# On Secure mmWave RSMA Systems††thanks: Manuscript received.

Hongjiang Lei, Sha Zhou, Xinhu Chen, Imran Shafique Ansari,
Yun Li, Gaofeng Pan, and Mohamed-Slim Alouini This work was supported by the
National Natural Science Foundation of China under Grant 61971080.
(Corresponding author: Hongjiang Lei.)Hongjiang Lei is with the School of
Communication and Information Engineering, Chongqing University of Posts and
Telecommunications, Chongqing 400065, China, also with Chongqing Key Lab of
Mobile Communications Technology, Chongqing 400065, China (e-mail:
leihj@cqupt.edu.cn).Sha Zhou and Xinhu Chen are with the School of
Communication and Information Engineering, Chongqing University of Posts and
Telecommunications, Chongqing 400065, China (e-mail:
cquptzhous@163.com;chenxhcqupt@163.com).Imran Shafique Ansari is with James
Watt School of Engineering, University of Glasgow, Glasgow G12 8QQ, United
Kingdom (e-mail: imran.ansari@glasgow.ac.uk).Yun Li is with Chongqing Key Lab
of Mobile Communications Technology, Chongqing 400065, China (e-mail:
liyun@cqupt.edu.cn).Gaofeng Pan is with the School of Cyberspace Science and
Technology, Beijing Institute of Technology, Beijing 100081, China (e-mail:
gaofeng.pan.cn@ieee.org).Mohamed-Slim Alouini is with CEMSE Division, King
Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900,
Saudi Arabia (e-mail: slim.alouini@kaust.edu.sa).

###### Abstract

Millimeter-wave (mmWave) communication is one of the effective technologies
for the next generation of wireless communications due to the enormous amount
of available spectrum resources. Rate splitting multiple access (RSMA) is a
powerful multiple access, interference management, and multi-user strategy for
designing future wireless networks. In this work, a multiple-input-single-
output mmWave RSMA system is considered wherein a base station serves two
users in the presence of a passive eavesdropper. Different eavesdropping
scenarios are considered corresponding to the overlapped resolvable paths
between the main and the wiretap channels under the considered transmission
schemes. The analytical expressions for the secrecy outage probability (SOP)
are derived respectively through the Gaussian–Chebyshev quadrature method.
Monte Carlo simulation results are presented to validate the correctness of
the derived analytical expressions and demonstrate the effects of system
parameters on the SOP of the considered mmWave RSMA systems.

###### Index Terms:

Millimeter-wave, rate splitting multiple access, uniform linear array, secrecy
outage probability.

##  I Introduction

###  I-A Background and Related Work

Nowadays, a growing number of electronic devices and various emerging
applications have entered our daily routines, which bring about significant
growth in the wireless data traffic of wireless networks and is likely to leap
10000 fold in the next 20 years [1]-[3]. To tackle this incredible growth,
millimeter-wave (mmWave) has become one of the most efficient resolutions due
to the plentiful underutilized spectrum resources [4]. Recently, mmWave
communication has received substantial attention, such as channel modelling
and estimation, beamforming strategy design, and performance analysis. The
authors in [5] provided a comprehensive overview of mathematical models and
analytical techniques of mmWave cellular systems. A baseline mathematical
method and analysis in blocking and substantial directionality aspects was
proposed and the result indicated that the ultra-dense deployments were more
available in mmWave systems. In [6], an adaptive algorithm was developed to
estimate the mmWave channel parameters exploiting the sparse scattering of the
channel and was extended to the case of the multi-path channel. The result
shows that the proposed low-complexity channel estimation algorithm achieves
more precoding gains and spectral efficiency than exhaustive algorithms. In
[7], a hybrid precoding/combining designs of full-duplex amplify-and-forward
mmWave relay systems was proposed. Their simulation indicated that the
proposed design is approaching an all-digital scheme. A downlink non-
orthogonal multiple access (NOMA) mmWave system was investigated in [8]. An
opportunistic beam-splitting NOMA scheme was proposed to inquire into the
antenna gain and the closed-form expressions of the coverage probability and
the ASR were derived.

In next-generation communication networks and beyond, interference management
becomes a fundamental problem for multi-user communications and multiple
access design [9]. Rate-splitting multiple access (RSMA) as a candidate
multiple access scheme has been proposed in [10] and has been recognized as a
powerful multiple access, interference management, and multi-user strategy
[11]. Based on the rate-splitting (RS) principle, RSMA not only displays more
flexibility in managing interference, i.e., partially decodes interference of
inter-user and partially treats interference as noise, but also unifies and
generalizes orthogonal multiple access (OMA), space-division multiple access,
NOMA, physical-layer multicasting [12], [13], [14]. A more flexible and
powerful cooperative scheme for a multiple-input-single-output (MISO)
broadcast channel scenario was proposed based on the three-node relay channel
in [15]. Their simulation results show that the proposed cooperative RS
strategy can obtain an explicit rate improvement than NOMA. In [16], linearly-
precoded 1-layer and multi-layer transmission strategies based on the RS were
investigated in a Non-Orthogonal unicast and multicast transmission system,
which the weighted sum rate and energy efficiency problems were solved by
weighted minimum mean square error algorithm and successive convex
approximation algorithm. The Numerical results indicated that the RS-assisted
transmission strategies are more efficient in spectrum and energy compared
with multi-user linear- precoding, OMA and NOMA in extensive user deployments
and network loads. Compared to rate region, sum rate, spectral and energy
efficiency improvement in terms of optimization, the benefit of RSMA for
performance analysis, such as throughput, ergodic sum rate (ESR), and outage
probability (OP), also needs to be explored. In [17], the downlink unmanned
aerial vehicle systems based on the RSMA scheme were investigated and the
closed-form expressions of the OP and throughput at each user were derived.
The performance of a multi-cell RSMA network was investigated in [18] wherein
the analytical expressions for ESR and spectral efficiency based on stochastic
geometry theory were derived. Their results showed that the power splitting
ratio between common and private streams significantly impacts performance.
Bansal A. et al. in [19] studied a novel Intelligent reflecting surface (IRS)
RS framework of multi-user communication system, the closed-form expressions
of OP for the far and near users were derived. The simulation results showed
that the proposed framework was superior to the decode-and-forward-RS, RS
without IRS, and IRS-NOMA scenarios.

Physical layer security (PLS), which utilizes the inherent characteristics of
wireless channels, is an exciting complement to sophisticated cryptographic
techniques [20]. Since the mmWave’s propagation characteristics, which support
large antenna arrays and highly directional transmission, can reduce the
leakage of confidential information. Thus, physical layer security in mmWave
communications has attracted considerable recent attention. Specifically, by
using a sectored model to analyze the beam pattern, the secrecy transmission
of a mmWave cellular network was investigated in [21], the secure connectivity
probability was studied and their results indicated the array pattern and
intensity of eavesdroppers are both important network parameters for improving
the secrecy performance. The author studied the physical layer security of
mmWave relaying networks In [22] wherein developed a new joint relay selection
and power allocation method in multiple eavesdroppers and relays, the closed-
form expressions of average secrecy rate (ASR) and secrecy outage probability
(SOP) were derived respectively and demonstrated the superiority of proposed
method. Ju et al. in [23] utilized the discrete angular domain channel (DADC)
model to analyze the secrecy performance of the mmWave MISO systems. Three
transmission schemes were proposed to improve the secure mmWave systems.
Further, they investigated the security mmWave MISO systems in the presence of
multiple randomly located eavesdroppers in [24]. In [25], the multiple input
multiple output DADC were developed in mmWave decode-and-forward relay
systems. The secrecy performance with three eavesdropping scenarios was
investigated by designing the corresponding beamforming strategies. Huang et
al. investigated the secure mmWave NOMA systems in which all the authorized
and unauthorized receivers were randomly located in [26]. Then, they proposed
two schemes to improve the secrecy transmission and the closed-form SOPs for
two beamforming schemes with varying eavesdropper detection abilities were
derived.

As a novel, general, and robust framework for the sixth generation mobile
communications, RSMA has a high potential to be used for security applications
[27]. In [28], two decoding strategies were adopted by a near user and a far
user to explore outage and secrecy outage performance. The closed-form
expressions of OP and tight approximations of SOP were derived, considering
four decoding combinations. The author in [29] proposed two RS schemes with
one-layer-successive interference cancellation (SIC) and two-layer-SIC wherein
there is an untrusted near user. The closed-form expressions of OP and SOP are
obtained to analyze outage performance. In [30], the secrecy performance of
RSMA in multi-user MISO systems was investigated and analytical expressions of
the ESR and ASR were obtained with zero-forcing precoding and minimum mean
square error approach. The result demonstrated that proposed power allocation
methods could offer inherent tradeoffs over the ESR and ASR.

In the RSMA scheme, the messages are split into common and private parts,
encoded into a single common stream and private streams, respectively. All the
common and private streams are superimposed, linearly precoded, and
transmitted simultaneously by the transmitter. By utilizing SIC technology,
the common stream is decoded firstly by treating all private streams as
interference, removing from the received signals, and the private stream is
decoded by treating the private streams for other users as interference. In
the mmWave systems with the DADC model, depending on the spatial correlation
of the users, the spatially resolvable paths are split into overlapped and
non-overlapped paths. As thus, the following questions naturally arise: Should
the common streams be transmitted on overlapped paths or all paths? What is
the performance of the mmWave systems with the RSMA scheme?  To answer these
questions, the performance of the mmWave RSMA systems was investigated, and
two beamforming transmission schemes were proposed in [32]. The closed-form
expressions for the exact and asymptotic OP with proposed schemes were derived
using stochastic geometry theory. The results demonstrated that RSMA helps
improve transmission reliability of the mmWave systems. TABLE I outlines the
typical works discussed.

TABLE I: Related Literature on RSMA or mmWave systems. Reference |  RSMA | mmWave |  Multi-antenna technology |  Channel Model | PLS |  Performance metric
---|---|---|---|---|---|---
[17] | ✓ |  |  | Nakagami-$m$ |  | OP
[18] | ✓ |  | Transmitter |  Rayleigh |  | ESR
[19] | ✓ |  |  | Rayleigh |  | OP
[21] |  | ✓ |  | Nakagami-$m$ | ✓ | SCP
[22] |  | ✓ | Transmitter | Rayleigh | ✓ | SOP
[23] |  | ✓ | Transmitter |  DADC | ✓ | SOP
[24] |  | ✓ | Transmitter |  DADC | ✓ | SOP
[25] |  | ✓ |  Transmitter & destination |  DADC | ✓ | SOP
[26] |  | ✓ | Transmitter |  DADC | ✓ | SOP
[28] | ✓ |  |  | Rayleigh | ✓ | OP and SOP
[29] | ✓ |  | Transmitter | Rayleigh | ✓ | OP and SOP
[30] | ✓ |  | Transmitter | Rayleigh | ✓ |  ESR & ASR
[32] | ✓ | ✓ | Transmitter |  DADC |  | OP
Our Work | ✓ | ✓ | Transmitter |  DADC | ✓ | SOP
|  |  |  |  |  |

###  I-B Motivation and Contributions

To the best of the authors’ knowledge, based on the open literature, there is
a lack of research focused on the following issue: Is it beneficial to
leverage the RSMA scheme in mmWave systems for security enhancement? Hence,
this work answers this question by analyzing the secrecy performance of the
mmWave RSMA systems. Different eavesdropping scenarios are considered
corresponding to the overlapped resolvable paths between the main and the
wiretap channels under the considered transmission schemes. Technically
speaking, it is much more challenging to obtain the analytical expression of
the SOP than that of the OP for the mmWave RSMA system since there are
multiple parameters (random variables) that must be considered. We summarize
the contributions of this work as follows.

  1. 1.

The secrecy performance of the MISO mmWave RSMA system is investigated wherein
the common and private streams are transmitted on the overlapping and non-
overlapping paths, respectively. Different scenarios are considered based on
the spatial correlation of the legitimate and illegitimate user and the
principle of the RSMA scheme. Subsequently, the analytical expressions for the
SOP are derived respectively through the Gaussian–Chebyshev quadrature method.

  2. 2.

We present Monte Carlo simulation results to validate the correctness of the
derived analytical expressions and demonstrate the effects of system
parameters on the SOP of the considered mmWave RSMA systems. The result shows
that the power allocation between the common and private streams and the
number of overlapped paths significantly affect the secrecy performance.

  3. 3.

Relative to [26] wherein the security mmWave NOMA systems were enhanced by the
proposed beamforming schemes, the security mmWave RSMA system is improved by
the proposed beamforming scheme in this work wherein the scenarios considered
in this work are more challenging.

  4. 4.

Relative to [28]-[31] wherein the secrecy performance of the RSMA systems with
internal untrusted users was investigated. In these scenarios, it was assumed
that the common message could always be decoded and only the private message
was wiretapped. Technically speaking, it is much more challenging to
investigate the secrecy performance of the RSMA systems with an external
eavesdropper than with an internal eavesdropper.

###  I-C Organization

The rest of this paper is organized as follows. Section II describes the
system model and beamforming scheme. The analytical expressions for the exact
SOP of mmWave RSMA systems is derived under different scenarios in Sections
III. The simulation results are presents to valid the analysis in Section IV.
Section V concludes this work. Table II lists the notations and symbols
utilized throughout this work.

