,OMA}}=\bar{R}_{\mathrm{s,OMA}}+\bar{R}_{\mathrm{w,OMA}}$
$\displaystyle=\sum_{k\in\left\\{\mathrm{s,w}\right\\}}\mathbb{E}\left\\{0.5\max\left\\{\log_{2}\left(\dfrac{1+\tilde{g}_{k}P}{1+\tilde{g}_{\mathrm{e}}P}\right),0\right\\}\right\\}.$
(15)

As the focus of this paper is on the NOMA-based system, we do not provide the
closed-form expression for $\bar{R}_{\mathrm{sum,OMA}}$. Next, we carry out
the asymptotic analysis $\left(I_{\mathrm{p}}\to\infty\right)$ of the ESSR for
the NOMA system.

##  IV Asymptotic ESSR for the NOMA System

######  Theorem 3.

An analytical expression for the asymptotic
$\left(I_{\mathrm{p}}\to\infty\right)$ ESSR for the NOMA system can be derived
as

$\bar{R}_{\mathrm{sum,asymp}}=\mathcal{I}_{\mathrm{s}}+\mathcal{I}_{\mathrm{w}},$
(16)

where the closed-form expressions for $\mathcal{I}_{\mathrm{s}}$ and
$\mathcal{I}_{\mathrm{w}}$ are given by (17) and (18), respectively, shown on
the next page.

$\displaystyle\mathcal{I}_{\mathrm{s}}=\log_{2}\left(e\right)a_{\mathrm{s}}\sum_{\ell\in\left\\{1,2,3\right\\}}\left[\hat{A}_{\ell}\Xi_{\ell}\mathbb{J}_{1}\left(a_{\mathrm{s}}\Xi_{\ell}\right)-\hat{A}_{\ell}\hat{\mathscr{C}}_{\mathrm{e}}\Xi_{\ell,\mathrm{e}}\mathbb{J}_{1}\left(a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}\right)-\hat{\mathcal{A}}_{\ell}\hat{C}_{\mathrm{e}}\Xi_{\ell,\mathrm{e}}\mathbb{J}_{1}\left(a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}\right)\right],$
(17)

$\displaystyle\mathcal{I}_{\mathrm{w}}=\log_{2}\left(e\right)\left[\check{A}_{2}\Xi_{2}\mathbb{J}_{1}\\!\left(\Xi_{2}\right)\\!-\\!\check{A}_{2}\check{\mathscr{C}}_{\mathrm{e}}\Xi_{2,\mathrm{e}}\mathbb{J}_{1}\\!\left(\Xi_{2,\mathrm{e}}\right)\\!+\\!\hat{A}_{2}a_{\mathrm{s}}\Xi_{2}\mathbb{J}_{1}\\!\left(a_{\mathrm{s}}\Xi_{2}\right)\\!-\\!\hat{A}_{2}\hat{\mathscr{C}}_{\mathrm{e}}a_{\mathrm{s}}\Xi_{2,\mathrm{e}}\mathbb{J}_{1}\\!\left(a_{\mathrm{s}}\Xi_{2,\mathrm{e}}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\left.\check{\mathcal{A}}_{2}\check{C}_{\mathrm{e}}\Xi_{2,\mathrm{e}}\mathbb{J}_{1}\left(\Xi_{2,\mathrm{e}}\right)-\hat{\mathcal{A}}_{2}\hat{C}_{\mathrm{e}}a_{\mathrm{s}}\Xi_{2,\mathrm{e}}\mathbb{J}_{1}\left(a_{\mathrm{s}}\Xi_{2,\mathrm{e}}\right)\right].$
(18)

###### Proof:

Note that for $I_{\mathrm{p}}\to\infty$ and finite $P_{\max}$, we have
$\Theta=\tfrac{I_{\mathrm{P}}}{P_{\max}}\to\infty$. Letting $\Theta\to\infty$
in (19) and (25), and then solving the integrals following a similar line of
argument as given in Appendix A, a closed-form expression for
$\bar{R}_{\mathrm{sum,asymp}}$ is given by (16). This concludes the proof. ∎

It can be noted from (16) – (18) that for large values of $I_{\mathrm{p}}$,
the ESSR of the NOMA is independent of the number of PRs as well as the
quality of the link between the ST and PRs. Also, the slope of the ESSR w.r.t
$I_{\mathrm{p}}$ becomes equal to zero for $I_{\mathrm{p}}\to\infty$.

