# Large-Scale Rate-Splitting Multiple Access in Uplink UAV Networks: Effective
Secrecy Throughput Maximization Under Limited Feedback Channel

Hamed Bastami, Hamid Behroozi, Majid Moradikia, Ahmed Abdelhadi,
Derrick Wing Kwan Ngand, , and Lajos Hanzo Hamed Bastami and Hamid Behroozi
are with the Department of Electrical Engineering, Sharif University of
Technology, Tehran, Iran, e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>Moradikia is with Department of Data Science
Worcester Polytechnic Institute, Worcester, Massachusetts, e-mail:
<EMAIL_ADDRESS>Ahmed Abdelhadi is with the Engineering Technology
Department at University of Houston<EMAIL_ADDRESS>Wing
Kwan N is with the School of Electrical Engineering and Telecommunications,
University of New South Wales, Sydney, Australia, e-mail:
<EMAIL_ADDRESS>Hanzo is with the University of Southampton,
Southampton SO17 1BJ, U.K, e-mail<EMAIL_ADDRESS>
###### Abstract

Unmanned aerial vehicles (UAVs) are capable of improving the performance of
next generation wireless systems. Specifically, UAVs can be exploited as
aerial base-stations (UAV-BS) for supporting legitimate ground users in remote
uncovered areas or in environments temporarily requiring high capacity.
However, their communication performance is prone to both channel estimation
errors and potential eavesdropping. Hence, we investigate the effective
secrecy throughput of the UAV-aided uplink, in which rate-splitting multiple
access (RSMA) is employed by each legitimate user for secure transmission
under the scenario of massive access. To maximize the effective network
secrecy throughput in the uplink, the transmission rate vs. power allocation
relationship is formulated as a max-min optimization problem, relying on
realistic imperfect channel state information (CSI) of both the legitimate
users and of the potential eavesdroppers ($Eves$). We then propose a novel
transformation of the associated probabilistic constraints for decoupling the
variables, so that our design problem can be solved by alternatively
activating the related block coordinate decent programming. In the model
considered, each user transmits a superposition of two messages to a UAV-BS,
each having different transmit power and the UAV-BS uses a successive
interference cancellation (SIC) technique to decode the received messages.
Given the non-convexity of the problem, it is decoupled into a pair of sub-
problems. In particular, we derive a closed form expression for the optimal
rate-splitting fraction of each user. Then, given the optimal rate-splitting
fraction of each user, the $\epsilon$-constrainted transmit power of each user
is calculated by harnessing sequential parametric convex approximation (SPCA)
programming. Finally, the optimal SIC order is determined by an exhaustive
search method. Our simulation results confirm that the scheme conceived
significantly improves the effective secrecy throughput compared to both the
existing orthogonal and non-orthogonal benchmarks as well as to the RSMA
scheme ignoring CSI uncertainty.

###### Index Terms:

Rate-splitting, physical layer security, effective network secrecy throughput,
imperfect CSIT, connection outage probability, secrecy outage probability,
worst-case optimization, uplink UAV networks.

##  I Introduction

In order to support the emerging Beyond 5G (B5G) system concept, unmanned
aerial vehicles (UAV) may be harnessed as air-borne base-station (BS),
particularly in areas of high tele-traffic density [1]-[6]. However, owing to
their LoS propagation UAV-BSs usually suffer from strong co-channel
interference. Although this problem can be potentially mitigated by the
sophisticated trajectory design of UAVs [1, 2], the degree of freedom attained
is typically inadequate to support the ever-growing terrestrial user
population. In this context, multiple access (MA) techniques play a crucial
role in fulfilling the high data rate, low latency, and massive connectivity
requirements, as the three most important Key Performance Indicators (KPI)s
for B5G [3]-[13].

Rate-splitting multiple access (RSMA) has attracted a great deal of interest,
as a key-enabling radio access technology capable of satisfying the massive
connectivity requirements of B5G [3, 5, 6], [9]-[13]. Briefly, RSMA is a
generalization of non-orthogonal multiple access (NOMA) and space-division
multiple access (SDMA) [9], that outperforms both schemes in terms of its
robustness and spectral efficiency. In a rate-splitting (RS) scheme, the
transmitted signals are split into two parts at the transmitter (Tx), namely
into a common message and a private message. Subsequently, by performing
successive interference cancellation (SIC) at the receiver (Rx), the capacity
region of the MA channel (MAC) can be approached. Inspired by this promising
MA framework, most of the RSMA-based literature considered the downlink (DL)
[3, 5, 6, 9, 10] even though the DL actually represents a broadcast scenario
and multiple access is only possible in the uplink (UL). In [11] a RS scheme
was designed for guaranteeing max-min fairness in UL-NOMA. As a further
advance, a cooperative rate-splitting (CRS) UL scheme was proposed in [12],
where each user broadcasts his/her signal during the first phase and receives
the transmitted signal of the other user, while during the second phase, each
user relays the other user’s message. Then, Yang et al. [13], proposed UL-RSMA
for maximizing the users’ sum-rate sum-rate by optimally sharing the total
transmit powers of both user-messages, while exhaustively searching for the
optimal decoding order at the SIC receiver. However, these contributions
stipulate the idealized simplifying assumption of having perfect channel state
information (CSI) for resource allocation design, which is not realistic in
practice. More particularly, in massive access scenarios in which a large
number of CSIs have to be reported to the BS using limited feedback having CSI
imperfections is unavoidable, resulting in link outage [8, 14]. More
importantly, none of the above-mentioned RSMA UL scenarios of [11]-[13] have
addressed the associated security concerns. In particular, the concurrent UL
transmissions of a massive number of messages over the same bandwidth
increases the risk of security breaches. To protect the confidentiality of the
transmitted signals, physical layer security (PLS) techniques can be exploited
for increasing the channel capacity difference between the legitimate and
eavesdropping links. Unfortunately, MA systems are particularly susceptible to
passive attacks, since the eavesdroppers ($Eves$) have more target users, they
can glean information from [5, 8, 10]. In this context, jamming aims for
confusing the potential $Eves$ by deliberately injecting specifically designed
artificial noise (AN) with the aid of beamforming [15, 16]. As a further
development, the authors of [17, 18] proposed a secure NOMA approach. However,
since the super-imposed non-orthogonal signals may be detected by SIC at
$Eve$, superposition potentially degrades the level of security. In fact,
after detecting the superimposed streams $Eve$ becomes capable of wiretapping
the rest of the embedded information, which is becoming less interference-
infested. By contrast, using an RSMA scheme, the common message plays the dual
roles of the desired message as well as that of the AN without the need for
assigning a portion of the limited transmit power to the AN [5, 6, 10].
However, the secure RSMA designs of [5, 6, 10] considered the DL scenario,
hence their results are not applicable to the UL due to the different nature
of the problems. To the best of our knowledge, at the time of writing, no
attention has been devoted to the integration of UL-RS with UAV-BS.
Furthermore, the robust and secure design of RSMA-aided UAV networks relying
on realistic imperfect CSI has not been investigated so far.

Given the knowledge gaps mentioned above, we consider a network in which the
legitimate users aim for communicating with a UAV-BS in the presence of
multiple passive $Eve$. In this UL scenario, each user employs RS, where the
corresponding message of each user is split into two parts. Then, each user
transmits a superposition of two messages having different power levels. To
realize massive connectivity, we assume furthermore that at the network
initialization a clustering process is accomplished by which the users are
divided into different non-overlapping groups. Furthermore, due to the limited
CSI feedback accuracy, a link outage may occur. Hence we introduce a maximum
tolerable connection outage probability (COP) constraint for quantifying its
impact on the system performance. It is worth mentioning that in contrast to
the RSMA downlink in [5, 10], where the authors considered the secure design
of the common streams, here we exploit a different strategy, where the
transmission rate and the power allocated to each part of the bipartite
messages is optimized in terms of Effective Network Secrecy Throughput (ENST)
maximization. ENST is a secrecy performance metric quantifying the average
secure throughput. More explicitly, when the reception reliability is
considered to be similarly important to the security, then this parameter is
considered. Mathematically, ENST is formulated as the product of the target
secrecy rate and the probability of successful reception as defined in [33,
Eq. (5)]. Against this background, our contributions are summarized as
follows:

  * •

In addition to the COP constraint, which captures the impact of link outages,
the secrecy outage probability (SOP) is tightly controlled to be under the
tolerable level under unknown CSI of the $Eve$. We then maximize the ENST,
subject to both COP and SOP constraints, as well as to the limited power
budget. In particular, we design the RS power allocation at the users as well
as the SIC-ordering corresponding to each cluster, so that the ENST is
maximized.

  * •

To deal with the resultant non-convex problem, we first derive a closed-form
expression for characterizing the COP and a tight approximation of the SOP
constraints. Then, we harness the two-tier block coordinate decent technique,
where the optimization variables are estimated successively in an iterative
manner. The first loop of this twin-tier approach maximizes the transmission
rates, leading to a closed-form optimal solution relying on the Lambert
$W$-function. By contrast, the second loop encounters some non-convexities,
which are tackled by the powerful sequential parametric convex approximation
(SPCA) method. The convex approximation of the non-convex factors are found
with the aid of the first-order Taylor expansion.

  * •

Our simulation results demonstrate that the proposed framework outperforms the
existing non-orthogonal benchmarks in terms of the ENST criterion.

TABLE I: Boldly and explicitly contrasting our contributions to the existing literature. References$\Rightarrow$ Keywords$\Downarrow$ |  Our Approach |  [1] |  [2] |  [3] |  [4] |  [5] |  [6] |  [7] |  [8] |  [9] |  [10] |  [11] |  [12] |  [13] |  [14] |  [15] |  [16] |  [17] |  [18]
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
UAV-BS |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |  |
UAV Trajectory Design |  |  $\checkmark$ |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
IUI Management |  |  $\checkmark$ |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |
IUI Cancellation |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |  |  |  $\checkmark$ |  |  |  |  |
SIC Ordering |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |  |  |  $\checkmark$ |  |  |  |  |
RSMA |  $\checkmark$ |  |  |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  |  |  |
NOMA |  |  |  |  |  |  |  |  |  $\checkmark$ |  |  |  |  |  |  |  |  |  $\checkmark$ |  $\checkmark$
SDMA |  |  |  |  |  |  |  |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |
DL |  |  |  |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  |  |  |  |  |  $\checkmark$ |  $\checkmark$
UL |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  |  |  |  $\checkmark$ |
Limited Feedback Error |  $\checkmark$ |  |  |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  |  |  |  |  |  $\checkmark$ |  |  |  |
Imperfect CSI |  $\checkmark$ |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  |  |  |  $\checkmark$ |  |  |  $\checkmark$ |
PLS |  $\checkmark$ |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  |  |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$
AN Design |  |  |  |  |  |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  |
Beamformer (precoder) Design |  $\checkmark$ |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  |  |  |  $\checkmark$ |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |  $\checkmark$
Power Allocation |  $\checkmark$ |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ |  |  $\checkmark$ |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  $\checkmark$ |
Known $Eve$ with Imperfect E-CSIT |  |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  |  |  |  $\checkmark$ |  |  |  |  |  |  |  $\checkmark$ |
Worst-Case Secrecy Rate Maximization |  |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  |  |  |  $\checkmark$ |  |  |  |  |  |  |  |
Max-Min Fairness |  |  |  |  |  |  $\checkmark$ |  $\checkmark$ |  |  |  |  $\checkmark$ |  $\checkmark$ |  |  |  |  |  |  |
Sum-Rate Maximization |  |  |  |  |  |  |  |  |  $\checkmark$ |  |  |  |  |  $\checkmark$ |  |  |  |  |
Known $Eve$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $\checkmark$ |  |
Unknown $Eve$ |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $\checkmark$ |  |  |
ICI Cancellation |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
ENST Maximization |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
COP Constraint |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
SOP Constraint |  $\checkmark$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |

Our contributions are boldly and explicitly contrasted to the state-of-the-art
at a glance in Table $1$. The rest of this paper is organized as follows. The
system model and channel definitions are provided in Section II. Section III
describes the signal representation and formulates our ENST maximization
problem. The proposed SPCA-based solution, the two-tier block coordinate
decent procedures and our complexity analysis are provided in Section IV. In
Section V, our simulation results are presented and the paper is concluded in
Section VI. Finally, the Appendices and Proofs of the claims are provided in
Section VII.

Notation: Vectors and matrices are denoted by lower-case and upper-case
boldface symbols, respectively; $\left(.\right)^{\mathrm{T}}$,
$\left(.\right)^{\mathrm{*}}$, $\left(.\right)^{\mathrm{H}}$, and
$\left(.\right)^{\mathrm{-1}}$ denote the transpose, conjugate, conjugate
transpose, and inverse of a matrix respectively; $\mathfrak{R}e(.)$ denotes
the real part of a complex variable, and $\mathfrak{I}m(.)$ the imaginary part
of a complex variable; We use $\mathbb{E}{\left\\{\cdot\right\\}}$ and
$\triangleq$ to denote the expectation operation and a definition,
respectively. A complex Gaussian random variable with mean $\mathit{\mu}$ and
variance $\sigma^{2}$ reads as
$\mathcal{C}\mathcal{N}\left(\mathit{\mu},\sigma^{2}\right)$, and
$\textrm{Exp}\left(\lambda\right)$, $\textrm{Beta}\left(\alpha,\beta\right)$,
and $\textrm{Gamma}\left(\gamma,\zeta\right)$ respectively denote the
exponential distribution with mean $\lambda$, beta-distribution with
parameters $\alpha$ and $\beta$, and gamma-distribution with shape $\gamma$
and rate $\zeta$. The principal branch of the Lambert $W$-function is defined
by $W_{0}\left(x\right)e^{W_{0}\left(x\right)}=x$ for $x\geq-\frac{1}{e}$ with
$W_{0}\left(x\right)\geq-1$ [19]; $\mathbf{I}_{N}$ denotes the $N\times N$
identity matrix; $\mathbb{R}^{N\text{\texttimes}1}$ and $\mathbb{C}^{N\times
1}$ denote the set of $N$-dimensional standard real and complex Gaussian
random variable, respectively; $\mathbb{C}^{N\times N}$ stands for an $N\times
N$ element standard complex Gaussian random matrix whose real and imaginary
parts are independent normally distributed random variables with a mean of
zero and variance $\frac{1}{2}$. The notations $\left[x\right]^{+}$ and
$\mathbb{P}\left(.\right)$ stand for $\textrm{max}{\left\\{x,0\right\\}}$ and
probability, respectively. The entry in the $i$-th row of a vector
$\mathbf{h}$ is represented by $\mathbf{h}\left[i\right]$. Furthermore,
$\mathbf{u}^{max}\left\\{\mathbf{A}\right\\}$ and
$\mathbf{v}^{max}\left\\{\mathbf{A}\right\\}$ denote the columns of
$\mathbf{U}_{\mathbf{A}}$ and $\mathbf{V}_{\mathbf{A}}$ corresponding to the
dominant singular value $\lambda^{max}\left\\{\mathbf{A}\right\\}$ of matrix
$\mathbf{A}$, respectively, i.e., the matrix $\mathbf{A}$ has a Singular Value
Decomposition (SVD) given by
$\mathbf{A}\triangleq\mathbf{U}_{\mathbf{A}}\mathbf{\mathbf{\Lambda}}_{\mathbf{A}}\mathbf{V}_{\mathbf{A}}$.
Finally, $\angle\left(\mathbf{u},\mathbf{v}\right)$ represents the angle
between vectors $\mathbf{v}$ and $\mathbf{u}$.

