# Proactive Monitoring via Jamming in
Fluid Antenna Systems

Junteng Yao, Tuo Wu, Xiazhi Lai, Ming Jin, _Member_ , _IEEE_ , Cunhua Pan,
_Senior Member_ , _IEEE_ ,
Maged Elkashlan, _Senior Member_ , _IEEE_ , and Kai-Kit Wong, _Fellow_ ,
_IEEE_ _(Corresponding authors: Tuo Wu and Cunhua Pan.)_ J. Yao and M. Jin are
with the Faculty of Electrical Engineering and Computer Science, Ningbo
University, Ningbo 315211, China (e-mail: {yaojunteng, jinming}@nbu.edu.cn).T.
Wu and M. Elkashlan are with the School of Electronic Engineering and Computer
Science at Queen Mary University of London, London E1 4NS, U.K. (e-mail:
{tuo.wu, maged.elkashlan}@qmul.ac.uk).X. Lai is with the School of Computer
Science, Guangdong University of Education, Guangzhou, Guangdong, China
(E-mail: xzlai@outlook.com).C. Pan is with the National Mobile Communications
Research Laboratory, Southeast University, Nanjing 210096, China. (e-mail:
cpan@seu.edu.cn).K.-K. Wong is with the Department of Electronic and
Electrical Engineering, University College London, WC1E 6BT London, U.K., and
also with the Yonsei Frontier Laboratory and the School of Integrated
Technology, Yonsei University, Seoul 03722, South Korea (e-mail: kat-
kit.wong@ucl.ac.uk).

###### Abstract

This paper investigates the efficacy of utilizing fluid antenna system (FAS)
at a legitimate monitor to oversee suspicious communication. The monitor
switches the antenna position to minimize its outage probability for enhancing
the monitoring performance. Our objective is to maximize the average
monitoring rate, whose expression involves the integral of the first-order
Marcum $Q$ function. The optimization problem, as initially posed, is non-
convex owing to its objective function. Nevertheless, upon substituting with
an upper bound, we provide a theoretical foundation confirming the existence
of a unique optimal solution for the modified problem, achievable efficiently
by the bisection search method. Furthermore, we also introduce a locally
closed-form optimal resolution for maximizing the average monitoring rate.
Empirical evaluations confirm that the proposed schemes outperform
conventional benchmarks considerably.

###### Index Terms:

Average monitoring rate, fluid antenna system, jamming, Marcum $Q$ function,
proactive monitoring.

##  I Introduction

In the realm of wireless network security, the endeavor to monitor and detect
suspicious transmissions has been a focal point of research and development.
Xu _et al._ [1] made significant strides in this domain by introducing
proactive monitoring as a strategy to bolster public security.

This pioneering work laid the foundation for subsequent advancements,
including the proposal of proactive monitoring schemes that leverage jamming
techniques in relay-assisted networks [2]. Recognizing the need for even more
efficient monitoring solutions, researchers began to explore reconfigurable
intelligent surface (RIS)-assisted proactive monitoring systems [3]. These
innovative systems take advantage of the RIS to drastically reduce the power
consumption of monitoring [4]. However, wireless communication is not static
in nature and many mission-critical applications are now based on finite
blocklength transmissions, posing new challenges for monitoring systems. In
response to this shift, investigations have been carried out to design and
optimize proactive monitoring systems tailored specifically for finite
blocklength transmissions, a topic that has garnered attention in [5, 6].
Furthermore, in the context of covert monitoring, Cheng _et al._ [7]
considered a covert monitoring system, where artificial noise was used.

The aforementioned studies primarily focused on monitors equipped with a
single antenna. In pursuit of enhancing monitoring performance, researchers
have delved into proactive monitoring schemes with multiple antennas [9, 8].
Although multi-antenna monitors can indeed offer a substantial improvement in
monitoring performance, they are often accompanied by elevated energy
consumption and increased hardware costs. These additional expenses primarily
stem from the utilization of multiple antennas and radio frequency (RF)
chains.

To increase space diversity for performance enhancement while keeping the
energy consumption low, a novel technique known as fluid antenna system (FAS)
was recently introduced in [10, 11]. FAS has the ability to flexibly change
the antenna position over $N$ preset ports, retrieving space diversity even in
a small space. Motivated by this, we propose to incorporate FAS into proactive
monitoring. This integration aims to reduce the outage probability of monitor
without incurring excessive energy consumption or hardware costs.

