# Movable Antennas-Assisted Secure Transmission Without Eavesdroppers’
Instantaneous CSI

Guojie Hu, Qingqing Wu, Senior Member, IEEE, Donghui Xu, Kui Xu, Member, IEEE,
Jiangbo Si, Senior Member, IEEE, Yunlong Cai, Senior Member, IEEE and Naofal
Al-Dhahir, Fellow, IEEE Guojie Hu and Donghui Xu are with the College of
Communication Engineering, Rocket Force University of Engineering, Xi’an
710025, China (email<EMAIL_ADDRESS>xdh_45@163.com). Qingqing Wu is with
the Department of Electronic Engineering, Shanghai Jiao Tong University,
Shanghai 200240, China (qingqingwu@sjtu.edu.cn). Kui Xu is with the College of
Communications Engineering, Army Engineering University of PLA, Nanjing
210007, China (lgdxxukui@sina.com). Jiangbo Si is with the Integrated Service
Networks Lab of Xidian University, Xi’an 710100, China (jbsi@xidian.edu.cn).
Yunlong Cai is with the College of Information Science and Electronic
Engineering, Zhejiang University, Hangzhou 310027, China (email:
ylcai@zju.edu.cn). Naofal Al-Dhahir is with the Department of Electrical and
Computer Engineering, The University of Texas at Dallas, Richardson, TX 75080
USA (aldhahir@utdallas.edu).

###### Abstract

Movable antenna (MA) technology is highly promising for improving
communication performance, due to its advantage of flexibly adjusting
positions of antennas to reconfigure channel conditions. In this paper, we
investigate MAs-assisted secure transmission under a legitimate transmitter
Alice, a legitimate receiver Bob and multiple eavesdroppers. Specifically, we
consider a practical scenario where Alice has no any knowledge about the
instantaneous non-line-of-sight component of the wiretap channel. Under this
setup, we evaluate the secrecy performance by adopting the secrecy outage
probability metric, the tight approximation of which is first derived by
interpreting the Rician fading as a special case of Nakagami fading and
concurrently exploiting the Laguerre series approximation. Then, we minimize
the secrecy outage probability by jointly optimizing the transmit beamforming
and positions of antennas at Alice. However, the problem is highly non-convex
because the objective includes the complex incomplete gamma function. To
tackle this challenge, we, for the first time, effectively approximate the
inverse of the incomplete gamma function as a simple linear model. Based on
this approximation, we arrive at a simplified problem with a clear structure,
which can be solved via the developed alternating projected gradient ascent
(APGA) algorithm. Considering the high complexity of the APGA, we further
design another scheme where the zero-forcing based beamforming is adopted by
Alice, and then we transform the problem into minimizing a simple function
which is only related to positions of antennas at Alice. Such problem is well-
solved via another projected gradient descent algorithm developed with a lower
complexity. As demonstrated by simulations, our proposed schemes achieve
significant performance gains compared to conventional schemes based on fixed-
position antennas.

###### Index Terms:

Movable antennas, secrecy outage probability, transmit beamforming, Laguerre
series approximation, projected gradient ascent/descent.

##  I Introduction

With the explosive growth of data traffic in 5G/B5G and future 6G, there will
be a higher risk that a large amount of private data can be overheard by
malicious eavesdroppers [1]. Physical-layer security (PLS), by exploiting the
inherent properties of wireless channels (for instance, random fading and
noise), is expected to realize substantial and fundamental secrecy for
wireless communication, regardless of how powerful the eavesdroppers’
computing ability is [2].

To improve the secrecy performance of PLS, numerous technologies have been
proposed, such as secure beamforming [3], cooperative jamming [4], artifical
noise [5] and channel-based secret key [6], among which secure beamforming
always plays an important role because it effectively improves/reduces the
receiving power gains at the legitimate receiver/malicious eavesdroppers by
carefully designing antenna weights.

Conventional secure beamforming schemes all rely on fixed-position antennas
(FPAs), where positions of antennas at the transmitter/receiver cannot be
changed [7]. This limitation of FPAs will result in a fundamental deficiency,
i.e., when the direction angles of the eavesdroppers are close to that of the
legitimate receiver, the correlation between the main channel and the wiretap
channel will become much higher. Then, beamforming with FPAs cannot
effectively distinguish between the legitimate receiver and the eavesdroppers,
and thus their signal receiving quality will be similar, leading to a poor
secrecy performance [8].

Recently, researchers have proposed a novel technology of movable antennas
(MAs) [9] or fluid antennas [10, 11, 12]. Specifically, employing MAs,
antennas (at the transmitter or the receiver) are connected to radio frequency
(RF) chains via flexible cables, and positions of antennas can be flexibly
adjusted in the specified region supported by the stepper motors or servos.
This feature of MAs allows for real-time reconfiguration of wireless channels
and thus brings an additional spatial degree of freedom (DoF). Therefore,
using MAs, even when the direction angles of the eavesdroppers and the
legitimate receiver are similar, their channel correlation can be proactively
reduced by optimizing antennas’ positions and then beamforming can unlock its
true advantages [13, 14]. Further, compared to other schemes such as i)
antenna selection [15], which requires more candidate antennas, RF chains,
expensive hardware cost and complex channel estimation, and ii) rotatable
uniform linear array [16], which only mechanically rotates the
transmit/receive array and cannot exploit spatial channel variation, MAs fully
utilize the channel variation in the continuous spatial region to achieve a
higher spatial diversity without no excessive demand for additional hardware
costs or algorithm overheads.

Motivated by these potential merits, early works have considered integrating
MAs into various system setups [9, 17, 14, 18, 19, 20, 21, 22]. For instance,
[9] and [17] first provided an overview of the promising applications and
hardware designs for MAs-aided communications, and concurrently the advantages
of employing MAs for signal power improvement, interference mitigation,
flexible beamforming and spatial multiplexing. Subsequently, [19] proposed the
field-response-based channel model, and provided theoretical analysis for the
maximum channel gain achieved by a single MA in both deterministic and
stochastic channels. In [20, 21, 22], compressed sensing, low-sample-size and
successive Bayesian reconstructor based channel estimations for MAs-aided
systems were presented. In [23, 24, 25], MAs-aided multiple-input multiple-
output (MIMO) system was investigated, where in addition to the covariance of
the transmit signals, positions of MAs at the transmitter/receiver are also
carefully optimized to enhance the capacity. In [26, 27, 28, 29], MAs-enabled
multiuser uplink/downlink was studied, which demonstrated that the total
transmit power (sum rate) of all users can be further reduced (improved) by
additionally optimizing positions of MA(s) at each user or the base station.
Recently, [30, 31] further proposed a novel six-dimensional movable antenna
(6DMA) architecture, where 3D positions and 3D rotations of antenna surfaces
can be concurrently optimized to fully exploit the spatial DoF for achieving
the greatest flexibility and then the highest capacity improvement. Besides,
MAs-enabled wideband communications, interference networks, symbiotic radio
communications, coordinated multi-point (CoMP) reception systems and over-the-
air computation were further considered in [32, 33, 34, 35, 36]. In addition
to these works, exploiting MAs for enhancing the secrecy performance has also
been studied in [37, 38, 39]. In detail, under the setups of multiple-input
single-output single/multiple eavesdropper(s) (MISOSE/MISOME), [37, 38]
considered the problem of maximizing the secrecy rate and minimizing the power
consumption via the optimizations of MAs’ positions and beamforming at the
transmitter. Under the setup of MIMOME, [39] intended to maximize the secrecy
rate by jointly optimizing the transmit precoding matrix, the artificial noise
covariance matrix and MAs’ positions.

In this paper, we study MAs-assisted secure transmission under the setup of
MISOME, where the private information is transmitted from the MAs-enabled
Alice to the single-antenna Bob, in the presence of multiple single-antenna
and colluding eavesdroppers. Unlike previously related works [37, 38, 39], we
consider the general Rician fading for the wiretap channel, for which Alice
does not know its instantaneous non-line-of-sight (NLoS) component and just
masters its statistical LoS component via channel environment measures, global
positioning systems and so on [40, 41, 42, 43]. Due to the randomness of the
NLoS component of the wiretap channel, the instantaneous secrecy rate cannot
be obtained anymore. Therefore, unlike [37, 38, 39], we adopt the secrecy
outage probability as the secrecy performance evaluation metric and aim to
reveal the advantages of MAs to improve the secrecy performance, even with
relatively statistical channel state information. Specifically, the main
contributions of this paper are summarized as follows.

  * •

We consider a pessimistic eavesdropping scenario where multiple eavesdroppers
aim to jointly decode the private information of Alice. This behavior makes it
very hard to derive the exact secrecy outage probability expression. To
overcome this challenge, we first regard the Rician fading as a special case
of Nakagami fading and concurrently exploit the Laguerre series approximation
to obtain a tight and closed-form secrecy outage probability, based on which
numerous insights are gained and discussed.

  * •

We minimize the secrecy outage probability by jointly optimizing the transmit
beamforming and positions of antennas at Alice. However, the problem is highly
non-convex since the objective includes the complex incomplete gamma function,
and more importantly, the shape and scale parameters of which all involve the
optimization variables. Facing this challenge, we introduce the slack variable
and place the original objective in the constraint. Afterwards, we propose a
novel linear model to effectively approximate the inverse of the incomplete
gamma function, based on which the problem can be successfully transformed
into a simpler one with a clear structure.

  * •

The simplified problem, although more concise, is still highly non-convex.
Addressing this challenge, we develop a general alternating projected gradient
ascent (APGA) algorithm to iteratively optimize the transmit beamforming and
positions of antennas at Alice to obtain a stationary solution. Furthermore,
considering the higher complexity of the APGA, we also propose a sub-optimal
scheme where the zero-forcing (ZF)-based transmit beamforming is adopted at
Alice to completely eliminate the averaged receiving power at eavesdroppers
via the LoS paths, according to which the problem can be simplified into
minimizing a simple function which is only related to positions of antennas at
Alice. Then, another projected gradient descent (PGD) algorithm is further
designed to effectively solve the problem with a lower complexity.

  * •

We conduct numerical simulations and provide several competitive benchmarks to
show the effectiveness of the proposed schemes. We reveal that, i) compared to
the conventional FPAs, MAs with careful optimizations of antennas’ positions
can achieve a significant performance gain by exploiting the additional
spatial DoF; ii) the ZF-based transmit beamforming works well when the
difference in direction angles between the main channel and the wiretap
channel is higher; iii) the secrecy outage probability decreases and then
converges to a constant with respect to (w.r.t.) the length of antennas’
movement region, indicating that only a limited span is enough to achieve the
optimal performance.

The rest of this paper is organized as follows. The system model and problem
formulation are presented in Sections II. The secrecy outage probability
analysis and some insights are detailed in Section III. The proposed novel
optimization framework for minimizing the secrecy outage probability is
provided in Section IV, and another competitive benchmark based on the ZF-
based transmit beamforming is given in Section V. The simulations are
presented in Section VI and the conclusions are shown in Section VII.

Figure 1: System model.

##  II System Model and Problem Formulation

###  II-A System Model

As illustrated in Fig. 1, this paper considers a downlink secure transmission
system consisting of a transmitter (Alice), a legitimate receiver (Bob), and
$M\geq 2$ eavesdroppers, denoted as
$\left\\{{{\rm{Ev}}{{\rm{e}}_{i}}}\right\\}_{i=1}^{M}$. Bob and
$\left\\{{{\rm{Ev}}{{\rm{e}}_{i}}}\right\\}_{i=1}^{M}$ are equipped with a
single antenna each, while Alice is equipped with a linear MA array consisting
of $N\geq 2$ antennas.111Note that Alice can also adopt the planar array.
Nevertheless, all conclusions obtained in this paper are not affected by this
extension. Therefore, we consider a simple setting for a clear presentation.
Since the linear array is assumed, we denote
${\left[{{x_{1}},{x_{2}},...,{x_{N}}}\right]^{T}}\buildrel\Delta\over{=}{\bf{x}}$
as the one-dimensional positions of $N$ MAs relative to the reference point
zero, where ${\left(\cdot\right)^{T}}$ is the transpose operation. Without
loss of generality, we set $0\leq{x_{1}}<{x_{2}}<...<{x_{N}}\leq L$, with $L$
denoting the total span for the movement of MAs. Note that unlike conventional
FPAs, positions of these $N$ MAs, i.e.,
$\left\\{{{x_{i}}}\right\\}_{i=1}^{N}$, can be flexibly adjusted in the
specific region to proactively cater to effective beamforming designs, such
that the secrecy performance can be further improved. The details are shown
later.

The $1\times N$ main channel vector and the $i$-th wiretap channel vector from
Alice to Bob and from Alice to ${\rm{Eve}}_{i}$, are respectively denoted as
${{\bf{h}}_{0}}({\bf{x}})$ and ${{\bf{h}}_{i}}({\bf{x}})$, $i=1,2,...,M$.
Unlike [37, 38, 39], we in this paper adopt the general Rician fading model
for the wiretap channel, because eavesdroppers may deliberately hide by the
trees and other structures to avoid being discovered and then there may exist
not only the LoS channel path, but also the NLoS channel paths. Therefore,
${{\bf{h}}_{i}}({\bf{x}})$, $\forall i=1,...,M$, has the following form

${{\bf{h}}_{i}}({\bf{x}})=\sqrt{\frac{{{K_{i}}{\beta_{i}}}}{{{K_{i}}+1}}}{\overline{\bf{h}}_{i}}({\bf{x}})+\sqrt{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}{\widetilde{\bf{h}}_{i}},$
(1)

where $K_{i}>0$ is the Rician K-factor, ${{\beta_{i}}}$ is the path loss
component, ${\widetilde{\bf{h}}_{i}}\in{{\mathbb{C}}^{1\times N}}$ is the NLoS
component, and entries of ${\widetilde{\bf{h}}_{i}}$ are independent and
identically distributed (i.i.d) circularly-symmetric complex Gaussian random
variables with zero mean and unit variance, i.e.,
${\widetilde{\bf{h}}_{i}}\sim{\cal C}{\cal N}\left({0,{{\bf{I}}_{N}}}\right)$.
In addition, ${\overline{\bf{h}}_{i}}({\bf{x}})$ is the LoS component, which
is related to positions of Alice’s antennas ${\bf{x}}$ and can be expressed
as222Actually, the instantaneous ${\widetilde{\bf{h}}_{i}}$ corresponding to
different ${\bf{x}}$ may be also different. However, as explained later, we in
this paper consider a practical limitation that Alice only masters the
statistical information of ${\widetilde{\bf{h}}_{i}}$. Therefore, from the
perspective of Alice, the statistical distribution of
${\widetilde{\bf{h}}_{i}}$ is the same for different ${\bf{x}}$.