##  II System Model

We consider the secure single-input multi-output (SIMO) uplink system, shown
in Fig. 1. There are $M$ clusters in the network considered, whose $m^{th}$
cluster includes $K_{m}$ number of single-antenna legitimate users gathered in
the set $\boldsymbol{U}_{m}\triangleq\left\\{U_{m,k}\right\\}$, $\forall
k\in\mathcal{K}_{m}\triangleq\left\\{1,\ldots,K_{m}\right\\}$, who aim for
transmitting to an $N_{t}$-antenna UAV-BS. We consider a massive access
setting, where $\sum_{m=1}^{M}K_{m}\gg M$. Meanwhile, $J$ number of non-
cooperative passive $N_{e}$-antenna eavesdroppers ($Eves$) gathered in the set
$\boldsymbol{E}\triangleq\left\\{E_{e,j}\right\\},\forall
j\in\mathcal{J}\triangleq\left\\{1,\ldots,J\right\\}$, manage in covert
wiretapping111In this paper we have focused on non-colluding $Eve$’s who try
to maximize their own SINR individually. The problem of colluding $Eve$s has
been left for future work. . Next, the channel models and the clustering
procedure operating under CSI error are described.

###  II-A Channel Definitions

Figure 1: The considered system model adopting RSMA

In our scenario, the $U_{m,k}\longrightarrow E_{e,j}$ channels are represented
by $\mathbf{q}_{m,j,k},\ \forall m,\ k,\ j$, while the legitimate channels
spanning from the terrestrial user to the UAV-BS, i.e.,
$U_{m,k}\longrightarrow UAV$, are denoted by $\mathbf{h}_{m,k}\ \forall m,k$.
The ground-to-air (G2A) channels are modeled by
$\mathbf{h}_{m,k}=\sqrt{PL\left(d_{m,k}\right)}\,\mathbf{f}_{m,k}$, where
$PL\left(d_{m,k}\right)\triangleq{d_{m,k}}^{-\alpha_{m,k}}$ represents the
large-scale fading, while $d_{m,k}$ and
$\mathbf{f}_{m,k}\sim\mathcal{CN}\left(0,\mathbf{I}_{N_{t}}\right)$ therein,
respectively, denote the G2A distance and the corresponding small-scale
fading. The path-loss exponent $\alpha_{m,k}$ obeys the probabilistic model
[20], which is appropriate for low-altitude UAVs comprised of both the LoS and
non-LoS components $\mathcal{L}_{m,k}$ and $\mathcal{N}_{m,k}$, given by:

$\alpha_{m,k}\triangleq\frac{\mathcal{L}_{m,k}-\mathcal{N}_{m,k}}{1+\lambda_{1}.\exp{\left[\lambda_{2}\left(\theta_{m,k}-\lambda_{1}\right)\right]}}+\mathcal{N}_{m,k},$
(1)

where $\theta_{m,k}$ denotes the elevation angle between the UAV and user
$U_{m,k}$, while $\lambda_{1}$ and $\lambda_{2}$ are the constants determined
by the propagation environment [20].

###  II-B Clustering Under CSI Error

Given the slowly time-varying nature of the $PL\left(d_{m,k}\right)$, we
assume that both the UAV as well as the users can estimate it perfectly.
However, due to the limited hardware complexity of the UAV, we assume that the
UAV only captures the angle-of-arrival (AoA) information of the user-UAV
channel, and even this AoA information is imperfect. To elaborate a little
further, first the UAV broadcasts a sequence of training symbols towards the
ground users, who aim for acquiring the knowledge of their own DL channels. In
general, given a sufficiently high transmit power, as well as a long training
sequence, legitimate users are capable of perfectly estimating their own
channels. More explicitly, all the users within the $m^{th}$ cluster have the
same AoA and thus we can construct a codebook $\mathcal{V}$, comprised of $M$
unit-norm vectors
$\left\\{\mathbf{v}_{m}\right\\}_{m=1}^{M}\in\mathbb{C}^{N_{t}\times 1}$. At
the network’s initialization, this codebook is randomly generated and made
known off-line to both the UAV and the users for example via the codebook
distribution regime of [22]. To convey the corresponding AoA to the UAV, each
$U_{m,k}$ quantizes its channel direction, i.e.,
$\widetilde{\mathbf{f}}_{m,k}\triangleq\frac{\mathbf{f}_{m,k}}{\left\|\mathbf{f}_{m,k}\right\|}$,
to the closest vector in terms of the chordal distance metric of 222The
optimal vector quantization strategy in multi-user uplink channels, even in
single-cell systems, is not known in general and is beyond the scope of our
work. [23, 24]:

$\displaystyle\hat{\mathbf{f}}_{m,k}$
$\displaystyle\triangleq\underset{\mathbf{v}_{m}\in\mathcal{V}}{\arg\max}\left|\widetilde{\mathbf{f}}_{m,k}^{H}\mathbf{v}_{m}\right|^{2}=\underset{\mathbf{v}_{m}\in\mathcal{V}}{\arg\max}\,\,\cos^{2}\left[\angle\left(\widetilde{\mathbf{f}}_{m,k},\mathbf{v}_{m}\right)\right].$
(2)

Accordingly, the users having the maximum chordal distance between their so-
obtained channel direction ${\widetilde{\mathbf{f}}}_{m,k}$ and
$\mathbf{v}_{m}$ are allocated to the $m^{th}$ cluster and the number of users
within each cluster, i.e., $K_{m}$, is also updated after the grouping. Now,
each user sends the corresponding codebook index back to the UAV using
$B\triangleq\left\lceil\log_{2}{M}\right\rceil$ bits through an error-free and
delay-free feedback channel. However, because of the limited feedback per
channel coherence block as well as the instability of the UAV platform, the
CSI of the main channel obtained at the UAV is imperfect. Thus, a quantization
error in the form of

$\widetilde{\mathbf{f}}_{m,k}=\cos\left(\phi_{m,k}\right)\hat{\mathbf{f}}_{m,k}+\sin\left(\phi_{m,k}\right)\mathbf{e}_{m,k},$
(3)

appears in the AoA estimates of users, where
$\mathbf{e}_{m,k}\in\mathbb{C}^{N_{t}\text{\texttimes}1}$ is the unit-norm
quantization error vector isotropically distributed in the null-space of
$\widetilde{\mathbf{f}}_{m,k}$, while
$\phi_{m,k}\triangleq\angle\left(\widetilde{\mathbf{f}}_{m,k},\mathbf{v}_{m}\right)$
of (3) represents the angle between $\widetilde{\mathbf{f}}_{m,k}$ and
$\mathbf{v}_{m}$, and $\sin^{2}\left(\phi_{m,k}\right)$ being a random
variable, whose variance is determined by $B$ [25]. To visualize the proposed
approach, the whole procedure is shown in Fig. 2, while further details will
be presented in the sequel.

##  III Signal Representation and Problem Formulation

In the RSMA uplink, each $U_{m,k}$ within the $m^{th}$ cluster transmits a
superposition code of two normalized sub-messages
$\left.s_{m,k,n}\right|_{n=1}^{2}$, i.e.,
$\mathbb{E}\left\\{\left|s_{m,k,n}\right|^{2}\right\\}=1$ , given by [26]:

$s_{m,k}=\stackrel{{\scriptstyle[}}{{n}}=1]{2}{\sum}\sqrt{p_{m,k,n}}s_{m,k,n},\,\,\forall
k\text{$\in$}\mathcal{K}_{m},$ (4)

where $p_{m,k,n},\,\forall n\in\left\\{1,2\right\\}$ corresponds to the
transmit power of $s_{m,k,n}\,\forall n\in\left\\{1,2\right\\}$. During the
uplink signal reception, the UAV relies on beamforming for discriminating the
signals received by suppressing the IUI.

Figure 2: The algorithmic procedure of the proposed method Figure 3: RSMA-
based BS structure in $M$-Cluster UL communications.

Therefore, as shown in Fig. 3, the signal received at the UAV, i.e.,
$y_{\mathrm{UAV}}$, is passed through $M$ different angularly selective
filters $\left\\{\mathbf{w}_{m}\right\\}_{m=1}^{M}\in\mathbb{C}^{N_{t}\times
1}$, distributed in $M$ branches. Accordingly, the $m^{th}$ signal (i.e., the
$m^{th}$ branch) received by the UAV and the received signal at $Eves$,
respectively denoted by $y_{UAV,m}$ and $y_{e,j}$, are formulated as follows:

$\displaystyle y_{\mathrm{UAV}}$
$\displaystyle=\stackrel{{\scriptstyle[}}{{i}}=1]{M}{\sum}\left(\stackrel{{\scriptstyle[}}{{k}}=1]{K_{i}}{\sum}\mathbf{h}_{i,k}s_{i,k}\right)+\mathbf{z},$
(5) $\displaystyle y_{\mathrm{UAV},m}$
$\displaystyle=\mathbf{w}_{m}^{H}\left(y_{\mathrm{UAV}}\right),$
$\displaystyle y_{e,j}$
$\displaystyle=\mathbf{w}_{m,j}^{H}\left(\mathbf{Q}_{m,j}\mathbf{s}_{m}+\mathbf{z}_{m,j}\right),\,\forall
j\text{$\in$}\mathcal{J},$ (6)

where
$\mathbf{Q}_{m,j}\triangleq\left[\mathbf{q}_{m,j,1},\mathbf{q}_{m,j,2},...,\mathbf{q}_{m,j,K}\right]$,
$\mathbf{s}_{m}\triangleq\left[s_{m,1},s_{m,2},...,s_{m,K}\right]^{T}$,
$z_{m}\triangleq\mathbf{w}_{m}^{H}\mathbf{z}_{m}\sim\mathcal{CN}\left(0,\sigma_{m}^{2}\right)$
and
$\mathbf{z}_{m,j}\sim\mathcal{CN}\left(0,\sigma_{e}^{2}\mathbf{I}_{N_{e}}\right)$
represent the additive white Gaussian noise (AWGN) due to the $m^{th}$ cluster
at the UAV and at the $j^{th}$ $Eve$, respectively.
${\color[rgb]{1,0,0}{\color[rgb]{0,0,0}\mathbf{w}_{m,j}\triangleq\mathbf{u}^{max}\left\\{\tilde{\mathcal{\mathbf{Q}}}_{m,j}\right\\}}}$
represents the MRC beamformer employed by $Eve$ where
$\tilde{\mathcal{\mathbf{Q}}}_{m,j}\triangleq\mathbf{Q}_{m,j}\left(\mathbf{Q}_{m,j}\right)^{H}$.
It is easy to show that
$y_{e,j}=\stackrel{{\scriptstyle[}}{{k}}=1]{K_{m}}{\sum}g_{m,k,j}s_{m,k}+\xi_{e,j},\,\forall
j\text{$\in$}\mathcal{J},$ where
$g_{m,k,j}\sim\mathcal{CN}\left(0,\sqrt{\lambda_{j}^{max}\left\\{\tilde{\mathcal{\mathbf{Q}}}_{m,j}\right\\}}\right)$
[21], and $\xi_{e,j}\sim\mathcal{CN}\left(0,\sigma_{e}^{2}\right)$. In terms
of the worst-case secrecy scenario, $Eve$ is assumed to be able to perfectly
estimate its corresponding CSI and no ICI is available to degrade the
performance of $Eve$. To force the ICI terms to zero, we resort the zero-
forcing (ZF) beamforming. More explicitly, upon relying on the codebook
$\mathcal{V}$ discussed earlier in Section I.B, $\mathbf{w}_{m}$ is chosen so
that we have $\mathbf{w}_{m}^{H}\mathbf{v}_{l}=0,\,\forall l\neq
m,\,l\text{$\in$}\left\\{1,2,...,M\right\\}$.

when considering the signal extracted from the $m^{th}$ cluster, the UAV
employs $2K_{m}$ number of SIC stages to suppress IUI as well as to decode all
transmitted messages in the set
$\mathcal{K}_{m,n}\triangleq\left\\{s_{m,k,n}\right\\}$ received from
$y_{\mathrm{UAV},m}$, as illustrated in Fig. 3. The decoding order of the
$m^{th}$ cluster at the UAV is denoted by a permutation
$\boldsymbol{\mathbf{\Phi}}_{m}$, which belongs to set $\mathbf{\Pi}_{m}$
defined as the set of all possible decoding orders of all $2K_{m}$ messages
arriving from $K_{m}$ users, which includes $\frac{2K_{m}!}{2^{K_{m}}}$
elements. Let $\mathrm{\Phi}_{m,k,n}$ represents the position of the message
$s_{m,k,n}$ in $\boldsymbol{\mathbf{\Phi}}_{m}$. Therefore, we can define
$\Phi_{m,k,n}=\left\\{\left.\left(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)\right|\left(k^{\prime},n^{\prime}\right)\succ\left(k,n\right)\right\\}$,
where the operator $\left(k^{\prime},n^{\prime}\right)\succ\left(k,n\right)$
indicates that $s_{m,k,n}$ has a higher decoding order than
$s_{m,k^{\prime},n^{\prime}}$ in $\boldsymbol{\mathbf{\Phi}}_{m}$ , i.e., the
UAV is scheduled to decode $s_{m,k^{\prime},n^{\prime}}$ after decoding and
cancelling out the effect of $s_{m,k,n}$.

