# Movable Frequency Diverse Array-Assisted Covert Communication With Multiple
Wardens

Zihao Cheng, Jiangbo Si, _Member, IEEE_ , Zan Li, _Senior Member, IEEE_ ,
Pengpeng Liu, Xiaoting Wang and Naofal Al-Dhahir, _Fellow, IEEE_

###### Abstract

The frequency diverse array (FDA) is highly promising for improving covert
communication performance by adjusting the frequency of each antenna at the
transmitter. However, when faced with the cases of multiple wardens and highly
correlated channels, FDA is limited by the frequency constraint and cannot
provide satisfactory covert performance. In this paper, we propose a novel
movable FDA (MFDA) antenna technology where positions of the antennas can be
dynamically adjusted in a given finite region. Specifically, we aim to
maximize the covert rate by jointly optimizing the antenna beamforming vector,
antenna frequency vector and antenna position vector. To solve this non-convex
optimization problem with coupled variables, we develop a two-stage
alternating optimization (AO) algorithm based on the block successive upper-
bound minimization (BSUM) method. Moreover, considering the challenge of
obtaining perfect channel state information (CSI) at multiple wardens, we
study the case of imperfect CSI. Simulation results demonstrate that MFDA can
significantly enhance covert performance compared to the conventional FDA. In
particular, when the frequency constraint is strict, MFDA can further increase
the covert rate by adjusting the positions of antennas instead of the
frequencies.

###### Index Terms:

Covert communication, Frequency diverse array, Movable antenna, Phased array.

##  I Introduction

Covert communication is a promising technology that has attracted extensive
attention in recent years. Many techniques have been proposed to enhance the
covert performance, where multi-antenna technique exploit the spatial degrees
of freedom (DoFs) in transmit antennas to focus the power of signals toward
the legitimate receiver and away from the warden by smartly designing the
beamformer [1, 2, 3, 4, 5]. In particular, based on the global perfect Channel
State Information (CSI), the signals can achieve spatial cancellation without
being detected by the warden [5]. It is worth mentioning that this beamforming
scheme for covert communication relies on phased array (PA) antennas. Hence,
the improvement of covert performance requires that the legitimate receiver
and the adversary is not locate in the same beam direction. However, in
realistic scenarios, the warden may approach the legitimate receiver and be in
the same beam direction, which leads to a degradation in covert performance.

Frequency diverse array (FDA) is a promising antenna technology used at the
multi-antenna transmitter in wireless communization systems [6, 7, 8, 9, 10].
Compared with conventional PA, FDA introduces a small frequency increment
across the array elements to produce distinct accumulating phase lags among
array antennas and forms a special beampattern related to distance and angle.
By tuning the antenna frequency vector (AFV) at the transmitter, FDA can
enable the transmitted signals to constructively superimpose at the desired
positions and destructively cancel at the undesired positions simultaneously
[8, 9, 10]. Especially in covert communication, FDA can achieve perfect
covertness by optimizing AFV with the instantaneous channel state information
(CSI) at the warden [11, 12]. Moreover, FDA develops an angle-range dependent
beampattern which helps decouple the correlated channels to enhance covert
performance. In particular, by optimizing AFV, the received signal levels of
the legitimate user and the warden can be distinguished clearly even if they
are in the same beam direction. In the multi-warden scenario, a larger
frequency increment is needed to decouple the correlated channels [13, 14].
However, the frequency increment is usually assumed to be much smaller than
the carrier frequency due to receiver restrictions which greatly limits FDA
performance.

It is worth mentioning that both FDA and PA are fixed-position arrays which
means that the distance between each antenna is fixed and no smaller than a
half-wavelength [15, 16, 17]. Recently, movable antenna (MA) technology was
proposed to achieve better communication performance where the antennas at the
transmitter are movable within a confined region [18, 19, 20, 21, 22, 23, 24,
25]. By smartly designing the antenna positions vector (APV), MA can achieve
higher spatial diversity gains by employing much fewer antennas compared with
the conventional antenna selection (AS) technology [18, 19, 20]. Specifically,
with known global instantaneous CSI, MA can achieve full array gain with null
steering [21]. Moreover, MA is capable of mitigating the interference from an
adversary when it is deployed at the receiver [26]. In a multi-user and multi-
interference scenario, MA can move in a three-dimensional (3D) region to
obtain higher rate gain from the users and reduce the impact from different
interference sources simultaneously. In addition, MA has great flexibility
which can be deployed in one-dimensional (1D) line, two-dimensional (2D)
surface and even six-dimensional (6D) region [22, 23, 24, 25].

In light of the above considerations, we propose a novel antenna technology,
namely movable frequency diverse array (MFDA), which can further exploit the
spatial DoFs in transmit antennas based on FDA. Our main contributions are
summarized as follows:

  * 1.

MFDA-aided covert communication with multiple wardens is firstly investigated
in this paper. Different from FDA, MFDA can not only adjust the frequency of
each antenna but also adjust the position, such that it achieves two-
dimensional security. Specifically, we jointly optimize the antenna
beamforming vector (ABV), AFV and APV at the transmitter to maximize the
covert rate subject to the given constraints. Moreover, we investigate the
effects of imperfect CSI at wardens on covert performance and the case of
perfect covertness.

  * 2.

Since the formulated optimization problem is non-convex with multiple coupled
variables, we propose a low-complexity two-stage alternating optimization (AO)
algorithm to solve this problem. Specifically, in the first stage, we
alternately optimize AFV and APV by utilizing the block successive upper-bound
minimization (BSUM) method. In the second stage, with optimized AFV and APV,
the optimization problem is convex and can be solved by standard convex
optimization solvers, e.g., CVX. For the case of imperfect CSI, we assume that
the exact coordinates of wardens are unknown, but the uncertain regions of
wardens are known. We first divide the regions into multiple discrete samples
and consider the sum of the channel correlations. Then, the proposed two-stage
AO algorithm can be utilized to solve the problem.

  * 3.

MFDA can achieve better covert performance than FDA and PA. In particular,
when the channels of the legitimate user and wardens are highly correlated,
FDA is limited by the frequency constraint and cannot provide satisfactory
performance. By contrast, MFDA can adjust the positions of antennas instead of
the frequencies to further improve covert performance. Moreover, for the case
of imperfect CSI, MFDA can achieve signal covertness in the uncertain regions.

The rest of this paper is organized as follows. Section II describes the MFDA-
aided covert system model and formulates the optimization problem. Section III
develops a two-stage AO algorithm to solve the problem with perfect CSI at
wardens. Section VI considers the case of imperfect CSI at wardens. Numerical
results are presented and discussed in Section V, and conclusions are drawn in
Section IV.

_Notations_ : Boldface uppercase and lowercase letters denote matrices and
vectors, respectively. Italic letters denote scalars; ${\bf{x}}^{\dagger}$
denotes the Hermitian of a vector $\bf{x}$. For a given matrix $\bf{A}$, we
denote its transpose, Hermitian and inverse by ${\bf{A}}^{T}$,
${\bf{A}}^{{\dagger}}$ and ${\bf{A}}^{-1}$, respectively. Denote
${\left\|.\right\|^{2}}$ as ${\ell_{2}}$-norm. $\mathbb{C}^{M\times N}$
denotes the complex $M\times N$ matrices.
$\left\langle{\cdot,\cdot}\right\rangle$ denotes the inner product.
${\bf{tr}}\left(\cdot\right)$ is trace operator of a matrix.
$\left\lfloor\cdot\right\rfloor$ and $\left\lceil\cdot\right\rceil$ are the
nearest integer towards $-\infty$ and $+\infty$, respectively. $c=a{\kern
1.0pt}\bmod{\kern 1.0pt}b$ means $c$ is the remainder of integer $a$ divided
by integer $b$.

##  II System Model

Figure 1: MFDA-aided covert communication system model

###  II-A System Description and Channel Model

In this paper, we consider a covert communication system with multiple wardens
where a 1D linear MFDA is deployed to assist in the communication from the
legitimate user Alice to Bob, while preventing being detected by $K$ wardens
Willies. The number of antennas at Alice is denoted by $M$, and both Bob and
Willies are assumed to be equipped with a single antenna. As shown in Fig.
1(a), a smart MFDA controller helps adjust the APV
${\bf{x}}\buildrel\Delta\over{=}{\left[{{x_{1}},...,{x_{M}}}\right]}\in{{\mathbb{C}}^{1\times
M}}$ and the AFV
${\bf{f}}\buildrel\Delta\over{=}{\left[{{f_{1}},...,{f_{M}}}\right]}\in{{\mathbb{C}}^{1\times
M}}$ at Alice via a separate wireless link. Compared with PA, MFDA introduces
a small frequency increment across the array antennas, such that the frequency
of the $m$-th antenna is given by

$\displaystyle\ {f_{m}}={f_{C}}+\Delta{f_{m}},m=1,...,M,$ (1)

where $f_{C}$ is the carrier frequency and $\Delta{f_{m}}$ is the introduced
frequency increment. We assume that $0\leq\Delta{f_{m}}\leq\Delta F$ and the
frequency increment is much smaller than the carrier frequency $\Delta
F\ll{f_{C}}$.

