# New Paradigm for Secure Full-Duplex Transmission: Movable Antenna-Aided
Multi-User Systems

Jingze Ding, , Zijian Zhou, , and Bingli Jiao This work was supported by the
National Natural Science Foundation of China under Grant 62171006. The
calculations were supported by the High-Performance Computing Platform of
Peking University. (Corresponding author: Zijian Zhou.)Jingze Ding, Zijian
Zhou, and Bingli Jiao are with the School of Electronics, Peking University,
Beijing 100871, China (e-mail<EMAIL_ADDRESS>{zjzhou1008,
jiaobl}@pku.edu.cn).

###### Abstract

In this paper, we investigate physical layer security (PLS) for full-duplex
(FD) multi-user systems. To simultaneously protect uplink (UL) and downlink
(DL) transmissions and ensure efficient use of time-frequency resources, we
consider a base station (BS) that operates in FD mode and enables to emit the
artificial noise (AN). Conventional fixed-position antennas (FPAs) at the BS
struggle to fully exploit spatial degrees of freedom (DoFs). Therefore, we
propose a new paradigm for secure FD multi-user systems, where multiple
transmit and receive movable antennas (MAs) are deployed at the BS to serve UL
and DL users and effectively counter the cooperative interception by multiple
eavesdroppers (Eves). Specifically, the MA positions, the transmit, receive,
and AN beamformers at the BS, and the UL powers are jointly optimized to
maximize the sum of secrecy rates (SSR). To solve the challenging non-convex
optimization problem with highly coupled variables, we propose an alternating
optimization (AO) algorithm. This algorithm decomposes the original problem
into three sub-problems, which are iteratively solved by the proposed multi-
velocity particle swarm optimization (MVPSO) and successive convex
approximation (SCA). Simulation results demonstrate that the proposed scheme
for MA-aided secure FD multi-user systems can significantly enhance security
performance compared to conventional FPA systems.

###### Index Terms:

Movable antenna (MA), physical layer security (PLS), full-duplex (FD),
alternating optimization (AO), particle swarm optimization (PSO).

##  I Introduction

In the era of ubiquitous connectivity, people extensively rely on wireless
networks for transmitting important and private information. Consequently,
security and privacy have become critical concerns for next-generation
wireless communications [1]. Over the last few decades, conventional security
mechanisms primarily focus on cryptographic encryption/decryption methods [2].
However, as threats evolve and become increasingly sophisticated, the
encryption/decryption-based methods will cause heavy computation and key
management costs. Thus, relying merely on these solutions is insufficient.
This has led to a growing interest in physical layer security (PLS), which is
based on information theory fundamentals and emphasizes the secrecy capacity
of the propagation channel [3, 4].

In particular, a pivotal technique for enhancing PLS is multi-antenna
transmission, which leverages spatial degrees of freedom (DoFs) [5]. To
improve legitimate channels while degrading eavesdropping channels,
significant efforts have been devoted to secure beamforming techniques in
various scenarios [6, 7, 8]. By exploiting the spatial diversity offered by
multiple antennas, these studies comprehensively validated the efficacy of
beamforming in improving security performances.

However, conventional multi-antenna systems typically utilize fixed-position
antennas (FPAs), which restricts their abilities to further exploit the
channel variations, especially in cases with a limited number of antennas. To
overcome this limitation, movable antenna (MA) [9] offers a practical and
innovative solution. The MA enables flexible movement within two or three-
dimensional region through a driver, such as a step motor along a slide
track[10]. Due to continuous movement, the MA can better utilize spatial DoFs
than the antenna selection (AS) [11] with discretely arranged antennas.
Specifically, for secure communication systems, the MAs are expected to move
to positions where the channel conditions are advantageous for legitimate
users but adverse for eavesdroppers (Eves), leading to an improved secrecy
rate. So far, numerous studies have demonstrated the advantages of MA-aided
systems over FPA systems. The pioneering work [9] established the field-
response channel model for MA-aided communication systems. Under the far-field
condition, the authors provided a methodology for calculating the channel
responses at various MA positions and analyzed the variations in channel gains
under deterministic and statistical channels. Moreover, the authors in [12]
investigated the channel capacity of multiple-input multiple-output (MIMO)
systems with MAs. Dealing with the challenges in multi-user uplink (UL)
communications, the authors in [13, 14] respectively considered the
utilization of MAs at the base station (BS) and user ends. Regarding multi-
user downlink (DL) communications, the authors in [15] modeled the motions of
MAs as discrete movements and jointly optimized the transmit beamformer and
the MA positions at the BS to minimize the total transmit power. Furthermore,
the mobility of antenna heralds a new era for flexible beamforming.
Specifically, the authors in [16, 17] investigated the MA array-aided
beamforming. The antenna position and weight vectors are jointly optimized to
achieve the full array gain and the null steering over the desired and
undesired directions, respectively. Moreover, the MA-aided secure
communication system has become a hot topic. Reference [18] showed the
performance improvements in multiple-input single-output (MISO) systems
achieved by the one-dimensional MA array over conventional FPAs. The authors
in [19] extended the movements of MAs to two dimensions and studied the secure
DL communication in a MISO system with a single-FPA user and a single-FPA Eve.
Furthermore, a more general MA-aided secure MIMO DL communication system was
investigated in [20]. Besides, without perfect channel state information (CSI)
of the Eves, the authors in [21, 22] jointly optimized the transmit beamformer
and MA positions to bolster systems’ securities. It is worth noting that
accurate CSI is paramount to ensuring the improvements of MA-aided
communication systems. Fortunately, recent works [23, 24] have proposed some
practical methods with low pilot overhead and computational complexity to
achieve satisfactory CSI estimation for MA-aided systems.

In addition, artificial noise (AN) is another effective technique to improve
the secrecy rate by interfering with Eves’ receptions [25]. Nevertheless, in
half-duplex (HD) mode, the AN transmitted by the BS solely shields the DL
users, leaving the UL users entirely vulnerable to the Eves, particularly when
each UL user is equipped with only a single FPA. It is noteworthy that the
full-duplex (FD) mode, wherein transmit and receive signals are superimposed
onto the same time-frequency resource block, naturally addresses this issue
while concurrently yielding gains in spectral efficiency. Consequently, the
resource allocations for secure FD multi-user systems employing AN have
garnered increasing attention. Reference [26] proposed the robust beamforming
and jamming methods using AN in a worst case, where an FD BS simultaneously
serves multiple UL and DL users with single FPA in the presence of a multi-FPA
Eve. The authors in [27] expanded the scenario to include multiple Eves with
single FPA. Additionally, the authors in [28] considered a more complex
scenario involving multiple UL users, DL users, and multi-FPA Eves and jointly
minimized the total transmit powers for achieving secure UL and DL
transmissions concurrently.

As previously mentioned, current researches on MA-aided communication systems
are confined to HD mode. The flexible movement of MA provides remarkable
advantages over conventional FPA in signal power enhancement, interference
management, and flexible beamforming. These advantages render MA exceptionally
well-suited for enhancing the signal of interest while suppressing different
interference, i.e., self-interference (SI), multi-user interference, and AN
interference, in secure FD multi-user systems [29]. On the other hand,
existing secure FD systems have solely considered FPAs, which limit the
exploitation of spatial DoFs, and their results are not directly applicable to
MA-aided systems with reconfigurable channels by optimizing antenna positions
[30]. As such, in this paper, we propose a new paradigm for secure FD multi-
user systems. The main contributions of this paper are summarized as follows.

  * 1)

We propose deploying the separate transmit and receive MAs at the FD BS to
simultaneously serve multiple UL and DL users while actively resisting
cooperative eavesdropping by multiple Eves. An optimization problem is
formulated to maximize the sum of secrecy rates (SSR) by jointly optimizing
the MA positions, the transmit, receive, and AN beamformers at the BS, and the
UL powers, subject to the constraints of maximum transmit powers of each user
and the BS, finite moving regions for MAs, and minimum inter-MA distance.

  * 2)

To solve the formulated non-convex optimization problem with highly coupled
variables, we propose an alternating optimization (AO) algorithm to decompose
the original problem into three sub-problems and iteratively solve them. In
particular, we first modify the particle swarm optimization (PSO) and propose
the multi-velocity PSO (MVPSO) for optimizing MA positions. Then, we
reformulate the SSR into a more tractable form and utilize the successive
convex optimization (SCA) to optimize the transmit and AN beamformers at the
BS and the UL powers. Finally, the closed-form expression for the optimal
receive beamformer is derived.

  * 3)

We conduct extensive simulations to evaluate the performance of the proposed
scheme for the MA-aided secure FD multi-user system. It is shown that compared
to the conventional HD BS with only FPAs, the proposed novel FD BS equipped
with MAs can excellently utilize spatial DoFs to resist the Eves and assist
the interference suppression. Meanwhile, the reuse of time-frequency resources
in FD mode and the AN released by the BS further improve the SSR. Moreover,
simulation results confirm that the proposed MVPSO algorithm effectively
avoids undesired local optimal solutions and the derived optimal receive
beamformer significantly improves the receptions of UL signals. Finally, we
evaluate the impact of the imperfect field-response information (FRI) on MA
positioning.

The rest of this paper is organized as follows. Section II introduces the
system model and the optimization problem for the proposed system. In Section
III, we propose the AO algorithm to solve the optimization problem. Next,
simulation results and discussions are provided in Section IV. Finally, this
paper is concluded in Section V.

Notation: $a/A$, $\mathbf{a}$, $\mathbf{A}$, and $\mathcal{A}$ denote a
scalar, a vector, a matrix, and a set, respectively.
$\left[\mathbf{a}\right]_{i}$ denotes the $i$-th element of the vector
$\mathbf{a}$. $\mathbf{A}\succeq 0$ indicates that $\mathbf{A}$ is a positive
semidefinite matrix. ${\left(\cdot\right)^{T}}$, ${\left(\cdot\right)^{H}}$,
$\left\|\cdot\right\|_{2}$, $\mathrm{Tr}\left\\{\cdot\right\\}$, and
$\mathrm{Rank}\left\\{\cdot\right\\}$ denote the transpose, conjugate
transpose, Euclidean norm, trace, and rank, respectively. $\odot$ represents
Hadamard product. $\left[x\right]^{+}$ stands for $\max\left\\{0,x\right\\}$.
$\mathbb{C}^{M\times N}$ and $\mathbb{R}^{M\times N}$ are the sets for complex
and real matrices of $M\times N$ dimensions, respectively. $\mathbf{I}_{N}$ is
the identity matrix of order $N$.
$\mathcal{CN}\left(\mathbf{0},\mathbf{\Lambda}\right)$ represents the
circularly symmetric complex Gaussian (CSCG) distribution with mean zero and
covariance matrix $\mathbf{\Lambda}$. $\mathcal{A}\backslash\mathcal{B}$
denotes the subtraction of set $\mathcal{B}$ from set $\mathcal{A}$. $\sim$
and $\triangleq$ stand for “distributed as” and “defined as”, respectively.

##  II System Model

Figure 1: Illustration of the proposed MA-aided secure FD multi-user system.
Figure 2: Illustration of the coordinates and spatial angles for transmit and
receive regions.

As shown in Fig. 1, we consider a secure FD multi-user system with an MA-aided
FD BS to serve $K_{\mathrm{U}}$ HD UL users and $K_{\mathrm{D}}$ HD DL users,
in the presence of $K_{\mathrm{E}}$ HD Eves. Each user or Eve is equipped with
a single FPA. The BS is equipped with $N_{\mathrm{t}}$ transmit MAs and
$N_{\mathrm{r}}$ receive MAs, which can move in the two-dimensional regions
for actively reconfiguring the channel conditions. The positions of the
${n_{\mathrm{t}}}$-th transmit MA and the ${n_{\mathrm{r}}}$-th receive MA are
described by their Cartesian coordinates, i.e.,
${\mathbf{t}_{{n_{\mathrm{t}}}}}={\left[{{x^{\mathrm{t}}_{n_{\mathrm{t}}}},{y^{\mathrm{t}}_{n_{\mathrm{t}}}}}\right]^{T}}\in{\mathcal{C}_{\mathrm{t}}}$
($1\leq{n_{\mathrm{t}}}\leq{N_{\mathrm{t}}}$) and
${\mathbf{r}_{{n_{\mathrm{r}}}}}={\left[{{x^{\mathrm{r}}_{n_{\mathrm{r}}}},{y^{\mathrm{r}}_{n_{\mathrm{r}}}}}\right]^{T}}\in{\mathcal{C}_{\mathrm{r}}}$
($1\leq{n_{\mathrm{r}}}\leq{N_{\mathrm{r}}}$), where
${\mathcal{C}_{\mathrm{t}}}$ and ${\mathcal{C}_{\mathrm{r}}}$ represent the
transmit and receive regions, respectively.

Define the collections of $N_{\mathrm{t}}$ transmit MAs and $N_{\mathrm{r}}$
receive MAs as
$\tilde{\mathbf{t}}={\left[{\mathbf{t}_{1}^{T},\cdots,\mathbf{t}_{{N_{\mathrm{t}}}}^{T}}\right]^{T}}\in{\mathbb{R}^{2{N_{\mathrm{t}}}\times
1}}$ and
$\tilde{\mathbf{r}}={\left[{\mathbf{r}_{1}^{T},\cdots,\mathbf{r}_{{N_{\mathrm{r}}}}^{T}}\right]^{T}}\in{\mathbb{R}^{2{N_{\mathrm{r}}}\times
1}}$, respectively. For MA-aided communication systems, the channel response
can be written as the function of MA positions [9]. Thus, the SI channel of
the BS, the channel from the BS to DL user $k_{\mathrm{D}}$
($k_{\mathrm{D}}\in\left\\{{1,\ldots,K_{\mathrm{D}}}\right\\}\triangleq\mathcal{K}_{\mathrm{D}}$),
the channel from the BS to Eve $k_{\mathrm{E}}$
($k_{\mathrm{E}}\in\left\\{{1,\ldots,K_{\mathrm{E}}}\right\\}\triangleq\mathcal{K}_{\mathrm{E}}$),
and the channel from UL user $k_{\mathrm{U}}$
($k_{\mathrm{U}}\in\left\\{{1,\ldots,K_{\mathrm{U}}}\right\\}\triangleq\mathcal{K}_{\mathrm{U}}$)
to the BS are denoted as
${\mathbf{H}_{\mathrm{SI}}}\left({\tilde{\mathbf{t}}},{\tilde{\mathbf{r}}}\right)\in{\mathbb{C}^{{N_{\mathrm{r}}}\times{N_{\mathrm{t}}}}}$,
$\mathbf{h}_{\mathrm{BD},{k_{\mathrm{D}}}}\left(\tilde{\mathbf{t}}\right)\in{\mathbb{C}^{{N_{\mathrm{t}}}\times
1}}$,
$\mathbf{h}_{\mathrm{BE},{k_{\mathrm{E}}}}\left(\tilde{\mathbf{t}}\right)\in{\mathbb{C}^{{N_{\mathrm{t}}}\times
1}}$, and
${\mathbf{h}_{\mathrm{UB},{k_{\mathrm{U}}}}\left(\tilde{\mathbf{r}}\right)\in{\mathbb{C}^{{N_{\mathrm{r}}}\times
1}}}$, respectively. Besides, we denote the channel from UL user
${k_{\mathrm{U}}}$ to DL user ${k_{\mathrm{D}}}$ as
${h_{\mathrm{UD},{k_{\mathrm{U}}},{k_{\mathrm{D}}}}}\in{\mathbb{C}^{1\times
1}}$ and to Eve ${k_{\mathrm{E}}}$ as
${h_{\mathrm{UE},{k_{\mathrm{U}}},{k_{\mathrm{E}}}}}\in{\mathbb{C}^{1\times
1}}$.