Therefore, in the SIC scheme of the RSMA uplink, the UAV first decodes and
subtracts the remodulated signals having higher decoding orders. i.e.,
$\left(k^{\prime},n^{\prime}\right)\in\mathcal{K}_{m,n}\setminus\left\\{{{\left(k,n\right)}\bigcup\Phi_{m,k,n}}\right\\}$,
then it decodes signal $s_{m,k,n}$, where the signal of the users in
$\Phi_{m,k,n}$ is treated as noise. According to the SIC protocol, the signal
of $s_{m,k,n}$ will be decoded prior to $s_{m,k^{\prime},n^{\prime}}$ if we
have $\left|\Phi_{m,k,n}\right|>\left|\Phi_{m,k^{\prime},n^{\prime}}\right|$,
where $|\,\mathcal{A}\,|$ is the cardinality of the set $\mathcal{A}$.
Accordingly, the signal-to-interference-plus-noise ratio (SINR) at the UAV
experienced upon detecting $s_{m,k,n}$, and denoted by $\rho_{m,k,n}$ is
formulated as (7),

$\displaystyle\rho_{m,k,n}$
$\displaystyle=\frac{p_{m,k,n}\left|\mathbf{w}_{m}^{H}\mathbf{h}_{m,k}\right|^{2}}{\underset{{}_{\left(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}}}{\sum}p_{m,k^{\prime},n^{\prime}}\left|\mathbf{w}_{m}^{H}\mathbf{h}_{m,k^{\prime}}\right|^{2}+\stackrel{{\scriptstyle[}}{{i}}=1,i\neq
m]{M}{\sum}\,\stackrel{{\scriptstyle[}}{{k}}^{\prime\prime}=1]{K_{i}}{\sum}P_{i,k^{\prime\prime}}\left|\mathbf{w}_{m}^{H}\mathbf{h}_{i,k^{\prime\prime}}\right|^{2}+\sigma_{m}^{2}}$
(7)

$=\frac{p_{m,k,n}\mathrm{PL}\left(d_{m,k}\right)\left|\mathbf{w}_{m}^{H}\mathbf{f}_{m,k}\right|^{2}}{\underset{\mathrm{IUI}}{\underbrace{\underset{{}_{\left(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}}}{\sum}p_{m,k^{\prime},n^{\prime}}\mathrm{PL}\left(d_{m,k^{\prime}}\right)\,\left|\mathbf{w}_{m}^{H}\mathbf{f}_{m,k^{\prime}}\right|^{2}}}+\underset{\mathrm{ICI}}{\underbrace{\stackrel{{\scriptstyle[}}{{i}}=1,i\neq
m]{M}{\sum}\,\stackrel{{\scriptstyle[}}{{k}}^{\prime\prime}=1]{K_{i}}{\sum}P_{i,k^{\prime\prime}}\mathrm{PL}\left(d_{i,k^{\prime\prime}}\right)\left\|\sin\left(\phi_{i,k^{\prime\prime}}\right)\mathbf{f}_{i,k^{\prime\prime}}\right\|^{2}\left|\mathbf{w}_{m}^{H}\mathbf{e}_{i,k^{\prime\prime}}\right|^{2}}}+\sigma_{m}^{2}}$

______________________________________________________________________________________________________

where the IUI and ICI terms are obtained by substituting (4) into (7).
Notably, after clustering in the presence of beamforming weight quantization
errors the ICI cannot be completely removed by the beamformer having the
weights of $\mathbf{w}_{m}$, thus a residual ICI term contaminates the
corresponding received signal of the $m^{th}$ cluster. In other words, the
beamformer weights $\mathbf{w}_{m}$ fail to perfectly null out the ICI due to
the limited feedback.

Furthermore, it is assumed that the $Eves$ have no information about
$\boldsymbol{\mathbf{\Phi}}_{m}$, hence they cannot perform SIC within a
cluster 333As a more challenging secrecy scenario, for comparison, in our
simulation we consider a scenario when $E_{e,j}$ can exploit the optimal SIC
decoding order and receives no CSI.. Consequently, from the perspective of
$E_{e,j}$, the received SINR of decoding $s_{m,k,n}$, while treating the other
ones as noise, is formulated as:

$\mu_{j,k,n}=\frac{p_{m,k,n}\mathrm{PL}\left(d_{m,k,j}\right)\left|g_{m,k,j}\right|^{2}}{\sum_{\left(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)}p_{m,k^{\prime},n^{\prime}}\mathrm{PL}\left(d_{m,k^{\prime},j}\right)\left|g_{m,k^{\prime},j}\right|^{2}+\sigma_{e}^{2}}.$
(8)

Given the SINRs in (7) and (8), the corresponding achievable rates are
respectively given by
$C_{m,k,n}\triangleq\log_{2}{\left(1+\rho_{k,m,n}\right)}$ and
$C_{j,k,n}\triangleq\log_{2}{\left(1+\mu_{j,m,n}\right)}$.

Remark 1. For ensuring reliable uplink communication, the transmission rate
$r_{m,k,n}$ of each $U_{m,k}$ should not exceed $C_{m,k,n}$, i.e.,
$C_{m,k,n}\geq r_{m,k,n}$. However, as a consequence of ICI and fading,
$r_{m,k,n}$ might violate this condition, hence leading to the link outages.
However, the COP, defined as the probability that a system is unable to
support the target transmission rate $C_{m,k,n}$, must be limited by the
maximum tolerable COP $\epsilon_{cop}\in\left(0,1\right)$, as follows:

$\textrm{COP}:\qquad
P_{m,k,n}^{CO}\triangleq\mathbb{P}\left\\{r_{m,k,n}>C_{m,k,n}\right\\}\leq\epsilon_{cop}.$
(9)

Remark 2. On the other hand, since the $U_{m,k}$ has no knowledge concerning
the CSIs of passive $Eves$ [28], the values of $\mu_{j,m,n}$ are unknown. A
beneficial secrecy policy in this situation is to adjust the redundancy rate
of $U_{m,k}$ [28], denoted by $D_{j,k,n}$, so that the COP limit of (9) is
satisfied. In other words, $D_{j,k,n}$ must not exceed $C_{j,k,n}$. To do so,
the SOP of $\epsilon_{sop}\in\left(0,1\right)$, satisfies: 444It should be
highlighted that, we considered the worst-case condition for ensuring both
reliability and security of each part of split messages. This implies that if
actual transmission rate of each split messages $s_{m,k,n}\,\forall
n\in\left\\{1,2\right\\}$ corresponding to $U_{m,k}$ can satisfy its target
transmission rate $C_{m,k,n}\,\forall n\in\left\\{1,2\right\\}$, as stipulated
in the COP condition of (9), we can guarantee that the per user basis
condition is also met. At the receiver, UAV-BS recover and merges each of
these two split messages corresponding to each user separately to retrieve
their original messages. Thus, the conditions described above, should be
satisfied for each part of message separately. We can use the same
justification for the SOP constraint (10).

$\textrm{SOP}:\qquad P_{j,k,n}^{SO}\triangleq\mathbb{P}\left\\{D_{j,k,n}\leq
C_{j,k,n}\right\\}\leq\epsilon_{sop}.$ (10)

Remark 3. Upon considering non-colluding $Eves$, the achievable secrecy rates
of $U_{m,k}$ where transmitting $s_{k,m,n}$ is limited by the worst-case $Eve$
scenario of $C_{m,k,n}^{sec}\triangleq$
$\underset{1\text{$\leq$}j\text{$\leq$}J}{\min}\left\\{\left[r_{m,k,n}\text{\textminus}D_{j,k,n}\right]^{+}\right\\}$.
On the other hand, while minimizing $P_{m,k,n}^{CO}$ would improve the
reliability, maximizing $C_{m,k,n}^{sec}$ will enhance the security upon
jointly considering both the reliability and security requirements of all
$\left.U_{m,k}\right|_{k=1}^{\mathcal{K}_{m}}$, we should rather maximize the
ENST, defined as
$C_{ENST}\triangleq\sum_{k=1}^{K_{m}}$$\sum_{n=1}^{2}\left(1-P_{m,k,n}^{CO}\right)\,C_{m,k,n}^{sec}$.

Based on the discussion in Remarks 1-3, while considering the limited power
budget imposed on each $U_{m,k}$, formulated as $\sum_{n=1}^{2}p_{m,k,n}\leq
P_{m,k}$, the optimization problem of the proposed secure RSMA-based uplink is
formulated as:

$\underset{\mathcal{S}}{\textrm{max}}\left(\underset{1\text{$\leq$}j\text{$\leq$}J}{\min}\left\\{\stackrel{{\scriptstyle[}}{{k}}=1]{K_{m}}{\sum}\stackrel{{\scriptstyle[}}{{n}}=1]{2}{\sum}\left(1\text{\textminus}P_{m,k,n}^{CO}\right)\left[r_{m,k,n}\text{\textminus}D_{j,k,n}\right]^{+}\right\\}\right)\,\,$
(11)

s.t.

$C_{1}:$ $P_{m,k,n}^{CO}\leq{\epsilon_{cop}},\,\,\,\forall\,k$,
---

$C_{2}:$ $P_{j,k,n}^{SO}\leq{\epsilon_{sop}},\,\,\forall\,k,j$,
---

$C_{3}:$ $\stackrel{{\scriptstyle[}}{{n}}=1]{2}{\sum}p_{m,k,n}\leq
P_{m,k},\,\,p_{m,k,n}\geq 0,\,\,\forall\,k,$
---

where $\mathcal{S}\triangleq\left\\{r_{m,k,n}\geq 0,\,D_{j,k,n}\geq
0,p_{m,k,n}\geq 0,\,\Phi_{m}\right\\}$. Due to the non-convex OF, as well as
the discontinuous variable ${\Phi}_{m}$, the problem in (11) represents a non-
convex mixed integer programming problem. In the next section, we derive
closed-form expressions both for the COP and SOP constraints, while
${\Phi}_{m}$ is obtained through an exhaustive search.

##  IV ENST Maximization Solution

In this section, we construct the overall algorithm for finding the optimal
solution of (11).

###  IV-A Handling the Probabilistic Constraints (11)-$C_{1}$ and (11)-$C_{2}$

We first intend to handle the COP constraint (11)-$C_{1}$ . In this regard, we
first insert (7) into (9), so that (11)-$C_{1}$ may be reformulated as (12)

$P_{m,k,n}^{CO}\triangleq\mathbb{P}\left\\{r_{m,k,n}>\log_{2}\left(1+\rho_{m,k,n}\right)\right\\}$
(12)
---
$=1-\exp\left(-\frac{\beta_{m,n,k}\sigma_{m}^{2}}{2}\right)\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\prod}\left(1+\lambda_{m,k^{\prime},n^{\prime}}^{-1}\frac{\beta_{m,n,k}}{2}\right)^{-1}\stackrel{{\scriptstyle[}}{{i}}=1,i\neq
m]{M}{\prod}\stackrel{{\scriptstyle[}}{{k}}^{\prime\prime}=1]{K_{i}}{\prod}\left(1+\lambda_{i,k^{\prime\prime}}^{-1}\frac{\beta_{m,n,k}}{2}\right)^{-1},\forall
k\text{$\in$}\mathcal{K}_{m}$

, where we have $\beta_{m,n,k}=2^{r_{m,k,n}}-1$,
$\lambda_{m,k^{\prime},n^{\prime}}\triangleq$$\frac{1}{2p_{m,k^{\prime},n^{\prime}}\mathrm{PL}\left(d_{m,k^{\prime}}\right)}$,
and
$\lambda_{i,k^{\prime\prime}}=$$\frac{2^{\frac{B}{N_{t}-1}}}{P_{i,k^{\prime\prime}}\mathrm{PL}\left(d_{i,k^{\prime\prime}}\right)}$
(Proof: See Appendix A).

Upon inserting (8) into (10), we may reformulate the SOP constraint
(11)-$C_{2}$ as (13),

$\displaystyle P_{j,k,n}^{SO}$
$\displaystyle\triangleq\mathbb{P}\left\\{D_{j,k,n}\leq\log_{2}\left(1+\mu_{j,k,n}\right)\right\\}=\exp\left(-\eta_{j,k,n}\kappa_{j,k,n}\sigma_{e}^{2}\right)\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\prod}\left(1+\eta_{j,k,n}\kappa_{j,k,n}\zeta_{j,k^{\prime},n^{\prime}}^{-1}\right)^{-1},$
(13)

______________________________________________________________________________________

where we have
$\eta_{j,k,n}=\frac{1}{p_{m,k,n}\mathrm{PL}\left(d_{m,k,j}\right)\lambda_{j}^{max}\left\\{\tilde{\mathcal{\mathbf{Q}}}_{m,j}\right\\}}$,
$\zeta_{j,k^{\prime},n^{\prime}}=\frac{1}{p_{m,k^{\prime},n^{\prime}}\mathrm{PL}\left(d_{m,k^{\prime},j}\right)\lambda_{j}^{max}\left\\{\tilde{\mathcal{\mathbf{Q}}}_{m,j}\right\\}}$,
and $\kappa_{j,k,n}=2^{D_{j,k,n}}-1$ (Proof: See Appendix B). On the other
hand, since $D_{j,k,n}$ independent of both $r_{k,m,n}$ and $p_{k,m,n}$ within
the OF, the maximization problem (11) over $D_{j,k,n}$ is equivalent to
minimizing $D_{j,k,n}$. To find a more conservative solution, we exploit that
$D_{j,k,n}$ appears both in the OF and in the SOP constraint (11), we have to
exploit a tighter constraint than (13) for obtaining the minimum value of
$D_{j,k,n}$, which is given by (14),

$D_{j,k,n}\leq\log_{2}\left(1+\frac{2K_{m}-1}{\eta_{j,k,n}\sigma_{e}^{2}}W_{0}\left(\frac{\eta_{j,k,n}\sigma_{e}^{2}}{2K_{m}-1}\left(\frac{\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\prod}\zeta_{j,k^{\prime},n^{\prime}}}{\eta_{j,k,n}}{\epsilon_{sop}}^{-1}\right)^{\frac{1}{2K_{m}-1}}\right)\right),$
(14)
---

where $W_{0}\left(x\right)$ is the Lambert $W$-function (Proof: See Appendix
C). However, it is still challenging to solve (11), since $r_{k,m,n}$ and
$p_{k,m,n}$ are coupled in the OF of (11). To arrive at a more tractable form,
the operations of maximization and the minimization can be swapped in (11).
Additionally, since $\left\\{D_{j,k,n}\right\\}_{j=1}^{M}$ are independent, we
can actually solve $J$ independent maximization problems and then simply
choose the minimum one. Furthermore, by exploiting the inequalities of
$\exp\left(-x\right)\leq\frac{1}{1+x}$ and $\frac{1}{1+x}\leq\frac{1}{x}$, we
can instead replace the lower bound and upper bound of the OF and of the COP
constraint (11), respectively. Accordingly, based on what was mentioned above,
a bound of the solution may be obtained as (15),

$\underset{1\text{$\leq$}j\text{$\leq$}J}{\min}\left(\underset{\mathcal{S}}{\max}\left\\{\stackrel{{\scriptstyle[}}{{k}}=1]{K_{m}}{\sum}\stackrel{{\scriptstyle[}}{{n}}=1]{2}{\sum}\exp\left(\frac{\beta_{m,n,k}}{2}\left[\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\sum}\lambda_{m,k^{\prime},n^{\prime}}^{-1}+\xi-\sigma_{m}^{2}\right]\right)\left[r_{m,k,n}\text{\textminus}D_{j,k,n}\right]^{+}\right\\}\right)\,\,$
(15)

s.t.