In this paper, we aim to maximize the average monitoring rate within the FAS.
The primary contributions of this paper can be summarized as follows:

  * •

We investigate proactive monitoring using FAS in which the legitimate monitor
is equipped with a single fluid antenna to monitor suspicious communication.
We maximize the average monitoring rate, which involves the integral of the
first-order Marcum $Q$ function.

  * •

The considered problem is non-convex due to the non-convex objective function.
Additionally, the expression of this objective function poses significant
computational challenges. To address this issue, we employ the upper bound of
the objective function, and provide a theoretical proof demonstrating the
existence of a unique optimal solution for the formulated problem. This
solution can be effectively obtained using a bisection search algorithm.

  * •

To further reduce the computational complexity, we also propose an approximate
expression of the objective function, and obtain a closed-form local-optimal
solution.

  * •

Our proposed schemes outperform greatly other benchmarking schemes in terms of
monitoring performance, as evidenced by the numerical results.

##  II System Model

Consider a legitimate surveillance FAS configuration comprising the following
components: a single-antenna suspicious source (SS), a single-antenna
suspicious destination (SD) and a legitimate monitor. The monitor is equipped
with a single fluid antenna for receiving signals from the SS and another
fixed antenna for transmitting a jamming signal to the SD. The legitimate
monitor dynamically switches the fluid antenna to the most favorable port over
$N$ evenly distributed ports in a linear space of size $W\lambda$, where
$\lambda$ represents the wavelength [10]. We assume that the monitor knows its
own jamming and can flawlessly cancel its self-interference [1, 6].
Additionally, we consider that any delays resulting from port switching can be
negligibly small, e.g., using pixel antennas [10, 11].

The channels linking the SS to the SD, the SS to the $k$-th port of the
monitor, and the monitor to the SD are denoted as $h$, $g_{k}$, and $f$,
respectively. In this paper, we assume that $h$ and $f$ obey Rayleigh fading
distribution, i.e., $h\sim\mathcal{CN}(0,\sigma_{h}^{2})$ and
$f\sim\mathcal{CN}(0,\sigma_{f}^{2})$, respectively.

Given the close proximity of the $N$ ports in the FAS, the channels, denoted
by $g_{k},\forall k$, are inherently correlated. Their mathematical
expressions can be defined as [12, 13]

$\displaystyle g_{k}=\mu g_{0}+(1-\mu)e_{k},\quad k=1,2,\dots,N,$ (1)

where $N\geq 2$. The channel parameter of a virtual reference port, $g_{0}$,
follows a complex Gaussian distribution denoted by
$\mathcal{CN}(0,\sigma_{g}^{2})$ with zero mean and variance $\sigma^{2}$. The
term $e_{k}$ is an independently and identically distributed (i.i.d.) random
variable, also distributed as $\mathcal{CN}(0,\sigma_{g}^{2})$. Additionally,
$\mu$ represents the correlation factor, which is given by [12, 13]

$\displaystyle\mu=\sqrt{2}\sqrt{{}_{1}F_{2}\left(\frac{1}{2};1;\frac{3}{2};-\pi^{2}W^{2}\right)-\frac{J_{1}(2\pi
W)}{2\pi W}}$ (2)

with ${}_{a}F_{b}$ being the generalized hypergeometric function and
$J_{1}(\cdot)$ as the first-order Bessel function of the first kind.

Given $|g_{0}|$, the probability density function (PDF) of $g_{k}$ can be
expressed as

$\displaystyle
f_{|g_{k}|\big{|}|g_{0}|}(r_{k}|r_{0})=\frac{2r_{k}}{\sigma_{g}^{2}(1-\mu^{2})}e^{-\frac{r_{k}^{2}+\mu^{2}r_{0}^{2}}{\sigma_{g}^{2}(1-\mu^{2})}}I_{0}\left(\frac{2\sigma_{g}\mu
r_{k}r_{0}}{\sigma_{g}^{2}(1-\mu^{2})}\right),$ (3)

where $I_{0}(u)$ denotes the modified Bessel function of the first kind and
order zero. Its series representation is given by [14]

$\displaystyle
I_{0}(z)=\sum_{k=0}^{\infty}\frac{z^{2k}}{2^{2k}k!\Gamma(k+1)}.$ (4)