${\overline{\bf{h}}_{i}}({\bf{x}})=\left[{{e^{j\frac{{2\pi}}{\lambda}{x_{1}}\sin{\theta_{i}}}},{e^{j\frac{{2\pi}}{\lambda}{x_{2}}\sin{\theta_{i}}}},...,{e^{j\frac{{2\pi}}{\lambda}{x_{N}}\sin{\theta_{i}}}}}\right],$
(2)

where ${{\theta_{i}}}$ is the angle of departure (AoD) from the linear array
at Alice to ${\rm{Eve}}_{i}$, and $\lambda$ is the carrier wavelength. Without
loss of generality, it is assumed that ${\theta_{i}}\neq{\theta_{j}}$,
$\forall i,j\in\left\\{{1,2,...,M}\right\\}$ and $i\neq j$.

On the other hand, unlike eavesdroppers, Bob can communicate with Alice
without any need for concealment. Hence, for simplification we consider the
LoS condition for the main channel. Therefore, ${{{\bf{h}}_{0}}({\bf{x}})}$ is
expressed as

${{\bf{h}}_{0}}({\bf{x}})=\sqrt{{\beta_{0}}}\left[{{e^{j\frac{{2\pi}}{\lambda}{x_{1}}\sin{\theta_{0}}}},{e^{j\frac{{2\pi}}{\lambda}{x_{2}}\sin{\theta_{0}}}},...,{e^{j\frac{{2\pi}}{\lambda}{x_{N}}\sin{\theta_{0}}}}}\right],$
(3)

where ${{\beta_{0}}}$ is the path loss component and ${{\theta_{0}}}$ is the
AoD from Alice to Bob.333Similar to [13], we assume the far-field condition
between Alice and Bob (eavesdroppers) because the size of the moving region
for all antennas is much smaller than the signal propagation distance. Then,
the AoDs do not change for different positions of antennas.

Based on the above analysis, the received signal at Bob is expressed as

${y_{0}}=\sqrt{{P_{a}}}{{\bf{h}}_{0}}({\bf{x}}){{\bf{w}}}{x_{s}}+{n_{0}},$ (4)

where ${\bf{w}}\in{{\mathbb{C}}^{N\times 1}}$ is the transmit beamforming
vector at Alice, with $\left\|{\bf{w}}\right\|=1$, $x_{s}$ is the Gaussian
distributed information bearing signal satisfying
${\mathbb{E}}\left[{{{\left|{x_{s}}\right|}^{2}}}\right]=1$, $n_{0}$ is the
additive white Gaussian noise with zero mean and variable ${\sigma^{2}}$, and
${{P_{a}}}$ is Alice’s power budget. Next, we will present the detailed proof
that why Alice should set its transmit power to the power budget value.

Based on (4), the receiving signal-to-noise ratio (SNR) at Bob is derived as

${\gamma_{0}}({\bf{w}},{\bf{x}})={P_{a}}{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|^{2}}/{\sigma^{2}}.$
(5)

Also, we can use the similar manner to derive the receiving SNR at
${\rm{Eve}}_{i}$ as

${\gamma_{i}}({\bf{w}},{\bf{x}})={P_{a}}{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|^{2}}/{\sigma^{2}}.$
(6)

We consider a pessimistic eavesdropping scenario where $M$ colluding
eavesdroppers can perform joint decoding for $x_{s}$. Hence, the effective
receiving SNR of the wiretap channel is expressed as

${\gamma_{e}}({\bf{w}},{\bf{x}})={P_{a}}\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}/{\sigma^{2}}}.$
(7)

###  II-B Secrecy Rate and Secrecy Outage Probability

The secrecy rate in PLS is denoted as ${C_{s}}={({C_{m}}-{C_{e}})^{+}}$, where
${(x)^{+}}\buildrel\Delta\over{=}\max(x,0)$, ${C_{m}}$ and ${C_{e}}$ are the
capacities of the main channel and the wiretap channel, respectively. Based on
the above analysis, $C_{m}$ and $C_{e}$ are given by

$\begin{split}{}{C_{m}}=&{\log_{2}}(1+{\gamma_{0}}({\bf{w}},{\bf{x}})),\\\
{C_{e}}=&{\log_{2}}(1+{\gamma_{e}}({\bf{w}},{\bf{x}})).\end{split}$ (8)

Generally, it is practical for Alice to obtain the CSI of the main channel.
Therefore, the rate of transmitted codewords at Alice can be adaptively set as
$C_{m}$. On the other hand, consider the challenging case where eavesdroppers
are passive and do not emit any signals, such that Alice has no any knowledge
about the random components of the wiretap channel, i.e.,
${\widetilde{\bf{h}}_{i}}$, $i=1,2,...,M$, but can only estimate the
statistical components of the wiretap channel, i.e., $K_{i}$, ${{\beta_{i}}}$
and ${\overline{\bf{h}}_{i}}({\bf{x}})$, $i=1,2,...,M$, via channel
environment measurements, global positioning systems and so on. Under this
setup, the secrecy outage probability, denoted as ${{\mathop{\rm
P}\nolimits}_{{\rm{out}}}}$, is usually exploited to characterize the security
metric. Specifically, ${{\mathop{\rm P}\nolimits}_{{\rm{out}}}}$ is defined as
the probability that the targeted PLS coding rate of Alice’s encoder, i.e.,
$R_{s}$, is larger than the secrecy rate $C_{s}$ [43]. Hence, based on (8),
${{\mathop{\rm P}\nolimits}_{{\rm{out}}}}$ is derived as

$\begin{split}{}&{{\rm{P}}_{{\rm{out}}}}=\Pr\left({{C_{s}}\leq{R_{s}}}\right)=\Pr\left({{C_{e}}\geq{C_{m}}-{R_{s}}}\right)\\\
=&\Pr\left({\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}\geq\frac{{{\sigma^{2}}}}{P_{a}}\left({\frac{{{P_{a}}{{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}{{{\sigma^{2}}{2^{{R_{s}}}}}}+\frac{1}{{{2^{{R_{s}}}}}}-1}\right)}\right).\end{split}$
(9)

$\begin{split}{}&{\rm{P}}_{{\rm{out}}}=1-{F_{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}}\left({\frac{{{\sigma^{2}}}}{P_{a}}\left({\frac{{{P_{a}}{{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}{{{\sigma^{2}}{2^{{R_{s}}}}}}+\frac{1}{{{2^{{R_{s}}}}}}-1}\right)}\right)\\\
=&1-\gamma\left({\underbrace{\frac{{{{\left({\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}\left({{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}\right)}^{2}}}}{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}\left({2{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}}_{{f_{1}}({\bf{w}},{\bf{x}})},\underbrace{\frac{{\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}\left({{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}\left({2{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}\frac{{{\sigma^{2}}}}{{{P_{a}}}}\left({\frac{{{P_{a}}{{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}{{{\sigma^{2}}{2^{{R_{s}}}}}}+\frac{1}{{{2^{{R_{s}}}}}}-1}\right)}_{{f_{2}}({\bf{w}},{\bf{x}})}}\right).\end{split}$
(19)

###  II-C Problem Formulation

In this paper, we aim to minimize the secrecy outage probability
${\rm{P}}_{\rm{out}}$, by jointly optimizing the beamforming vector ${\bf{w}}$
and antennas’ positions ${\bf{x}}$ at Alice. The optimization problem can be
formulated as

$\displaystyle({\rm{P1}}):{\rm{}}\mathop{\min}\limits_{{{\bf{w}},{\bf{x}}}}\
{\rm{P}}_{\rm{out}}$ (${\rm{10a}}$) $\displaystyle\
{\rm{s.t.}}\quad\left\|{\bf{w}}\right\|=1,$ (${\rm{10b}}$)
$\displaystyle\quad\quad\ \ \ {\bf{x}}\in{\cal C},$ (${\rm{10c}}$)

where ${\cal C}$ in (10c) denotes the feasible moving region for $N$ antennas
at Alice. Similar to [27], in light of the following two aspects: i) there
should exist a minimum distance, denoted as $d_{{\rm{min}}}$, between any two
adjacent MAs to avoid the coupling effect, i.e.,
$\left|{{x_{i}}-{x_{j}}}\right|\geq{d_{\min}}$, $\forall i\neq j$; ii) the
movement span should be the same for each antenna, we can set ${\cal C}$ as
${\cal
C}\buildrel\Delta\over{=}\left\\{{{x_{i}}\in[{F_{i}},{G_{i}}]}\right\\}_{i=1}^{N}$,
where

$\begin{split}{}{F_{i}}=&\frac{{L-(N-1){d_{\min}}}}{N}(i-1)+(i-1){d_{\min}},\\\
{G_{i}}=&\frac{{L-(N-1){d_{\min}}}}{N}i+(i-1){d_{\min}},\end{split}$ (11)

based on which we have
$0={F_{1}}<{G_{1}}<{F_{2}}<{G_{2}}<...<{F_{N}}<{G_{N}}=L$ and
${G_{i}}-{F_{i}}=\frac{{L-(N-1){d_{\min}}}}{N}$, $\forall i=1,...,N$.

Remark 1: Given ${\bf{w}}$ and ${\bf{x}}$: i) the exact value of
${\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|^{2}}$ is known and $R_{s}$
is a constant; ii) however,
$\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}$
is totally random because Alice does not know the instantaneous values of
${\widetilde{\bf{h}}_{i}}$, $i=1,2,...,M$. Therefore, to solve (P1), next we
need to first derive the cumulative distribution function (CDF) of
$\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}$
to pave the way for obtaining the expression of ${\rm{P}}_{{\rm{out}}}$.

##  III Analysis About the Expression of ${\rm{P}}_{{\rm{out}}}$

###  III-A The CDF of
$\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}$

Given ${\bf{w}}$ and ${\bf{x}}$, based on (1) we can expand
$\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}$
as

$\begin{split}{}&\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}\\\
=&\sum\nolimits_{i=1}^{M}{{{\left|{\underbrace{\sqrt{\frac{{{K_{i}}{\beta_{i}}}}{{{K_{i}}+1}}}{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}_{{\rm{deterministic,\
denoted\ as}}{{\
\rm{X}}_{i}}}+\underbrace{\sqrt{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}{{\widetilde{\bf{h}}}_{i}}{\bf{w}}}_{{\rm{random,\
denoted\ as}}{{\ \rm{Y}}_{i}}}}\right|}^{2}}}.\end{split}$ (12)

Observing (12), each ${{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}$, $i=1,2,...,M$,
consists of a deterministic component ${\rm{X}}_{i}$ and a circularly-
symmetric complex Gaussian random component ${\rm{Y}}_{i}$ (since ${\bf{w}}$
is independent of ${{{\widetilde{\bf{h}}}_{i}}}$). Hence, it is concluded that
${\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}$ follows a Rician
distribution, where the total power (denoted as ${\widetilde{Z}_{i}}$) and the
Rician K-factor (denoted as ${\widetilde{K}_{i}}$) are

$\begin{split}{}{\widetilde{Z}_{i}}=&{\left|{{{\rm{X}}_{i}}}\right|^{2}}+{\mathbb{E}}\left[{{{\left|{{{\rm{Y}}_{i}}}\right|}^{2}}}\right]\mathop{=}\limits^{(a)}\frac{{{K_{i}}{\beta_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}{{{K_{i}}+1}}+\frac{{{\beta_{i}}}}{{{K_{i}}+1}},\\\
{\widetilde{K}_{i}}=&\frac{{{{\left|{{{\rm{X}}_{i}}}\right|}^{2}}}}{{{\mathbb{E}}\left[{{{\left|{{{\rm{Y}}_{i}}}\right|}^{2}}}\right]}}={K_{i}}{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|^{2}},\end{split}$
(13)

and the equality $\mathop{=}\limits^{(a)}$ is established because
${\mathbb{E}}\left[{{{\left|{{{\rm{Y}}_{i}}}\right|}^{2}}}\right]=\frac{{{\beta_{i}}}}{{{K_{i}}+1}}{\mathbb{E}}\left[{{{\left|{{{\widetilde{\bf{h}}}_{i}}{\bf{w}}}\right|}^{2}}}\right]=\frac{{{\beta_{i}}}}{{{K_{i}}+1}}$.

It is well-known that the CDF of the Rician random variable involves the
standard Marcum-Q function, which is not favorable for subsequent analysis. To
overcome this challenge, same as [44], we can conveniently regard the Rician
fading as a special case of Nakagami fading, based on which the probability
density function (PDF) of
${{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}$ can be tightly
approximated as

$\begin{split}{}{f_{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}(x)=&{\left({\frac{{\widetilde{{m_{i}}}}}{{\widetilde{{Z_{i}}}}}}\right)^{\widetilde{{m_{i}}}}}\frac{{{x^{\widetilde{{m_{i}}}-1}}}}{{\Gamma(\widetilde{{m_{i}}})}}{e^{-\frac{{\widetilde{{m_{i}}}}}{{\widetilde{{Z_{i}}}}}x}},\end{split}$
(14)

where ${\widetilde{{m_{i}}}}$ is the Nakagami fading parameter denoted by
$\widetilde{{m_{i}}}={({\widetilde{K}_{i}}+1)^{2}}/(2{\widetilde{K}_{i}}+1)$.
Since the PDF of ${{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}$
is known, its expectation and variance are easily derived as
${\mathbb{E}}\left[{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}\right]=\widetilde{{Z_{i}}}$
and
${\mathbb{V}}\left[{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}\right]=\frac{{{{\widetilde{{Z_{i}}}}^{2}}}}{{\widetilde{{m_{i}}}}}$,
respectively.

Although the tight and simple-structure PDF of each
${{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}$ is derived
successfully, the PDF of the sum of each
${{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}$, $i=1,2,...,M$, is
still hard to be obtained. To make process, we now approximate the PDF of
${\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}$
as a Gamma distribution by exploiting the Laguerre series approximation [45],
i.e.,

${f_{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}}(x)\approx\frac{1}{{\Gamma(\mu){\vartheta^{\mu}}}}{x^{\mu-1}}{e^{-\frac{x}{\vartheta}}},$
(15)

where $\mu$ and $\vartheta$ are the shape parameter and scale parameter, which
are defined as

$\mu\buildrel\Delta\over{=}\frac{{{{\left({{\mathbb{E}}\left[{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}\right]}\right)}^{2}}}}{{{\mathbb{V}}\left[{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}\right]}},\vartheta\buildrel\Delta\over{=}\frac{{{\mathbb{V}}\left[{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}\right]}}{{{\mathbb{E}}\left[{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}\right]}},$
(16)

and
${{\mathbb{E}}\left[{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}\right]}$
and
${{\mathbb{V}}\left[{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}\right]}$
are computed as

$\begin{split}{}&{\mathbb{E}}\left[{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}\right]=\sum\nolimits_{i=1}^{M}{{\mathbb{E}}\left[{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}\right]}=\sum\nolimits_{i=1}^{M}{\widetilde{{Z_{i}}}},\\\
&{\mathbb{V}}\left[{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}\right]=\sum\nolimits_{i=1}^{M}{{\mathbb{V}}\left[{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}\right]}=\sum\nolimits_{i=1}^{M}{\frac{{{{\widetilde{{Z_{i}}}}^{2}}}}{{\widetilde{{m_{i}}}}}}.\end{split}$
(17)

Figure 2: Comparison of the exact CDF of
${\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}$
via Monto Carlo and the derived approximation via (18), where $N=5$, $M=3$,
${\bf{x}}={[0,0.5\lambda,\lambda,1.5\lambda,2\lambda]^{T}}$,
${\bf{w}}={{\bf{w}}_{1}}/\left\|{{{\bf{w}}_{1}}}\right\|$ with
${{\bf{w}}_{1}}={\left[{1+j,2+j3,2+j,3-j,4+j5}\right]^{T}}$,
$\left\\{{{\beta_{i}}}\right\\}_{i=1}^{M}=1$, ${\theta_{1}}=\pi/6$,
${\theta_{2}}=\pi/4$, ${\theta_{3}}=\pi/10$ and
$\left\\{{{K_{i}}}\right\\}_{i=1}^{M}=K$.