$C_{1}:$
$\xi\exp\left(-\frac{\beta_{m,n,k}\sigma_{m}^{2}}{2}\right)\beta_{m,n,k}^{-A}\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\prod}2\lambda_{m,k^{\prime},n^{\prime}}^{-1}\leq{\epsilon_{cop}},\,\,\,\forall\,k,n$
---

$C_{2}:$
$D_{j,k,n}\leq\log_{2}\left(1+\frac{2K_{m}-1}{\eta_{j,k,n}\sigma_{e}^{2}}W_{0}\left(\frac{\eta_{j,k,n}\sigma_{e}^{2}}{2K_{m}-1}\left(\frac{\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\prod}\zeta_{j,k^{\prime},n^{\prime}}}{\eta_{j,k,n}}{\epsilon_{sop}}^{-1}\right)^{\frac{1}{2K_{m}-1}}\right)\right),\,\,\forall\,k,j,n$
---

$C_{3}:$ $\stackrel{{\scriptstyle[}}{{n}}=1]{2}{\sum}p_{m,k,n}\leq
P_{m,k},\,\,p_{m,k,n}\geq 0,\,\,\forall\,k,$
---

where we have $A\triangleq\left|\Phi_{m,k,n}\right|$$+M+K_{i}$,
$\xi\triangleq\stackrel{{\scriptstyle[}}{{i}}=1,i\neq
m]{M}{\prod}\stackrel{{\scriptstyle[}}{{k}}^{\prime\prime}=1]{K_{i}}{\prod}2\lambda_{i,k^{\prime\prime}}^{-1}$.

Now, we can exploit the block coordinate decent technique, where
$\left\\{p_{m,k,1},D_{j,k,1},\ p_{m,k,2},D_{j,k,2}\right\\}_{k=1}^{K_{M}}$ and
the $\left\\{r_{m,k,1}\text{,$r_{m,k,2}$}\right\\}_{k=1}^{K_{M}}$ are found
successively in an iterative manner. In particular, the $l^{th}$ iteration of
the algorithm is constituted by separately maximizing the criterion with
respect to each of
$\left\\{r_{m,k,1}\text{,$r_{m,k,2}$}\right\\}_{k=1}^{K_{M}}$ and
$\left\\{p_{m,k,1},D_{j,k,1},\ p_{m,k,2},D_{j,k,2}\right\\}_{k=1}^{K_{M}}$,
while keeping the other one fixed. Given this perspective, we first update
$\left\\{r_{m,k,1}\text{,$r_{m,k,2}$}\right\\}_{k=1}^{K_{M}}$, while assuming
that $\left\\{p_{m,k,1},D_{j,k,1},\
p_{m,k,2},D_{j,k,2}\right\\}_{k=1}^{K_{M}}$ are fixed values which results in
the following optimization problem:

$\underset{1\text{$\leq$}j\text{$\leq$}J}{\min}\left(\underset{\left\\{r_{m,k,1},r_{m,k,2}\right\\}}{\max}\left\\{\stackrel{{\scriptstyle[}}{{k}}=1]{K_{m}}{\sum}\stackrel{{\scriptstyle[}}{{n}}=1]{2}{\sum}\exp\left(\Xi_{m,n}^{r}\frac{\beta_{m,n,k}}{2}\right)r_{m,k,n}\right\\}\right),\,\,$
(16)

s.t.

$\xi\exp\left(-\frac{\beta_{m,n,k}\sigma_{m}^{2}}{2}\right)\beta_{m,n,k}^{-A}\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\prod}\left(2\lambda_{m,k^{\prime},n^{\prime}}^{*^{-1}}\right)\leq{\epsilon_{cop}},$
---

$\forall\,k,n,$ where
$\Xi_{m,n}^{r}\triangleq$$\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\sum}\lambda_{m,k^{\prime},n^{\prime}}^{*^{-1}}$$+\xi-\sigma_{m}^{2}$.

It is easy to check that in terms of
$\left\\{r_{m,k,1}\text{,$r_{m,k,2}$}\right\\}_{k=1}^{K_{M}}$, while the OF of
(16) is an increasing function, while constraint (16)-$C_{1}$ is a decreasing
one. Hence, the closed form expression of
$\left\\{r_{m,k,1}\text{,$r_{m,k,2}$}\right\\}_{k=1}^{K_{M}}$ may obtained
when the inequality constraint (16)-$C_{1}$ is active at the optimum. Hence,
the optimal point of (16), i.e.,
$\left\\{r_{m,k,1}^{*}\text{,$r_{m,k,2}$}^{*}\right\\}_{k=1}^{K_{M}}$, may be
found by solving the following equation:

$\exp\left(-\frac{\sigma_{m}^{2}\beta_{m,n,k}}{2}\right)\beta_{m,n,k}^{-A}=\frac{{\epsilon_{cop}}}{\xi}\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\prod}\left(\frac{\lambda_{m,k^{\prime},n^{\prime}}^{*}}{2}\right).$
(17)

By doing so, followed by some algebraic manipulations, together with the help
of the principal branch of the Lambert $W$-function, the optimum is formulated
as follows:

$r_{m,k,n}^{*}=\log_{2}\left(1+\frac{2A}{\sigma_{m}^{2}}W_{0}\left(\left[\xi{\epsilon_{cop}}^{-1}\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\prod}\left(2\lambda_{m,k^{\prime},n^{\prime}}^{*^{-1}}\right)\right]^{\frac{1}{A}}\right)\right).$
(18)

$\underset{1\text{$\leq$}j\text{$\leq$}J}{\min}\left(\underset{\left\\{p_{m,k,1},D_{j,k,1},\
p_{m,k,2},D_{j,k,2}\right\\}_{k=1}^{K_{M}}}{\max}\left\\{\mathcal{F}_{j}\right\\}\right),\,\,$
(19)

s.t.

$C_{1}:$
$\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\sum}\log_{2}\left(p_{m,k^{\prime},n^{\prime}}\right)\leq\psi_{k},$
---

$C_{3}:$ $r_{m,k,n}^{*}\geq D_{j,k,n},$
---

$C_{4}:$ $\stackrel{{\scriptstyle[}}{{n}}=1]{2}{\sum}p_{m,k,n}\leq
P_{m,k},\,p_{m,k,n}\geq 0,$
---

$C_{2}:$
$D_{j,k,n}\leq\log_{2}\left(1+\frac{2K_{m}-1}{\eta_{j,k,n}\sigma_{e}^{2}}W_{0}\left(\frac{\eta_{j,k,n}\sigma_{e}^{2}}{2K_{m}-1}\left(\frac{\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\prod}\zeta_{j,k^{\prime},n^{\prime}}}{\eta_{j,k,n}}{\epsilon_{sop}}^{-1}\right)^{\frac{1}{2K_{m}-1}}\right)\right),\,\,\forall\,k,j,n$
---

Now, assuming
$\left\\{r_{m,k,1}^{*}\text{,$r_{m,k,2}$}^{*}\right\\}_{k=1}^{K_{m}}$ to be
fixed values, we can update $\left\\{p_{m,k,1},D_{j,k,1},\
p_{m,k,2},D_{j,k,2}\right\\}_{k=1}^{K_{M}}$, which result in the following
equivalent transformation of (16) in the log-domain as (19), where
$\mathcal{F}_{j}$ $\triangleq$
$\stackrel{{\scriptstyle[}}{{k}}=1]{K_{m}}{\sum}\stackrel{{\scriptstyle[}}{{n}}=1]{2}{\sum}\beta_{m,n,k}^{*}$
$\left[\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\sum}\mathrm{PL}\left(d_{m,k^{\prime}}\right)p_{m,k^{\prime},n^{\prime}}\right]$$+\ln\left[r_{m,k,n}^{*}\text{\textminus}D_{j,k,n}\right]$,
$\gamma\triangleq 2^{-2\left|\Phi_{m,k,n}\right|}$ ${\epsilon_{cop}}\xi^{-1}$
$\exp\left(\frac{\beta_{m,n,k}^{*}\sigma_{m}^{2}}{2}\right)$
$\beta_{m,n,k}^{*^{\left(\left|\Phi_{m,k,n}\right|+M+K_{i}\right)}}$, and
$\psi_{k}\triangleq\log_{2}\left({\gamma}\right)-$$\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\sum}\log_{2}\left(\mathrm{PL}\left(d_{m,k^{\prime}}\right)\right)$.

Note that the newly added constraint (19)-$C_{3}$ arises from the fact that
the point-wise maximum operator $\left[\,\right]^{+}$ within the OF of (15)
leads to non-convexity. Thus, by adding (19)-$C_{3}$ we are equivalently
stating that the OF must be non-negative at the optimum and then we can simply
remove $\left[\,\right]^{+}$. Now, since the OF in (19) is constituted by the
sum of convex and affine functions with respect to
$\left\\{p_{m,k,1},D_{j,k,1},\ p_{m,k,2},D_{j,k,2}\right\\}_{k=1}^{K_{M}}$, it
is a convex function. However, problem (19) is still non-convex because of the
non-convex constraint (19)-$C_{2}$ . To circumvent the non-convexity, we
harness the SPCA of [5, 15], where the non-convex factor is approximated by
its first-order Taylor expansion at each iteration. Given this perspective, in
the following we attempt to circumvent the non-convexity imposed by the
fractional form and the logarithmic function within (19)-$C_{2}$ by
introducing some auxiliary variables, namely
$\left\\{\theta_{j,k,n},\varrho_{j,k,n},\nu_{j,k,n},\vartheta_{j,k,n}\right\\}$.
Following the classic variable transformation approach, the constraint
(19)-$C_{2}$ can be decomposed into $\forall\,k,j,n,$:

Algorithm 1 : Secure resource allocation proposed for the RSMA uplink

For ${\Phi}_{m}$ $\in$ $\mathbf{\Pi}_{m}$ do:

Call Function Outer_Loop

End.

Obtain the optimal
solution$\biggl{\\{}p_{m,k,1}^{*},D_{j,k,1}^{*},p_{m,k,2}^{*}$
$,D_{j,k,2}^{*}\biggr{\\}}_{k=1}^{K_{m}}$ ,
$\left\\{r_{m,k,1}^{*},r_{m,k,2}^{*}\right\\}_{k=1}^{K_{m}}$ and optimal
decoding order ${\Phi}_{m}^{*}$=${\Phi}_{m}$ with the highest OF.

=============================================

Function Outer_Loop

Step 1: Initialize the maximum number of iterations $Q_{max}$, $T_{max}$ and
the maximum tolerance $\epsilon$.

Step 2: Initialize
$\left\\{r_{m,k,1}^{*}{}^{\left[0\right]},r_{m,k,2}^{*}{}^{\left[0\right]}\right\\}$
and the outer iteration index $q=0$.

While
$\left(\left|r_{m,k,n}^{*}{}^{\left[q+1\right]}-r_{m,k,n}^{*}{}^{\left[q\right]}\right|\geq\epsilon\,\textrm{or}\,q\leq
Q_{max}\right)$ $\forall\,k,n$, do:

Step 3: Call the Function Inner_Loop with
$\left\\{r_{m,k,1}^{*}{}^{\left[q\right]},r_{m,k,2}^{*}{}^{\left[q\right]}\right\\}$
to obtain the $\epsilon$-constraint solution
$\left\\{p_{m,k,1}^{*},D_{j,k,1}^{*},p_{m,k,2}^{*},D_{j,k,2}^{*}\right\\}$.
Step 4: Update $r_{m,k,n}^{*}{}^{\left[q+1\right]}$ in (18). Step 5: Goto Step
3.
---

end while.

Step 6: Return the $\epsilon$-constraint solution
$\left\\{p_{m,k,1}^{*},D_{j,k,1}^{*},p_{m,k,2}^{*},D_{j,k,2}^{*}\right\\}$,
$r_{m,k,1}^{*}=r_{m,k,1}^{*}{}^{\left[q+1\right]}$ and
$r_{m,k,2}^{*}=r_{m,k,2}^{*}{}^{\left[q+1\right]}$ .

end.

=============================================

Function Inner_Loop
$\left(\left\\{r_{m,k,1}^{*}{}^{\left[q+1\right]},r_{m,k,2}^{*}{}^{\left[q+1\right]}\right\\}\right)$

Step 1: Initialize the inner iteration index $t=0$,
$\left\\{p_{m,k,1}^{\left[0\right]},D_{j,k,1}^{\left[0\right]},p_{m,k,2}^{\left[0\right]},D_{j,k,2}^{\left[0\right]}\right\\}$.

