# Cognitive Radio-Inspired Rate-Splitting Multiple Access for Semi-Grant-Free
Transmissions

Hongwu Liu, Kyeong Jin Kim, Theodoros A. Tsiftsis, Bruno Clerckx,
Kyung Sup Kwak, and H. Vincent Poor H. Liu is with the School of Information
Science and Electrical Engineering, Shandong Jiaotong University, Jinan
250357, China (e-mail: liuhongwu@sdjtu.edu.cn).K. J. Kim is with Mitsubishi
Electric Research Laboratories, Cambridge, MA 02139 USA (e-mail:
kkim@merl.com).T. A. Tsiftsis is with the School of Intelligent Systems
Science and Engineering, Jinan University, Zhuhai 519070, China (e-mail:
theo_tsiftsis@jnu.edu.cn).B. Clerckx is with the Department of Electrical and
Electronic Engineering, Imperial College London, London, UK (e-mail:
b.clerckx@imperial.ac.uk).K. S. Kwak is with Inha University, Incheon, South
Korea (email: kskwak@inha.ac.kr).H. V. Poor is with the Department of
Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544
USA (e-mail: poor@princeton.edu).

###### Abstract

In this paper, we propose a cognitive radio-inspired rate-splitting multiple
access (CR-RSMA) scheme to assist semi-grant-free (SGF) transmissions in which
a grant-based user (GBU) and multiple grant-free users (GFUs) access the base-
station (BS) by sharing the same resource block. Using the cognitive radio
principle, the GBU and admitted GFU are treated as the primary and secondary
users, respectively, and rate-splitting is applied at the admitted GFU to
realize SGF transmissions. The admitted GFU’s transmit power allocation,
target rate allocation, and successive interference cancellation decoding
order at the BS are jointly optimized to attain the maximum achievable rate
for the admitted GFU without deteriorating the GBU’s outage performance
compared to orthogonal multiple access. Due to the extended non-outage zone,
CR-RSMA-assised SGF (CR-RSMA-SGF) transmissions achieve a lower outage
probability than SGF transmissions assisted by cognitive radio-inspired non-
orthogonal multiple access. Exact expressions and asymptotic analysis for the
admitted GFU’s outage probability are derived to evaluate the system
performance achieved by CR-RSMA-SGF transmissions. The superior outage
performance and full multiuser diversity gain achieved by CR-RSMA-SGF
transmissions are verified by the analytical and simulation results.

###### Index Terms:

Rate-splitting, multiple access, grant-free transmissions, outage probability.

##  I Introduction

The proliferation of Internet-of-Things (IoT) applications, intelligent
robots, and Industry 4.0 networks is resulting in unprecedented need for
massive and spectrally efficient connections. As part of the sixth-generation
(6G) evolution roadmap, extremely reliable and low-latency communication
(ERLLC), further-enhanced mobile broadband (FeMBB), and ultra-massive machine-
type communication (umMTC) have been proposed to enable envisioned
heterogeneous 6G applications [1]. For ubiquitous scenarios of umMTC and
ERLLC, where traditional grant-based (GB) transmissions are impractical due to
the corresponding excessive signaling overhead and massive computational
resource consumption, grant-free (GF) transmissions have been proposed,
attracting considerable interest from academia and industry [2, 3, 4]. A main
advantage of GF transmissions is that terminals can access the network without
engaging in lengthy handshaking, whereas the amount of handshaking signaling
can exceed the amount of data sent by terminals in GB transmissions. Although
GF transmissions are a promising solution for massive connectivity, GB
transmissions cannot be overlooked due to the stringent quality of service
(QoS) requirements of existing grant-based GB users (GBUs). For this reason,
semi-grant-free (SGF) transmissions have been proposed to accommodate the
coexistence of GB and GF transmissions, which results in higher spectral
efficiency than admitting only GBUs or GF users (GFUs) [5]. To
opportunistically admit GFUs to resource blocks occupied by GBUs, non-
orthogonal multiple access (NOMA)-assisted SGF (NOMA-SGF) transmissions have
been proposed, which eliminates complex handshaking processes for admitting
GFUs [5, 6]. By using NOMA, user collisions caused by the GFUs’ contention can
be resolved by spectrum sharing with the aid of successive interference
cancellation (SIC).

To avoid system performance degradation of a GBU in NOMA-SGF transmissions,
the GFU’s contention must be managed appropriately taking into account the
GBU’s QoS requirements. Specifically, outage performance experienced by the
GBU should be comparable to that of its orthogonal multiplex access (OMA)
counterpart [5, 6]. In [5], a distributed contention protocol was proposed to
guarantee that a fixed number of GFUs are admitted. It was seen that this
open-loop protocol suffered from user collisions at a rate similar to that
seen in pure GF transmissions. In [6], the received GBU’s signal power at the
base station (BS) was used to determine an interference threshold, which was
then broadcast to GFUs to facilitate distributed contentions. Considering that
existing power-domain NOMA schemes rely on superposition coding and transmit
power allocation, advanced SIC can be used to maintain a high transmission
reliability while preventing the GBU from unnecessary awareness of the GFUs’
contention [6, 7, 8]. In [9], hybrid SIC was proposed to decode the desired
signals based on their relative power levels, i.e., the admitted GFU’s signal
can be decoded at the first or second stage of SIC to attain the allowed
maximum achievable rate, which can be used in conjunction with transmit power
allocation to further improve the reliability of NOMA-SGF transmissions [10].

When viewed in the context of underlay cognitive radio (CR), the paired NOMA
users can be regarded as a primary user (PU) and a secondary user (SU),
respectively [11], which signifies that GB and GF transmissions in NOMA-SGF
systems can be treated as primary and secondary transmissions, respectively.
Since CR-inspired NOMA (CR-NOMA) with SIC can achieve only a subset on the
capacity region boundary of uplink multiple access channels (MACs), the outage
performance of SU in this setting can result in decreased user fairness [12].
To avoid deterioration in the PU’s outage performance, both PU and SU can be
allowed to access the same BS simultaneously in CR-NOMA, and a new hybrid SIC
was proposed to improve the achievable rate of the SU in such a system [13].
To achieve arbitrary points on the capacity region boundary of uplink MACs,
rate-splitting multiple access (RSMA) proposed in [14] can also be implemented
in a CR-inspired way, which can extend the non-outage zone for CR-NOMA [12].
In [15], the authors proposed a CR-inspired RSMA (CR-RSMA) scheme to improve
the SU’s outage performance by admitting both PU and SU simultaneously to the
same BS. However, the admission contention from multiple GFUs was not taken
into account in CR-RSMA and the effects of the GFUs’ contention on CR-RSMA-
assisted SGF (CR-RSMA-SGF) transmissions are still not known.

###  I-A Related Work

_1) RSMA:_ RSMA has received significant attention recently due to its
capability to improve the spectral/energy efficiencies, robustness,
reliability, and latency of downlink and uplink multi-user transmissions [16,
17]. In downlink RSMA, the users’ messages are split into common and private
streams using available channel state information at the transmitter (CSIT)
[18]. By treating inter-user interference flexibly, i.e., interference can be
partially decoded and partially treated as noise, downlink RSMA not only
bridges space division multiple access (SDMA) and NOMA, but also achieves
superior system performance, so that it is regarded as a promising enabling
technology for 6G new radio (NR) [19, 20, 21, 22] (see also [16] and
references therein).

However, RSMA and the corresponding applications for uplink MACs are still in
their infancy. Being essentially different from downlink RSMA, which decodes
common and private streams partially at a receiver to balance the decoding
performance and complexity, the split data streams from all the users are
fully decoded at the BS using SIC in uplink RSMA [23, 24, 25, 26]. In [23],
the sum-rate maximization problem was investigated for uplink RSMA in which
the proportional rate constraints among users were considered. To enhance the
outage performance of uplink NOMA, several rate-splitting schemes were
proposed in [24, 25, 26]. For single-input multiple-output (SIMO) NOMA, uplink
rate-splitting was investigated to guarantee max-min user fairness in [26].
Nevertheless, an exhaustive search was needed to find the optimal SIC decoding
order and optimal power allocation for uplink RSMA [23, 26]. Using the CR
principle, adaptive power allocation and rate-splitting were proposed to
improve the outage performance and user fairness for uplink NOMA in [12].

Recently, uplink RSMA has been applied to realize physical layer network
slicing for ultra-reliable and low-latency (URLLC) and enhanced mobile
broadband communications (eMMB) [27]. Also, RSMA has been used to support
URLLC and eMMB in 6G NR downlink transmissions [28]. In [29], the outage
performance of uplink RSMA was investigated taking into account all possible
SIC decoding orders. In [30], cooperative RSMA was proposed to realize uplink
user cooperation. In [31] and [32], rate-splitting schemes were designed for
uplink aerial networks and satellite communications, respectively. Rate-
splitting applications for cell-free machine-type communications and device-
to-device fog radio access networks were investigated in [33] and [34],
respectively.

_2) NOMA-SGF transmissions:_ To control the number of admitted GFUs for NOMA-
SGF transmissions, two contention mechanisms, namely open-loop contention and
distributed contention protocols, have been proposed in [5], in which an
interference temperature alike channel gain threshold was broadcast to aid the
GFUs’ contention. Furthermore, a dynamic interference threshold proportional
to the received GBU’s signal power was proposed in [7] to determine the
admitted GFUs, which results in reduced GFUs’ interference compared to the
open-loop contention. The authors in [6] and [9] proposed CR-NOMA with hybrid
SIC decoding order to enhance transmission reliability of the admitted GFU,
meanwhile ensuring that the GBU experiences the same outage performance as in
OMA. An ergodic rate analysis was provided for CR-NOMA-assisted SGF (CR-NOMA-
SGF) transmissions in [35]. To efficiently leverage the capability of CR-NOMA
to achieve the capacity region of uplink MACs, adaptive power allocation with
hybrid and fixed SIC decoding orders were proposed in [8] and [10]. Further,
adaptive power allocation was proposed in [36] for CR-NOMA-SGF transmissions
to improve the outage performance and sum rate. In [37], a downlink collided-
preamble feedback was applied to facilitate massive multiple-input multiple-
output assisted SGF random access. For tactile IoT networks that use NOMA-SGF
transmissions, joint power allocation and sub-channel assignment was
investigated in [38]. In [39], a user barring scheme was proposed to increase
the average arrival rate for NOMA-SGF transmissions with multiple transmit
power levels. In [40] and [41], multi-agent deep reinforcement learning
(MADRL) was applied to optimize the transmit power for NOMA-SGF and NOMA-
assisted GF (NOMA-GF) transmissions. Moreover, MADRL was used to optimize
transmit power allocation, sub-channel assignment and reflection beamforming
for an intelligent reflecting surface (IRS) aided NOMA-SGF system [42].

###  I-B Motivation and Contributions

Although NOMA-SGF transmissions can accommodate the coexistence of GB and GF
transmissions, this system achieves only a subset of the uplink MAC capacity
region as noted above. In contrast, the full capacity region of uplink MACs
can be achieved by using RSMA [14]. However, to maximize the sum-rate and
ensure user fairness, exhaustive search was used to determine the optimal SIC
decoding order and optimal power allocation for uplink RSMA, which is
computationally prohibitive [23, 26]. For delay-limited transmissions, the
authors in [43] proposed a CR-inspired transmit power allocation to aid rate-
splitting, in which target rates of the SU’s data streams were chosen
heuristically.

On the other hand, the recently proposed CR-RSMA can improve the SU’s outage
performance [15] and reduce the task-offloading latency for mobile edge
computing [44] by optimizing the transmit power and target rate allocations.
Motivated by the capacity region achieved by SGF transmissions, we propose a
new CR-RSMA that adopts RSMA and CR principles to assist SGF transmissions,
with the aim of improving the system performance under the uplink RSMA
framework. In CR-RSMA-SGF transmissions, rate-splitting is conducted at the
admitted GFU, while the optimal transmit power and target rate allocations at
the admitted GFU and SIC decoding order at the BS are jointly optimized using
the CR principle, which improves the admitted GFU’s outage performance
significantly.