Figure 2: Comparison of ESR for NOMA and OMA systems for $P_{\max}=50$ dB.
Figure 3: Comparison of ESSR for NOMA and OMA systems. The number in the
parentheses denotes $P_{\max}$ in dB. Figure 4: Effect of channel estimation
error on the ESSR. The numbers in the parentheses denote
$\left(I_{\mathrm{p}},P_{\max}\right)$ in dB. Figure 5: Effect of the number
of PRs on the ESSR of the NOMA system. The number in the parentheses denotes
$P_{\max}$ in dB.

##  V Results and Discussions

In this section, we present the numerical and analytical results for the ESSR
of NOMA and OMA systems. We consider a system with $d_{\mathrm{n}}=30$ m,
$d_{\mathrm{f}}=100$ m, $d_{\mathrm{e}}=150$ m,
$\left\\{d_{1},d_{2},d_{3},d_{4}\right\\}=\left\\{200,205,210,215\right\\}$ m,
$\sigma_{\epsilon}^{2}=2\times 10^{-5}$, and $\alpha=2$, unless stated
otherwise. In the case of NOMA, we use the bisection method to find the value
of $a_{\mathrm{s}}$ such that $\bar{R}_{\mathrm{s}}=\bar{R}_{\mathrm{s,OMA}}$,
and then the remaining power fraction $\left(a_{\mathrm{w}}\right)$ is
allocated to the weak user. In each figure, the legend indicates the
numerically obtained results, while the solid curves indicate the analytical
results.

Fig. 2 shows a performance comparison of the ESR of the NOMA and OMA systems.
It can be observed from the figure that the ESR of $\mathrm{U_{s}}$ in the
NOMA system is equal to that in the case of the OMA system (by virtue of the
power allocation scheme), whereas the ESR of $\mathrm{U_{w}}$ in the NOMA
system is higher that that of the OMA system. This result confirms that NOMA
helps in maintaining the secrecy rate of the strong user while significantly
enhancing the secrecy performance of the weak user as compared to OMA. We also
show the asymptotic $\left(I_{\mathrm{p}}\to\infty\right)$ ESR for the NOMA
system, which matches well with the exact ESR for large $I_{\mathrm{p}}$.
Moreover, a close agreement between the numerical and analytical results (for
the case of NOMA) verifies the correctness of the derived closed-form
expressions.

Fig. 3 shows a comparison of the ESSR for the NOMA and OMA systems. Note that
the NOMA system outperforms its OMA-based counterpart by achieving a
significantly higher ESSR. The ESSR for the NOMA/OMA system first increases
with an increase of the value of $I_{\mathrm{p}}$ (which we refer to the
interference-constrained regime) and then saturates for larger value of
$I_{\mathrm{p}}$ (which we refer to as the power-constrained regime).
Interestingly, in the interference-constrained regime, the ESSR of the NOMA
system remains the same irrespective of the value of $P_{\max}$, whereas, in
the power-constrained regime, the ESSR is independent of the value of
$I_{\mathrm{p}}$. A similar behavior is evident for the OMA system. It is
noteworthy that for the system parameters considered in Fig. 3,
$\min\left(P_{\max},\tfrac{I_{\mathrm{p}}}{\mathbb{E}\left\\{\tilde{g}_{\mathrm{p}}\right\\}}\right)=\min\left(P_{\max},\tfrac{I_{\mathrm{p}}}{\tilde{\Omega}_{\mathrm{p}}}\right)$
becomes constant $\left(=P_{\max}\right)$ w.r.t $I_{p}$ at
$I_{\mathrm{p}}\gtrsim 0$ dB for $P_{\max}=40$ dB, at $I_{\mathrm{p}}\gtrsim
10$ dB for $P_{\max}=50$ dB, and at $I_{\mathrm{p}}\gtrsim 20$ dB for
$P_{\max}=60$ dB. Therefore, the ESSR for both the NOMA and OMA systems
becomes constant w.r.t. $I_{\mathrm{p}}$ (power-constrained regime) at
$I_{\mathrm{p}}\gtrsim\left\\{0,10,20\right\\}$ dB for
$P_{\max}=\left\\{40,50,60\right\\}$ dB.