The SS broadcasts a signal, denoted as $x_{s}$ with
$\mathbb{E}[|x_{s}|^{2}]=1$. The received signals at the SD and the $k$-th
port of the legitimate monitor are, respectively, expressed as

$\displaystyle y_{d}$
$\displaystyle=\sqrt{p_{s}}hx_{s}+\sqrt{p_{m}}fx_{m}+n_{d},$ (5)
$\displaystyle y_{m,k}$ $\displaystyle=\sqrt{p_{s}}g_{k}x_{s}+n_{m},$ (6)

where $p_{s}$ represents the transmission power of the SS; $p_{m}$ denotes the
power of the jamming signal; $x_{m}$ represents the jamming signal with
$\mathbb{E}[|x_{m}|^{2}]=1$ sent from the monitor,
$n_{d}\sim\mathcal{CN}(0,\sigma_{d}^{2})$ and
$n_{m}\sim\mathcal{CN}(0,\sigma_{m}^{2})$ denote the additive Gaussian noise
at the SD and the monitor, respectively.

Therefore, the signal-to-interference-plus-noise ratio (SINR) at the SD and
the signal-to-noise ratio (SNR) at the $k$-th port of the monitor can be
expressed as

$\displaystyle\gamma_{d}$
$\displaystyle=\frac{p_{s}|h|^{2}}{p_{m}|f|^{2}+\sigma^{2}_{d}},$ (7)
$\displaystyle\gamma_{m,k}$
$\displaystyle=\frac{p_{s}|g_{k}|^{2}}{\sigma^{2}_{m}},$ (8)

respectively.

We assume that the monitor can promptly switch the fluid antenna to the most
favorable port, and the maximum of $|g_{k}|$ can be expressed as

$\displaystyle|g_{\max}|=\max\\{|g_{1},|g_{2}|,\dots,|g_{N}|\\}.$ (9)

As a result, the SNR of the monitor is found as

$\displaystyle\gamma_{m}$
$\displaystyle=\frac{p_{s}|g_{\max}|^{2}}{\sigma^{2}_{m}}.$ (10)

In accordance with [15], for a given transmission rate $R$, the outage
probability at the SD is defined as

$\displaystyle\mathbb{P}^{\mathrm{out}}_{d}$
$\displaystyle=\mathbb{P}(\log_{2}(1+\gamma_{d})<R)=1-\frac{\lambda_{2}}{\lambda_{1}+\lambda_{2}}e^{-\lambda_{1}\sigma^{2}_{h}\gamma_{th}},$
(11)

where $\gamma_{th}=2^{R}-1$, $\lambda_{1}=\frac{1}{\sigma^{2}_{h}p_{s}}$, and
$\lambda_{2}=\frac{1}{\sigma^{2}_{f}\gamma_{th}p_{s}}$.

According to [12, (16)], the outage probability at the monitor for a given
transmission rate $R$ is expressed as

$\displaystyle\mathbb{P}^{\mathrm{out}}_{m}=$
$\displaystyle\mathbb{P}(\log_{2}(1+\gamma_{m})<R)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}e^{-t}$
$\displaystyle\times\left[1-Q_{1}\left(\sqrt{\frac{2\mu^{2}}{1-\mu^{2}}}\sqrt{t},\sqrt{\frac{2}{1-\mu^{2}}}\sqrt{\frac{\gamma_{th}}{\Gamma}}\right)\right]^{N}dt,$
(12)

in which $\Gamma=p_{s}\sigma^{2}_{g}/\sigma^{2}_{m}$ and $Q_{1}(\cdot,\cdot)$
represents the first-order Marcum $Q$-function. The average monitoring rate at
the legitimate monitor can therefore be determined by [1]

$R_{m}=R(1-\mathbb{P}^{\mathrm{out}}_{m}).$ (13)

Our objective is to optimize $p_{m}$ for maximizing the average monitoring
rate at the legitimate monitor. This leads to the following optimization
problem:

$\displaystyle\max_{0\leq p_{m}\leq p^{\max}_{m}}\ \ R_{m}\ \ \mbox{s.t.}\
\mathbb{P}^{\mathrm{out}}_{d}=\delta,$ (14)

where $p^{\max}_{m}$ represents the maximum transmission power of the jamming
signal, and $\delta>0$ denotes the target outage probability at the SD.
Problem (14) is a non-convex problem due to the non-convex objective function,
which is challenging.