While
$\left(\left|\mathcal{F}_{j}^{\left[t+1\right]}\right.\text{\textminus}\left.\mathcal{F}_{j}^{\left[t\right]}\right|\geq\delta_{I}\,\textrm{or}\,t\leq
T_{max}\right)$ do:

Step 2: Find the $\epsilon$-constraint solution
$\left\\{p_{m,k,1}^{\left[t+1\right]},D_{j,k,1}^{\left[t+1\right]},p_{m,k,2}^{\left[t+1\right]},D_{j,k,2}^{\left[t+1\right]}\right\\}$
of the following problem for given
$\left\\{p_{m,k,1}^{\left[t\right]},D_{j,k,1}^{\left[t\right]},p_{m,k,2}^{\left[t\right]},D_{j,k,2}^{\left[t\right]}\right\\}$,
and $r_{m,k,n}^{*}{}^{\left[m\right]}$
$\left\\{p_{m,k,1}^{\left[t+1\right]},D_{j,k,1}^{\left[t+1\right]},p_{m,k,2}^{\left[t+1\right]},D_{j,k,2}^{\left[t+1\right]}\right\\}=\textrm{Solving}\,\eqref{eq:30},\,\,$
Step 3: Update $\mathcal{F}_{j}^{\left[t+1\right]}$. Step 4: Goto Step 2.
---

end while.

Step 5: Return
$\left\\{p_{m,k,1}^{*},D_{j,k,1}^{*},p_{m,k,2}^{*},D_{j,k,2}^{*}\right\\}=\left\\{p_{m,k,1}^{\left[t+1\right]},D_{j,k,1}^{\left[t+1\right]},p_{m,k,2}^{\left[t+1\right]},D_{j,k,2}^{\left[t+1\right]}\right\\}$
.

end

$D_{j,k,n}\leq\log_{2}\left(1+\theta_{j,k,n}\right),\,\,\forall\,k,j,n,$ (20)
$\theta_{j,k,n}\geq\frac{\mathrm{PL}\left(d_{m,k,j}\right)\left(2K_{m}-1\right)}{\sigma_{e}^{2}}p_{m,k,n}\varrho_{j,k,n},$
(21) $\varrho_{j,k,n}\geq W_{0}\left(\nu_{j,k,n}\right),$ (22)
$\nu_{j,k,n}p_{m,k,n}^{2K_{m}}\geq\frac{\sigma_{e}^{2}}{\mathrm{PL}\left(d_{m,k,j}\right)^{2K_{m}}\left(2K_{m}-1\right)}\vartheta_{j,k,n},$
(23)
$\vartheta_{j,k,n}\geq\left(\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\prod}\zeta_{j,k^{\prime},n^{\prime}}{\epsilon_{sop}}^{-1}\right)^{\frac{1}{2K_{m}-1}}.$
(24)

Now, since the non-convexity still persists within (20)-(24), we attempt to
approximate the non-convex factor at each iteration by its first-order Taylor
expansion at the $t^{st}$ SPCA iteration. Following this approach, the affine
approximation becomes straightforward for each of (20)-(24). Hence , we can
replace each non-convex constraint by its affine approximation, and thus the
equivalent convex form of (19)-$C_{2}$ at the $t^{st}$ SPCA iteration may be
formulated as:

$2^{D_{j,k,n}}\leq 1+\theta_{j,k,n},$ (25)
$\theta_{j,k,n}\geq\frac{\mathrm{PL}\left(d_{m,k,j}\right)\left(2K_{m}-1\right)}{\sigma_{e}^{2}}\varTheta^{[t]}\left(p_{m,k,n},\varrho_{j,k,n}\right),$
(26) $\varrho_{j,k,n}\geq
W_{0}\left(\nu_{j,k,n}^{[t]}\right)\left(\nu_{j,k,n}^{[t]}\left(1-W_{0}\left(\nu_{j,k,n}^{[t]}\right)\right)\right)^{-1}\left(\nu_{j,k,n}-\nu_{j,k,n}^{[t]}\right),$
(27)
$\Psi^{[t]}\left(\nu_{j,k,n},p_{m,k,n}^{2K_{m}}\right)\geq\frac{\sigma_{e}^{2}}{\mathrm{PL}\left(d_{m,k,j}\right)^{2K_{m}}\left(2K_{m}-1\right)}\vartheta_{j,k,n},$
(28)
$\log\left(\vartheta_{j,k,n}\right)\geq\frac{1}{\left(2K_{m}-1\right)}\left(\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\sum}\Lambda^{[t]}\left(\zeta_{j,k^{\prime},n^{\prime}}\right)-\log\left({\epsilon_{sop}}\right)\right),$
(29)

$\forall\,k,j,n,$ respectively,
$\varTheta^{[t]}\left(p_{m,k,n},\varrho_{j,k,n}\right)\triangleq$
$\frac{1}{4}(p_{m,k,n}+\varrho_{j,k,n})^{2}$
$+\frac{1}{4}(p_{m,k,n}^{[t]}-\varrho_{j,k,n}^{[t]})^{2}$
$-\frac{1}{2}(p_{m,k,n}^{[t]}-\varrho_{j,k,n}^{[t]})$
$\left(p_{m,k,n}-\varrho_{j,k,n}\right),$
$\varGamma^{[m]}\left(D_{j,k,n}\right)\triangleq 2^{D_{j,k,n}^{[t]}}$
$\left[1+\ln(2)\left(D_{j,k,n}-D_{j,k,n}^{[t]}\right)\right]$,
$\Lambda^{[t]}\left(\zeta_{j,k^{\prime},n^{\prime}}\right)$ $\triangleq$
$\log\left(\zeta_{j,k^{\prime},n^{\prime}}^{[t]}\right)$
$+\frac{\zeta_{j,k^{\prime},n^{\prime}}-\zeta_{j,k^{\prime},n^{\prime}}^{[t]}}{\zeta_{j,k^{\prime},n^{\prime}}^{[t]}}$,
and $\Psi^{[t]}\left(\nu_{j,k,n},p_{m,k,n}^{2K_{m}}\right)$ $\triangleq$
$\nu_{j,k,n}^{[t]}$ $\left(p_{m,k,n}^{[t]}\right)^{2K_{m}}+$
$\left(p_{m,k,n}^{[t]}\right)^{\left(2K_{m}-1\right)}$
$\left[\left(p_{m,k,n}^{[t]}\right),\right.$ $\left(2K_{m}-1\right)$
$\left.\nu_{j,k,n}^{[t]}\right]$ $\times$
$\left[\nu_{j,k,n}-\nu_{j,k,n}^{[t]},\right.$
$\left.p_{m,k,n}-p_{m,k,n}^{[t]}\right]^{T}$.

In order to arrive at (27) from (22), we have exploited the fact that since
$W_{0}\left(x\right)$ is concave over the interval of
$\left(-e^{-1},\infty\right)$ and positive over $\left(1,\infty\right)$, upon
using the first order Taylor expansion of $W_{0}\left(x\right)$ we have
$W_{0}\left(\nu_{j,k,n}\right)\leq W_{0}$ $\left(\nu_{j,k,n}^{[t]}\right)$
$\left[\nu_{j,k,n}^{[t]}\right.$
$\left.\left(1-W_{0}\left(\nu_{j,k,n}^{[t]}\right)\right)\right]^{-1}$
$\left(\nu_{j,k,n}-\nu_{j,k,n}^{[t]}\right)$.

###  IV-B Overall Solution of the Original Problem (11)

Note that due to the decoding order constraint $\left\\{\Phi_{m}\right\\}$, it
is challenging to find the optimal solution of problem (11). To solve this
problem, we first fix the decoding order $\Phi_{m}$ to obtain the optimal
triplet $\left\\{r_{k,m,n}^{\ast}\geq 0,\right.$ $D_{j,k,n}^{\ast}\geq 0,$
$\left.p_{m,k,n}^{\ast}\geq 0\right\\}$ and then exhaustively search the
entire set to find the optimal ${\Phi}_{m}^{\ast}$555Although it would also be
beneficial to look for an optimal SIC-ordering [5, 6], as the UAV is assumed
to only have access to the AoA and distances, but not to the small-scale
fading parameters, we have not performed the SIC-ordering here and left it for
future works. It has been shown in [5] that for an algorithm including $N$
initial points, the exhaustive SIC ordering for a $K$ users uplink-RSMA
imposes a tolerable computational complexity of
$\mathcal{O}\left(2^{K}+NK^{3}\left(2K!\right)/2^{K}\right)$.. Upon assuming a
fixed ${\Phi}_{m}$, we conceive a two-tier iterative algorithm for attaining
the overall $\epsilon$-constraint solution
$\left\\{p_{m,k,1}^{*},D_{j,k,1}^{*},p_{m,k,2}^{*},D_{j,k,2}^{*}\right\\}$ in
two different tiers. More explicitly, using the approximations obtained in
(25)-(29), together with $\left\\{r_{k,m,1}^{\ast},r_{k,m,2}^{\ast}\right\\}$
gleaned from the outer tier, the $\left(t+1\right)^{st}$ iteration of the
inner tier solves the following equivalent convex form of problem (19) for
finding the $\epsilon$-constraint solution as (30),

$\underset{1\text{$\leq$}j\text{$\leq$}J}{\min}\left(\underset{\mathbf{X}}{\max}\left\\{\mathcal{F}_{j}\right\\}\right),\,\,$
(30)

s.t.

$C_{1}:$
$\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\sum}\log_{2}\left(p_{m,k^{\prime},n^{\prime}}\right)\leq\psi_{k},\,\forall\,k,n$,
---

$C_{2-1}:$ $\varGamma^{[t]}\left(D_{j,k,n}\right)\geq
1+\theta_{j,k,n},\,\,\forall\,k,j,n$,
---

$C_{2-2}:$
$\theta_{j,k,n}\geq\frac{\mathrm{PL}\left(d_{m,k,j}\right)\left(2K_{m}-1\right)}{\sigma_{e}^{2}}\varTheta^{[t]}\left(p_{m,k,n},\varrho_{j,k,n}\right),\,\,\forall\,k,j,n$,
$C_{2-3}:$ $\varrho_{j,k,n}\geq
W_{0}\left(\nu_{j,k,n}^{[t]}\right)\left(\nu_{j,k,n}^{[t]}\left(1-W_{0}\left(\nu_{j,k,n}^{[t]}\right)\right)\right)^{-1}\left(\nu_{j,k,n}-\nu_{j,k,n}^{[t]}\right),\,\,\forall\,k,j,n$,
$C_{2-4}:$
$\Psi^{[t]}\left(\nu_{j,k,n},p_{m,k,n}^{2K_{m}}\right)\geq\frac{\sigma_{e}^{2}}{\mathrm{PL}\left(d_{m,k,j}\right)^{2K_{m}}\left(2K_{m}-1\right)}\vartheta_{j,k,n},\,\,\forall\,k,j,n$,
$C_{2-5}:$
$\log\left(\vartheta_{j,k,n}\right)\geq\frac{1}{\left(2K_{m}-1\right)}\left(\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\sum}\Lambda^{[t]}\left(\zeta_{j,k^{\prime},n^{\prime}}\right)-\log\left({\epsilon_{sop}}\right)\right),\,\,\forall\,k,j,n$,
---

$C_{3}:$ $r_{m,k,n}^{*}\geq D_{j,k,n},\,\,\forall\,k,n$, $C_{4}:$
$\stackrel{{\scriptstyle[}}{{n}}=1]{2}{\sum}p_{m,k,n}\leq
P_{m,k},\,\,\forall\,k$.
---

where $\mathbf{X}\triangleq\left(\boldsymbol{x},\left\\{p_{m,k,1},D_{j,k,1},\
p_{m,k,2},D_{j,k,2}\right\\}_{k=1}^{K_{M}}\right)$, and
$\boldsymbol{x}\triangleq\left\\{\theta_{j,k,n},\varrho_{j,k,n},\nu_{j,k,n},\vartheta_{j,k,n},\nu_{j,k,n}\right\\}$.
Upon the $\epsilon$-constraint point
$\left\\{p_{m,k,1}^{*},D_{j,k,1}^{*},p_{m,k,2}^{*},D_{j,k,2}^{*}\right\\}$
found by the inner loop, the $\left(q+1\right)^{st}$ iteration of the outer
loop finds $\epsilon$-constraint solution
$\left\\{r_{m,k,1}^{*}{}^{\left[q+1\right]},r_{m,k,2}^{*}{}^{\left[q+1\right]}\right\\}$,
given by (31),

$\displaystyle{\left\\{p_{m,k,1}^{*},D_{j,k,1}^{*},p_{m,k,2}^{*},D_{j,k,2}^{*}\right\\}}\,\textrm{obtained\,from}\,\eqref{eq:30}$
$\displaystyle\Downarrow\,\,\,\Uparrow$
$\displaystyle\textrm{update\,}\,r_{m,k,n}^{*}{}^{\left[q+1\right]}=\log_{2}\left(1+\frac{2A}{\sigma_{m}^{2}}W_{0}\left(\left[\xi{\epsilon_{cop}}^{-1}\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\prod}\left(2\lambda_{m,k^{\prime},n^{\prime}}^{*^{-1}}\right)\right]^{\frac{1}{A}}\right)\right),$
(31)

______________________________________________________________________________________________________

where the superscript “$*$” represents the final iteration of the inner loop.
Since the proposed method consists of two layers of iterations, the stopping
criterion of each layer depends on the relative change of the two consecutive
of values. Therefore, the outer loop proceeds to the next iteration and runs
until
$\left|r_{m,k,n}^{*}{}^{\left[q+1\right]}-r_{m,k,n}^{*}{}^{\left[q\right]}\right|\leq\epsilon$
is met or the maximum affordable number of iterations $Q_{max}$ is reached. To
find the $\epsilon$-constraint solution
$\left\\{p_{m,k,1}^{*},p_{m,k,2}^{*},D_{j,k,n}^{*}\right\\}$ , the problem
((30)) is solved using the classic SPCA in another iterative process of the
inner loop. In particular, the inner iterations are continued until the
stopping criterion of $\left|\mathcal{F}_{j}^{\left[t+1\right]}\right.$ −
$\left.\mathcal{F}_{j}^{\left[t\right]}\right|\leq\delta_{I}$666___________________________________________________________
Note that $\mathcal{F}_{j}^{\left[t\right]}$ represents the value of
$\mathcal{F}_{j}$ at iteration $t^{th}$ . is satisfied at the
$\left(t+1\right)^{st}$ iteration or the maximum affordable number of
iterations $T_{max}$ is reached. The proposed two-tier scheme is presented in
Algorithm I.