In Fig. 4, we investigate the effect of channel estimation error on the ESSR
of the NOMA and OMA systems. It can be noted from the figure that the ESSR for
both the NOMA and OMA systems decreases with an increase in the value of
$\sigma_{\epsilon}^{2}$. This implies that a decrease in the accuracy of
channel estimation has an adverse effect on the system performance. The effect
of the interference-constrained regime is also clearly evident from the
figure, as the value of ESSR at $I_{\mathrm{p}}=0$ dB is the same for
$P_{\max}=50$ dB and $P_{\max}=60$ dB. However, for a large value of
$I_{\mathrm{p}}$ (10 dB), the benefit of a larger $P_{\max}$ can be noted from
the figure.

Fig. 5 shows the effect of the number and distance of PRs on the ESSR of the
NOMA system. In this figure, three different cases are studied. In Case I, we
consider 2 PRs both located at a distance of 200 m from the ST; in Case II, we
consider 4 PRs located at $\left\\{200,205,210,215\right\\}$ m from the ST,
and in Case III, we consider 10 PRs, all located at a distance of 200 m.
Moreover, for all of the cases, we consider two different values of transmit
power: $P_{\max}=40$ dB and 60 dB. It is evident from the figure that in the
interference-constrained regime (i.e., for small values of $I_{\mathrm{p}}$),
an increase in the number of PRs results in a reduction of the ESSR. However,
in this regime, no benefit is observed from a higher $P_{\max}$ budget. On the
other hand, in the power-constrained regime (i.e., for large values of
$I_{p}$), the ESSR for all the cases becomes the same, irrespective of the
number/location of PRs. This occurs due to the reason that in the power-
constrained regime, the secrecy performance becomes independent of the ST-PR
link quality as well as the interference constraint at the PRs. However, a
larger $P_{\max}$ turns out to be clearly beneficial in this regime.

##  VI Conclusion

In this paper, we have presented the ergodic secrecy rate analysis of a two-
user downlink NOMA system in an underlay spectrum sharing scenario consisting
of multiple primary-user receivers in the presence of channel estimation
error. Exact and asymptotic closed-form expressions for the ESSR of the NOMA
system were derived. Our results confirmed that the NOMA system outperforms
its OMA-based counterpart in terms of the ESSR. More interestingly, the
results confirmed that no benefit is obtained in terms of the ESSR from a
higher power budget at the ST in the interference-constrained regime, whereas,
in the power-constrained regime, the ESSR becomes independent of the number of
PRs as well as the quality of the ST-PR links. The results also showed that a
larger channel estimation error results in a reduced secrecy rate. Finally,
the asymptotic analysis demonstrated that the ESSR of the NOMA system becomes
independent of the value of peak tolerable interference and therefore, the
slope of the ESSR w.r.t. the peak tolerable interference becomes zero.