The main contributions of this paper are summarized as follows:

  * •

A new CR-RSMA is proposed to allow simultaneous access of the GBU and GFU to
the BS without deteriorating the outage performance of the GBU compared to
OMA. Applying CR-inspired rate-splitting at the admitted GFU, the SIC decoding
order, transmit power allocation, and target rate allocation are jointly
optimized for the admitted GFU to obtain the maximum achievable rate. By
exploiting the full capacity region achieved by RSMA, the non-outage zone of
CR-RSMA-SGF transmissions is extended significantly compared to CR-NOMA-SGF
transmissions.

  * •

With respect to the capacity region, the extended non-outage zone and reduced
outage zone are described for CR-RSMA-SGF transmissions in comparison with the
CR-NOMA-SGF transmissions. Due to the extended non-outage zone, additional
target rate pairs containing the higher target rates for the admitted GFU can
be supported by CR-RSMA-SGF transmissions, so that the outage performance of
the admitted GFU is significantly improved when CR-RSMA is applied.

  * •

We derive an exact expression for the outage probability of the admitted GFU
and its approximation in the high signal-to-noise ratio (SNR) region. These
analytical results reveal that the full multiuser diversity gain is achieved
by CR-RSMA-SGF transmissions. Moreover, benefiting from the obtained maximum
achievable rate for the admitted GFU, the multiuser diversity can be achieved
in a wider target rate regions than those of the existing CR-NOMA-SGF
transmissions.

  * •

Various computer simulation results are presented to verify the accuracy of
the derived analytical results and high SNR approximations. The superior
outage performance of the admitted GFU achieved by CR-RSMA-SGF transmissions
is verified by these simulation results. The impact of the target rate and
number of GFUs on the outage performance of the admitted GFU are revealed.

The remainder of this paper is organized as follows: Section II presents the
system model and CR-RSMA-SGF transmissions, respectively; In Section III, the
outage performance of the admitted GFU achieved by CR-RSMA-SGF transmissions
is analyzed and the high SNR approximation for the outage probability is
derived; In Section IV, simulation results are presented for corroborating the
superior outage performance of CR-RSMA-SGF transmissions, and Section V
summarizes this work.

##  II System Model and CR-RSMA Transmissions

###  II-A System Model

We assume that the SGF transmission system consists of multiple GBUs and
multiple GFUs. To prevent GF transmissions from generating too much
interference to GBUs, all the GFUs and GBUs are divided into multiple groups,
each of which contains $K$ GFUs and one GBU. In each group, we assume that
only one out of $K$ GFUs is paired with the GBU for simultaneous uplink
transmissions, whereas multiple GFUs from different groups can be
simultaneously admitted through wireless resource allocation among the groups.

Without loss of generality, we consider SGF transmissions within a single
group, in which the GBU and the $k$th GFU are denoted by $U_{0}$ and
$U_{\tilde{k}}\in\\{U_{\tilde{1}},U_{\tilde{2}},\cdots,U_{\tilde{K}}\\}$,
respectively. The channel coefficients from $U_{0}$ and $U_{\tilde{k}}$ to the
BS are denoted by $h_{0}$ and $h_{\tilde{k}}$
(${\tilde{k}}={\tilde{1}},{\tilde{2}},\cdots,{\tilde{K}}$), respectively,
which are modeled as independent and identically distributed (i.i.d.) circular
symmetric complex Gaussian random variables with zero mean and unit variance.
We assume that the channels follow a quasi-static fading, i.e., the channel
coefficients remain constant during a single transmission block and can vary
from one transmission block to another independently. Moreover, the channel
gains can be ordered as

$\displaystyle|h_{1}|^{2}\leq|h_{2}|^{2}\leq\cdots\leq|h_{K}|^{2},$ (1)

where the subscripts denote the ordered indices. It should be noted that the
above ordering information is unavailable to the $K$ GFUs and BS. In the
considered system, we assume that the $K$ GFUs have the knowledge of their own
channel state information (CSI) and the admitted GFU’s CSI is not required to
be known at the BS prior to the SGF transmissions. In addition, the BS has
acquired the information of the GBU’s CSI and transmit power.

Without loss of generality, we assume that the $\ell$th GFU $U_{\ell}$ is
admitted to transmit ($1\leq\ell\leq K$). For each block of transmissions,
$U_{\ell}$ can split its message signal $\bar{x}_{\ell}$ into two parts
$\bar{x}_{{\ell,1}}$ and $\bar{x}_{{\ell,2}}$. Corresponding to the message
signals $\bar{x}_{\ell}$, $\bar{x}_{{\ell,1}}$, and $\bar{x}_{{\ell,2}}$, the
generated transmit signals at $U_{\ell}$ are denoted by $x_{\ell}$,
$x_{{\ell,1}}$, and $x_{{\ell,2}}$, respectively. In each transmission block,
$U_{0}$ and $U_{\ell}$ simultaneously transmit their signals to the BS. Then,
the received signal at the BS can be written as:

$\displaystyle y=\sqrt{P_{0}}h_{0}x_{0}+\sqrt{\alpha
P_{s}}h_{\ell}x_{{\ell,1}}+\sqrt{(1-\alpha)P_{s}}h_{\ell}x_{{\ell,2}}+w,$ (2)

where $P_{0}$ and $P_{s}$ denote the transmit power of $U_{0}$ and $U_{\ell}$,
respectively, $x_{0}$ is the transmit signal of $U_{0}$, $\alpha$ is the
transmit power allocation factor at $U_{\ell}$ satisfying $0\leq\alpha\leq 1$,
and $w$ is additive white Gaussian noise (AWGN) at the BS with zero mean and
unit variance. We assume that each transmit signal
$\tilde{x}\in\\{x_{0},x_{\ell},x_{{\ell,1}},x_{{\ell,2}}\\}$ is coded by an
independent Gaussian code book and satisfies
${\mathbb{E}}\\{|\tilde{x}|^{2}\\}=1$, where ${\mathbb{E}}\\{\cdot\\}$ is the
expectation operator.

At the BS, the decoding order $x_{{\ell,1}}\to x_{0}\to x_{{\ell,2}}$ is
adopted in SIC to recover the signal $x_{\ell,1}$, $x_{0}$, and $x_{\ell,2}$,
sequentially. According to the uplink RSMA principle [14], the SIC decoding
order $x_{{\ell,1}}\to x_{0}\to x_{{\ell,2}}$ ensures that the full capacity
region boundary of MACs can be achieved. Then, the received signal-to-
interference-plus-noise ratio (SINR) (or signal-to-noise ratio (SNR)) for
decoding $x_{{\ell,1}}$, $x_{0}$, and $x_{{\ell,2}}$ can be expressed as
follows:

$\displaystyle\gamma_{\ell,1}=\frac{\alpha
P_{s}|h_{\ell}|^{2}}{P_{0}|h_{0}|^{2}+(1-\alpha)P_{s}|h_{\ell}|^{2}+1},$ (3)
$\displaystyle\gamma_{0}=\frac{P_{0}|h_{0}|^{2}}{(1-\alpha)P_{s}|h_{\ell}|^{2}+1},$
(4)

and

$\displaystyle\gamma_{\ell,2}=(1-\alpha)P_{s}|h_{\ell}|^{2}.$ (5)

For the admitted $U_{0}$ and $U_{\ell}$, the achievable rate are given by
$R_{0}=\log_{2}(1+\gamma_{0})$ and $R_{{\ell}}=R_{\ell,1}+R_{\ell,2}$,
respectively, with $R_{\ell,1}=\log_{2}(1+\gamma_{\ell,1})$ and
$R_{\ell,2}=\log_{2}(1+\gamma_{\ell,2})$ representing the achievable rates to
transmit $x_{\ell,1}$ and $x_{\ell,2}$, respectively.

_Remark 1_ : By simply interchanging $x_{{\ell,1}}$ and $x_{{\ell,2}}$, all
possible SIC decoding orders at the BS can be categorized into three types as
$x_{{\ell,1}}\to x_{{\ell,2}}\to x_{0}$, $x_{0}\to x_{{\ell,1}}\to
x_{{\ell,2}}$, and $x_{{\ell,1}}\to x_{0}\to x_{{\ell,2}}$. Using the
logarithmic product law, we can readily prove that applying $x_{{\ell,1}}\to
x_{{\ell,2}}\to x_{0}$ is equal to applying $x_{\ell}\to x_{0}$ in the sense
of attaining the achievable rates $R_{\ell}$ and $R_{0}$. In such a case,
rate-splitting is unnecessary, so does the decoding order $x_{0}\to
x_{{\ell,1}}\to x_{{\ell,2}}$. As it will be seen in the next subsection, NOMA
with the SIC decoding orders $x_{\ell}\to x_{0}$ and $x_{0}\to x_{\ell}$
cannot approach the full capacity region boundary of MACs. Thus, the decoding
order $x_{{\ell,1}}\to x_{0}\to x_{{\ell,2}}$ is used in the CR-RSMA-SGF
transmissions with respect to its capability to achieve the full capacity
region boundary.

###  II-B CR-RSMA-SGF Transmissions

For the considered CR-RSMA-SGF transmissions, to guarantee that the GBU
$U_{0}$ achieves the same outage performance as in OMA, a CR analogous
interference threshold is broadcast to the $K$ GFUs to aid user contention.
Let $\hat{R}_{0}$, $\hat{R}_{s}$, $\hat{R}_{s,1}$, and $\hat{R}_{s,2}$ denote
the target rates to transmit $x_{0}$, $x_{\ell}$, $x_{\ell,1}$, and
$x_{\ell,2}$, respectively. In addition, we define
$\hat{R}_{s}=\hat{R}_{s,1}+\hat{R}_{s,2}$, $\hat{R}_{s,1}=\beta\hat{R}_{s}$,
and $\hat{R}_{s,2}=(1-\beta)\hat{R}_{s}$, where $0\leq\beta\leq 1$ is the
target rate allocation factor. With respect to the SIC decoding order
$x_{{\ell,1}}\to x_{0}\to x_{{\ell,2}}$, the signal $x_{0}$ can be decoded
correctly only when the following constraints are satisfied, i.e.,

$\displaystyle
R_{\ell,1}\geq\hat{R}_{s,1}{\text{~{}~{}and~{}~{}}}R_{0}\geq\hat{R}_{0}.$ (6)

Under the constraints in (6), the GBU achieves the same outage performance as
in OMA when $\Pr\left(R_{\ell,1}\geq\hat{R}_{s,1}\right)=1$ and

$\displaystyle\Pr\left(R_{0}\geq\hat{R}_{0}\right)=\Pr\left(\log_{2}(1+P_{0}|h_{0}|^{2})\geq\hat{R}_{0}\right).$
(7)

By substituting (4) into (7), the equality in (7) holds under the condition of

$\displaystyle(1-\alpha)P_{s}|h_{\ell}|^{2}\leq\hat{\tau},$ (8)

where $\hat{\tau}=\tfrac{P_{0}|h_{0}|^{2}}{2^{\hat{R}_{0}}-1}-1$. The equation
(8) indicates that the interference power caused by the transmission of
$\sqrt{(1-\alpha)P_{s}}h_{\ell}x_{{\ell,2}}$ cannot surpass $\hat{\tau}$ to
ensure the correct decoding of $x_{0}$ at the second stage of SIC. By assuming
that $\Pr\left(R_{\ell,1}\geq\hat{R}_{s,1}\right)=1$ is guaranteed in the CR-
RSMA-SGF transmissions, as it will be explained later in this subsection, the
interference threshold to be broadcast by the BS is determined as [6]:

$\displaystyle\tau=\max\left\\{0,\hat{\tau}\right\\}.$ (9)

The admission procedure of the CR-RSMA-SGF transmissions is presented as
follows:

  * •

The BS broadcasts the pilot signals to assist the users to estimate CSI.

  * •

$U_{0}$ feeds back its CSI and $P_{0}$ to the BS.

  * •

The BS calculates $\tau$ according to (9) and broadcasts it to the $K$ GFUs.

  * •

Each GFU calculates the achievable rate using its own CSI and the
corresponding optimal $\alpha^{*}$ and $\beta^{*}$, which will be provided in
the later part of this subsection.