Substituting (17) into (16) and then substituting (16) into (15), the CDF of
${\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}$
can be derived as

$\begin{split}{}{F_{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}}(x)=&\int_{0}^{x}{{f_{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}}(t)dt}\\\
=&\gamma\left({\frac{{{{\left({\sum\nolimits_{i=1}^{M}{\widetilde{{Z_{i}}}}}\right)}^{2}}}}{{\sum\nolimits_{i=1}^{M}{\frac{{{{\widetilde{{Z_{i}}}}^{2}}}}{{\widetilde{{m_{i}}}}}}}},\frac{{\sum\nolimits_{i=1}^{M}{\widetilde{{Z_{i}}}}}}{{\sum\nolimits_{i=1}^{M}{\frac{{{{\widetilde{{Z_{i}}}}^{2}}}}{{\widetilde{{m_{i}}}}}}}}x}\right),\end{split}$
(18)

where $\gamma(a,x)=\frac{1}{{\Gamma(a)}}\int_{0}^{x}{{t^{a-1}}{e^{-t}}dt}$ is
the lower incomplete gamma function.

As a supplement, we further compare the obtained
${F_{\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}}(x)$
in (18) and the exact CDF (via Monto Carlo) as shown in Fig. 2, from which it
is clear that our derived approximation is very tight and thus can be
effectively exploited for subsequent optimization analysis.

###  III-B The Expression of ${\rm{P}}_{{\rm{out}}}$

Based on (9) and the CDF of
${\sum\nolimits_{i=1}^{M}{{{\left|{{{\bf{h}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}$
derived in (18), and by further expanding each ${\widetilde{{Z_{i}}}}$ and
${\widetilde{{m_{i}}}}$, $i=1,2,...,M$, we can finally derive
${\rm{P}}_{{\rm{out}}}$ as in (19).

Insight 1: The secrecy outage probability ${\rm{P}}_{{\rm{out}}}$ is
monotonically decreasing w.r.t. the transmit power of Alice and then converges
to a constant when the power budget $P_{a}$ tends to infinity.

###### Proof.

Observing (19), it is clear that ${{f_{1}}({\bf{w}},{\bf{x}})}$ is not related
to $P_{a}$, while ${{f_{2}}({\bf{w}},{\bf{x}})}$ can be expanded as

$\begin{split}{}&{f_{2}}({\bf{w}},{\bf{x}})=\underbrace{\frac{{\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}\left({{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}\left({2{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}\frac{{{{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}{{{2^{{R_{s}}}}}}}_{{\rm{constant}}}\\\
&+\underbrace{\frac{{\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}\left({{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}\left({2{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}\frac{{{\sigma^{2}}}}{{{P_{a}}}}\left({\frac{1}{{{2^{{R_{s}}}}}}-1}\right)}_{{\rm{related}}\
{\rm{to}}\ {{P}_{{a}}}},\end{split}$ (20)

from (20) it is easy to determine that ${{f_{2}}({\bf{w}},{\bf{x}})}$ is
monotonically increasing w.r.t. $P_{a}$ (since
$\frac{1}{{{2^{{R_{s}}}}}}-1<0$). Further, note that
$\gamma\left({{f_{1}}({\bf{w}},{\bf{x}}),{f_{2}}({\bf{w}},{\bf{x}})}\right)$
is monotonically increasing w.r.t. ${{f_{2}}({\bf{w}},{\bf{x}})}$. Hence, it
is always favorable to increase the transmit power of Alice to increase
$\gamma\left({{f_{1}}({\bf{w}},{\bf{x}}),{f_{2}}({\bf{w}},{\bf{x}})}\right)$
so as to decrease
${\rm{P}}_{{\rm{out}}}=1-\gamma\left({{f_{1}}({\bf{w}},{\bf{x}}),{f_{2}}({\bf{w}},{\bf{x}})}\right)$.
This is why we set the transmit power of Alice to its power budget in Section
II.

In addition, when ${P_{a}}\to\infty$, it has
$\frac{{{\sigma^{2}}}}{{{P_{a}}}}\to 0$, leading to

$\begin{split}{}{f_{2}}({\bf{w}},{\bf{x}})\to\frac{{\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}\left({{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}\left({2{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}\frac{{{{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}{{{2^{{R_{s}}}}}},\end{split}$
(21)

which is not related to $P_{a}$ anymore. Thus, when ${P_{a}}\to\infty$,
${\rm{P}}_{{\rm{out}}}$ will converge to a constant. This completes the proof.
∎

Insight 2: In the special case of $\left\\{{{K_{i}}}\right\\}_{i=1}^{M}=0$,
the wiretap channel suffers from Rayleigh fading and is totally random to
Alice. Then, since there is no any useful information about the wiretap
channel that can be exploited, to minimize ${\rm{P}}_{{\rm{out}}}$, Alice
should just maximize the power gain of the main channel, i.e., maximize the
term ${{{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|}^{2}}}$.

###### Proof.

By substituting $\left\\{{{K_{i}}}\right\\}_{i=1}^{M}=0$ into (19),
${\rm{P}}_{{\rm{out}}}$ can be simplified as
${\rm{P}}_{{\rm{out}}}=1-\gamma\left({\frac{{{{\left({\sum\nolimits_{i=1}^{M}{{\beta_{i}}}}\right)}^{2}}}}{{\sum\nolimits_{i=1}^{M}{\beta_{i}^{2}}}},\frac{{\sum\nolimits_{i=1}^{M}{{\beta_{i}}}}}{{\sum\nolimits_{i=1}^{M}{\beta_{i}^{2}}}}\frac{{{\sigma^{2}}}}{{{P_{a}}}}\left({\frac{{{P_{a}}{{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}{{{\sigma^{2}}{2^{{R_{s}}}}}}+\frac{1}{{{2^{{R_{s}}}}}}-1}\right)}\right)$,
based on which the conclusion in Insight 2 can be easily obtained. This
completes the proof. ∎

Based on the above analysis, in the case of
$\left\\{{{K_{i}}}\right\\}_{i=1}^{M}=0$, we should just consider the
following optimization problem to minimize ${\rm{P}}_{{\rm{out}}}$:

$\displaystyle({\rm{P2}}):{\rm{}}\mathop{\max}\limits_{{{\bf{w}},{\bf{x}}}}\
{{{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|}^{2}}}$ (${\rm{22a}}$)
$\displaystyle\ {\rm{s.t.}}\quad(10{\rm{b}}),(10{\rm{c}}).$ (${\rm{22b}}$)

Problem (P2) can be easily solved by noting that given any ${\bf{x}}\in{{\cal
C}}$, the optimal ${\bf{w}}$ is in the form of maximum-ratio transmission
(MRT), i.e.,
${{\bf{w}}^{*}}={\bf{h}}_{0}^{H}({\bf{x}})/\left\|{{\bf{h}}_{0}^{H}({\bf{x}})}\right\|$,
where ${\left(\cdot\right)^{H}}$ is the conjugate transpose operation. Based
on this, we have
${\left|{{{\bf{h}}_{0}}({\bf{x}}){{\bf{w}}^{*}}}\right|^{2}}={\left|{\frac{{{{\bf{h}}_{0}}({\bf{x}}){\bf{h}}_{0}^{H}({\bf{x}})}}{{\left\|{{\bf{h}}_{0}^{H}({\bf{x}})}\right\|}}}\right|^{2}}={\beta_{0}}N$
given any ${\bf{x}}\in{{\cal C}}$. That is to say, in the case of
$\left\\{{{K_{i}}}\right\\}_{i=1}^{M}=0$, positions of antennas at Alice can
be arbitrary in the specified region and Alice just employs the MRT-based
beamforming for maximizing the receiving SNR of Bob.444Although the MRT-based
beamforming is optimal when $\left\\{{{K_{i}}}\right\\}_{i=1}^{M}=0$, it will
achieve undesirable performance under the general case of
$\left\\{{{K_{i}}}\right\\}_{i=1}^{M}>0$ as pointed out in numerous literature
and also shown in subsequent simulations. Therefore, we will not conduct
further theoretical analysis on this scheme.

Insight 3: In the special case of
$\left\\{{{K_{i}}}\right\\}_{i=1}^{M}\to\infty$, i.e., the wiretap channel
only consists of the LoS component and is totally deterministic to Alice,
absolute secrecy can be guaranteed given $N\geq M+1$.

###### Proof.

When $\left\\{{{K_{i}}}\right\\}_{i=1}^{M}\to\infty$,
${{\bf{h}}_{i}}({\bf{x}})$ is simplified as
${\beta_{i}}{\overline{\bf{h}}_{i}}({\bf{x}})$, $i=1,2,...,M$. Then, given
$N\geq M+1$, Alice can employ the simple ZF-based transmit beamforming
(denoted as ${{{\bf{w}}_{{\rm{ZF}}}}}$) to completely eliminate the receiving
power gain at each eavesdropper, i.e.,
${\sum\nolimits_{i=1}^{M}{\left|{{\beta_{i}}{{\overline{\bf{h}}}_{i}}({\bf{x}}){{\bf{w}}_{{\rm{ZF}}}}}\right|}^{2}}=0$.
Then, based on (9) it is clear that
${P_{{\rm{out}}}}=\left({0\geq\frac{{{\sigma^{2}}}}{{{P_{a}}}}\left({\frac{{{P_{a}}{{\left|{{{\bf{h}}_{0}}({\bf{x}}){{\bf{w}}_{{\rm{ZF}}}}}\right|}^{2}}}}{{{\sigma^{2}}{2^{{R_{s}}}}}}+\frac{1}{{{2^{{R_{s}}}}}}-1}\right)}\right)=0$.
This completes the proof. ∎

##  IV Novel Optimization Design to (P1)

We now focus on how to efficiently solve (P1). Based on (19), minimizing
${\rm{P}}_{{\rm{out}}}$ is equivalent to maximizing
$\gamma\left({{f_{1}}({\bf{w}},{\bf{x}}),{f_{2}}({\bf{w}},{\bf{x}})}\right)$.
Therefore, (P1) can be reformulated as

$\displaystyle({\rm{P3}}):{\rm{}}\mathop{\max}\limits_{{{\bf{w}},{\bf{x}}}}\
\gamma\left({{f_{1}}({\bf{w}},{\bf{x}}),{f_{2}}({\bf{w}},{\bf{x}})}\right)$
(${\rm{23a}}$) $\displaystyle\ {\rm{s.t.}}\quad(10{\rm{b}}),(10{\rm{c}}).$
(${\rm{23b}}$)

In most literature, for the objective $\gamma\left({a,b}\right)$, the
optimization variables are just involved in the parameter $a$ or $b$. Then,
maximizing $\gamma\left({a,b}\right)$ can be conveniently transformed into
minimizing $a$ or maximizing $b$, since $\gamma\left({a,b}\right)$ is
monotonically decreasing and increasing w.r.t. $a$ and $b$, respectively. With
this way, the complex incomplete gamma function can be removed from the
objective and then the optimization becomes much easier. However, observing
the objective of (P3), the variables ${\bf{w}}$ and ${\bf{x}}$ are involved in
both $a$ and $b$, which indicates that the conventional method is not suitable
anymore and we have to seek new methods to solve (P3).

Figure 3: ${\gamma^{-1}}(\varepsilon,{f_{1}}({\bf{w}},{\bf{x}}))$ and the
proposed linear approximation model w.r.t. ${f_{1}}({\bf{w}},{\bf{x}})$ under
different $\varepsilon$. Figure 4: The parameters $\kappa(\varepsilon)$ and
$\rho(\varepsilon)$ w.r.t. $\varepsilon$.

###  IV-A A Novel Optimization Framework

To proceed, we now introduce the slack variable $\varepsilon$ and equivalently
express (P3) as

$\displaystyle({\rm{P3.1}}):{\rm{}}\mathop{\max}\limits_{{{\bf{w}},{\bf{x}}},\varepsilon}\
\varepsilon$ (${\rm{24a}}$) $\displaystyle\ {\rm{s.t.}}\quad\
\gamma\left({{f_{1}}({\bf{w}},{\bf{x}}),{f_{2}}({\bf{w}},{\bf{x}})}\right)\geq\varepsilon,$
(${\rm{24b}}$) $\displaystyle\quad\quad\ \ \ (10{\rm{b}}),(10{\rm{c}}).$
(${\rm{24c}}$)

Note that the constraint (24b) is equivalent to

$\begin{split}{}{f_{2}}({\bf{w}},{\bf{x}})\geq{\gamma^{-1}}\left({\varepsilon,{f_{1}}({\bf{w}},{\bf{x}})}\right),\end{split}$
(25)

where ${\gamma^{-1}}(\varepsilon,{f_{1}}({\bf{w}},{\bf{x}}))$, with
$\varepsilon\in[0,1]$, is the inverse of the lower incomplete gamma function.

Figure 5: The achievable performance of the proposed algorithm under two
typical cases.

Compared to the complex
$\gamma\left({{f_{1}}({\bf{w}},{\bf{x}}),{f_{2}}({\bf{w}},{\bf{x}})}\right)$,
${\gamma^{-1}}\left({\varepsilon,{f_{1}}({\bf{w}},{\bf{x}})}\right)$ is only
related to ${{f_{1}}({\bf{w}},{\bf{x}})}$ when $\varepsilon$ is given. This
motivates us to plot the curves of
${\gamma^{-1}}(\varepsilon,{f_{1}}({\bf{w}},{\bf{x}}))$ w.r.t.
${f_{1}}({\bf{w}},{\bf{x}})$ under different $\varepsilon\in[0,1]$ as shown in
Fig. 3 (solid lines). From Fig. 3, we observe that
${\gamma^{-1}}(\varepsilon,{f_{1}}({\bf{w}},{\bf{x}}))$ almost increases
linearly w.r.t. ${f_{1}}({\bf{w}},{\bf{x}})$ for any $\varepsilon$, especially
when ${f_{1}}({\bf{w}},{\bf{x}})$ lies in a larger region. Based on this fact,
the simple linear model (dashed lines in Fig. 3), i.e.,
$\kappa(\varepsilon){f_{1}}({\bf{w}},{\bf{x}})+\rho(\varepsilon)$, can be
exploited to tightly approximate
${\gamma^{-1}}(\varepsilon,{f_{1}}({\bf{w}},{\bf{x}}))$, where
$\kappa(\varepsilon)$ and $\rho(\varepsilon)$ are related to the parameter
$\varepsilon$ and are determined based on the criterion of minimum mean square
error. In particular, the “nlinfit” function in Matlab can be used to find
$\kappa(\varepsilon)$ and $\rho(\varepsilon)$. We further plot
$\kappa(\varepsilon)$ and $\rho(\varepsilon)$ w.r.t. $\varepsilon$ as shown in
Fig. 4, from which $\kappa(\varepsilon)$ and $\rho(\varepsilon)$ are both
monotonically increasing w.r.t. $\varepsilon$. In addition,
$\kappa(\varepsilon)>0$, $\forall\varepsilon\in[0,1]$, while
$\rho(\varepsilon)$ can be positive or negative when $\varepsilon\in[0,1]$.