##  Appendix A Proof of Theorem 1

Using (1), we have

$\displaystyle\bar{R}_{\mathrm{s}}=\mathbb{E}\left\\{\max\left\\{\log_{2}\left(1+\hat{g}_{\mathrm{s}}P\right)-\log_{2}\left(1+\hat{g}_{\mathrm{e}}P\right),0\right\\}\right\\}$
$\displaystyle=$
$\displaystyle\int_{y=0}^{\Theta}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\\!\log_{2}\left(1\\!+\\!P_{\max}x\right)f_{\hat{g}_{\mathrm{s}}}\\!\left(x\right)\mathscr{F}_{\hat{g}_{\mathrm{e}}}\\!\left(\sigma_{\epsilon}^{2},x\right)f_{\tilde{g}_{\mathrm{p}}}\\!\left(y\right)\mathrm{d}x\mathrm{d}y$
$\displaystyle+$
$\displaystyle\int_{y=\Theta}^{\infty}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\\!\log_{2}\left(1\\!+\\!I_{\mathrm{p}}\dfrac{x}{y}\right)f_{\hat{g}_{\mathrm{s}}}\\!\left(x\right)\mathscr{F}_{\hat{g}_{\mathrm{e}}}\\!\left(\sigma_{\epsilon}^{2},x\right)f_{\tilde{g}_{\mathrm{p}}}\\!\left(y\right)\mathrm{d}x\mathrm{d}y$
$\displaystyle-$
$\displaystyle\int_{y=0}^{\Theta}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\log_{2}\left(1+P_{\max}x\right)f_{\hat{g}_{\mathrm{e}}}\left(x\right)\mathcal{F}_{\hat{g}_{\mathrm{s}}}\left(x\right)f_{\tilde{g}_{\mathrm{p}}}\left(y\right)\mathrm{d}x\mathrm{d}y$
$\displaystyle-$
$\displaystyle\int_{y=\Theta}^{\infty}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\log_{2}\left(1+I_{\mathrm{p}}\dfrac{x}{y}\right)f_{\hat{g}_{\mathrm{e}}}\left(x\right)\mathcal{F}_{\hat{g}_{\mathrm{s}}}\left(x\right)f_{\tilde{g}_{\mathrm{p}}}\left(y\right)\mathrm{d}x\mathrm{d}y$
$\displaystyle\triangleq$
$\displaystyle\mathscr{I}_{\mathrm{s,1}}+\mathscr{I}_{s,2}-\mathscr{I}_{s,3}-\mathscr{I}_{s,4}.$
(19)

Solving for $\mathscr{I}_{\mathrm{s,1}}$, using the expressions for
$f_{\hat{g}_{\mathrm{s}}}\left(x\right),\mathscr{F}_{\hat{g}_{\mathrm{e}}}\left(\sigma_{\epsilon}^{2},x\right)$
and $f_{\tilde{g}_{\mathrm{p}}}\left(y\right)$ yields

$\displaystyle\mathscr{I}_{\mathrm{s,1}}=-\log_{2}\left(e\right)\sum_{\ell\in\left\\{1,2,3\right\\}}\sum_{\boldsymbol{\eta}\in\Phi}\hat{A}_{\ell}B_{\boldsymbol{\eta}}\kappa_{\boldsymbol{\eta}}$
$\displaystyle\times$
$\displaystyle\left[\int_{y=0}^{\Theta}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\ln\left(1\\!+\\!P_{\max}x\right)\exp\\!\left(\\!\dfrac{-x}{a_{\mathrm{s}}\Xi_{\ell}}\\!\right)\exp\left(-B_{\boldsymbol{\eta}}y\right)\mathrm{d}x\mathrm{d}y\right.$
$\displaystyle-$
$\displaystyle\left.\hat{\mathscr{C}}_{\mathrm{e}}\\!\\!\int_{y=0}^{\Theta}\\!\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\\!\\!\ln\left(1\\!+\\!P_{\max}x\right)\exp\\!\left(\\!\dfrac{-x}{a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}}\\!\right)\\!\exp\left(-B_{\boldsymbol{\eta}}y\right)\mathrm{d}x\mathrm{d}y\right]$
$\displaystyle=$
$\displaystyle-\log_{2}\left(e\right)\sum_{\ell\in\left\\{1,2,3\right\\}}\sum_{\boldsymbol{\eta}\in\Phi}\hat{A}_{\ell}\kappa_{\boldsymbol{\eta}}\left\\{1-\exp\left(-B_{\boldsymbol{\eta}}\Theta\right)\right\\}$
$\displaystyle\qquad\times\left[\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\ln\left(1\\!+\\!P_{\max}x\right)\exp\\!\left(\\!\dfrac{-x}{a_{\mathrm{s}}\Xi_{\ell}}\\!\right)\mathrm{d}x\right.$
$\displaystyle\qquad-\left.\hat{\mathscr{C}}_{\mathrm{e}}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\ln\left(1\\!+\\!P_{\max}x\right)\exp\\!\left(\\!\dfrac{-x}{a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}}\\!\right)\mathrm{d}x\right].$
(20)