Without loss of generality, we define the first antenna of Alice as the origin
of the coordinate system. Thus, the coordinates of Bob and the $k$-th Willie
are denoted by $\left({{r_{b}},{\theta_{b}}}\right)$ and
$\left({{r_{{w_{k}}}},{\theta_{{w_{k}}}}}\right)$, respectively. To
characterize the theoretical performance gain brought by the MFDA, we assume
that $\left({{r_{b}},{\theta_{b}}}\right)$ and
$\left({{r_{{w_{k}}}},{\theta_{{w_{k}}}}}\right)$ are perfectly known by Alice
and all channels are line-of-sight (LoS) channels as in [27, 28, 29]. In
addition, the channels of Alice$\to$Bob and Alice$\to$ the $k$-th Willie are
defined as

$\displaystyle\
{{\bf{h}}_{ab}}={\rm{Lfs}}\left({{r_{b}}}\right)\left[{{h_{1}}\left({t,{r_{b}},{\theta_{b}}}\right),...,{h_{M}}\left({t,{r_{b}},{\theta_{b}}}\right)}\right]$
(2)

and

$\displaystyle\
{{\bf{h}}_{a{w_{k}}}}={\rm{Lfs}}\left({{r_{{w_{k}}}}}\right)\left[{{h_{1}}\left({t,{r_{{w_{k}}}},{\theta_{{w_{k}}}}}\right),...,{h_{M}}\left({t,{r_{{w_{k}}}},{\theta_{{w_{k}}}}}\right)}\right],$
(3)

where
${h_{m}}\left({t,r,\theta}\right)={e^{-j2\pi{f_{m}}\left[{t-\frac{{r-{x_{m}}\sin\theta}}{c}}\right]}}$,
$\forall m$. $c$ is the speed of light and ${\rm{Lfs}}\left(\cdot\right)$ is
the path loss factor.

###  II-B Covert Constraint

We observe from (2) and (3) that the channels of MFDA are dependent on time
$t$. Thus, the ABV at Alice is assumed to be time-varying to adapt to the
variable channels which are defined as

$\displaystyle\
{\bf{w}}\left(t\right)={\left[{{w_{1}}\left(t\right),...,{w_{M}}\left(t\right)}\right]^{T}}.$
(4)

From Willies’ perspective, a binary hypothesis testing problem should be
considered under two events: ${\cal{H}}_{0}$ denotes the null hypothesis when
Alice does not transmit information, and ${\cal{H}}_{1}$ denotes the
alternative hypothesis when Alice transmits information. Thus, the received
signal at the $k$-th Willie is given by

$\displaystyle\
{y_{{w_{k}}}}=\left\\{{\begin{array}[]{*{20}{l}}{{{\bf{n}}_{{w_{k}}}}}&{{{\cal{H}}_{0}}}\\\
{{{\bf{h}}_{{aw_{k}}}}{\bf{w}}\left(t\right)+{{\bf{n}}_{{w_{k}}}}}&{{{\cal{H}}_{1}},}\end{array}}\right.$
(7)

where ${{\bf{n}}_{{w_{k}}}}$ is the additive white Gaussian noise (AWGN) at
the $k$-th Willie with mean zero and variance $\sigma_{{w_{k}}}^{2}$, i.e.,
${{\bf{n}}_{{w_{k}}}}\sim{\cal{CN}}\left({0,\sigma_{{w_{k}}}^{2}}\right)$. To
determine whether Alice transmits information or not, the $k$-th Willie needs
to distinguish the probability curves of ${y_{w_{k}}}$ under ${\cal{H}}_{0}$
and ${\cal{H}}_{1}$. Based on (7), let ${p_{k,0}}\left({{y_{w_{k}}}}\right)$
and ${p_{k,1}}\left({{y_{w_{k}}}}\right)$ denote the likelihood functions of
the received signals at the $k$-th Willie under ${\cal{H}}_{0}$ and
${\cal{H}}_{1}$, respectively. Now, ${p_{k,0}}\left({{y_{w_{k}}}}\right)$ and
${p_{k,1}}\left({{y_{w_{k}}}}\right)$ are given by

$\displaystyle\
{p_{k,0}}\left({{y_{{w_{k}}}}}\right)=\frac{1}{{\pi{\lambda_{k,0}}}}\exp\left({-\frac{{{{\left|{{y_{{w_{k}}}}}\right|}^{2}}}}{{{\lambda_{k,0}}}}}\right)$
(8)

and

$\displaystyle\
{p_{k,1}}\left({{y_{w_{k}}}}\right)=\frac{1}{{\pi{{\lambda_{k,1}}}}}\exp\left({-\frac{{{{\left|{{y_{w_{k}}}}\right|}^{2}}}}{{{{\lambda_{k,1}}}}}}\right),$
(9)

where ${\lambda_{k,0}}\buildrel\Delta\over{=}\sigma_{w_{k}}^{2}$ and
${\lambda_{k,1}}\buildrel\Delta\over{=}{\left|{{{\bf{h}}_{{aw_{k}}}}{\bf{w}}\left(t\right)}\right|^{2}}+\sigma_{{w_{k}}}^{2}$.

In general, the detection error probability (DEP) ${\xi}$ metric is used to
evaluate covert performance. For an optimal detector at the $k$-th Willie,
${\xi_{k}}$ is given by [30, 31]

$\displaystyle\
\xi_{k}=1-{{\cal{V}}_{T}}\left({{{p_{k,0}}},{{p_{k,1}}}}\right),$ (10)

where ${{\cal{V}}_{T}}\left({{{p_{k,0}}},{p_{k,1}}}\right)$ is the total
variation between ${{p_{k,0}}}\left({{y_{w_{k}}}}\right)$ and
${p_{k,1}}\left({{y_{w_{k}}}}\right)$. Since computing
${{\cal{V}}_{T}}\left({{{p_{k,0}}},{p_{k,1}}}\right)$ is intractable, we adopt
Pinsker’s inequality to obtain the following upper bound [32]

$\displaystyle\
{V_{T}}\left({{{p_{k,0}}},{p_{k,1}}}\right)\leq\sqrt{\frac{1}{2}{\cal{D}}\left({{{p_{k,0}}}||{p_{k,1}}}\right)},$
(11)

where ${{\cal{D}}\left({{{p_{k,0}}}||{p_{k,1}}}\right)}$ is the Kullback-
Leibler (KL) divergence, which is given by

$\displaystyle\ {\cal{D}}\left({{p_{k,0}}||{p_{k,1}}}\right)$
$\displaystyle=\int_{-\infty}^{+\infty}{{p_{k,0}}\left({{y_{{w_{k}}}}}\right)}\ln\frac{{{p_{k,0}}\left({{y_{{w_{k}}}}}\right)}}{{{p_{k,1}}\left({{y_{{w_{k}}}}}\right)}}dy$
$\displaystyle=\ln\frac{{{\lambda_{k,1}}}}{{{\lambda_{k,0}}}}+\frac{{{\lambda_{k,0}}}}{{{\lambda_{k,1}}}}-1.$
(12)

Then, the covert constraint at the $k$-th Willie can be written as

$\displaystyle\ {\cal{D}}\left({{p_{k,0}}||{p_{k,1}}}\right)\leq
2{\epsilon^{2}},$ (13)

where $\epsilon$ is a small value and predetermined to ensure satisfactory
covertness. According to (II-B),
$y\left(x\right)=\ln\left(x\right)+\frac{1}{x}-1$ for $x\geq 1$ is a
monotonically increasing function. Thus, the covert constraint of the $k$-th
Willie can be simplified as

$\displaystyle\
\frac{{{\lambda_{k,1}}}}{{{\lambda_{k,0}}}}\leq{y^{-1}}\left({2{\epsilon^{2}}}\right),$
(14)

where ${y^{-1}}\left(x\right)$ is the inverse function of $y\left(x\right)$.

###  II-C Problem Formulation

The received signal at Bob can be written as

$\displaystyle\
{{\bf{y}}_{b}}\left(t\right)={{\bf{h}}_{ab}}{\bf{w}}\left(t\right)+{{\bf{n}}_{b}},$
(15)

where ${{\bf{n}}_{b}}$ is the AWGN at Bob with mean zero and variance
$\sigma_{b}^{2}$, i.e.,
${{\bf{n}}_{b}}\sim{\cal{CN}}\left({0,\sigma_{b}^{2}}\right)$. The covert rate
at Bob is given by

$\displaystyle\
{R_{b}}\left(t\right)=\log\left({1+\sigma_{b}^{-2}{{\left|{{\bf{h}}_{ab}{\bf{w}}\left(t\right)}\right|}^{2}}}\right).$
(16)

To improve covert performance, we aim to maximize the covert rate by jointly
optimizing ABV, APV and AFV at Alice, while avoiding being detected by
different Willies. The optimization problem can be formulated as follows

$\displaystyle\mathop{\max}\limits_{\left\\{{{\bf{w}}\left(t\right),{\bf{x}},{\bf{f}}}\right\\}}{\rm{}}{R_{b}}\left(t\right)$
(17a)
$\displaystyle{s.t.}\quad{\left\|{\bf{w}}\left(t\right)\right\|^{2}}\leq{P_{max}},$
(17b) $\displaystyle\quad\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (17c)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{min}},\forall m,$ (17d)
$\displaystyle\quad\quad{f_{C}}\leq{f_{m}}\leq{f_{C}}+\Delta F,\forall m,$
(17e)
$\displaystyle\quad\quad{\left|{{{\bf{h}}_{{aw_{k}}}}{\bf{w}}\left(t\right)}\right|^{2}}\leq\sigma_{{w_{k}}}^{2}\left({{y^{-1}}\left({2{\epsilon^{2}}}\right)-1}\right),k=1,...,K,$
(17f)

where (17b) is the maximum power constraint and ${P_{max}}$ is the maximum
transmit power. (15c) is the 1D linear range constraint which ensures that
MFDA is movable within the predetermined region $\left[{0,{D_{\max}}}\right]$
and ${{D_{\max}}}$ is the maximum movable distance of MFDA. Constraint (17d)
aims to avoid antenna coupling and ${D_{min}}$ is the minimum distance between
any adjacent antennas at the MFDA transmitter. Besides, (17e) is the frequency
constraint of MFDA, and (17f) is the covert constraint for different Willies.

##  III Proposed Algorithm for Optimization Problem

We observe that problem (15) contains three optimized variables which are
tightly coupled making the problem challenging to solve. Usually, the AO
algorithm is utilized to solve this problem [33, 34]. However, when there are
multiple variables, the complexity of the AO algorithm increases significantly
and the convergence accuracy decreases. To avoid these problems, we propose a
low-complexity two-stage AO algorithm to solve problem (15).