In a given time slot, the BS transmits ${K_{\mathrm{D}}}$ independent
information streams to the ${K_{\mathrm{D}}}$ DL users. For DL user
${k_{\mathrm{D}}}$, the information stream can be expressed as
${\mathbf{x}_{k_{\mathrm{D}}}}={\mathbf{w}_{k_{\mathrm{D}}}}{s_{k_{\mathrm{D}}}^{\mathrm{D}}}$,
where ${s_{k_{\mathrm{D}}}^{\mathrm{D}}}$ denote the DL information with
normalized power and
${\mathbf{w}_{k_{\mathrm{D}}}}\in{\mathbb{C}^{{N_{\mathrm{t}}}\times 1}}$ is
the corresponding beamformer. To ensure security, the BS also generates an AN
vector ${\mathbf{v}}\in{\mathbb{C}^{{N_{\mathrm{t}}}\times 1}}$ and transmits
it with DL information to interfere with Eves’ malicious interception.
Therefore, the DL signals can be expressed as

$\mathbf{x}=\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}{{\mathbf{x}_{k_{\mathrm{D}}}}}+\mathbf{v}\in{\mathbb{C}^{{N_{\mathrm{t}}}\times
1}}.$ (1)

$\displaystyle{\mathbf{y}^{\mathrm{U}}}=\underbrace{\sum\limits_{{k_{\mathrm{U}}}\in\mathcal{K}_{\mathrm{U}}}{{\mathbf{h}_{\mathrm{UB},{k_{\mathrm{U}}}}}\left(\tilde{\mathbf{r}}\right)\sqrt{{p_{k_{\mathrm{U}}}}}{s_{k_{\mathrm{U}}}^{\mathrm{U}}}}}_{\mathrm{UL\
information}}+\underbrace{{\mathbf{H}_{\mathrm{SI}}}\left(\tilde{\mathbf{t}},\tilde{\mathbf{r}}\right)\sum\limits_{{k_{\mathrm{D}}}\in\mathcal{K}_{\mathrm{D}}}{{\mathbf{x}_{k_{\mathrm{D}}}}}+{\mathbf{H}_{\mathrm{SI}}}\left(\tilde{\mathbf{t}},\tilde{\mathbf{r}}\right)\mathbf{v}}_{\mathrm{Self-
interference}}+{\mathbf{n}^{\mathrm{U}}}.$ (2)
$\displaystyle{y_{k_{\mathrm{D}}}^{\mathrm{D}}}=\underbrace{\mathbf{h}_{\mathrm{BD},{k_{\mathrm{D}}}}^{H}\left(\tilde{\mathbf{t}}\right){\mathbf{x}_{k_{\mathrm{D}}}}}_{\mathrm{DL\
desired\
information}}+\underbrace{\sum\limits_{i\in{\mathcal{K}_{\mathrm{D}}\backslash\left\\{{k_{\mathrm{D}}}\right\\}}}{\mathbf{h}_{\mathrm{BD},{k_{\mathrm{D}}}}^{H}\left(\tilde{\mathbf{t}}\right){\mathbf{x}_{i}}}}_{\mathrm{DL\
multiuser\
interference}}+\underbrace{\mathbf{h}_{\mathrm{BD},{k_{\mathrm{D}}}}^{H}\left(\tilde{\mathbf{t}}\right)\mathbf{v}}_{\mathrm{Artificial\
noise}}+\underbrace{\sum\limits_{{k_{\mathrm{U}}}\in{\mathcal{K}_{\mathrm{U}}}}{h_{\mathrm{UD},{k_{\mathrm{U}}},{k_{\mathrm{D}}}}\sqrt{{p_{k_{\mathrm{U}}}}}}{s_{k_{\mathrm{U}}}^{\mathrm{U}}}}_{\mathrm{Co-
channel\ interference}}+{n_{k_{\mathrm{D}}}^{\mathrm{D}}}.$ (3)
$\displaystyle{y_{k_{\mathrm{E}}}^{\mathrm{E}}}=\underbrace{\sum\limits_{{k_{\mathrm{D}}}\in\mathcal{K}_{\mathrm{D}}}{\mathbf{h}_{\mathrm{BE},{k_{\mathrm{E}}}}^{H}\left(\tilde{\mathbf{t}}\right){\mathbf{x}_{k_{\mathrm{D}}}}}}_{\mathrm{DL\
information}}+\underbrace{\mathbf{h}_{\mathrm{BE},{k_{\mathrm{E}}}}^{H}\left(\tilde{\mathbf{t}}\right)\mathbf{v}}_{\mathrm{Artificial\
noise}}+\underbrace{\sum\limits_{{k_{\mathrm{U}}}\in\mathcal{K}\mathrm{U}}{{h_{\mathrm{UE},{k_{\mathrm{U}}},{k_{\mathrm{E}}}}}\sqrt{{p_{k_{\mathrm{U}}}}}{s_{k_{\mathrm{U}}}^{\mathrm{U}}}}}_{\mathrm{UL\
information}}+{n_{k_{\mathrm{E}}}^{\mathrm{E}}}.$ (4)

Then, the receive signals at the BS, DL user ${k_{\mathrm{D}}}$, and Eve
${k_{\mathrm{E}}}$ are respectively given by (2), (3), and (4) at the bottom
of the next page, where ${s_{k_{\mathrm{U}}}^{\mathrm{U}}}$ and
${p_{k_{\mathrm{U}}}}$ are the UL information with normalized power and the
transmit power of UL user ${k_{\mathrm{U}}}$, respectively.
${\mathbf{n}^{\mathrm{U}}}\sim\mathcal{CN}\left({\mathbf{0},\sigma_{\mathrm{U}}^{2}{\mathbf{I}_{{N_{\mathrm{r}}}}}}\right)$,
${n_{k_{\mathrm{D}}}^{\mathrm{D}}}\sim\mathcal{CN}\left({0,\sigma_{\mathrm{D},{k_{\mathrm{D}}}}^{2}}\right)$,
and
${n_{k_{\mathrm{E}}}^{\mathrm{E}}}\sim\mathcal{CN}\left({0,\sigma_{\mathrm{E},{k_{\mathrm{E}}}}^{2}}\right)$
represent the additive white Gaussian noise (AWGN) at the BS, DL user
${k_{\mathrm{D}}}$, and Eve ${k_{\mathrm{E}}}$, respectively.

###  II-A Channel Model

We consider quasi-static block-fading channels and concentrate on one specific
fading block with the multi-path channel components at any location in the
regions given as fixed [19]. As shown in Fig. 2, define the elevation and
azimuth AoDs and AoAs as
${\theta^{\mathrm{t}},\phi^{\mathrm{t}}}\in\left[{0,\pi}\right]$ and
${\theta^{\mathrm{r}},\phi^{\mathrm{r}}}\in\left[{0,\pi}\right]$,
respectively. Based on the field-response channel model [9], we establish the
channel responses involving the MA, i.e., SI channel
${\mathbf{H}_{\mathrm{SI}}}\left({\tilde{\mathbf{t}}},{\tilde{\mathbf{r}}}\right)$,
UL channel
${\mathbf{h}_{\mathrm{UB},k_{\mathrm{U}}}}\left({\tilde{\mathbf{r}}}\right)$,
and DL channels
${\mathbf{h}_{\mathrm{BD},k_{\mathrm{D}}}}\left({\tilde{\mathbf{t}}}\right)$
and
${\mathbf{h}_{\mathrm{BE},k_{\mathrm{E}}}}\left({\tilde{\mathbf{t}}}\right)$,
as follows.

####  II-A1 SI channel

Let ${{L}_{\mathrm{SI}}^{\mathrm{t}}}$ and ${{L}_{\mathrm{SI}}^{\mathrm{r}}}$
denote the numbers of transmit and receive paths, respectively. The difference
of the signal propagation distance for the $j$-th ($1\leq
j\leq{{L}_{\mathrm{SI}}^{\mathrm{t}}}$) transmit path between the MA position
${\mathbf{t}_{{n_{\mathrm{t}}}}}$ and the origin of the transmit region, i.e.,
$O_{\mathrm{t}}$ in Fig. 2, can be expressed as
$\rho_{\mathrm{SI},j}^{\mathrm{t}}\left({{\mathbf{t}_{{n_{\mathrm{t}}}}}}\right)=x_{{n_{\mathrm{t}}}}^{\mathrm{t}}\sin\theta_{\mathrm{SI},j}^{\mathrm{t}}\cos\phi_{\mathrm{SI},j}^{\mathrm{t}}+y_{{n_{\mathrm{t}}}}^{\mathrm{t}}\cos\theta_{\mathrm{SI},j}^{\mathrm{t}}$.
Denoting $\lambda$ as the carrier wavelength, the phase difference is
calculated by
$\frac{{2\pi}}{\lambda}\rho_{\mathrm{SI},j}^{\mathrm{t}}\left({{\mathbf{t}_{{n_{\mathrm{t}}}}}}\right)$.
Thus, the transmit field-response vector (FRV), which characterizes the phase
differences of ${{L}_{\mathrm{SI}}^{\mathrm{t}}}$ transmit paths, are obtained
as

${\mathbf{g}_{\mathrm{SI}}}\left({{\mathbf{t}_{{n_{\mathrm{t}}}}}}\right)={\left[{{e^{\mathrm{j}\frac{{2\pi}}{\lambda}\rho_{\mathrm{SI},1}^{\mathrm{t}}\left({{\mathbf{t}_{{n_{\mathrm{t}}}}}}\right)}},\ldots,{e^{\mathrm{j}\frac{{2\pi}}{\lambda}\rho_{\mathrm{SI},L_{\mathrm{SI}}^{\mathrm{t}}}^{\mathrm{t}}\left({{\mathbf{t}_{{n_{\mathrm{t}}}}}}\right)}}}\right]^{T}}\in{\mathbb{C}^{{L_{\mathrm{SI}}^{\mathrm{t}}}\times
1}}.$ (5)

Similarly, the receive FRV is given by

${\mathbf{f}_{\mathrm{SI}}}\left({{\mathbf{r}_{{n_{\mathrm{r}}}}}}\right)={\left[{{e^{\mathrm{j}\frac{{2\pi}}{\lambda}\rho_{\mathrm{SI},1}^{\mathrm{r}}\left({{\mathbf{r}_{{n_{\mathrm{r}}}}}}\right)}},\ldots,{e^{\mathrm{j}\frac{{2\pi}}{\lambda}\rho_{\mathrm{SI},L_{\mathrm{SI}}^{\mathrm{r}}}^{\mathrm{r}}\left({{\mathbf{r}_{{n_{\mathrm{r}}}}}}\right)}}}\right]^{T}}\in{\mathbb{C}^{{L_{\mathrm{SI}}^{\mathrm{r}}}\times
1}},$ (6)

which represents the phase differences of ${{L}_{\mathrm{SI}}^{\mathrm{r}}}$
receive paths, where
$\rho_{\mathrm{SI},i}^{\mathrm{r}}\left({{\mathbf{r}_{{n_{\mathrm{r}}}}}}\right)=x_{{n_{\mathrm{r}}}}^{\mathrm{r}}\sin\theta_{\mathrm{SI},i}^{\mathrm{r}}\cos\phi_{\mathrm{SI},i}^{\mathrm{r}}+y_{{n_{\mathrm{r}}}}^{\mathrm{r}}\cos\theta_{\mathrm{SI},i}^{\mathrm{r}}$
($1\leq i\leq{{L}_{\mathrm{SI}}^{\mathrm{r}}}$) is the difference of the
signal propagation distance for the $i$-th receive path between the MA
position ${\mathbf{r}_{{n_{\mathrm{r}}}}}$ and the origin of the receive
region, i.e., $O_{\mathrm{r}}$ in Fig. 2.

Moreover, define the path-response matrix (PRM) as
$\mathbf{\Sigma}\in{\mathbb{C}^{{{L}_{\mathrm{SI}}^{\mathrm{r}}}\times{{L}_{\mathrm{SI}}^{\mathrm{t}}}}}$,
where the entry in the $i$-th row and $j$-th column represent the channel
response between the $j$-th transmit path and the $i$-th receive path from
$O_{\mathrm{t}}$ to $O_{\mathrm{r}}$. As a result, the SI channel matrix is
obtained as

${\mathbf{H}_{\mathrm{SI}}}\left({\tilde{\mathbf{t}}},{\tilde{\mathbf{r}}}\right)={\mathbf{F}_{\mathrm{SI}}}{\left({\tilde{\mathbf{r}}}\right)^{H}}\mathbf{\Sigma}{\mathbf{G}_{\mathrm{SI}}}\left({\tilde{\mathbf{t}}}\right),$
(7)

where
$\mathbf{F}_{\mathrm{SI}}\left({\tilde{\mathbf{r}}}\right)=\left[\mathbf{f}_{\mathrm{SI}}\left({\mathbf{r}_{1}}\right),\cdots,\mathbf{f}_{\mathrm{SI}}\left({\mathbf{r}_{N_{\mathrm{r}}}}\right)\right]\in{\mathbb{C}^{{L_{\mathrm{SI}}^{\mathrm{r}}}\times
N_{\mathrm{r}}}}$ and
$\mathbf{G}_{\mathrm{SI}}\left({\tilde{\mathbf{t}}}\right)=\left[\mathbf{g}_{\mathrm{SI}}\left({\mathbf{t}_{1}}\right),\cdots,\mathbf{g}_{\mathrm{SI}}\left({\mathbf{t}_{N_{\mathrm{t}}}}\right)\right]\in{\mathbb{C}^{{L_{\mathrm{SI}}^{\mathrm{t}}}\times
N_{\mathrm{t}}}}$ are the field-response matrices (FRMs) of $N_{\mathrm{t}}$
transmit MAs and $N_{\mathrm{r}}$ receive MAs, respectively.