Computing the integrals above using integration by parts and with some
algebraic manipulations, we arrive at the closed-form expression for
$\mathscr{I}_{\mathrm{s,1}}$ is given by (3). Similarly, for
$\mathscr{I}_{\mathrm{s,3}}$, we have

$\displaystyle\mathscr{I}_{\mathrm{s,3}}=-\log_{2}\left(e\right)\hat{C}_{\mathrm{e}}\sum_{\ell\in\left\\{1,2,3\right\\}}\sum_{\boldsymbol{\eta}\in\Phi}\hat{\mathcal{A}}_{\ell}B_{\boldsymbol{\eta}}\kappa_{\boldsymbol{\eta}}$
$\displaystyle\times\int_{y=0}^{\Theta}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\\!\ln\left(1\\!+\\!P_{\max}x\right)\exp\left(\\!\dfrac{-x}{a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}}\\!\right)\exp\left(-B_{\boldsymbol{\eta}}y\right)\mathrm{d}x\mathrm{d}y$
$\displaystyle=-\log_{2}\left(e\right)\hat{C}_{\mathrm{e}}\sum_{\ell\in\left\\{1,2,3\right\\}}\sum_{\boldsymbol{\eta}\in\Phi}\hat{\mathcal{A}}_{\ell}\kappa_{\boldsymbol{\eta}}\left\\{1-\exp\left(-B_{\boldsymbol{\eta}}\Theta\right)\right\\}$
$\displaystyle\qquad\qquad\times\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\ln\left(1\\!+\\!P_{\max}x\right)\exp\left(\\!\dfrac{-x}{a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}}\\!\right)\mathrm{dx}.$
(21)

A closed-form expression for (21) can be obtained by following the steps
similar to those used to obtain $\mathscr{I}_{\mathrm{s,1}}$, and is given in
(5). Now solving for $\mathscr{I}_{\mathrm{s,4}}$, we have

$\displaystyle\mathscr{I}_{s,4}=-\log_{2}\left(e\right)\hat{C}_{\mathrm{e}}\sum_{\ell\in\left\\{1,2,3\right\\}}\sum_{\boldsymbol{\eta}\in\Phi}\hat{\mathcal{A}}_{\ell}B_{\boldsymbol{\eta}}\kappa_{\boldsymbol{\eta}}\int_{y=\Theta}^{\infty}$
$\displaystyle\left[\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\ln\left(1+\dfrac{I_{\mathrm{p}}}{y}x\right)\exp\left(\dfrac{-x}{a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}}\right)\mathrm{d}x\right]\exp\left(-B_{\boldsymbol{\eta}}y\right)\mathrm{d}y.$
(22)

Integrating the preceding expression w.r.t. $x$ using integration by parts and
using a few algebraic manipulations yields