Remark 2: From Fig. 3, the accuracy of the proposed approximation mainly
depends on the value of ${f_{1}}({\bf{w}},{\bf{x}})$. Specifically, when
${f_{1}}({\bf{w}},{\bf{x}})$ is larger than two, the proposed linear model
almost approaches the numerical data; while there exists the slight deviation
between ${\gamma^{-1}}\left({\varepsilon,{f_{1}}({\bf{w}},{\bf{x}})}\right)$
and $\kappa(\varepsilon){f_{1}}({\bf{w}},{\bf{x}})+\rho(\varepsilon)$ when
${f_{1}}({\bf{w}},{\bf{x}})$ does not exceed one. Fortunately, observing (19),
we can derive that

$\begin{split}{}{f_{1}}({\bf{w}},{\bf{x}})>&\frac{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}{{\left({{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}^{2}}}}}{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}\left({2{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}\\\
>&\frac{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}\left({2{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}\left({2{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}>1.\end{split}$
(26)

Hence, it is concluded that the proposed linear model can achieve a pretty
good fitting result.

Armed with the above analysis, we now can provide an approximate problem for
(P3.1), i.e.,

$\displaystyle({\rm{P3.2}}):{\rm{}}\mathop{\max}\limits_{{{\bf{w}},{\bf{x}}},\varepsilon}\
\varepsilon$ (${\rm{27a}}$) $\displaystyle\ {\rm{s.t.}}\quad\
{f_{2}}({\bf{w}},{\bf{x}})\geq\kappa(\varepsilon){f_{1}}({\bf{w}},{\bf{x}})+\rho(\varepsilon),$
(${\rm{27b}}$) $\displaystyle\quad\quad\ \ \ (10{\rm{b}}),(10{\rm{c}}).$
(${\rm{27c}}$)

Since $\kappa(\varepsilon)$ and $\rho(\varepsilon)$ have no closed-form
expressions, it is still hard to directly solve (P3.2). Because of this
reason, the following steps should be followed to find the optimal solution of
(P3.2):

1) Use bisection to search $\varepsilon$, with $\varepsilon\in[0,1]$;

2) Given each $\varepsilon$, check the feasibility of the following problem:

$\displaystyle({\rm{P3.3}}):{\rm{Find}}\ \ {\bf{w}},{\bf{x}}$ (${\rm{28a}}$)
$\displaystyle\ {\rm{s.t.}}\quad\ (27{\rm{b}}),(27{\rm{c}}).$ (${\rm{28b}}$)

Specifically, if (P3.3) is feasible, the optimal solution of (P3.2) must be no
smaller than the given $\varepsilon$. Thus, in the next search, the updated
$\varepsilon$ should be larger than the given $\varepsilon$ of the last round.
Otherwise, if (P3.3) is infeasible, the optimal solution of (P3.2) must not be
larger than the given $\varepsilon$. Thus, in the next search, the updated
$\varepsilon$ should be smaller than the given $\varepsilon$ of the last
round. In addition, to judge that whether (P3.3) is feasible or not, it is
equivalent to considering the following problem:

$\displaystyle({\rm{P3.4}}):{\rm{}}\mathop{\max}\limits_{{{\bf{w}},{\bf{x}}}}\
{f_{2}}({\bf{w}},{\bf{x}})-\kappa(\varepsilon){f_{1}}({\bf{w}},{\bf{x}})-\rho(\varepsilon)$
(${\rm{29a}}$) $\displaystyle\ {\rm{s.t.}}\quad\ (10{\rm{b}}),(10{\rm{c}}).$
(${\rm{29b}}$)

Given $\varepsilon$, if the optimal output of (P3.4) is not larger than zero,
(P3.3) is infeasible. Otherwise, (P3.3) is feasible.

Algorithm 1 Steps for Solving (P3.2)

1:Input: $\kappa(\varepsilon)$ and $\rho(\varepsilon)$,
$\varepsilon=0:\tau:1$, $\tau$ is the accuracy.

2:Initialize: ${\varepsilon_{1}}=0.5$, ${\varepsilon_{L}}=0$ and
${\varepsilon_{R}}=1$.

3:Repeat:

4: Set $\varepsilon={\varepsilon_{1}}$ and solve (P3.3).

5: If (P3.3) is infeasible, i.e., the optimal output of (P3.4) is smaller than
zero, update
${\varepsilon_{R}}={\varepsilon_{1}};{\varepsilon_{1}}=({\varepsilon_{L}}+{\varepsilon_{1}})/2$;

6: Otherwise, update
${\varepsilon_{L}}={\varepsilon_{1}};{\varepsilon_{1}}=({\varepsilon_{1}}+{\varepsilon_{R}})/2$.

7:Until: ${\varepsilon_{1}}$ converges to a stationary value.

8:Output: ${{\bf{w}}_{{\varepsilon_{1}}}}$, ${{\bf{x}}_{{\varepsilon_{1}}}}$
and the resulted secrecy outage probability
${{\rm{P}}_{{\rm{out}}}}=1-\gamma\left({{f_{1}}({{\bf{w}}_{{\varepsilon_{1}}}},{{\bf{x}}_{{\varepsilon_{1}}}}),{f_{2}}({{\bf{w}}_{{\varepsilon_{1}}}},{{\bf{x}}_{{\varepsilon_{1}}}})}\right)$.

Armed with the above analysis, we can finally list the detailed steps for
solving (P3.2) as shown in Algorithm 1. After obtaining the stationary
${\varepsilon_{1}}$ of (P3.2) and the corresponding ${\bf{w}}$ (denoted as
${{\bf{w}}_{{\varepsilon_{1}}}}$) and ${\bf{x}}$ (denoted as
${{\bf{x}}_{{\varepsilon_{1}}}}$), the actual secrecy outage probability can
be computed as
${{\rm{P}}_{{\rm{out}}}}=1-\gamma\left({{f_{1}}({{\bf{w}}_{{\varepsilon_{1}}}},{{\bf{x}}_{{\varepsilon_{1}}}}),{f_{2}}({{\bf{w}}_{{\varepsilon_{1}}}},{{\bf{x}}_{{\varepsilon_{1}}}})}\right)$.

$\begin{split}{}{f_{3}}({\bf{w}},{\bf{x}})=&\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}\left({{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)\frac{{{\sigma^{2}}}}{{{P_{a}}}}\left({\frac{{{P_{a}}{{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}{{{\sigma^{2}}{2^{{R_{s}}}}}}+\frac{1}{{{2^{{R_{s}}}}}}-1}\right)}\\\
&-\kappa(\varepsilon){\left({\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}\left({{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}\right)^{2}}-\rho(\varepsilon)\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}\left({2{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}.\end{split}$
(31)

Remark 3: Before implementing Algorithm 1, we have to list
${\left\\{{\kappa(\varepsilon),\rho(\varepsilon)}\right\\}_{\varepsilon=0:\tau:1}}$
based on numerical simulations. However, we must emphasize that
${\left\\{{\kappa(\varepsilon),\rho(\varepsilon)}\right\\}_{\varepsilon=0:\tau:1}}$
are not related to
$\left\\{{{f_{1}}({\bf{w}},{\bf{x}}),{f_{2}}({\bf{w}},{\bf{x}})}\right\\}$.
Thus,
${\left\\{{\kappa(\varepsilon),\rho(\varepsilon)}\right\\}_{\varepsilon=0:\tau:1}}$
are just computed only once and then saved forever for handling any different
$\left\\{{{f_{1}}({\bf{w}},{\bf{x}}),{f_{2}}({\bf{w}},{\bf{x}})}\right\\}$.

Remark 4: In Algorithm 1, the bisection method is used to search each
$\varepsilon$, leading to the corresponding search complexity as
$\log\left({1/\tau}\right)$. Furthermore, compared to the original problem
(P3.1), the complexity of solving (P3.4) would be much lower due to its clear
structure. We will verify this fact via two examples as shown subsequently.

Example 1: Let us first focus on a simple problem, i.e.,
$\mathop{\max}\limits_{x}\gamma\left({x+1,2\sqrt{x}}\right),{\rm{s.t}}.,x\in[0,2]$.
As $\gamma\left({a,b}\right)$ is monotonically increasing w.r.t. $b$ but
decreasing w.r.t. $a$, it is hard to find the closed-form and optimal $x$ that
maximizes $\gamma\left({x+1,2\sqrt{x}}\right)$. Hence, the optimal solution
should be obtained via the one-dimensional search. If the search accuracy is
also $\tau$, the whole complexity is ${\cal{O}}\left({1/{\tau}}\right)$.
Nevertheless, using Algorithm 1, given $\varepsilon$, the objective of (P3.4)
becomes $2\sqrt{x}-\kappa(\varepsilon)(x+1)-\rho(\varepsilon)$ which is
clearly concave and thus can be efficiently solved using existing solvers,
e.g., CVX. This indicates that the whole complexity of implementing Algorithm
1 is only ${\cal O}\left({\log(1/\tau)\times{1^{3.5}}}\right)={\cal
O}\left({\log(1/\tau)}\right)$, where ${\cal O}\left({{1^{3.5}}}\right)$ is
the complexity of running (P3.4) given $\varepsilon$.

Example 2: Let us now focus on a more complex problem, i.e.,
$\mathop{\max}\limits_{x,y}\gamma\left({2x+1.6y+2.1,2.1\sqrt{x}+1.8\sqrt{y}+0.2}\right)$,
${\rm{s}}.{\rm{t}}.,x,y\in[0,2]$. Due to the same reason mentioned above, the
optimal solution should be obtained via the two-dimensional search with the
complexity of ${\cal{O}}\left({1/{\tau^{2}}}\right)$. Nevertheless, using
Algorithm 1, given $\varepsilon$, the objective of (P3.4) becomes
$2.1\sqrt{x}+1.8\sqrt{y}+0.2-\kappa(\varepsilon)(2x+1.6y+2.1)-\rho(\varepsilon)$
and is still concave. Hence, the whole complexity of implementing Algorithm 1
is only ${\cal{O}}\left({\log(1/\tau)\times{2^{3.5}}}\right)$, where ${\cal
O}\left({{2^{3.5}}}\right)$ is the complexity of running (P3.4) given
$\varepsilon$.

Via Examples 1 and 2, it is evident that our proposed novel algorithm can
reduce the implementation complexity significantly. For instance, when
$\tau=0.01$, the complexities of the exhaustive search and Algorithm 1 for
Example 1 are ${\cal{O}}\left({100}\right)$ and ${\cal{O}}\left({4.6}\right)$,
and for Example 2 are ${\cal{O}}\left({100^{2}}\right)$ and
${\cal{O}}\left({50}\right)$, respectively. More importantly, our proposed
algorithm can also achieve the pretty good performance. This is verified in
Fig. 5, from which it is clear that the sub-optimal solution found by
Algorithm 1 almost approaches the optimal solution found via the exhaustive
search.

###  IV-B The APGA for Solving (P3.4)

Via the above analysis, the key step in Algorithm 1 is to solve (P3.4) given
each $\varepsilon$. To proceed, we first express the objective of (P3.4) as

$\begin{split}&{f_{2}}({\bf{w}},{\bf{x}})-\kappa(\varepsilon){f_{1}}({\bf{w}},{\bf{x}})-\rho(\varepsilon)\\\
=&\frac{{{f_{3}}({\bf{w}},{\bf{x}})}}{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}\left({2{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}},\end{split}$
(30)

where ${{f_{3}}({\bf{w}},{\bf{x}})}$ is shown in (31).

Note that we just need to determine that whether the optimal output of (P3.4)
is larger than zero or not. Keeping this in mind and since
$\frac{1}{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}\left({2{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}}}>0$,
(P3.4) can be transformed into the following problem with a clear structure:

$\displaystyle({\rm{P3.5}}):{\rm{}}\mathop{\max}\limits_{{{\bf{w}},{\bf{x}}}}\
{{f_{3}}({\bf{w}},{\bf{x}})}$ (${\rm{32a}}$) $\displaystyle\ {\rm{s.t.}}\quad\
(10{\rm{b}}),(10{\rm{c}}).$ (${\rm{32b}}$)

Then, we can conclude that, given $\varepsilon$, if the optimal output of
(P3.5) is not larger than zero, (P3.3) is infeasible. Otherwise, (P3.3) is
feasible.

Problem (P3.5) is highly non-convex because its objective is much complex,
which is neither convex or concave w.r.t. ${\bf{w}}$ and ${\bf{x}}$, and thus
may not be solved via standard convex optimization techniques. Responding to
this difficulty, we in this paper try to exploit the APGA method [46] to solve
(P3.5), since the APGA handles the unconstrained or constrained problems well
and is not sensitive to concavity or convexity of the objective.

Algorithm 2 The APGA for Solving (P3.5)

1:Input: ${{\bf{w}}^{(0)}}$, with $\left\|{{{\bf{w}}^{(0)}}}\right\|=1$,
${{\bf{x}}^{(0)}}\in{\cal C}$, $t\leftarrow 0$, $\delta>0$ and $\chi<1$.

2:Repeat:

3: Repeat: $\%\%\%\%\%$ The steps for updating ${\bf{w}}$

4:
${{\bf{w}}^{(t+1)}}={{\bf{w}}^{(t)}}+\delta{\nabla_{\bf{w}}}{f_{3}}({{\bf{w}}^{(t)}},{{\bf{x}}^{(t)}})$;

5:
${{\bf{w}}^{(t+1)}}={{\bf{w}}^{(t+1)}}/\left\|{{{\bf{w}}^{(t+1)}}}\right\|$;

6: if
${f_{3}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t)}})<{Q_{\delta}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t)}};{{\bf{w}}^{(t)}})$,
then

7: $\delta\leftarrow\chi\delta$,

8: end

9: Until:
${f_{3}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t)}})\geq{Q_{\delta}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t)}};{{\bf{w}}^{(t)}})$.