####  II-A2 UL and DL Channels

Since the users and Eves are equipped with FPAs, the FRMs only present at the
receive end for UL channel and the transmit end for DL channel. Denote
$L_{\mathrm{UB},k_{\mathrm{U}}}^{\mathrm{r}}$,
$L_{\mathrm{BD},k_{\mathrm{D}}}^{\mathrm{t}}$, and
$L_{\mathrm{BE},k_{\mathrm{E}}}^{\mathrm{t}}$ as the numbers of the receive
paths from UL user $k_{\mathrm{U}}$ to the BS and the transmit paths from the
BS to DL user $k_{\mathrm{D}}$ and to Eve $k_{\mathrm{E}}$, respectively.
Then, the UL and DL channels can be written as

$\displaystyle{\mathbf{h}_{\mathrm{UB},k_{\mathrm{U}}}}\left({\tilde{\mathbf{r}}}\right)$
$\displaystyle={\mathbf{F}_{\mathrm{UB},k_{\mathrm{U}}}}{\left({\tilde{\mathbf{r}}}\right)^{H}}{\mathbf{p}_{\mathrm{UB},k_{\mathrm{U}}}},$
(8)
$\displaystyle{\mathbf{h}_{\mathrm{BD},k_{\mathrm{D}}}}\left({\tilde{\mathbf{t}}}\right)$
$\displaystyle={{\mathbf{G}_{\mathrm{BD},k_{\mathrm{D}}}}\left({\tilde{\mathbf{t}}}\right)}^{H}{\mathbf{p}_{\mathrm{BD},k_{\mathrm{D}}}},$
(9)
$\displaystyle{\mathbf{h}_{\mathrm{BE},k_{\mathrm{E}}}}\left({\tilde{\mathbf{t}}}\right)$
$\displaystyle={{\mathbf{G}_{\mathrm{BE},k_{\mathrm{E}}}}\left({\tilde{\mathbf{t}}}\right)}^{H}{\mathbf{p}_{\mathrm{BE},k_{\mathrm{E}}}},$
(10)

where
${\mathbf{F}_{\mathrm{UB},k_{\mathrm{U}}}}\left({\tilde{\mathbf{r}}}\right)\in{\mathbb{C}^{{L_{\mathrm{UB},k_{\mathrm{U}}}^{\mathrm{r}}}\times
N_{\mathrm{r}}}}$ is the receive FRM for UL channel of UL user
$k_{\mathrm{U}}$,
${\mathbf{G}_{\mathrm{BD},k_{\mathrm{D}}}}\left({\tilde{\mathbf{t}}}\right)\in{\mathbb{C}^{{L_{\mathrm{BD},k_{\mathrm{D}}}^{\mathrm{t}}}\times
N_{\mathrm{t}}}}$ and
${\mathbf{G}_{\mathrm{BE},k_{\mathrm{E}}}}\left({\tilde{\mathbf{t}}}\right)\in{\mathbb{C}^{{L_{\mathrm{UE},k_{\mathrm{E}}}^{\mathrm{t}}}\times
N_{\mathrm{t}}}}$ are the corresponding transmit FRMs for DL channels of DL
user $k_{\mathrm{D}}$ and Eve $k_{\mathrm{E}}$.
${\mathbf{p}_{\mathrm{UB},k_{\mathrm{U}}}}\in{\mathbb{C}^{{L_{\mathrm{UB},k_{\mathrm{U}}}^{\mathrm{r}}}\times
1}}$ and
${\mathbf{p}_{\mathrm{BD},k_{\mathrm{D}}}}\in{\mathbb{C}^{{L_{\mathrm{BD},k_{\mathrm{D}}}^{\mathrm{t}}}\times
1}}$,
${\mathbf{p}_{\mathrm{BE},k_{\mathrm{E}}}}\in{\mathbb{C}^{{L_{\mathrm{BE},k_{\mathrm{E}}}^{\mathrm{t}}}\times
1}}$ are the path-response vectors (PRVs), which respectively represent the
channel responses from UL user $k_{\mathrm{U}}$ to $O_{\mathrm{r}}$ at the BS
and from $O_{\mathrm{t}}$ at the BS to DL user $k_{\mathrm{D}}$ and Eve
$k_{\mathrm{E}}$. Since
${\mathbf{F}_{\mathrm{UB},k_{\mathrm{U}}}}\left({\tilde{\mathbf{r}}}\right)$
and
${\mathbf{G}_{\mathrm{BD},k_{\mathrm{D}}}}\left({\tilde{\mathbf{t}}}\right)$,
${\mathbf{G}_{\mathrm{BE},k_{\mathrm{E}}}}\left({\tilde{\mathbf{t}}}\right)$
have the similar structures as
${\mathbf{F}_{\mathrm{SI}}}\left({\tilde{\mathbf{r}}}\right)$ and
${\mathbf{G}_{\mathrm{SI}}}\left({\tilde{\mathbf{t}}}\right)$, we can modify
the calculations of
${\mathbf{F}_{\mathrm{SI}}}\left({\tilde{\mathbf{r}}}\right)$ and
${\mathbf{G}_{\mathrm{SI}}}\left({\tilde{\mathbf{t}}}\right)$ to obtain
${\mathbf{F}_{\mathrm{UB},k_{\mathrm{U}}}}\left({\tilde{\mathbf{r}}}\right)$
and
${\mathbf{G}_{\mathrm{BD},k_{\mathrm{D}}}}\left({\tilde{\mathbf{t}}}\right)$,
${\mathbf{G}_{\mathrm{BE},k_{\mathrm{E}}}}\left({\tilde{\mathbf{t}}}\right)$
by replacing

$\left\\{{L_{\mathrm{SI}}^{\mathrm{r}},\left\\{{\theta_{\mathrm{SI},i}^{\mathrm{r}}}\right\\}_{i=1}^{L_{\mathrm{SI}}^{\mathrm{r}}},\left\\{{\phi_{\mathrm{SI},i}^{\mathrm{r}}}\right\\}_{i=1}^{L_{\mathrm{SI}}^{\mathrm{r}}}}\right\\}$
(11)

with

$\left\\{{L_{\mathrm{UB},k_{\mathrm{U}}}^{\mathrm{r}},\left\\{{\theta_{\mathrm{UB},k_{\mathrm{U}},i}^{\mathrm{r}}}\right\\}_{i=1}^{L_{\mathrm{UB},k_{\mathrm{U}}}^{\mathrm{r}}},\left\\{{\phi_{\mathrm{UB},k_{\mathrm{U}},i}^{\mathrm{r}}}\right\\}_{i=1}^{L_{\mathrm{UB},k_{\mathrm{U}}}^{\mathrm{r}}}}\right\\},$
(12)

and replacing

$\left\\{{L_{\mathrm{SI}}^{\mathrm{t}},\left\\{{\theta_{\mathrm{SI},j}^{\mathrm{t}}}\right\\}_{j=1}^{L_{\mathrm{SI}}^{\mathrm{t}}},\left\\{{\phi_{\mathrm{SI},j}^{\mathrm{t}}}\right\\}_{j=1}^{L_{\mathrm{SI}}^{\mathrm{t}}}}\right\\}$
(13)

with

$\displaystyle\left\\{{L_{\mathrm{BD},k_{\mathrm{D}}}^{\mathrm{t}},\left\\{{\theta_{\mathrm{BD},k_{\mathrm{D}},j}^{\mathrm{t}}}\right\\}_{j=1}^{L_{\mathrm{BD},k_{\mathrm{D}}}^{\mathrm{t}}},\left\\{{\phi_{\mathrm{BD},k_{\mathrm{D}},j}^{\mathrm{t}}}\right\\}_{j=1}^{L_{\mathrm{BD},k_{\mathrm{D}}}^{\mathrm{t}}}}\right\\},$
(14)
$\displaystyle\left\\{{L_{\mathrm{BE},k_{\mathrm{E}}}^{\mathrm{t}},\left\\{{\theta_{\mathrm{BE},k_{\mathrm{E}},j}^{\mathrm{t}}}\right\\}_{j=1}^{L_{\mathrm{BE},k_{\mathrm{E}}}^{\mathrm{t}}},\left\\{{\phi_{\mathrm{BE},k_{\mathrm{E}},j}^{\mathrm{t}}}\right\\}_{j=1}^{L_{\mathrm{BE},k_{\mathrm{E}}}^{\mathrm{t}}}}\right\\}.$
(15)

###  II-B Problem Formulation

$\displaystyle{\gamma_{k_{\mathrm{U}}}^{\mathrm{U}}}$
$\displaystyle=\frac{{{{\left|{\mathbf{b}_{{k_{\mathrm{U}}}}^{H}{\mathbf{h}_{\mathrm{UB},{k_{\mathrm{U}}}}\left(\tilde{\mathbf{r}}\right)}}\right|}^{2}}{p_{{k_{\mathrm{U}}}}}}}{{\sum\limits_{i\in{\mathcal{K}_{\mathrm{U}}}\backslash\left\\{{k_{\mathrm{U}}}\right\\}}{{{\left|{\mathbf{b}_{{k_{\mathrm{U}}}}^{H}{\mathbf{h}_{\mathrm{UB},i}\left(\tilde{\mathbf{r}}\right)}}\right|}^{2}}}{p_{i}}+\rho\left({{{\left|{\mathbf{b}_{{k_{\mathrm{U}}}}^{H}{\mathbf{H}_{\mathrm{SI}}\left(\tilde{\mathbf{t}},\tilde{\mathbf{r}}\right)}\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}{{\mathbf{w}_{{k_{\mathrm{D}}}}}}}\right|}^{2}}+{{\left|{\mathbf{b}_{{k_{\mathrm{U}}}}^{H}{\mathbf{H}_{\mathrm{SI}}\left(\tilde{\mathbf{t}},\tilde{\mathbf{r}}\right)}\mathbf{v}}\right|}^{2}}}\right)+\left\|{{\mathbf{b}_{{k_{\mathrm{U}}}}}}\right\|_{2}^{2}\sigma_{\mathrm{U}}^{2}}}$
$\displaystyle\mathop{=}\limits^{\left({{\alpha_{1}}}\right)}\frac{{{{\left|{{{\tilde{h}}_{\mathrm{UB},{k_{\mathrm{U}}}}}}\right|}^{2}}{p_{{k_{\mathrm{U}}}}}}}{{\sum\limits_{i\in{\mathcal{K}_{\mathrm{U}}}\backslash\left\\{{k_{\mathrm{U}}}\right\\}}{{{\left|{{{\tilde{h}}_{\mathrm{UB},i}}}\right|}^{2}}{p_{i}}}+\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}\mathrm{Tr}\left\\{{{{\mathbf{W}_{{k_{\mathrm{D}}}}}}{{\widetilde{\mathbf{H}}}_{\mathrm{SI},{k_{\mathrm{U}}}}}}\right\\}+\mathrm{Tr}\left\\{{\mathbf{V}{{\widetilde{\mathbf{H}}}_{\mathrm{SI},{k_{\mathrm{U}}}}}}\right\\}+\left\|{{\mathbf{b}_{{k_{\mathrm{U}}}}}}\right\|_{2}^{2}\sigma_{\mathrm{U}}^{2}}}.$
(16) $\displaystyle{\gamma_{k_{\mathrm{D}}}^{\mathrm{D}}}$
$\displaystyle=\frac{{{{\left|{\mathbf{h}_{\mathrm{BD},{k_{\mathrm{D}}}}^{H}\left(\tilde{\mathbf{t}}\right){\mathbf{w}_{{k_{\mathrm{D}}}}}}\right|}^{2}}}}{{\sum\limits_{i\in{\mathcal{K}_{\mathrm{D}}}\backslash\left\\{{k_{\mathrm{D}}}\right\\}}{{{\left|{\mathbf{h}_{\mathrm{BD},{k_{\mathrm{D}}}}^{H}\left(\tilde{\mathbf{t}}\right){\mathbf{w}_{i}}}\right|}^{2}}}+{{\left|{\mathbf{h}_{\mathrm{BD},{k_{\mathrm{D}}}}^{H}\left(\tilde{\mathbf{t}}\right)\mathbf{v}}\right|}^{2}}+\sum\limits_{{k_{\mathrm{U}}}\in{\mathcal{K}_{\mathrm{U}}}}{{{\left|{{h_{\mathrm{UD},{k_{\mathrm{U}}},{k_{\mathrm{D}}}}}}\right|}^{2}}{p_{{k_{\mathrm{U}}}}}}+\sigma_{\mathrm{D},{k_{\mathrm{D}}}}^{2}}}$
$\displaystyle\mathop{=}\limits^{\left({{\alpha_{2}}}\right)}{\frac{{\mathrm{Tr}\left\\{{{\mathbf{W}_{{k_{\mathrm{D}}}}}{\mathbf{H}_{\mathrm{BD},{k_{\mathrm{D}}}}}}\right\\}}}{{\sum\limits_{i\in{\mathcal{K}_{\mathrm{D}}}\backslash\left\\{{k_{\mathrm{D}}}\right\\}}{\mathrm{Tr}\left\\{{{\mathbf{W}_{i}}{\mathbf{H}_{\mathrm{BD},{k_{\mathrm{D}}}}}}\right\\}}+\mathrm{Tr}\left\\{{\mathbf{V}{\mathbf{H}_{\mathrm{BD},{k_{\mathrm{D}}}}}}\right\\}+\sum\limits_{{k_{\mathrm{U}}}\in{\mathcal{K}_{\mathrm{U}}}}{{{\left|{{h_{\mathrm{UD},{k_{\mathrm{U}}},{k_{\mathrm{D}}}}}}\right|}^{2}}{p_{{k_{\mathrm{U}}}}}}+\sigma_{\mathrm{D},{k_{\mathrm{D}}}}^{2}}}}.$
(17) $\displaystyle\gamma_{k_{\mathrm{U}}}^{\mathrm{E}-\mathrm{U}}$
$\displaystyle=\sum\limits_{{k_{\mathrm{E}}}\in{\mathcal{K}_{\mathrm{E}}}}{\frac{{{{\left|{{h_{\mathrm{UE},{k_{\mathrm{U}}},{k_{\mathrm{E}}}}}}\right|}^{2}}{p_{{k_{\mathrm{U}}}}}}}{{{{\left|{\mathbf{h}_{\mathrm{BE},{k_{\mathrm{E}}}}^{H}\left({\tilde{\mathbf{t}}}\right)\mathbf{v}}\right|}^{2}}+\sigma_{\mathrm{E},{k_{\mathrm{E}}}}^{2}}}}\mathop{=}\limits^{\left({{\alpha_{3}}}\right)}{\frac{{\sum\limits_{{k_{\mathrm{E}}}\in{\mathcal{K}_{\mathrm{E}}}}{{{\left|{{h_{\mathrm{UE},{k_{\mathrm{U}}},{k_{\mathrm{E}}}}}}\right|}^{2}}{p_{{k_{\mathrm{U}}}}}\prod\limits_{i\in{\mathcal{K}_{\mathrm{E}}}\backslash\left\\{{k_{\mathrm{E}}}\right\\}}{\left(\mathrm{Tr}\left\\{{\mathbf{V}{\mathbf{H}_{\mathrm{BE},i}}}\right\\}+\sigma_{\mathrm{E},i}^{2}\right)}}}}{{\prod\limits_{{k_{\mathrm{E}}}\in{\mathcal{K}_{\mathrm{E}}}}{\left(\mathrm{Tr}\left\\{{\mathbf{V}{\mathbf{H}_{\mathrm{BE},{k_{\mathrm{E}}}}}}\right\\}+\sigma_{\mathrm{E},{k_{\mathrm{E}}}}^{2}\right)}}}}.$
(18) $\displaystyle\gamma_{k_{\mathrm{D}}}^{\mathrm{E}-\mathrm{D}}$
$\displaystyle=\sum\limits_{{k_{\mathrm{E}}}\in{\mathcal{K}_{\mathrm{E}}}}{\frac{{{{\left|{\mathbf{h}_{\mathrm{BE},{k_{\mathrm{E}}}}^{H}\left({\tilde{\mathbf{t}}}\right){\mathbf{w}_{{k_{\mathrm{D}}}}}}\right|}^{2}}}}{{{{\left|{\mathbf{h}_{\mathrm{BE},{k_{\mathrm{E}}}}^{H}\left({\tilde{\mathbf{t}}}\right)\mathbf{v}}\right|}^{2}}+\sigma_{\mathrm{E},{k_{\mathrm{E}}}}^{2}}}}\mathop{=}\limits^{\left({{\alpha_{4}}}\right)}{\frac{{\sum\limits_{{k_{\mathrm{E}}}\in{\mathcal{K}_{\mathrm{E}}}}{\mathrm{Tr}\left\\{{{\mathbf{W}_{{k_{\mathrm{D}}}}}{\mathbf{H}_{\mathrm{BE},{k_{\mathrm{E}}}}}}\right\\}\prod\limits_{i\in{\mathcal{K}_{\mathrm{E}}}\backslash\left\\{{k_{\mathrm{E}}}\right\\}}{\left(\mathrm{Tr}\left\\{{\mathbf{V}{\mathbf{H}_{\mathrm{BE},i}}}\right\\}+\sigma_{\mathrm{E},i}^{2}\right)}}}}{{\prod\limits_{{k_{\mathrm{E}}}\in{\mathcal{K}_{\mathrm{E}}}}{\left(\mathrm{Tr}\left\\{{\mathbf{V}{\mathbf{H}_{\mathrm{BE},{k_{\mathrm{E}}}}}}\right\\}+\sigma_{\mathrm{E},{k_{\mathrm{E}}}}^{2}\right)}}}}.$
(19)