$\displaystyle\mathscr{I}_{\mathrm{s,4}}=-\log_{2}\left(e\right)\hat{C}_{\mathrm{e}}\sum_{\ell\in\left\\{1,2,3\right\\}}\sum_{\boldsymbol{\eta}\in\Phi}\hat{\mathcal{A}}_{\ell}B_{\boldsymbol{\eta}}\kappa_{\boldsymbol{\eta}}a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}\int_{y=\Theta}^{\infty}$
$\displaystyle\times\left[\exp\left(\dfrac{-\sigma_{\epsilon}^{2}}{a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}}\right)\ln\left(1+\dfrac{I_{\mathrm{p}}\sigma_{\epsilon}^{2}}{y}\right)+\exp\left(\dfrac{y}{a_{\mathrm{s}}I_{\mathrm{p}}\Xi_{\ell,\mathrm{e}}}\right)\right.$
$\displaystyle\left.\times\mathrm{E}_{1}\left(\dfrac{1}{a_{\mathrm{s}}\Xi_{\ell,\mathrm{e}}}\left(\dfrac{y}{I_{\mathrm{p}}}+\sigma_{\epsilon}^{2}\right)\right)\right]\exp\left(-B_{\boldsymbol{\eta}}y\right)\mathrm{d}y.$
(23)

Now solving the integral in (23) w.r.t $y$ using integration by parts, a
closed-form expression for $\mathscr{I}_{\mathrm{s,4}}$ is given in (6). Next,
for $\mathscr{I}_{\mathrm{s,2}},$ we have

$\displaystyle\mathscr{I}_{\mathrm{s,}2}\\!=\\!-\log_{2}\left(e\right)\\!\\!\\!\sum_{\ell\in\left\\{1,2,3\right\\}}\\!\sum_{\boldsymbol{\eta}\in\Phi}\\!\hat{A}_{\ell}B_{\boldsymbol{\eta}}\kappa_{\boldsymbol{\eta}}\int_{y=\Theta}^{\infty}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\ln\left(\\!1+\dfrac{I_{\mathrm{p}}}{y}x\\!\right)$
$\displaystyle\times\exp\left(\dfrac{-x}{a_{\mathrm{s}}\Xi_{\ell}}\right)\left\\{1-\hat{\mathscr{C}}_{\mathrm{e}}\exp\left(\dfrac{-x}{a_{\mathrm{s}}\tilde{\Omega}_{\mathrm{e}}}\right)\right\\}\exp\left(-B_{\boldsymbol{\eta}}y\right)\mathrm{d}x\mathrm{d}y.$
(24)

Solving the integrals above by following the steps similar to those used for
$\mathscr{I}_{\mathrm{s,4}}$, a closed-form expression for
$\mathscr{I}_{\mathrm{s,2}}$ is given by (4). This concludes the proof.