  * •

Through the distributed contention, the GFU that obtains the maximum
achievable rate is admitted by the BS.

With respect to the capability of uplink RSMA to achieve the full capacity
region boundary of MACs, the design goal of the CR-RSMA-SGF transmissions is
to maximize the achievable rate for the admitted GFU meanwhile guaranteeing
that the GBU achieves the same outage performance as in OMA. With respect to
all the possible CSI realizations, the rate-splitting operation and associated
$\alpha^{*}$ and $\beta^{*}$ of the CR-RSMA-SGF transmissions are jointly
designed as follows:

_Case I:_ _$0 <P_{s}|h_{K}|^{2}\leq\tau$._ In this case, the maximum
interference level caused by admitting an arbitrary GFU is not greater than
the interference threshold $\tau$. When $U_{\ell}$ is admitted, the maximum
achievable rate is given by $R_{\ell}=R_{\ell,2}=\log_{2}(1+\gamma_{\ell,2})$
considering that $x_{\ell,2}$ is interference-freely decoded at the last stage
of SIC. In other words, $U_{\ell}$ will allocate all of $P_{s}$ to transmit
$x_{\ell}$ by setting $x_{\ell,2}=x_{\ell}$. Consequently, $U_{K}$, which has
the greatest channel gain among all the GFUs, is admitted to attain the
maximum achievable rate $R_{K}=R_{K,2}=\log_{2}(1+\gamma_{K,2})$. In this
case, the optimal transmit power and target rate allocation factors are
respectively given by

$\displaystyle\alpha^{*}=0{\text{~{}~{}and~{}~{}}}\beta^{*}=0,$ (10)

where $(\cdot)^{*}$ denotes the optimal solution for the corresponding
parameter.

Since only $x_{K,2}$ ($x_{K,2}=x_{K}$) is transmitted, the SIC decoding order
$x_{K,1}\to x_{0}\to x_{K,2}$ degenerates to $x_{0}\to x_{K}$. Consequently,
the achievable rate of the admitted GFU can be expressed as

$\displaystyle R_{K}^{(\rm I)}=\log_{2}\left(1+P_{s}|h_{K}|^{2}\right).$ (11)

_Case II:_ _$0 <\tau<P_{s}|h_{1}|^{2}$ or
$P_{s}|h_{k}|^{2}<\tau<P_{s}|h_{k+1}|^{2}$ with $k=1,2,\ldots,K-1$._ In this
case, it can be seen that $\hat{\tau}>0$. Then, a GFU $U_{\ell}$ whose channel
gain is larger than $\hat{\tau}$ will be admitted, where $k+1\leq\ell\leq K$.
Due to $\hat{\tau}>0$, the GBU’s signal $x_{0}$ can be correctly decoded at
the second stage of SIC ($x_{\ell,1}\to x_{0}\to x_{\ell,2}$) subject to the
constraints $R_{\ell,1}>\hat{R}_{\ell,1}$ and
$R_{\ell,2}\leq\log_{2}(1+\hat{\tau})$. Thus, the admitted GFU’s achievable
rate $R_{\ell}=R_{\ell,1}+R_{\ell,2}$ can be maximized by first maximizing
$R_{\ell,2}=\log_{2}(1+\gamma_{\ell,2})$, which is obtained as
$R_{\ell,2}=\log_{2}(1+\hat{\tau})$ by setting $\gamma_{\ell,2}=\hat{\tau}$.
The corresponding optimal transmit power allocation factor is given by

$\displaystyle\alpha^{*}=1-\frac{\hat{\tau}}{P_{s}|h_{\ell}|^{2}}.$ (12)

By setting the target rate $\hat{R}_{\ell,2}=\log_{2}(1+\hat{\tau})$ to
transmit $x_{\ell,2}$, the optimal target rate allocation factor is given by

$\displaystyle\beta^{*}=1-\frac{\log_{2}(1+\hat{\tau})}{\hat{R}_{s}}.$ (13)

Furthermore, $R_{\ell,1}$ can be maximized by setting $\ell=K$ considering
that $U_{K}$ has the greatest channel gain, i.e., the achievable rate for the
GFU is maximized by admitting $U_{K}$. Accordingly, the optimal transmit power
and target rate allocation factors are respectively given by $\alpha^{*}(K)$
and $\beta^{*}$, and the achievable rate in this case can be written as

$\displaystyle R_{K}^{({\rm II})}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle R_{K,1}^{({\rm II})}+R_{K,2}^{({\rm II})},$ (14)

where

$\displaystyle R_{K,1}^{({\rm II})}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\log_{2}\left(1+\frac{P_{s}|h_{K}|^{2}-\hat{\tau}}{P_{0}|h_{0}|^{2}+\hat{\tau}+1}\right)$
(15)

and

$\displaystyle R_{K,2}^{({\rm II})}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\log_{2}(1+\hat{\tau}).$ (16)

Since the received SINR/SNR $\gamma_{0}(\alpha^{*}(K))$ and
$\gamma_{K,2}(\alpha^{*}(K))$ provide the necessary conditions for the correct
decoding of $x_{0}$ and $x_{K,2}$ in SIC processing $x_{{K,1}}\to x_{0}\to
x_{{K,2}}$, only the failure decoding of $x_{K,1}$ can result in error
propagation in SIC. To avoid this error propagation, $R_{K,1}^{({\rm
II})}\geq\hat{R}_{s,1}$ is required to decode $x_{K,1}$ correctly or
equivalently $R_{K}^{({\rm II})}\geq\hat{R}_{s}$; Otherwise, both $U_{0}$ and
$U_{K}$ encounter outage. Therefore, in the proposed CR-RSMA-SGF
transmissions, $U_{K}$ is permitted to transmit only when $R_{K,1}^{({\rm
II})}\geq\hat{R}_{s,1}$; Otherwise, $U_{K}$ keeps silence while $U_{0}$ is
transmitting alone to the BS.

_Case III:_ _$\tau=0$._ In this case, it can be seen that $\hat{\tau}<0$, so
that $x_{0}$ cannot be correctly decoded due to a weak channel gain
$|h_{0}|^{2}$. To avoid error propagation in SIC processing $x_{\ell,1}\to
x_{0}\to x_{\ell,2}$ caused by the failure decoding of $x_{0}$, the admitted
GFU is chosen not to transmit $x_{\ell,2}$ rather to transmit $x_{\ell}$ by
setting $x_{\ell,1}=x_{\ell}$ and using the whole $P_{s}$. Therefore, the
optimal transmit power and target rate allocation factors are given by

$\displaystyle\alpha^{*}=1{\text{~{}~{}and~{}~{}}}\beta^{*}=1.$ (17)

Since $U_{K}$ has the greatest channel gain among the GFUs, it is admitted to
attain the maximum achievable rate as

$\displaystyle R_{K}^{(\rm
III)}=\log_{2}\left(1+\frac{P_{s}|h_{K}|^{2}}{P_{0}|h_{0}|^{2}+1}\right)$ (18)

and the corresponding SIC decoding order $x_{K,1}\to x_{0}\to x_{K,2}$
degenerates to $x_{K}\to x_{0}$.

_Remark 1_ : Due to the channel gain ordering
$|h_{1}|^{2}\leq|h_{2}|^{2}\leq\cdots\leq|h_{K}|^{2}$, the maximum achievable
rates in Cases I, II, and III for the admitted GFU are always attained by
admitting $U_{K}$, no matter which decoding order is applied in SIC. Since the
channel coefficients of all the $K$ GFUs follow i.i.d. complex Gaussian
distribution, all the GFUs have the equal probability to have the greatest
channel gain, or equivalently, to be admitted. Thus, the admission probability
for all the $K$ GFUs are equal in the proposed CR-RSMA-SGF transmissions.

_Remark 2_ : In Cases I and III, the CR-NOMA-SGF transmissions also admit
$U_{K}$ and apply the SIC decoding order $x_{0}\to x_{K}$ and $x_{K}\to
x_{0}$, respectively, to obtain the maximum achievable rates [6].
Nevertheless, in Case II, the CR-NOMA-SGF transmissions admit $U_{K}$ (or
$U_{k}$) subject to $P_{s}|h_{1}|^{2}>\tau$ (or
$P_{s}|h_{k}|^{2}<\tau<P_{s}|h_{k+1}|^{2}$). For the CR-NOMA-SGF
transmissions, the achievable rate of the admitted GFU in Case II is given by
[6]

$\displaystyle R_{\rm NOMA}^{(\rm
II)}=\left\\{{\begin{array}[]{*{20}{c}}{R_{K}^{(\rm
III)},}&{0<\tau<P_{s}|h_{1}|^{2}}\\\ {\max\big{\\{}R_{k}^{(\rm I)},R_{K}^{(\rm
III)}\big{\\}},}&{P_{s}|h_{k}|^{2}<\tau<P_{s}|h_{k+1}|^{2}}\end{array}}\right.,$
(21)

where $k=1,2,\ldots,K-1$. Since $R_{K}^{(\rm II)}>R_{K}^{(\rm III)}$ and
$R_{K}^{(\rm II)}>R_{k}^{(\rm I)}$ ($k=1,2,\ldots,K-1$) hold for $\tau>0$, we
have $R_{K}^{({\rm II})}>R_{\rm NOMA}^{(\rm II)}$, i.e., the CR-RSMA-SGF
transmissions always achieve a larger achievable rate for the admitted GFU
than that of the CR-NOMA-SGF transmissions in Case II.

_Remark 3_ : In Case II, $U_{K}$ only transmits its signal when
$R_{K,1}^{({\rm II})}\geq\hat{R}_{s,1}$, which prevents error propagation in
SIC processing $x_{K,1}\to x_{0}\to x_{K,2}$. When $R_{K,1}^{({\rm
II})}<\hat{R}_{s,1}$, $U_{K}$ keeps silence and only the GBU $U_{0}$
transmits. Thus, the GBU $U_{0}$ does not encounter outage when the CR-RSMA-
SGF transmissions operate in Case II. As such, the outage events occur to the
transmissions of $x_{0}$ in Cases I and III are the same as in OMA, and the
GBU $U_{0}$ achieves the same outage performance as in OMA when the CR-RSMA-
SGF transmissions are applied.

##  III Outage Performance Analysis

In the CR-RSMA-SGF transmissions, since the GBU’s outage performance is
guaranteed to be the same as in OMA, we mainly focus on the outage performance
of the admitted GFU. In this section, we first introduce the non-outage zone
to clarify the advantage of the CR-RSMA-SGF transmissions. Then, we derive the
analytical expression for the outage probability in closed-form and
investigate the asymptotic outage performance in the high SNR region.

For the uplink MACs in which the two-user $U_{0}$ and $U_{K}$ are admitted
simultaneously, the non-outage zone is defined by

$\displaystyle\bar{\cal{O}}\triangleq\left\\{\\{\hat{R}_{0},\hat{R}_{s}\\}\big{|}~{}\hat{R}_{0}\leq
R_{0},\hat{R}_{s}\leq R_{K}\right\\}.$ (22)

If a target rate pair $\\{\hat{R}_{0},\hat{R}_{s}\\}$ lies in the non-outage
zone, both $U_{0}$ and $U_{K}$ can attain the achievable rates
$R_{0}\geq\hat{R}_{0}$ and $R_{K}\geq\hat{R}_{K}$ such that neither $U_{0}$
nor $U_{K}$ encounters the outage event. Otherwise, both $U_{0}$ and $U_{K}$
are in outage. Corresponding to (22), the set
${\cal{O}}\triangleq\left\\{\\{\hat{R}_{0},\hat{R}_{s}\\}\big{|}~{}\hat{R}_{0}>R_{0},\hat{R}_{s}>R_{K}\right\\}$
is called the outage zone when both $U_{0}$ and $U_{K}$ are admitted
simultaneously.

Figure 1: Non-outage zones achieved by different schemes ($P_{0}|h_{0}|^{2}=8$
dB, $P_{s}|h_{K}|^{2}=15$ dB).