##  III Proactive Monitoring using an Upper Bound

Here, we begin by deriving an upper bound of the objective function of (14).
Subsequently, we prove that this optimization problem possesses a unique
solution, which can be effectively determined using the bisection search
method.

According to (4), it can be seen that a lower bound of the
$f_{|g_{k}|||g_{0}|}(r_{k}|r_{0})$ can be found by

$\displaystyle f_{|g_{k}|||g_{0}|}(r_{k}|r_{0})=$
$\displaystyle\frac{2r_{k}}{\sigma_{g}^{2}(1-\mu^{2})}e^{-\frac{r_{k}^{2}+\mu^{2}r_{0}^{2}}{\sigma_{g}^{2}(1-\mu^{2})}}.$
(15)

Considering $|g_{0}|$, it is evident that $|g_{1}|,\dots,|g_{N}|$ are mutually
independent. This allows us to deduce the joint PDF of $|g_{1}|,\dots,|g_{N}|$
as

$\displaystyle f_{|g_{1}|,\dots,|g_{N}|||g_{0}|}(r_{1},\dots,r_{N}|r_{0})=$
$\displaystyle\prod_{k=1}^{N}\frac{2r_{k}}{\sigma_{g}^{2}(1-\mu^{2})}e^{-\frac{r_{k}^{2}+\mu^{2}r_{0}^{2}}{\sigma_{g}^{2}(1-\mu^{2})}}.$
(16)

Also, from (16), the joint PDF of $|g_{0}|,\dots,|g_{N}|$ is written as

$\displaystyle f_{|g_{0}|,\dots,|g_{N}|}(r_{0},\dots,r_{N})=$
$\displaystyle\frac{2r_{0}}{\sigma_{g}^{2}}e^{-\frac{r_{0}^{2}}{\sigma_{g}^{2}}}\prod_{k=1}^{N}\frac{2r_{k}}{\sigma_{g}^{2}(1-\mu^{2})}e^{-\frac{r_{k}^{2}+\mu^{2}r_{0}^{2}}{\sigma_{g}^{2}(1-\mu^{2})}}.$
(17)

Now, using (16), the lower bound of $\mathbb{P}^{\mathrm{out}}_{m}$ can be
written as

$\displaystyle\hat{\mathbb{P}}^{\mathrm{out}}_{m}=$
$\displaystyle\mathbb{P}\left(|g_{0}|<\infty,|g_{1}|<\sqrt{\frac{\gamma_{th}\sigma_{g}^{2}}{\Gamma}},\dots,|g_{N}|<\sqrt{\frac{\gamma_{th}\sigma_{g}^{2}}{\Gamma}}\right)$
$\displaystyle=$
$\displaystyle\eta\left[1-e^{-\frac{\gamma_{th}}{\Gamma(1-\mu^{2})}}\right]^{N},$
(18)

where $\eta=\frac{1-\mu^{2}}{1+(N-1)\mu^{2}}$. Therefore, an upper bound of
the average monitoring rate $R_{m}$ can be found as

$\hat{R}_{m}=R(1-\hat{\mathbb{P}}^{\mathrm{out}}_{m}).$ (19)

As a result, Problem (14) can be reformulated by

$\displaystyle\max_{0\leq p_{m}\leq p^{\max}_{m}}\ \ \hat{R}_{m}\ \
\mbox{s.t.}\ \mathbb{P}^{\mathrm{out}}_{d}=\delta.$ (20)

Based on [1], it is established that $R$ and $p_{m}$ are in one-to-one
correspondence subject to the constraint
$\mathbb{P}^{\mathrm{out}}_{d}=\delta$ in Problem (20). Consequently, the
constraint $0\leq p_{m}\leq p^{\max}_{m}$ can be equivalently expressed as

$R^{\min}\leq R\leq R^{\max}$ (21)

with

$\displaystyle R^{\min}$
$\displaystyle=\log_{2}\left(1+\frac{p_{s}\sigma^{2}_{h}}{\sigma^{2}_{d}}\mathcal{W}\left(\frac{\sigma^{2}_{d}e^{\frac{\sigma^{2}_{h}}{p_{m}\sigma^{2}_{f}}}}{\sigma^{2}_{f}p_{m}(1-\delta)}\right)-\frac{\sigma^{2}_{h}p_{s}}{p_{m}\sigma^{2}_{f}}\right),$
(22) $\displaystyle R^{\max}$
$\displaystyle=\log_{2}\left(1-\frac{p_{s}\sigma^{2}_{h}\ln(1-\delta)}{\sigma^{2}_{d}}\right),$
(23)

where $\mathcal{W}(z)$ represents the Lambert $\mathcal{W}$ function.