TABLE II: Notation and Symbols Notation | Description
---|---
${\Omega_{l}}$ | The index set of resolvable paths at $l$
${{\mathbf{g}}_{l}}$ | The complex gain vector between $S$ and $l$
${\mathbf{U}}$ | The spatially orthogonal basis
$L_{l}$ | The number of resolvable paths of $l$
${\Delta_{l,o}}$ | The angular range between the $o$th and $(o+1)$th paths at $l$
${\omega_{l,o}}$ | The width between the $o$th and $(o+1)$th paths at $l$
${\Omega_{c}}$ | The index set of overlapped paths between $U_{1}$ and $U_{2}$
${\Omega_{i,p}}$ | The index set of non-overlapping paths at $U_{i}$
${\Omega_{ec}}$ | The index set of overlapped paths among $E$ and ${\Omega_{c}}$
${\Omega_{e,pi}}$ | The index set of overlapped paths between $E$ and ${\Omega_{i,p}}$
${\tau_{c}}$ | The power allocation coefficient for $s_{c}$
${\tau_{i}}$ | The power allocation coefficient for $s_{i}$
$R_{1,c}^{\mathrm{th}}$ | The secrecy rate threshold for the common messages
$R_{1,p}^{\mathrm{th}}$ | The secrecy rate threshold for the private messages
$G_{c,d}^{a,b}\left[\cdot\right]$ | Meijer’s $G$-function
$H_{,:,:,}^{,:,:,}\left[\cdot\right]$ | Extended generalized bivariate Fox’s $H$-function
$\Gamma\left(x\right)$ | Gamma function
|

##  II System Model

###  II-A System Model

Figure 1: A downlink mmWave RSMA system consists a base station ($S$), two
user ($U_{1}$ and $U_{2}$), and a external passive eavesdropper ($E$).

As shown in Fig. 1, a downlink mmWave RSMA system is considered where the base
station $\left(S\right)$ communicates with two users denoted by ${U_{1}}$ and
${U_{2}}$ 111 Similar to [10], [12], [14], the RSMA system with two users is
considered in this work. As such, the results in this paper can serve as a
benchmark for the performance of such systems. The performance of RSMA systems
with multiple users will be part of our future work.  while an external
passive eavesdropper denoted by $E$ attempt to intercept the information. $S$
is equipped with ${N_{s}}$ antennas and both legitimate and illegitimate
receivers are equipped with a single antenna.

Similar to [32], the message, ${W_{i}}$ $\left({i=1,2}\right)$, is split into
${W_{i,c}}$ and ${W_{i,p}}$. ${W_{1,c}}$ and ${W_{c,2}}$ are encoded together
into ${s_{c}}$, which is a common stream decoded by both users. At the same
time, ${W_{1,p}}$ and ${W_{p,2}}$ are encoded into ${s_{1}}$ and ${s_{2}}$,
respectively. The transmitted signal from $S$ is given by

${\mathbf{x}}={{\mathbf{w}}_{c}}{\sqrt{P{\tau_{c}}}}{s_{c}}+{{\mathbf{w}}_{1}}\sqrt{{{P{\tau_{1}}}}}{s_{1}}+{{\mathbf{w}}_{2}}\sqrt{{{P{\tau_{2}}}}}{s_{2}},$
(1)

where ${{\mathbf{w}}_{c}}$, ${{\mathbf{w}}_{1}}$, and ${{\mathbf{w}}_{2}}$ are
unit vectors that denote the beamforming direction, $P$ signifies the transmit
power, and ${\tau_{c}}$ and ${\tau_{i}}$ denote the power allocation
coefficient for $s_{c}$ and $s_{i}$, respectively.

The polar coordinate system is established with the position of $S$ as the
origin. The position of ${U_{1}}$ and ${U_{2}}$ are expressed as
$\left({{r_{1}},{\theta_{1}}}\right)$ and
$\left({{r_{2}},{\theta_{2}}}\right)$ respectively, where ${{\theta_{i}}}$ is
the center angle of angles of departure (AODs) of $U_{i}$’s paths, i.e.,
${\theta_{i}}=\frac{{{\theta_{i,\min}}+{\theta_{i,\max}}}}{2}$. The channels
of between $S$ and the receiver $l$ is expressed as [23]-[25]

${{\mathbf{h}}_{l}}=\sqrt{\frac{{{N_{s}}P\left({{r_{l}}}\right)}}{{{L_{l}}}}}{{\mathbf{g}}_{l}}{{\mathbf{U}}^{H}},$
(2)

where $l\in\left\\{{{1},{2},e}\right\\}$,
$P\left({{r_{l}}}\right)=r_{0}{r_{l}}^{-\alpha}$,
${r_{0}}={10^{-\frac{{{\beta_{L}}}}{{10}}}}$,
${\beta_{L}}=3.66+24.3{\log_{10}}\left({{f_{c}}}\right)$, ${f_{c}}=28$ GHz
[35], $r_{l}$ denotes the distance between the transmitter and the receiver
$l$, $\alpha$ signifies the path loss exponent, $L_{l}$ denotes the number of
resolvable paths of the receiver $l$, ${\mathbf{U}}$ is the spatially
orthogonal basis,
${{\mathbf{g}}_{l}}=\left[{{g_{l,1}},{g_{l,2}},\ldots,{g_{l,{N_{s}}}}}\right]$.
It is assumed that the AODs of the receiver $l$’ paths is distributed within
the angular range $\left[{{\theta_{l,\min}},{\theta_{l,\max}}}\right]$, where
${g_{l,n}}\sim CN\left({0,1}\right)$ if
${\theta_{l,n}}\in\left[{{\theta_{l,\min}},{\theta_{l,\max}}}\right]$
otherwise ${g_{l,n}}=0$ [23]. To facilitate analysis, we assumed that
${\theta_{1}}=-{\theta_{2}}$ and ${L_{1}}={L_{2}}={L_{e}}=L$. We define sets
${\Omega_{l}}=\left\\{{{I_{l,o}}\left|{{I_{l,o}}\in{Z^{+}}}\right.,{\Psi_{{I_{l,o}}}}\in\left[{\sin\left({{\theta_{l,\min}}}\right),\sin\left({{\theta_{l,\max}}}\right)}\right],}\right.{I_{l,1}}<{I_{l,2}}$$\left.{<\cdots<{I_{l,L}}}\right\\}$.
So we have
${{{\mathrm{th}}{\Omega}}_{i}}=\left\\{{\frac{{{N_{{\mathrm{th}}{s}}}\mp{L_{{\mathrm{th}}{c}}}}}{2}\pm
1,\frac{{{N_{{\mathrm{th}}{s}}}\mp{L_{{\mathrm{th}}{c}}}}}{2}\pm
2,\cdots,\frac{{{N_{{\mathrm{th}}{s}}}\mp{L_{{\mathrm{th}}{c}}}}}{2}\pm
L}\right\\}$, where ${\Omega_{l,o}}=\frac{{{N_{s}}\mp{L_{c}}}}{2}\pm o$.
Define the angular range
${\Delta_{l,o}}=\left[{\arcsin\left({{\Psi_{\frac{{{N_{s}}\mp{L_{c}}}}{2}\pm
o}}}\right),\arcsin\left({{\Psi_{\frac{{{N_{s}}\mp{L_{c}}}}{2}\pm
o+1}}}\right)}\right]$ and the width
${\omega_{l,o}}=\arcsin\left({{\Psi_{\frac{{{N_{s}}\mp{L_{c}}}}{2}\pm
o+1}}}\right)-\arcsin\left({{\Psi_{\frac{{{N_{s}}\mp{L_{c}}}}{2}\pm
o}}}\right)$, where ${\Delta_{l,o}}$ (${1\leq o<L}$) describes the angular
range of user $l$ between the $o$th and $\left({o+1}\right)$th spatially
resolvable paths, ${\omega_{l,o}}$ denote the width between the $o$th and
$\left({o+1}\right)$th spatially resolvable paths.. Then we have
$\left|{{\Omega_{1}}}\right|=\left|{{\Omega_{2}}}\right|=\left|{{\Omega_{e}}}\right|=L$.

Based on the spatial correlation of all receivers, the spatially resolvable
paths are divided into overlapped and non-overlapping paths [23]. ${L_{c}}$
and $L_{p}$ are defined as the number of the overlapped and non-overlapping
paths between $U_{1}$ and $U_{2}$, respectively. Thus, we have
${\Omega_{c}}={\Omega_{1}}\cap{\Omega_{2}}$,
${\Omega_{1,p}}={\Omega_{1}}-{\Omega_{c}}$,
${\Omega_{2,p}}={\Omega_{2}}-{\Omega_{c}}$.
$\left|{{\Omega_{c}}}\right|={L_{c}}$, and
$\left|{{\Omega_{1,p}}}\right|=\left|{{\Omega_{2,p}}}\right|={L_{p}}$.
Similarly, ${L_{ec}}$, ${L_{e1}}$, ${L_{e2}}$ denote the number of the
overlapped paths between ${\Omega_{e}}$, and ${\Omega_{c}}$, ${\Omega_{1,p}}$,
${\Omega_{2,p}}$, respectively. Thus, we have
${\Omega_{ec}}={\Omega_{c}}\cap{\Omega_{e}}$ and
${\Omega_{e,p1}}={\Omega_{1,p}}\cap{\Omega_{e}}$,
${\Omega_{e,p2}}={\Omega_{2,p}}\cap{\Omega_{e}}$,
$\left|{{\Omega_{e,c}}}\right|={L_{ec}}$,
$\left|{{\Omega_{e,p1}}}\right|={L_{e1}}$,
$\left|{{\Omega_{e,p2}}}\right|={L_{e2}}$.

In this work, $s_{c}$ is transmitted on the overlapped paths and $s_{i}$ is
transmitted on their non-overlapping paths to eliminate the inter-user
interference of private signals by utilizing the spatial correlation of two
users’ channels [32]. Similar to [23] and [32], it is assumed that the perfect
channel state information (CSI) of the legitimate receivers is available at
$S$ , the beamforming for $s_{c}$ is expressed as ${{\mathbf{w}}_{c}}={\cal
S}\left({{\mathbf{U}},{\Omega_{c}}}\right){\in^{{N_{s}}\times{L_{c}}}}$, where
${\cal S}\left({{\mathbf{B}},{D}}\right)$ is utilized to generate a new matrix
with columns selected from ${\mathbf{B}}$ based on the selected column index
set ${D}$, and ${{\Omega_{c}}}$ denotes the index sets of common resolvable
paths between the receiver ${U_{1}}$ and ${U_{2}}$. Similarly, the beamforming
for ${s_{i}}$ is expressed as
${{\mathbf{w}}_{i}}=S\left({{\mathbf{U}},{\Omega_{i,p}}}\right){\in^{{N_{s}}\times{L_{p}}}}$,
where ${{\Omega_{i,p}}}$ denotes the index set of the non-overlapping paths of
receiver ${U_{i}}$. Thus, the transmitted signal from $S$ is rewritten as

${\mathbf{x}}={{\mathbf{w}}_{c}}\sqrt{P{\tau_{c}}}{{\mathbf{s}}_{c}}+{{\mathbf{w}}_{1}}\sqrt{P{\tau_{1}}}{{\mathbf{s}}_{1}}+{{\mathbf{w}}_{2}}\sqrt{P{\tau_{2}}}{{\mathbf{s}}_{2}},$
(3)

where ${{\mathbf{s}}_{c}}\in{{\mathbb{C}}^{{L_{c}}\times 1}}$ and
${{\mathbf{s}}_{i}}\in{{\mathbb{C}}^{{L_{p}}\times 1}}$.

###  II-B Signal to Interference Plus Noise Ratio

According to the RSMA principle, ${U_{i}}$ decodes ${s_{c}}$ firstly by
treating all the other signals as noise. Then the instantaneous signal-to-
interference and noise ratio (SINR) of decoding ${s_{c}}$ at ${U_{i}}$ is
${\gamma_{i,c}}=\frac{{{\delta_{i,c}}{{\left\|{{{\mathbf{g}}_{i,c}}}\right\|}^{2}}}}{{{\delta_{i}}{{\left\|{{{\mathbf{g}}_{{i,p}}}}\right\|}^{2}}+1}}$,
where ${\delta_{i,c}}=\frac{{\delta{\tau_{c}}}}{{r_{i}^{\alpha}}}$,
${\delta_{i}}=\frac{{\delta{\tau_{i}}}}{{r_{i}^{\alpha}}}$,
$\rho=\frac{P}{{\sigma^{2}}}$, and $\delta=\frac{{{N_{s}}\rho{r_{0}}}}{L}$.
After performing SIC, i.e., $s_{c}$ is re-encoded, precoded, and removed from
the received signal, the SINR of decoding $s_{i}$ at ${U_{i}}$ is obtained as
${\gamma_{i,p}}={\delta_{i}}{\left\|{{{\mathbf{g}}_{{i,p}}}}\right\|^{2}}$.

(a) ${\theta_{e,\min}}\in{\Delta_{L-{L_{e1}}}}$.

(b) ${\theta_{e,\min}}\in{\Delta_{L-{L_{p}}-{L_{ec}}}}$.

(c) ${\theta_{e,\max}}\in{\Delta_{{L_{ec}}+{L_{e1}}}}$.

(d) ${\theta_{e,\max}}\in{\Delta_{{L_{ec}}}}$.

Figure 2: Scenarios for the different number of the overlapped overlapped
paths between $U_{i}$ and $E$ with $L=5$ and ${L_{c}}=3$.