###  IV-C Complexity Analysis

In Algorithm 1, the major complexity lies in solving problem (30). According
to Algorithm 1, a globally near-optimal solution of problem (30) is obtained
via solving a series of convex problems with different initial points and
decoding order strategies. Considering that the dimension of the variables in
problem (30) is $\mathcal{L}_{m}=\text{5$\left(1+J\right)$}K_{m}$, the worst-
case complexity order of solving the convex problem in Step 2 of inner-loop by
using the standard interior point method is given by
$\mathcal{O}\left(\left(\frac{\mathcal{L}_{m}-1}{2}\right)^{3}\right)$ [32,
Pages 487, 569]. Since each cluster consists of $K_{m}$ users and each user
transmits a superposition of two messages (there are $2K_{m}$ messages for the
$m^{th}$ cluster), the decoding order set $\mathbf{\Pi}_{m}$ consists of
$\frac{\left(2K_{m}\right)!}{2^{K_{m}}}$ elements. Therefore, the total
complexity of solving problem (30) at each iteration is given by
$\mathcal{O}\left(\left(\frac{\mathcal{L}_{m}-1}{2}\right)^{3}\frac{\left(2K_{m}\right)!}{2^{K_{m}}}\right)$.
In practice, we consider small $K_{m}$ to reduce the SIC complexity, so that
the computational complexity of Algorithm 1 remains practical. To deal with a
large number of users, we can increase the number of clusters and the users
can be classified into different clusters, each having a small number of
users.

###  IV-D Convergence Analysis

In this section we establish a convergence analysis for the SPCA algorithm.
Since the original problem (15) is non-convex, it is not possible to prove
convergence to a global minimum but rather convergence to $\mathrm{KKT}$
points under some regularity conditions. We use the following simple and
technical lemmas will be used in the convergence proof. For simplicity we
define $\varOmega\triangleq\textrm{feasible\,set\,of }$ (15),
$\varOmega^{\left[t\right]}\triangleq\textrm{feasible\,set\,of }$
(30)777____________________________________________________________ In this
paper, we consider the standard form of generic optimization problem as follow
[36]:  $\begin{array}[]{cc}\mathit{\textrm{min}\>\>}f(\mathbf{x})\\\
\textrm{s.t.}\>\mathit{c_{j}(\mathbf{x)\leq 0}},&\forall\,j=1,2,...,m\\\
\mathbf{x}\in\mathbb{R}^{n}\end{array}$ where
$\mathit{f(\mathbf{x})\>\textrm{and}\>c_{j}(\mathbf{x}),\>\forall\,j=1,...,m}$
are all continuously differentiable objective and constraint functions over
$\mathit{\mathbb{R}^{n}}$, respectively. Also, we assume that the function
$\mathit{f(\mathbf{x})}$ and the last $\mathit{m-p}$ constraint functions
$\mathit{c_{p+1}(\mathbf{x}),...,c_{m}(\mathbf{x})}$ ($\mathit{p\leq m}$) are
convex over $\mathit{\mathit{\mathbb{R}^{n}}}$. Therefore, the “non-convex
part” of the problem is due to the nonconvexity of the first $\mathit{p}$
constraint functions $\mathit{c_{1}(\mathbf{x}),...,c_{p}(\mathbf{x})}$. The
case $\mathit{p=m}$ corresponds to the case when all the constraints are non-
convex. In addition, suppose that for every $j=1,...,p$, $c_{j}(\mathbf{x})$
has a continuous convex upper estimate function
$C_{j}:\mathbb{R}^{n}\times\mathbb{Y}\rightarrow\mathbb{R}$, specifically,
assume that there exists a set $\mathbb{Y}\subseteq\mathbb{R}^{r}$ ($r$ is a
positive integer), such that $c_{j}(\mathbf{x})\leq
C_{j}(\mathbf{x},\boldsymbol{\rho}),$ $\forall\,$$\mathbf{x}$
$\in\mathbb{R}^{n},\>\forall\,\boldsymbol{\rho}\in\mathbb{Y},$ where for a
fixed $\boldsymbol{\rho}$ the function $C_{j}(\mathbf{.},\boldsymbol{\rho})$
is convex and continuously differentiable. The basic idea of SPCA is that at
each iteration $i$, we replace each non-convex functions
$c_{j}(\mathbf{x}),\>\forall\,j=1,...,p$ by the upper convex approximation
function $C_{j}(\mathbf{x},\boldsymbol{\rho})$ for some appropriately chosen
parameter vector $\boldsymbol{\rho}$. Thus, at step $i\>(i\geq 1)$ we need to
solve the following equivalent convex problem:
$\qquad\>\quad\begin{array}[]{cc}\mathit{\textrm{min}\>\>}f(\mathbf{x})\\\
\textrm{s.t.}\>\mathit{C_{j}(\mathbf{x},\boldsymbol{\rho})\leq
0},&\forall\,j=1,2,...,p\\\ \>\mathit{c_{i}(\mathbf{x)\leq
0}},&\quad\quad\>\>\forall\,j=p+1,p+2,...,m\\\
\mathbf{x}\in\mathbb{R}^{n}\end{array}$ for $t^{th}$ iteration.

######  Lemma 1.

Let $\mathcal{D}$ : $\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a strictly convex
and differentiable function on a nonempty convex set
$S\subseteq\mathbb{R}^{n}$. Then $\mathcal{D}$ is strongly convex on the set
$S$.

###### Proof:

See [34]. ∎

######  Lemma 2.

Suppose $\left\\{{\mathbf{X}^{\left[t\right]}}\right\\}$ be the sequence
generated by the SPCA method. Then for every $t\geq 0$: i).
$\varOmega^{\left[t\right]}$ $\subseteq\varOmega,$ ii).
${\mathbf{x}^{\left[t\right]}}$ $\in$ $\varOmega^{\left[t\right]}$ $\cap$
$\varOmega^{\left[t+1\right]}$,  iii). ${\mathbf{X}^{\left[t\right]}}$ is a
feasible point of (15),  iv).
$\mathcal{F}_{j}^{\left[t+1\right]}\,\leq\,\mathcal{F}_{j}^{\left[t\right]}$.

###### Proof:

See [34]. ∎

######  Lemma 3.

The sequence $\left\\{\mathcal{F}_{j}^{\left[t\right]}\right\\}$ converges.

###### Proof:

See [34]. ∎

Recall that a feasible solution $\mathbf{X}^{\logof}$ of a optimization
problem is $\mathit{regular}$ if the set of gradients of the active
constraints at $\mathbf{X}^{\logof}$ is linearly independent [35]. If
${\mathbf{X}^{\left[t\right]}}$ converges to a regular point
$\mathbf{X}^{\logof}$, then $\mathbf{X}^{\logof}$ is a $\mathrm{KKT}$ point of
problem (15). By Lemma2 it follows that the strictly convex objective function
$\mathcal{F}_{j}$ is also strongly convex on the convex feasible set
$\varOmega^{\left[t+1\right]}$. In particular, there exists $\vartheta>0$ such
that for all $k\geq 0$ we have:

$\displaystyle\mathcal{F}_{j}^{\left[t\right]}-\mathcal{F}_{j}^{\left[t+1\right]}\geq\left(\mathbf{X}^{\left[t\right]}-\mathbf{X}^{\left[t+1\right]}\right)^{\text{T}}\nabla\mathcal{F}_{j}^{\left[t+1\right]}+\vartheta\left\|\mathbf{X}^{\left[t\right]}-\mathbf{X}^{\left[t+1\right]}\right\|^{2},$
(32)

since ${\mathbf{X}^{\left[t\right]}}$ is a feasible point of (30) (by Lemma
3), and ${\mathbf{X}^{\left[t+1\right]}}$ is its optimum, then from the
optimality conditions for $\left(t+1\right)^{th}$ iteration of (30) (see [36,
proposition 2.1.2]), we obtain
$\left(\mathbf{X}^{\left[t\right]}-\mathbf{X}^{\left[t+1\right]}\right)^{\text{T}}\nabla\mathcal{F}_{j}^{\left[t+1\right]}\geq
0$, which combined with (32) yields:

$\mathcal{F}_{j}^{\left[t\right]}-\mathcal{F}_{j}^{\left[t+1\right]}\geq\vartheta\left\|\mathbf{X}^{\left[t\right]}-\mathbf{X}^{\left[t+1\right]}\right\|^{2}$
(33)

By Lemma 3, the sequence $\left\\{\mathcal{F}_{j}^{\left[t\right]}\right\\}$
converges and thus the inequality (33) implies that
$\left\|\mathbf{X}^{\left[t\right]}-\mathbf{X}^{\left[t+1\right]}\right\|\rightarrow
0$. Let be
$\mathbf{X}^{\diamondsuit}\triangleq\left({\color[rgb]{1,0,0}\boldsymbol{x}}^{\diamondsuit},\left\\{p_{m,k,1}^{\diamondsuit},D_{j,k,1}^{\diamondsuit},p_{m,k,2}^{\diamondsuit},D_{j,k,2}^{\diamondsuit}\right\\}_{k=1}^{K_{M}}\right)$
an accumulation point of the sequence
$\left\\{\mathbf{X}^{\left[t\right]}\right\\}$, we will show that
$\mathbf{X}^{\diamondsuit}$ is a $\mathrm{KKT}$ point. Since
$\mathbf{X}^{\diamondsuit}$ is an accumulation point of
$\left\\{\mathbf{X}^{\left[t\right]}\right\\}$, there exists a subsequence
$\left\\{\mathbf{X}^{\left[t_{n}\right]}\right\\}$ such that
$\mathbf{X}^{\left[t_{n}\right]}\rightarrow\mathbf{X}^{\diamondsuit}$ when the
iteration number $n\rightarrow\infty$. Regarding the limit point
$\mathbf{X}^{\diamondsuit}$ , we can make the following statement.

######  Corollary 4.

The accumulation point $\mathbf{X}^{\diamondsuit}$ of the sequence
$\left\\{\mathbf{X}^{\left[t\right]}\right\\}$ generated by the proposed SPCA
method is a KKT point of the (30).

###### Proof:

We know from [35] that there exist Lagrangian multipliers $\lambda_{i}^{*}$
together with the accumulation point $\mathbf{X}^{\diamondsuit}$ that satisfy
the following $\mathrm{KKT}$’s necessary and sufficient condition for
optimality of convex problem [35, Sec 5.5], at (34)
$\left\\{\gamma_{i}\right\\}_{i=1}^{8}$ denote the lagrangian multipliers of
problem (30). If we choose $\gamma_{i}=\lambda_{i}$ for $i=1,....,8$ we
conclude that the point $\mathbf{X}^{\diamondsuit}$ also satisfy so, we proved
that if the sequence $\left\\{\mathbf{X}^{\left[t\right]}\right\\}$ generated
by the SPCA method converges to a regular point $\mathbf{X}^{\diamondsuit}$,
then $\mathbf{X}^{\diamondsuit}$ is a $\mathrm{KKT}$ point of the SPCA problem
(30). It has already been shown that the point $\mathbf{X}^{\diamondsuit}$ is
a $\mathrm{KKT}$ point (stationary point) of the SPCA problem (30). This
stationary point cannot be saddle point, since the objective function
$\mathcal{F}_{j}$ is strictly convex function and twice-continuously
differentiable in the variable $\mathbf{X}$. By a simple contradiction method,
we can also show that the point $\mathbf{X}$ cannot be a local maximum [35].

$\nabla\mathcal{F}_{j}+\gamma_{1}\nabla\left(\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\sum}\log_{2}\left(p_{m,k^{\prime},n^{\prime}}\right)-\psi_{k}\right)+\gamma_{2}\nabla\left(1+\theta_{j,k,n}-\varGamma^{[t]}\left(D_{j,k,n}\right)\right)+\gamma_{3}\nabla\Biggl{(}\frac{\mathrm{PL}\left(d_{m,k,j}\right)\left(2K_{m}-1\right)}{\sigma_{e}^{2}}$

$+\varTheta^{[t]}\left(p_{m,k,n},\varrho_{j,k,n}\right)-\theta_{j,k,n}\Biggr{)}+\gamma_{4}\nabla\left(W_{0}\left(\nu_{j,k,n}^{[t]}\right)\left(\nu_{j,k,n}^{[t]}\left(1-W_{0}\left(\nu_{j,k,n}^{[t]}\right)\right)\right)^{-1}\left(\nu_{j,k,n}-\nu_{j,k,n}^{[t]}\right)-\varrho_{j,k,n}\right)+$

$\gamma_{5}\nabla\left(\frac{\sigma_{e}^{2}}{\mathrm{PL}\left(d_{m,k,j}\right)^{2K_{m}}\left(2K_{m}-1\right)}\vartheta_{j,k,n}-\Psi^{[t]}\left(\nu_{j,k,n},p_{m,k,n}^{2K_{m}}\right)\right)+\gamma_{6}\nabla\left(D_{j,k,n}-r_{m,k,n}^{*}\right)+$

$\gamma_{7}\nabla\Biggl{(}\frac{1}{\left(2K_{m}-1\right)}\left(\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\sum}\Lambda^{[t]}\left(\zeta_{j,k^{\prime},n^{\prime}}\right)-\log\left({\epsilon_{sop}}\right)\right)-\log\left(\vartheta_{j,k,n}\right)\Biggr{)}+\gamma_{8}\nabla\left(\stackrel{{\scriptstyle[}}{{n}}=1]{2}{\sum}p_{m,k,n}-P_{m,k}\right)=0.$
(34)

______________________________________________________________________________________________________

∎

##  V Simulation Results

In this section, we evaluate the secure transmission performance of the
proposed algorithm through simulations. Each point in the figures is obtained
by averaging over $150$ simulation trials. Unless otherwise specified, the
simulation setup is as follows throughout this section. In the scenario
investigated a UAV hovers above the users to provide communication services.
Explicitly, the UAV has a coverage radius of $R_{UAV}=800\,\textrm{m}$ and
altitude of $H_{i}=140\,\textrm{m}$. The path-loss model in the UAV network
includes both LoS and non-LoS links associated with the path-loss exponents of
$\mathcal{L}_{m,k}=2\,$ and $\mathcal{N}_{m,k}=3.5$, respectively. There are
$M=3$ clusters, $100$ users associated with
$P_{i,k}=\frac{P}{\stackrel{{\scriptstyle[}}{{m}}=1]{M}{\sum}K_{m}}\,$, where
$P$ is the total power budget, and $J=3$ $Eve$s randomly distributed in the
whole system between $1\,\textrm{m}$ and $800\,\textrm{m}$. Once the large-
scale fading parameters are generated, they are assumed to be known and fixed
throughout the simulations. The small-scale fading vectors of all users and
$Eve$s are independently generated according to
$\mathcal{CN}(0,\,\mathbf{I}_{N_{t}})$. The noise power at each user and
eavesdropper is set to $\sigma_{m}^{2}=\sigma_{e}^{2}=0\,\textrm{dB}$.
Moreover, we set $T_{max}=Q_{max}=20$, $N_{t}=5$,
$\epsilon_{cop}=\epsilon_{sop}=0.1$, and the maximum threshold value used for
the termination of Algorithm 1 is set to $\delta_{I}=10^{-2}$. The maximum
tolerance of $\epsilon=10^{-3}$ is assumed for the termination criterion used
in Algorithm 1.