The achievable rate of UL user $k_{\mathrm{U}}$ is given by
$R_{{k_{\mathrm{U}}}}^{\mathrm{U}}={\log_{2}}\left({1+\gamma_{{k_{\mathrm{U}}}}^{\mathrm{U}}}\right)$,
where $\gamma_{{k_{\mathrm{U}}}}^{\mathrm{U}}$ is the receive signal-to-
interference-plus-noise ratio (SINR) and given by (16) at the bottom of the
next page. Here,
$\mathbf{b}_{{k_{\mathrm{U}}}}\in{\mathbb{C}^{{N_{\mathrm{r}}}\times 1}}$ is
the receive beamformer of UL user $k_{\mathrm{U}}$, $0<\rho\ll 1$ is the SI
loss coefficient representing the path loss and the SI cancellation in analog
and digital domains. Besides, the achievable rate of DL user $k_{\mathrm{D}}$
is given by
$R_{{k_{\mathrm{D}}}}^{\mathrm{D}}={\log_{2}}\left({1+\gamma_{{k_{\mathrm{D}}}}^{\mathrm{D}}}\right)$,
where $\gamma_{{k_{\mathrm{D}}}}^{\mathrm{D}}$ is the receive SINR and given
by (17) at the bottom of the next page.

To ensure secure communications, we consider a worst-case assumption. In
particular, we assume that each Eve separately eavesdrops on UL and DL
transmissions [27] and can cancel all multi-user interference before decoding
the information of a desired user [28]. Furthermore, $K_{\mathrm{E}}$ Eves aim
to collaborate in processing the confidential information [18]. Thus, under
these assumptions, the achievable rates of Eves eavesdropping on UL user
$k_{\mathrm{U}}$ and DL user $k_{\mathrm{D}}$ are respectively given by
$R_{{k_{\mathrm{U}}}}^{\mathrm{E-U}}={{\log}_{2}}\left({1+\gamma_{k_{\mathrm{U}}}^{\mathrm{E}-\mathrm{U}}}\right)$
and
$R_{{k_{\mathrm{D}}}}^{\mathrm{E-D}}={{\log}_{2}}\left({1+\gamma_{k_{\mathrm{D}}}^{\mathrm{E}-\mathrm{D}}}\right)$,
where $\gamma_{k_{\mathrm{U}}}^{\mathrm{E}-\mathrm{U}}$ and
$\gamma_{k_{\mathrm{D}}}^{\mathrm{E}-\mathrm{D}}$ are the receive SINRs and
given by (18) and (19) at the bottom of the next page. For the convenience of
subsequent derivations, we rewrite the receive SINRs by the equalities marked
by $\left({\alpha_{1}}\right)$-$\left({\alpha_{4}}\right)$ in (16)-(19), where
$\mathbf{W}_{k_{\mathrm{D}}}=\mathbf{w}_{k_{\mathrm{D}}}\mathbf{w}_{k_{\mathrm{D}}}^{H}\in{\mathbb{C}^{{N_{\mathrm{t}}}\times{N_{\mathrm{t}}}}}$,
$\mathbf{V}=\mathbf{vv}^{H}\in{\mathbb{C}^{{N_{\mathrm{t}}}\times{N_{\mathrm{t}}}}}$,
${{\tilde{h}}_{\mathrm{UB},{k_{\mathrm{U}}}}}=\mathbf{b}_{{k_{\mathrm{U}}}}^{H}{\mathbf{h}_{\mathrm{UB},{k_{\mathrm{U}}}}\left(\tilde{\mathbf{r}}\right)}\in{\mathbb{C}^{1\times
1}}$,
${{\tilde{\mathbf{h}}}_{\mathrm{SI},{k_{\mathrm{U}}}}}=\sqrt{\rho}{\mathbf{H}_{\mathrm{SI}}}^{H}\left({\tilde{\mathbf{t}},\tilde{\mathbf{r}}}\right)\mathbf{b}_{{k_{\mathrm{U}}}}\in{\mathbb{C}^{{N_{\mathrm{t}}}\times
1}}$,
${\widetilde{\mathbf{H}}_{\mathrm{SI},{k_{\mathrm{U}}}}}={{\tilde{\mathbf{h}}}_{\mathrm{SI},{k_{\mathrm{U}}}}}\tilde{\mathbf{h}}_{\mathrm{SI},{k_{\mathrm{U}}}}^{H}\in{\mathbb{C}^{{N_{\mathrm{t}}}\times{N_{\mathrm{t}}}}}$,
${\mathbf{H}_{\mathrm{BD},k_{\mathrm{D}}}}={\mathbf{h}_{\mathrm{BD},k_{\mathrm{D}}}\left({\tilde{\mathbf{t}}}\right)}\mathbf{h}_{\mathrm{BD},k_{\mathrm{D}}}^{H}\left({\tilde{\mathbf{t}}}\right)\in{\mathbb{C}^{{N_{\mathrm{t}}}\times{N_{\mathrm{t}}}}}$,
and
${\mathbf{H}_{\mathrm{BE},k_{\mathrm{E}}}}={\mathbf{h}_{\mathrm{BE},k_{\mathrm{E}}}\left({\tilde{\mathbf{t}}}\right)}\mathbf{h}_{\mathrm{BE},k_{\mathrm{E}}}^{H}\left({\tilde{\mathbf{t}}}\right)\in{\mathbb{C}^{{N_{\mathrm{t}}}\times{N_{\mathrm{t}}}}}$.

In the paper, we focus on maximizing the SSR of the users, i.e.,

$R_{\mathrm{SSR}}=\sum\limits_{{k_{\mathrm{U}}}\in{\mathcal{K}_{\mathrm{U}}}}{\left[R_{{k_{\mathrm{U}}}}^{\mathrm{U}}-R_{{k_{\mathrm{U}}}}^{\mathrm{E-U}}\right]^{+}}+\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}{\left[R_{{k_{\mathrm{D}}}}^{\mathrm{D}}-R_{{k_{\mathrm{D}}}}^{\mathrm{E-D}}\right]^{+}},$
(20)

by jointly optimizing the MA positions, $\tilde{\mathbf{t}}$ and
$\tilde{\mathbf{r}}$, the transmit, AN, and receive beamformers at the BS,
${\mathbf{w}_{{k_{\mathrm{D}}}}}$, $\mathbf{v}$, and
$\mathbf{b}_{k_{\mathrm{U}}}$, and the UL powers, ${p_{{k_{\mathrm{U}}}}}$.
The corresponding optimization problem is formulated as follows.

$\displaystyle\mathop{\mathrm{maximize}}\limits_{\tilde{\mathbf{t}},\tilde{\mathbf{r}},{\mathbf{w}_{{k_{\mathrm{D}}}}},\mathbf{v},{p_{{k_{\mathrm{U}}}}},{\mathbf{b}_{{k_{\mathrm{U}}}}}}\quad{R_{\mathrm{SSR}}}$
(21)
$\displaystyle\mathrm{s.t.}\quad\text{C1}:\left\|{{\mathbf{b}_{{k_{\mathrm{U}}}}}}\right\|_{2}^{2}=1,\
\forall{k_{\mathrm{U}}}\in\mathcal{K}_{\mathrm{U}},$ $\displaystyle\hskip
23.00006pt\text{C2}:\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}{\mathrm{Tr}\left\\{\mathbf{w}_{k_{\mathrm{D}}}\mathbf{w}_{k_{\mathrm{D}}}^{H}\right\\}}+\mathrm{Tr}\left\\{\mathbf{v}\mathbf{v}^{H}\right\\}\leq
P_{\max}^{\mathrm{D}},$ $\displaystyle\hskip
23.00006pt\text{C3}:0\leq{p_{{k_{\mathrm{U}}}}}\leq
P_{\max,{k_{\mathrm{U}}}}^{\mathrm{U}},\
\forall{k_{\mathrm{U}}}\in\mathcal{K}_{\mathrm{U}},$ $\displaystyle\hskip
23.00006pt\text{C4}:\tilde{\mathbf{t}}\in{\mathcal{C}_{\mathrm{t}}},\
\tilde{\mathbf{r}}\in{\mathcal{C}_{\mathrm{r}}},$ $\displaystyle\hskip
23.00006pt\text{C5}:{\left\|{{\mathbf{t}_{a}}-{\mathbf{t}_{\tilde{a}}}}\right\|_{2}}\geq
D,1\leq a\neq{\tilde{a}}\leq{N_{\mathrm{t}}},$ $\displaystyle\hskip
23.00006pt\text{C6}:{\left\|{{\mathbf{r}_{b}}-{\mathbf{r}_{\tilde{b}}}}\right\|_{2}}\geq
D,1\leq b\neq{\tilde{b}}\leq{N_{\mathrm{r}}}.$

Here, constraint C1 normalizes the receive beamformer.
$P_{\max}^{\mathrm{D}}>0$ and $P_{\max,{k_{\mathrm{U}}}}^{\mathrm{U}}>0$ in
constraints C2 and C3 are the maximum transmit powers of the BS and UL user
${k_{\mathrm{U}}}$, respectively. Constraint C4 limits the ranges of MA
movements. Constraints C5 and C6 ensure that minimum inter-MA distance $D$ at
the BS for practical implementation. The operator ${\left[\cdot\right]^{+}}$
has no impact on the optimization and will be omitted in the subsequent
derivations.

Note that problem (21) is a highly non-convex optimization problem.
Specifically, the non-convexity of the objective function and the minimum
inter-MA distance constraints C5 and C6, along with the couplings between the
optimization variables, make the optimization problem particularly
intractable. To the best of our knowledge, existing optimization tools cannot
be directly applied to obtain the globally optimal solution. Thus, we propose
an AO algorithm to solve problem (21) in the next section.

##  III Proposed Solution

In this section, we propose an AO algorithm to address problem (21), which has
high couplings between the optimization variables. Indeed, AO is a widely
applicable methodology that decomposes the original problem into several sub-
problems and iteratively solves each one while holding the optimization
variables in other sub-problems fixed [12, 20, 26, 28]. Specifically, we
decompose problem (21) into three sub-problems, i.e., iteratively optimizing
$\left\\{{\tilde{\mathbf{t}},\tilde{\mathbf{r}}}\right\\}$,
$\left\\{{{\mathbf{w}_{{k_{\mathrm{D}}}}},\mathbf{v},{p_{{k_{\mathrm{U}}}}}}\right\\}$,
and ${\mathbf{b}_{{k_{\mathrm{U}}}}}$.

###  III-A Sub-Problem 1: Optimize
$\left\\{{\tilde{\mathbf{t}},\tilde{\mathbf{r}}}\right\\}$ with Given
$\left\\{{{\mathbf{w}_{{k_{\mathrm{D}}}}},\mathbf{v},{p_{{k_{\mathrm{U}}}}}}\right\\}$
and ${\mathbf{b}_{{k_{\mathrm{U}}}}}$

With the given
$\left\\{{{\mathbf{w}_{{k_{\mathrm{D}}}}},\mathbf{v},{p_{{k_{\mathrm{U}}}}}}\right\\}$
and ${\mathbf{b}_{{k_{\mathrm{U}}}}}$, the SSR can be expressed as a function
of $\tilde{\mathbf{t}}$ and $\tilde{\mathbf{r}}$. Therefore, sub-problem 1 can
be formulated as

$\displaystyle\mathop{\mathrm{maximize}}\limits_{\tilde{\mathbf{t}},\tilde{\mathbf{r}}}\quad{R_{\mathrm{SSR}}\left(\tilde{\mathbf{t}},\tilde{\mathbf{r}}\right)}$
(22) $\displaystyle\mathrm{s.t.}\quad{\text{C4, C5, C6}}.$

The conventional alternating position optimization (APO) [12], which
iteratively fixes the other MAs while moving only one, may converge to an
undesired local optimal solution because the given positions of other MAs
narrow the optimization space of the current MA to a tiny region [13]. To
address this problem, we propose the MVPSO algorithm, which is an effective
improvement of the standard PSO [29, 30] by replacing the single velocity of
each particle in the iterations with multiple candidate velocities, to
simultaneously optimize the positions of all transmit and receive MAs. The
details of MVPSO are presented in the following.

Algorithm 1 Multi-Velocity Particle Swarm Optimization

0: Set initial MVPSO parameters, i.e., $N$, $Q$, $A$, $D$,
$\tau_{\mathrm{t}}$, $\tau_{\mathrm{r}}$, $\bar{I}$, $\bar{J}$, $c_{1}$, and
$c_{2}$.