In this work, it is assumed that $E$ is interested only in $U_{1}$’s message
222 Since the analysis for both users is interchangeable, the secrecy
performance of the scenarios in which $E$ is interested in $U_{2}$’s message
can be obtained based on the results of this manuscript. . Moreover, for
tractability of analysis, ${\Delta_{1,o}}$ (${1\leq o<L}$) is represented by
${\Delta_{o}}$, the users’ locations are assumed to be fixed in this work 333
The secrecy performance of scenarios wherein the users are randomly
distributed also can be easily obtained based on [32, (7)] and the results of
this work. . Depending on the relative positions of $U_{1}$ and $E$, as shown
in Fig. 2, there are four scenarios for the SINR of $E$ 444 Except for the
four scenarios considered here, there is another scenario wherein there is no
overlapped path between $E$ and $U_{1}$ and both the common and private
messages are secure. , where ${\Delta_{L-{L_{e1}}}}$ signifies the angular
range of $U_{1}$ between the $\left({L-{L_{e1}}}\right)$th and
$\left({L-{L_{e1}}+1}\right)$th spatially resolvable paths, similarly,
${\Delta_{L-{L_{p}}-{L_{ec}}}}$ signifies the angular range of $U_{1}$ between
the $\left({L-{L_{p}}-{L_{ec}}}\right)$th and
$\left({L-{L_{p}}-{L_{ec}}}+1\right)$th spatially resolvable paths.

1) Scenario I: ${\theta_{e,\min}}\in{\Delta_{L-{L_{e1}}}}$

As show in Fig. 2(a), only the private stream of $U_{1}$ is wiretapped by $E$,
which denotes ${\gamma_{e,c}^{\mathrm{I}}}=0$. The SINR of $E$ in
eavesdropping private stream of $U_{1}$ is given as
${\gamma_{e,p1}^{\mathrm{I}}}={\delta_{e1}}{\left\|{{{\mathbf{g}}_{e,p1}}}\right\|^{2}},$
where ${\delta_{e1}}=\delta{\tau_{1}}r_{e}^{-\alpha}$.

2) Scenario II: ${\theta_{e,\min}}\in{\Delta_{L-{L_{p}}-{L_{ec}}}}$

This scenario is shown in Fig. 2(b), where in both common and private streams
of $U_{1}$ are eavesdropped by $E$. More Specifically, all non-overlapping
resolvable paths of $U_{1}$ is wiretapped and overlapped paths of $U_{1}$ is
partially wiretapped. Like [33], it is assumed that illegitimate has the same
decoding capability as the legitimate users555 There is another eavesdropping
scenario, called the worst-case security scenario, considered in many works,
such as [33, 34]. Specifically, by utilizing specific detection techniques,
the data stream received can be distinguished by the eavesdropper by
subtracting interference generated by the superposed signals from each other.
This assumption has been utilized in a reasonable amount of literature focused
on the physical layer security of NOMA systems. In fact, this assumption
simplified the analysis but overestimated the eavesdropper’s multi-user
decodability. . According to the RSMA principle, $E$ decodes common streams of
$U_{1}$ firstly by treating all the other signals as noise, and then decodes
private streams of $U_{1}$ . Thus, the SINR of $E$ in eavesdropping both the
common stream and private stream of $U_{1}$ are given as

${\gamma_{e,c}^{\mathrm{II}}}=\frac{{{\delta_{ec}}{{\left\|{{{\mathbf{g}}_{e,c}}}\right\|}^{2}}}}{{{\delta_{e1}}{{\left\|{{{\mathbf{g}}_{e,p1}}}\right\|}^{2}}+1}},$
(4)
${\gamma_{e,p1}^{\mathrm{II}}}={\delta_{e1}}{\left\|{{{\mathbf{g}}_{e,p1}}}\right\|^{2}},$
(5)

respectively, where ${\delta_{ec}}=\delta{\tau_{c}}r_{e}^{-\alpha}$.

3) Scenario III: ${\theta_{e,\max}}\in{\Delta_{{L_{ec}}+{L_{e1}}}}$

As show in Fig. 2(c), all the overlapped paths and part of the non-overlapping
paths for $U_{1}$ is intercepted. Similarly, the $E$ firstly decodes common
streams and then decodes private streams of $U_{1}$. Thus, the SINR of $E$ in
eavesdropping the common stream and private stream of $U_{1}$ are given as

${\gamma_{e,c}^{\mathrm{III}}}=\frac{{{\delta_{ec}}{{\left\|{{{\mathbf{g}}_{e,c}}}\right\|}^{2}}}}{{{\delta_{e1}}{{\left\|{{{\mathbf{g}}_{e,p1}}}\right\|}^{2}}+{\delta_{e2}}{{\left\|{{{\mathbf{g}}_{e,p2}}}\right\|}^{2}}+1}},$
(6)

and

${\gamma_{e,p1}^{\mathrm{III}}}=\frac{{{\delta_{e1}}{{\left\|{{{\mathbf{g}}_{e,p1}}}\right\|}^{2}}}}{{{\delta_{e2}}{{\left\|{{{\mathbf{g}}_{e,p2}}}\right\|}^{2}}+1}},$
(7)

respectively, where ${\delta_{e2}}=\delta{\tau_{2}}r_{e}^{-\alpha}$.

4) Scenario IV: ${\theta_{e,\max}}\in{\Delta_{{L_{ec}}}}$

This case is shown in Fig. 2(d) wherein only the common stream of $U_{1}$ is
eavesdropped by $E$, which denotes ${\gamma_{e,p1}^{\mathrm{IV}}}=0$ and

${\gamma_{e,c}^{\mathrm{IV}}}=\frac{{{\delta_{ec}}{{\left\|{{{\mathbf{g}}_{e,c}}}\right\|}^{2}}}}{{{\varpi_{0}}{\delta_{e2}}{{\left\|{{{\mathbf{g}}_{e,p2}}}\right\|}^{2}}+1}}.$
(8)

To facilitate analysis, we define
${X_{lq}}={\left\|{{{\mathbf{g}}_{l,q}}}\right\|^{2}}$
$\left({\left({l,q}\right)\in\left\\{{\left({i,c}\right),\left({i,p}\right),\left({e,c1}\right),\left({e,p1}\right),\left({e,p2}\right)}\right\\}}\right)$.
The PDF and CDF of ${X_{l,q}}$ are expressed as

${f_{{X_{lq}}}}\left(x\right)=\frac{{{e^{-x}}{x^{{\kappa_{lq}}-1}}}}{{\Gamma\left({{\kappa_{lq}}}\right)}},$
(9)

and

${F_{{X_{lq}}}}\left(x\right)=1-{e^{-x}}\sum\limits_{t=0}^{{\kappa_{lq}}-1}{\frac{{{x^{t}}}}{{t!}}},$
(10)

respectively, where ${\kappa_{1c}}={\kappa_{2c}}={L_{c}}$,
${\kappa_{1p}}={\kappa_{2p}}={L_{p}}$, ${\kappa_{ec1}}={L_{ec}}$,
${\kappa_{ep1}}={L_{e1}}$, and ${\kappa_{ep2}}={L_{e2}}$.

##  III Secrecy Outage Probability Analysis

In this work, $U_{1}$ is secure only when both the common and private stream
are secrecy. Thus, the SOP of $U_{1}$ in the $j$th
$\left({j\in\left\\{{{\mathrm{I}},{\mathrm{II}},{\mathrm{III}},{\mathrm{IV}}}\right\\}}\right)$
scenario is expressed as

${P_{{\mathrm{sop}},1}^{j}}=1-\underbrace{\Pr\left\\{{C_{1,c}^{{\mathrm{s}},j}>R_{1,c}^{{\mathrm{th}}},C_{1,p}^{\mathrm{s},j}>R_{1,p}^{{\mathrm{th}}}}\right\\}}_{\buildrel\Delta\over{=}{P_{{\mathrm{scp}},1}^{j}}},$
(11)

where
${C_{1,c}^{\mathrm{s},j}}={\left[{{{\log}_{2}}\left({1+{\gamma_{1,c}}}\right)-{{\log}_{2}}\left({1+{\gamma_{e,c}^{j}}}\right)}\right]^{+}}$
denotes instantaneous secrecy capacity of common streams intended to $U_{1}$,
${C_{1,p}^{\mathrm{s},j}}={\left[{{{\log}_{2}}\left({1+{\gamma_{1,p}}}\right)-{{\log}_{2}}\left({1+{\gamma_{e,p1}^{j}}}\right)}\right]^{+}}$
denotes instantaneous secrecy capacity of private streams transmitted to the
$U_{1}$, ${\left[x\right]^{+}}=\max\left\\{{x,0}\right\\}$, and
$R_{1,c}^{\mathrm{th}}$ and $R_{1,p}^{\mathrm{th}}$ signify the secrecy rate
threshold for the common and private messages, respectively, and
${P_{{\mathrm{scp}},1}^{j}}$ denotes the secrecy connection probability (SCP),
which is the complementary of SOP. It should be noted that, when $E$ can only
wiretap the common streams or private streams, the SCP of $U_{1}$ is
degenerated to
${P_{{{\mathrm{scp}}},1}^{j}}=\Pr\left\\{{{C_{1,c}^{{\mathrm{s}},j}}>{R_{1,c}^{\mathrm{th}}}}\right\\}$
or
${P_{{{\mathrm{scp}}},1}^{j}}=\Pr\left\\{{{C_{1,p}^{{\mathrm{s}},j}}>{R_{1,p}^{\mathrm{th}}}}\right\\}$
respectively.

1) Scenario I: ${\theta_{e,\min}}\in{\Delta_{L-{L_{e1}}}}$

In this case, $E$ only eavesdrop on private streams of $U_{1}$. Based on (9),
(10) and (11), utilizing [36, (3.351.3)],
${P_{{\mathrm{scp}},1}^{\mathrm{I}}}$ is obtained as

$\displaystyle{P_{{\mathrm{scp}},1}^{\mathrm{I}}}$
$\displaystyle=\Pr\left\\{{{{\log}_{2}}\left({\frac{{1+{\gamma_{1,p}}}}{{1+{\gamma_{e,p1}^{\mathrm{I}}}}}}\right)>{R_{1,p}^{\mathrm{th}}}}\right\\}$
(12) $\displaystyle=\Pr\left\\{{{X_{1p}}>{A_{1}}{X_{ep1}}+{A_{2}}}\right\\}$
$\displaystyle={{\mathbb{E}}_{{X_{ep1}}}}\left[{{e^{-\left({{A_{1}}{X_{ep1}}+{A_{2}}}\right)}}\sum\limits_{t=0}^{{L_{p}}-1}{\frac{{{{\left({{A_{1}}{X_{ep1}}+{A_{2}}}\right)}^{t}}}}{{t!}}}}\right]$
$\displaystyle=\sum\limits_{t=0}^{{L_{p}}-1}{\sum\limits_{n=0}^{t}{\frac{{{e^{-{A_{2}}}}A_{1}^{n}A_{2}^{t-n}}}{{n!\left({t-n}\right)!\left({{L_{e1}}-1}\right)!}}}}$
$\displaystyle\times\int_{0}^{\infty}{{e^{-\left({{A_{1}}+1}\right)x}}}{x^{{L_{e1}}+n-1}}dx$
$\displaystyle=\sum\limits_{t=0}^{{L_{p}}-1}{\sum\limits_{n=0}^{t}{\frac{{{e^{-{A_{2}}}}A_{1}^{n}A_{2}^{t-n}\left({{L_{e1}}+n-1}\right)!}}{{n!\left({t-n}\right)!\left({{L_{e1}}-1}\right)!{{\left({{A_{1}}+1}\right)}^{{L_{e1}}+n}}}}}},$

where ${\mathbb{E}}\left[.\right]$ denotes the expectation operation,
${\Theta_{1,p}}={2^{{R_{1,p}^{\mathrm{th}}}}}$,
${A_{1}}=\frac{{{\Theta_{1,p}}{\delta_{e1}}}}{{{\delta_{1}}}}=\frac{{{\Theta_{1,p}}r_{1}^{\alpha}}}{{r_{e}^{\alpha}}}$,
and ${A_{2}}=\frac{{{\Theta_{1,p}}-1}}{{{\delta_{1}}}}$.