Figure 4: The Convergence of Algorithm 1, black-line refer to
$\mathcal{F}_{j}^{\left[t\right]}$ and red-line refer to
$r_{m,k,n}^{*}{}^{\left[q\right]}$ for different values of $P$.

Figure 5: ENST versus $J$ for different values of $\epsilon_{cop}$ and
with/without optimal $Eve$ SIC ordering.

Figure 6: ENST versus $\epsilon_{cop}$ for different values of $P$.

Firstly, in Fig. 6 we demonstrate the convergence of Algorithm $1$ is solving
(18) and (19) for different values of $P$. The convergence of the inner loop
of Algorithm 1 in terms of updating $\mathcal{F}_{j}^{\left[t\right]}$ is
shown by black-lines. It is observed that the inner loop converges within $6$
iterations for different values of $P$, which corroborates the convergence of
(30). When fixing the number of users, the power $P_{i,k}$ allocated to each
user increases upon increasing $P$. As a result, the OF value of
$\mathcal{F}_{j}^{\left[t\right]}$ in (30) increases as $P$ increases. On the
other hand, the convergence of the outer loop of Algorithm $1$ is
characterized by the red-lines. Observe that the algorithm used for solving
(18) converges after a maximum of $10$ iterations under different values of
$P$. Fig. 6 shows the ENST of the proposed scheme versus $J$ for different
values of $\epsilon_{cop}$. Naturally, upon increasing $J$, the performance
degrades due to having more $Eves$ in the system, but using a higher
$\epsilon_{cop}$ would increase ENST and compensate for the performance loss.
Additionally, as another important observation, without performing SIC at Eve
(ESIC), the Eve’s rate is increased, resulting in ENST enhancement.

To show the performance advantages of the proposed scheme by employing the
RSMA scheme, we compare it to both power-domain (PD) NOMA and TDMA. In PD-
NOMA, the UAV first decodes the messages of users having high channel gains
and then decodes the messages of users with low channel gains by subtracting
the interference imposed by the previously decoded high-gain user. In TDMA,
each user will be assigned a fraction of time to use the whole bandwidth. Let
$\alpha_{m,k}=\frac{1}{K_{m}}$ be the fraction of time allocated to $U_{m,k}$.
Then the data rate of $U_{m,k}$ becomes
$C_{m,k}^{TDMA}=\alpha_{m,k}\log_{2}\left(1+\frac{p_{m,k,n}\left|\mathbf{w}_{m}^{H}\mathbf{h}_{m,k}\right|^{2}}{\sigma_{m}^{2}}\right)$.
Observe from Fig. 6 that RSMA always achieves a better performance than PD-
NOMA and TDMA. Moreover, the ENST gain of the proposed scheme over TDMA
becomes more prominent as $P$ and $\epsilon_{cop}$ increases.

Figure 7: ENST versus $\epsilon_{sop}$ for different values of $P$ and
$\epsilon_{cop}$.

Figure 8: ENST versus $P$ for different values of $\epsilon_{cop}$ and SIC
method.

Figure 9: ENST versus $\epsilon_{cop}$ for different values of $N_{e}$.

The corresponding ENST versus $\epsilon$ plot is provided in Fig. 9, where the
proposed scheme achieves a significantly higher ENST upon increasing $P$,
$\epsilon_{sop}$ and $\epsilon_{cop}$. A heuristic explanation of this
phenomenon is that increasing both the connection and secrecy outage threshold
tends to relax the constraints of (30), and decrease the lower bound of
(19)-$C_{2}$, which in turn increases the ENST.

Finally, to show the importance of considering SIC ordering as well as
imperfect CSIT, we compare the proposed scheme for both optimal SIC (OSIC) and
Sub-optimal SIC (SSIC) ordering. Additionally, we also consider RSMA with
perfect CSIT. To make a fair comparison, we simulate all schemes under the
same security requirement. The ENST versus $P$ trends recorded for different
values of $\epsilon_{cop}$ are provided in Fig. 9, where the proposed scheme
always achieves significantly higher ENST than RSMA ignoring CSIT uncertainty.
Fig. 9 suggests that using bigger $\epsilon_{cop}$ would increase the ENST and
mitigate the performance loss of SSIC.

Finally, Fig. 9 illustrates the ENST versus $N_{e}$. This figure indicates
that increasing the number of the receive antennas at $Eve$, the system’s
secrecy performance is degraded due to $Eve$’s improved ability to eavesdrop
and infer from common message. Interestingly, our proposed scheme still shows
considerable robustness against a multiple antenna-aided $Eve$, hence we can
achieve non-zero ENST.

##  VI Conclusions

In this article, we proposed secure RSMA uplink transmission under imperfect
CSIT for a UAV-BS network, in which RSMA is employed by each legitimate users
for secure transmission under large-scale uplink access. To characterize the
performance of this system, an efficient block coordinate decent algorithm was
proposed for maximizing the effective network secrecy throughput under the
constraints of secrecy and the reliability outage probabilities and transmit
power budget constraints. To solve this problem, we derived the closed-form
optimal RS rate expression of each user. Then, the $\epsilon$-constraint
transmit power of each user was calculated by the classic SPCA technique under
a given decoding order and then the optimal decoding order was found by an
exhaustive search method. Numerical results demonstrated that the proposed
algorithm significantly improves the effective network secrecy throughput
compared to the PD-NOMA and TDMA benchmarks, as well as to the RSMA
transmission ignoring CSIT uncertainty.

##  Appendix A DERIVATION OF (12)

Based on (9), $P_{m,k,n}^{CO}$ can be expressed as $A1$,

$\mathbb{P}\left\\{2^{r_{m,k,n}}-1>\frac{p_{m,k,n}\mathrm{PL}\left(d_{m,k}\right)\left|\mathbf{w}_{m}^{H}\mathbf{f}_{m,k}\right|^{2}}{\sum_{\left(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}}p_{m,k^{\prime},n^{\prime}}\mathrm{PL}\left(d_{m,k^{\prime}}\right)\,\left|\mathbf{w}_{m}^{H}\mathbf{f}_{m,k^{\prime}}\right|^{2}+\stackrel{{\scriptstyle[}}{{i}}=1,i\neq m]{M}{\sum}\,\stackrel{{\scriptstyle[}}{{k}}^{\prime\prime}=1]{K_{i}}{\sum}P_{i,k^{\prime\prime}}\mathrm{PL}\left(d_{i,k^{\prime\prime}}\right)\left\|\sin\left(\phi_{i,k^{\prime\prime}}\right)\mathbf{f}_{i,k^{\prime\prime}}\right\|^{2}\left|\mathbf{w}_{m}^{H}\mathbf{e}_{i,k^{\prime\prime}}\right|^{2}+\sigma_{m}^{2}}\right\\}$ |
---|---

=$\mathbb{P}\left\\{2^{r_{m,k,n}}-1>\frac{p_{m,k,n}\mathrm{PL}\left(d_{m,k}\right)\mathcal{X}_{k}}{\sum_{\left(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}}p_{m,k^{\prime},n^{\prime}}\mathrm{PL}\left(d_{m,k^{\prime}}\right)\mathcal{X}_{k^{\prime}}+\stackrel{{\scriptstyle[}}{{i}}=1,i\neq m]{M}{\sum}\,\stackrel{{\scriptstyle[}}{{k}}^{\prime\prime}=1]{K_{i}}{\sum}P_{i,k^{\prime\prime}}\mathrm{PL}\left(d_{i,k^{\prime\prime}}\right)\mathcal{Y}_{i,k^{\prime\prime}}+\sigma_{m}^{2}}\right\\},$ | $\left(A1\right)$
---|---

where we have
$\mathcal{X}_{k}\triangleq\left|\mathbf{w}_{m}^{H}\mathbf{f}_{m,k}\right|^{2}\geq
0$,
$\mathcal{Y}_{i,k"}\triangleq\left\|\mathbf{f}_{i,k^{\prime\prime}}\right\|^{2}\sin^{2}\left(\phi_{i,k^{\prime\prime}}\right)\left|\mathbf{w}_{m}^{H}\mathbf{e}_{i,k^{\prime\prime}}\right|^{2}\geq
0$. Otherwise, $P_{m,k}^{CO}$ is always one. Furthermore, based on the
independence of the interference terms [7, eq 24], the variables
$\left\\{\mathcal{X}_{k}\right\\}$ and
$\left\\{\mathcal{Y}_{i,k^{\prime\prime}}\right\\}$ $\forall
i,k,k^{\prime\prime}$ are indeed independent.

To obtain a closed-form expression of $P_{m,k}^{CO}$ , we first provide the
probability density function (pdf) of $\mathcal{X}_{k}$ . Recall that
$\mathbf{n}_{m,k}\sim\mathcal{CN}(0,\,\mathbf{I}_{N_{t}})$ and
$\widetilde{\mathbf{f}}_{m,k}\triangleq\frac{\mathbf{f}_{m,k}}{\left\|\mathbf{f}_{m,k}\right\|}$,
$\mathcal{X}_{k}$ can be rewritten as
$\mathcal{X}_{k}=\left\|\mathbf{f}_{m,k}\right\|^{2}\left|\mathbf{w}_{m}^{H}\widetilde{\mathbf{f}}_{m,k}\right|^{2}$.
Since the normalized beamformer weights $\mathbf{w}_{m}$ are determined by
$\left\\{\mathbf{v}_{i}\right\\}_{i=1,\,i\neq m}^{M}$ according to
$\mathbf{w}_{m}^{H}\mathbf{v}_{l}=0,\,\forall l\neq m,$ and
$\left\\{\mathbf{v}_{i}\right\\}_{i=1,\,i\neq m}^{M}$ are independent of
$\mathbf{f}_{m,k}$, the vectors $\mathbf{f}_{m,k}$ and $\mathbf{w}_{m}$ are
also independent. As a result, $\mathbf{f}_{m,k}$ and $\mathbf{w}_{m}$ are
independent unit-norm vectors in the $N_{t}$-dimensional space. Based on [27,
Lemma 1], the square inner product between two independent unit-norm random
vectors is Beta distributed with shape parameters of $\left(1,N_{t}-1\right)$,
i.e., we have
$X_{1}\triangleq\left|\mathbf{w}_{m}^{H}\widetilde{\mathbf{f}}_{m,k}\right|^{2}$
$\sim\textrm{Beta}\left(1,N_{t}-1\right)$ and its pdf is
$f_{X_{1}}\left(x_{1}\right)=\frac{\left(1-x_{1}\right)^{N_{t}-2}}{\textrm{Be}\left(1,N_{t}-1\right)}$,
$x_{1}\in\left[0,1\right]$ [29, eq 8.380]. On the other hand, since we have
$\mathbf{f}_{m,k}\sim\mathcal{CN}(0,\,\mathbf{I}_{N_{t}})$,
$X_{2}\triangleq\left\|\mathbf{f}_{m,k}\right\|^{2}$ is distributed as a chi-
squared r.v. with $2N_{t}$ degrees of freedom as $\chi_{2N_{t}}^{2}$, and its
pdf is
$f_{X_{2}}\left(x_{2}\right)=\frac{x_{2}^{N_{t}-1}e^{-\frac{x_{2}}{2}}}{2^{N_{t}}\varGamma\left(N_{t}\right)}$,
$x_{2}\geq 0$, where $\varGamma\left(x\right)$ is the Gamma function [50, eq.
8.310]. Since $\mathcal{X}_{k}=X_{1}X_{2}$, and $X_{1}$ and $X_{2}$ are
independent, the pdf of $\mathcal{X}_{k}$ is given by:

$\displaystyle f_{\mathcal{X}_{k}}\left(x\right)$
$\displaystyle=\int_{x}\frac{1}{\left|x_{2}\right|}f_{X_{2}}\left(x_{2}\right)f_{X_{1}}\left(\frac{x}{x_{2}}\right)dx_{2}$
$\displaystyle=\frac{\int_{x}^{+\infty}\left(x_{2}-x\right)^{N_{t}-2}e^{-\frac{x_{2}}{2}}dx_{2}}{\textrm{Be}\left(1,N_{t}-1\right)2^{N_{t}}\varGamma\left(N_{t}\right)}$
$\displaystyle=\frac{1}{2}e^{-\frac{x}{2}},x\geq
0=\textrm{Exp}(\frac{1}{2}).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(A2)$

Thus, $\mathcal{X}_{k}\sim\textrm{Exp}(\frac{1}{2})$ is exponentially
distributed with rate $\lambda_{k}=\frac{1}{2}$. Now, we derive the pdf of
$\mathcal{Y}_{i,k^{\prime\prime}}$. The cumulative distribution function of
$\varOmega_{i,k"}\triangleq\sin^{2}\left(\phi_{i,k^{\prime\prime}}\right)$ is
given by [30]:

$F_{\varOmega_{i,k"}}\left(\varpi\right)=\begin{cases}\begin{array}[]{c}1\end{array},&\textrm{if}\,\,\,0\leq\varpi\leq
2^{-\frac{B}{N_{t}-1}},\,\left(A3\right)\\\
2^{B}\left(\varpi\right)^{N_{t}-1},&\textrm{if}\,\,\,\varpi\geq
2^{-\frac{B}{N_{t}-1}}.\end{cases}$