0: $\mathbf{u}$.

1: Initialize the positions and velocities of $N$ particles as
$\mathbf{U}^{(0)}$ and $\mathbf{Z}^{(0)}$, respectively;

2: Calculate the fitness value of each particle by (24);

3: Initialize the personal best position
$\mathbf{u}_{\mathrm{pbest},n}=\mathbf{u}_{n}^{(0)}$ and the global best
position
$\mathbf{u}_{\mathrm{gbest}}=\arg\mathop{\max}\limits_{\mathbf{u}_{n}^{(0)}}\left\\{{\mathcal{F}\left({\mathbf{u}_{1}^{(0)}}\right),\cdots,\mathcal{F}\left({\mathbf{u}_{N}^{(0)}}\right)}\right\\}$;

4: for $q=1:1:Q$ do

5: for $n=1:1:N$ do

6: Initialize the velocity components matrix
${\mathbf{\Psi}_{n}^{\left(q\right)}}$;

7: for $\bar{i}=1:1:\bar{I}$ do

8: Initialize the combination weight vector ${\mathbf{c}_{n,\bar{i}}}$;

9: Calculate the candidate velocity $\mathbf{z}_{n,\bar{i}}^{\left(q\right)}$
by (28) and the candidate position $\mathbf{u}_{n,\bar{i}}^{\left(q\right)}$
by (25);

10: Calculate the fitness value of each candidate position by (24);

11: end for

12: Update the position by (30) and the corresponding velocity;

13: if
$\mathcal{F}\left({\mathbf{u}_{n}^{(q)}}\right)>\mathcal{F}\left({\mathbf{u}_{\mathrm{pbest},n}}\right)$
then

14: Update $\mathbf{u}_{\mathrm{pbest},n}=\mathbf{u}_{n}^{(q)}$;

15: end if

16: if
$\mathcal{F}\left({\mathbf{u}_{n}^{(q)}}\right)>\mathcal{F}\left({\mathbf{u}_{\mathrm{gbest}}}\right)$
then

17: Update $\mathbf{u}_{\mathrm{gbest}}={\mathbf{u}_{n}^{(q)}}$;

18: end if

19: end for

20: end for

21: return $\mathbf{u}=\mathbf{u}_{\mathrm{gbest}}$.

We first randomly initialize the positions and velocities of $N$ particles as
${\mathbf{U}^{\left(0\right)}}=\left\\{{\mathbf{u}_{1}^{\left(0\right)},\ldots,\mathbf{u}_{N}^{\left(0\right)}}\right\\}\in{\mathbb{R}^{2\left(N_{\mathrm{t}}+N_{\mathrm{r}}\right)\times
N}}$ and
${\mathbf{Z}^{\left(0\right)}}=\left\\{{\mathbf{z}_{1}^{\left(0\right)},\ldots,\mathbf{z}_{N}^{\left(0\right)}}\right\\}\in{\mathbb{R}^{2\left(N_{\mathrm{t}}+N_{\mathrm{r}}\right)\times
N}}$, respectively, where each particle’s position represents a possible
solution for the antenna position vector, i.e.,
$\mathbf{u}_{n}^{\left(0\right)}={\left[{{{\tilde{\mathbf{t}}}^{T}},{{\tilde{\mathbf{r}}}^{T}}}\right]^{T}}\in{\mathbb{R}^{2\left(N_{\mathrm{t}}+N_{\mathrm{r}}\right)\times
1}}$ ($1\leq n\leq N$). Without loss of generality, we assume that each moving
region is a square with size $A\times A$. Each element in
${\mathbf{U}^{\left(0\right)}}$ obeys the uniform distribution over the real-
number interval $\left[{-\frac{A}{2},\frac{A}{2}}\right]$ to ensure that the
initial positions of MAs do not exceed the corresponding moving regions, i.e.,
constraint C4 holds. Then, the personal best position of the $n$-th particle
$\mathbf{u}_{\mathrm{pbest},n}$ are initialized as
$\mathbf{u}_{n}^{\left(0\right)}$ and the global best position
$\mathbf{u}_{\mathrm{gbest}}$ is selected based on the fitness function. After
completing the initialization, the processing procedures of the MVPSO
algorithm are summarized in Algorithm 1. Let $Q$ denote the maximum number of
iterations, the introduction of Algorithm 1 is given as follows.

####  III-A1 Define Fitness Function

Considering constraints C5 and C6 on the minimum inter-MA distance, we first
define a penalty function as

$\displaystyle\mathcal{P}\left({\mathbf{u}_{n}^{\left(q\right)}}\right)$
$\displaystyle=\tau_{\mathrm{t}}\sum\limits_{a=1}^{{N_{\mathrm{t}}}-1}{\sum\limits_{\tilde{a}=a+1}^{{N_{\mathrm{t}}}}{\delta\left({{{\left\|{{\mathbf{t}_{a}}-{\mathbf{t}_{\tilde{a}}}}\right\|}_{2}}<D}\right)}}$
$\displaystyle+\tau_{\mathrm{r}}\sum\limits_{b=1}^{{N_{\mathrm{r}}}-1}{\sum\limits_{\tilde{b}=b+1}^{{N_{\mathrm{r}}}}{\delta\left({{{\left\|{{\mathbf{r}_{b}}-{\mathbf{r}_{\tilde{b}}}}\right\|}_{2}}<D}\right)}},$
(23)

where ${\mathbf{u}_{n}^{\left(q\right)}}$ is the position of the $n$-th
particle in the $q$-th ($1\leq q\leq Q$) iteration.
${\delta\left(\cdot\right)}$ is an indicator function, equaling 1 when the
condition within the bracket is true, otherwise, it equals 0.
$\tau_{\mathrm{t}}$ and $\tau_{\mathrm{r}}$ are the positive penalty factors
utilized to regulate the severity of the penalty. Assume that the best
position has the largest fitness value. Based on this given penalty function,
for maximizing the SSR, the fitness function is defined as

$\mathcal{F}\left({\mathbf{u}_{n}^{\left(q\right)}}\right)={R_{\mathrm{SSR}}}\left({\mathbf{u}_{n}^{\left(q\right)}}\right)-\mathcal{P}\left({\mathbf{u}_{n}^{\left(q\right)}}\right),$
(24)

where the values of $\tau_{\mathrm{t}}$ and $\tau_{\mathrm{r}}$ consistently
maintain
${R_{\mathrm{SSR}}}\left({\mathbf{u}_{n}^{\left(q\right)}}\right)-\left(\tau_{\mathrm{t}}+\tau_{\mathrm{r}}\right)<0$.
Thus, the penalty function can push the particles to satisfy the minimum
inter-MA distance. In other words, with the progression of iterations,
$\mathcal{P}\left({\mathbf{u}_{n}^{\left(q\right)}}\right)$ will converge to
zero.

####  III-A2 Update Positions and Velocities

The candidate positions of the $n$-th particle in the $q$-th iteration are
updated by ${\bar{I}}$ candidate velocities, i.e.,

$\mathbf{u}_{n,\bar{i}}^{\left(q\right)}=\mathcal{B}\left\\{{\mathbf{u}_{n}^{\left({q-1}\right)}+\mathbf{z}_{n,\bar{i}}^{\left(q\right)}}\right\\},$
(25)

where $\mathbf{z}_{n,\bar{i}}^{\left(q\right)}$ is the $\bar{i}$-th
($0<\bar{i}<\bar{I}$) candidate velocity, which will be specified later.
$\mathcal{B}\left\\{\mathbf{u}\right\\}$ is a function, that projects each
entry of the vector $\mathbf{u}$ to the nearest boundary if it exceeds the
feasible region, to satisfy constraint C4, i.e.,

${\left[{\mathcal{B}\left\\{\mathbf{u}\right\\}}\right]_{i}}=\left\\{\begin{array}[]{rcl}\frac{A}{2},&\mathrm{if}&{{\left[\mathbf{u}\right]_{i}}>\frac{A}{2}},\\\
{\left[\mathbf{u}\right]_{i}},&\mathrm{if}&{-\frac{A}{2}\leq{\left[\mathbf{u}\right]_{i}}\leq\frac{A}{2}},\\\
-\frac{A}{2},&\mathrm{if}&{{\left[\mathbf{u}\right]_{i}}<-\frac{A}{2}}.\\\
\end{array}\right.$ (26)

For standard PSO, the inertia weight in each particle’s velocity decreases
with the number of iterations in the interval
$\left[\omega_{\mathrm{min}},\omega_{\mathrm{max}}\right]$, i.e.,

$\omega={\omega_{\max}}-\frac{{\left({{\omega_{\max}}-{\omega_{\min}}}\right)q}}{Q}.$
(27)

Generally, a small $\omega$ leads to local exploitation for optimal solutions
within the current region, whereas a large $\omega$ signifies that the
particles can globally explore to evade undesired local optimal solutions
[31]. Hence, the standard PSO lacks exploitation in the early iterations and
exploration in the late iterations. Based on the aforementioned observations,
we modify the standard PSO and propose the MVPSO, in which each particle can
select the optimal velocity from multiple candidate velocities in each
iteration.

The ${\bar{I}}$ candidate velocities in (25) are generated by the weighted
combinations of ${\bar{J}}$ velocity components, i.e.,

$\mathbf{z}_{n,\bar{i}}^{\left(q\right)}={\mathbf{\Psi}_{n}^{\left(q\right)}}{\mathbf{c}_{n,\bar{i}}},$
(28)

where ${\mathbf{c}_{n,\bar{i}}}\in{\mathbb{R}^{\bar{J}\times 1}}$ is a
constant vector of the $n$-th particle, which represents the combination
weights for generating the $\bar{i}$-th candidate velocity.
${\mathbf{\Psi}_{n}^{\left(q\right)}}=\left[{\boldsymbol{\psi}_{n,1}^{\left(q\right)},\ldots,\boldsymbol{\psi}_{n,\bar{J}}^{\left(q\right)}}\right]\in{\mathbb{R}^{2\left(N_{\mathrm{t}}+N_{\mathrm{r}}\right)\times\bar{J}}}$
is the collection of velocity components for the $n$-th particle and
$\boldsymbol{\psi}_{n,\bar{j}}^{\left(q\right)}\in{\mathbb{R}^{2\left(N_{\mathrm{t}}+N_{\mathrm{r}}\right)\times
1}}$ represents the $\bar{j}$-th ($1\leq\bar{j}\leq\bar{J}$) velocity
component, which is calculated as

$\displaystyle\boldsymbol{\psi}_{n,\bar{j}}^{\left(q\right)}=\omega_{\bar{j}}\boldsymbol{\psi}_{n}^{\left({q-1}\right)}$
$\displaystyle+{c_{1}}{\mathbf{e}_{1}}\odot\left({{\mathbf{u}_{\mathrm{pbest},n}}-\mathbf{u}_{n}^{\left({q-1}\right)}}\right)$
$\displaystyle+{c_{2}}{\mathbf{e}_{2}}\odot\left({{\mathbf{u}_{\mathrm{gbest}}}-\mathbf{u}_{n}^{\left({q-1}\right)}}\right),$
(29)

where $c_{1}$ and $c_{2}$ are the personal and global learning factors that
push each particle toward the personal and global best positions,
respectively. To reduce the possibility of converging to an undesired local
optimal solution, two random vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$,
with uniformly distributed entries in the range $\left[0,1\right]$, are
utilized. $\omega_{\bar{j}}$ is the inertia weight of the $\bar{j}$-th
velocity component. In general, the combination weights of different velocity
components can be configured to simultaneously accommodate exploitation and
exploration. However, the velocity components in
${\mathbf{\Psi}_{n}^{\left(q\right)}}$ may not cover the optimal velocity but
contain the velocity components with biased local and biased global search
behaviors. In other words, both the $\bar{J}$ velocity components themselves
and their weighted combinations can serve as the candidate velocities. Thus,
the constant vector ${\mathbf{c}_{n,\bar{i}}}$ is introduced to control the
combination of candidate velocities. It is worth noting that the standard PSO
can be regarded as a special case of the proposed MVPSO by only setting one
candidate velocity. Similar multi-candidate approaches have been employed to
address optimization problems in communications, e.g., [14].

Finally, the position of the $n$-th particle in the $q$-th iteration is
selected from $\bar{I}$ candidate solutions which can achieve the maximum
fitness value in (24), i.e.,

$\mathbf{u}_{n}^{\left(q\right)}=\mathop{\arg\max}\limits_{\mathbf{u}_{n,\bar{i}}^{\left(q\right)}}\left\\{{\mathcal{F}\left({\mathbf{u}_{n,1}^{\left(q\right)}}\right),\ldots,\mathcal{F}\left({\mathbf{u}_{n,\bar{I}}^{\left(q\right)}}\right)}\right\\}.$
(30)

The corresponding velocity is updated to the candidate velocity associated
with the selected solution. The generation and selection of each particle’s
position and velocity are presented in lines 6-12.

####  III-A3 Update Personal and Global Best Positions

After obtaining the particles’ positions, the personal and global best
positions are updated if the fitness value at the current position exceeds the
personal and global best fitness values, respectively. The corresponding
pseudo-code is shown in lines 13-18. After $Q$ iterations, an optimized
solution for the antenna position vector is obtained by the global best
position, i.e., line 21.

###  III-B Sub-Problem 2: Optimize
$\left\\{{{\mathbf{w}_{{k_{\mathrm{D}}}}},\mathbf{v},{p_{{k_{\mathrm{U}}}}}}\right\\}$
with Given $\left\\{{\tilde{\mathbf{t}},\tilde{\mathbf{r}}}\right\\}$ and
${\mathbf{b}_{{k_{\mathrm{U}}}}}$

Based on the rule of the logarithmic function, we define $f$ and $g$, which
are given by (31) and (32) at the bottom of the next page, and rewrite the SSR
as $R_{\mathrm{SSR}}=f-g$.