2) Scenario II: ${\theta_{e,\min}}\in{\Delta_{L-{L_{p}}-{L_{ec}}}}$

In this case, the private streams is eavesdropped completely while the common
streams is partly eavesdropped. Thus, the SCP of $U_{1}$ is expressed as

$\displaystyle{P_{{\mathrm{scp}},1}^{\mathrm{II}}}$
$\displaystyle=\Pr\left\\{{{{\log}_{2}}\left({\frac{{1+{\gamma_{1,c}}}}{{1+{\gamma_{e,c}^{\mathrm{II}}}}}}\right)>{R_{1,c}^{\mathrm{th}}},}\right.$
(13)
$\displaystyle\left.{{{\log}_{2}}\left({\frac{{1+{\gamma_{1,p}}}}{{1+{\gamma_{e,p1}^{\mathrm{II}}}}}}\right)>{R_{1,p}^{\mathrm{th}}}}\right\\}$
$\displaystyle=\Pr\left\\{{\frac{{1+\frac{{{\delta_{1,c}}{X_{1c}}}}{{{X_{1}}}}}}{{1+\frac{{{\delta_{ec}}{X_{ec}}}}{{{X_{2}}}}}}>{\Theta_{1,c}},\frac{{{X_{1}}}}{{{X_{2}}}}>{\Theta_{1,p}}}\right\\}$
$\displaystyle=\int_{1}^{\infty}{\int_{{{X}_{2}}{\Theta_{1,p}}}^{\infty}{{\Delta_{1}}{f_{{{X}_{1}}}}\left({{{X}_{1}}}\right)d{{X}_{1}}{f_{{{X}_{2}}}}\left({{{X}_{2}}}\right)d{{X}_{2}}}},$

where
${\Delta_{1}}=\int_{{{X}_{1}}{\eta_{1}}}^{\infty}{{F_{{X_{ec}}}}\left({\left({\frac{y}{{{{X}_{1}}{\eta_{2}}}}-\frac{{{\eta_{1}}}}{{{\eta_{2}}}}}\right){{X}_{2}}}\right){f_{{X_{1c}}}}\left(y\right)dy}$,
${{X}_{1}}=1+{\delta_{1}}{X_{1p}}$, ${{X}_{2}}=1+{\delta_{e1}}{X_{ep1}}$,
${\eta_{1}}=\frac{{{\Theta_{1,c}}-1}}{{{\delta_{1,c}}}}$,
${\eta_{2}}=\frac{{{\Theta_{1,c}}{\delta_{ec}}}}{{{\delta_{1,c}}}}$. Based on
(9) and (10) and utilizing [36, (3.351.2)], ${\Delta_{1}}$ is obtained as

$\displaystyle{\Delta_{1}}$
$\displaystyle={{\bar{F}}_{{X_{1c}}}}\left({{{X}_{1}}{\eta_{1}}}\right)$ (14)
$\displaystyle-\sum\limits_{t=0}^{{L_{ec}}-1}{\sum\limits_{m=0}^{t}{\sum\limits_{k=0}^{m+{L_{c}}-1}{\frac{{{A_{3}}{X}_{1}^{{L_{c}}}{X}_{2}^{t}{e^{-{{X}_{1}}{\eta_{1}}}}}}{{{{\left({{{X}_{2}}+{{X}_{1}}{\eta_{2}}}\right)}^{m+{L_{c}}-k}}}}}}},$

where
${A_{3}}=\frac{{{{\left({-1}\right)}^{t-m}}\left({m+{L_{c}}-1}\right)!{\eta_{1}}^{t-m+k}\eta_{2}^{m+{L_{c}}-k-t}}}{{k!m!\left({t-m}\right)!\Gamma\left({{L_{c}}}\right)}}$
and ${{\bar{F}}_{{X_{1c}}}}\left(x\right)=1-{F_{{X_{1c}}}}\left(x\right)$.
Define ${X_{I}}=1+\varepsilon X$, $\left({I=1,2}\right)$, the PDF of ${X_{I}}$
is obtained as

${f_{{X_{I}}}}\left(x\right)=\frac{{{e^{-\frac{{x-1}}{\varepsilon}}}{{\left({x-1}\right)}^{{\kappa_{X}}-1}}}}{{{\varepsilon^{{\kappa_{X}}}}\left({{\kappa_{X}}-1}\right)!}}.$
(15)

Then, ${P_{{\mathrm{scp}},1}^{\mathrm{II}}}$ is obtained as

$\displaystyle{P_{{\mathrm{scp}},1}^{\mathrm{II}}}$
$\displaystyle=\sum\limits_{t=0}^{{L_{c}}-1}{\frac{{\delta_{1}^{-{L_{p}}}\eta_{1}^{t}\delta_{e1}^{-{L_{e1}}}{\Delta_{2}}}}{{t!\left({{L_{p}}-1}\right)!\left({{L_{e1}}-1}\right)!}}}$
(16)
$\displaystyle-\sum\limits_{t=0}^{{L_{ec}}-1}{\sum\limits_{m=0}^{t}{\sum\limits_{k=0}^{m+{L_{c}}-1}{\sum\limits_{n=0}^{{L_{c}}}{A_{3}}{A_{4}}{\Delta_{3}}}}},$

where
${A_{4}}=\frac{{\left({{L_{c}}}\right)!{e^{-{\eta_{1}}}}\delta_{1}^{-{L_{p}}}\delta_{e1}^{-{L_{e1}}}}}{{n!\left({{L_{c}}-n}\right)!\left({{L_{p}}-1}\right)!\left({{L_{e1}}-1}\right)!}}$,
${\Delta_{2}}=\int_{1}^{\infty}{\int_{y{\Theta_{1,p}}}^{\infty}{{e^{-x{\eta_{1}}}}{x^{t}}{e^{-\frac{{x-1}}{{{\delta_{1}}}}}}{{\left({x-1}\right)}^{{L_{p}}-1}}dx}}{e^{-\frac{{y-1}}{{{\delta_{e1}}}}}}{\left({y-1}\right)^{{L_{e1}}-1}}dy$
and
${\Delta_{3}}=\int_{1}^{\infty}{\int_{y{\Theta_{1,p}}-1}^{\infty}{\frac{{{e^{-\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)z}}{z^{n+{L_{p}}-1}}dz}}{{{{\left({z{\eta_{2}}+y+{\eta_{2}}}\right)}^{m+{L_{c}}-k}}}}}\frac{{{y^{t}}{{\left({y-1}\right)}^{{L_{e1}}-1}}dy}}{{{e^{\frac{{y-1}}{{{\delta_{e1}}}}}}}}}$.

Utilizing [36, (3.351.2), (3.351.3)], ${\Delta_{2}}$ is obtained as (17),
shown at the top of this page, where
${A_{5}}=\frac{{\left({m+{L_{p}}-1}\right)!t!{e^{\frac{1}{{{\delta_{1}}}}-{\Theta_{1,p}}\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)}}{{\left({-1}\right)}^{k-i}}\Theta_{1,p}^{i}}}{{m!\left({t-m}\right)!{{\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)}^{m+{L_{p}}-k}}\left({k-i}\right)!j!\left({i-j}\right)!}}$.
Similarly, based on (16) and utilizing [36, (7.811.5)] and (25.4.39), we
obtain ${\Delta_{3}}$ as (18), shown at the top of next page, where
${A_{6}}=\frac{{\pi\sqrt{{b_{1}}-b_{1}^{2}}t!\left({n+{L_{p}}}\right)!\Theta_{1,p}^{j}{{\left({{\Theta_{1,p}}-1}\right)}^{n+{L_{p}}-j}}}}{{Ib_{1}^{{a_{3}}}q!\left({t-q}\right)!j!\left({n+{L_{p}}-j!}\right){e^{\left({\frac{{{\eta_{1}}}}{{{b_{1}}}}+\frac{1}{{{b_{1}}{\delta_{1}}}}}\right)\left({{\Theta_{1,p}}-1}\right)}}}}$,
${b_{1}}=\frac{1}{2}\left({1+\cos\frac{{\left({2i-1}\right)\pi}}{{2I}}}\right)$,
$G_{p,q}^{m,n}\left[\cdot\right]$ is the Meijer’s $G$-function as defined by
[36, (9.301)], $I$ is the summation items that reflect accuracy versus
complexity, ${a_{1}}=n+{L_{p}}+1-m-{L_{c}}+k$,
${a_{2}}=j+q+{L_{e1}}-m-{L_{c}}+k$,
${b_{2}}={\eta_{2}}\left({{\Theta_{1,p}}-1}\right)+{b_{1}}+{b_{1}}{\eta_{2}}$
and ${b_{3}}=\frac{{{b_{2}}}}{{{\eta_{2}}{\Theta_{1,p}}+{b_{1}}}}$.

$\displaystyle{\Delta_{2}}$
$\displaystyle=\sum\limits_{m=0}^{t}{\frac{{t!{e^{-{\eta_{1}}}}}}{{m!\left({t-m}\right)!}}}\int_{1}^{\infty}{\int_{y{\Theta_{1,p}}-1}^{\infty}{{e^{-\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)z}}{z^{m+{L_{p}}-1}}dz}}{e^{-\frac{{y-1}}{{{\delta_{e1}}}}}}{\left({y-1}\right)^{{L_{e1}}-1}}dy$
(17)
$\displaystyle=\sum\limits_{m=0}^{t}{\sum\limits_{k=0}^{m+{L_{p}}-1}{\sum\limits_{i=0}^{k}{\sum\limits_{j=0}^{i}{{A_{5}}}}}}\int_{0}^{\infty}{{e^{-\left({{\Theta_{1,p}}\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)+\frac{1}{{{\delta_{e1}}}}}\right)z}}{z^{j+{L_{e1}}-1}}dz}$
$\displaystyle=\sum\limits_{m=0}^{t}{\sum\limits_{k=0}^{m+{L_{p}}-1}{\sum\limits_{i=0}^{k}{\sum\limits_{j=0}^{i}{{A_{5}}}}}}\left({j+{L_{e1}}-1}\right)!{\left({{\Theta_{1,p}}\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)+\frac{1}{{{\delta_{e1}}}}}\right)^{-j-{L_{e1}}}}$
$\displaystyle{\Delta_{3}}$
$\displaystyle=\sum\limits_{i=1}^{I}{\frac{{\pi\sqrt{{b_{1}}-b_{1}^{2}}}}{{Ib_{1}^{{a_{1}}}}}}\int_{1}^{\infty}{\frac{{{e^{-\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)\frac{{y{\Theta_{1,p}}-1}}{{{b_{1}}}}}}{{\left({y{\Theta_{1,p}}-1}\right)}^{n+{L_{p}}}}{y^{t}}{{\left({y-1}\right)}^{{L_{e1}}-1}}}}{{{{\left({{\eta_{2}}\left({y{\Theta_{1,p}}-1}\right)+{b_{1}}y+{b_{1}}{\eta_{2}}}\right)}^{m+{L_{c}}-k}}{e^{\frac{{y-1}}{{{\delta_{e1}}}}}}}}dy}$
(18)
$\displaystyle=\sum\limits_{i=1}^{I}{\sum\limits_{j=0}^{n+{L_{p}}}{\sum\limits_{q=0}^{t}{{A_{6}}}}}\int_{0}^{\infty}{\frac{{{e^{-\left({\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)\frac{{{\Theta_{1,p}}}}{{{b_{1}}}}+\frac{1}{{{\delta_{e1}}}}}\right)u}}{u^{j+q+{L_{e1}}-1}}}}{{{{\left({\left({{\eta_{2}}{\Theta_{1,p}}+{b_{1}}}\right)u+{b_{2}}}\right)}^{m+{L_{c}}-k}}}}dy}$
$\displaystyle=\sum\limits_{i=1}^{I}{\sum\limits_{j=0}^{n+{L_{p}}}{\sum\limits_{q=0}^{t}{\frac{{{A_{6}}{b_{3}}^{{a_{2}}}G_{1,2}^{2,1}\left[{\left({\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)\frac{{{\Theta_{1,p}}}}{{{b_{1}}}}+\frac{1}{{{\delta_{e1}}}}}\right){b_{3}}\left|{{}_{-{a_{2}},0}^{1-j-q-{L_{e1}}}}\right.}\right]}}{{{{\left({{\eta_{2}}{\Theta_{1,p}}+{b_{1}}}\right)}^{m+{L_{c}}-k}}\Gamma\left({m+{L_{c}}-k}\right)}}}}}$

3) Scenario III: ${\theta_{e,\max}}\in{\Delta_{{L_{ec}}+{L_{e1}}}}$

In this case, the common streams is eavesdropped completely while the private
streams is eavesdropped partly. Thus, ${P_{{\mathrm{scp}},1}^{\mathrm{III}}}$
is expressed as

$\displaystyle{P_{{\mathrm{scp}},1}^{\mathrm{III}}}$ (19)
$\displaystyle=\Pr\left\\{{{{\log}_{2}}\left({\frac{{1+{\gamma_{1,c}}}}{{1+{\gamma_{e,c}^{\mathrm{III}}}}}}\right)>{R_{1,c}^{\mathrm{th}}},{{\log}_{2}}\left({\frac{{1+{\gamma_{1,p}}}}{{1+{\gamma_{e,p1}^{\mathrm{III}}}}}}\right)>{R_{1,p}^{\mathrm{th}}}}\right\\}$
$\displaystyle=\Pr\left\\{{{X_{ec}}<\left({\frac{{{X_{1c}}}}{{{{X}_{1}}{\eta_{2}}}}-\frac{{{\eta_{1}}}}{{{\eta_{2}}}}}\right)\left({{{X}_{3}}+{\delta_{e1}}{X_{ep1}}}\right),{X_{ep1}}<}\right.$
$\displaystyle\;\quad\;\quad\left.{\left({\frac{{{X_{1}}}}{{{\Theta_{1,p}}}}-1}\right)\frac{{{X_{3}}}}{{{\delta_{e1}}}},}{{X_{1c}}>{{X}_{1}}{\eta_{1}},{{X}_{1}}>{\Theta_{1,p}},{{X}_{3}}>1}\right\\}$
$\displaystyle={{\mathbb{E}}_{{X_{1c}},{{X}_{1}},{{X}_{3}}}}\left[{{\Delta_{4}}}\right],$

where ${{X}_{3}}=1+{\delta_{e2}}{X_{ep2}}$ and
${\Delta_{4}}=\int_{0}^{\frac{{Q{{X}_{3}}}}{{{\delta_{e1}}}}}{{F_{{X_{ec}}}}\left({P\left({{{X}_{3}}+{\delta_{e1}}y}\right)}\right){f_{{X_{ep1}}}}\left(y\right)dy}$,
$P=\frac{{{X_{1c}}}}{{{{X}_{1}}{\eta_{2}}}}-\frac{{{\eta_{1}}}}{{{\eta_{2}}}}$,
and $Q=\frac{{{{X}_{1}}}}{{{\Theta_{1,p}}}}-1$. Based on (9) and (10) and
utilizing [36, (3.351.1)], ${\Delta_{4}}$ is obtained as