10: Repeat: $\%\%\%\%\%$ The steps for updating ${\bf{x}}$

11:
${{\bf{x}}^{(t+1)}}={{\bf{x}}^{(t)}}+\delta{\nabla_{\bf{x}}}{f_{3}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t)}})$;

12: ${{\bf{x}}^{(t+1)}}={{\cal B}}\left\\{{{{\bf{x}}^{(t+1)}},{{\cal
C}}}\right\\}$;

13: if
${f_{3}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t+1)}})<{Q_{\delta}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t+1)}};{{\bf{x}}^{(t)}})$,
then

14: $\delta\leftarrow\chi\delta$,

15: end

16: Until:
${f_{3}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t+1)}})\geq{Q_{\delta}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t+1)}};{{\bf{x}}^{(t)}})$.

17: $t\leftarrow t+1$;

18:Until: ${f_{3}}({{\bf{w}}^{(t)}},{{\bf{x}}^{(t)}})$ converges to a
stationary point.

$\begin{split}{}{\nabla_{\bf{w}}}{f_{3}}({\bf{w}},{\bf{x}})=&\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}{K_{i}}\overline{\bf{h}}_{i}^{H}({\bf{x}}){{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\frac{{{\sigma^{2}}}}{{{P_{a}}}}\left({\frac{{{P_{a}}{{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}{{{\sigma^{2}}{2^{{R_{s}}}}}}+\frac{1}{{{2^{{R_{s}}}}}}-1}\right)\\\
+&\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}\left({{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}\frac{{{\bf{h}}_{0}^{H}({\bf{x}}){{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}}{{{2^{{R_{s}}}}}}-2\kappa(\varepsilon)\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}\left({{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}\\\
\times&\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}{K_{i}}\overline{\bf{h}}_{i}^{H}({\bf{x}}){{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}-2\rho(\varepsilon)\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}{K_{i}}\overline{\bf{h}}_{i}^{H}({\bf{x}}){{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}.\end{split}$
(35)
$\begin{split}{}{\nabla_{\bf{x}}}{f_{3}}({\bf{w}},{\bf{x}}){\rm{}}&=\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}{K_{i}}\left({-{{\bf{W}}_{i}}(2{\bf{C}}{{\bf{g}}_{i}}+2{\bf{D}}{{\bf{q}}_{i}})-{{\bf{S}}_{i}}(2{\bf{C}}{{\bf{q}}_{i}}-2{\bf{D}}{{\bf{g}}_{i}})}\right)}\frac{{{\sigma^{2}}}}{{{P_{a}}}}\left({\frac{{{P_{a}}{{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|}^{2}}}}{{{\sigma^{2}}{2^{{R_{s}}}}}}+\frac{1}{{{2^{{R_{s}}}}}}-1}\right)\\\
&+\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}\left({{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}{\beta_{0}}\frac{{-{{\bf{W}}_{0}}(2{\bf{C}}{{\bf{g}}_{0}}+2{\bf{D}}{{\bf{q}}_{0}})-{{\bf{S}}_{0}}(2{\bf{C}}{{\bf{q}}_{0}}-2{\bf{D}}{{\bf{g}}_{0}})}}{{{2^{{R_{s}}}}}}\\\
&-2\kappa(\varepsilon)\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}\left({{K_{i}}{{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|}^{2}}+1}\right)}\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}{K_{i}}\left({-{{\bf{W}}_{i}}(2{\bf{C}}{{\bf{g}}_{i}}+2{\bf{D}}{{\bf{q}}_{i}})-{{\bf{S}}_{i}}(2{\bf{C}}{{\bf{q}}_{i}}-2{\bf{D}}{{\bf{g}}_{i}})}\right)}\\\
&-2\rho(\varepsilon)\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}{K_{i}}\left({-{{\bf{W}}_{i}}(2{\bf{C}}{{\bf{g}}_{i}}+2{\bf{D}}{{\bf{q}}_{i}})-{{\bf{S}}_{i}}(2{\bf{C}}{{\bf{q}}_{i}}-2{\bf{D}}{{\bf{g}}_{i}})}\right)}.\end{split}$
(36)

We now present the key steps of applying the APGA for solving (P3.5) as shown
in Algorithm 2, where given ${{\bf{x}}^{(t)}}$, Steps 3$-$9 aim to update
${{\bf{w}}^{(t)}}$ as ${{\bf{w}}^{(t+1)}}$, and given ${{\bf{w}}^{(t+1)}}$,
Steps 10$-$16 aim to update ${{\bf{x}}^{(t)}}$ as ${{\bf{x}}^{(t+1)}}$. These
two procedures are implemented alternately until the objective converges to a
stationary value. More specifically, Step 4 is used to update
${{\bf{w}}^{(t)}}$ without restraint, where $\delta$ is the step size and
${\nabla_{\bf{w}}}{f_{3}}({{\bf{w}}^{(t)}},{{\bf{x}}^{(t)}})$ is the gradient
of ${f_{3}}({\bf{w}},{\bf{x}})$ w.r.t. ${\bf{w}}$ at
$({{\bf{w}}^{(t)}},{{\bf{x}}^{(t)}})$. Step 5 is used to guarantee that
${{\bf{w}}^{(t+1)}}$ always satisfies the constraint in (10b), i.e.,
$\left\|{{{\bf{w}}^{(t+1)}}}\right\|=1$. Subsequently, we need to further
examine whether the inequality in Step 6 is established or not, where
${Q_{\delta}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t)}};{{\bf{w}}^{(t)}})$ is a
quadratic model of ${f_{3}}({\bf{w}},{\bf{x}})$ around ${{\bf{w}}^{(t)}}$ and
can be expressed as [46]

$\begin{split}&{Q_{\delta}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t)}};{{\bf{w}}^{(t)}})={f_{3}}({{\bf{w}}^{(t)}},{{\bf{x}}^{(t)}})\\\
+&\left\langle{{\nabla_{\bf{w}}}{f_{3}}({{\bf{w}}^{(t)}},{{\bf{x}}^{(t)}}),{{\bf{w}}^{(t+1)}}-{{\bf{w}}^{(t)}}}\right\rangle-\frac{1}{\delta}{\left\|{{{\bf{w}}^{(t+1)}}-{{\bf{w}}^{(t)}}}\right\|^{2}},\end{split}$
(33)

with
$\left\langle{{\bf{x}},{\bf{y}}}\right\rangle=2{\mathbb{R}}({{\bf{x}}^{T}}{\bf{y}})$.
If the inequality in Step 6 is not feasible, we need to shrink the step size
via Step 7 and then re-implement Steps 4$-$8 until the condition in Step 9 is
established. Otherwise, if the inequality in Step 6 is feasible, the update
for ${{\bf{w}}^{(t+1)}}$ can be stopped and we can continue to update
${\bf{x}}$ in the next, for which the details are similar and thus are omitted
here for brevity. The key point we should emphasize is that the update for
${{\bf{x}}^{(t+1)}}$ via the projection function ${{\cal
B}}\left\\{\cdot\right\\}$ in Step 12 is to guarantee that solutions for MAs’
positions in each iteration always satisfy the constraint in (10c), where the
projection function, based on the nearest-distance rule, can be easily derived
as:

$\begin{split}{\cal B}\left\\{{{{\bf{x}}^{(t+1)}},{\cal
C}}\right\\}\triangleright
x_{i}^{(t+1)}=\left\\{{\begin{array}[]{*{20}{c}}{{F_{i}}}&{x_{i}^{(t+1)}<{F_{i}}}\\\
{x_{i}^{(t+1)}}&{{F_{i}}\leq x_{i}^{(t+1)}\leq{G_{i}}}\\\
{{G_{i}}}&{x_{i}^{(t+1)}>{G_{i}}}\end{array}}\right..\end{split}$ (34)

To successfully implement Algorithm 2, we have to further derive the
expressions of ${\nabla_{\bf{w}}}{f_{3}}({\bf{w}},{\bf{x}})$ and
${\nabla_{\bf{x}}}{f_{3}}({\bf{w}},{\bf{x}})$, as shown in Corollary 1.

Corollary 1: ${\nabla_{{{\bf{w}}}}}{f_{3}}({\bf{w}},{\bf{x}})$ and
${\nabla_{{{\bf{x}}}}}{f_{3}}({\bf{w}},{\bf{x}})$ can be derived as in (35)
and (36), where
${\bf{C}}={\bf{u}}{{\bf{u}}^{T}}+{\bf{z}}{{\bf{z}}^{T}},{\bf{D}}={\bf{u}}{{\bf{z}}^{T}}-{\bf{z}}{{\bf{u}}^{T}}$,
${\bf{u}}$ and ${\bf{z}}$ denote the real part and imaginary part of
${\bf{w}}$, respectively, i.e., ${\bf{w}}={\bf{u}}+j{\bf{z}}$. In addition,
for each $i=0,1,...,M$,
${{\bf{g}}_{i}}={\left[{{g_{1,i}},{g_{2,i}},...,{g_{N,i}}}\right]^{T}}$,
${{\bf{q}}_{i}}={\left[{{q_{1,i}},{q_{2,i}},...,{q_{N,i}}}\right]^{T}}$, with

$\begin{split}{g_{n,i}}=\cos\left({\frac{{2\pi}}{\lambda}{x_{n}}\sin{\theta_{i}}}\right),{q_{n,i}}=-\sin\left({\frac{{2\pi}}{\lambda}{x_{n}}\sin{\theta_{i}}}\right),\end{split}$

and

$\begin{split}{{\bf{W}}_{i}}=&{\rm{diag}}\left({\left\\{{\frac{{2\pi}}{\lambda}\sin{\theta_{i}}\sin\left({\frac{{2\pi}}{\lambda}{x_{n}}\sin{\theta_{i}}}\right)}\right\\}_{n=1}^{N}}\right),\\\
{{\bf{S}}_{i}}=&{\rm{diag}}\left({\left\\{{\frac{{2\pi}}{\lambda}\sin{\theta_{i}}\cos\left({\frac{{2\pi}}{\lambda}{x_{n}}\sin{\theta_{i}}}\right)}\right\\}_{n=1}^{N}}\right).\end{split}$

###### Proof.

Corollary 1 can be proved by resorting to the differentiation method and
further noting that:

$\begin{split}{\nabla_{\bf{w}}}{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|^{2}}=&\overline{\bf{h}}_{i}^{H}({\bf{x}}){\overline{\bf{h}}_{i}}({\bf{x}}){\bf{w}},i=1,2,...,M,\\\
{\nabla_{\bf{w}}}{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|^{2}}=&{\bf{h}}_{0}^{H}{{\bf{h}}_{0}}{\bf{w}},\\\
\end{split}$

and

$\begin{split}{\nabla_{\bf{x}}}{\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|^{2}}\mathop{=}\limits^{(b)}&-{{\bf{W}}_{i}}(2{\bf{C}}{{\bf{g}}_{i}}+2{\bf{D}}{{\bf{q}}_{i}})\\\
&-{{\bf{S}}_{i}}(2{\bf{C}}{{\bf{q}}_{i}}-2{\bf{D}}{{\bf{g}}_{i}}),i=1,2,...M,\\\
{\nabla_{\bf{x}}}{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|^{2}}\mathop{=}\limits^{(c)}&{\beta_{0}}(-{{\bf{W}}_{0}}(2{\bf{C}}{{\bf{g}}_{0}}+2{\bf{D}}{{\bf{q}}_{0}})\\\
-&{{\bf{S}}_{0}}(2{\bf{C}}{{\bf{q}}_{0}}-2{\bf{D}}{{\bf{g}}_{0}})),\end{split}$

where equalities $\mathop{=}\limits^{(b)}$ and $\mathop{=}\limits^{(c)}$ are
established based on conclusions of our previous work [38]. ∎

Convergence analysis: According to [46], since in each iteration, the suitable
step size is chosen based on the backtracking line search to strictly
guarantee that the objective is no smaller than the quadratic model, the
objective will be strictly increasing w.r.t. the number of iterations.
Moreover, ${f_{3}}({\bf{w}},{\bf{x}})$ has its upper bound since the values of
${{{\left|{{{\bf{h}}_{0}}({\bf{x}}){\bf{w}}}\right|}^{2}}}$ and
${\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){\bf{w}}}\right|^{2}}$,
$i=1,2,...,M$, are all bounded given the constraints in (10b) and (10c).
Therefore, the objective value will converge to a stationary point.

The complexity of Algorithm 2: To simplify the analysis while still keeping a
good approximation to the complexity of Algorithm 2, we focus on the number of
complex multiplications required per iteration. Based on this rule and note
that given ${{\bf{w}}^{(t)}}$ and ${{\bf{x}}^{(t)}}$,
${\nabla_{\bf{w}}}{f_{3}}({{\bf{w}}^{(t)}},{{\bf{x}}^{(t)}})$ in Step 4 is
computed only once, and the complexity mainly comes from computing
$\overline{\bf{h}}_{i}^{H}({{\bf{x}}^{(t)}}){\overline{\bf{h}}_{i}}({{\bf{x}}^{(t)}}){{\bf{w}}^{(t)}}$,
$i=1,2,...,M$, and
${\bf{h}}_{0}^{H}({{\bf{x}}^{(t)}}){{\bf{h}}_{0}}({{\bf{x}}^{(t)}}){{\bf{w}}^{(t)}}$,
which is about ${\cal O}({N^{2}}M)$. While
${f_{3}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t)}})$ and
${Q_{\delta}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t)}};{{\bf{w}}^{(t)}})$ in Step 6
are computed frequently with the update of the step size in Step 7 (which
results in the update of ${{\bf{w}}^{(t+1)}}$ in Step 5). Focusing on the
expression of ${f_{3}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t)}})$, since the
complexity of computing
${\left|{{{\bf{h}}_{0}}({{\bf{x}}^{(t)}}){{\bf{w}}^{(t+1)}}}\right|^{2}}$ and
${\left|{{{\overline{\bf{h}}}_{i}}({{\bf{x}}^{(t)}}){{\bf{w}}^{(t+1)}}}\right|^{2}}$,
$i=1,2,...,M$, is about ${\cal O}(N)$, the complexity of computing
${f_{3}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t)}})$ would be ${{\cal
O}}\left({{N}M}\right)$. Also, focusing on the expression of
${Q_{\delta}}({{\bf{w}}^{(t+1)}},{{\bf{x}}^{(t)}};{{\bf{w}}^{(t)}})$, the
complexity of obtaining
$\left\langle{{\nabla_{\bf{w}}}{f_{3}}({{\bf{w}}^{(t)}},{{\bf{x}}^{(t)}}),{{\bf{w}}^{(t+1)}}-{{\bf{w}}^{(t)}}}\right\rangle$
is only ${\cal O}(N)$ since
${\nabla_{\bf{w}}}{f_{3}}({{\bf{w}}^{(t)}},{{\bf{x}}^{(t)}})$ is already
known. Then, further denote the number of iterations to obtain a feasible step
size for updating ${\bf{w}}$ as ${{I_{\delta,{\bf{w}}}}}$, the complexity of
Steps 3$-$9 is about ${\cal O}({N^{2}}M+{I_{\delta,{\bf{w}}}}(NM+N))$. Using
the same analysis, the complexity of Steps 10$-$16 can be derived as ${\cal
O}({N^{2}}M+{I_{\delta,{\bf{x}}}}(NM+N))$, where ${{I_{\delta,{\bf{x}}}}}$ is
the number of iterations to obtain a feasible step size for updating
${\bf{x}}$. Hence, the total complexity of Algorithm 2 is about ${\cal
O}\left({{I_{{\rm{out}}}}({N^{2}}M+({I_{\delta,{\bf{w}}}}+{I_{\delta,{\bf{x}}}})(NM+N))}\right)$,
with ${{I_{{\rm{out}}}}}$ denoting the number of iterations for alternately
optimizing ${\bf{w}}$ and ${\bf{x}}$.