$\displaystyle f$
$\displaystyle=\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}{{{\log}_{2}}\left({{{\sum\limits_{k_{\mathrm{D}}\in{\mathcal{K}_{\mathrm{D}}}}{\mathrm{Tr}\left\\{{{\mathbf{W}_{k_{\mathrm{D}}}}{\mathbf{H}_{\mathrm{BD},{k_{\mathrm{D}}}}}}\right\\}}+\mathrm{Tr}\left\\{{\mathbf{V}{\mathbf{H}_{\mathrm{BD},{k_{\mathrm{D}}}}}}\right\\}+\sum\limits_{{k_{\mathrm{U}}}\in{\mathcal{K}_{\mathrm{U}}}}{{{\left|{{h_{\mathrm{UD},{k_{\mathrm{U}}},{k_{\mathrm{D}}}}}}\right|}^{2}}{p_{{k_{\mathrm{U}}}}}}+\sigma_{\mathrm{D},{k_{\mathrm{D}}}}^{2}}}}\right)}$
$\displaystyle+\sum\limits_{{k_{\mathrm{U}}}\in{\mathcal{K}_{\mathrm{U}}}}{{{\log}_{2}}\left({{\sum\limits_{k_{\mathrm{U}}\in{\mathcal{K}_{\mathrm{U}}}}{{{\left|{{{\tilde{h}}_{\mathrm{UB},{k_{\mathrm{U}}}}}}\right|}^{2}}{p_{k_{\mathrm{U}}}}}+\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}\mathrm{Tr}\left\\{{{{\mathbf{W}_{{k_{\mathrm{D}}}}}}{{\widetilde{\mathbf{H}}}_{\mathrm{SI},{k_{\mathrm{U}}}}}}\right\\}+\mathrm{Tr}\left\\{{\mathbf{V}{{\widetilde{\mathbf{H}}}_{\mathrm{SI},{k_{\mathrm{U}}}}}}\right\\}+\left\|{{\mathbf{b}_{{k_{\mathrm{U}}}}\left(\tilde{\mathbf{r}}\right)}}\right\|_{2}^{2}\sigma_{\mathrm{U}}^{2}}}\right)}$
$\displaystyle+\left(K_{\mathrm{D}}+K_{\mathrm{U}}\right){{{\log}_{2}}\left({{\prod\limits_{{k_{\mathrm{E}}}\in{\mathcal{K}_{\mathrm{E}}}}{\left(\mathrm{Tr}\left\\{{\mathbf{V}{\mathbf{H}_{\mathrm{BE},{k_{\mathrm{E}}}}}}\right\\}+\sigma_{\mathrm{E},{k_{\mathrm{E}}}}^{2}\right)}}}\right)}.$
(31) $\displaystyle g$
$\displaystyle=\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}{{{\log}_{2}}\left({{{\sum\limits_{i\in{\mathcal{K}_{\mathrm{D}}}\backslash\left\\{{k_{\mathrm{D}}}\right\\}}{\mathrm{Tr}\left\\{{{\mathbf{W}_{i}}{\mathbf{H}_{\mathrm{BD},{k_{\mathrm{D}}}}}}\right\\}}+\mathrm{Tr}\left\\{{\mathbf{V}{\mathbf{H}_{\mathrm{BD},{k_{\mathrm{D}}}}}}\right\\}+\sum\limits_{{k_{\mathrm{U}}}\in{\mathcal{K}_{\mathrm{U}}}}{{{\left|{{h_{\mathrm{UD},{k_{\mathrm{U}}},{k_{\mathrm{D}}}}}}\right|}^{2}}{p_{{k_{\mathrm{U}}}}}}+\sigma_{\mathrm{D},{k_{\mathrm{D}}}}^{2}}}}\right)}$
$\displaystyle+\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}{{{\log}_{2}}\left({{\prod\limits_{{k_{\mathrm{E}}}\in{\mathcal{K}_{\mathrm{E}}}}{\left(\mathrm{Tr}\left\\{{\mathbf{V}{\mathbf{H}_{\mathrm{BE},{k_{\mathrm{E}}}}}}\right\\}+\sigma_{\mathrm{E},{k_{\mathrm{E}}}}^{2}\right)}}}+{{\sum\limits_{{k_{\mathrm{E}}}\in{\mathcal{K}_{\mathrm{E}}}}{\mathrm{Tr}\left\\{{{\mathbf{W}_{{k_{\mathrm{D}}}}}{\mathbf{H}_{\mathrm{BE},{k_{\mathrm{E}}}}}}\right\\}\prod\limits_{i\in{\mathcal{K}_{\mathrm{E}}}\backslash\left\\{{k_{\mathrm{E}}}\right\\}}{\left(\mathrm{Tr}\left\\{{\mathbf{V}{\mathbf{H}_{\mathrm{BE},i}}}\right\\}+\sigma_{\mathrm{E},i}^{2}\right)}}}}\right)}$
$\displaystyle+\sum\limits_{{k_{\mathrm{U}}}\in{\mathcal{K}_{\mathrm{U}}}}{{{\log}_{2}}\left({{\sum\limits_{i\in{\mathcal{K}_{\mathrm{U}}}\backslash\left\\{{k_{\mathrm{U}}}\right\\}}{{{\left|{{{\tilde{h}}_{\mathrm{UB},i}}}\right|}^{2}}{p_{i}}}+\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}\mathrm{Tr}\left\\{{{{\mathbf{W}_{{k_{\mathrm{D}}}}}}{{\widetilde{\mathbf{H}}}_{\mathrm{SI},{k_{\mathrm{U}}}}}}\right\\}+\mathrm{Tr}\left\\{{\mathbf{V}{{\widetilde{\mathbf{H}}}_{\mathrm{SI},{k_{\mathrm{U}}}}}}\right\\}+\left\|{{\mathbf{b}_{{k_{\mathrm{U}}}}\left(\tilde{\mathbf{r}}\right)}}\right\|_{2}^{2}\sigma_{\mathrm{U}}^{2}}}\right)}$
$\displaystyle+\sum\limits_{{k_{\mathrm{U}}}\in{\mathcal{K}_{\mathrm{U}}}}{{{\log}_{2}}\left({{\prod\limits_{{k_{\mathrm{E}}}\in{\mathcal{K}_{\mathrm{E}}}}{\left(\mathrm{Tr}\left\\{{\mathbf{V}{\mathbf{H}_{\mathrm{BE},{k_{\mathrm{E}}}}}}\right\\}+\sigma_{\mathrm{E},{k_{\mathrm{E}}}}^{2}\right)}}}+{{\sum\limits_{{k_{\mathrm{E}}}\in{\mathcal{K}_{\mathrm{E}}}}{{{\left|{{h_{\mathrm{UE},{k_{\mathrm{U}}},{k_{\mathrm{E}}}}}}\right|}^{2}}{p_{{k_{\mathrm{U}}}}}\prod\limits_{i\in{\mathcal{K}_{\mathrm{E}}}\backslash\left\\{{k_{\mathrm{E}}}\right\\}}{\left(\mathrm{Tr}\left\\{{\mathbf{V}{\mathbf{H}_{\mathrm{BE},i}}}\right\\}+\sigma_{\mathrm{E},i}^{2}\right)}}}}\right)}.$
(32)

Thus, with given $\left\\{{\tilde{\mathbf{t}},\tilde{\mathbf{r}}}\right\\}$
and ${\mathbf{b}_{{k_{\mathrm{U}}}}}$, sub-problem 2 can be formulated as

$\displaystyle\mathop{\mathrm{maximize}}\limits_{{\mathbf{W}_{{k_{\mathrm{D}}}}},\mathbf{V},{p_{{k_{\mathrm{U}}}}}}\quad
f-g$ (33) $\displaystyle\mathrm{s.t.}\quad\text{C3,
C7}:\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}{\mathrm{Tr}\left\\{\mathbf{W}_{k_{\mathrm{D}}}\right\\}}+\mathrm{Tr}\left\\{\mathbf{V}\right\\}\leq
P_{\max}^{\mathrm{D}}$ $\displaystyle\hskip
23.00006pt\text{C8}:{\mathbf{W}_{k_{\mathrm{D}}}}\succeq\mathbf{0},\mathbf{V}\succeq\mathbf{0},\
\forall{k_{\mathrm{D}}}\in\mathcal{K}_{\mathrm{D}},$ $\displaystyle\hskip
23.00006pt\text{C9}:\mathrm{Rank}\left\\{{\mathbf{W}_{k_{\mathrm{D}}}}\right\\}\leq
1,\mathrm{Rank}\left\\{\mathbf{V}\right\\}\leq 1,\
\forall{k_{\mathrm{D}}}\in\mathcal{K}_{\mathrm{D}}.$

Problem (33) is also non-convex due to the objective function and the rank
constraint C9. Note that $f$ and $g$ are concave functions, and thus the
objective function in (33) is a difference-of-concave function. Therefore, the
SCA [32] is applied to obtain a sub-optimal solution. Specifically, define the
maximum number of iterations for SCA as $M$. In the $m$-th ($1\leq m\leq M$)
iteration, we construct a global overestimate of $g$ for a given feasible
point
$\left({{\mathbf{W}_{k_{\mathrm{D}}}^{\left(m\right)}},{\mathbf{V}^{\left(m\right)}}},p_{k_{\mathrm{U}}}^{\left(m\right)}\right)$
by the first-order Taylor expansion, i.e.,
$\widetilde{g}\left({{\mathbf{W}_{{k_{\mathrm{D}}}}},\mathbf{V},{p_{{k_{\mathrm{U}}}}}\left|{\mathbf{W}_{{k_{\mathrm{D}}}}^{\left(m\right)},{\mathbf{V}^{\left(m\right)}},p_{{k_{\mathrm{U}}}}^{\left(m\right)}}\right.}\right)$,
which is given by (III-B) at the bottom of the next page, where
${\nabla_{{\mathbf{W}_{{k_{\mathrm{D}}}}}}}{g}$, ${\nabla_{\mathbf{V}}}{g}$,
and ${\nabla_{{p_{{k_{\mathrm{U}}}}}}}{g}$ denote the gradients of function
$g$ with respect to ${\mathbf{W}_{{k_{\mathrm{D}}}}}$, $\mathbf{V}$, and
${p_{{k_{\mathrm{U}}}}}$, respectively.

$\displaystyle{g}\left({{\mathbf{W}_{{k_{\mathrm{D}}}}},\mathbf{V},{p_{{k_{\mathrm{U}}}}}}\right)\leq{g}\left({\mathbf{W}_{{k_{\mathrm{D}}}}^{\left(m\right)},{\mathbf{V}^{\left(m\right)}},p_{{k_{\mathrm{U}}}}^{\left(m\right)}}\right)+\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}{\mathrm{Tr}\left\\{{{{\left({{\nabla_{{\mathbf{W}_{{k_{\mathrm{D}}}}}}}{g}\left({\mathbf{W}_{{k_{\mathrm{D}}}}^{\left(m\right)},{\mathbf{V}^{\left(m\right)}},p_{{k_{\mathrm{U}}}}^{\left(m\right)}}\right)}\right)}^{H}}\left({{\mathbf{W}_{{k_{\mathrm{D}}}}}-\mathbf{W}_{{k_{\mathrm{D}}}}^{\left(m\right)}}\right)}\right\\}}$
$\displaystyle+\mathrm{Tr}\left\\{{{{\left({{\nabla_{\mathbf{V}}}{g}\left({\mathbf{W}_{{k_{\mathrm{D}}}}^{\left(m\right)},{\mathbf{V}^{\left(m\right)}},p_{{k_{\mathrm{U}}}}^{\left(m\right)}}\right)}\right)}^{H}}\left({\mathbf{V}-{\mathbf{V}^{\left(m\right)}}}\right)}\right\\}+\sum\limits_{{k_{\mathrm{U}}}\in{\mathcal{K}_{\mathrm{U}}}}{\mathrm{Tr}\left\\{{{{\left({{\nabla_{{p_{{k_{\mathrm{U}}}}}}}{g}\left({\mathbf{W}_{{k_{\mathrm{D}}}}^{\left(m\right)},{\mathbf{V}^{\left(m\right)}},p_{{k_{\mathrm{U}}}}^{\left(m\right)}}\right)}\right)}^{H}}\left({{p_{{k_{\mathrm{U}}}}}-p_{{k_{\mathrm{U}}}}^{\left(m\right)}}\right)}\right\\}}$
$\displaystyle\triangleq\widetilde{{g}}\left({{\mathbf{W}_{{k_{\mathrm{D}}}}},\mathbf{V},{p_{{k_{\mathrm{U}}}}}\left|{\mathbf{W}_{{k_{\mathrm{D}}}}^{\left(m\right)},{\mathbf{V}^{\left(m\right)}},p_{{k_{\mathrm{U}}}}^{\left(m\right)}}\right.}\right).$
(34)

Subsequently, for a given feasible point
$\left({{\mathbf{W}_{k_{\mathrm{D}}}^{\left(m\right)}},{\mathbf{V}^{\left(m\right)}}},p_{k_{\mathrm{U}}}^{\left(m\right)}\right)$
in the $m$-th iteration, a lower bound of the maximization problem in (33) can
be obtained by solving the following optimization problem.

$\displaystyle\mathop{\mathrm{maximize}}\limits_{{\mathbf{W}_{{k_{\mathrm{D}}}}},\mathbf{V},{p_{{k_{\mathrm{U}}}}}}\quad\widetilde{F}\left({\mathbf{W}_{{k_{\mathrm{D}}}}},\mathbf{V},{p_{{k_{\mathrm{U}}}}}\right)$
(35) $\displaystyle\mathrm{s.t.}\quad\text{C3, C7, C8, C9},$

where
$\widetilde{F}\left({\mathbf{W}_{{k_{\mathrm{D}}}}},\mathbf{V},{p_{{k_{\mathrm{U}}}}}\right)$
is defined as
$\widetilde{F}\left({\mathbf{W}_{{k_{\mathrm{D}}}}},\mathbf{V},{p_{{k_{\mathrm{U}}}}}\right)\triangleq
f-\widetilde{{g}}\left({{\mathbf{W}_{{k_{\mathrm{D}}}}},\mathbf{V},{p_{{k_{\mathrm{U}}}}}\left|{\mathbf{W}_{{k_{\mathrm{D}}}}^{\left(m\right)},{\mathbf{V}^{\left(m\right)}},p_{{k_{\mathrm{U}}}}^{\left(m\right)}}\right.}\right)$.
Note that the persistent non-convexity of problem (35) stems from the rank-one
constraint C9. Thus, the semidefinite relaxation (SDR) is adopted to relax
constraint C9 by removing it. Then, the relaxed version of problem (35) can be
optimally solved with the aid of standard convex solvers such as CVX. Besides,
the tightness of the rank relaxation is verified in the following theorem.

###### Theorem 1

If $P_{\max}^{\mathrm{D}}>0$, the optimal beamforming matrices
$\mathbf{W}_{k_{\mathrm{D}}}$ and $\mathbf{V}$, which satisfy
$\mathrm{Rank}\left\\{\mathbf{W}_{k_{\mathrm{D}}}\right\\}\leq 1$ and
$\mathrm{Rank}\left\\{\mathbf{V}\right\\}\leq 1$, can always be obtained.

###### Proof:

Please refer to [32, Appendix A]. ∎

Then, the relaxed version of problem (35) is iteratively solved until the
increase of
$\widetilde{F}\left({\mathbf{W}_{{k_{\mathrm{D}}}}},\mathbf{V},{p_{{k_{\mathrm{U}}}}}\right)$
is less than the predefined convergence threshold
${\varepsilon_{{\mathrm{SCA}}}}$ or the maximum number of iterations $M$ is
reached. Finally, the optimized UL powers $p_{k_{\mathrm{U}}}$ are outputted,
and the optimized beamformers $\mathbf{w}_{k_{\mathrm{D}}}$ and $\mathbf{v}$
are obtained by the eigenvalue decomposition.