##  Appendix B Proof of Theorem 2

Using (9), we have

$\displaystyle\bar{R}_{\mathrm{w}}=\mathbb{E}\left\\{\max\left\\{\log_{2}\left(\dfrac{1+\left(\tilde{g}_{{}_{\mathrm{w}}}+\sigma_{\epsilon}^{2}\right)P}{1+\left(a_{\mathrm{s}}\tilde{g}_{\mathrm{{}_{\mathrm{w}}}}+\sigma_{\epsilon}^{2}\right)P}\right)\right.\right.$
$\displaystyle\qquad\qquad\qquad-\left.\left.\log_{2}\left(\dfrac{1+\left(\tilde{g}_{{}_{\mathrm{e}}}+\sigma_{\epsilon}^{2}\right)P}{1+\left(a_{\mathrm{s}}\tilde{g}_{{}_{\mathrm{e}}}+\sigma_{\epsilon}^{2}\right)P}\right),0\right\\}\right\\}$
$\displaystyle=\left[\int_{y=0}^{\Theta}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\\!\\!\log_{2}\left(1\\!+\\!xP_{\max}\right)f_{\check{g}_{{}_{\mathrm{w}}}}\\!\left(x\right)\mathscr{F}_{\check{g}_{{}_{\mathrm{e}}}}\\!\left(x\right)f_{\tilde{g}_{{}_{\mathrm{p}}}}\\!\left(y\right)\mathrm{d}x\mathrm{d}y\right.$
$\displaystyle\left.-\int_{y=0}^{\Theta}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\\!\\!\log_{2}\left(1\\!+\\!xP_{\max}\right)f_{\hat{g}_{{}_{\mathrm{w}}}}\\!\left(x\right)\mathscr{F}_{\hat{g}_{{}_{\mathrm{e}}}}\\!\left(x\right)f_{\tilde{g}_{{}_{\mathrm{p}}}}\\!\left(y\right)\mathrm{d}x\mathrm{d}y\right]$
$\displaystyle+\left[\int_{y=\Theta}^{\infty}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\\!\\!\log_{2}\left(1+I_{\mathrm{p}}\dfrac{x}{y}\right)f_{\check{g}_{{}_{\mathrm{w}}}}\\!\left(x\right)\mathscr{F}_{\check{g}_{{}_{\mathrm{e}}}}\\!\left(x\right)f_{\tilde{g}_{{}_{\mathrm{p}}}}\\!\left(y\right)\mathrm{d}x\mathrm{d}y\right.$
$\displaystyle\left.-\int_{y=\Theta}^{\infty}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\\!\\!\log_{2}\left(1+I_{\mathrm{p}}\dfrac{x}{y}\right)f_{\hat{g}_{{}_{\mathrm{w}}}}\\!\left(x\right)\mathscr{F}_{\hat{g}_{{}_{\mathrm{e}}}}\\!\left(x\right)f_{\tilde{g}_{{}_{\mathrm{p}}}}\\!\left(y\right)\mathrm{d}x\mathrm{d}y\right]$
$\displaystyle-\left[\int_{y=0}^{\Theta}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\\!\\!\log_{2}\left(1\\!+\\!xP_{\max}\right)f_{\check{g}_{{}_{\mathrm{e}}}}\\!\left(x\right)\mathcal{F}_{\check{g}_{{}_{\mathrm{w}}}}\\!\left(x\right)f_{\tilde{g}_{{}_{\mathrm{p}}}}\\!\left(y\right)\mathrm{d}x\mathrm{d}y\right.$
$\displaystyle\left.-\int_{y=0}^{\Theta}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\\!\\!\log_{2}\left(1\\!+\\!xP_{\max}\right)f_{\hat{g}_{{}_{\mathrm{e}}}}\\!\left(x\right)\mathcal{F}_{\hat{g}_{{}_{\mathrm{w}}}}\\!\left(x\right)f_{\tilde{g}_{{}_{\mathrm{p}}}}\\!\left(y\right)\mathrm{d}x\mathrm{d}y\right]$
$\displaystyle-\left[\int_{y=\Theta}^{\infty}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\\!\\!\log_{2}\left(1+I_{\mathrm{p}}\dfrac{x}{y}\right)f_{\check{g}_{{}_{\mathrm{e}}}}\\!\left(x\right)\mathcal{F}_{\check{g}_{{}_{\mathrm{w}}}}\\!\left(x\right)f_{\tilde{g}_{{}_{\mathrm{p}}}}\\!\left(y\right)\mathrm{d}x\mathrm{d}y\right.$
$\displaystyle\left.-\int_{y=\Theta}^{\infty}\int_{x=\sigma_{\epsilon}^{2}}^{\infty}\\!\\!\\!\\!\\!\log_{2}\left(1+I_{\mathrm{p}}\dfrac{x}{y}\right)f_{\hat{g}_{{}_{\mathrm{e}}}}\\!\left(x\right)\mathcal{F}_{\hat{g}_{{}_{\mathrm{w}}}}\\!\left(x\right)f_{\tilde{g}_{{}_{\mathrm{p}}}}\\!\left(y\right)\mathrm{d}x\mathrm{d}y\right]$
$\displaystyle\triangleq\mathscr{I}_{\mathrm{w,1}}+\mathscr{I}_{\mathrm{w,2}}-\mathscr{I}_{\mathrm{w,3}}-\mathscr{I}_{\mathrm{w,4}}.$
(25)

Following a similar line of argument as given in Appendix A, closed-form
expressions for
$\mathscr{I}_{\mathrm{w,1}},\mathscr{I}_{\mathrm{w,2}},\mathscr{I}_{\mathrm{w,3}}$
and $\mathscr{I}_{\mathrm{w,4}}$ are given by (11) – (14). This completes the
proof.

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