###  III-A Optimizing ${\bf{w}}\left(t\right)$ with Given ${\bf{x}}$ and
${\bf{f}}$

According to (16), problem (15) can be reformulated as

$\displaystyle\mathop{\max}\limits_{{{\bf{w}}\left(t\right)}}\quad{{\bf{w}}^{\dagger}}\left(t\right){{\bf{H}}_{ab}}{\bf{w}}\left(t\right)$
(18a)
$\displaystyle{s.t.}\quad{\left\|{\bf{w}}\left(t\right)\right\|^{2}}\leq{P_{max}},$
(18b)
$\displaystyle\quad\quad{{\bf{w}}^{\dagger}}\left(t\right){{\bf{H}}_{a{w_{k}}}}{\bf{w}}\left(t\right)\leq\sigma_{{w_{k}}}^{2}\left({{y^{-1}}\left({2{\epsilon^{2}}}\right)-1}\right),k=1,...,K,$
(18c)

where ${{\bf{H}}_{ab}}={\bf{h}}_{ab}^{\dagger}{{\bf{h}}_{ab}}$ and
${{\bf{H}}_{a{w_{k}}}}={\bf{h}}_{a{w_{k}}}^{\dagger}{{\bf{h}}_{a{w_{k}}}}$.
Note that
${{\bf{w}}^{\dagger}}\left(t\right){{\bf{H}}_{ab}}{\bf{w}}\left(t\right)=\rm{tr}\left({{{\bf{H}}_{ab}}{\bf{w}}\left(t\right){{\bf{w}}^{\dagger}}\left(t\right)}\right)$.
We define the matrix
${\bf{W}}\left(t\right)={\bf{w}}\left(t\right){{\bf{w}}^{\dagger}}\left(t\right)$
which implies that ${\rm{rank}}\left({\bf{W}}\right)=1$, and the rank-1
constraint is non-convex which can be relaxed by the semi-definite relaxation
(SDR) method. Then, problem (16) can be converted to

$\displaystyle\mathop{\max}\limits_{{\bf{W}}\left(t\right)}\quad\rm{tr}\left({{{\bf{H}}_{ab}}{\bf{W}}\left(t\right)}\right)$
(19a)
$\displaystyle{s.t.}\quad\rm{tr}\left({{\bf{W}}\left(t\right)}\right)\leq{P_{\max}},$
(19b)
$\displaystyle\quad\quad{\rm{tr}}\left({{{\bf{H}}_{{\rm{a}}{{\rm{w}}_{k}}}}{\bf{W}}\left({\rm{t}}\right)}\right)\leq\sigma_{{w_{k}}}^{2}\left({{y^{-1}}\left({2{\epsilon^{2}}}\right)-1}\right),k=1,...,K.$
(19c)

Problem (17) is a semi-definite program (SDP) problem which can be efficiently
solved by convex solvers such as CVX [35]. We observe that the optimal
solution ${{\bf{W}}^{opt}}\left(t\right)$ may not satisfy the rank-1
constraint. Thus, we apply the Gaussian randomization method to recover
${{\bf{w}}^{opt}}\left(t\right)$ approximately [36].

###  III-B Optimizing ${\bf{x}}$ with Given ${\bf{w}}\left(t\right)$ and
${\bf{f}}$

It does not appear possible to derive an expression for the optimal ABV in
closed form from problem (17). Hence, there is no simplified optimization
object when optimizing the APV and AFV with given ABV. Fortunately, we note
that FDA can make the channels of the legitimate and illegitimate users as
orthogonal as possible to minimize the channel correlations by optimizing the
AFV [37, 11, 27, 29]. Inspired by this observation, we assume that MFDA aims
to minimize the sum of the inner product between the channels of Bob and
Willies. In particular, the ideal case is
${{\bf{h}}_{ab}}\perp{{\bf{h}}_{aw_{k}}}$, $k=1,..,K$. Then, the optimization
problem in this case can be formulated as

$\displaystyle\mathop{\min}\limits_{{{\bf{x}}}}{\rm{}}\sum\limits_{k=1}^{K}{{{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{a{w_{k}}}}}\right\rangle}\right|}^{2}}}$
(20a) $\displaystyle{s.t.}\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (20b)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{min}},\forall m.$ (20c)

According to (2) and (3), (20a) can be written as

$\displaystyle\sum\limits_{k=1}^{K}{{{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{a{w_{k}}}}}\right\rangle}\right|}^{2}}}$
$\displaystyle=\sum\limits_{k=1}^{K}{{{\left|{\sum\limits_{m=1}^{M}{{e^{j2\pi{f_{m}}\left({{x_{m}}\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\right)}}}}\right|}^{2}}}$
$\displaystyle=\sum\limits_{k=1}^{K}{\sum\limits_{m=1}^{M}{\sum\limits_{n=1}^{M}{{e^{j2\pi\left({\left({{f_{m}}{x_{m}}-{f_{n}}{x_{n}}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\right)}}}}}$
$\displaystyle=KM+\sum\limits_{k=1}^{K}{\sum\limits_{m=1}^{M}{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}\cos\left[{2\pi\left(\begin{array}[]{l}\left({{f_{m}}{x_{m}}-{f_{n}}{x_{n}}}\right)\\\
\times\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\\\
+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}\end{array}\right)}\right]}}.$
(24)

Next, problem (18) can be reformulated as

$\displaystyle\mathop{\min}\limits_{{{\bf{x}}}}{\sum\limits_{k=1}^{K}{\sum\limits_{m=1}^{M}{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}\cos\left[{2\pi\left(\begin{array}[]{l}\left({{f_{m}}{x_{m}}-{f_{n}}{x_{n}}}\right)\\\
\times\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\\\
+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}\end{array}\right)}\right]}}}$
(25d) $\displaystyle{s.t.}\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (25e)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{min}},\forall m.$ (25f)

Problem (20) is clearly non-convex. Thus, we resort to the BSUM method to
solve it. BSUM solves the problem by formulating a new convex function to
approach the optimization target. Specifically, in the $s$-th iteration of
BSUM, we first optimize ${{{x_{m}}}}$ where $m=s{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\rm{mod}}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}M$, and obtain the optimal position $x_{m}^{opt}$. Then, we update
${{\bf{x}}^{s-1}}$ with $x_{m}^{opt}$, i.e.,
${{\bf{x}}^{s}}={{\bf{x}}^{s-1}}$. In the $s$-th iteration, the optimization
problem can be written as

$\displaystyle\mathop{\min}\limits_{{x_{m}}}\sum\limits_{k=1}^{K}{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}\cos\left[{2\pi\left(\begin{array}[]{l}\left({{f_{m}}{x_{m}}-{f_{n}}{x_{n}}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\\\
+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}\end{array}\right)}\right]}$
(26c) $\displaystyle{s.t.}\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (26d)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{min}},\forall m.$ (26e)

Next, we define the non-convex function in (26c) as

$\displaystyle\
{y_{k,m,n}}\left(x\right)=\cos\left[{2\pi\left(\begin{array}[]{l}\left({{f_{m}}{x}-{f_{n}}{x_{n}}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\\\
+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}\end{array}\right)}\right].$
(29)

Following the BSUM method, the objective function (29) is approximated by the
upper-bound quadratic function ${u_{k,m,n}}\left({{x}}\right)$, which is
defined by

$\displaystyle\
{u_{k,m,n}}\left(x\right)={A_{k,m,n}}{\left({x-{B_{k,m,n}}}\right)^{2}}+{C_{k,m,n}},$
(30)

where ${A_{k,m,n}}\in\mathbb{R}$, ${A_{k,m,n}}>0$,
${{B_{k,m,n}}}\in\mathbb{R}$ and ${C_{k,m,n}}\in\mathbb{R}$ are the parameters
of the new quadratic function. For a given point ${x_{m}^{s-1}}$ in (29), the
approximate function (30) should satisfy the following constraints:

$\displaystyle\
\left\\{{\begin{array}[]{*{20}{l}}{{u_{k,m,n}}\left({x_{m}^{s-1};x_{n}^{s-1}}\right)={y_{k,m,n}}\left({x_{m}^{s-1};x_{n}^{s-1}}\right)}\\\
{u_{k,m,n}^{{}^{\prime}}\left({x_{m}^{s-1};x_{n}^{s-1}}\right)=y_{k,m,n}^{{}^{\prime}}\left({x_{m}^{s-1};x_{n}^{s-1}}\right)}\\\
{{u_{k,m,n}}\left({{B_{k,m,n}};x_{n}^{s-1}}\right)\in\left\\{{1,-1}\right\\}}\\\
-1\leq{y_{k,m,n}}\left({{B_{k,m,n}};x_{n}^{s-1}}\right)\leq
1,\end{array}}\right.$ (35)

where
$y_{k,m,n}^{{}^{\prime}}\left({x_{m}^{s-1};x_{n}^{s-1}}\right)=-2\pi{f_{m}}\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\times\\\
\sin\left[{2\pi\left({\left({{f_{m}}x_{m}^{s-1}-{f_{n}}{x_{n}}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\right)}\right]$.
Next, we can obtain the parameters of the new quadratic function in (30) by
utilizing Lemma 1.

_Lemma 1:_ If
$y_{k,m,n}^{{}^{\prime}}\left({x_{m}^{s-1};x_{n}^{s-1}}\right)\neq 0$, we have
(39).