After obtaining $R^{\min}$ and $R^{\max}$, (20) is recast as

$\displaystyle\max_{R}\ \ \hat{R}_{m}\ \ \mbox{s.t.}\ R^{\min}\leq R\leq
R^{\max}.$ (24)

To solve (24), we let $x=2^{R}-1$ and have $R=\log_{2}(1+x)$. The objective
function $\hat{R}_{m}$ can be rewritten as

$\displaystyle
F(x)=\log_{2}(1+x)\left(1-\eta\left(1-e^{-\frac{x}{\Gamma(1-\mu^{2})}}\right)^{N}\right).$
(25)

Upon calculating the first-order partial derivative of $F(x)$ with respect to
$x$, and performing subsequent mathematical manipulations, we obtain the
following results:

$\displaystyle\frac{\partial F(x)}{\partial x}=$ $\displaystyle h(x)-g(x),$
(26)

where

$\displaystyle h(x)=$ $\displaystyle\frac{1}{\ln
2(1+x)}\left(1-\eta\left(1-e^{-\frac{x}{\Gamma(1-\mu^{2})}}\right)^{N}\right),$
(27) $\displaystyle g(x)=$
$\displaystyle\frac{N\log_{2}(1+x)}{\Gamma(1-\mu^{2})}\left(\eta\left(1-e^{-\frac{x}{\Gamma(1-\mu^{2})}}\right)^{N-1}e^{-\frac{x}{\Gamma(1-\mu^{2})}}\right).$
(28)

To continue, we have the following lemma.

_Lemma 1_ : $F(x)$ is a monotonically increasing function over the interval
$x\in[0,x^{o}]$, and monotonically decreasing function over the interval
$x\in(x^{o},+\infty]$, where $x^{o}=2^{R^{o}}-1$ and $R^{o}$ is the solution
of the equation $\frac{\partial\hat{R}_{m}}{\partial R}=0$.

###### Proof:

See Appendix A. ∎

According to Lemma 1, we can ascertain that Problem (20) possesses a
distinctive optimal solution, which is given by

$\displaystyle\min\\{\max\\{R^{\min},R^{o}\\},R^{\max}\\},$ (29)

where $R^{o}$ can be determined through utilizing the bisection search method
within the interval $R\in[R^{\min},R^{\max}]$.

_Complexity Analysis_ : The computational complexity of solving Problem (20)
is

$\mathcal{O}\left(\log_{2}\left(\frac{R^{\max}-R^{\min}}{\epsilon}\right)\right),$
(30)

where $\epsilon$ denotes the accuracy of bisection search.

##  IV A Closed-Form Solution to Proactive Monitoring

In this section, we aim to reduce the computational complexity for achieving
proactive monitoring by proposing a closed-form solution.

Considering $0\leq
Q_{1}\left(\sqrt{\frac{2\mu^{2}}{1-\mu^{2}}}\sqrt{t},\sqrt{\frac{2}{1-\mu^{2}}}\sqrt{\frac{\gamma_{th}}{\Gamma}}\right)\leq
1$, we can obtain

$\left[1-Q_{1}\left(\sqrt{\frac{2\mu^{2}}{1-\mu^{2}}}\sqrt{t}\sqrt{\frac{2}{1-\mu^{2}}}\sqrt{\frac{\gamma_{th}}{\Gamma}}\right)\right]^{N}\\\
\approx
1-NQ_{1}\left(\sqrt{\frac{2\mu^{2}}{1-\mu^{2}}}\sqrt{t},\sqrt{\frac{2}{1-\mu^{2}}}\sqrt{\frac{\gamma_{th}}{\Gamma}}\right).$
(31)

Therefore, $\mathbb{P}^{\mathrm{out}}_{m}$ can be approximated by

$\displaystyle\tilde{\mathbb{P}}^{\mathrm{out}}_{m}=$ $\displaystyle
1-Ne^{-\frac{\gamma_{th}}{\Gamma}}.$ (32)

The approximated average monitoring rate at the legitimate monitor is denoted
as

$\tilde{R}_{m}=R(1-\tilde{\mathbb{P}}^{\mathrm{out}}_{m})=NRe^{-\frac{\gamma_{th}}{\Gamma}}.$
(33)

This expression (33) reveals that the approximated average monitoring rate is
essentially $N$ times $Re^{-\frac{\gamma_{th}}{\Gamma}}$. This is consistent
with the average monitoring rate of the conventional single antenna scheme
presented in [1].