$\displaystyle{\Delta_{4}}$
$\displaystyle=1-{e^{-\frac{{Q{{X}_{3}}}}{{{\delta_{e1}}}}}}\sum\limits_{t=0}^{{L_{e1}}-1}{\frac{1}{{t!}}{{\left({\frac{{Q{{X}_{3}}}}{{{\delta_{e1}}}}}\right)}^{t}}}$
(20)
$\displaystyle-\sum\limits_{t=0}^{{L_{ec}}-1}{\sum\limits_{m=0}^{t}{\frac{{{B_{1}}{P^{t}}{X}_{3}^{t-m}{e^{-P{{X}_{3}}}}}}{{{{\left({P{\delta_{e1}}+1}\right)}^{m+{L_{e1}}}}}}}}$
$\displaystyle+\sum\limits_{t=0}^{{L_{ec}}-1}{\sum\limits_{m=0}^{t}{\sum\limits_{k=0}^{m+{L_{e1}}-1}{\frac{{{B_{1}}{P^{t}}{{\left({\frac{{Q{{X}_{3}}}}{{{\delta_{e1}}}}}\right)}^{k}}{e^{-\frac{{{{X}_{1}}}}{{{\Theta_{1,p}}}}P{{X}_{3}}}}}}{{k!{X}_{3}^{m-t}{e^{\frac{{Q{{X}_{3}}}}{{{\delta_{e1}}}}}}{{\left({P{\delta_{e1}}+1}\right)}^{m+{L_{e1}}-k}}}}}}},$

where
${B_{1}}=\frac{{\delta_{e1}^{m}\left({m+{L_{e1}}-1}\right)!}}{{m!\left({t-m}\right)!\Gamma\left({{L_{e1}}}\right)}}$.
Substituting (20) into (19), ${P_{{\mathrm{scp}},1}^{\mathrm{III}}}$ is
obtained as

$\displaystyle{P_{{\mathrm{scp}},1}^{\mathrm{III}}}$
$\displaystyle={\Delta_{5}}-\sum\limits_{t=0}^{{L_{e1}}-1}{\frac{1}{{t!}}{\Delta_{6}}}-\sum\limits_{t=0}^{{L_{ec}}-1}{\sum\limits_{m=0}^{t}{{\Delta_{7}}}}$
(21)
$\displaystyle+\sum\limits_{t=0}^{{L_{ec}}-1}{\sum\limits_{m=0}^{t}{\sum\limits_{k=0}^{m+{L_{e1}}-1}{{\Delta_{8}}}}},$

where
${\Delta_{5}}=\int_{1}^{\infty}{\int_{{\Theta_{1,p}}}^{\infty}{\int_{y{\eta_{1}}}^{\infty}{{f_{{X_{1c}}}}}\left(x\right)dx{f_{{{X}_{1}}}}\left(y\right)dy{f_{{{X}_{3}}}}\left(z\right)dz}}$,
${\Delta_{6}}={{\mathbb{E}}_{{X_{1c}},{{X}_{1}},{{X}_{3}}}}\left[{{e^{-\left({\frac{{{{X}_{1}}}}{{{\Theta_{1,p}}}}-1}\right)\frac{{{{X}_{3}}}}{{{\delta_{e1}}}}}}{{\left({\left({\frac{{{{X}_{1}}}}{{{\Theta_{1,p}}}}-1}\right)\frac{{{{X}_{3}}}}{{{\delta_{e1}}}}}\right)}^{t}}}\right]$,
${\Delta_{7}}={{\mathbb{E}}_{{X_{1c}},{{X}_{1}},{{X}_{3}}}}\left[{\frac{{{B_{1}}{X}_{3}^{t-m}{{\left({\frac{{{X_{1c}}}}{{{{X}_{1}}{\eta_{2}}}}-\frac{{{\eta_{1}}}}{{{\eta_{2}}}}}\right)}^{t}}{e^{-\left({\frac{{{X_{1c}}}}{{{{X}_{1}}{\eta_{2}}}}-\frac{{{\eta_{1}}}}{{{\eta_{2}}}}}\right){{X}_{3}}}}}}{{{{\left({\left({\frac{{{X_{1c}}}}{{{{X}_{1}}{\eta_{2}}}}-\frac{{{\eta_{1}}}}{{{\eta_{2}}}}}\right){\delta_{e1}}+1}\right)}^{m+{L_{e1}}}}}}}\right]$,
and
${\Delta_{8}}={{\mathbb{E}}_{{X_{1c}},{{X}_{1}},{{X}_{3}}}}\left[{\frac{{{B_{1}}{{\left({\left({\frac{{{X_{1}}}}{{{\Theta_{1,p}}}}-1}\right)\frac{{{X_{3}}}}{{{\delta_{e1}}}}}\right)}^{k}}{{\left({\frac{{{X_{1c}}}}{{{X_{1}}{\eta_{2}}}}-\frac{{{\eta_{1}}}}{{{\eta_{2}}}}}\right)}^{t}}}}{{{{\left({\left({\frac{{{X_{1c}}}}{{{X_{1}}{\eta_{2}}}}-\frac{{{\eta_{1}}}}{{{\eta_{2}}}}}\right){\delta_{e1}}+1}\right)}^{m+{L_{e1}}-k}}}}}\right.$
$\times\left.{\frac{{{e^{-\frac{{{X_{1}}}}{{{\Theta_{1,p}}}}\left({\frac{{{X_{1c}}}}{{{X_{1}}{\eta_{2}}}}-\frac{{{\eta_{1}}}}{{{\eta_{2}}}}}\right){X_{3}}}}}}{{k!X_{3}^{m-t}{e^{\left({\frac{{{X_{1}}}}{{{\Theta_{1,p}}}}-1}\right)\frac{{{X_{3}}}}{{{\delta_{e1}}}}}}}}}\right]$.

Utilizing [36, (3.351.2)], ${\Delta_{5}}$ is obtained as

$\displaystyle{\Delta_{5}}$
$\displaystyle=\int_{1}^{\infty}{\int_{{\Theta_{1,p}}}^{\infty}{{{\bar{F}}_{{X_{1c}}}}\left({y{\eta_{1}}}\right){f_{{{X}_{1}}}}\left(y\right)dy{f_{{{X}_{3}}}}\left(z\right)dz}}$
(22)
$\displaystyle=\sum\limits_{t=0}^{{L_{c}}-1}{\frac{{\eta_{1}^{t}}}{{t!}}}\int_{{\Theta_{1,p}}}^{\infty}{\frac{{{e^{-y{\eta_{1}}}}{y^{t}}{e^{-\frac{{y-1}}{{{\delta_{1}}}}}}{{\left({y-1}\right)}^{{L_{p}}-1}}}}{{\delta_{1}^{{L_{p}}}\left({{L_{p}}-1}\right)!}}dy}$
$\displaystyle=\sum\limits_{t=0}^{{L_{c}}-1}{\sum\limits_{m=0}^{t}{\frac{{\eta_{1}^{t}{e^{-{\eta_{1}}}}\Gamma\left({m+{L_{p}},\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)\left({{\Theta_{1,p}}-1}\right)}\right)}}{{\delta_{1}^{{L_{p}}}m!\left({t-m}\right)!\left({{L_{p}}-1}\right)!{{\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)}^{m+{L_{p}}}}}}}}.$

Based on (10) and (15) and utilizing [36, (3.351.3), (7.811.5)],
${\Delta_{6}}$ is obtained as

$\displaystyle{\Delta_{6}}$
$\displaystyle=\sum\limits_{m=0}^{{L_{c}}-1}{\sum\limits_{n=0}^{m}{\sum\limits_{i=0}^{{L_{p}}-1}{{B_{2}}}}}\int_{1}^{\infty}{\int_{0}^{\infty}{{e^{-\left({z+\frac{{{\Theta_{1,p}}{\delta_{e1}}}}{{{\delta_{1}}}}+{\Theta_{1,p}}{\delta_{e1}}{\eta_{1}}}\right)u}}}}$
(23)
$\displaystyle\times{u^{t+n+i}}du{z^{t}}\frac{{{e^{-\frac{{z-1}}{{{\delta_{e2}}}}}}{{\left({z-1}\right)}^{{L_{e2}}-1}}}}{{\delta_{e2}^{{L_{e2}}}\left({{L_{e2}}-1}\right)!}}dz$
$\displaystyle=\sum\limits_{m=0}^{{L_{c}}-1}{\sum\limits_{n=0}^{m}{\sum\limits_{i=0}^{{L_{p}}-1}{\sum\limits_{j=0}^{t}{\frac{{{B_{2}}\left({t+n+i}\right)!t!{\Xi_{1}}}}{{j!\left({t-j}\right)!\delta_{e2}^{{L_{e2}}}\left({{L_{e2}}-1}\right)!}}}}}},$

where
${B_{2}}=\frac{{{{\left({{\Theta_{1,p}}-1}\right)}^{{L_{p}}-1-i}}\Theta_{1,p}^{m+1+i}\delta_{e1}^{i+n+1}\eta_{1}^{m}}}{{i!\left({{L_{p}}-1-i}\right)!n!\left({m-n}\right)!\delta_{1}^{{L_{p}}}{e^{{\eta_{1}}{\Theta_{1,p}}+\frac{{{\Theta_{1,p}}-1}}{{{\delta_{1}}}}}}}}$,
${\Xi_{1}}=\frac{{{{\left({1+{\eta_{3}}}\right)}^{{a_{4}}-{a_{3}}}}}}{{\Gamma\left({{a_{3}}}\right)}}G_{1,2}^{2,1}\left[{\frac{{\left({1+{\eta_{3}}}\right)}}{{{\delta_{e2}}}}\left|{{}_{{a_{3}}-{a_{4}},0}^{1-{a_{4}}}}\right.}\right]$,
${a_{3}}=t+n+i+1$, ${a_{4}}=j+{L_{e2}}$, and
${\eta_{3}}=\frac{{{\Theta_{1,p}}{\delta_{e1}}}}{{{\delta_{1}}}}+{\Theta_{1,p}}{\delta_{e1}}{\eta_{1}}$.
Based on (9) and (15), ${\Delta_{7}}$ is obtained as

$\displaystyle{\Delta_{7}}$
$\displaystyle={B_{1}}{{\mathbb{E}}_{{{X}_{1}},{{X}_{3}}}}\left[{\frac{{X_{3}^{t-m}}}{{\Gamma\left({{L_{c}}}\right)}}\int_{{X_{1}}{\eta_{1}}}^{\infty}{\frac{{{{\left({\frac{x}{{{X_{1}}{\eta_{2}}}}-\frac{{{\eta_{1}}}}{{{\eta_{2}}}}}\right)}^{t}}{e^{-x}}{x^{{L_{c}}-1}}}}{{{{\left({\left({\frac{x}{{{X_{1}}{\eta_{2}}}}-\frac{{{\eta_{1}}}}{{{\eta_{2}}}}}\right){\delta_{e1}}+1}\right)}^{m+{L_{e1}}}}}}}}\right.$
(24)
$\displaystyle\times\left.{{e^{-\left({\frac{x}{{{X_{1}}{\eta_{2}}}}-\frac{{{\eta_{1}}}}{{{\eta_{2}}}}}\right){X_{3}}}}dx}\right]$
$\displaystyle=\sum\limits_{n=0}^{{L_{c}}-1}{{B_{3}}}{{\mathbb{E}}_{{{X}_{1}},{{X}_{3}}}}{\left[{\frac{{X_{3}^{t-m}X_{1}^{{L_{c}}}{e^{-{X_{1}}{\eta_{1}}}}}}{{\Gamma\left({{L_{c}}}\right)}}}\right.}$
$\displaystyle\times{\left.{\int_{0}^{\infty}{\frac{{{e^{-\left({{X_{3}}+{X_{1}}{\eta_{2}}}\right)u}}{u^{n+t}}}}{{{{\left({u{\delta_{e1}}+1}\right)}^{m+{L_{e1}}}}}}du}}\right]}$
$\displaystyle=\sum\limits_{n=0}^{{L_{c}}-1}{\frac{{{B_{\mathrm{3}}}\delta_{1}^{-{L_{p}}}\delta_{e2}^{-{L_{e2}}}{\Xi_{\mathrm{2}}}}}{{\Gamma\left({{L_{c}}}\right)\left({{L_{p}}-1}\right)!\left({{L_{e2}}-1}\right)!}}},$

where
${B_{3}}=\frac{{{B_{1}}\left({{L_{c}}-1}\right)!\eta_{1}^{{L_{c}}-1-n}\eta_{2}^{n+1}}}{{n!\left({{L_{c}}-1-n}\right)!}}$
and
${\Xi_{\mathrm{2}}}=\int_{1}^{\infty}{\frac{{{\nabla_{1}}{z^{t-m}}{{\left({z-1}\right)}^{{L_{e2}}-1}}}}{{{e^{\frac{{z-1}}{{{\delta_{e2}}}}}}}}}dz$,
${\nabla_{1}}=\int_{{\Theta_{1,p}}}^{\infty}{\int_{0}^{\infty}{\frac{{{e^{-\left({z+y{\eta_{2}}}\right)u}}{u^{n+t}}}}{{{{\left({u{\delta_{e1}}+1}\right)}^{m+{L_{e1}}}}}}du}}\frac{{{y^{{L_{c}}}}{{\left({y-1}\right)}^{{L_{p}}-1}}}}{{{e^{y{\eta_{1}}}}{e^{\frac{{y-1}}{{{\delta_{1}}}}}}}}dy$.
Then, utilizing [36, (7.811.5)] and (25.4.39), ${\nabla_{1}}$ is obtained as