###  III-C Sub-Problem 3: Optimize ${\mathbf{b}_{{k_{\mathrm{U}}}}}$ with
Given $\left\\{{\tilde{\mathbf{t}},\tilde{\mathbf{r}}}\right\\}$ and
$\left\\{{{\mathbf{w}_{{k_{\mathrm{D}}}}},\mathbf{v},{p_{{k_{\mathrm{U}}}}}}\right\\}$

With given $\left\\{{\tilde{\mathbf{t}},\tilde{\mathbf{r}}}\right\\}$ and
$\left\\{{{\mathbf{w}_{{k_{\mathrm{D}}}}},\mathbf{v},{p_{{k_{\mathrm{U}}}}}}\right\\}$,
maximizing the receive SINR ${\gamma_{k_{\mathrm{U}}}^{\mathrm{U}}}$ of each
UL user ${k_{\mathrm{U}}}$ with beamformer ${\mathbf{b}_{{k_{\mathrm{U}}}}}$
yields the maximization of the SSR. Specifically, let
$\mathbf{w}=\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}{{\mathbf{w}_{{k_{\mathrm{D}}}}}}\in{\mathbb{C}^{{N_{\mathrm{t}}}\times
1}}$ and
$\mathbf{W}=\mathbf{ww}^{H}\in\mathbb{C}^{{N_{\mathrm{t}}}\times{N_{\mathrm{t}}}}$,
we can obtain the optimal receive beamformer ${\mathbf{b}_{{k_{\mathrm{U}}}}}$
by solving the following optimization problem.

$\displaystyle\mathop{\mathrm{maximize}}\limits_{\mathbf{b}_{k_{\mathrm{U}}}}\quad\frac{\mathbf{b}_{k_{\mathrm{U}}}^{H}{\mathbf{h}_{\mathrm{UB},k_{\mathrm{U}}}}\left({\tilde{\mathbf{r}}}\right)\mathbf{h}_{\mathrm{UB},{k_{\mathrm{U}}}}^{H}\left({\tilde{\mathbf{r}}}\right){\mathbf{b}_{k_{\mathrm{U}}}}}{\mathbf{b}_{k_{\mathrm{U}}}^{H}{\mathbf{A}_{k_{\mathrm{U}}}}{\mathbf{b}_{k_{\mathrm{U}}}}}$
(36) $\displaystyle\mathrm{s.t.}\quad\text{C1},$

where
${\mathbf{A}_{k_{\mathrm{U}}}}\in\mathbb{C}^{{N_{\mathrm{r}}}\times{N_{\mathrm{r}}}}$
is defined as follows.

$\displaystyle\mathbf{A}_{k_{\mathrm{U}}}$
$\displaystyle=\sum\limits_{i\in{\mathcal{K}_{\mathrm{U}}}\backslash\left\\{{k_{\mathrm{U}}}\right\\}}{\mathbf{h}_{\mathrm{UB},i}\left({\tilde{\mathbf{r}}}\right)\mathbf{h}_{\mathrm{UB},i}^{H}\left({\tilde{\mathbf{r}}}\right){p_{i}}}$
$\displaystyle+\rho\left({{\mathbf{H}_{\mathrm{SI}}}\left({\tilde{\mathbf{t}},\tilde{\mathbf{r}}}\right)\left({\mathbf{W}+\mathbf{V}}\right)\mathbf{H}_{\mathrm{SI}}^{H}\left({\tilde{\mathbf{t}},\tilde{\mathbf{r}}}\right)}\right)+\sigma_{\mathrm{U}}^{2}{\mathbf{I}_{{N_{\mathrm{r}}}}}.$
(37)

The optimal solution of problem (36) is given by [32]

$\mathbf{b}_{k_{\mathrm{U}}}=\frac{{{\mathbf{A}_{k_{\mathrm{U}}}^{-1}}{\mathbf{h}_{\mathrm{UB},{k_{\mathrm{U}}}}}\left({\tilde{\mathbf{r}}}\right)}}{{{{\left\|{{\mathbf{A}_{k_{\mathrm{U}}}^{-1}}{\mathbf{h}_{\mathrm{UB},{k_{\mathrm{U}}}}}\left({\tilde{\mathbf{r}}}\right)}\right\|}_{2}}}}.$
(38)

After obtaining the solutions of three sub-problems, the proposed AO algorithm
iteratively solves the three sub-problems until the increase of
$R_{\mathrm{SSR}}$ is less than the threshold $\varepsilon_{{\mathrm{AO}}}$ or
the maximum number of iterations for AO is reached.

###  III-D Convergence and Complexity Analysis

The convergence and computational complexity of the overall AO algorithm are
analyzed as follows. Define the index and maximum number of iterations for AO
as $c$ and $C$, respectively, where $1\leq c\leq C$. The convergence is
ensured by the following inequality

$\displaystyle
R_{\mathrm{SSR}}\left(\tilde{\mathbf{t}}^{\left(c\right)},\tilde{\mathbf{r}}^{\left(c\right)},{\mathbf{w}_{{k_{\mathrm{D}}}}^{\left(c\right)}},\mathbf{v}^{\left(c\right)},{p_{{k_{\mathrm{U}}}}^{\left(c\right)}},{\mathbf{b}_{{k_{\mathrm{U}}}}^{\left(c\right)}}\right)$
$\displaystyle\mathop{\geq}\limits^{\left({{\alpha_{5}}}\right)}$
$\displaystyle
R_{\mathrm{SSR}}\left(\tilde{\mathbf{t}}^{\left(c\right)},\tilde{\mathbf{r}}^{\left(c\right)},{\mathbf{w}_{{k_{\mathrm{D}}}}^{\left(c\right)}},\mathbf{v}^{\left(c\right)},{p_{{k_{\mathrm{U}}}}^{\left(c\right)}},{\mathbf{b}_{{k_{\mathrm{U}}}}^{\left(c-1\right)}}\right)$
$\displaystyle\mathop{\geq}\limits^{\left({{\alpha_{6}}}\right)}$
$\displaystyle
R_{\mathrm{SSR}}\left(\tilde{\mathbf{t}}^{\left(c\right)},\tilde{\mathbf{r}}^{\left(c\right)},{\mathbf{w}_{{k_{\mathrm{D}}}}^{\left(c-1\right)}},\mathbf{v}^{\left(c-1\right)},{p_{{k_{\mathrm{U}}}}^{\left(c-1\right)}},{\mathbf{b}_{{k_{\mathrm{U}}}}^{\left(c-1\right)}}\right)$
$\displaystyle\mathop{\geq}\limits^{\left({{\alpha_{7}}}\right)}$
$\displaystyle
R_{\mathrm{SSR}}\left(\tilde{\mathbf{t}}^{\left(c-1\right)},\tilde{\mathbf{r}}^{\left(c-1\right)},{\mathbf{w}_{{k_{\mathrm{D}}}}^{\left(c-1\right)}},\mathbf{v}^{\left(c-1\right)},{p_{{k_{\mathrm{U}}}}^{\left(c-1\right)}},{\mathbf{b}_{{k_{\mathrm{U}}}}^{\left(c-1\right)}}\right),$

where the inequality marked by $\left({{\alpha_{5}}}\right)$ holds because
${\mathbf{b}_{{k_{\mathrm{U}}}}^{\left(c\right)}}$ is the optimal receive
beamformer for maximizing the SINR of each UL user. The inequality marked by
$\left({{\alpha_{6}}}\right)$ holds because
${\mathbf{w}_{{k_{\mathrm{D}}}}^{\left(c\right)}}$,
$\mathbf{v}^{\left(c\right)}$, and ${p_{{k_{\mathrm{U}}}}^{\left(c\right)}}$
are the optimized transmit and AN beamformers and UL powers by the SCA. The
inequality marked by $\left({{\alpha_{7}}}\right)$ holds because
$\tilde{\mathbf{t}}^{\left(c\right)}$ and
$\tilde{\mathbf{r}}^{\left(c\right)}$ are the optimized positions of MAs
searched by the proposed MVPSO. As a result, the SSR is non-decreasing during
the iterations in the AO algorithm. Meanwhile, due to the finite communication
resources, the SSR is always bounded. As such, the convergence of the overall
algorithm is guaranteed. Moreover, the convergences are verified by the
simulations in Section IV.

The computational complexity of the AO mainly arises from the search process
of the MVPSO algorithm, the iterations of the AO and SCA algorithms, and the
calculation of the receive beamformer. For the MVPSO in Algorithm 1, the
complexities of calculating
${R_{\mathrm{SSR}}}\left({\mathbf{u}_{n}^{\left(q\right)}}\right)$ and
$\mathcal{P}\left({\mathbf{u}_{n}^{\left(q\right)}}\right)$ in (24) are
${o_{1}}\triangleq\mathcal{O}\left({N_{\mathrm{t}}}\left({L_{\mathrm{SI}}^{\mathrm{t}}{N_{\mathrm{r}}}+\sum\limits_{{k_{\mathrm{D}}}\in{\mathcal{K}_{\mathrm{D}}}}{L_{\mathrm{BD},{k_{\mathrm{D}}}}^{\mathrm{t}}}+\sum\limits_{{k_{\mathrm{E}}}\in{\mathcal{K}_{\mathrm{E}}}}{L_{\mathrm{BE},{k_{\mathrm{E}}}}^{\mathrm{t}}}}\right)\right.$
$\left.+{N_{\mathrm{r}}}\left({L_{\mathrm{SI}}^{\mathrm{t}}L_{\mathrm{SI}}^{\mathrm{r}}+\sum\limits_{{k_{\mathrm{U}}}\in{\mathcal{K}_{\mathrm{U}}}}{L_{\mathrm{UB},{k_{\mathrm{U}}}}^{\mathrm{r}}}}\right)\right)$
and ${o_{2}}\triangleq\mathcal{O}\left({{N_{\mathrm{t}}}\choose
2}+{{N_{\mathrm{r}}}\choose 2}\right)$, respectively. Thus, the total
complexity of Algorithm 1 is
$\mathcal{O}\left(QN\bar{I}\left(o_{1}+o_{2}\right)\right)$. For the SCA
algorithm, let $M_{\mathrm{SCA}}$ denote the number of iterations, the
complexity is
$\mathcal{O}\left(M_{\mathrm{SCA}}\left(\left(K_{\mathrm{D}}+1\right)N_{\mathrm{t}}^{3.5}+K_{\mathrm{U}}^{3.5}\right)\right)$
due to solving the SDR problem iteratively [32]. For calculating the receive
beamformer, the complexity is
$\mathcal{O}\left(K_{\mathrm{U}}N_{\mathrm{r}}^{3}\right)$ due to the matrix
inversion in (38). Based on the above analyses, the computational complexity
of the overall algorithm is
$\mathcal{O}\left(C_{\mathrm{AO}}\left(QN\bar{I}\left(o_{1}+o_{2}\right)+M_{\mathrm{SCA}}\left(\left(K_{\mathrm{D}}+1\right)N_{\mathrm{t}}^{3.5}+K_{\mathrm{U}}^{3.5}\right)\right.\right.$
$\left.\left.+K_{\mathrm{U}}N_{\mathrm{r}}^{3}\right)\right)$, where
$C_{\mathrm{AO}}$ is the number of iterations for AO.

##  IV Simulation Results

In this section, we present the simulation results to evaluate the performance
of the proposed system.

###  IV-A Simulation Setup and Benchmark Schemes

TABLE I: Simulation Parameters Parameter | Description | Value
---|---|---
$N$ | Number of antennas | 6
$A\times A$ | Moving region size | $4\lambda\times 4\lambda$
$D$ | Minimum inter-MA distance | $\lambda/2$
$L$ | Number of channel paths | $6$
$\rho$ | SI loss coefficient | -90dB
$\rho_{0}$ | Path loss at the reference distance | -40dB
$\alpha$ | Path loss exponent | 2.8
$\sigma_{\mathrm{U}}^{2}$, $\sigma_{\mathrm{D},k_{\mathrm{D}}}^{2}$, $\sigma_{\mathrm{E},k_{\mathrm{E}}}^{2}$ | Average noise powers | -90dBm
$P^{\mathrm{D}}_{\mathrm{max}}$ | Maximum DL transmit power | 40dBm
$P^{\mathrm{U}}_{\mathrm{max}}$ | Maximum UL transmit power | 10dBm
$K_{\mathrm{D}}$, $K_{\mathrm{U}}$, $K_{\mathrm{E}}$ | Numbers of users/Eves | 4
$N$ | Number of particles | 100
$Q$, $M$, $C$ | Maximum numbers of iterations | 100
$\tau_{\mathrm{t}}$, $\tau_{\mathrm{r}}$ | Penalty factors | 10
$c_{1}$, $c_{2}$ | Personal and global learning factors | 1.4
$\varepsilon_{\mathrm{SCA}}$, $\varepsilon_{\mathrm{AO}}$ | Convergence thresholds | $10^{-3}$

In the simulation, the UL users, DL users, and Eves are randomly and uniformly
distributed in a cell centered on the FD BS with a radius of 600 meters (m).
The FD BS is equipped with the same number of transmit and receive antennas,
i.e., $N_{\mathrm{t}}=N_{\mathrm{r}}\triangleq N$. We adopt the geometry
channel model [9, 12], where the numbers of all transmit and receive paths are
identical, i.e.,
$L_{\mathrm{SI}}^{\mathrm{t}}=L_{\mathrm{SI}}^{\mathrm{r}}=L_{\mathrm{BD},k_{\mathrm{D}}}^{\mathrm{t}}=L_{\mathrm{BE},k_{\mathrm{E}}}^{\mathrm{t}}=L_{\mathrm{UB},k_{\mathrm{U}}}^{\mathrm{r}}\triangleq
L$. In this way, the PRM $\mathbf{\Sigma}$ of the SI channel is a diagonal
matrix, where each diagonal element follows the CSCG distribution
$\mathcal{CN}\left(0,\frac{\rho}{L}\right)$. For UL and DL channels, each
element in PRVs follows the CSCG distribution
$\mathcal{CN}\left(0,\frac{\rho_{0}d_{k}^{-\alpha}}{L}\right)$, where
$\rho_{0}$ represents the path loss at the reference distance of 1 m, $\alpha$
is the path loss exponent, and $d_{k}$ denotes the propagation distance from
the BS to DL user/Eve $k$ or from UL user $k$ to the BS. The elevation and
azimuth AoDs and AoAs are assumed to be the independent and identically
distributed random variables within the interval $\left[0,\pi\right]$. Unless
specified otherwise, the default simulation parameters are set as shown in
Table I.