Hence, we have
$\left\|\mathbf{f}_{i,k"}\right\|^{2}\sin^{2}\left(\phi_{i,k"}\right)\sim\textrm{Gamma}\left(N_{t}-1,2^{-\frac{B}{N_{t}-1}}\right)$,
which is gamma distributed with a shape parameter of $N_{t}-1$ and scale
parameter of $2^{-\frac{B}{N_{t}-1}}$ [30, Lemma 1]. On the other hand,
$\mathbf{e}_{i,k"}$ is a unit vector that has the same distribution as
$\mathbf{f}_{i,k"}$. Moreover, the unit vector $\mathbf{w}_{m}$ is isotropic
within the $\left(N_{t}-1\right)$-dimensional hyperplane and independent of
$\mathbf{e}_{i,k"}$. Based on [25, Lemma 2], we have
$\left|\mathbf{w}_{m}^{H}\mathbf{e}_{i,k"}\right|^{2}\sim\textrm{Beta}\left(1,N_{t}\text{\textminus}2\right)$.
Therefore the product
$\mathcal{Y}_{i,k"}=\textrm{Beta}\left(1,N_{t}\text{\textminus}2\right)\times\textrm{Gamma}\left(N_{t}-1,2^{-\frac{B}{N_{t}-1}}\right)$
is exponentially distributed as [31, Lemma 1],
$\mathcal{Y}_{i,k"}=\left\|\mathbf{f}_{i,k"}\right\|^{2}\sin^{2}\left(\phi_{i,k"}\right)\left|\mathbf{w}_{m}^{H}\mathbf{e}_{i,k"}\right|^{2}\sim\textrm{Exp}\left(2^{\frac{B}{N_{t}-1}}\right)$.
If we define the new variables
$\mathcal{\bar{X}}_{m,k^{\prime},n^{\prime}}\sim\textrm{Exp}\left(\lambda_{m,k^{\prime},n^{\prime}}=\frac{1}{2p_{m,k^{\prime},n^{\prime}}\mathrm{PL}\left(d_{m,k^{\prime}}\right)}\right)$,
$\mathcal{\bar{\mathcal{Y}}}_{i,k^{\prime\prime}}\sim\textrm{Exp}\left(\lambda_{i,k"}=\frac{2^{\frac{B}{N_{t}-1}}}{P_{i,k"}\mathrm{PL}\left(d_{i,k^{\prime\prime}}\right)}\right)$
, then the $P_{m,k,n}^{CO}$ can be expressed as $\left(A4\right)$,

$\displaystyle P_{m,k,n}^{CO}$
$\displaystyle=1-\mathbb{P}\left\\{\beta_{m,n,k}\leq\frac{\mathcal{\bar{X}}_{m,k,n}}{\sum_{\left(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}}\mathcal{\bar{X}}_{m,k^{\prime},n^{\prime}}+\stackrel{{\scriptstyle[}}{{i}}=1,i\neq
m]{M}{\sum}\,\stackrel{{\scriptstyle[}}{{k}}"=1]{K_{i}}{\sum}\mathcal{\bar{\mathcal{Y}}}_{i,k"}+\sigma_{m}^{2}}\right\\}$
$\displaystyle=1-\mathbb{P}\left\\{\mathcal{\bar{X}}_{m,k,n}\geq\beta_{m,n,k}\left(\sum_{\left(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}}\mathcal{\bar{X}}_{m,k^{\prime},n^{\prime}}+\stackrel{{\scriptstyle[}}{{i}}=1,i\neq
m]{M}{\sum}\,\stackrel{{\scriptstyle[}}{{k}}^{\prime\prime}=1]{K_{i}}{\sum}\mathcal{\bar{\mathcal{Y}}}_{i,k^{\prime\prime}}+\sigma_{m}^{2}\right)\right\\}$
$\displaystyle\overset{(a)}{=}1-\mathbb{E}\left\\{\exp\left(-\frac{\beta_{m,n,k}\left(\sum_{\left(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}}\mathcal{\bar{X}}_{m,k^{\prime},n^{\prime}}+\stackrel{{\scriptstyle[}}{{i}}=1,i\neq
m]{M}{\sum}\,\stackrel{{\scriptstyle[}}{{k}}^{\prime\prime}=1]{K_{i}}{\sum}\mathcal{\bar{\mathcal{Y}}}_{i,k^{\prime\prime}}+\sigma_{m}^{2}\right)}{2}\right)\right\\}$
$\displaystyle\overset{(b)}{=}1-\mathbb{E}\left\\{\exp\left(-\frac{\beta_{m,n,k}\sigma_{m}^{2}}{2}\right)\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\prod}\mathcal{L}_{\mathcal{\bar{X}}_{m,k^{\prime},n^{\prime}}}\left(\frac{\beta_{m,n,k}}{2}\right)\stackrel{{\scriptstyle[}}{{i}}=1,i\neq
m]{M}{\prod}\stackrel{{\scriptstyle[}}{{k}}^{\prime\prime}=1]{K_{i}}{\prod}\mathcal{L}_{\mathcal{\bar{\mathcal{Y}}}_{i,k^{\prime\prime}}}\left(\frac{\beta_{m,n,k}}{2}\right)\right\\}$
$\displaystyle=1-\exp\left(-\frac{\beta_{m,n,k}\sigma_{m}^{2}}{2}\right)\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}]{}{\prod}\stackrel{{\scriptstyle[}}{{i}}=1,i\neq
m]{M}{\prod}\stackrel{{\scriptstyle[}}{{k}}"=1]{K_{i}}{\prod}\left(\frac{\lambda_{m,k^{\prime},n^{\prime}}}{\lambda_{m,k^{\prime},n^{\prime}}+\frac{\beta_{m,n,k}}{2}}\right)\left(\frac{\lambda_{i,k^{\prime\prime}}}{\lambda_{i,k^{\prime\prime}}+\frac{\beta_{m,n,k}}{2}}\right),\,\,\,\,\,\,\,\,\,\,\,\,\,\left(A4\right)$

______________________________________________________________________________________________________

where
$\mathcal{L}_{X}\left(s\right)\triangleq\mathbb{E}\left\\{e^{-sx}\right\\}$ is
the Laplace transform. Step $(a)$ is reached by conditioning on the aggregate
interference $\sum_{\left(k^{\prime},n^{\prime}\right)\in\Phi_{m,k,n}}$
$\mathcal{\bar{X}}_{m,k^{\prime},n^{\prime}}$ $+$
$\stackrel{{\scriptstyle[}}{{i}}=1,i\neq
m]{M}{\sum}\,\stackrel{{\scriptstyle[}}{{k}}^{\prime\prime}=1]{K_{i}}{\sum}\mathcal{\bar{\mathcal{Y}}}_{i,k^{\prime\prime}}$
and $(b)$ by the independence of the interference terms.

##  Appendix B DERIVATION OF (13)

Based on (10), $P_{j,k,n}^{SO}$ can be expressed as $\left(B1\right)$,

$\displaystyle P_{j,k,n}^{SO}$
$\displaystyle=\mathbb{P}\left\\{2^{D_{j,k,n}}-1<\frac{p_{m,k,n}\mathrm{PL}\left(d_{m,k,j}\right)\left|g_{m,k,j}\right|^{2}}{\sum_{\left(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)}p_{m,k^{\prime},n^{\prime}}\mathrm{PL}\left(d_{m,k^{\prime},j}\right)\left|g_{m,k^{\prime},j}\right|^{2}+\sigma_{e}^{2}}\right\\}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left(B1\right)$
$\displaystyle=\mathbb{P}\left\\{2^{D_{j,k,n}}-1<\frac{\mathcal{W}_{j,k,n}}{\sum_{\left(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)}\mathcal{U}_{j,k^{\prime},n^{\prime}}+\sigma_{e}^{2}}\right\\},\,\,$

______________________________________________________________________________________________________

where $\mathcal{W}_{j,k,n}\triangleq
p_{m,k,n}\mathrm{PL}\left(d_{m,k,j}\right)\lambda_{j}^{max}\left\\{\tilde{\mathcal{\mathbf{Q}}}_{m,j}\right\\}\left|g_{m,k,j}\right|^{2}$
and $\mathcal{U}_{j,k^{\prime},n^{\prime}}\triangleq
p_{m,k^{\prime},n^{\prime}}\mathrm{PL}\left(d_{m,k^{\prime},j}\right)\lambda_{j}^{max}\left\\{\tilde{\mathcal{\mathbf{Q}}}_{m,j}\right\\}\left|g_{m,k^{\prime},j}\right|^{2}$
are independent exponential r.v obeying the distribution of
$\mathcal{W}_{j,k,n}\sim\textrm{Exp}\left(\eta_{j,k,n}=\frac{1}{p_{m,k,n}\mathrm{PL}\left(d_{m,k,j}\right)\lambda_{j}^{max}\left\\{\tilde{\mathcal{\mathbf{Q}}}_{m,j}\right\\}}\right)$
and
$\mathcal{U}_{k^{\prime},n}\sim\textrm{Exp}\left(\zeta_{j,k^{\prime},n^{\prime}}=\frac{1}{p_{m,k^{\prime},n^{\prime}}\mathrm{PL}\left(d_{m,k^{\prime},j}\right)\lambda_{j}^{max}\left\\{\tilde{\mathcal{\mathbf{Q}}}_{m,j}\right\\}}\right)$.
Then the $P_{j,k,n}^{SO}$ can be expressed as $\left(B2\right)$,

$\displaystyle P_{j,k,n}^{SO}$
$\displaystyle=\mathbb{P}\left\\{\mathcal{W}_{j,k,n}>\kappa_{j,k,n}\left(\sum_{\left(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)}\mathcal{U}_{j,k^{\prime},n^{\prime}}+\sigma_{e}^{2}\right)\right\\}\overset{(a)}{=}\mathbb{P}\left\\{\exp\left(-\eta_{j,k,n}\kappa_{j,k,n}\left(\sum_{\left(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)}\mathcal{U}_{j,k^{\prime},n^{\prime}}+\sigma_{e}^{2}\right)\right)\right\\},\,\,\,\left(B2\right)$
$\displaystyle\overset{(b)}{=}\mathbb{E}\left\\{\exp\left(-\eta_{j,k,n}\kappa_{j,k,n}\sigma_{e}^{2}\right)\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\prod}\mathcal{L}_{\mathcal{U}_{k^{\prime},n}}\left(\eta_{j,k,n}\kappa_{j,k,n}\right)\right\\}=\exp\left(-\eta_{j,k,n}\kappa_{j,k,n}\sigma_{e}^{2}\right)\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\prod}\left(1+\eta_{j,k,n}\kappa_{j,k,n}\zeta_{j,k^{\prime},n^{\prime}}^{-1}\right)^{-1},$

______________________________________________________________________________________________________

where $\kappa_{j,k,n}\triangleq 2^{D_{j,k,n}}-1$.

##  Appendix C DERIVATION OF (14)

Upon using the inequality of $\frac{1}{1+x}\leq\frac{1}{x}$ we have
$\left(C1\right)$ and $\left(C2\right)$.

$\exp\left(-\eta_{j,k,n}\kappa_{j,k,n}\sigma_{e}^{2}\right)\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\prod}\left(1+\eta_{j,k,n}\kappa_{j,k,n}\zeta_{j,k^{\prime},n^{\prime}}^{-1}\right)^{-1}\overset{}{\leq}\Psi\left(\kappa_{j,k,n}\right)\leq{\epsilon_{sop}},$ | $\left(C1\right)$
---|---

$\Psi\left(\kappa_{j,k,n}\right)\triangleq\exp\left(-\eta_{j,k,n}\kappa_{j,k,n}\sigma_{e}^{2}\right)\kappa_{j,k,n}^{-\left(2K_{m}-1\right)}\frac{\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\prod}\zeta_{j,k^{\prime},n^{\prime}}}{\eta_{j,k,n}}.$ | $\left(C2\right)$
---|---

$\Psi^{{}^{\prime}}\left(\kappa_{j,k,n}\right)=-\kappa_{j,k,n}^{-\left(2K_{m}-1\right)}\left(\eta_{j,k,n}\sigma_{e}^{2}+\frac{\left(2K_{m}-1\right)}{\kappa_{j,k,n}}\right)<0,\,\,\kappa_{j,k,n}>0,$ | $\left(C3\right)$
---|---

$\exp\left(-\eta_{j,k,n}\kappa_{j,k,n}\sigma_{e}^{2}\right)\kappa_{j,k,n}^{-\left(2K_{m}-1\right)}\frac{\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\prod}\zeta_{j,k^{\prime},n^{\prime}}}{\eta_{j,k,n}}={\epsilon_{sop}}.$ | $\left(C4\right)$
---|---

$\frac{\eta_{j,k,n}\sigma_{e}^{2}}{\left(2K_{m}-1\right)}\kappa_{j,k,n}\exp\left(-\frac{\eta_{j,k,n}\sigma_{e}^{2}}{\left(2K_{m}-1\right)}\kappa_{j,k,n}\right)=\frac{\eta_{j,k,n}\sigma_{e}^{2}}{\left(2K_{m}-1\right)}\left(\frac{\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\prod}\zeta_{j,k^{\prime},n^{\prime}}}{\eta_{j,k,n}}{\epsilon_{sop}}^{-1}\right)^{\frac{1}{\left(2K_{m}-1\right)}}.$ | $\left(C5\right)$
---|---

$W_{0}\left(\frac{\eta_{j,k,n}\sigma_{e}^{2}}{\left(2K_{m}-1\right)}\left(\frac{\stackrel{{\scriptstyle[}}{{\left}}(k^{\prime},n^{\prime}\right)\neq\left(k,n\right)]{}{\prod}\zeta_{j,k^{\prime},n^{\prime}}}{\eta_{j,k,n}}{\epsilon_{sop}}^{-1}\right)^{\frac{1}{\left(2K_{m}-1\right)}}\right)=\frac{\eta_{j,k,n}\sigma_{e}^{2}}{\left(2K_{m}-1\right)}\kappa_{j,k,n}.$ | $\left(C6\right)$
---|---

The first-order derivative of $\Psi\left(\kappa_{j,k,n}\right)$ is given by
$\left(C3\right)$, which is negative. By exploiting the decreasing nature of
$\Psi\left(\kappa_{j,k,n}\right)$ and its lower-bounded nature in
$\left(C2\right)$, the minimum value of $D_{j,k,n}$, is obtained by solving
the $\left(C4\right)$. By straightforward algebra, ($C4$) can be further
reformulated as $\left(C5\right)$. Finally, with the help of the principal
branch of the Lambert $W$-function [19], ($C5$) is rewritten as
$\left(C6\right)$. Re-arranging the terms in ($C6$) leads to (14).

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