The optimization problem is thus expressed as

$\displaystyle\max_{0\leq p_{m}\leq p^{\max}_{m}}\ \
Re^{-\frac{\gamma_{th}}{\Gamma}}\ \ \mbox{s.t.}\
\mathbb{P}^{\mathrm{out}}_{d}=\delta.$ (34)

Analogous to _Lemma 1_ , we find that $Re^{-\frac{\gamma_{th}}{\Gamma}}$ first
increases and then decreases concerning $R$. The closed-form solution for
$\frac{\partial\tilde{R}_{m}}{\partial R}=0$ is therefore provided by

$\displaystyle\bar{R}^{\star}=\frac{1}{\ln 2}\mathcal{W}\left(\Gamma\right).$
(35)

Thus, the optimal solution to Problem (34) becomes

$\displaystyle\min\\{\max\\{R^{\min},\bar{R}^{\star}\\},R^{\max}\\}.$ (36)

_Complexity Analysis_ : The computational complexity of solving Problem (34)
is negligible.

##  V Numerical Results

In our simulations, we make several key assumptions and parameter choices to
set the conditions for investigation. We assume that the channel variances are
$\sigma_{h}^{2}=1$, such that the link between SS and SD offers superior
signal quality compared to the links between SD and the monitor, as well as
between the monitor and SD, where
$\sigma_{g}^{2}=\sigma_{f}^{2}=\sigma^{2}\leq 1$. Additionally, we set
$\sigma^{2}_{d}=\sigma^{2}_{m}=1$ for the additive Gaussian noises at both the
SD and the monitor. The signal power at the SS is established as
$p_{s}/\sigma^{2}=20$ dB, while the maximum transmission power of the jamming
signal is capped at $p_{m}^{\max}=30$ dB. Our target outage probability at the
SD is $\delta=0.05$, and we employ a parameter value of $W=5$, as referenced
in [10]. These assumptions and parameter selections collectively define the
conditions for our simulation scenarios.

In Fig. 1, the results are provided for the true and approximate average
monitoring rates across various values of $p_{m}$, considering a scenario with
$N=8$ and $\sigma^{2}/\sigma^{2}_{h}=-18$ dB. In the legend, “Approximate in
(19)” corresponds to $\hat{R}_{m}$ in (19), “Approximate in (33)” represents
$\tilde{R}_{m}$ in (33), while “True in (13)” represents $R_{m}$ in (13). The
results in Fig. 1 reveal that the optimal jamming powers for the monitor are
$17$ dB, $24$ dB, and $19$ dB for “Optimum in (19)”, “Optimum in (33)”, and
“Optimum in (13)”, respectively.

Figure 1: True and approximate average monitoring rates versus $p_{m}$, where
$N=8$ and $\sigma^{2}/\sigma^{2}_{h}=-18$ dB.

In Fig. 2, we examine the impact of $\sigma^{2}/\sigma^{2}_{h}$ on the average
monitoring rate. In the legend, “Upper bound” means that the optimal solution
of Problem (14) is obtained through exhaustive search, while “Proposed
jamming” denotes the optimal solution of Problem (24), “Proposed closed-form
solution” denotes the optimal solution of Problem (34), “Constant jamming”
represents proactive monitoring with constant jamming power, where
$p_{m}=P^{\max}_{m}=30$ dB, “Passive monitoring” represents proactive
monitoring without jamming. Furthermore, “Conventional jamming” represents the
proactive monitoring with a conventional single antenna in [1].