$\displaystyle{\nabla_{1}}$
$\displaystyle=\sum\limits_{k=0}^{{L_{c}}}{\frac{{\left({{L_{c}}}\right)!\delta_{e1}^{-{a_{5}}}{e^{-{\eta_{1}}}}}}{{k!\left({{L_{c}}-k}\right)!\Gamma\left({m+{L_{e1}}}\right)}}}$
(25)
$\displaystyle\times\int_{{\Theta_{1,p}}-1}^{\infty}{G_{1,2}^{2,1}\left[{\frac{{z+\lambda{\eta_{2}}+{\eta_{2}}}}{{{\delta_{e1}}}}\left|{{}_{-{a_{5}},0}^{-n-t}}\right.}\right]}\frac{{{\lambda^{k+{L_{p}}-1}}}}{{{e^{\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)\lambda}}}}d\lambda$
$\displaystyle=\sum\limits_{k=0}^{{L_{c}}}{\frac{{\left({{L_{c}}}\right)!\delta_{e1}^{-{a_{5}}}{e^{-{\eta_{1}}}}}}{{k!\left({{L_{c}}-k}\right)!\Gamma\left({m+{L_{e1}}}\right)}}}$
$\displaystyle\times\int_{0}^{\frac{1}{{{\Theta_{1,p}}-1}}}{G_{1,2}^{2,1}\left[{\frac{{\left({z+{\eta_{2}}}\right)u+{\eta_{2}}}}{{{\delta_{e1}}u}}\left|{{}_{-{a_{5}},0}^{-n-t}}\right.}\right]}\frac{{{e^{-\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)\frac{1}{u}}}}}{{{u^{k+{L_{p}}+1}}}}du$
$\displaystyle=\sum\limits_{k=0}^{{L_{c}}}{\sum\limits_{i=1}^{N}{{B_{4}}}}G_{1,2}^{2,1}\left[{\frac{{{b_{4}}z+\left({{b_{4}}+1}\right){\eta_{2}}}}{{{\delta_{e1}}{b_{4}}}}\left|{{}_{-{a_{5}},0}^{-n-t}}\right.}\right],$

where ${a_{5}}=n+t+1-m-{L_{e1}}$,
${B_{4}}=\frac{{\pi\left({{L_{c}}}\right)!\delta_{e1}^{-{a_{5}}}{e^{-{\eta_{1}}}}\sqrt{\frac{1}{{{\Theta_{1,p}}-1}}-{b_{4}}}}}{{Nk!\left({{L_{c}}-k}\right)!\Gamma\left({m+{L_{e1}}}\right){e^{\left({{\eta_{1}}+\frac{1}{{{\delta_{1}}}}}\right)\frac{1}{{{b_{4}}}}}}{{\left({{b_{4}}}\right)}^{k+{L_{p}}+\frac{1}{2}}}}}$,
${b_{4}}=\frac{{1+\cos\frac{{\left({2i-1}\right)\pi}}{{2N}}}}{{2\left({{\Theta_{1,p}}-1}\right)}}$
and $N$ is the summation items that reflect accuracy versus complexity. Based
on (25) and utilizing (25.4.45), ${\Xi_{\mathrm{2}}}$ is obtained as (26),
shown at the top of next page, where $V$ is the summation items that reflect
accuracy versus complexity, ${t_{u}}$ is the $u$th zero of Leguerre
polynomials and ${w_{u}}$ is Gaussian weight, which are given in Table (25.9)
of [37]. With the same method, ${\Delta_{8}}$ is obtained as

$\displaystyle{\Xi_{\mathrm{2}}}$
$\displaystyle=\sum\limits_{k=0}^{{L_{c}}}{\sum\limits_{i=1}^{N}{\sum\limits_{j=0}^{t-m}{\frac{{\left({t-m}\right)!{B_{4}}}}{{j!\left({t-m-j!}\right)}}}}}\int_{0}^{\infty}{G_{1,2}^{2,1}\left[{\frac{{{b_{4}}f+{b_{4}}+\left({{b_{4}}+1}\right){\eta_{2}}}}{{{\delta_{e1}}{b_{4}}}}\left|{{}_{-{a_{5}},0}^{-n-t}}\right.}\right]{e^{-\frac{f}{{{\delta_{e2}}}}}}{f^{j+{L_{e2}}-1}}}df$
(26)
$\displaystyle=\sum\limits_{k=0}^{{L_{c}}}{\sum\limits_{i=1}^{N}{\sum\limits_{j=0}^{t-m}{\frac{{\left({t-m}\right)!{B_{4}}\delta_{e2}^{j+{L_{e2}}}}}{{j!\left({t-m-j!}\right)}}}}}\sum\limits_{u=1}^{V}{{w_{u}}}G_{1,2}^{2,1}\left[{\frac{{{b_{4}}{\delta_{e2}}{t_{u}}+{b_{4}}+\left({{b_{4}}+1}\right){\eta_{2}}}}{{{\delta_{e1}}{b_{4}}}}\left|{{}_{-{a_{5}},0}^{-n-t}}\right.}\right]t_{u}^{j+{L_{e2}}-1}$
$\displaystyle=\sum\limits_{k=0}^{{L_{c}}}{\sum\limits_{i=1}^{N}{\sum\limits_{j=0}^{t-m}{\sum\limits_{u=1}^{V}{\frac{{\left({t-m}\right)!{B_{4}}\delta_{e2}^{j+{L_{e2}}}{w_{u}}}}{{j!\left({t-m-j!}\right)}}t_{u}^{j+{L_{e2}}-1}G_{1,2}^{2,1}\left[{\frac{{{b_{4}}{\delta_{e2}}{t_{u}}+{b_{4}}+\left({{b_{4}}+1}\right){\eta_{2}}}}{{{\delta_{e1}}{b_{4}}}}\left|{{}_{-{a_{5}},0}^{-n-t}}\right.}\right]}}}}$

$\displaystyle{\Delta_{8}}$
$\displaystyle=\sum\limits_{n=0}^{{L_{c}}-1}{\frac{{{B_{3}}\delta_{1}^{-{L_{p}}}\delta_{e2}^{-{L_{e2}}}{\Xi_{\mathrm{3}}}}}{{k!\Gamma\left({{L_{c}}}\right)\left({{L_{p}}-1}\right)!\left({{L_{e2}}-1}\right)!}}},$
(27)

where
${\Xi_{\mathrm{3}}}=\int_{1}^{\infty}{\frac{{{\nabla_{2}}{{\left({z-1}\right)}^{{L_{e2}}-1}}}}{{{z^{m-t}}{e^{\frac{{z-1}}{{{\delta_{e2}}}}}}}}dz}$,
${\nabla_{2}}=\int_{{\Theta_{1,p}}}^{\infty}{\int_{0}^{\infty}{\frac{{{e^{-\left({\frac{y}{{{\Theta_{1,p}}}}z+y{\eta_{2}}}\right)u}}{u^{n+t}}}}{{{{\left({u{\delta_{e1}}+1}\right)}^{m+{L_{e1}}-k}}}}du}\frac{{{e^{-\left({\frac{y}{{{\Theta_{1,p}}}}-1}\right)\frac{z}{{{\delta_{e1}}}}}}{y^{{L_{c}}}}{{\left({y-1}\right)}^{{L_{p}}-1}}}}{{{{\left({\left({\frac{y}{{{\Theta_{1,p}}}}-1}\right)\frac{z}{{{\delta_{e1}}}}}\right)}^{-k}}{e^{y{\eta_{1}}}}{e^{\frac{{y-1}}{{{\delta_{1}}}}}}}}dy}$.
By utilizing [36, (7.811.5)] and (25.4.45), ${\nabla_{2}}$ is obtained as as
(28), shown at the top of next page, where
${B_{5}}=\frac{{\left({{L_{c}}}\right)!\left({{L_{p}}-1}\right)!\Theta_{1,p}^{{L_{c}}+j+1}{{\left({{\Theta_{1,p}}-1}\right)}^{{L_{p}}-1-j}}{e^{-\left({{\Theta_{1,p}}{\eta_{1}}+\frac{{{\Theta_{1,p}}-1}}{{{\delta_{1}}}}}\right)}}}}{{i!\left({{L_{c}}-i}\right)!j!\left({{L_{p}}-1-j}\right)!\delta_{e1}^{k+n+t+1}\Gamma\left({m+{L_{e1}}-k}\right)}}$,
${a_{6}}=m+{L_{e1}}-k-n-t-1$,
${b_{5}}={\delta_{e1}}{\Theta_{1,p}}{\eta_{1}}+{\delta_{e1}}\frac{{{\Theta_{1,p}}}}{{{\delta_{1}}}}$,
$K$ is the summation items that reflect accuracy versus complexity,
${{u_{v}}}$ is the $v$th zero of Leguerre polynomials and ${{q_{v}}}$ is
Gaussian weight, which are given in Table (25.9) of [37]. Based on (28) and
utilizing (25.4.45), $\Xi_{3}$ is obtained as

$\displaystyle{\nabla_{2}}$
$\displaystyle=\sum\limits_{i=0}^{{L_{c}}}{\sum\limits_{j=0}^{{L_{p}}-1}{{B_{5}}}}{z^{k}}\int_{0}^{\infty}{G_{1,2}^{2,1}\left[{\left({z+{\Theta_{1,p}}{\eta_{2}}}\right)\frac{{\left({\lambda+1}\right)}}{{{\delta_{e1}}}}\left|{{}_{{a_{6}},0}^{-n-t}}\right.}\right]}\frac{{{\lambda^{i+j+k}}}}{{{e^{\left({\frac{z}{{{\delta_{e1}}}}+{\Theta_{1,p}}{\eta_{1}}+\frac{{{\Theta_{1,p}}}}{{{\delta_{1}}}}}\right)\lambda}}}}d\lambda$
(28)
$\displaystyle=\sum\limits_{i=0}^{{L_{c}}}{\sum\limits_{j=0}^{{L_{p}}-1}{\sum\limits_{v=1}^{K}{\frac{{{q_{v}}{B_{5}}\delta_{e1}^{i+j+k+1}{z^{k}}}}{{{{\left({z+{b_{5}}}\right)}^{i+j+k+1}}}}}}}u_{v}^{i+j+k}G_{1,2}^{2,1}\left[{\left({z+{\Theta_{1,p}}{\eta_{2}}}\right)\left({\frac{{{u_{v}}}}{{z+{b_{5}}}}+\frac{1}{{{\delta_{e1}}}}}\right)\left|{{}_{{a_{6}},0}^{-n-t}}\right.}\right]$

$\displaystyle{\Xi_{\mathrm{3}}}$
$\displaystyle=\sum\limits_{i=0}^{{L_{c}}}{\sum\limits_{j=0}^{{L_{p}}-1}{\sum\limits_{v=1}^{K}{\sum\limits_{q=0}^{k+t-m}{\sum\limits_{u=1}^{D}{B_{6}}{\varepsilon_{u}}}}}}$
(29)
$\displaystyle\times{\frac{{G_{1,2}^{2,1}\left[{\left({\frac{{{s_{u}}{\delta_{e2}}}}{{{\delta_{e1}}}}+\frac{{{u_{v}}\left({{\Theta_{1,p}}{\eta_{2}}-{b_{5}}}\right)}}{{{\delta_{e2}}{s_{u}}+1+{b_{5}}}}+{b_{6}}}\right)\left|{{}_{{a_{6}},0}^{-n-t}}\right.}\right]}}{{{{\left({{\delta_{e2}}{s_{u}}+1+{b_{5}}}\right)}^{i+j+k+1}}\delta_{e2}^{-q-{L_{e2}}}s_{u}^{1-q-{L_{e2}}}}}},$

where
${B_{6}}=\frac{{\left({k+t-m}\right)!{q_{v}}{B_{5}}\delta_{e1}^{i+j+k+1}u_{v}^{i+j+k}}}{{q!\left({k+t-m-q}\right)}}$,
${b_{6}}={u_{v}}+\frac{{1+{\Theta_{1,p}}{\eta_{2}}}}{{{\delta_{e1}}}}$, $D$ is
the summation items that reflect accuracy versus complexity, ${s_{u}}$ is the
$u$th zero of Leguerre polynomials and ${\varepsilon_{u}}$ is Gaussian weight,
which are given in Table (25.9) of [37].