The total complexity of Algorithm 1: Armed with the above analysis, we can
finally compute the total complexity for solving the initial problem (P3.2) as
${\cal
O}\left({\log(1/\tau){I_{{\rm{out}}}}({N^{2}}M+({I_{\delta,{\bf{w}}}}+{I_{\delta,{\bf{x}}}})(NM+N))}\right)$.

Algorithm 3 The PGD for Solving (P4)

1:Input: ${{\bf{x}}^{(0)}}\in{\cal C}$, $t\leftarrow 0$, $\delta>0$ and
$\chi<1$.

2: Repeat: $\%\%\%\%\%$ The steps for updating ${\bf{x}}$

3:
${{\bf{x}}^{(t+1)}}={{\bf{x}}^{(t)}}-\delta{\nabla_{\bf{x}}}\Theta({{\bf{x}}^{(t)}})$;

4: ${{\bf{x}}^{(t+1)}}={\cal B}\left\\{{{{\bf{x}}^{(t+1)}},{\cal
C}}\right\\}$;

5: if
$\Theta({{\bf{x}}^{(t+1)}})<{Q_{\delta}}({{\bf{x}}^{(t+1)}};{{\bf{x}}^{(t)}})$,
then

6: $\delta\leftarrow\chi\delta$,

7: end

8: Until:
$\Theta({{\bf{x}}^{(t+1)}})\geq{Q_{\delta}}({{\bf{x}}^{(t+1)}};{{\bf{x}}^{(t)}})$.

9: $t\leftarrow t+1$;

10:Until: $\Theta({{\bf{x}}^{(t)}})$ converges to a stationary point.

##  V A Competitive Benchmark

In this section, we further present a competitive benchmark for handling (P3).
This benchmark still employs the MA technique to reap the additional spatial
DoF and concurrently adopts the ZF-based transmit beamforming due to that: i)
it can effectively decrease the averaged signal power leaked to eavesdroppers;
ii) it enjoys low implementation complexities especially when the number of
antennas at Alice is larger.

Specifically, employing the ZF-based beamforming ${{{\bf{w}}_{{\rm{ZF}}}}}$
and without eavesdroppers’ instantaneous CSI, Alice aims to force the
receiving power gain at $\left\\{{{\rm{Ev}}{{\rm{e}}_{i}}}\right\\}_{i=1}^{M}$
via the LoS paths to zero, i.e.,
${\left|{{{\overline{\bf{h}}}_{i}}({\bf{x}}){{\bf{w}}_{{\rm{ZF}}}}}\right|^{2}}=0$,
$i=1,2,...,M$, and concurrently maximize the receiving power gain at Bob. Note
that this operation requires $N\geq M+1$.

Given positions of antennas at Alice as ${\bf{x}}$, ${{{\bf{w}}_{{\rm{ZF}}}}}$
can be expressed as
${{\bf{w}}_{{\rm{ZF}}}}({\bf{x}})={\bf{w}}({\bf{x}})/\left\|{{\bf{w}}({\bf{x}})}\right\|$
[13], with

$\begin{split}{\bf{w}}({\bf{x}})=&\left[{{{\bf{I}}_{N}}-{\bf{H}}({\bf{x}}){{\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)}^{-1}}{{\bf{H}}^{H}}({\bf{x}})}\right]{\bf{h}}_{0}^{H}({\bf{x}}),\\\
{\bf{H}}({\bf{x}})=&{[{\overline{\bf{h}}_{1}}({\bf{x}});{\overline{\bf{h}}_{2}}({\bf{x}});...;{\overline{\bf{h}}_{M}}({\bf{x}})]^{H}}\in{{\mathbb{C}}^{N\times
M}},\end{split}$ (37)

and ${\left(\cdot\right)^{-1}}$ denotes the inverse operation. Based on the
known ${{{\bf{w}}_{{\rm{ZF}}}}}$, the objective of (P3) can be simplified as
$\gamma\left({{\psi_{1}},{\psi_{2}}\left({\frac{{{P_{a}}{{\left|{{{\bf{h}}_{0}}({\bf{x}}){{\bf{w}}_{{\rm{ZF}}}}({\bf{x}})}\right|}^{2}}}}{{{\sigma^{2}}{2^{{R_{s}}}}}}+\frac{1}{{{2^{{R_{s}}}}}}-1}\right)}\right)$,
where
${\psi_{1}}=\frac{{{{\left({\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}}\right)}^{2}}}}{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}}}}$
and
${\psi_{2}}=\frac{{\sum\nolimits_{i=1}^{M}{\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}}}{{\sum\nolimits_{i=1}^{M}{{{\left({\frac{{{\beta_{i}}}}{{{K_{i}}+1}}}\right)}^{2}}}}}\frac{{{\sigma^{2}}}}{{{P_{a}}}}$.

Observing the above simplified objective, to minimize the secrecy outage
probability, it is obviously equivalent to maximizing
${{{\left|{{{\bf{h}}_{0}}({\bf{x}}){{\bf{w}}_{{\rm{ZF}}}}({\bf{x}})}\right|}^{2}}}$,
which can be further expanded as

$\begin{split}&{\left|{{{\bf{h}}_{0}}({\bf{x}}){{\bf{w}}_{{\rm{ZF}}}}({\bf{x}})}\right|^{2}}\\\
=&{\left|{{{\bf{h}}_{0}}({\bf{x}})\left[{{{\bf{I}}_{N}}-{\bf{H}}({\bf{x}}){{\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)}^{-1}}{{\bf{H}}^{H}}({\bf{x}})}\right]{\bf{h}}_{0}^{H}({\bf{x}})}\right|}\\\
=&{\beta_{0}}N-\underbrace{{{\bf{h}}_{0}}({\bf{x}}){\bf{H}}({\bf{x}}){{\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)}^{-1}}{{\bf{H}}^{H}}({\bf{x}}){\bf{h}}_{0}^{H}({\bf{x}})}_{\Theta({\bf{x}})}.\end{split}$
(38)

Therefore, to maximize
${{{\left|{{{\bf{h}}_{0}}({\bf{x}}){{\bf{w}}_{{\rm{ZF}}}}({\bf{x}})}\right|}^{2}}}$,
it is equivalent to minimizing ${\Theta({\bf{x}})}$, leading to the
optimization problem as

$\displaystyle({\rm{P4}}):{\rm{}}\mathop{\min}\limits_{{{\bf{x}}}}\
{\Theta({\bf{x}})}$ (${\rm{39a}}$) $\displaystyle\
{\rm{s.t.}}\quad(10{\rm{c}}).$ (${\rm{39b}}$)

$\begin{split}{}\frac{{\partial\Theta({\bf{x}})}}{{\partial{x_{n}}}}=&\frac{{\partial{\bf{h}}({\bf{x}})}}{{\partial{x_{n}}}}{\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)^{-1}}{{\bf{h}}^{H}}({\bf{x}})+{\bf{h}}({\bf{x}}){\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)^{-1}}\frac{{\partial{{\bf{h}}^{H}}({\bf{x}})}}{{\partial{x_{n}}}}\\\
&+{\bf{h}}({\bf{x}})\left[{-{{\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)}^{-1}}\left[{\frac{{\partial{{\bf{H}}^{H}}({\bf{x}})}}{{\partial{x_{n}}}}{\bf{H}}({\bf{x}})+{{\bf{H}}^{H}}({\bf{x}})\frac{{\partial{\bf{H}}({\bf{x}})}}{{\partial{x_{n}}}}}\right]{{\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)}^{-1}}}\right]{{\bf{h}}^{H}}({\bf{x}}).\end{split}$
(42)

Since ${\Theta({\bf{x}})}$ is neither convex or concave w.r.t. ${\bf{x}}$, we
now exploit the PGD to solve (P4), the key steps of which are similar to those
shown in Algorithm 2. Thus, we here provide a brief description as presented
in Algorithm 3, where ${\nabla_{\bf{x}}}\Theta({{\bf{x}}^{(t)}})$ is the
gradient of $\Theta({\bf{x}})$ w.r.t. ${\bf{x}}$ at ${{\bf{x}}^{(t)}}$, and
${Q_{\delta}}({{\bf{x}}^{(t+1)}};{{\bf{x}}^{(t)}})$ is a quadratic model of
$\Theta({\bf{x}})$ around ${{\bf{x}}^{(t)}}$ and can be expressed as

$\begin{split}&{Q_{\delta}}({{\bf{x}}^{(t+1)}};{{\bf{x}}^{(t)}})=\Theta({{\bf{x}}^{(t)}})\\\
+&\left\langle{{\nabla_{\bf{x}}}\Theta({{\bf{x}}^{(t)}}),{{\bf{x}}^{(t+1)}}-{{\bf{x}}^{(t)}}}\right\rangle-\frac{1}{\delta}{\left\|{{{\bf{x}}^{(t+1)}}-{{\bf{x}}^{(t)}}}\right\|^{2}}.\end{split}$
(40)

We now focus on deriving the closed-form ${\nabla_{\bf{x}}}\Theta({\bf{x}})$
as follows. Specifically, based on the known expressions of
${{\bf{h}}_{0}}({\bf{x}})$ and ${\bf{H}}({\bf{x}})$ shown in (37), we denote

$\begin{split}{\bf{h}}({\bf{x}})\buildrel\Delta\over{=}&{{\bf{h}}_{0}}({\bf{x}}){\bf{H}}({\bf{x}})\\\
=&\sqrt{{\beta_{0}}}{\left[{\begin{array}[]{*{20}{c}}{\sum\nolimits_{i=1}^{N}{{e^{j\frac{{2\pi}}{\lambda}{x_{i}}(\sin{\theta_{0}}-\sin{\theta_{1}})}}}}\\\
\vdots\\\
{\sum\nolimits_{i=1}^{N}{{e^{j\frac{{2\pi}}{\lambda}{x_{i}}(\sin{\theta_{0}}-\sin{\theta_{M}})}}}}\end{array}}\right]^{T}},\end{split}$
(41)

and thus
$\Theta({\bf{x}})={\bf{h}}({\bf{x}}){\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)^{-1}}{{\bf{h}}^{H}}({\bf{x}})$.

Corollary 2:
${\nabla_{\bf{x}}}\Theta({\bf{x}})={\left[{\frac{{\partial\Theta({\bf{x}})}}{{\partial{x_{1}}}},\frac{{\partial\Theta({\bf{x}})}}{{\partial{x_{2}}}},...,\frac{{\partial\Theta({\bf{x}})}}{{\partial{x_{N}}}}}\right]^{T}}$,
where ${\frac{{\partial\Theta({\bf{x}})}}{{\partial{x_{n}}}}}$, $n=1,2,...,N$,
is derived in (42).

###### Proof.

Note that

$\begin{split}\frac{{\partial\Theta({\bf{x}})}}{{\partial{x_{n}}}}=&\frac{{\partial{\bf{h}}({\bf{x}})}}{{\partial{x_{n}}}}{\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)^{-1}}{{\bf{h}}^{H}}({\bf{x}})\\\
&+{\bf{h}}({\bf{x}})\frac{{\partial{{\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)}^{-1}}}}{{\partial{x_{n}}}}{{\bf{h}}^{H}}({\bf{x}})\\\
&+{\bf{h}}({\bf{x}}){\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)^{-1}}\frac{{\partial{{\bf{h}}^{H}}({\bf{x}})}}{{\partial{x_{n}}}},\end{split}$
(43)

and since $d{{\bf{A}}^{-1}}=-{{\bf{A}}^{-1}}(d{\bf{A}}){{\bf{A}}^{-1}}$, we
can obtain

$\begin{split}&\frac{{\partial{{\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)}^{-1}}}}{{\partial{x_{n}}}}\\\
=&-{\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)^{-1}}\frac{{\partial\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)}}{{\partial{x_{n}}}}{\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)^{-1}},\end{split}$
(44)

where

$\begin{split}\frac{{\partial\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)}}{{\partial{x_{n}}}}=\frac{{\partial{{\bf{H}}^{H}}({\bf{x}})}}{{\partial{x_{n}}}}{\bf{H}}({\bf{x}})+{{\bf{H}}^{H}}({\bf{x}})\frac{{\partial{\bf{H}}({\bf{x}})}}{{\partial{x_{n}}}}.\end{split}$
(45)

Substituting (45) into (44) and further substituting (44) into (43), we can
obtain the result in (42). This completes the proof. ∎

Given ${\bf{x}}$, to compute the value of
$\frac{{\partial\Theta({\bf{x}})}}{{\partial{x_{n}}}}$, we need to derive the
expressions of $\frac{{\partial{\bf{h}}({\bf{x}})}}{{\partial{x_{n}}}}$
($\frac{{\partial{{\bf{h}}^{H}}({\bf{x}})}}{{\partial{x_{n}}}}={\left[{\frac{{\partial{\bf{h}}({\bf{x}})}}{{\partial{x_{n}}}}}\right]^{H}}$)
and ${\frac{{\partial{\bf{H}}({\bf{x}})}}{{\partial{x_{n}}}}}$
($\frac{{\partial{{\bf{H}}^{H}}({\bf{x}})}}{{\partial{x_{n}}}}={\left[{\frac{{\partial{\bf{H}}({\bf{x}})}}{{\partial{x_{n}}}}}\right]^{H}}$)
as

$\begin{split}&\frac{{\partial{\bf{h}}({\bf{x}})}}{{\partial{x_{n}}}}\\\
=&\sqrt{{\beta_{0}}}\frac{{2\pi}}{\lambda}{\left[{\begin{array}[]{*{20}{c}}{(\sin{\theta_{0}}-\sin{\theta_{1}}){e^{j\left[{\frac{{2\pi}}{\lambda}{x_{n}}(\sin{\theta_{0}}-\sin{\theta_{1}})+\frac{\pi}{2}}\right]}}}\\\
\vdots\\\
{(\sin{\theta_{0}}-\sin{\theta_{M}}){e^{j\left[{\frac{{2\pi}}{\lambda}{x_{n}}(\sin{\theta_{0}}-\sin{\theta_{M}})+\frac{\pi}{2}}\right]}}}\end{array}}\right]^{T}},\end{split}$
(46)

and

$\begin{split}\frac{{\partial{\bf{H}}({\bf{x}})}}{{\partial{x_{n}}}}=\frac{{2\pi}}{\lambda}\left[{\begin{array}[]{*{20}{c}}{{{\bf{0}}^{1\times
M}}}\\\ {...}\\\ {{\bf{b}}({x_{n}})}\\\ {...}\\\ {{{\bf{0}}^{1\times
M}}}\end{array}}\right],\end{split}$ (47)

with

$\begin{split}{\bf{b}}({x_{n}})={\left[{\begin{array}[]{*{20}{c}}{\sin{\theta_{1}}{e^{-j\left({\frac{{2\pi}}{\lambda}{x_{n}}\sin{\theta_{1}}+\frac{\pi}{2}}\right)}}}\\\
\vdots\\\
{\sin{\theta_{M}}{e^{-j\left({\frac{{2\pi}}{\lambda}{x_{n}}\sin{\theta_{M}}+\frac{\pi}{2}}\right)}}}\end{array}}\right]^{T}}.\end{split}$
(48)

Convergence analysis: Similarly, since the suitable step size is chosen based
on the backtracking line search to guarantee that the objective is not larger
than the quadratic model, the objective will be strictly decreasing w.r.t. the
number of iterations. In addition, obviously $\Theta({\bf{x}})$ has its lower
bound as zero, at which the maximum receiving power gain is achieved at Bob,
i.e.,
${\left|{{{\bf{h}}_{0}}({\bf{x}}){{\bf{w}}_{{\rm{ZF}}}}({\bf{x}})}\right|^{2}}={\beta_{0}}N-\Theta({\bf{x}})={\beta_{0}}N$.
Based on the above analysis, the objective value of (P4) will converge to a
stationary point via Algorithm 3.