Observing from Fig. 2, we see that the performance of “Proposed jamming”
scheme is nearly identical to that of the “Upper bound” scheme. Additionally,
it is evident that our proposed proactive monitoring schemes consistently
outperform the “Conventional jamming” scheme. Notably, as depicted in Fig. 2,
when $\sigma^{2}/\sigma^{2}_{h}$ is high, the monitoring performance of our
proposed schemes closely matches that of the “Passive monitoring” scheme. This
convergence occurs because, with a high $\sigma^{2}/\sigma^{2}_{h}$ ratio, the
optimal jamming power is $p_{m}=0$.

Finally, in Fig. 3, we demonstrate the impact of $N$ on the average monitoring
rate with $\sigma^{2}/\sigma^{2}_{h}=-18$ dB. As seen from Fig. 3, the
monitoring efficacy of our proposed proactive monitoring schemes enhances as
$N$ increases. This is due to the monitor’s outage probability decreasing as
$N$ goes up. It is noteworthy that the curve for the “Constant jamming” scheme
levels off when $N>4$, indicating little to no improvement in the outage
probability as $N$ further increases.

Figure 2: Average monitoring rate versus $\sigma^{2}/\sigma^{2}_{h}$;
performance comparison of different schemes, where $N=8$. Figure 3: Average
monitoring rate versus $N$; performance comparison of different schemes, where
$\sigma^{2}/\sigma^{2}_{h}=-18$ dB.

##  VI Conclusion

In this paper, we focused on scenarios in which the monitor is equipped with a
single fluid antenna for performance enhancement. We introduced two proactive
monitoring schemes involving the design of jamming strategies. Our results
consistently demonstrated that our proposed scheme surpasses other schemes,
including proactive monitoring with a conventional single antenna, passive
monitoring without jamming, and proactive monitoring with constant jamming
power schemes.

##  Appendix A Proof of Lemma 1

From (27) and (28), we know that $h(0)=1$ and $g(0)=0$, and $h(x)$ and $g(x)$
are both larger than 0 when $x>0$.

Taking the first-order partial derivative of $h(x)$ with respect to $x$, we
can obtain

$\frac{\partial h(x)}{\partial x}=-\frac{1}{\ln
2(1+x)^{2}}\left(1-\eta\left(1-e^{-\frac{x}{\Gamma(1-\mu^{2})}}\right)^{N}\right)\\\
-\frac{N}{\ln 2(1+x)\Gamma(1-\mu^{2})}\\\
\times\left(\eta\left(1-e^{-\frac{x}{\Gamma(1-\mu^{2})}}\right)^{N-1}e^{-\frac{x}{\Gamma(1-\mu^{2})}}\right).$
(37)

We can readily know that $h(x)$ is a monotonically decreasing function with
respect to $x$ when $x\geq 0$.

Similarly, taking the first-order partial derivative of $\log_{2}(1+x)g(x)$
with respect to $x$, we can obtain

$\frac{\partial g(x)}{\partial x}=\frac{N}{\ln 2(1+x)\Gamma(1-\mu^{2})}\\\
\times\left(\eta\left(1-e^{-\frac{x}{\Gamma(1-\mu^{2})}}\right)^{N-1}e^{-\frac{x}{\Gamma(1-\mu^{2})}}\right)\\\
+\log_{2}(1+x)\left(1-e^{-\frac{x}{\Gamma(1-\mu^{2})}}\right)^{N-2}e^{-\frac{x}{\Gamma(1-\mu^{2})}}\\\
\times(N-2+e^{-\frac{x}{\Gamma(1-\mu^{2})}}).$ (38)

We can readily know that $g(x)$ is a monotonically increasing function with
respect to $x$ when $x\geq 0$ and $N\geq 2$. In our considered proactive
monitoring system, we have $x\geq 0$ and $N\geq 2$. As a result, we can
conclude that $h(x)$ is a monotonically decreasing function with respect to
$x$ and $g(x)$ is a monotonically increasing function.

Moreover, we have $h(0)=1$ and $g(0)=0$, and $h(+\infty)=0$ and
$g(+\infty)=+\infty$. $h(x)-g(x)$ is larger than 0 over the interval
$x\in[0,x^{o})$, and smaller than 0 over the interval $x\in(x^{o},+\infty]$,
where $x^{o}=2^{R^{o}}-1$ and $R^{o}$ is the solution of $h(R^{o})=g(R^{o})$.
Therefore, we can obtain that $F(x)$ is a monotonically increasing function
over the interval $x\in[0,x^{o}]$, and monotonically decreasing function over
the interval $x\in(x^{o},+\infty]$, which concludes the proof.