$\displaystyle\
\left\\{{\begin{array}[]{*{20}{l}}{A_{k,m,n}}=\frac{{-\pi{f_{m}}\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}}}{{x_{m}^{s-1}-{B_{k,m,n}}}}\sin\left[{2\pi\left({\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\right)}\right]\\\
{{B_{k,m,n}}=\frac{c}{{{f_{m}}\left({\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}\right)}}\left[{\frac{{1+2\left({\left\lceil{\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\right\rceil-1}\right)}}{2}-\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\right]+\frac{{{f_{n}}x_{n}^{s-1}}}{{{f_{m}}}}}\\\
{C_{k,m,n}}=\cos\left[{2\pi\left({\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\right)}\right]-{A_{k,m,n}}{\left({x_{m}^{s-1}-{B_{k,m,n}}}\right)^{2}}.\end{array}}\right.$
(39)

If $y_{k,m,n}^{{}^{\prime}}\left({x_{m}^{s-1};x_{n}^{s-1}}\right)=0$, we have

$\displaystyle\
\left\\{{\begin{array}[]{*{20}{l}}{{A_{k,m,n}}=-2{{\left({\pi{f_{m}}\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}}\right)}^{2}}{C_{k,m,n}}}\\\
{{B_{k,m,n}}=x_{m}^{s-1}}\\\
{{C_{k,m,n}}={y_{k,m,n}}\left({x_{m}^{s-1};x_{n}^{s-1}}\right)\in\left\\{{-1,1}\right\\}.}\end{array}}\right.$
(43)

The proof of Lemma 1 is presented in Appendix A. Therefore problem (21a) can
be replaced by

$\displaystyle\
\mathop{\min}\limits_{{x_{m}}}\sum\limits_{k=1}^{K}{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}{{A_{k,m,n}}{{\left({{x_{m}}-{B_{k,m,n}}}\right)}^{2}}+{C_{k,m,n}}}}.$
(44)

It is clear that the optimal solution of problem (21) is given by

$\displaystyle\
x^{opt}=\frac{{\sum\limits_{k=1}^{K}{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}{{A_{k,m,n}}{B_{k,m,n}}}}}}{{\sum\limits_{k=1}^{K}{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}{{A_{k,m,n}}}}}}.$ (45)

According to the constraints (26d) and (26e), the optimal position of the
$m$-th antenna is given by

$\displaystyle\
x_{m}^{opt}=\left\\{{\begin{array}[]{*{20}{l}}{x_{m-1}^{s}+{D_{\min}}}&{{x^{opt}}\leq
x_{m-1}^{s}+{D_{\min}}}\\\
{{x^{opt}}}&{x_{m-1}^{s}+{D_{\min}}<{x^{opt}}\leq{D_{upper}}}\\\
{{D_{upper}}}&{{D_{upper}}<{x^{opt}}},\end{array}}\right.$ (49)

where ${D_{upper}}={D_{\max}}-\left({M-m}\right){D_{\min}}$. The convergence
of BSUM is verified in Fig. 2.

###  III-C Optimizing ${\bf{f}}$ with Given ${\bf{w}}\left(t\right)$ and
${\bf{x}}$

With given ${\bf{w}}\left(t\right)$ and ${\bf{x}}$, the optimization problem
can be formulated as

$\displaystyle\mathop{\min}\limits_{\bf{f}}\sum\limits_{k=1}^{K}{\sum\limits_{m=1}^{M}{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}{\cos}\left[{2\pi\left({{\tau_{k,m}}{f_{m}}-{\tau_{k,n}}{f_{n}}}\right)}\right]}}$
(50a) $\displaystyle{s.t.}\quad{f_{C}}\leq{f_{m}}\leq{f_{C}}+\Delta F,\forall
m,$ (50b)

where
${\tau_{k,m}}={x_{m}}\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}$.
Problem (30) is non-convex which can be solved by the BSUM method developed in
the previous subsection. Thus, we define the non-convex function in (50a) as

$\displaystyle\
{y_{k,m,n}}\left(f\right)=\cos\left[{2\pi\left({{\tau_{k,m}}{f}-{\tau_{k,n}}{f_{n}}}\right)}\right].$
(51)

The new quadratic function is the same as (30). Then, the parameters of the
new quadratic function can be obtained by utilizing Lemma 2.

_Lemma 2:_ If
$y_{k,m,n}^{{}^{\prime}}\left({f_{m}^{s-1};f_{n}^{s-1}}\right)\neq 0$ and
${r_{b}}\geq{r_{{w_{k}}}}$, we have

$\displaystyle\
\left\\{{\begin{array}[]{*{20}{l}}{A_{k,m,n}}&=\frac{{-\pi{\tau_{k,m}}\sin\left[{2\pi\left({f_{m}^{s-1}{\tau_{k,m}}-f_{n}^{s-1}{\tau_{k,n}}}\right)}\right]}}{{f_{m}^{s-1}-{B_{k,m,n}}}}\\\
{B_{k,m,n}}&=\frac{{1+2\left\lfloor{\left({f_{m}^{s-1}{\tau_{k,m}}-f_{n}^{s-1}{\tau_{k,n}}}\right)}\right\rfloor}}{{2{\tau_{k,m}}}}+\frac{{f_{n}^{s-1}{\tau_{k,n}}}}{{{\tau_{k,m}}}}\\\
{C_{k,m,n}}&=\cos\left[{2\pi\left({f_{m}^{s-1}{\tau_{k,m}}-f_{n}^{s-1}{\tau_{k,n}}}\right)}\right]\\\
&-{A_{k,m,n}}{\left({f_{m}^{s-1}-{B_{k,m,n}}}\right)^{2}}.\end{array}}\right.$
(56)

If $y_{k,m,n}^{{}^{\prime}}\left({f_{m}^{s-1};f_{n}^{s-1}}\right)\neq 0$ and
${r_{b}}<{r_{{w_{k}}}}$, we have

$\displaystyle\
\left\\{{\begin{array}[]{*{20}{l}}{A_{k,m,n}}&=\frac{{-\pi{\tau_{k,m}}\sin\left[{2\pi\left({f_{m}^{s-1}{\tau_{k,m}}-f_{n}^{s-1}{\tau_{k,n}}}\right)}\right]}}{{f_{m}^{s-1}-{B_{k,m,n}}}}\\\
{B_{k,m,n}}&=-\frac{{1+2\left\lfloor{\left({f_{n}^{s-1}{\tau_{k,n}}-f_{m}^{s-1}{\tau_{k,m}}}\right)}\right\rfloor}}{{2{\tau_{k,m}}}}+\frac{{f_{n}^{s-1}{\tau_{k,n}}}}{{{\tau_{k,m}}}}\\\
{C_{k,m,n}}&=\cos\left[{2\pi\left({f_{m}^{s-1}{\tau_{k,m}}-f_{n}^{s-1}{\tau_{k,n}}}\right)}\right]\\\
&-{A_{k,m,n}}{\left({f_{m}^{s-1}-{B_{k,m,n}}}\right)^{2}}.\end{array}}\right.$
(61)

If $y_{k,m,n}^{{}^{\prime}}\left({f_{m}^{s-1};f_{n}^{s-1}}\right)=0$, we have

$\displaystyle\
\left\\{{\begin{array}[]{*{20}{l}}{{A_{k,m,n}}=-2{\pi^{2}}\tau_{k,m}^{2}{C_{k,m,n}}}\\\
{{B_{k,m,n}}=f_{m}^{s-1}}\\\
{{C_{k,m,n}}={y_{k,m,n}}\left({f_{m}^{s-1};f_{n}^{s-1}}\right)\in\left\\{{-1,1}\right\\}},\end{array}}\right.$
(65)

where
$y_{k,m,n}^{{}^{\prime}}\left(f\right)=-2\pi{\tau_{k,m}}\sin\left[{2\pi\left({{\tau_{k,m}}f-{\tau_{k,n}}{f_{n}}}\right)}\right]$.
Due to the similar mathematical proof as Lemma 1, we omit the proof of Lemma 2
here for brevity. Then, the optimal frequency of the $m$-th antenna is given
by

$\displaystyle\
f_{m}^{opt}=\left\\{{\begin{array}[]{*{20}{l}}{{f_{C}}}&{{f_{opt}}<{f_{C}}}\\\
{{f^{opt}}}&{{f_{C}}\leq{f^{opt}}<{f_{C}}+\Delta F}\\\ {{f_{C}}+\Delta
F}&{{f_{C}}+\Delta F\leq{f^{opt}}},\end{array}}\right.$ (69)

where ${{f^{opt}}}$ can be obtained from (45) with the parameters in Lemma 2.

###  III-D Overall Two-Stage AO Algorithm

According to problems (20) and (30), it is clear that ABV cannot affect the
optimization for APV or AFV. It means that ABV is not coupled with APV and
AFV, but APV and AFV are coupled with each other. Thus, we propose a two-stage
AO algorithm with low complexity. Specifically, the first stage aims to
alternately optimize APV and AFV by utilizing the BSUM method which is
summarized in Algorithm 1.

Algorithm 1 Optimizing ${\bf{x}}$ by BSUM

1: Initialization: Initializing ${\bf{f}}$ via Algorithm 1 .

2: Set: $s=0$.

3: Repeat:a) $s=s+1$, and $m=s{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\rm{mod}}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\rm{M}}$.b) The new
quadratic function ${u_{k,m,n}}\left({{x}}\right)$ can be obtained by Lemma 1.
Then, the optimal position of $m$-th antenna can be obtained by (49).c) We
update ${{\bf{x}}^{s-1}}$ with $x_{m}^{opt}$ and obtain ${{\bf{x}}^{s}}$.

4: Until
$\left|\begin{array}[]{l}\sum\limits_{k=1}^{K}{{{\left|{\left\langle{{{\bf{h}}_{ab}}\left({{{\bf{x}}^{s}},{\bf{f}}}\right),{{\bf{h}}_{a{w_{k}}}}\left({{{\bf{x}}^{s}},{\bf{f}}}\right)}\right\rangle}\right|}^{2}}}\\\
-\sum\limits_{k=1}^{K}{{{\left|{\left\langle{{{\bf{h}}_{ab}}\left({{{\bf{x}}^{s-1}},{\bf{f}}}\right),{{\bf{h}}_{a{w_{k}}}}\left({{{\bf{x}}^{s-1}},{\bf{f}}}\right)}\right\rangle}\right|}^{2}}}\end{array}\right|\leq\varsigma$
is satisfied where $\varsigma$ is a small convergence value.