In Fig. 1, an example of the non-outage zone achieved by the CR-RSMA-SGF
transmissions is illustrated. For the comparison purpose, the non-outage zones
achieved by the CR-NOMA-SGF transmissions are also illustrated in which the
SIC decoding orders $x_{0}\to x_{K}$ and $x_{K}\to x_{0}$ are respectively
utilized [6]. In this example, we assume $0<\tau<P_{s}|h_{1}|^{2}$. From Fig.
1, we can see that the non-outage zones achieved by the CR-NOMA-SGF
transmissions using the SIC decoding orders $x_{0}\to x_{K}$ and $x_{K}\to
x_{0}$ are identical to those achieved by the CR-RSMA-SGF transmissions in
Cases I and III, respectively. Thus, in Cases I and III, the CR-RSMA-SGF
transmissions can achieve the same outage performance as that of the CR-NOMA-
SGF transmissions. Nevertheless, the CR-NOMA-SGF transmissions cannot prevent
$U_{0}$ and $U_{K}$ from being in outage when the target rate pair
$\\{\hat{R}_{0},\hat{R}_{K}\\}$ lies in the triangle ABC, which is beyond the
non-outage zone of the CR-NOMA-SGF transmissions. Fortunately, the points on
the line AB in Fig. 1 can be achieved by rate-splitting. Specifically, an
additional non-outage zone, the triangle ABC, is achieved when the CR-RSMA-SGF
transmissions operate in Case II. Thus, the non-outage zone of the CR-RSMA-SGF
transmissions is extended compared to that of the CR-NOMA-SGF transmissions,
which shows that the CR-RSMA-SGF transmissions can support more target rate
pairs than that of the CR-NOMA-SGF transmissions.

With respect to the operations of the CR-RSMA-SGF transmissions in Cases I,
II, and III, the outage probability of the admitted GFU can be written as

$\displaystyle P_{\rm out}=P_{\rm out}^{(\rm I)}+P_{\rm out}^{(\rm II)}+P_{\rm
out}^{(\rm III)},$ (23)

where $P_{\rm out}^{(\rm I)}$, $P_{\rm out}^{(\rm II)}$, and $P_{\rm
out}^{(\rm III)}$ respectively denote the probability of that $U_{K}$
encounters outage in Cases I, II, and III, which can be expressed as

$\displaystyle P_{\rm out}^{(\rm
I)}=\Pr\left(0<P_{s}|h_{K}|^{2}<\tau,R_{K}^{(\rm I)}<\hat{R}_{s}\right),$ (24)
$\displaystyle P_{\rm out}^{(\rm II)}=\sum\limits_{k=0}^{K-1}{P_{\rm
out}^{(\rm II,\it{k})}},$ (25)

and

$\displaystyle P_{\rm out}^{(\rm III)}=\Pr\left(\tau=0,R_{K}^{(\rm
III)}<\hat{R}_{s}\right)$ (26)

with

$\displaystyle P_{\rm out}^{(\rm
II,0)}=\Pr\left(0<\tau<P_{s}|h_{1}|^{2},R_{K}^{(\rm II)}<\hat{R}_{s}\right)$
(27)

and for $k=1,2,\ldots,K-1$,

$\displaystyle P_{\rm out}^{(\rm
II,\it{k})}=\Pr\left(P_{s}|h_{k}|^{2}<\tau<P_{s}|h_{k+1}|^{2},R_{K}^{(\rm
II)}<\hat{R}_{s}\right).$ (28)

The following theorem provides an exact expression for the admitted GFU’s
outage probability achieved by the CR-RSMA-SGF transmissions.

###### Theorem 1

Assume that $K\geq 2$, the outage probability of the admitted GFU is given by

$\displaystyle P_{\rm out}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K(K-1)}\sum\limits_{n=0}^{K}\binom{K}{n}(-1)^{n}~{}\mu_{1}\nu(0,\mu_{2})$
(29)
$\displaystyle+\sum\limits_{k=1}^{K-2}\varphi_{k}\sum\limits_{m=0}^{K-k}\binom{K\\!-\\!k}{m}(-1)^{m}\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}e^{\frac{n}{P_{s}}}\mu_{3}\nu(n,\mu_{4})$
$\displaystyle+\frac{\varphi_{0}}{K-1}\sum\limits_{n=0}^{K-1}\binom{K\\!-\\!1}{n}(-1)^{n}e^{\frac{n}{P_{s}}}\left(e^{\frac{1}{P_{s}}}\nu(n,\mu_{5})-e^{-\frac{\epsilon_{0}+\epsilon_{s}+\epsilon_{0}\epsilon_{s}}{P_{s}}}\nu(n,\mu_{6})\right)$
$\displaystyle+\sum\limits_{n=0}^{K}\binom{K}{n}(-1)^{n}e^{\frac{n}{P_{s}}}\nu(n,0)+\left(1-e^{-\eta_{s}}\right)^{K}e^{-\eta_{0}(1+\epsilon_{s})}$
$\displaystyle+\sum\limits_{n=0}^{K}\binom{K}{n}(-1)^{n}e^{-n\eta_{s}}\frac{1-e^{-(1+n\eta_{s}P_{0})\eta_{0}}}{1+n\eta_{s}P_{0}},$

where $\epsilon_{0}\triangleq 2^{\hat{R}_{0}}-1$, $\epsilon_{s}\triangleq
2^{\hat{R}_{s}}-1$, $\eta_{0}\triangleq\frac{\epsilon_{0}}{P_{0}}$,
$\eta_{s}\triangleq\frac{\epsilon_{s}}{P_{s}}$,
$\mu_{1}=e^{\frac{K-n(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}}$,
$\mu_{2}=\frac{K-n}{P_{s}\eta_{0}}-\frac{nP_{0}}{P_{s}}$,
$\mu_{3}=e^{\frac{K-k-m\left(1+\epsilon_{0}\right)\left(1+\epsilon_{s}\right)}{P_{s}}}$,
$\mu_{4}=\frac{K-k-m}{P_{s}\eta_{0}}-\frac{mP_{0}}{P_{s}}$,
$\mu_{5}=\frac{1}{P_{s}\eta_{0}}$, $\mu_{6}=-\frac{P_{0}}{P_{s}}$,
$\varphi_{0}=\frac{K!}{(K-2)!}$, $\varphi_{k}=\frac{K!}{k!(K-k)!}$ for $1\leq
k\leq K-2$, and

$\displaystyle\nu(n,\mu)=\left\\{{\begin{array}[]{*{20}{c}}{\epsilon_{s}\eta_{0}},&{{\rm
if~{}~{}}\mu=-1-\frac{n}{P_{s}\eta_{0}}},\\\
{\frac{e^{-\eta_{0}\left(\frac{n}{P_{s}\eta_{0}}+\mu+1\right)}-e^{-\eta_{0}(1+\epsilon_{s})\left(\frac{n}{P_{s}\eta_{0}}+\mu+1\right)}}{\frac{n}{P_{s}\eta_{0}}+\mu+1}},&{\text{otherwise}}.\end{array}}\right.$
(32)

###### Proof:

See Appendix A. ∎

Figure 2: Outage and non-outage zones achieved by different schemes
($P_{0}|h_{0}|^{2}=5$ dB, $P_{s}|h_{K}|^{2}=4$ dB).

_Remark 4:_ In deriving the expressions for $P_{\rm out}^{(\rm I)}$ and
$P_{\rm out}^{(\rm II)}$, as derived in Appendix A, all the outage events
reflect the fact that the upper bound on the channel gain $|h_{K}|^{2}$ should
be greater than the lower bound on a specific GFU’s channel gain, which is
always and naturally guaranteed by $|h_{0}|^{2}<\eta_{0}(1+\epsilon_{s})$
without imposing additional constraints on $\epsilon_{0}$ and $\epsilon_{s}$,
so that the derived expression for the outage probability is applicable to all
the feasible $\epsilon_{0}$ and $\epsilon_{s}$. On the contrary, for the CR-
NOMA-SGF transmissions, the analytical expression for the outage probability
in [6] is applicable only for $\epsilon_{0}\epsilon_{s}<1$. As an example, the
outage and non-outage zones achieved by the CR-RSMA-SGF and CR-NOMA-SGF
transmissions are illustrated in Fig. 2. It can be seen that the outage zone
constrained by $\epsilon_{0}\epsilon_{s}<1$ is a small portion of the whole
outage zone for the CR-NOMA-SGF transmissions and a similar phenomenon happens
to the CR-RSMA-SGF transmissions as well. Therefore, the analytical results
provided in Theorem 1 are more general due to its applicability to all the
feasible values of $\epsilon_{0}$ and $\epsilon_{s}$. In contrast to the CR-
RSMA-SGF transmissions, the CR-NOMA-SGF transmissions result in a worse outage
performance due to the extended outage zone.

###### Theorem 2

Assuming that $K\geq 2$, the outage probability experienced by the admitted
GFU can be approximated in the high SNR region as follows:

$\displaystyle P_{\rm out}$ $\displaystyle\\!\\!\\!\approx\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}\epsilon_{0}(1+\epsilon_{0})^{K}}{P_{s}^{K+1}K(K-1)}\sum\limits_{n=0}^{K}\binom{K}{n}\frac{(-1)^{n}}{n+1}\left((1+\epsilon_{s})^{K+1}-(1+\epsilon_{s})^{K-n}\right)$
(33)
$\displaystyle+\frac{\varphi_{k}\epsilon_{0}(1+\epsilon_{0})^{K-k}(-1)^{k}}{P_{s}^{K+1}}\sum\limits_{m=0}^{K-k}\binom{K-k}{m}(-1)^{m}(1+\epsilon_{s})^{K-k-m}$
$\displaystyle\times\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}\frac{(1+\epsilon_{s})^{m+n+1}-1}{m+n+1}+\frac{\varphi_{0}\epsilon_{0}\epsilon_{s}^{K}(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}^{K+1}K(K-1)}$
$\displaystyle-\frac{\varphi_{0}\epsilon_{s}^{K}(\epsilon_{0}^{-1}+1)(K(1+\epsilon_{s})+1)}{P_{s}^{K+1}K(K-1)(K+1)}+\frac{\epsilon_{0}\epsilon_{s}^{K+1}}{(K+1)P_{s}^{K+1}}+\frac{\epsilon_{s}^{K}}{P_{s}^{K}}-\frac{\epsilon_{0}\epsilon_{s}^{K}(1+\epsilon_{s})}{P_{s}^{K+1}}$
$\displaystyle+\frac{\epsilon_{s}^{K}\left((1+\epsilon_{0})^{K+1}-1\right)}{P_{s}^{K+1}(K+1)}-\frac{\epsilon_{s}^{K}\left((\epsilon_{0}(K+1)-1)(1+\epsilon_{0})^{K+1}+1\right)}{P_{s}^{K+2}(K+2)(K+1)}.$

###### Proof:

See Appendix B. ∎

From the results in Theorem 2, we can see that there is one term in (33) being
proportional to $\frac{1}{P_{s}^{K}}$, while the other terms are proportional
to $\frac{1}{P_{s}^{K+1}}$ or $\frac{1}{P_{s}^{K+2}}$. Therefore, we have the
following corollary.

###### Corollary 1

Assuming that $K\geq 2$ , the admitted GFU’s outage probability can be further
approximated as $\frac{\epsilon_{s}^{K}}{P_{s}^{K}}$ in the high SNR region.
Thus, a diversity gain of $K$ is achieved by the CR-RSMA-SGF transmissions.