## References

  * [1] J. Xu, L. Duan, and R. Zhang, “Proactive eavesdropping via jamming for rate maximization over Rayleigh fading channels,” _IEEE Wireless Commun. Lett._ , vol. 5, no. 1, pp. 80-83, Feb. 2016.
  * [2] D. Hu, Q. Zhang, P, Yang, and J. Qin, “Proactive monitoring via jamming in amplify-and-forward relay networks,” _IEEE Signal Process. Lett._ , vol. 24, no. 11, pp. 1714-1718, Nov. 2017.
  * [3] J. Yao, T. Wu, Q. Zhang, and J. Qin, “Proactive monitoring via passive reflection using intelligent reflecting surface,” _IEEE Commun. Lett._ , vol. 24, no. 9, pp. 1909-1913, Sep. 2020.
  * [4] G. Hu, J. Si, Y. Cai, and N. Al-Dhahir, “Intelligent reflecting surface assisted proactive eavesdropping over suspicious broadcasting communication with statistical CSI,” _IEEE Trans. Veh. Technol._ , vol. 71, no. 4, pp. 4483-4488, Apr. 2022.
  * [5] D. Xu and H. Zhu, “Proactive eavesdropping via jamming over short packet suspicious communications with finite blocklength,” _IEEE Trans. Commun._ , vol. 70, no. 11, pp. 7505-7519, Nov. 2022.
  * [6] J. Yao, Q. Zhang, and J. Qin, “Proactive monitoring via jamming with finite blocklength transmissions,” _IEEE Trans. Veh. Technol._ , vol. 71, no. 1, pp. 1020-1025, Jan. 2020.
  * [7] Z. Cheng et al., “Covert surveillance via proactive eavesdropping under channel uncertainty,” _IEEE Trans. Commun._ , vol. 69, no. 6, pp. 4024-4037, Jun. 2021.
  * [8] C. Zhong, X. Jiang, F. Qu, and Z. Zhang, “Multi-antenna wireless legitimate surveillance systems: Design and performance analysis,” _IEEE Trans. Wireless Commun._ , vol. 16, no. 7, pp. 4585-4599, Jul. 2017.
  * [9] H. Cai, Q. Zhang, Q. Li, and J. Qin, “Proactive monitoring via jamming for rate maximization over MIMO rayleigh fading channels,” _IEEE Commun. Lett._ , vol. 21, no. 9, pp. 2021-2024, Sep. 2017.
  * [10] K.-K. Wong, A. Shojaeifard, K.-F. Tong, and Y. Zhang, “Fluid antenna system,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 3, pp. 1950-1962, Mar 2021.
  * [11] K.-K. Wong and K.-F. Tong, “Fluid antenna multiple access,” _IEEE Trans. Wireless Commun._ , vol. 21, no. 7, pp. 4801-4815, Jul. 2022.
  * [12] K. K. Wong, K. F. Tong, Y. Chen and Y. Zhang, “Closed-form expressions for spatial correlation parameters for performance analysis of fluid antenna systems,” IET Elect. Lett., vol. 58, no. 11, pp. 454–457, Apr. 2022.
  * [13] K.-K. Wong, D. Morales-Jimenez, K.-F. Tong, and C.-B. Chae, “Slow fluid antenna multiple access,” _IEEE Trans. Commun._ , vol. 71, no. 5, pp. 2831-2846, May 2023.
  * [14] I. S. Gradshteyn and I. M. Ryzhik, _Table of Integrals, Series, and Products, 7th ed._ San Diego, CA: Academic, 2007
  * [15] X. Tang, P. Ren, and Z. Han, “Combating full-duplex active eavesdropper: A game-theoretic perspective,” in _Proc. IEEE Int. Conf. Commun. (ICC)_ , Kuala Lumpur, Malaysia, May 2016, pp. 1-6.
  * [16] M. K. Simon and M.-S. Alouini, “Exponential-type bounds on the generalized Marcum Q-function with application to error probability analysis over fading channels” _IIEEE Trans. on Commun._ , vol. 48, pp. 359-366, Mar. 2000.
  * [17] S. Boyd and L. Vandenberghe, _Convex Optimization_. Cambridge, U.K.: Cambridge Univ. Press, 2004.