4) Scenario IV: ${\theta_{e,\max}}\in{\Delta_{{L_{ec}}}}$

In this case, $E$ only eavesdrop on common streams. It should noted that
${{\gamma_{1,c}}}$ and $\gamma_{e,c}^{\mathrm{IV}}$ are independent of each
other in this case. Based on (9) and (10) and utilizing [36, (3.351.3)], the
CDF of ${\gamma_{1,c}}$ and PDF of ${\gamma_{e,c}^{\mathrm{IV}}}$ are obtained
as

$\displaystyle{F_{{\gamma_{1,c}}}}\left(x\right)$
$\displaystyle=1-\sum\limits_{t=0}^{{L_{c}}-1}{\sum\limits_{m=0}^{t}{{\varpi_{1,c}}{x^{t}}{e^{-\frac{x}{{{\delta_{1,c}}}}}}{{\left({\frac{{{\delta_{1}}}}{{{\delta_{1,c}}}}x+1}\right)}^{-m-{L_{p}}}}}},$
(30)
$\displaystyle{f_{{\gamma_{e,c}}}}\left(y\right)=\frac{1}{{{\delta_{ec}}}}\sum\limits_{s=0}^{{L_{ec}}-1}{\sum\limits_{n=0}^{s}{\left({s+1-n}\right){\varpi_{e,c}}{y^{s}}{e^{-\frac{y}{{{\delta_{ec}}}}}}{{\left({\frac{{{\delta_{e2}}}}{{{\delta_{ec}}}}y+1}\right)}^{-n-{L_{e2}}}}}}$
(31)
$\displaystyle+\frac{{{\delta_{e2}}}}{{{\delta_{ec}}}}\sum\limits_{s=0}^{{L_{ec}}-1}{\sum\limits_{n=0}^{s}{\left({s+1-n}\right)\left({n+{L_{e2}}}\right){\varpi_{e,c}}{y^{s}}{e^{-\frac{y}{{{\delta_{ec}}}}}}{{\left({\frac{{{\delta_{e2}}}}{{{\delta_{ec}}}}y+1}\right)}^{-n-{L_{e2}}-{\mathrm{1}}}}}}$
$\displaystyle-\sum\limits_{s=0}^{{L_{ec}}-2}{\sum\limits_{n=0}^{s+1}{\frac{1}{{{\delta_{ec}}}}{\varpi_{e,c}}\left({s+1}\right){y^{s}}{e^{-\frac{y}{{{\delta_{ec}}}}}}{{\left({\frac{{{\delta_{e2}}}}{{{\delta_{ec}}}}y+1}\right)}^{-n-{L_{e2}}}}}},$

respectively, where
${\varpi_{1,c}}=\frac{{{\delta_{1}}^{m}\left({m+{L_{p}}-1}\right)!}}{{m!\delta_{1,c}^{t}\left({{L_{p}}-1}\right)!\left({t-m}\right)!}}$
and
${\varpi_{e,c}}=\frac{{\delta_{e2}^{n}\left({n+{L_{e2}}-1}\right)!}}{{n!\delta_{ec}^{s}\left({{L_{e2}}-1}\right)!\left({s+1-n}\right)!}}$.

To facilitate the following analysis, we define

$\displaystyle\phi\left({{c_{1}},{c_{2}},{c_{3}},{c_{4}},{c_{5}},{c_{6}}}\right)=\int_{0}^{\infty}{\frac{{{e^{-{c_{1}}y}}{y^{{c_{2}}}}}}{{{{\left({{c_{3}}y+1}\right)}^{{c_{4}}}}{{\left({{c_{5}}y+1}\right)}^{{c_{6}}}}}}dy}.$
(32)

By utilizing [38, (10), (11)], [36, (9.31.5)], and [39, (1.2)] in turn, we
obtain

$\displaystyle\phi\left({{c_{1}},{c_{2}},{c_{3}},{c_{4}},{c_{5}},{c_{6}}}\right)=\frac{{{{\left({{c_{1}}}\right)}^{-{c_{2}}}}}}{{\Gamma\left({{c_{4}}}\right)\Gamma\left({{c_{6}}}\right)}}$
(33)
$\displaystyle\times\int_{0}^{\infty}{G_{0,1}^{1,0}\left[{{c_{1}}y\left|{{}_{{c_{2}}}^{-}}\right.}\right]G_{1,1}^{1,1}\left[{{c_{3}}y\left|{{}_{0}^{1-{c_{4}}}}\right.}\right]G_{1,1}^{1,1}\left[{{c_{5}}y\left|{{}_{0}^{1-{c_{6}}}}\right.}\right]}dy$
$\displaystyle=\frac{{{{\left({{c_{1}}}\right)}^{-{c_{2}}}}}}{{\Gamma\left({{c_{4}}}\right)\Gamma\left({{c_{6}}}\right){c_{1}}}}$
$\displaystyle\times
H_{1,0:1,1:1,1}^{1,0:1,1:1,1}\left({{}_{-}^{{c_{2}}+1}\left|{{}_{0}^{1-{c_{4}}}\left|{{}_{0}^{1-{c_{6}}}\left|{\frac{{{c_{3}}}}{{{c_{1}}}},\frac{{{c_{5}}}}{{{c_{1}}}}}\right.}\right.}\right.}\right),$

where
${H_{c,d:p,r:\alpha,\beta}^{b,0:m,n:\gamma,\varepsilon}\left[\cdot\right]}$ is
the extended generalized bivariate Fox’s H-function as defined by [39,
(2.57)].

The SCP of $U_{1}$ in this scenario, ${P_{{\mathrm{scp}},1}^{\mathrm{IV}}}$ is
obtained as

$\displaystyle{P_{{\mathrm{scp}},1}^{\mathrm{IV}}}$
$\displaystyle=\Pr\left\\{{\frac{{1+{\gamma_{1,c}}}}{{1+{\gamma_{e,c}^{\mathrm{IV}}}}}>{\Theta_{1,c}}}\right\\}$
(34)
$\displaystyle=\int_{0}^{\infty}{{{\bar{F}}_{{\gamma_{1,c}}}}\left({{\Theta_{1,c}}y+{\Theta_{1,c}}-1}\right)}{f_{{\gamma_{e,c}^{\mathrm{IV}}}}}\left(y\right)dy$
$\displaystyle=\sum\limits_{t=0}^{{L_{c}}-1}{\sum\limits_{m=0}^{t}{{\varpi_{1,c}}}}\sum\limits_{s=0}^{{L_{ec}}-1}{\sum\limits_{n=0}^{s}{{\varpi_{e,c}}}}$
$\displaystyle\times\left({s+1-n}\right)\left({\frac{{{\Delta_{9}}}}{{{\delta_{ec}}}}+\frac{{{\delta_{e2}}}}{{{\delta_{ec}}}}\left({n+{L_{e2}}}\right){\Delta_{10}}}\right)$
$\displaystyle-\sum\limits_{t=0}^{{L_{c}}-1}{\sum\limits_{m=0}^{t}{{\varpi_{1,c}}}}\sum\limits_{s=0}^{{L_{ec}}-2}{\sum\limits_{n=0}^{s+1}{\frac{{{\varpi_{e,c}}}}{{{\delta_{ec}}}}\left({s+1}\right)}}{\Delta_{9}},$

where

$\displaystyle{\Delta_{9}}$
$\displaystyle={e^{-\frac{{{\Theta_{1,c}}-1}}{{{\delta_{1,c}}}}}}\sum\limits_{o=0}^{t}{\frac{{\Theta_{1,c}^{o}t!}}{{o!\left({t-o}\right)!}}}{\left({{\Theta_{1,c}}-1}\right)^{t-o}}$
(35)
$\displaystyle\times{\left({\frac{{{\delta_{1}}}}{{{\delta_{1,c}}}}\left({{\Theta_{1,c}}-1}\right)+1}\right)^{-m-{L_{p}}}}\phi\left({\frac{{{\Theta_{1,c}}}}{{{\delta_{1,c}}}}+\frac{1}{{{\delta_{ec}}}},s+o,}\right.$
$\displaystyle\left.{\frac{{{\delta_{1}}{\Theta_{1,c}}}}{{{\delta_{1}}\left({{\Theta_{1,c}}-1}\right)+{\delta_{1,c}}}},m+{L_{p}},\frac{{{\delta_{e2}}}}{{{\delta_{ec}}}},n+{L_{e2}}}\right),$

and

$\displaystyle{\Delta_{10}}$
$\displaystyle={e^{-\frac{{{\Theta_{1,c}}-1}}{{{\delta_{1,c}}}}}}\sum\limits_{o=0}^{t}{\frac{{\Theta_{1,c}^{o}t!}}{{o!\left({t-o}\right)!}}}{\left({{\Theta_{1,c}}-1}\right)^{t-o}}$
(36)
$\displaystyle\times{\left({\frac{{{\delta_{1}}}}{{{\delta_{1,c}}}}\left({{\Theta_{1,c}}-1}\right)+1}\right)^{-m-{L_{p}}}}\phi\left({\frac{{{\Theta_{1,c}}}}{{{\delta_{1,c}}}}+\frac{1}{{{\delta_{ec}}}},s+o,}\right.$
$\displaystyle\left.{\frac{{{\delta_{1}}{\Theta_{1,c}}}}{{{\delta_{1}}\left({{\Theta_{1,c}}-1}\right)+{\delta_{1,c}}}},m+{L_{p}},\frac{{{\delta_{e2}}}}{{{\delta_{ec}}}},n+{L_{e2}}+1}\right).$

The analytical expressions provided in this section are complicated since many
factors affect the secrecy performance of $U_{1}$, specifically, the transmit
power, power allocation coefficient, the target data rate, and relative
locations between all the legitimate users and illegitimate receivers.

##  IV Numerical Results

This section presents simulation and numerical results to verify the secrecy
outage performance of mmWave RSMA systems with the considered beamforming
scheme. The noise power is set at ${\sigma^{2}}=-71$ dBm, and the path-loss
model is set as $\alpha=4.14$ [21, 25, 35]. In all the figures, ‘Sim’ and
‘Ana’ denote the simulation and numerical results, respectively.

(a) ${P_{{\mathrm{sop}},1}^{\mathrm{I}}}$ for varying $r_{1}$ with
${L_{e1}}=2$.

(b) ${P_{{\mathrm{sop}},1}^{\mathrm{II}}}$ for varying $r_{1}$ with
${L_{ec}}=2$ and ${L_{e1}}=L_{p}$.

(c) ${P_{{\mathrm{sop}},1}^{\mathrm{III}}}$ for varying $r_{1}$ with
${L_{e1}}=1$, ${L_{ec}}=L_{c}$ and ${L_{e2}}=L-{L_{ec}}-{L_{e1}}$.

(d) ${P_{{\mathrm{sop}},1}^{\mathrm{IV}}}$ for varying $r_{1}$ with
${L_{ec}}=2$ and ${L_{e2}}=L-{L_{ec}}$.

Figure 3: SOP versus the $P$ with $\tau_{c}=\tau_{1}=\tau_{2}=\frac{1}{3}$,
$r_{e}=30$, $L=8$, $L_{c}=4$, and
$R_{{1,c}}^{{\mathrm{th}}}=R_{1,p}^{\mathrm{th}}=0.1$.

Fig. 3 demonstrates the impact of $P$ for varying $r_{1}$ on SOP. In Fig.
3(a), one can easily observe that the secrecy outage performance of the mmWave
RSMA systems is enhanced while increasing $P$. There is a floor for the SOP,
which is independent of $P$, which has been testified in [40] and stated in
Remark 1. Furthermore, the SOP with lower $r_{1}$ outperforms that with larger
$r_{1}$ since lower $r_{1}$ denotes weak path loss on $U_{1}$. One interesting
result is found from Fig.s 3(b) \- 3(d) that the secrecy outage performance of
$U_{1}$ initially decreases as $P$ increases and then increases to a constant.
The reason is given as follows. In Fig. 3(b), part of the common stream and
all the private streams are wiretapped, and decoding the common stream is the
bottleneck in the scenarios with lower $r_{1}$ and decoding the private stream
is the bottleneck in the scenarios with larger $r_{1}$. However, in Fig. 3(c),
all of the common stream and the private stream are wiretapped, and decoding
the private stream is the bottleneck in all the scenarios. In Fig. 3(d), only
part of the common stream is wiretapped. Moreover, it can be found that the
SOP floor is independent of $r_{1}$ in the scenarios where the bottleneck is
decoding the common stream. Based on all the subfigures in Fig. 3, we also
find that the effect from the location of $E$ on the difference between the
SOP of $U_{1}$ with different $r_{1}$ is weakening.

(a) ${P_{{\mathrm{sop}},1}^{\mathrm{I}}}$ for varying $L_{c}$ with
${L_{e1}}=2$.

(b) ${P_{{\mathrm{sop}},1}^{\mathrm{II}}}$ for varying $L_{c}$ with
${L_{ec}}=2$, ${L_{e1}}=L_{p}$.

(c) ${P_{{\mathrm{sop}},1}^{\mathrm{III}}}$ for varying $L_{c}$ with
${L_{e1}}=1$, ${L_{ec}}=L_{c}$, ${L_{e2}}=L-{L_{ec}}-{L_{e1}}$.

(d) ${P_{{\mathrm{sop}},1}^{\mathrm{IV}}}$ for varying $L_{c}$ with
${L_{ec}}=2$, ${L_{e2}}=L-{L_{ec}}$.

Figure 4: SOP versus the $P$ with $N_{s}=50$,
$\tau_{c}=\tau_{1}=\tau_{2}=\frac{1}{3}$, $r_{1}=15$, $r_{e}=25$, $L=9$, and
$R_{1,c}^{\mathrm{th}}=R_{1,p}^{\mathrm{th}}=0.1$.