When the considered system consists of a single GFU and GBU, the two users are
paired directly. Then, the GFU’s outage probability achieved by the CR-RSMA-
SGF transmissions can be written as

$\displaystyle P_{\rm out}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\Pr\left(0<P_{s}|h_{1}|^{2}\leq\tau,R_{1}^{(\rm
I)}<\hat{R}_{s}\right)+\Pr\left(0<\tau<P_{s}|h_{1}|^{2},R_{1}^{(\rm
II)}<\hat{R}_{s}\right)$ (34) $\displaystyle+\Pr\left(\tau=0,R_{1}^{(\rm
III)}<\hat{R}_{s}\right).$

###### Corollary 2

Assume that $K=1$, the admitted GFU’s outage probability is given by

$\displaystyle P_{\rm out}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
1-e^{-\frac{\epsilon_{0}+\epsilon_{s}+\epsilon_{0}\epsilon_{s}}{P_{s}}}\nu(0,\mu_{6})-e^{-\eta_{s}-\eta_{0}(1+\epsilon_{s})}-\frac{e^{-\eta_{s}}(1-e^{-\eta_{0}-\epsilon_{0}\eta_{s}})}{1+P_{0}\eta_{s}}$
(35)

and its approximation in the high SNR region is given by

$\displaystyle P_{\rm out}$ $\displaystyle\\!\\!\\!\approx\\!\\!\\!$
$\displaystyle\epsilon_{s}P_{s}^{-1}.$ (36)

###### Proof:

A proof can be found in [15]. ∎

_Remark 7:_ Corollaries 1 and 2 demonstrate that the CR-RSMA-SGF transmissions
ensure an achievable diversity gain proportional to the number of the GFUs
without resulting in an outage floor.

##  IV Simulation Results

In this section, we present simulation results to verify the accuracy of the
analytical results and the superior outage performance of the CR-RSMA-SGF
transmissions. For the purpose of comparisons, the admitted GFU’s outage
probabilities achieved by the CR-NOMA-SGF transmissions [6] and the CR-NOMA-
SGF transmissions with power control (CR-NOMA-SGF-PC) [13] are also presented.
In the simulation, all the channel coefficients $h_{\tilde{k}}$s are randomly
generated according to i.i.d. circular symmetric complex Gaussian random
variables with zero mean and unit variance and the transmission data rate is
measured in bits per channel use (BPCU).

Figure 3: Outage probability comparison of the SGF schemes with
$\hat{R}_{0}=2.5$ BPCU, $\hat{R}_{s}=1.5$ BPCU, and $P_{s}=\frac{P_{0}}{15}$.
Figure 4: Outage probability comparison of the SGF schemes with
$\hat{R}_{0}=3$ BPCU, $\hat{R}_{s}=3$ BPCU, and the fixed $P_{0}=15$ dB.

The outage performance achieved by the CR-RSMA-SGF transmissions is compared
with those of the CR-NOMA-SGF and CR-NOMA-SGF-PC schemes in Fig. 3, where we
set $\hat{R}_{0}=2.5$ BPCU, $\hat{R}_{s}=1.5$ BPCU, and
$P_{s}=\frac{P_{0}}{15}$, i.e., the transmit SNR of $U_{0}$ is 11.76 dB higher
than that of the GFUs. From Fig. 3, we can see that the CR-RSMA-SGF
transmissions achieve the smallest outage probabilities among the three
schemes for both $K=1$ and $K=5$. The results in Fig. 3 also verify the
accuracy of the derived analytical expressions. As transmit SNR increases, the
outage probabilities achieved by both the CR-RSMA-SGF and CR-NOMA-SGF-PC
schemes decrease monotonically, whereas an outage probability floor occurs for
the CR-NOMA-SGF transmissions in the high SNR regime. Although CR-NOMA-SGF-PC
scheme avoids the outage probability floor, the achieved outage probability is
still higher than that of the CR-RSMA-SGF transmissions. Therefore, the
results in Fig. 3 verify that the CR-RSMA-SGF transmissions achieves the
superior outage performance compared to the CR-NOMA-SGF and CR-NOMA-SGF-PC
schemes.

Figure 5: Accuracy of the derived analytical expressions.

In Fig. 4, we investigate the outage performance achieved by the SGF schemes
for various transmit powers. In particular, we set a fixed transmit power
$P_{0}=15$ dB and vary $P_{s}$ from 0 dB to 45 dB, which reflects that the
GFUs can have greater and lower transmit SNRs than that of $U_{0}$. The
results in Fig. 4 also verify the accuracy of the derived analytical
expressions. From Fig. 4, we can see that the CR-RSMA-SGF transmissions
achieve the smallest outage probabilities in the whole SNR region. As transmit
SNR increases, the outage probability achieved by the CR-RSMA-SGF
transmissions decreases monotonically for both $K=1$ and $K=5$. Also, the CR-
NOMA-SGF transmissions achieve the worst outage probabilities in the whole SNR
region. Especially in the high SNR region, the outage probability floor occurs
for the CR-NOMA-SGF transmissions. The results in Fig. 4 verify that the
superior outage performance of the CR-RSMA-SGF transmissions is irrespective
of whether the GFUs are stronger users or weaker users compared to $U_{0}$.

Figure 6: Impact of the target rate on the outage probability. Figure 7:
Impact of the number of the GFUs on the outage probability ($\hat{R}_{0}=1.5$
BPCU and $\hat{R}_{s}=2$ BPCU).

In Fig. 5, we examine the accuracy of the derived analytical expressions for
the outage probability. In Fig. 5, we set $\hat{R}_{0}=2$ BPCU,
$\hat{R}_{s}=1.5$ BPCU, and $P_{s}=\frac{P_{0}}{15}$. For the “approximation
I”, we use the expression provided in Theorem 2. For the “approximation II”,
we use the expressions provided in Corollaries 1 and 2 for $K>2$ and $K=1$,
respectively. The curves in Fig. 5 verify that the accuracy of derived
analytical expression in Theorem 1. For the approximated expressions, we can
see that the curves of the “approximation I” match well with the exact results
in the high SNR region, whereas the curves of the “approximation II” matches
well with the exact results in the high SNR region only for small $K$ values.
For large $K$ values ($K=4$ in this example), a gap exists between the curves
of the “approximation II” and exact results. The reason for this phenomenon is
that we ignore the terms being proportional to $\frac{1}{P_{s}^{K+1}}$ and
$\frac{1}{P_{s}^{K+2}}$ in the expression in Corollary 1.

The impact of the target rate on the outage probability is investigated in
Fig. 6. For the simulation results in Fig. 6, we set $P_{0}=10$ dB, $P_{s}=15$
dB, and $\hat{R}_{0}=\hat{R}_{s}$. It is seen in Fig. 6 that for given
transmit SNR values, the CR-RSMA-SGF transmissions achieve the smallest outage
probabilities in the considered whole target rate region. As the target rate
increases, the outage probability values achieved by all the SGF schemes
increase and approach 1.

The impact of the number of the GFUs on the outage probability is investigated
in Fig. 7. For the simulation corresponding to Fig. 7, we set
{$\hat{R}_{0}=1.5$ BPCU, $\hat{R}_{s}=2$ BPCU} and consider two cases of
transmit SNR settings {$P_{0}=20$ dB, $P_{s}=10$ dB} and {$P_{0}=20$ dB,
$P_{s}=10$ dB}. As $K$ increases, the outage probabilities achieved by the
three SGF schemes decreases. Furthermore, Fig. 7 verifies that the multiuser
diversity can be achieved by the CR-RSMA-SGF transmissions. Thus, among the
three SGF schemes, the CR-RSMA-SGF transmissions achieve the smallest outage
probability for different numbers of the GFUs.

##  V Conclusions

In this paper, we have proposed a new CR-RSMA to improve the outage
performance of SGF transmissions. By applying rate-splitting under the CR
principle, the CR-RSMA-SGF transmissions can effectively utilize the transmit
power to attain the maximum achievable rate for the admitted GFU. Without
introducing intolerable interference to the GBU, the CR-RSMA-SGF transmissions
have significantly extended the non-outage zone compared to the CR-NOMA-SGF
transmissions. We have derived exact and approximated expressions for the
outage probability of the admitted GFU and revealed that the full multiuser
diversity gain can be achieved. Simulation results have verified the superior
outage performance achieved by the CR-RSMA-SGF transmissions.

## Appendix A: A proof of Theorem 1

To derive the admitted GFU’s outage probability, we evaluate the probability
terms $P_{\rm out}^{(\rm I)}$, $P_{\rm out}^{(\rm II)}$, and $P_{\rm
out}^{(\rm III)}$, respectively. In Case II, $P_{\rm out}^{(\rm
II)}=\sum\nolimits_{k=0}^{K-1}P_{\rm out}^{(\rm II,\it{k})}$ is derived by
evaluating the different $P_{\rm out}^{(\rm II,\it{k})}$ taking into account
the corresponding order statistics.

###  V-A Evaluation of $P_{\rm out}^{(\rm II,0)}$

To evaluate $P_{\rm out}^{(\rm II,0)}$, let us introduce $S_{0}$ as

$\displaystyle
S_{0}=\Pr\left(|h_{1}|^{2}>\frac{P_{0}\epsilon_{0}^{-1}|h_{0}|^{2}-1}{P_{s}},|h_{K}|^{2}<\frac{(1+\epsilon_{s})(1+\epsilon_{0})-(1+P_{0}|h_{0}|^{2})}{P_{s}}\right).$
(A.1)

Then, $P_{\rm out}^{(\rm II,0)}$ in (27) can be rewritten as

$\displaystyle P_{\rm out}^{(\rm II,0)}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\mathop{\mathbb{E}}\limits_{\eta_{0}<|h_{0}|^{2}<\eta_{0}(1+\epsilon_{s})+\frac{\epsilon_{s}}{P_{0}}}\left\\{S_{0}\right\\},$
(A.2)

where $\mathop{\mathbb{E}}\\{\cdot\\}$ stands for the expectation operation,
$\epsilon_{0}=2^{\hat{R}_{0}}-1$, $\epsilon_{s}=2^{\hat{R}_{s}}-1$, and
$\eta_{0}=\frac{\epsilon_{0}}{P_{0}}$. In (A.2), the expectation on $S_{0}$ is
conducted over
$\eta_{0}<|h_{0}|^{2}<\eta_{0}(1+\epsilon_{s})+\frac{\epsilon_{s}}{P_{0}}$
taking into account $\hat{\tau}>0$ and
$\frac{(1+\epsilon_{s})(1+\epsilon_{0})-(1+P_{0}|h_{0}|^{2})}{P_{s}}$ $>0$.
Since the upper bound on $|h_{K}|^{2}$ should be larger than the lower bound
on $|h_{1}|^{2}$, the expectation in (A.2) should consider the hidden
constraint $|h_{0}|^{2}<\eta_{0}(1+\epsilon_{s})$ as well, so that $P_{\rm
out}^{(\rm II,0)}$ is rewritten as

$\displaystyle P_{\rm out}^{(\rm
II,0)}=\mathop{\mathbb{E}}\limits_{\eta_{0}<|h_{0}|^{2}<\eta_{0}(1+\epsilon_{s})}\left\\{S_{0}\right\\}.$
(A.3)

The joint probability density function (PDF) of the order statistics
$|h_{1}|^{2}$ and $|h_{K}|^{2}$ is given by [45]

$\displaystyle
f_{|h_{1}|^{2},|h_{K}|^{2}}(x,y)=\varphi_{0}e^{-x}(e^{-x}-e^{-y})^{K-2}e^{-y},$
(A.4)

where $x<y$ and $\varphi_{0}=\frac{K!}{(K-2)!}$. Then, $S_{0}$ can be
evaluated as follows:

$\displaystyle S_{0}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\varphi_{0}\sum\limits_{i=0}^{K-2}\binom{K\\!-\\!2}{i}(-1)^{i}\int\nolimits_{\frac{\eta_{0}^{-1}|h_{0}|^{2}-1}{P_{s}}}^{\frac{(1+\epsilon_{s})(1+\epsilon_{0})-(1+P_{0}|h_{0}|^{2})}{P_{s}}}e^{-(K-i-1)x}$
(A.5)
$\displaystyle\times\int\nolimits_{x}^{\frac{(1+\epsilon_{s})(1+\epsilon_{0})-(1+P_{0}|h_{0}|^{2})}{P_{s}}}e^{-(i+1)y}dydx$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\varphi_{0}\sum\limits_{i=0}^{K-2}\binom{K\\!-\\!2}{i}\frac{(-1)^{i}}{i+1}$
$\displaystyle\times\left(\frac{\tilde{\mu}_{3}e^{-\tilde{\mu}_{4}|h_{0}|^{2}}-\tilde{\mu}_{5}e^{-\tilde{\mu}_{6}|h_{0}|^{2}}}{K}-\frac{\tilde{\mu}_{1}e^{-\tilde{\mu}_{2}|h_{0}|^{2}}-\tilde{\mu}_{5}e^{-\tilde{\mu}_{6}|h_{0}|^{2}}}{K-i-1}\right),$

where
$\tilde{\mu}_{1}=e^{\frac{K-(1+i)(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}}$,
and $\tilde{\mu}_{2}=\frac{K-i-1}{P_{s}\eta_{0}}-\frac{P_{0}(1+i)}{P_{s}}$,
$\tilde{\mu}_{3}=e^{\frac{K}{P_{s}}}$,
$\tilde{\mu}_{4}=\frac{K}{P_{s}\eta_{0}}$,
$\tilde{\mu}_{5}=e^{-\frac{K(\eta_{0}+\eta_{s}+\eta_{0}\eta_{s})}{P_{s}}}$,
and $\tilde{\mu}_{6}=-\frac{KP_{0}}{P_{s}}$, and
$\eta_{s}=\frac{\epsilon_{s}}{P_{s}}$.