The second stage is to obtain the optimal ABV with the optimized APV and AFV.
The overall algorithm is summarized in Algorithm 2.

Algorithm 2 Overall two-stage AO algorithm

1: Initialization: Initializing ${{{\bf{w}}_{opt}}}$ via (17b).

2: Set: $i=0$.

3: Repeat:a) $i=i+1$.b) Optimizing ${\bf{x}}$ and ${\bf{f}}$ alternately by
BSUM.c) Obtaining ${{{\bf{w}}_{opt}}}$ with the optimized ${{\bf{x}}^{s}}$ and
${{\bf{f}}^{s}}$ by solving problem (17).

4: Until
$\left|{{R_{b}}\left({{{\bf{w}}_{opt}},{{\bf{x}}^{s}},{{\bf{f}}^{s}}}\right)-{R_{b}}\left({{{\bf{w}}_{opt}},{{\bf{x}}^{s-1}},{{\bf{f}}^{s-1}}}\right)}\right|\leq\varsigma$
is satisfied.

The complexity of Algorithm 1 is ${\cal{O}}\left({{I_{BSUM}}}\right)$ where
${{I_{BSUM}}}$ denotes the number of iterations required for BSUM convergence.
The overall complexity of Algorithm 2 is
${\cal{O}}\left({{I_{AO}}\left({{I_{BSUM}}+{I_{BSUM}}}\right)}+{M^{3.5}}\log\left({\frac{1}{\varsigma}}\right)\right)$
where ${{I_{AO}}}$ denotes the number of iterations required for AO algorithm
convergence.

###  III-E Special Case: Perfect Covert Communication

Based on the perfect CSI at Willies, the perfect covertness may be achieved
which means that no signal power can be detected by Willies, i.e.,
${{\bf{w}}^{\dagger}}\left(t\right){{\bf{H}}_{a{w_{k}}}}{\bf{w}}\left(t\right)=0,k=1,...,K,$.
The optimization problem in this case can be formulated as

$\displaystyle\mathop{\max}\limits_{\left\\{{{\bf{w}}\left(t\right),{\bf{x}},{\bf{f}}}\right\\}}{\rm{}}{{\bf{w}}^{\dagger}}\left(t\right){{\bf{H}}_{ab}}{\bf{w}}\left(t\right)$
(70a)
$\displaystyle{s.t.}\quad{\left\|{\bf{w}}\left(t\right)\right\|^{2}}\leq{P_{max}},$
(70b) $\displaystyle\quad\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (70c)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{min}},\forall m,$ (70d)
$\displaystyle\quad\quad{f_{C}}\leq{f_{m}}\leq{f_{C}}+\Delta F,\forall m,$
(70e)
$\displaystyle\quad\quad{{\bf{w}}^{\dagger}}\left(t\right){{\bf{H}}_{a{w_{k}}}}{\bf{w}}\left(t\right)=0,k=1,...,K.$
(70f)

Problem (36) can be solved by the two-stage AO algorithm proposed in the
previous subsection. With given ${\bf{f}}$ and ${\bf{x}}$, problem (36) can be
rewritten as

$\displaystyle\mathop{\max}\limits_{{{\bf{w}}\left(t\right)}}{\rm{}}{{\bf{w}}^{\dagger}}\left(t\right){{\bf{H}}_{ab}}{\bf{w}}\left(t\right)$
(71a)
$\displaystyle{s.t.}\quad{\left\|{\bf{w}}\left(t\right)\right\|^{2}}\leq{P_{max}},$
(71b)
$\displaystyle\quad\quad{{\bf{w}}^{\dagger}}\left(t\right){{\bf{H}}_{a{w_{k}}}}{\bf{w}}\left(t\right)=0,k=1,...,K.$
(71c)

The optimal solution to problem (37) is given as [38]

$\displaystyle\
{{\bf{w}}_{opt}}\left(t\right)=\sqrt{{P_{\max}}}\frac{{\left({{I_{M}}-{\bf{H}}_{{\bf{aw}}}^{\dagger}{{\left({{{\bf{H}}_{{\bf{aw}}}}{\bf{H}}_{{\bf{aw}}}^{\dagger}}\right)}^{-1}}{{\bf{H}}_{{\bf{aw}}}}}\right){\bf{h}}_{ab}^{\dagger}}}{{\left\|{\left({{I_{M}}-{\bf{H}}_{{\bf{aw}}}^{\dagger}{{\left({{{\bf{H}}_{{\bf{aw}}}}{\bf{H}}_{{\bf{aw}}}^{\dagger}}\right)}^{-1}}{{\bf{H}}_{{\bf{aw}}}}}\right){\bf{h}}_{ab}^{\dagger}}\right\|}},$
(72)

where
${{\bf{H}}_{{\bf{aw}}}}=\left[{{{\bf{h}}_{a{w_{1}}}};...;{{\bf{h}}_{a{w_{K}}}}}\right]$.
The proof is presented in Appendix B. Then, with given
${{\bf{w}}_{opt}}\left(t\right)$, the optimal ${\bf{f}}$ and ${\bf{x}}$ can be
obtained using BSUM. The optimization process in this case is the same as that
in the previous section and we omit it here for brevity.

##  IV Imperfect CSI at Eve

In this section, we consider a more practical scenario where Alice has
imperfect CSIs knowledge about Willies. According to (2) and (3), we conclude
that the imperfect CSIs at Willies are mainly caused by an estimation error in
Willies’ ranges and angles. Without loss of generality, we assume that the
exact coordinates of Willies are unknown, but the uncertain regions for
Willies are known. For the $k$-th Willie, the uncertain region is denoted as
${{\cal{R}}_{{w_{k}}}}\buildrel\Delta\over{=}\left[{{r_{{w_{k}}}}-\Delta{r_{{w_{k}}}},{r_{{w_{k}}}}+\Delta{r_{{w_{k}}}}}\right]$
and
${{\cal{A}}_{{w_{k}}}}\buildrel\Delta\over{=}\left[{{\theta_{{w_{k}}}}-\Delta{\theta_{{w_{k}}}},{\theta_{{w_{k}}}}+\Delta{\theta_{{w_{k}}}}}\right]$,
where ${{\Delta r}_{{w_{k}}}}$ and ${\Delta{\theta_{{w_{k}}}}}$ are the
estimation errors in range and angle, respectively. Since
${{\cal{R}}_{{w_{k}}}}$ and ${{\cal{A}}_{{w_{k}}}}$ are continuous sets, we
propose to uniformly generate some discrete samples over the sets which are
given as
${r_{k,l}}={r_{{w_{k}}}}-\Delta{r_{{w_{k}}}}+\left({l-1}\right)\frac{{2\Delta{r_{{w_{k}}}}}}{{L-1}},l\in\left\\{{1,2,...,L}\right\\}$
and
${\theta_{k,l}}={\theta_{{w_{k}}}}-\Delta{\theta_{{w_{k}}}}+\left({l-1}\right)\frac{{2\Delta{\theta_{{w_{k}}}}}}{{L-1}}$
where $L$ denotes the number of the samples. The optimization problem can be
formulated as

$\displaystyle\mathop{\max}\limits_{\left\\{{{\bf{w}}\left(t\right),{\bf{x}},{\bf{f}}}\right\\}}{\rm{}}{{R_{b}}\left(t\right)}$
(73a)
$\displaystyle{s.t.}\quad{\left\|{\bf{w}}\left(t\right)\right\|^{2}}\leq{P_{max}},$
(73b) $\displaystyle\quad\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (73c)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{0}},\forall m,$ (73d)
$\displaystyle\quad\quad{f_{C}}\leq{f_{m}}\leq{f_{C}}+\Delta F,\forall m.$
(73e)
$\displaystyle\quad\quad\begin{array}[]{l}{\left|{{{\bf{h}}_{a{w_{k}}}}\left({{r_{k,l}},{\theta_{k,l}}}\right){\bf{w}}\left(t\right)}\right|^{2}}\leq\sigma_{{w_{k}}}^{2}\left({{y^{-1}}\left({2{\epsilon^{2}}}\right)-1}\right),\\\
k=1,...,K,l=1,...,L.\end{array}$ (73h)

Problem (39) is a non-convex function with coupled variables which can be
solved by our proposed two-stage AO algorithm.

###  IV-A Optimizing ${\bf{w}}\left(t\right)$ with Given ${\bf{x}}$ and
${\bf{f}}$

For given APV and AFV, problem (39) is converted to

$\displaystyle\mathop{\max}\limits_{{\bf{W}}\left(t\right)}\quad\rm{tr}\left({{{\bf{H}}_{ab}}{\bf{W}}\left(t\right)}\right)$
(74a)
$\displaystyle{s.t.}\quad\rm{tr}\left({{\bf{W}}\left(t\right)}\right)\leq{P_{\max}},$
(74b)
$\displaystyle\quad\quad\begin{array}[]{l}{\rm{tr}}\left({{{\bf{H}}_{{\rm{a}}{{\rm{w}}_{k}}}}{\bf{W}}\left({\rm{t}}\right)}\right)\leq\sigma_{{w_{k}}}^{2}\left({{y^{-1}}\left({2{\epsilon^{2}}}\right)-1}\right),\\\
k=1,...,K,l=1,...,L.\end{array}$ (74e)

Problem (40) is a semi-definite program (SDP) problem which is efficiently
solved by CVX. If the rank-1 constraint is not satisfied, the Gussian
randomization method is applied to recover ${{\bf{w}}^{opt}}$ approximately.