The complexity of Algorithm 3: We first analyze the complexity of computing
$\frac{{\partial\Theta({\bf{x}})}}{{\partial{x_{n}}}}$, $n=1,2,...,N$.
Specifically, note that computing
${{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})$,
$\frac{{\partial{{\bf{H}}^{H}}({\bf{x}})}}{{\partial{x_{n}}}}{\bf{H}}({\bf{x}})$
and
$\frac{{\partial{\bf{H}}({\bf{x}})}}{{\partial{x_{n}}}}{{\bf{H}}^{H}}({\bf{x}})$
requires the complexities of ${{\cal O}}(N{M^{2}})$, ${{\cal O}}(N{M^{2}})$
and ${{\cal O}}(M{N^{2}})$, respectively. Further, for a matrix
${\bf{A}}\in{{\mathbb{C}}^{M\times M}}$, computing ${{\bf{A}}^{-1}}$ requires
the complexity of ${{\cal O}}({M^{3}})$. Based on the above analysis, the
complexity of computing
$\frac{{\partial{{\left({{{\bf{H}}^{H}}({\bf{x}}){\bf{H}}({\bf{x}})}\right)}^{-1}}}}{{\partial{x_{n}}}}$
is about ${{\cal O}}(N{M^{2}}+{M^{3}})$, according to which it is easy to
derive the complexity of computing
$\frac{{\partial\Theta({\bf{x}})}}{{\partial{x_{n}}}}$ as ${{\cal
O}}(N{M^{2}}+{M^{3}})$. Therefore, obtaining
${\nabla_{\bf{x}}}\Theta({\bf{x}})$ requires the complexity of ${{\cal
O}}({N^{2}}{M^{2}}+N{M^{3}})$. Next, using the similar analysis method in the
above for handing Algorithm 2, we can directly derive the total complexity for
solving (P4) as ${{\cal
O}}\left({{I_{{\rm{out}}}}({N^{2}}{M^{2}}+N{M^{3}}+{I_{\delta,{\bf{x}}}}(N{M^{2}}+{M^{3}}))}\right)$,
where ${I_{\delta,{\bf{x}}}}$ is the number of iterations to obtain a feasible
step size and ${I_{{\rm{out}}}}$ is the number of iterations until the
objective converges to a stationary value.

##  VI Simulation Results

In this section, we provide numerical results to demonstrate the effectiveness
of our proposed algorithms for minimizing the secrecy outage probability.
Unless otherwise stated, the minimum distance between any two MAs is set as
${d_{\min}}=\lambda/2$, where $\lambda$ is set as one for simplification and
without affecting the obtained results, the Rician K-factors for all
eavesdropping channels are set to be the same , i.e.,
$\left\\{{{K_{i}}}\right\\}_{i=1}^{M}=K$, and the targeted PLS coding rate of
Alice’s encoder is set as $R_{s}=3$ bits/s/Hz. In addition, we set the large-
scale fading power of the main channel and the eavesdropping channels as
$\left\\{{{\beta_{i}}}\right\\}_{i=0}^{M}=1$. Here, such power gains are
normalized over the receiver noise power so that we can conveniently set the
noise power as ${\sigma^{2}}=1$.

Figure 6: Convergence behavior of the proposed APGA for handing (P3.5), where
${P_{a}}=25$ dB, $N=5$, $M=1$, $L=4$, $K=4$, ${\theta_{0}}=\frac{\pi}{4}$ and
${\theta_{1}}=\frac{{1.1\pi}}{4}$. Figure 7: Convergence behavior of the
proposed PGD for handling (P4), where ${P_{a}}=15$ dB, $N=5$, $M=2$, $L=4$,
$K=4$ and ${\theta_{0}}=\frac{\pi}{4}$.

###  VI-A Convergence Analysis of the APGA and the PGD for Handing (P3.5) and
(P4)

We first illustrate the convergence behavior of the proposed APGA for solving
(P3.5) as shown Fig. 6, from which it is observed that: i) given each
$\varepsilon=0.51,0.52,0.53$, the objective of (P3.5) always converges to a
stationary value after hundreds of iterations, thus verifying the
effectiveness of our proposed algorithm; ii) as $\varepsilon$ increases, the
stationary value of the objective of (P3.5) will decrease accordingly, and
this behavior can be explained based on (30). That is to say, as $\varepsilon$
increases, $\kappa(\varepsilon){f_{1}}({\bf{w}},{\bf{x}})+\rho(\varepsilon)$
will increase by observing Fig. 3. Thus,
${f_{2}}({\bf{w}},{\bf{x}})-\kappa(\varepsilon){f_{1}}({\bf{w}},{\bf{x}})-\rho(\varepsilon)$
will inevitably decrease w.r.t. $\varepsilon$; iii) note that Algorithm 1 aims
to determine the maximum value of $\varepsilon$ that makes the optimal output
of (P3.5) not smaller than zero. Based on this conclusion, it is easy to
determine that the optimal $\varepsilon$ of (P3.2) is 0.52. Then, the obtained
secrecy outage probability in this case is
${P_{{\rm{out}}}}=1-\varepsilon=0.48$.

We now illustrate the convergence behavior of the proposed PGD for solving
(P4) as shown in Fig. 7, from which it is observed that: i) given different
combinations of ${\theta_{1}}$ and ${\theta_{2}}$, the objective of (P4)
always converges to a stationary value after about five iterations, which
indicates that Algorithm 3 enjoys a low complexity; ii) the stationary
objective of (P4), i.e., $\Theta({\bf{x}})$, becomes smaller when
${\theta_{1}}$ and ${\theta_{2}}$ become larger. For instance, when
${\theta_{1}}=\frac{{1.7\pi}}{4}$ and ${\theta_{2}}=\frac{{1.8\pi}}{4}$, the
stationary value of $\Theta({\bf{x}})$ approaches zero; while when
${\theta_{1}}=\frac{{1.5\pi}}{4}$ and ${\theta_{2}}=\frac{{1.6\pi}}{4}$, the
stationary value of $\Theta({\bf{x}})$ is even greater than 0.5. This
phenomenon is due to that in this case (${\theta_{0}}=\frac{\pi}{4}$), when
${\theta_{1}}$ and ${\theta_{2}}$ increases, the correlation between the main
channel and the wiretap channel becomes smaller, and thus $\Theta({\bf{x}})$
should become smaller such that the receiving power gain at Bob via the ZF-
based beamforming, i.e., ${\beta_{0}}N-\Theta({\bf{x}})$, becomes larger.

###  VI-B Performance Comparison

We now formally present the achievable performance of our proposed algorithms.
For simplification, we denote the schemes designed in Algorithm 1 and
Algorithm 3 as MA+OB and MA+ZF, respectively. In addition, for comprehensive
comparisons, we further consider the following five benchmarks:

  * •

RAP+OB: Randomly generate ${\bf{x}}$ satisfying (10c) for 100 independent
realizations. Given each realization for ${\bf{x}}$, obtain the secrecy outage
probability by only optimizing ${\bf{w}}$ using the PGA mentioned in Algorithm
2. Finally, compute 100 results corresponding to different ${\bf{x}}$ and
select the minimum value.

  * •

RAP+ZF: Compared to RAP+OB, the only difference is that given each realization
for ${\bf{x}}$, ${\bf{w}}$ is designed by adopting the ZF-based beamforming
shown in (37).

  * •

FPA+OB: Compared to RAP+OB, the only difference is that positions of antennas
at Alice is fixed as ${\bf{x}}={[0,0.5,1,...,(N-1)2]^{T}}$.

  * •

FPA+ZF: Positions of antennas of Alice is fixed as
${\bf{x}}={[0,0.5,1,...,(N-1)2]^{T}}$, and ${\bf{w}}$ is designed by adopting
the ZF-based beamforming shown in (37).

  * •

MA+MRT: Alice adopts the MRT-based beamforming and concurrently determines
positions of antennas at itself using the PGA mentioned in Algorithm 2.

Fig. 8 first illustrates the secrecy outage probability w.r.t. the Rician-
factor $K$, from which it is observed: i) when $K=0$, i.e., the wiretap
channel suffers from the special case of Rayleigh fading, as revealed in
Insight 2, positions of antennas at Alice can be arbitrary and Alice just
adopts the MRT-based beamforming to achieve the optimal performance. Hence,
obviously MA+OB, RAP+OB, FPA+OB and MA+MRT can concurrently achieve the
optimal solution. While due to the inappropriate beamforming, MA+ZF, RAP+ZF
and FPA+ZF achieve a undesirable performance; ii) when $K>0$, the secrecy
outage probability achieved by MA+OB, RAP+OB and FPA+OB first increases and
then decreases. This is due to that when $K$ increases from zero to a positive
number, there will exist the sudden increase for the LoS component of the
wiretap channel, and so does the averaged receiving power at eavesdroppers via
the LoS paths. While when $K$ continues to increase, for the wiretap channel,
the LoS component becomes dominant and the NLoS component gradually vanishes.
Hence, the averaged receiving power at eavesdroppers via the NLoS paths will
become smaller. At the same time, even the LoS paths are dominant, the
averaged receiving power at eavesdroppers via them can also be reduced via the
joint design of ${\bf{w}}$ and ${\bf{x}}$. For MA+ZF, RAP+ZF and FPA+ZF, since
the averaged receiving power at eavesdroppers via the LoS paths always equals
zero regardless of how $K$ is, while such power via the NLoS paths reduces as
$K$ increases, the achievable secrecy outage probability of these schemes will
decrease w.r.t. $K$; iii) due to the random set for positions of antennas at
Alice, RAP+OB (RAP+ZF) results in a higher secrecy outage probability compared
to MA+OB (MA+ZF); iv) for FPA+OB and FPA+ZF, since antennas’ positions at
Alice are fixed, i.e., no additional spatial DoF can be exploited, the
correlation between the main channel and the wiretap channel cannot be
reduced, which thus results in the larger secrecy outage probability. For
instance, when $K=7$, compared to MA+OB, FPA+OB results in the $30\%$
performance loss; v) even the ZF-based beamforming can completely eliminate
the averaged receiving power at eavesdroppers via the LoS paths, such
beamforming also results in a smaller power gain at Bob, especially when the
AoDs ${\theta_{0}}$ and ${\theta_{1}}$ are close to each other. Hence,
obviously MA+ZF (RAP+ZF) is slightly inferior to MA+OB (RAP+OB); vi) since the
MRT-based beamforming just cares about the averaged receiving power at Bob and
totally ignores such power at eavesdroppers, MA+MRT will achieve the worst
performance when $K>0$; vii) as $K$ increases to infinity, except for MA+MRT,
predictably the secrecy outage probability achieved by the other schemes will
decrease and then approach zero. The reasons have been explained in Insight 3
and are not repeated here.

Figure 8: Secrecy outage probability versus the Rician-factor $K$, where
$P_{a}=15$ dB, $N=5$, $M=1$, $L=3$, ${\theta_{0}}=\frac{\pi}{4}$ and
${\theta_{1}}=\frac{{1.1\pi}}{4}$. Figure 9: Secrecy outage probability versus
the correlation (${\theta_{1}}/{\theta_{0}}$) between the main channel and the
wiretap channel, where $P_{a}=15$ dB, $N=5$, $M=1$, $L=3$, $K=10$ and
${\theta_{0}}=\frac{\pi}{4}$. Figure 10: Secrecy outage probability versus the
length of antennas’ movement region $L$, where $P_{a}=15$ dB, $N=5$, $M=1$,
$K=4$, ${\theta_{0}}=\frac{\pi}{4}$ and ${\theta_{1}}=\frac{{1.1\pi}}{4}$.
Figure 11: Secrecy outage probability versus the power budget ($P_{a}$) of
Alice, where $N=5$, $M=1$, $L=4$, $K=4$ and ${\theta_{0}}=\frac{\pi}{4}$, and
${\theta_{1}}=\frac{{1.1\pi}}{4}$. Figure 12: Secrecy outage probability
versus the number of antennas ($N$) at Alice, where $P_{a}=15$ dB, $M=1$,
$L=4$, $K=4$, ${\theta_{0}}=\frac{\pi}{4}$, and
${\theta_{1}}=\frac{{1.1\pi}}{4}$. Figure 13: Secrecy outage probability
versus the number of eavesdroppers $M$, where $P_{a}=25$ dB, $N=8$, $L=5.5$,
$K=4$, ${\theta_{0}}=\frac{\pi}{4}$, ${\theta_{1}}=\frac{{1.15\pi}}{4}$,
${\theta_{2}}=\frac{{1.35\pi}}{4}$, ${\theta_{3}}=\frac{{1.4\pi}}{4}$,
${\theta_{4}}=\frac{{1.45\pi}}{4}$, ${\theta_{5}}=\frac{{1.5\pi}}{4}$,
${\theta_{6}}=\frac{{1.55\pi}}{4}$ and ${\theta_{7}}=\frac{{1.6\pi}}{4}$.