Next, we introduce a term $\nu(i,\mu)$ as follows:

$\displaystyle\nu(i,\mu)$ $\displaystyle\\!\\!\\!\triangleq\\!\\!\\!$
$\displaystyle\mathop{\mathbb{E}}\limits_{\eta_{0}<|h_{0}|^{2}<\eta_{0}(1+\epsilon_{s})}\left\\{e^{-\left(\frac{i}{P_{s}\eta_{0}}+\mu\right)|h_{0}|^{2}}\right\\}$
(A.8) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}e^{-\left(\frac{i}{P_{s}\eta_{0}}+\mu+1\right)x}dx$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\left\\{{\begin{array}[]{*{20}{c}}{\epsilon_{s}\eta_{0}},&{{\rm
if~{}~{}}\mu=-1-\frac{i}{P_{s}\eta_{0}}},\\\
{\frac{e^{-\eta_{0}\left(\frac{i}{P_{s}\eta_{0}}+\mu+1\right)}-e^{-\eta_{0}(1+\epsilon_{s})\left(\frac{i}{P_{s}\eta_{0}}+\mu+1\right)}}{\frac{i}{P_{s}\eta_{0}}+\mu+1}},&{\text{
otherwise}}.\end{array}}\right.$

By substituting (A.8) into (A.5), $P_{\rm out}^{(\rm II,0)}$ can be evaluated
as

$\displaystyle P_{\rm out}^{(\rm II,0)}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\varphi_{0}\sum\limits_{i=0}^{K-2}\binom{K\\!-\\!2}{i}\frac{(-1)^{i}}{i+1}\left(\frac{\tilde{\mu}_{3}\nu(0,\tilde{\mu}_{4})-\tilde{\mu}_{5}\nu(0,\tilde{\mu}_{6})}{K}-\frac{\tilde{\mu}_{1}\nu(0,\tilde{\mu}_{2})-\tilde{\mu}_{5}\nu(0,\tilde{\mu}_{6})}{K-i-1}\right)$
(A.9) $\displaystyle\\!\\!\\!\mathop{=}\limits^{(a)}\\!\\!\\!$
$\displaystyle\\!-\frac{\varphi_{0}}{K\\!-\\!1}\\!\sum\limits_{n=0}^{K-1}\\!\binom{K\\!-\\!1}{n}\\!(-1)^{n}\\!\left(\frac{\tilde{\mu}_{3}\nu(0,\tilde{\mu}_{4})\\!-\\!\tilde{\mu}_{5}\nu(0,\tilde{\mu}_{6})}{K}\\!-\\!\frac{\mu_{1}\nu(0,\mu_{2})\\!-\\!\tilde{\mu}_{5}\nu(0,\tilde{\mu}_{6})}{K-n}\right)\\!,~{}~{}~{}~{}~{}$

where $\mu_{1}=e^{\frac{K-n(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}}$ and
$\mu_{2}=\frac{K-n}{P_{s}\eta_{0}}-\frac{P_{0}n}{P_{s}}$. In step (a) of
(A.9), we have applied $\binom{K-2}{i}=\binom{K-1}{i+1}\frac{i+1}{K-1}$,
replaced $n=i+1$, and added the term for $n=0$ without changing the summation
since
$\frac{\tilde{\mu}_{3}\nu(0,\tilde{\mu}_{4})\\!-\\!\tilde{\mu}_{5}\nu(0,\tilde{\mu}_{6})}{K}\\!-\\!\frac{\mu_{1}\nu(0,\mu_{2})\\!-\\!\tilde{\mu}_{5}\nu(0,\tilde{\mu}_{6})}{K-n}=0$
when $n=0$.

By eliminating the terms that are independent of $n$ using
$\sum\nolimits_{n=0}^{K-1}\binom{K-1}{n}(-1)^{n}=0$, (A.9) can be simplified
as:

$\displaystyle P_{\rm out}^{(\rm II,0)}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K-1}\sum\limits_{n=0}^{K-1}\binom{K\\!-\\!1}{n}(-1)^{n}~{}\frac{\mu_{1}\nu(0,\mu_{2})-\tilde{\mu}_{5}\nu(0,\tilde{\mu}_{6})}{K-n}$
(A.10) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K(K-1)}\sum\limits_{n=0}^{K}\binom{K}{n}(-1)^{n}\left(\mu_{1}\nu(0,\mu_{2})-\tilde{\mu}_{5}\nu(0,\tilde{\mu}_{6})\right),$

where the term for $n=K$ is added without changing the summation since
$\mu_{1}\nu(0,\mu_{2})-\tilde{\mu}_{5}\nu(0,\tilde{\mu}_{6})=0$ when $n=K$.
Again, by eliminating the terms that are independent of $n$ using
$\sum\nolimits_{n=0}^{K}\binom{K}{n}(-1)^{n}=0$, $P_{\rm out}^{(\rm II,0)}$
can be further simplified as:

$\displaystyle P_{\rm out}^{(\rm II,0)}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K(K-1)}\sum\limits_{n=0}^{K}\binom{K}{n}(-1)^{n}~{}\mu_{1}\nu(0,\mu_{2}).$
(A.11)

###  V-B Evaluation of $P_{\rm out}^{(\rm II,\it{k})}$ with $1\leq k\leq K-2$

When $1\leq k\leq K-2$, three order statistics, $h_{k}$, $h_{k+1}$, and
$h_{K}$, are involved in $P_{\rm out}^{(\rm II,\it{k})}$, which can be
rewritten as:

$\displaystyle P_{\rm out}^{(\rm II,\it{k})}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\Pr\left(|h_{0}|^{2}>\eta_{0},|h_{k}|^{2}<\frac{\tau}{P_{s}},|h_{k+1}|^{2}>\frac{\tau}{P_{s}},R_{{}_{{\rm
II},K}}<\hat{R}_{k}\right)$ (A.12) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\mathop{\mathbb{E}}\limits_{\eta_{0}<|h_{0}|^{2}<\eta_{0}(1+\epsilon_{s})}\\{S_{k}\\},$

where $S_{k}$ is defined by

$\displaystyle S_{k}$ $\displaystyle\\!\\!\\!\triangleq\\!\\!\\!$
$\displaystyle\Pr\left(|h_{k}|^{2}<\frac{P_{0}\epsilon_{0}^{-1}|h_{0}|^{2}-1}{P_{s}},|h_{k+1}|^{2}>\frac{P_{0}\epsilon_{0}^{-1}|h_{0}|^{2}-1}{P_{s}},\right.$
(A.13)
$\displaystyle~{}~{}~{}~{}~{}\left.|h_{K}|^{2}<\frac{(1+\epsilon_{s})(1+\epsilon_{0})-(1+P_{0}|h_{0}|^{2})}{P_{s}}\right).$

In (A.12), the expectation is taken over
$\eta_{0}<|h_{0}|^{2}<\eta_{0}(1+\epsilon_{s})$ considering the relationship
between the upper and lower bounds on the channel gains.

The joint PDF of three order statistics, $h_{k}$, $h_{k+1}$, and $h_{K}$, is
given by [45]

$\displaystyle f_{|h_{k}|^{2},|h_{k+1}|^{2},|h_{K}|^{2}}(x,y,z)$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\tilde{\varphi}_{k}e^{-x}(1-e^{-x})^{k-1}e^{-y}(e^{-y}-e^{-z})^{K-k-2}e^{-z}$
(A.14) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\tilde{\varphi}_{k}\\!\\!\sum\limits_{i=0}^{\\!K\\!-k\\!-\\!2\\!}\\!\\!\binom{K\\!-\\!k\\!-\\!2}{i}\\!(-1)^{i}e^{-x}(1-e^{-x})^{k-1}e^{-(K-k-i-1)y}e^{-(i+1)z},~{}~{}~{}~{}~{}~{}~{}$

where $x\leq y\leq z$ and $\tilde{\varphi}_{k}=\frac{K!}{(k-1)!(K-k-2)!}$.
Using (A.14), $S_{k}$ can be expressed in terms of $|h_{0}|^{2}$ as follows:

$\displaystyle S_{k}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\tilde{\varphi}_{k}\\!\sum\limits_{i=0}^{K-k-2}\\!\\!\binom{K\\!-\\!k\\!-\\!2}{i}(-1)^{i}\int\nolimits_{0}^{\frac{\eta_{0}^{-1}|h_{0}|^{2}-1}{P_{s}}}e^{-x}(1-e^{-x})^{k-1}$
(A.15)
$\displaystyle\times\int\nolimits_{\frac{\eta_{0}^{-1}|h_{0}|^{2}-1}{P_{s}}}^{\frac{(1+\epsilon_{s})(1+\epsilon_{0})-(1+P_{0}|h_{0}|^{2})}{P_{s}}}e^{-(K-k-i-1)y}\int\nolimits_{y}^{\frac{(1+\epsilon_{s})(1+\epsilon_{0})-(1+P_{0}|h_{0}|^{2})}{P_{s}}}e^{-(i+1)z}dzdydx.~{}~{}~{}~{}$

After some algebraic manipulations, $S_{k}$ can be further evaluated as
follows:

$\displaystyle S_{k}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\tilde{\varphi}_{k}\\!\sum\limits_{i=0}^{K-k-2}\\!\\!\binom{K\\!-\\!k\\!-\\!2}{i}\sum\limits_{\ell=0}^{k}\binom{k}{\ell}\frac{(-1)^{\ell+i}e^{\frac{\ell}{P_{s}}}e^{-\frac{\ell|h_{0}|^{2}}{P_{s}\eta_{0}}}}{k(i+1)}$
(A.16)
$\displaystyle\times\left(\frac{\tilde{\mu}_{1}e^{-\tilde{\mu}_{2}|h_{0}|^{2}}-\tilde{\mu}_{5}e^{-\tilde{\mu}_{6}|h_{0}|^{2}}}{K-k}-\frac{\tilde{\mu}_{3}e^{-\tilde{\mu}_{4}|h_{0}|^{2}}-\tilde{\mu}_{5}e^{-\tilde{\mu}_{6}|h_{0}|^{2}}}{K-k-i-1}\right),$

where $\tilde{\mu}_{1}=e^{\frac{K-k}{P_{s}}}$,
$\tilde{\mu}_{2}=\frac{K-k}{P_{s}\eta_{0}}$,
$\tilde{\mu}_{3}=e^{\frac{K-k-(1+i)\left(1+\epsilon_{0}\right)\left(1+\epsilon_{s}\right)}{P_{s}}}$,
$\tilde{\mu}_{4}=\frac{K-k-i-1}{P_{s}\eta_{0}}-\frac{(1+i)P_{0}}{P_{s}}$,
$\tilde{\mu}_{5}=e^{-\frac{(K-k)\left(\epsilon_{0}+\epsilon_{s}+\epsilon_{0}\epsilon_{s}\right)}{P_{s}}}$,
and $\tilde{\mu}_{6}=-\frac{(K-k)P_{0}}{P_{s}}$.