Fig. 4 presents the impact of $P$ for varying $L_{c}$ on the SOP of $U_{1}$.
In Fig. 4(a), one can observe the SOP increases as $L_{c}$ increases, which is
easy to follow since more significant $L_{c}$ leads to less $L_{P}$, thus,
${\gamma_{1,p}}$ become smaller and transmissions become more vulnerable to
intercept. In Figs. 4(b) and 4(c), one can observe that SOP initially
decreases and then increases in the lower-$P$ region as $L_{c}$ increases,
indicating an optimal $L_{c}$ in the lower-$P$ region to minimize the SOP of
$U_{1}$. This is because, in the scenarios with the lower $L_{c}$, the SOP of
$U_{1}$ depends on both the SINR/SNR of the common and private signals and
decoding the common stream is the RSMA system’s bottleneck; thus, the larger
the $L_{c}$, the larger the ${\gamma_{1,c}}$, and the smaller SOP. In the
scenarios with the larger $L_{c}$, decoding the private signals is the RSMA
system’s bottleneck, and the more significant $L_{c}$ leads to the smaller
${\gamma_{1,p}}$ and the larger SOP. In Fig. 4(d), it can be observed that the
secrecy performance of $U_{1}$ is improved as $L_{c}$ increases since larger
$L_{c}$ leads to ${\gamma_{1,c}}$.

(a) ${P_{{\mathrm{sop}},1}^{\mathrm{I}}}$ for varying $P$ with ${L_{e1}}=2$.

(b) ${P_{{\mathrm{sop}},1}^{\mathrm{II}}}$ for varying $P$ with ${L_{ec}}=2$,
${L_{e1}}=L_{p}$.

(c) ${P_{{\mathrm{sop}},1}^{\mathrm{III}}}$ for varying $P$ with ${L_{e1}}=1$,
${L_{ec}}=L_{c}$, ${L_{e2}}=L-{L_{ec}}-{L_{e1}}$.

(d) ${P_{{\mathrm{sop}},1}^{\mathrm{IV}}}$ for varying $P$ with ${L_{ec}}=2$,
${L_{e2}}=L-{L_{ec}}$.

Figure 5: SOP versus the $\tau_{1}$ and $\tau_{2}$
($\tau_{c}=1-\tau_{1}-\tau_{2}$) with $N_{s}=50$, $r_{1}=15$, $L=8$,
$L_{c}=4$, and $R_{1,c}^{\mathrm{th}}=R_{1,p}^{\mathrm{th}}=0.1$.

Fig. 5 demonstrates the impact of $P$ with varying $r_{e}$ on the SOP of
$U_{1}$. We observe that SOP decreases as $r_{e}$ increases; this is because
the path loss of $U_{1}$ is the main factor affecting SOP. In the
larger-$r_{e}$ region, SOP decreases as $P$ increases in Fig. 5(a) while SOP
initially decreases and then increases as $P$ increases Figs. 5(b) \- 5(d)
with the same reason as in Figs. 3(a) \- 3(d). In the low-$r_{e}$ region, we
observe that SOP decreases as $P$ increases in all the cases, which denotes
that more power can improve security performance.

(a) ${P_{{\mathrm{sop}},1}^{\mathrm{I}}}$ for varying $P$ with ${L_{e1}}=2$.

(b) ${P_{{\mathrm{sop}},1}^{\mathrm{II}}}$ for varying $P$ with ${L_{ec}}=2$,
${L_{e1}}=L_{p}$.

(c) ${P_{{\mathrm{sop}},1}^{\mathrm{III}}}$ for varying $P$ with ${L_{e1}}=1$,
${L_{ec}}=L_{c}$, ${L_{e2}}=L-{L_{ec}}-{L_{e1}}$.

(d) ${P_{{\mathrm{sop}},1}^{\mathrm{IV}}}$ for varying $P$ with ${L_{ec}}=2$,
${L_{e2}}=L-{L_{ec}}$.

Figure 6: SOP versus the $\tau_{c}$ with $N_{s}=50$, $\tau_{1}=\tau_{2}$,
$r_{1}=15$, $r_{e}=25$, $L=8$, $L_{c}=4$, and
$R_{1,c}^{\mathrm{th}}=R_{1,p}^{\mathrm{th}}=0.1$.

Fig. 6 plots the impact of $\tau_{c}$ with varying $P$ on the SOP of $U_{1}$.
We observe that SOP increases as $\tau_{c}$ in Fig. 6(a), which is easy to
follow since less power is allocated to private streams as increasing
$\tau_{c}$. In Figs. 6(b) and 6(c), one can observe SOP initially decreases
and then increases as $\tau_{c}$ increases, which denotes that there is an
optimal $\tau_{c}$ to minimize the SOP. This is because in the
lower-$\tau_{c}$ region, decoding the common streams is the bottleneck. Thus,
increasing $\tau_{c}$ will enhance the secrecy outage performance. In the
larger-$\tau_{c}$ region, decoding the private streams will be the bottleneck.
However, the power allocated to the private stream decreases, so the SOP
deteriorates. Moreover, the optimal $\tau_{c}$ is relative to $P$ and $L_{c}$.
Based on Fig. 6(d), the secrecy performance of $U_{1}$ improves with the
increasing $\tau_{c}$ since the SINR of common streams at $U_{1}$ increases as
$\tau_{c}$.

(a) ${P_{{\mathrm{sop}},1}^{\mathrm{II}}}$ for varying $R_{1,p}^{\mathrm{th}}$
with ${L_{ec}}=2$, ${L_{e1}}=L_{p}$.

(b) ${P_{{\mathrm{sop}},1}^{\mathrm{III}}}$ for varying
$R_{1,p}^{\mathrm{th}}$ with ${L_{e1}}=1$, ${L_{ec}}=L_{c}$,
${L_{e2}}=L-{L_{ec}}-{L_{e1}}$.

(c) ${P_{{\mathrm{sop}},1}^{\mathrm{II}}}$ for varying $R_{1,c}^{\mathrm{th}}$
with ${L_{ec}}=2$, ${L_{e1}}=L_{p}$.

(d) ${P_{{\mathrm{sop}},1}^{\mathrm{III}}}$ for varying
$R_{1,c}^{\mathrm{th}}$ with ${L_{e1}}=1$, ${L_{ec}}=L_{c}$,
${L_{e2}}=L-{L_{ec}}-{L_{e1}}$.

Figure 7: SOP versus the $\tau_{c}$ with $N_{s}=50$, $\tau_{1}=\tau_{2}$,
$r_{1}=15$, $r_{e}=25$, $L=8$, $L_{c}=4$, and
$R_{1,c}^{\mathrm{th}}=R_{1,p}^{\mathrm{th}}=0.1$.

Fig. 7 presents the impact of $\tau_{c}$ with varying
$R_{1,p}^{\mathrm{th}}$/$R_{1,c}^{\mathrm{th}}$ on the SOP of $U_{1}$. We
observe that SOP decreases as $R_{1,p}^{\mathrm{th}}$/$R_{1,c}^{\mathrm{th}}$
decreases, which is easy to follow since a more significant targeted secrecy
rate denotes higher security requirements. Moreover, the optimal $\tau_{c}$
depends on $R_{1,p}^{\mathrm{th}}$/$R_{1,c}^{\mathrm{th}}$. The smaller
$R_{1,p}^{\mathrm{th}}$/$R_{1,c}^{\mathrm{th}}$, the larger optimal
$\tau_{c}$, which is due to the low requirements and more power can be
allocated to the stream to be the system’s bottleneck. In Figs. 7(a) and 7(b),
in the large-$\tau_{c}$ region private signals is the bottleneck, so
$R_{1,p}^{\mathrm{th}}$ has a more pronounced impact on the SOP. However, in
Figs. 7(c) and 7(d), in the lower-$\tau_{c}$ region common signals is the
bottleneck, so $R_{1,c}^{\mathrm{th}}$ has a more pronounced impact on SOP.

(a) ${P_{{\mathrm{sop}},1}^{\mathrm{I}}}$ with ${L_{e1}}=2$.

(b) ${P_{{\mathrm{sop}},1}^{\mathrm{II}}}$ with ${L_{ec}}=2$,
${L_{e1}}=L_{p}$.

(c) ${P_{{\mathrm{sop}},1}^{\mathrm{III}}}$ with ${L_{e1}}=1$,
${L_{ec}}=L_{c}$, ${L_{e2}}=L-{L_{ec}}-{L_{e1}}$.

(d) ${P_{{\mathrm{sop}},1}^{\mathrm{IV}}}$ with ${L_{ec}}=2$,
${L_{e2}}=L-{L_{ec}}$.

Figure 8: SOP versus the $\tau_{1}$ and $\tau_{2}$ with $N_{s}=50$,
$P=10{\mathrm{th}}{dB}$, $r_{1}=15$, $r_{e}=25$, $L=8$, $L_{c}=4$, and
$R_{1,c}^{\mathrm{th}}=R_{1,p}^{\mathrm{th}}=0.1$.

To evaluate the performance of the RSMA-based mmWave systems, NOMA-based
mmWave systems is utilized as the benchmark in this work, which can be found
in Fig. 8, wherein the SOP of $U_{1}$ for varying $\tau_{c}$, $\tau_{1}$, and
$\tau_{2}$ is presented. Like [28], $P_{{\mathrm{sop}},1}^{\min}$ and
$P_{{\mathrm{sop}},1}^{{\mathrm{NOMA}},{{\min}}}$ represent the least
achievable SOP of RSMA and NOMA systems, respectively. We observe that
$P_{{\mathrm{sop}},1}^{\min}$ successively increases and then decreases from
Figs. 8(a) \- Fig. 8(d). The underlying reason is that the overlapped
resolvable paths between $U_{1}$ and $E$ become larger initially, and then
eavesdroppers can wiretap more confidential messages. Then, the overlapped
resolvable paths between $U_{1}$ and $E$ become smaller and also interfered
with by messages of $U_{2}$; thus, the secrecy performance of the mmWave RSMA
system is improved. Fig. 8(a) indicated that in the case of significant
differences of $L_{c}$ and $L_{e1}$, more power should be allocated to $s_{1}$
to improve secrecy performance. In Figs. 8(b) \- 8(d), one can observe that
the secrecy performance of common streams is easier to become a bottleneck;
thus, larger $\tau_{c}$ and more minor $\tau_{1}$ can achieve better secrecy
performance. In Figs. 8(c) and 8(d), a more significant fraction of power
should be allocated to $s_{2}$ relative to $s_{1}$ to confuse the
eavesdropper. If $\tau_{c}$ or $\tau_{1}$ in Figs. 8(a) \- 8(c) is close to
zero, the SOP would be close to 1 since with a low value of the users’ SINRs
are not sufficient for decoding the common stream or private stream; When
$\tau_{2}$ equals zero, the RSMA system will degenerate as a NOMA system. We
observe that the $P_{{\mathrm{sop}},1}^{\min}$ is less than or equal to
$P_{{\mathrm{sop}},1}^{{\mathrm{NOMA}},{{\min}}}$, the underlying reason in
Figs. 8(b) \- 8(c) is that common streams are easier to become a bottleneck.
More power is allocated to $s_{1}$ in NOMA systems than RSMA systems, and the
secrecy performance of common streams worsens as $\tau_{1}$ increases. In Fig.
8(d), when $\tau_{1}$ or $\tau_{2}$ is equal to zero, the RSMA system would
degenerate as a NOMA system, one can observe that
$P_{{\mathrm{sop}},1}^{\min}$ at $\tau_{2}=0$ is less than
$P_{{\mathrm{sop}},1}^{{\mathrm{NOMA}},{{\min}}}$. In comparison,
$P_{{\mathrm{sop}},1}^{\min}$ at $\tau_{1}=0$ is larger than
$P_{{\mathrm{sop}},1}^{{\mathrm{NOMA}},{{\min}}}$, the reason is that as
$\tau_{1}=0$ and ${\tau_{2}}\neq 0$ directly makes ${\gamma_{1,c}}$ increase
and ${\gamma_{e,c}^{\mathrm{IV}}}$ decrease.

Based on the results, some new insights are obtained as follows.

  1. 1.

In RSMA mmWave systems, both common and private streams can be the bottleneck
of the security.

  2. 2.

There is an optimal transmit power to minimize the SOP of RSMA-based mmWave
systems and the optimal transmit power depends on the overlapped paths and
distances in the considered systems.

  3. 3.

There is an power allocation coefficient to minimize the SOP of RSMA-based
mmWave systems and the optimal power allocation coefficient depends on many
parameters, such as, the transmit power, target secrecy rates of common
streams and private streams, relative location of all the receivers.

  4. 4.

Relative to Scenario I and Scenario II, allocating more power to $s_{2}$ than
$s_{1}$ in Scenario III and Scenario IV can enhance $U_{1}$’s security.

##  V Conclusion

In this paper, the secure transmissions considering multipath propagation for
mmWave RSMA MISO systems were analyzed. Based on the spatial correlation of
the users and the eavesdropper, different eavesdropping scenarios were
considered to investigate the secrecy performance, and then the analytical
expressions of the SOP for four scenarios were derived. Our results
illustrated the effects of overlapped paths between receivers, the power
allocation coefficient of RSMA users, and channel parameters on the SOP of
RSMA-based mmWave systems. This work considered that the illegitimate user has
the same decode order according to the framework of the RSMA system. Other
potential threats or more diverse eavesdropping strategies will be interesting
work and part of the future work. Moreover, the results of this work
demonstrate an optimal power allocation coefficient to minimize the SOP of
RSMA systems. Thus, allocating power reasonably to the common and private
streams is vital for the secure transmissions of RSMA mmWave systems. It is
challenging to obtain the analytical expression for the optimal power
allocation coefficient to maximize the secrecy performance of RSMA systems.
Optimizing the secrecy performance of RSMA mmWave systems will be conducted in
our future work.

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