###  IV-B Optimizing ${\bf{x}}$ with Given ${\bf{w}}\left(t\right)$ and
${\bf{f}}$.

Considering the coordinates uncertainty at Willies, we aim to minimize the sum
of the channel correlations between the Alice$\to$Bob link and all
Alice$\to$Willies links with Willies located at the different samples in the
uncertainty region. The problem is formulated as

$\displaystyle\mathop{\min}\limits_{{{\bf{x}}}}{\rm{}}\sum\limits_{k=1}^{K}{\sum\limits_{i=1}^{L}{\sum\limits_{j=1}^{L}{{{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{a{w_{k}}}}\left({{r_{k,i}},{\theta_{k,j}}}\right)}\right\rangle}\right|}^{2}}}}}$
(75a) $\displaystyle{s.t.}\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (75b)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{min}},\forall m.$ (75c)

According to (III-B), problem (41) is rewritten as

$\displaystyle\mathop{\min}\limits_{{{\bf{x}}}}{\rm{}}\sum\limits_{k=1}^{K}{\sum\limits_{i=1}^{L}{\sum\limits_{j=1}^{L}{\sum\limits_{m=1}^{M}{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}{\cos}\left[{2\pi\left({\begin{array}[]{*{20}{l}}{\left({{f_{m}}{x_{m}}-{f_{n}}{x_{n}}}\right)}\\\
{\times\frac{{\sin{\theta_{k,j}}-\sin{\theta_{b}}}}{c}}\\\
{+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{k,i}}}}{c}}\end{array}}\right)}\right]}}}}$
(76d) $\displaystyle{s.t.}\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (76e)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{min}},\forall m.$ (76f)

Problem (42) is non-convex and is solved by BSUM. According to (44), (76d) is
replaced by

$\displaystyle\
\mathop{\min}\limits_{{x_{m}}}\sum\limits_{k=1}^{K}{\sum\limits_{i=1}^{L}{\sum\limits_{j=1}^{L}{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}{{A_{k,m,n}}{{\left({{x_{m}}-{B_{k,m,n}}}\right)}^{2}}+{C_{k,m,n}}}}}},$
(77a)

where ${{A_{k,m,n}}}$, ${{B_{k,m,n}}}$ and ${{C_{k,m,n}}}$ are obtained using
Lemma 1. Then, the optimal solution of (77a) is given by

$\displaystyle\
{x^{opt}}=\frac{{\sum\limits_{k=1}^{K}{\sum\limits_{i=1}^{L}{\sum\limits_{j=1}^{L}{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}{{A_{k,m,n}}{B_{k,m,n}}}}}}}}{{\sum\limits_{i=1}^{L}{\sum\limits_{j=1}^{L}{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}{{A_{k,m,n}}}}}}}.$ (78)

Considering the constraints (76e) and (76f), the optimal position of the
$m$-th antenna is given in (49).

###  IV-C Optimizing ${\bf{f}}$ with Given ${\bf{w}}\left(t\right)$ and
${\bf{x}}$.

Similar to problem (41), the optimization problem in this case is formulated
as

$\displaystyle\mathop{\min}\limits_{{{\bf{f}}}}{\rm{}}\sum\limits_{k=1}^{K}{\sum\limits_{i=1}^{L}{\sum\limits_{j=1}^{L}{{{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{a{w_{k}}}}\left({{r_{k,i}},{\theta_{k,j}}}\right)}\right\rangle}\right|}^{2}}}}}$
(79a) $\displaystyle{s.t.}\quad{f_{C}}\leq{f_{m}}\leq{f_{C}}+\Delta F,\forall
m.$ (79b)

We propose to utilize the BSUM method to solve this problem. According to
(44), (79a) can be replaced by

$\displaystyle\
\mathop{\min}\limits_{{f_{m}}}\sum\limits_{k=1}^{K}{\sum\limits_{i=1}^{L}{\sum\limits_{j=1}^{L}{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}{{A_{k,m,n}}{{\left({{x_{m}}-{B_{k,m,n}}}\right)}^{2}}+{C_{k,m,n}}}}}},$
(80a)

where ${{A_{k,m,n}}}$, ${{B_{k,m,n}}}$ and ${{C_{k,m,n}}}$ are obtained using
Lemma 2. Then, the optimal solution of (80a) is given as (78). Considering the
constraint (79b), the optimal frequency of the $m$-th antenna is given as
(69).

_Remark 1:_ According to (III-B), it is clear that
${{{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{a{w_{k}}}}\left({{r_{k,i}},{\theta_{k,j}}}\right)}\right\rangle}\right|}^{2}}}$
is not monotonic for ${r_{k,i}}$ and ${\theta_{k,j}}$, which means that we
cannot directly find the worst-case coordinates for Willie in the estimated
region $\left({\cal{R}},{\cal{A}}\right)$. To address this issue, we utilize
the discretization method to simplify the region to some samples which can
effectively deal with the position uncertainty in FDA [11]. It is worth
mentioning that more samples can make the results accurate, but it also leads
to higher complexity of the optimization.

##  V Numerical Results

In this section, numerical results for MFDA-aided covert communication are
presented. Unless otherwise specified, the maximum transmission power is
${P_{max}}=10$dBm and the carrier frequency is ${f_{C}}=10$GHz. The noise
variances at Bob and Willies are $\sigma^{2}_{b}=-100$dBm and
$\sigma_{{w_{k}}}^{2}=-100$dBm, $k=1,...,K$. The position coordinate of Bob is
$\left({{r_{b}},{\theta_{b}}}\right)=\left({1000{\rm{m}},{{30}^{\circ}}}\right)$.
We consider four Willies, i.e., $K=4$, and the cases of low correlation
channel and high correlation channel in this paper. In the case of low
correlation channel, Willies are not close to Bob and located at
$\left({{r_{{w_{1}}}},{\theta_{{w_{1}}}}}\right)=\left({1050{\rm{m}},{{10}^{\circ}}}\right)$,
$\left({{r_{{w_{2}}}},{\theta_{{w_{2}}}}}\right)=\left({950{\rm{m}},{{20}^{\circ}}}\right)$,
$\left({{r_{{w_{3}}}},{\theta_{{w_{3}}}}}\right)=\left({1100{\rm{m}},{{40}^{\circ}}}\right)$
and
$\left({{r_{{w_{4}}}},{\theta_{{w_{4}}}}}\right)=\left({900{\rm{m}},{{50}^{\circ}}}\right)$,
respectively. In the case of high correlation channel, Willies are close to
Bob and located at
$\left({{r_{{w_{1}}}},{\theta_{{w_{1}}}}}\right)=\left({1010{\rm{m}},{{26}^{\circ}}}\right)$,
$\left({{r_{{w_{2}}}},{\theta_{{w_{2}}}}}\right)=\left({990{\rm{m}},{{28}^{\circ}}}\right)$,
$\left({{r_{{w_{3}}}},{\theta_{{w_{3}}}}}\right)=\left({1020{\rm{m}},{{32}^{\circ}}}\right)$
and
$\left({{r_{{w_{4}}}},{\theta_{{w_{4}}}}}\right)=\left({980{\rm{m}},{{34}^{\circ}}}\right)$,
respectively. The minimum antenna separation distance is
${D_{0}}=\frac{\lambda}{2}$ where $\lambda=\frac{c}{{{f_{C}}}}$ is the
wavelength of MFDA. The movable antenna distance is ${D_{\max}}=30\lambda$ and
the frequency increment range is $\Delta F=10$ MHz. Moreover, the path loss
factor is given as [36]

$\displaystyle\
{\rm{Lfs}}\left(r\right)=C{\left({\frac{r}{R}}\right)^{-\alpha}},$ (81)

where $C=-30$dBm is the path loss at the reference distance $R=1{\rm{m}}$, $r$
is the individual link distance and $\alpha$ is the path loss exponent where
${\alpha_{ab}}=2$ and ${\alpha_{ae}}=3$.

For comparison, the covert rates of three benchmark strategies are also
investigated as follows: 1) PA, i.e., only optimizing the ABV at Alice. 2)
FDA, i.e., optimizing the ABV and AFV at Alice. 3) Perfect covertness, i.e.,
${{\bf{w}}^{\dagger}}\left(t\right){{\bf{H}}_{a{w_{k}}}}{\bf{w}}\left(t\right)=0,k=1,...,K$.
4) Upper bound, i.e.,
$R_{b}={\log}\left({1+\sigma_{b}^{-2}{\rm{Lf}}{{\rm{s}}^{2}}\left({{r_{b}}}\right){P_{\max}}M}\right)$.

(a) BSUM algorithm for APV (b) BSUM algorithm for AFV

Figure 2: Typical converging traces of the BSUM for optimizing APV and AFV
with $M=10$.

Figure 2(a) demonstrates the convergence of the BSUM algorithm for optimizing
APV where each trace starting from random initial APV and channel inner
product is denoted as
$\sum\limits_{k=1}^{K}{{{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{a{w_{k}}}}}\right\rangle}\right|}^{2}}}$.
The channel inner product is a metric for measuring the channel correlation
between the legitimate channel and illegitimate channels. In particular,
$\sum\limits_{k=1}^{K}{{{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{a{w_{k}}}}}\right\rangle}\right|}^{2}}}=0$
means that ${{\bf{h}}_{ab}}\perp{{\bf{h}}_{aw_{k}}}$, $k=1,..,K$, which is the
ideal case for optimizing the APV and AFV. We observe that BSUM takes about
800 iterations to converge and reach the minimum values. Moreover, the
convergence behaviour for optimizing AFV is illustrated in Figure 7(b).

(a) Low correlation channel (b) High correlation channel

Figure 3: Covert rate ${R_{b}}$ versus number of antennas $M$.