By substituting (A.16) into (A.12), $P_{\rm out}^{(\rm II,\it{k})}$ can be
evaluated as:

$\displaystyle P_{\rm out}^{(\rm II,\it{k})}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\tilde{\varphi}_{k}\\!\sum\limits_{i=0}^{K-k-2}\\!\\!\binom{K\\!-\\!k\\!-\\!2}{i}\frac{(-1)^{i}}{k(i+1)}\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}e^{\frac{n}{P_{s}}}$
(A.17)
$\displaystyle\times\left(\frac{\tilde{\mu}_{1}\nu(n,\tilde{\mu}_{2})-\tilde{\mu}_{5}\nu(n,\tilde{\mu}_{6})}{K-k}-\frac{\tilde{\mu}_{3}\nu(n,\tilde{\mu}_{4})-\tilde{\mu}_{5}\nu(n,\tilde{\mu}_{6})}{K-k-i-1}\right)$
$\displaystyle\\!\\!\\!\mathop{=}\limits^{(a)}\\!\\!\\!$
$\displaystyle\frac{-\tilde{\varphi}_{k}}{k(K-k-1)}\\!\sum\limits_{m=0}^{K\\!-\\!k\\!-\\!1}\\!\\!\binom{K-k-1}{m}(-1)^{m}\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}e^{\frac{n}{P_{s}}}$
$\displaystyle\times\left(\frac{\tilde{\mu}_{1}\nu(n,\tilde{\mu}_{2})-\tilde{\mu}_{5}\nu(n,\tilde{\mu}_{6})}{K-k}-\frac{\mu_{3}\nu(n,\mu_{4})-\tilde{\mu}_{5}\nu(n,\tilde{\mu}_{6})}{K-k-m}\right),$

where
$\mu_{3}=e^{\frac{K-k-m\left(1+\epsilon_{0}\right)\left(1+\epsilon_{s}\right)}{P_{s}}}$
and $\mu_{4}=\frac{K-k-m}{P_{s}\eta_{0}}-\frac{mP_{0}}{P_{s}}$. In step (a) of
(A.17), we have replaced $\binom{K-k-2}{i}$ with
$\binom{K-k-1}{i+1}\frac{i+1}{K-k-1}$, applied $m=i+1$, and added term for
$m=0$ without changing the value of $P_{\rm out}^{(\rm II,\it{k})}$ since
$\frac{\tilde{\mu}_{1}\nu(n,\tilde{\mu}_{2})-\tilde{\mu}_{5}\nu(n,\tilde{\mu}_{6})}{K-k}-\frac{\mu_{3}\nu(n,\mu_{4})-\tilde{\mu}_{5}\nu(n,\tilde{\mu}_{6})}{K-k-m}=0$
when $m=0$.

Furthermore, some terms in (A.17) involving $\tilde{\mu}_{1}$,
$\tilde{\mu}_{2}$, $\tilde{\mu}_{5}$, and $\tilde{\mu}_{6}$ but being
independent of $m$ can be further eliminated since
$\sum\nolimits_{m=0}^{k}\binom{k}{m}(-1)^{m}=0$, while $\tilde{\mu}_{1}$,
$\tilde{\mu}_{2}$, $\tilde{\mu}_{5}$, and $\tilde{\mu}_{6}$ are not functions
of $m$. The simplification can be expressed as follows:

$\displaystyle P_{\rm out}^{(\rm II,\it{k})}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\tilde{\varphi}_{k}}{k(K-k-1)}\\!\sum\limits_{m=0}^{K-k-1}\\!\\!\binom{K\\!-\\!k\\!-\\!1}{m}(-1)^{m}\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}e^{\frac{n}{P_{s}}}\frac{\mu_{3}\nu(n,\mu_{4})-\tilde{\mu}_{5}\nu(n,\tilde{\mu}_{6})}{K-k-m}$
(A.18) $\displaystyle\\!\\!\\!\mathop{=}\limits^{(a)}\\!\\!\\!$
$\displaystyle\varphi_{k}\sum\limits_{m=0}^{K-k}\binom{K\\!-\\!k}{m}(-1)^{m}\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}e^{\frac{n}{P_{s}}}(\mu_{3}\nu(n,\mu_{4})-\tilde{\mu}_{5}\nu(n,\tilde{\mu}_{6}))$
$\displaystyle\\!\\!\\!\mathop{=}\limits^{(b)}\\!\\!\\!$
$\displaystyle\varphi_{k}\sum\limits_{m=0}^{K-k}\binom{K\\!-\\!k}{m}(-1)^{m}\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}e^{\frac{n}{P_{s}}}\mu_{3}\nu(n,\mu_{4}),$

where step (a) follows by absorbing $K-k-1$ into the binomial coefficients
without changing the summation and step (b) follows by using
$\sum\nolimits_{m=0}^{k}\binom{k}{m}(-1)^{m}=0$ and $\tilde{\mu}_{5}$ and
$\tilde{\mu}_{6}$ that are not functions of $m$.

###  V-C Evaluation of $P_{\rm out}^{(\rm II,{\it{K}}-1)}$

The probability term $P_{\rm out}^{(\rm II,{\it{K}}-1)}$ can be expressed as:

$\displaystyle P_{\rm out}^{(\rm II,{\it{K}}-1)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\mathop{\mathbb{E}}\limits_{|h_{0}|^{2}>\eta_{0}}\left\\{\Pr\left(|h_{K-1}|^{2}<\frac{P_{0}\epsilon_{0}^{-1}|h_{0}|^{2}-1}{P_{s}},|h_{K}|^{2}>\frac{P_{0}\epsilon_{0}^{-1}|h_{0}|^{2}-1}{P_{s}},\right.\right.$
(A.19)
$\displaystyle\left.\left.|h_{K}|^{2}<\frac{(1+\epsilon_{s})(1+\epsilon_{0})-(1+P_{0}|h_{0}|^{2})}{P_{s}}\right)\right\\}.$

By extracting the hidden constraint on the upper and lower bounds on
$|h_{K}|^{2}$ from (A.19), i.e.,
$\frac{P_{0}\epsilon_{0}^{-1}|h_{0}|^{2}-1}{P_{s}}<\frac{(1+\epsilon_{s})(1+\epsilon_{0})-(1+P_{0}|h_{0}|^{2})}{P_{s}}$,
$P_{\rm out}^{(\rm II,{\it{K}}-1)}$ can be rewritten as follows:

$\displaystyle P_{\rm out}^{(\rm II,{\it{K}}-1)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\mathop{\mathbb{E}}\limits_{\eta_{0}<|h_{0}|^{2}<(1+\epsilon_{s})\eta_{0}}\\{S_{K-1}\\},$
(A.20)

where $S_{K-1}$ denotes probability inside the expectation in (A.19). There
are two order statistics $h_{K-1}$ and $h_{K}$ involving in $S_{K-1}$ with the
joint PDF [45]

$\displaystyle
f_{|h_{K-1}|^{2},|h_{K}|^{2}}(x,y)=\varphi_{0}e^{-x}(1-e^{-x})^{K-2}e^{y},$
(A.21)

where $x\leq y$. Using (A.21), $S_{K-1}$ can be evaluated as follows:

$\displaystyle
S_{K-1}=\varphi_{0}\sum\limits_{n=0}^{K-1}\binom{K\\!-\\!1}{n}\frac{(-1)^{n}e^{\frac{n}{P_{s}}}e^{-\frac{n|h_{0}|^{2}}{P_{s}\eta_{0}}}}{K-1}\left(e^{\frac{1}{P_{s}}}e^{-\mu_{5}|h_{0}|^{2}}-e^{-\frac{\epsilon_{0}+\epsilon_{s}+\epsilon_{0}\epsilon_{s}}{P_{s}}}e^{-\mu_{6}|h_{0}|^{2}}\right),~{}~{}$
(A.22)

where $\mu_{5}=\frac{1}{P_{s}\eta_{0}}$ and $\mu_{6}=-\frac{P_{0}}{P_{s}}$.
Using the expression in (A.8), $P_{\rm out}^{(\rm II,{\it{K}}-1)}$ can be
derived as follows:

$\displaystyle P_{\rm out}^{(\rm
II,{\it{K}}-1)}=\frac{\varphi_{0}}{K-1}\sum\limits_{n=0}^{K-1}\binom{K\\!-\\!1}{n}(-1)^{n}e^{\frac{n}{P_{s}}}\left(e^{\frac{1}{P_{s}}}\nu(n,\mu_{5})-e^{-\frac{\epsilon_{0}+\epsilon_{s}+\epsilon_{0}\epsilon_{s}}{P_{s}}}\nu(n,\mu_{6})\right).$
(A.23)

###  V-D Evaluation of $P_{\rm out}^{(\rm I)}$ and $P_{\rm out}^{(\rm III)}$

In Case I, the determination of $P_{\rm out}^{(\rm I)}$ involves two
independent random variables $|h_{0}|^{2}$ and $|h_{K}|^{2}$. Recalling the
expression in (24), $P_{\rm out}^{(\rm I)}$ can be rewritten as follows:

$\displaystyle P_{\rm out}^{(\rm
I)}=\mathop{\mathbb{E}}\limits_{|h_{0}|^{2}>\eta_{0}}\left\\{\Pr\left(|h_{K}|^{2}<\frac{\eta_{0}^{-1}|h_{0}|^{2}-1}{P_{s}},|h_{K}|^{2}<\frac{\epsilon_{s}}{P_{s}}\right)\right\\}.$
(A.24)

By comparing $\frac{\eta_{0}^{-1}|h_{0}|^{2}-1}{P_{s}}$ and
$\frac{\epsilon_{s}}{P_{s}}$, it can be seen that
$\frac{\eta_{0}^{-1}|h_{0}|^{2}-1}{P_{s}}<\frac{\epsilon_{s}}{P_{s}}$ if
$|h_{0}|^{2}<\eta_{0}(1+\epsilon_{s})$; Otherwise,
$\frac{\eta_{0}^{-1}|h_{0}|^{2}-1}{P_{s}}>\frac{\epsilon_{s}}{P_{s}}$. Thus,
$P_{\rm out}^{(\rm I)}$ can be evaluated as follows:

$\displaystyle P_{\rm out}^{(\rm I)}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\mathop{\mathbb{E}}\limits_{\eta_{0}<|h_{0}|^{2}<\eta_{0}(1+\epsilon_{s})}\left\\{\Pr\left(|h_{K}|^{2}<\frac{\eta_{0}^{-1}|h_{0}|^{2}-1}{P_{s}}\right)\right\\}+\mathop{\mathbb{E}}\limits_{|h_{0}|^{2}>\eta_{0}(1+\epsilon_{s})}\left\\{\Pr\left(|h_{K}|^{2}<\eta_{s}\right)\right\\}$
(A.25) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}\left(1-e^{-\frac{\eta_{0}^{-1}x-1}{P_{s}}}\right)^{K}e^{-x}dx+\left(1-e^{-\eta_{s}}\right)^{K}e^{-\eta_{0}(1+\epsilon_{s})}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\sum\limits_{n=0}^{K}\binom{K}{n}(-1)^{n}e^{\frac{n}{P_{s}}}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}e^{-\left(\frac{n}{P_{s}\eta_{0}}+1\right)x}dx+\left(1-e^{-\eta_{s}}\right)^{K}e^{-\eta_{0}(1+\epsilon_{s})}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\sum\limits_{n=0}^{K}\binom{K}{n}(-1)^{n}e^{\frac{n}{P_{s}}}\nu(n,0)+\left(1-e^{-\eta_{s}}\right)^{K}e^{-\eta_{0}(1+\epsilon_{s})}.$

Similarly to $P_{\rm out}^{(\rm I)}$, $P_{\rm out}^{(\rm III)}$ is a function
of two independent random variables $|h_{0}|^{2}$ and $|h_{K}|^{2}$. In Case
III, the achievable rate is $R_{K}^{(\rm
III)}=\log_{2}\left(1+\frac{P_{s}|h_{K}|^{2}}{P_{0}|h_{0}|^{2}+1}\right)$ and
$\tau=0$. Then, $P_{\rm out}^{(\rm III)}$ can be evaluated as follows:

$\displaystyle P_{\rm out}^{(\rm III)}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\Pr\left(|h_{0}|^{2}<\eta_{0},\log_{2}\left(1+\frac{P_{s}|h_{K}|^{2}}{P_{0}|h_{0}|^{2}+1}\right)<\hat{R}_{s}\right)$
(A.26) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\sum\limits_{n=0}^{K}\binom{K}{n}(-1)^{n}e^{-n\eta_{s}}\frac{1-e^{-(1+n\eta_{s}P_{0})\eta_{0}}}{1+n\eta_{s}P_{0}}.$

By combing the obtained expressions for $P_{\rm out}^{(\rm I)}$, $P_{\rm
out}^{(\rm II)}$, and $P_{\rm out}^{(\rm III)}$, we arrive at (29).