In this section, the proposed scheme is termed as “MA-MVPSO-FD”. For the MVPSO
algorithm, we set $\bar{I}=3$ candidate velocities, which are derived from the
weighted combinations of $\bar{J}=3$ velocity components. Specifically, the
inertia weights for these three velocity components are respectively set as
$\omega_{1}=0.9$, $\omega_{2}=0.75$, and $\omega_{3}=0.4$ to balance both the
exploration and exploitation of the particles [31]. Moreover, the combination
weight vectors for all particles in (28) are set as
$\mathbf{c}_{n,1}=\left[1,0,0\right]^{T}$,
$\mathbf{c}_{n,2}=\left[0,1,0\right]^{T}$, and
$\mathbf{c}_{n,3}=\left[0,0,1\right]^{T}$. Besides, to fully demonstrate the
advantages of the proposed scheme, we define the following benchmark schemes.

  * 1)

FPA-FD: The BS is equipped with the transmit and receive FPA-based uniform
planar arrays (UPAs) with $N_{\mathrm{t}}$ and $N_{\mathrm{r}}$ antennas
spaced by $\frac{\lambda}{2}$, respectively.

  * 2)

AS-FD: The BS is equipped with the transmit and receive FPA-based UPAs with
$2N_{\mathrm{t}}$ and $2N_{\mathrm{r}}$ antennas spaced by
$\frac{\lambda}{2}$, respectively. In a single AO iteration, $N_{\mathrm{t}}$
transmit antennas and $N_{\mathrm{r}}$ receive antennas are selected via
exhaustive searches to maximize the SSR.

  * 3)

MA-APO-FD: The transmit and receive regions are discretized into multiple
grids of size $\frac{\lambda}{10}\times\frac{\lambda}{10}$. The MA positions
are determined by the APO method, i.e., with other MAs fixed, the optimal
position of the current MA, which meets constraints C5 and C6, is identified
via exhaustive searches.

  * 4)

MA-MVPSO-FD-ZF: This scheme employs the zero-forcing (ZF) receive beamformer
instead of (38).

  * 5)

MA-MVPSO-FD-NoAN: This scheme is similar to the proposed scheme, but omits the
AN.

  * 6)

MA-PSO-FD: This scheme optimizes the MA positions by the standard PSO
algorithm.

  * 7)

MA-MVPSO-HD: This scheme configures the BS to operate in the time-division HD
mode. Thus, the SSR is penalized due to the half communication time compared
to FD mode.

###  IV-B Convergence Evaluations of Proposed Algorithms

(a)

(b)

Figure 3: Convergence evaluations of the proposed (a) MVPSO and (b) AO
algorithms.

In Fig. 3a, the convergence of the proposed MVPSO is evaluated with different
moving region sizes and path numbers. As can be observed, the three SSRs
increase with the number of iterations and tend towards stable values within
100 iterations, validating the convergence performance. Additionally, to
verify the effectiveness of the proposed penalty function in (III-A1), we also
illustrate the variations of the penalty values with the number of iterations.
We can observe that the larger the moving region size, the fewer iterations
are needed for the penalty value to reach zero. The three penalty values
remain zero after 5 iterations, which ensures that the minimum inter-MA
distance constraints C5 and C6 are satisfied.

Besides, the convergence evaluation of the overall AO algorithm is shown in
Fig. 3b. With different moving region sizes, numbers of paths, and numbers of
MAs, the SSRs increase with the number of AO iterations and converge within 20
iterations, substantiating the previous discussions on the convergence of the
AO algorithm in Section III-D.

###  IV-C Channel Power Gains under MVPSO

(a)

(b)

(c)

(d)

Figure 4: Channel power gains (dB) (a) between the transmit MA of the BS and
the single FPA of the DL user, (b) between the transmit MA of the BS and the
single FPA of the Eve, (c) between the single FPA of the UL user and the
receive MA of the BS, and (d) between the transmit MA and the receive MA of
the BS.

To further investigate the impact of antenna position optimization via the
MVPSO on altering channel conditions, one realization of the channel power
gains (in dB) versus the MA position is illustrated in Fig. 4. For ease of
presentation, the numbers of transmit MA, receive MA, DL user, UL user, and
Eve are set as 1, i.e., $N=K_{\mathrm{D}}=K_{\mathrm{U}}=K_{\mathrm{E}}=1$. It
is shown that due to the prominent small-scale fading in the spatial domain,
for the DL and UL channels, each user and Eve has its unique channel gain map
in the transmit or receive regions (see Figs. 4a-4c). For the SI channel,
there are also corresponding channel gain maps in the transmit and receive
regions before and after antenna position optimization (see Fig. 4d). As can
be observed, for transmit MA, its position is initialized at the origin of the
transmit region, with the channel power gains for the DL user, Eve, and the
receive MA being -120.58 dB, -112.70 dB, and -100.17 dB, respectively. By
employing the proposed MVPSO, the channel power gain of the optimized transmit
MA’s position for the DL user increases to -114.65 dB, while the channel power
gains for the Eve and the receive MA decrease to -129.50 dB and -129.73 dB,
respectively. Besides, for the receive MA, the optimized position, compared to
the initial position, results in an increase of 1.77 dB in channel power gain
for the UL user and a decrease of 24.38 dB in channel power gain for the
transmit MA. These results indicate that even small movements of MAs can lead
to significant changes in channel responses. Furthermore, antenna position
optimization via the MVPSO comprehensively balances maximizing the channel
power gains for the UL and DL users while minimizing those for the Eve and SI.

###  IV-D Performance Comparison with Benchmark Schemes

In this sub-section, we compare the performance of the proposed scheme with
benchmark schemes.

(a)

(b)

(c)

Figure 5: Sum of secrecy rates versus (a) normalized moving region size, (b)
number of antennas, and (c) number of Eves

Fig. 5 illustrates the variations in the SSRs for different schemes. It can be
observed from Fig. 5a that, the SSRs of all MA-based schemes increase with the
normalized moving region size and gradually converge to stable values. This is
because a larger moving region allows the MA to further explore spatial DoFs,
but the limited number of paths restricts the diversity gains from increasing
indefinitely. The fixed antenna configurations in the AS and FPA schemes
render their performances unaffected by the moving region size, and the AS
scheme can leverage spatial DoFs to some extent to improve the SSR compared to
the FPA scheme. Furthermore, the proposed MVPSO outperforms the standard PSO,
which in turn surpasses the conventional APO scheme. As the moving region size
increases, the performance gap between the MVPSO and the PSO widens, while the
gap between the PSO and the APO narrows. This is because the APO scheme
maintains the fixed positions for other MAs while optimizing the current MA
position, thereby disregarding the interdependencies among the MA positions.
The standard PSO overcomes this limitation by optimizing all MA positions
concurrently to prevent undesired sub-optimal solutions. However, its
inability to balance exploitation and exploration hampers its effectiveness in
discovering the optimal MA positions for the larger moving regions. Therefore,
the MVPSO scheme stands out for its capability to simultaneously explore and
exploit throughout the entire iteration process. Besides, the noise
amplification caused by the ZF beamforming may deteriorate the SINRs of the UL
users. Thus, the ZF scheme results in an approximately 10% decrease in the SSR
compared to the proposed optimal receive beamformer. Moreover, the results
demonstrate that the AN and the FD mode effectively enhance the system’s PLS
performance. Unlike the HD systems, in FD mode, the transmission of AN can
simultaneously safeguard both UL and DL users, and the reuse of spectrum
further augments the SSR.

In Fig. 5b, the SSRs of different schemes are compared versus the number of
antennas. The APO and AS schemes are omitted when $N>8$ due to high
computational complexities. For example, considering $N=10$ MAs in the
$4\lambda\times 4\lambda$ transmit and receive regions, the APO scheme
necessitates calculating $40\times 40\times 20=32000$ channel responses in a
single AO iteration, and the number of total selections for the AS scheme is
${40\choose 20}>10^{11}$, which are hard to undertake. As the number of
antennas increases, all schemes exhibit increases in the SSRs due to the
enhanced spatial diversity gain and beamforming gain. We can find that the SSR
growth in the FPA scheme is slower compared to the MA-based schemes.
Specifically, the increase in the number of antennas from 5 to 11 brings the
18.44% and 22.34% increases in the SSRs for the FPA and the proposed MA-based
scheme, respectively. This is because for the MA-based schemes, increasing the
number of antennas allows for better exploitation of spatial DoFs by extensive
movements within the designated regions. Consistently, the proposed scheme
maintains optimal performance with varying numbers of antennas. Besides, we
note that as the number of antennas increases, the APO scheme gradually
approaches the SSR level nearly equivalent to that of the PSO scheme. This
reveals that the standard PSO with a single velocity fails to fully search for
the optimal solutions when the dimensions of the optimization variables, i.e.,
the number of antennas, are large, thus underscoring the necessity and
effectiveness of the proposed MVPSO.

Considering the worst-case scenario of the cooperative eavesdropping by
multiple Eves who can completely cancel multi-user interference, the
relationships between the performances of different systems and the number of
Eves are depicted in Fig. 5c. Intuitively, the SSRs of all schemes decrease as
the number of Eves increases. The reasons are as follows. With more Eves, the
SINRs of cooperative UL and DL eavesdropping (as seen in the summation terms
of (18) and (19)) increase. This coordinated effort undermines the systems’
abilities to maintain secure communications, thereby diminishing the SSRs of
all schemes. Fortunately, the proposed scheme still surpasses other benchmark
schemes. In addition, we note that the SSR of the NoAN scheme experiences a
steep decline with an increasing number of Eves, approaching the performance
of the HD scheme even at $K_{\mathrm{E}}=8$. This demonstrates that due to the
reuse of time-frequency resources in FD systems, the AN with effective
beamforming can interfere with Eves’ receptions while simultaneously
protecting the UL and DL transmissions. Thus, AN is indispensable for secure
FD transmission.

(a)

(b)

Figure 6: Sums of (a) UL and (b) DL secrecy rates versus SI loss coefficient.

Subsequently, we investigate the impact of the SI loss coefficient on system
performance. Figs. 6a and 6b respectively present the sums of UL and DL
secrecy rates versus SI loss coefficient without the HD scheme. For the SI
loss coefficient $\rho$, a smaller value in dB indicates a more powerful
capability for SI cancellation. It can be observed that as $\rho$ decreases,
the sums of UL and DL secrecy rates of all FD-based schemes increase and
decrease, respectively. This is because, with the improved SI cancellation,
the FD BS can reliably demodulate the UL users’ information. Thus, the user
scheduling policy allows for UL transmissions, resulting in an increased UL
secrecy rate. However, for DL users, the undesired co-channel interference
introduced by UL transmissions decreases their received SINRs, thus reducing
the DL secrecy rate. Additionally, in case of weak SI cancellation, i.e.,
$-20\mathrm{dB}<\rho<-40\mathrm{dB}$, we observe the dominance of DL
transmissions due to the severe SI affecting the receptions of UL signals.
Consequently, without UL transmissions, the performance of the ZF scheme
closely mirrors that of the proposed scheme. On the contrary, when SI
cancellation is robust, i.e., $\rho<-80\mathrm{dB}$, both UL and DL secrecy
rates approach saturation. This is primarily because the power of the residual
SI becomes negligible compared to other interference perceived at the BS,
rendering further improvements in SI cancellation ineffective in yielding
significant gains. Furthermore, we notice that the UL secrecy rate of the APO
scheme slightly surpasses that of the proposed scheme, but this improvement
comes at the sacrifice of the DL secrecy rate. This indicates that, compared
to the APO scheme, the proposed scheme more effectively leverages the DL
transmission capability to maximize the sum of UL and DL secrecy rates.

(a)

(b)

(c)

Figure 7: Impact of the (a) PRM, (b) AoD, and (c) AoA errors on the
performances of the proposed schemes

In the above discussions, we assume that the BS possesses the perfect
knowledge of the FRIs, i.e., the PRMs of UL, DL, and SI channels, the AoDs of
DL and SI channels, and the AoAs of UL and SI channels. Due to noise, limited
training overhead, and multiple Eves, obtaining perfect FRIs is challenging in
practice. Therefore, Fig. 7 separately evaluates the impact of PRM, AoD, and
AoA errors on system performances. The descriptions of the FRI errors
reference [13, 14, 20]. In each sub-figure, the remaining two types of FRIs
are assumed to be perfectly known. To focus on the impact of imperfect FRIs on
MA position optimization, the MA positions are determined by the estimated
FRIs, while the calculations of the beamformers and UL powers are based on the
actual channel responses [14]. We neglect the FPA scheme because it does not
involve antenna position optimization. From Fig. 7, it can be observed that
the SSRs decrease with the normalized variance of PRM error and maximum AoD
and AoA errors. Specifically, the proposed scheme experiences the performance
losses of 14.65%, 17.93%, and 0.61% due to PRM, AoD, and AoA errors,
respectively. Compared to PRM and AoD errors, AoA error has a less detrimental
effect on performances. The reasons are as follows. Because the AoA error only
affects the calculations of UL and SI channels, the low UL powers make the
deviations between the estimated and actual AoAs have an insignificant impact
on UL transmissions. Thus, the MA position optimization based on the estimated
AoA can still maintain the excellent performance. Besides, we find that the
APO scheme is less sensitive to PRM and AoD errors compared to the proposed
scheme. This is because the APO scheme discretizes the moving regions into
multiple grids, where the center of each grid serves as a candidate position.
This discretization restricts MAs to discerning channel variations solely
between different grids. In addition, the small PRM and AoD errors do not
significantly impact the relative channel gains between different grids.
Ultimately, the PRM and AoD errors have a minor impact on the APO scheme, yet
the discretized movements fail to fully utilize spatial DoFs. Hence, in
practical applications, it is necessary to strike a balance between fully
exploiting spatial DoFs and mitigating the sensitivity to imperfect FRIs.

##  V conclusion

In this paper, we proposed a new paradigm for secure FD multi-user systems,
where an FD BS equipped with MAs simultaneously serves multiple UL and DL
users while protecting their private information against multiple cooperative
Eves. We formulated an optimization problem to maximize the SSR by jointly
optimizing the MA positions, the transmit, receive, and AN beamformers at the
BS, and the UL powers. To tackle this non-convex problem, an AO algorithm was
proposed to decompose the original problem into three sub-problems and
iteratively solve them. Specifically, the MA positions are updated by the
proposed MVPSO, where multiple candidate velocities are applied to maximize
the SSR. Additionally, the transmit and AN beamformers, along with UL powers,
are optimized via the SCA algorithm. Moreover, the optimal receive beamformer
is derived as a closed-form solution. Simulation results showed that compared
to the conventional FPAs, the flexible movements of MAs can fully utilize the
spatial DoFs to benefit the channel gains of legitimate users and reduce the
channel gains of SI and Eves simultaneously. Besides, the results revealed the
superiority of the proposed MVPSO in determining the MA positions. In
addition, the AN emitted by the FD BS provides protections for both UL and DL
users, a capability that cannot be achieved with an HD BS. Moreover, the
impact of the discrepancies between the estimated and actual FRIs on system
performances was evaluated, which provides valuable references for the
applications of the proposed scheme in practice.

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