Figure 3 depicts the covert rate $R_{b}$ versus the number of antennas $M$ for
different strategies. We observe that $R_{b}$ increases with $M$, because that
brings more spatial DoFs. Besides, the covert rate of MFDA and FDA are larger
than that of PA. This is due to the fact that PA only optimizes the ABV at
Alice which results in a worse covert performance than other strategies as
expected. In the case of the low correlation channel, the covert rate of MFDA
is the same as that of FDA. However, in the case of the the high correlation
channel, MFDA outperforms FDA. This is because FDA can only adjust the
frequency, which is easily limited by the frequency constraint in a highly
correlated channel scenario. In particular, MFDA can jointly adjust the
frequency and position to achieve two-dimensional covertness and effectively
handle different channel scenarios. Moreover, the perfect covertness strategy
needs to ensure that Willies do not detect any signal power, thus cannot
achieve larger covert rate due to the strict covert constraints.

(a) Maximum distance (b) Frequency increment range

Figure 4: Covert rate ${R_{b}}$ versus maximum distance ${D_{\max}}$ and
frequency increment range $\Delta F/{f_{c}}$.

Figure 4 depicts the covert rate $R_{b}$ for different frequency increment
range $\Delta F/{f_{C}}$ and movable antenna maximum distance ${D_{\max}}$. It
is clear that larger ${D_{\max}}$ and $\Delta F$ can provide more selectivity
to optimize APV and AFV which can improve the covert rate correspondingly. In
practice, due to the device limitation or complex channel state, both APV and
AFV will have constraints, such that relying on separately optimizing either
one of them cannot achieve the best covert performance. Our proposed MFDA can
jointly optimize APV and AFV to realize two-dimensional covertness which leads
to a larger covert rate than FDA and PA. In particular, when the AFV
constraint is strict, MFDA can optimize APV to further enhance covert
performance instead of optimizing AFV.

(a) Low correlation channel (b) High correlation channel

Figure 5: Typical beampatterns for MFDA with $M=20$.

Figure 5 illustrates the beampattern with the optimal ABV, APV and AFV. We
observe that MFDA can produce a “dot-$shape$” beampattern. In particular, the
power gain is maximized at Bob’s coordinate and minimized at the different
Willies’ coordinates simultaneously. In the case of the low correlation
channel, this focal point is a single-maximum value in the range-angle
distribution diagram. However, in the case of the high correlation channel, it
is clear that the power gain is scattered on the beampattern, and only a
partial power gain is concentrated at Bob’s coordinate. Thus, this means that
highly correlated legitimate and illegitimate channels can degrade the covert
performance achieved by MFDA.

###  V-A MFDA with Imperfect CSI at Willies

In this subsection, we consider the case of imperfect CSI at Willies and
investigate the effects of the uncertainties in angle and range on covert
rate. Larger uncertainty needs more discrete samples to evaluate covert
performance which makes the optimization more complex. Hence, we only consider
three different Willies for simplicity, i.e., K=3. They are located at
$\left({{r_{{w_{1}}}},{\theta_{{w_{1}}}}}\right)=\left({1100{\rm{m}},{{10}^{\circ}}}\right)$,
$\left({{r_{{w_{2}}}},{\theta_{{w_{2}}}}}\right)=\left({850{\rm{m}},{{20}^{\circ}}}\right)$
and
$\left({{r_{{w_{3}}}},{\theta_{{w_{3}}}}}\right)=\left({1150{\rm{m}},{{40}^{\circ}}}\right)$,
respectively.

(a) BSUM algorithm for APV (b) BSUM algorithm for AFV

Figure 6: Typical converging traces of the BSUM for optimizing APV and AFV
with $M=10$ and $L=3$. $\Delta{r_{{w_{k}}}}=10\rm{m}$ and
$\Delta{\theta_{{w_{k}}}}={1^{\circ}}$, k=1,…,K.

Figure 6 demonstrates the convergence of the BSUM algorithm for optimizing APV
and AFV where each trace starts from random initial APV and AFV. The channel
inner product is denoted as
$\sum\limits_{k=1}^{K}{\sum\limits_{i=1}^{L}{\sum\limits_{j=1}^{L}{{{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{a{w_{k}}}}\left({{r_{k,i}},{\theta_{k,j}}}\right)}\right\rangle}\right|}^{2}}}}}$
in this case. Compared with Fig. 2, BSUM takes more iterations, about 8000, to
converge due to the imperfect CSI at Willies. Besides, the channel inner
products do not converge to zero by optimizing APV and AFV. This implies that
larger uncertainties in angle and range will increase the burden of optimizing
APV and AFV via BSUM.

Figure 7: Covert rate $R_{b}$ versus number of antennas $M$.

Figure 7 depicts the covert rate $R_{b}$ versus the number of antennas $M$.
The uncertain regions in angle and range are denoted as
${R_{{w_{k}}}}\buildrel\Delta\over{=}\left[{{r_{{w_{k}}}}-\frac{{L-1}}{2}\Delta
r_{{w_{k}}}^{{}^{\prime}},{r_{{w_{k}}}}+\frac{{L-1}}{2}\Delta
r_{{w_{k}}}^{{}^{\prime}}}\right]$ and
${\theta_{{w_{k}}}}\buildrel\Delta\over{=}\left[{{\theta_{{w_{k}}}}-\frac{{L-1}}{2}\Delta\theta_{{w_{k}}}^{{}^{\prime}},{\theta_{{w_{k}}}}+\frac{{L-1}}{2}\Delta\theta_{{w_{k}}}^{{}^{\prime}}}\right]$,
respectively. $\Delta r_{{w_{k}}}^{{}^{\prime}}=10{\rm{m}}$ and
$\Delta\theta_{{w_{k}}}^{{}^{\prime}}={1^{\circ}}$ reflect the accuracy of
discrete samples. To ensure that the signal power cannot be detected by any
samples within the uncertain regions, MFDA needs to decrease the correlations
of the legitimate channel and all possible illegitimate channels by optimizing
AFV and APV. However, the uncertain regions of different Willies increase with
$L$, making it increasingly difficult to decrease the correlations, which
leads to a degradation in covert performance. As expected, MFDA can
effectively decrease the performance loss caused by imperfect Willies CSI by
utilizing the frequency dimension and the position dimension compared to FDA
and PA.

(a) Same uncertainty, $\Delta{r_{{w_{k}}}}=20\rm{m}$ and
$\Delta{\theta_{{w_{k}}}}={2^{\circ}}$, k=1,…,3. (b) Different uncertainty,
$\Delta{r_{{w_{k}}}}=k\times 10{\rm{m}}$ and
$\Delta{\theta_{{{\rm{w}}_{k}}}}{=k}\times{{\rm{1}}^{\circ}}$, k=1,…,3.

Figure 8: Typical beampatterns for MFDA with $M=20$.

Figure 8 illustrates the beampatterns achieved by MFDA. We observe that MFDA
can maximize the received rate at Bob, while ensuring that Willies in the
uncertain area cannot detect the signal power. Moreover, MFDA can achieve the
regional covert communication by jointly optimizing ABV, AFV and APV based on
the sizes of different uncertain regions.

##  VI Conclusion

In this paper, we proposed a novel antenna technology MFDA to enhance covert
performance in covert communication with multiple wardens. Specifically, we
maximized the covert rate by jointly optimizing the ABV, APV and AFV at the
transmitter, subject to the given constraints. Then, a two-stage AO algorithm
with lower complexity was developed to solve the optimization problem.
Moreover, we considered the cases of perfect covertness and imperfect CSI at
wardens. Simulation results demonstrated that MFDA introduces a new dimension
which can further improve covert performance based on conventional FDA.
Moreover, MFDA has a strong channel identification ability to decouple the
highly correlated channel, and can effectively reduce the performance loss
caused by CSI estimation error.

##  Appendix A Proof of Lemma 1

###  A-A Case 1

In this case, we have
$y_{k,m,n}^{{}^{\prime}}{\left({x_{m}^{s-1};x_{n}^{s-1}}\right)}>0$,
${y_{k,m,n}}\left({{B_{k,m,n}};x_{n}^{s-1}}\right)=-1$ and
${B_{k,m,n}}<x_{m}^{s-1}$. The relation
$-1<{y_{k,m,n}}{\left({x_{m}^{s-1};x_{n}^{s-1}}\right)}<1$ holds if and only
if there exists an integer ${\kappa_{1}}\geq 0$ such that we have (92) where
$\left(a\right)$ is due to ${B_{k,m,n}}<x_{m}^{s-1}$ and $\left(b\right)$ is
because that ${\kappa_{1}}$ is an integer solution. Owing to
${y_{k,m,n}}\left({{B_{k,m,n}};x_{n}^{s-1}}\right)=-1$, we have

$\displaystyle\
\begin{array}[]{*{20}{l}}{-1<\cos\left[{2\pi\left({\begin{array}[]{*{20}{l}}{\left({{f_{m}}{x_{m}}-{f_{n}}{x_{n}}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\end{array}}\right)}\right]<1}\\\
{\begin{array}[]{*{20}{l}}{\mathop{\Rightarrow}\limits^{\left(a\right)}2\pi{\kappa_{1}}+\pi<2\pi\left({\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\right)<2\pi{\kappa_{1}}+2\pi}\\\
{\Rightarrow\left(\begin{array}[]{l}\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\\\
+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}\end{array}\right)-1<{\kappa_{1}}<\left(\begin{array}[]{l}\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\\\
+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}\end{array}\right)-\frac{1}{2}}\\\
{\mathop{\Rightarrow}\limits^{\left(b\right)}{\kappa_{1}}=\left\lceil{\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\right\rceil-1,}\end{array}}\end{array}$
(92)

$\displaystyle\
2\pi\left(\begin{array}[]{l}\left({{f_{m}}{B_{k,m,n}}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\\\
+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}\end{array}\right)=\pi+2{\kappa_{1}}\pi.$
(95)

Thus, ${B_{k,m,n}}$ is given by

$\displaystyle\ {B_{k,m,n}}$
$\displaystyle=\frac{c}{{{f_{m}}\left({\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}\right)}}$
$\displaystyle\times\left({\frac{{1+2{\kappa_{1}}}}{2}-\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\right)+\frac{{{f_{n}}x_{n}^{s-1}}}{{{f_{m}}}}.$
(96)