## Appendix B: A proof of Theorem 2

Considering that $P_{\rm out}^{(\rm I)}$, $P_{\rm out}^{(\rm II)}$, and
$P_{\rm out}^{(\rm III)}$ depend on $k$, the high SNR approximations for
$P_{\rm out}^{(\rm I)}$, $P_{\rm out}^{(\rm II)}$, and $P_{\rm out}^{(\rm
III)}$ will be derived separately in the following subsections.

###  V-A High SNR approximation for $P_{\rm out}^{(\rm II,0)}$

Based on the derived closed-form expression in (A.11), $P_{\rm out}^{(\rm
II,0)}$ can be rewritten as follows:

$\displaystyle P_{\rm out}^{(\rm II,0)}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K(K-1)}\sum\limits_{n=0}^{K}\binom{K}{n}(-1)^{n}\mu_{1}\nu(0,\mu_{2}).$
(B.1) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K(K-1)}\sum\limits_{n=0}^{K}\binom{K}{n}(-1)^{n}\mu_{1}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}e^{-(\mu_{2}+1)x}dx.$

By applying the approximation $e^{-x}=1-x$ for $x\to 0$ and using the
definitions of $\mu_{1}$ and $\mu_{2}$, as $P_{0}=P_{s}\to\infty$, $P_{\rm
out}^{(\rm II,0)}$ can be approximated as follows:

$\displaystyle P_{\rm out}^{(\rm II,0)}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K(K-1)}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}\sum\limits_{n=0}^{K}\binom{K}{n}(-1)^{n}e^{-\frac{n(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}}e^{n(\epsilon_{0}^{-1}+1)x}dx$
(B.2) $\displaystyle\\!\\!\\!\mathop{=}\limits^{(a)}\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K(K-1)}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}\left(1-e^{-\left(\frac{(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}-(\epsilon_{0}^{-1}+1)x\right)}\right)^{K}dx$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K(K-1)}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}\left(\frac{(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}-\left(\epsilon_{0}^{-1}+1\right)x\right)^{K}dx,$

where step (a) is obtained by applying
$\sum\nolimits_{n}^{K}\binom{K}{n}(-1)^{n}a^{n}=(1-a)^{K}$. By applying the
binomial expansion, $P_{\rm out}^{(\rm II,0)}$ can be further simplified as
follows:

$\displaystyle P_{\rm out}^{(\rm II,0)}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K(K-1)}\sum\limits_{n=0}^{K}\binom{K}{n}\left(\frac{(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}\right)^{K-n}(-1)^{n}\left(\epsilon_{0}^{-1}+1\right)^{n}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}x^{n}dx$
(B.3) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}\epsilon_{0}(1+\epsilon_{0})^{K}}{P_{s}^{K+1}K(K-1)}\sum\limits_{n=0}^{K}\binom{K}{n}\frac{(-1)^{n}}{n+1}\left((1+\epsilon_{s})^{K+1}-(1+\epsilon_{s})^{K-n}\right).$

###  V-B High SNR approximation for $P_{\rm out}^{(\rm II,\it{k})}$ with
$1\leq k\leq K-2$

Reflecting to the derivations made in Appendix A, $P_{\rm out}^{(\rm
II,\it{k})}$ can be rewritten as follows:

$\displaystyle P_{\rm out}^{(\rm II,\it{k})}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\varphi_{k}\sum\limits_{m=0}^{K-k}\binom{K\\!-\\!k}{m}(-1)^{m}\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}e^{\frac{n}{P_{s}}}\mu_{3}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon{{}_{k}})}e^{-\left(\frac{n}{P{{}_{s}}\eta_{0}}+\mu_{4}+1\right)x}dx.$
(B.4)

By applying the approximation $e^{-x}\approx 1-x$ for $x\to 0$ and using the
definitions of $\mu_{3}$ and $\mu_{4}$, as $P_{0}=P_{s}\to\infty$, $P_{\rm
out}^{(\rm II,\it{k})}$ can be approximated as follows:

$\displaystyle P_{\rm out}^{(\rm II,\it{k})}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\varphi_{k}\sum\limits_{m=0}^{K-k}\binom{K\\!-\\!k}{m}(-1)^{m}\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}e^{\frac{n}{P_{s}}}e^{-\frac{m\left(1+\epsilon_{0}\right)\left(1+\epsilon_{s}\right)}{P_{s}}}$
(B.5)
$\displaystyle\times\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon{{}_{k}})}e^{-\left(\frac{n}{P{{}_{k}}\eta_{0}}-\frac{m}{P_{s}\eta_{0}}-m\right)x}dx.$

Since $\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}a^{n}=(1-a)^{k}$, $P_{\rm
out}^{(\rm II,\it{k})}$ can be further expressed as follows:

$\displaystyle P_{\rm out}^{(\rm II,\it{k})}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\varphi_{k}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}\sum\limits_{m=0}^{K-k}\binom{K\\!-\\!k}{m}(-1)^{m}\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}$
(B.6) $\displaystyle\times
e^{-n\left(\frac{(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}-(\epsilon_{0}^{-1}+1)x\right)}e^{-n\left(\epsilon_{0}^{-1}x-P_{s}^{-1}\right)}dx$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\varphi_{k}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon{{}_{k}})}\left(1-e^{-\left(\frac{(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}-(\epsilon_{0}^{-1}+1)x\right)}\right)^{K-k}\left(1-e^{-\left(\epsilon_{0}^{-1}x-P_{s}^{-1}\right)}\right)^{k}dx$
$\displaystyle\\!\\!\\!\mathop{=}\limits^{(a)}\\!\\!\\!$
$\displaystyle\varphi_{k}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon{{}_{k}})}\left(\frac{(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}-(\epsilon_{0}^{-1}+1)x\right)^{K-k}\left(\epsilon_{0}^{-1}x-P_{s}^{-1}\right)^{k}dx,$

where step (a) follows high SNR approximations. Applying the binomial
expansions to (B.6), the high SNR approximation for $P_{\rm out}^{(\rm
II,\it{k})}$ can be obtained as follows:

$\displaystyle P_{\rm out}^{(\rm II,\it{k})}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{k}\epsilon_{0}(1+\epsilon_{0})^{K-k}(-1)^{k}}{P_{s}^{K+1}}\sum\limits_{m=0}^{K-k}\binom{K-k}{m}(-1)^{m}(1+\epsilon_{s})^{K-k-m}$
(B.7)
$\displaystyle\times\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}\frac{(1+\epsilon_{s})^{m+n+1}-1}{m+n+1}.$

###  V-C High SNR approximation for $P_{\rm out}^{(\rm II,{\it{K}}-1)}$

First, we rewrite $P_{\rm out}^{(\rm II,{\it{K}}-1)}$ as follows:

$\displaystyle P_{\rm out}^{(\rm II,{\it{K}}-1)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K-1}\sum\limits_{n=0}^{K-1}\binom{K\\!-\\!1}{n}(-1)^{n}e^{\frac{n}{P_{s}}}\left(e^{\frac{1}{P_{s}}}\nu(n,\mu_{5})-e^{-\frac{\epsilon_{0}+\epsilon_{s}+\epsilon_{0}\epsilon_{s}}{P_{s}}}\nu(n,\mu_{6})\right)$
(B.8) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K-1}\sum\limits_{n=0}^{K-1}\binom{K\\!-\\!1}{n}(-1)^{n}e^{\frac{n}{P_{s}}}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon{{}_{k}})}\left(e^{\frac{1}{P_{s}}}e^{-\left(\frac{n}{P{{}_{k}}\eta_{0}}+\mu_{5}+1\right)x}\right.$
$\displaystyle\left.-e^{-\frac{\epsilon_{0}+\epsilon_{s}+\epsilon_{0}\epsilon_{s}}{P_{s}}}e^{-\left(\frac{n}{P{{}_{k}}\eta_{0}}+\mu_{6}+1\right)x}\right)dx.$

Since $\sum\limits_{n=0}^{k}\binom{k}{n}(-1)^{n}=0$, $P_{\rm out}^{(\rm
II,{\it{K}}-1)}$ can be expressed as follows:

$\displaystyle P_{\rm out}^{(\rm II,{\it{K}}-1)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K-1}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}\left(e^{\frac{1}{P_{s}}-(\epsilon_{0}^{-1}+1)x}-e^{-\frac{\epsilon_{0}+\epsilon_{s}+\epsilon_{0}\epsilon_{s}}{P_{s}}}\right)\left(1-e^{-n\left(\epsilon_{0}^{-1}x-P_{s}^{-1}\right)}\right)^{K-1}dx.~{}~{}~{}~{}$
(B.9)

By applying the approximations $e^{-x}=1-x$ for $x\to 0$, as
$P_{0}=P_{s}\to\infty$, $P_{\rm out}^{(\rm II,{\it{K}}-1)}$ can be
approximated as follows:

$\displaystyle P_{\rm out}^{(\rm II,{\it{K}}-1)}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K-1}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}\left(\frac{(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}-(\epsilon_{0}^{-1}+1)x\right)\left(\epsilon_{0}^{-1}x-P_{s}^{-1}\right)^{K-1}dx$
(B.10) $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K-1}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}\epsilon_{0}^{-(K-1)}\left(\frac{(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}-(\epsilon_{0}^{-1}+1)x\right)\left(x-\frac{\epsilon_{0}}{P_{s}}\right)^{K-1}dx$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}}{K-1}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}\epsilon_{0}^{-(K-1)}\frac{(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}}\left(x-\frac{\epsilon_{0}}{P_{s}}\right)^{K-1}dx$
$\displaystyle-\frac{\varphi_{0}}{K-1}\int\nolimits_{\eta_{0}}^{\eta_{0}(1+\epsilon_{s})}\epsilon_{0}^{-(K-1)}(\epsilon_{0}^{-1}+1)x\left(x-\frac{\epsilon_{0}}{P_{s}}\right)^{K-1}dx$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\varphi_{0}\epsilon_{0}\epsilon_{s}^{K}(1+\epsilon_{0})(1+\epsilon_{s})}{P_{s}^{K+1}K(K-1)}-\frac{\varphi_{0}\epsilon_{s}^{K}(\epsilon_{0}^{-1}+1)(K(1+\epsilon_{s})+1)}{P_{s}^{K+1}K(K-1)(K+1)}.$

Following similar algebraic manipulations as for deriving the high SNR
approximation of $P_{\rm out}^{({\rm II},k)}$ ($1\leq k\leq K-1$), $P_{\rm
out}^{(\rm I)}$ and $P_{\rm out}^{(\rm III)}$ can be approximated as

$\displaystyle P_{\rm out}^{(\rm
I)}=\frac{\epsilon_{0}\epsilon_{s}^{K+1}}{P_{s}^{K+1}(K+1)}+\frac{\epsilon_{s}^{K}}{P_{s}^{K}}-\frac{\epsilon_{0}\epsilon_{s}^{K}(1+\epsilon_{s})}{P_{s}^{K+1}}$
(B.11)

and

$\displaystyle P_{\rm out}^{(\rm
III)}=\frac{\epsilon_{s}^{K}\left((1+\epsilon_{0})^{K+1}-1\right)}{P_{s}^{K+1}(K+1)}-\frac{\epsilon_{s}^{K}\left((\epsilon_{0}(K+1)-1)(1+\epsilon_{0})^{K+1}+1\right)}{P_{s}^{K+2}(K+2)(K+1)},$
(B.12)

respectively.

By combing (B.3), (B.7), (B.10), (B.11), and (B.12), the high SNR
approximation for $P_{\rm out}$ is obtained as (33).

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