Fig. 9 presents the secrecy outage probability w.r.t. the correlation between
the main channel and the wiretap channel, where the correlation is denoted as
${\theta_{1}}/{\theta_{0}}$. We can observe that, as
${\theta_{1}}/{\theta_{0}}$ increases, i.e., the correlation between two kinds
of channels becomes smaller, the secrecy outage probability achieved by all
schemes obviously reduces. Further, the performance of MA+ZF (RAP+ZF) almost
approaches that of MA+OB (RAP+OB) when ${\theta_{1}}/{\theta_{0}}$ is larger
than $1.25$. Therefore, it is concluded that when the correlation between the
main channel and the wiretap channel reduces, considering the implementation
complexity, it is enough to just adopt the ZF-based beamforming. In addition,
we can similarly observe that, by exploiting the additional spatial DoF
resulting from antennas’ flexible movements, MA+OB (MA+ZF) achieves an
excellent performance compared to FPA+OB (FPA+ZF).

Fig. 10 shows the secrecy outage probability w.r.t. the length of antennas’
movement region $L$, from which we can observe that, as $L$ increases, there
will exist a larger and more flexible space that can be exploited for
deploying Alice’s antennas to fully decrease the correlation between the main
channel and the wiretap channel. Therefore, obviously the secrecy outage
probability achieved by MA+OB, RAP+OB, MA+ZF, RAP+ZF and MA+MRT will become
smaller w.r.t. $L$. Further, as $L$ increases, due to the smaller correlation,
the ZF-based transmit beamforming will achieve the pretty good performance,
indicating that MA+ZF can be adopted in a larger $L$ region. In addition, as
$L$ continues to increase, the secrecy outage probability achieved by MA+OB,
RAP+OB, MA+ZF, RAP+ZF and MA+MRT will converge to a constant. This phenomenon
indicates that it is not necessary to expand $L$ indefinitely and only a
limited span is enough to achieve the optimal performance.

Fig. 11 presents the secrecy outage probability w.r.t. the power budget
($P_{a}$) of Alice, from which we can observe that, as $P_{a}$ increases, the
secrecy outage probability achieved by all schemes will first decrease and
then converge to a constant. The reasons have been explained in Insight 1 and
thus are omitted here.

Fig. 12 illustrates the secrecy outage probability w.r.t. the number of
antennas at Alice ($N$), from which we can observe that, as $N$ increases,
Alice could implement more powerful beamforming to reduce and increase the
averaged receiving power at eavesdroppers and Bob, respectively. Therefore,
the secrecy performance of all schemes (except for MA+MRT) becomes better. In
addition, we can intuitively see that when achieving the same secrecy outage
probability, e.g., 0.62, FPA+OB will employ additional four antennas (so does
the number of RF chains) as compared to MA+OB. Therefore, the hardware costs
can be effectively reduced by flexibly optimizing positions of antennas at
Alice.

Finally, Fig. 13 shows the secrecy outage probability w.r.t. the number of
eavesdroppers ($M$). We can see that as $M$ increases, more eavesdroppers can
jointly decode the signal transmitted by Alice, leading to a better receiving
power gain at eavesdroppers. Therefore, the secrecy performance achieved by
all schemes obviously deteriorates.

##  VII Conclusion

In this paper, secure transmission empowered by movable antennas technology is
investigated, under the setup where Alice does not master the instantaneous
CSI of the wiretap channel. The tight distribution of the receiving power
gains at eavesdroppers is first obtained using the Laguerre series
approximations, according to which the approximate expression of the secrecy
outage probability is successfully derived. Afterwards, we formulate the
problem of minimizing the secrecy outage probability by jointly optimizing the
transmit beamforming and antennas’ positions at Alice. Unfortunately, the
objective includes the complex incomplete gamma function, which greatly
hinders the optimization process. To tackle this, we effectively approximate
the inverse of the incomplete gamma function as a simple linear model,
according to which the problem can be transformed into a simple one. Then, we
develop an alternating projected gradient ascent algorithm and further a
projected gradient descent algorithm to tackle the problem with different
implementation complexities. It is shown by simulation results the
effectiveness of our proposed schemes compared to numerous benchmarks.

## References

  * [1] V.-L. Nguyen, P.-C. Lin, B.-C. Cheng, R.-H. Hwang, and Y.-D. Lin, “Security and privacy for 6G: A survey on prospective technologies and challenges,” _IEEE Commun. Surveys Tuts._ , vol. 23, no. 4, pp. 2384–2428, 2021.
  * [2] M. Bloch and et.al., “An overview of information-theoretic security and privacy: Metrics, limits and applications,” _IEEE J. Sel. Areas Inf. Theory._ , vol. 2, no. 1, pp. 5–22, 2021.
  * [3] J. Li and A. P. Petropulu, “On beamforming solution for secrecy capacity of mimo wiretap channels,” in _2011 IEEE GLOBECOM Workshops (GC Wkshps)_ , 2011, pp. 889–892.
  * [4] K. Cao, B. Wang, H. Ding, L. Lv, R. Dong, T. Cheng, and F. Gong, “Improving physical layer security of uplink NOMA via energy harvesting jammers,” _IEEE Trans. Inf. Forensics Security._ , vol. 16, pp. 786–799, 2021.
  * [5] S. Fang, G. Chen, Z. Abdullah, and Y. Li, “Intelligent omni surface-assisted secure MIMO communication networks with artificial noise,” _IEEE Commun. Lett._ , vol. 26, no. 6, pp. 1231–1235, 2022.
  * [6] K. Ren, H. Su, and Q. Wang, “Secret key generation exploiting channel characteristics in wireless communications,” _IEEE Wireless Commun._ , vol. 18, no. 4, pp. 6–12, 2011.
  * [7] Z. Xiao, L. Zhu, Y. Liu, P. Yi, R. Zhang, X.-G. Xia, and R. Schober, “A survey on millimeter-wave beamforming enabled UAV communications and networking,” _IEEE Commun. Surveys Tuts._ , vol. 24, no. 1, pp. 557–610, 2022.
  * [8] W. Xu, Z. Peng, and S. Jin, “On secrecy of a multi-antenna system with eavesdropper in close proximity,” _IEEE Signal Process. Lett._ , vol. 22, no. 10, pp. 1525–1529, 2015.
  * [9] L. Zhu, W. Ma, and R. Zhang, “Movable antennas for wireless communication: Opportunities and challenges,” _IEEE Commun. Mag._ , pp. 1–7, 2023.
  * [10] K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Fluid antenna systems,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 3, pp. 1950–1962, 2021.
  * [11] K. K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Performance limits of fluid antenna systems,” _IEEE Commun. Lett._ , vol. 24, no. 11, pp. 2469–2472, 2020.
  * [12] K.-K. Wong and K.-F. Tong, “Fluid antenna multiple access,” _IEEE Trans. Wireless Commun._ , vol. 21, no. 7, pp. 4801–4815, 2022.
  * [13] L. Zhu, W. Ma, and R. Zhang, “Movable-antenna array enhanced beamforming: Achieving full array gain with null steering,” _IEEE Commun. Lett._ , vol. 27, no. 12, pp. 3340–3344, 2023.
  * [14] G. Hu, Q. Wu, D. Xu, K. Xu, J. Si, Y. Cai, and N. Al-Dhahir, “Intelligent reflecting surface-aided wireless communication with movable elements,” _IEEE Wireless Commun. Lett._ , pp. 1–1, 2024.
  * [15] M. Gharavi-Alkhansari and A. Gershman, “Fast antenna subset selection in MIMO systems,” _IEEE Trans. Signal Process._ , vol. 52, no. 2, pp. 339–347, 2004.
  * [16] H. Do, N. Lee, and A. Lozano, “Reconfigurable ULAs for line-of-sight mimo transmission,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 5, pp. 2933–2947, 2021.
  * [17] J. Zheng, J. Zhang, H. Du, D. Niyato, S. Sun, B. Ai, and K. B. Letaief, “Flexible-position MIMO for wireless communications: Fundamentals, challenges, and future directions,” _IEEE Wireless Commun._ , pp. 1–9, 2024\.
  * [18] H. Qin, W. Chen, Z. Li, Q. Wu, N. Cheng, and F. Chen, “Antenna positioning and beamforming design for fluid antenna-assisted multi-user downlink communications,” _IEEE Wireless Commun. Lett._ , pp. 1–1, 2024.
  * [19] L. Zhu, W. Ma, and R. Zhang, “Modeling and performance analysis for movable antenna enabled wireless communications,” _IEEE Trans. Wireless Commun._ , pp. 1–1, 2023.
  * [20] W. Ma, L. Zhu, and R. Zhang, “Compressed sensing based channel estimation for movable antenna communications,” _IEEE Commun. Lett._ , vol. 27, no. 10, pp. 2747–2751, 2023.
  * [21] H. Xu, G. Zhou, K.-K. Wong, W. K. New, C. Wang, C.-B. Chae, R. Murch, S. Jin, and Y. Zhang, “Channel estimation for FAS-assisted multiuser mmwave systems,” _IEEE Commun. Lett._ , vol. 28, no. 3, pp. 632–636, 2024.
  * [22] Z. Zhang, J. Zhu, L. Dai, and R. W. H. Jr, “Successive bayesian reconstructor for channel estimation in fluid antenna systems,” 2024, [Online] Available: https://arxiv.org/abs/2312.06551.
  * [23] W. Ma, L. Zhu, and R. Zhang, “MIMO capacity characterization for movable antenna systems,” _IEEE Trans. Wireless Commun._ , pp. 1–1, 2023.
  * [24] Y. Ye, L. You, J. Wang, H. Xu, K.-K. Wong, and X. Gao, “Fluid antenna-assisted MIMO transmission exploiting statistical CSI,” _IEEE Commun. Lett._ , vol. 28, no. 1, pp. 223–227, 2024.
  * [25] X. Chen, B. Feng, Y. Wu, D. W. Kwan Ng, and R. Schober, “Joint beamforming and antenna movement design for moveable antenna systems based on statistical csi,” in _GLOBECOM 2023 - 2023 IEEE Global Communications Conference_ , 2023, pp. 4387–4392.
  * [26] X. Pi, L. Zhu, Z. Xiao, and R. Zhang, “Multiuser communications with movable-antenna base station via antenna position optimization,” in _2023 IEEE Globecom Workshops (GC Wkshps)_ , 2023, pp. 1386–1391.
  * [27] G. Hu, Q. Wu, K. Xu, J. Ouyang, J. Si, Y. Cai, and N. Al-Dhahir, “Fluid antennas-enabled multiuser uplink: A low-complexity gradient descent for total transmit power minimization,” _IEEE Commun. Lett._ , vol. 28, no. 3, pp. 602–606, 2024.
  * [28] S. Yang, W. Lyu, B. Ning, Z. Zhang, and C. Yuen, “Flexible precoding for multi-user movable antenna communications,” _IEEE Wireless Commun. Lett_ , pp. 1–1, 2024.
  * [29] Y. Wu, D. Xu, D. W. K. Ng, W. Gerstacker, and R. Schober, “Movable antenna-enhanced multiuser communication: Optimal discrete antenna positioning and beamforming,” 2023, [Online] Available: https://arxiv.org/abs/2308.02304.
  * [30] X. Shao, R. Zhang, Q. Jiang, and R. Schober, “6D movable antenna enhanced wireless network via discrete position and rotation optimization,” 2024, [Online] Available: https://arxiv.org/abs/2403.17122.
  * [31] X. Shao, Q. Jiang, and R. Zhang, “6D movable antenna based on user distribution: Modeling and optimization,” 2024, [Online] Available: https://arxiv.org/abs/2403.08123.
  * [32] L. Zhu, W. Ma, Z. Xiao, and R. Zhang, “Performance analysis and optimization for movable antenna aided wideband communications,” 2024, [Online] Available: https://arxiv.org/abs/2401.08974.
  * [33] H. Wang, Q. Wu, and W. Chen, “Movable antenna enabled interference network: Joint antenna position and beamforming design,” 2024, [Online] Available: https://arxiv.org/abs/2403.13573.
  * [34] B. Lyu, H. Liu, W. Hong, S. Gong, and F. Tian, “Primary rate maximization in movable antennas empowered symbiotic radio communications,” 2024, [Online] Available: https://arxiv.org/abs/2403.14943.
  * [35] G. Hu, Q. Wu, J. Ouyang, K. Xu, Y. Cai, and N. Al-Dhahir, “Movable-antenna-array-enabled communications with CoMP reception,” _IEEE Commun. Lett_ , pp. 1–1, 2024.
  * [36] D. Zhang, S. Ye, M. Xiao, K. Wang, M. D. Renzo, and M. Skoglund, “Fluid antenna array enhanced over-the-air computation,” _IEEE Wireless Commun. Lett_ , pp. 1–1, 2024.
  * [37] Z. Cheng, N. Li, J. Zhu, X. She, C. Ouyang, and P. Chen, “Enabling secure wireless communications via movable antennas,” in _ICASSP 2024 - 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2024, pp. 9186–9190.
  * [38] G. Hu, Q. Wu, K. Xu, J. Si, and N. Al-Dhahir, “Secure wireless communication via movable-antenna array,” _IEEE Signal Process. Lett._ , vol. 31, pp. 516–520, 2024.
  * [39] J. Tang, C. Pan, Y. Zhang, H. Ren, and K. Wang, “Secure MIMO communication relying on movable antennas,” 2024, [Online] Available: https://arxiv.org/abs/2403.04269.
  * [40] H.-M. Wang, J. Bai, and L. Dong, “Intelligent reflecting surfaces assisted secure transmission without eavesdropper’s CSI,” _IEEE Signal Process. Lett._ , vol. 27, pp. 1300–1304, 2020.
  * [41] M. Li, X. Tao, N. Li, H. Wu, and J. Xu, “Secrecy energy efficiency maximization in UAV-enabled wireless sensor networks without eavesdropper’s CSI,” _IEEE Internet Things J._ , vol. 9, no. 5, pp. 3346–3358, 2022.
  * [42] B. Li, M. Zhang, Y. Rong, and Z. Han, “Artificial noise-aided secure relay communication with unknown channel knowledge of eavesdropper,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 5, pp. 3168–3179, 2021.
  * [43] Y. Liu, Z. Su, C. Zhang, and H.-H. Chen, “Minimization of secrecy outage probability in reconfigurable intelligent surface-assisted mimome system,” _IEEE Trans. Wireless Commun._ , vol. 22, no. 2, pp. 1374–1387, 2023.
  * [44] S. Yan and R. Malaney, “Location-based beamforming for enhancing secrecy in rician wiretap channels,” _IEEE Trans. Wireless Commun._ , vol. 15, no. 4, pp. 2780–2791, 2016.
  * [45] S. Primak, V. Kontorovich, and V. Lyandres, “Stochastic methods and their applications to communications: Stochastic differential equations approach,” _Hoboken, NJ, USA: Wiley_ , 2004.
  * [46] N. S. Perovic, L.-N. Tran, M. D. Renzo, and M. F. Flanagan, “On the maximum achievable sum-rate of the RIS-aided MIMO broadcast channel,” _IEEE Trans. Signal Process._ , vol. 70, pp. 6316–6331, 2022.