According to (35), we can obtain ${k_{m,n}}$ and ${\delta_{m,n}}$ through
simple mathematical manipulations which are given by

$\displaystyle\ {A_{k,m,n}}$
$\displaystyle=\frac{{-\pi{f_{m}}\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}}}{{x_{m}^{s-1}-{B_{k,m,n}}}}$
$\displaystyle\times\sin\left[{2\pi\left(\begin{array}[]{l}\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\\\
+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}\end{array}\right)}\right]$
(99)

and

$\displaystyle\ {C_{k,m,n}}$
$\displaystyle=\cos\left[{2\pi\left(\begin{array}[]{l}\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\\\
+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}\end{array}\right)}\right]$
(102) $\displaystyle-{A_{k,m,n}}{\left({x_{m}^{s-1}-{B_{k,m,n}}}\right)^{2}}.$
(103)

###  A-B Case 2

In this case, we have
$y_{k,m,n}^{{}^{\prime}}{\left({x_{m}^{s-1};x_{n}^{s-1}}\right)}>0$,
${y_{k,m,n}}\left({{B_{k,m,n}};x_{n}^{s-1}}\right)=-1$ and
$x_{m}^{s-1}<{B_{k,m,n}}$. The relation
$-1<{y_{k,m,n}}{\left({x_{m}^{s-1};x_{n}^{s-1}}\right)}<1$ holds if and only
if there exists an integer ${\kappa_{2}}\geq 0$ such that we have (114).
According to (92), we note that ${\kappa_{1}}={\kappa_{2}}$. Hence, we can
obtain ${A_{k,m,n}}$, ${B_{k,m,n}}$ and ${C_{k,m,n}}$ as in (A-A), (A-A) and
(102), respectively.

$\displaystyle\
\begin{array}[]{*{20}{l}}{-1<\cos\left[{2\pi\left({\begin{array}[]{*{20}{l}}{\left({{f_{m}}{x_{m}}-{f_{n}}{x_{n}}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\end{array}}\right)}\right]<1}\\\
{\begin{array}[]{*{20}{l}}{\mathop{\Rightarrow}\limits^{\left(a\right)}2\pi{\kappa_{2}}<2\pi\left({\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\right)<2\pi{\kappa_{2}}+\pi}\\\
{\Rightarrow\left(\begin{array}[]{l}\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\\\
+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}\end{array}\right)-\frac{1}{2}<{\kappa_{2}}<\left(\begin{array}[]{l}\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\\\
+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}\end{array}\right)}\\\
{\mathop{\Rightarrow}{\kappa_{2}}=\left\lfloor{\left({{f_{m}}x_{m}^{s-1}-{f_{n}}x_{n}^{s-1}}\right)\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\right\rfloor,}\end{array}}\end{array}$
(114)

###  A-C Case 3

In this case, we have
$y_{k,m,n}^{{}^{\prime}}{\left({x_{m}^{s-1};x_{n}^{s-1}}\right)}=0$ and
${y_{k,m,n}}{\left({x_{m}^{s-1};x_{n}^{s-1}}\right)}=-1$ which means that
${B_{k,m,n}}=x_{m}^{s-1}$ and ${C_{k,m,n}}=-1$. To find ${A_{k,m,n}}$, we
first introduce a small parameter $\kappa_{3}$ whose value approaches zero,
i.e., ${\kappa_{3}}\to 0$. Then, at the point ${x_{m}^{s-1}+{\kappa_{3}}}$, we
have
$y_{k,m,n}^{{}^{\prime}}\left({x_{m}^{s-1}+{\kappa_{3}};x_{n}^{s-1}}\right)=u_{k,m,n}^{{}^{\prime}}\left({x_{m}^{s-1}+{\kappa_{3}};x_{n}^{s-1}}\right)$.
We can obtain

$\displaystyle\
\begin{array}[]{*{20}{l}}{{A_{k,m,n}}=\frac{{-\pi{f_{m}}\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}}}{{{\kappa_{3}}}}}\\\
{\times\sin\left[{2\pi\left({\begin{array}[]{*{20}{l}}{\left({{f_{m}}\left({x_{m}^{s-1}+{\kappa_{3}}}\right)-{f_{n}}x_{n}^{s-1}}\right)}\\\
{\times\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}}\\\
{+\left({{f_{m}}-{f_{n}}}\right)\frac{{{r_{b}}-{r_{{w_{k}}}}}}{c}}\end{array}}\right)}\right]}\\\
{\mathop{\Rightarrow}\limits^{\left(c\right)}\frac{{\pi{f_{m}}\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\sin\left({2\pi\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}{f_{m}}{\kappa_{3}}}\right)}}{{{\kappa_{3}}}}.}\end{array}$
(121)

where $\left(c\right)$ is due to the sum difference of product method [39]. It
is worth noting that $\mathop{\lim}\limits_{{\kappa_{3}}\to
0}\frac{{\pi{f_{m}}\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}\sin\left({2\pi\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}{f_{m}}{\kappa_{3}}}\right)}}{{{\kappa_{3}}}}=2{\left({\pi{f_{m}}\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}}\right)^{2}}$,
thus we have
${A_{k,m,n}}=2{\left({\pi{f_{m}}\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}}\right)^{2}}$.

###  A-D Case 4

In this case, we have
$y_{k,m,n}^{{}^{\prime}}{\left({x_{m}^{s-1};x_{n}^{s-1}}\right)}=0$ and
${y_{k,m,n}}\left({x_{m}^{s-1};x_{n}^{s-1}}\right)=1$ which means that
${B_{k,m,n}}=x_{m}^{s-1}$ and ${C_{k,m,n}}=1$. Similar to case 3, we have
$y_{k,m,n}^{{}^{\prime}}\left({x_{m}^{s-1}+{\kappa_{3}};x_{n}^{s-1}}\right)=u_{k,m,n}^{{}^{\prime}}\left({x_{m}^{s-1}+{\kappa_{3}};x_{n}^{s-1}}\right)$,
such that we have
${A_{k,m,n}}=-2{\left({\pi{f_{m}}\frac{{\sin{\theta_{{w_{k}}}}-\sin{\theta_{b}}}}{c}}\right)^{2}}$.

## Appendix B

According to (71c), ${\bf{w}}\left(t\right)$ needs to lie in the null space of
${{\bf{H}}_{{\bf{aw}}}}$. Thus, it takes the form
${\bf{w}}=\left({{I_{M}}-{\bf{H}}_{{\bf{aw}}}^{\dagger}{{\left({{{\bf{H}}_{{\bf{aw}}}}{\bf{H}}_{{\bf{aw}}}^{\dagger}}\right)}^{-1}}{{\bf{H}}_{{\bf{aw}}}}}\right){{\bf{w}}_{0}}$
where ${{\bf{w}}_{0}}\in{{\mathbb{C}}^{M}}$. Next, we note that
${\left|{{{\bf{h}}_{ab}}{\bf{w}}}\right|^{2}}={\left|{{{\bf{h}}_{ab}}\left({{I_{M}}-{\bf{H}}_{{\bf{aw}}}^{\dagger}{{\left({{{\bf{H}}_{{\bf{aw}}}}{\bf{H}}_{{\bf{aw}}}^{\dagger}}\right)}^{-1}}{{\bf{H}}_{{\bf{aw}}}}}\right){{\bf{w}}_{0}}}\right|^{2}}\leq{\left\|{{{\bf{h}}_{ab}}\left({{I_{M}}-{\bf{H}}_{{\bf{aw}}}^{\dagger}{{\left({{{\bf{H}}_{{\bf{aw}}}}{\bf{H}}_{{\bf{aw}}}^{\dagger}}\right)}^{-1}}{{\bf{H}}_{{\bf{aw}}}}}\right)}\right\|^{2}}\left\|{{{\bf{w}}_{0}}}\right\|$,
and the equality holds when
${{{\bf{h}}_{ab}}\left({{I_{M}}-{\bf{H}}_{{\bf{aw}}}^{\dagger}{{\left({{{\bf{H}}_{{\bf{aw}}}}{\bf{H}}_{{\bf{aw}}}^{\dagger}}\right)}^{-1}}{{\bf{H}}_{{\bf{aw}}}}}\right)}$
and ${{{\bf{w}}_{0}}}$ are collinear. Note that
${\left\|{\bf{w}}\right\|^{2}}\leq{P_{max}}$, we can obtain that
${{\bf{w}}_{0}}=\frac{{\sqrt{{P_{\max}}}{\bf{h}}_{ab}^{\dagger}}}{{\left\|{\left({{I_{M}}-{\bf{H}}_{{\bf{aw}}}^{\dagger}{{\left({{{\bf{H}}_{{\bf{aw}}}}{\bf{H}}_{{\bf{aw}}}^{\dagger}}\right)}^{-1}}{{\bf{H}}_{{\bf{aw}}}}}\right){\bf{h}}_{ab}^{\dagger}}\right\|}}$.
The optimal ABV can be written as
${{\bf{w}}_{opt}}=\sqrt{{P_{\max}}}\frac{{\left({{I_{M}}-{\bf{H}}_{{\bf{aw}}}^{\dagger}{{\left({{{\bf{H}}_{{\bf{aw}}}}{\bf{H}}_{{\bf{aw}}}^{\dagger}}\right)}^{-1}}{{\bf{H}}_{{\bf{aw}}}}}\right){\bf{h}}_{ab}^{\dagger}}}{{\left\|{\left({{I_{M}}-{\bf{H}}_{{\bf{aw}}}^{\dagger}{{\left({{{\bf{H}}_{{\bf{aw}}}}{\bf{H}}_{{\bf{aw}}}^{\dagger}}\right)}^{-1}}{{\bf{H}}_{{\bf{aw}}}}}\right){\bf{h}}_{ab}^{\dagger}}\right\|}}$.

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