Analysis and Mitigating Methods for Jamming in the Optical Reconfigurable Intelligent Surfaces-Assisted Dual-Hop FSO Communication Systems

Abstract

In this paper, we present a study investigating the impact of jamming in a Dual-Hop free-space optical (FSO) communication system assisted by reconfigurable intelligent surfaces (RIS) in the presence of a malicious jammer. We analyze the combined effects of atmospheric turbulence (AT), pointing error (PE), and angle of arrival (AoA) fluctuation of unmanned aerial vehicles (UAVs). Closed-form expressions for the overall average bit error rate (ABER) are derived while considering these impairments. To mitigate the jamming effect, we explore a Single-Input Multiple-Output (SIMO) FSO system and derive the end-to-end Average Bit Error Rate (ABER) under various jamming scenarios. Additionally, we conduct a comprehensive study by examining different placements of the malicious UAV jammer and RIS, drawing insightful conclusions on system performance. The analytically derived expressions are validated through Monte Carlo simulations.

1. Introduction

With the advancement of society towards full automation and remote management systems, there is a significant demand for network systems capable of handling a large number of transceivers [1]. Primary attributes of fifth-generation and future communication systems consist of robust security protocols, ultra-low latency (as low as 0.1 milliseconds), and remarkably high data transfer capacities (up to 1 Terabit per second) [2,3]. FSO technology is capable of providing very high data transmission rates, high security, and low latency compared to radio frequency (RF) technology, offering advantages in these aspects [4,5]. Furthermore, FSO offers advantages such as unlicensed spectrum usage and low installation costs [3,4]. These features make FSO an optimal technological solution for the imminent advent of fifth-generation and the subsequent sixth generation of wireless communication systems [6]. However, due to the nature of FSO as a line-of-sight (LoS) technology, it is highly susceptible to AT [7], PE [8], and other factors [9], which make long-distance communication challenging [3,10]. Numerous methodologies have been developed to enhance FSO communication [7,8,9].

In recent times, RIS have emerged as a promising innovation in the research field. Their objective is to mitigate LOS limitations for FSO links through active control over the direction of reflected beams [11,12]. RIS modules are passive components that operate autonomously without the need for a dedicated power source [13]. Since RIS only needs to reflect incident waves in the desired direction, there is no need for analog-to-digital/digital-to-analog converters or power amplifier circuits, thereby avoiding the impact of receiver noise [14,15]. These advantages make RIS a cost-effective, easy-to-deploy, and energy-efficient relay that can be used in wireless RF and FSO communication systems [16,17]. In [12], Najafi et al. investigated the use of intelligent reflecting surfaces to relax the LoS requirement of FSO systems. Naik et al. in [8] implemented an RIS-assisted FSO system in a smart city scenario and studied the impact of signal blockage and Gamma-Gamma (G-G) turbulence on the system performance. In [18], Ndjiongue et al. gave an analysis of RIS-Based Terrestrial-FSO Link Over G-G Turbulence With Distance and Jitter Ratios. In [19], Huang, J. et al. have presented a feasible multi-branch wireless optical communication setup utilizing optical reconfigurable intelligent surfaces. In [18,20], the authors introduced a unified end-to-end Probability Density Function (PDF) expression for a terrestrial FSO RIS system considering the combined impact of G-G atmospheric turbulence, pointing error on the S-IRS/IRS-D links, pointing error ratios, and the placement of the RIS. In summary, RIS integration in FSO communication indeed offers several compelling advantages that directly address the mentioned challenges:

(1)

Mitigation of Atmospheric Transmission Losses: By leveraging RIS, we can manipulate the propagation path of signals, thereby reducing losses incurred due to atmospheric transmission [19]. RIS facilitates directing signals towards the intended receiver, effectively minimizing transmission distances and associated losses within the atmosphere [12].

(2)

Compensation for Pointing Errors: RIS provides the capability to adjust phase and amplitude during signal propagation, offering compensation for pointing errors. Through the intelligent modulation capabilities of RIS, we can achieve more precise signal alignment, thus enhancing system stability and performance [18,20].

However, FSO systems are mainly vulnerable to malicious jamming signals due to three key factors. (1) In FSO communications, the selection of operational wavelengths is very limited. Some operational wavelengths for FSO transmission include 830 nm, 1300 nm, and 1550 nm [21]. Considering eye safety and lower optical losses, 1550 nm is the most commonly used operational wavelength in many FSO-based applications [21]. Consequently, it is relatively easy for jammer to determine the operational wavelength of dedicated FSO communication links; (2) To mitigate the impact of fading effects such as AT and PE on FSO systems, it is common practice to increase the receiver aperture Field-of-View (FoV) [22]. Enlarging the receiver FOV also makes it feasible to jam optical beams. Many terrestrial applications, such as storage area networks and enterprise connectivity, require aperture averaging and a wide FoV to mitigate the effects of building sway, seismic activity, or physical obstruction, even when the transceiver is stationary [23]. In the satellite communication, a wider FoV is particularly crucial due to the mobility of transceivers and their significant distance from each other [24]; (3) In FSO communication, which belongs to LoS communication, communication antennas are typically installed at the top of buildings along streets or in elevated and easily accessible locations to avoid obstructions [21]. Consequently, malicious jammers may easily identify the locations for jamming.

Researchers have proposed analytical models and methods to mitigate malicious jamming, analyze its impact on the communication performance of FSO systems. In [25] the authors investigated the impact of pulse jamming on FSO systems, focusing primarily on the system’s Bit Error Rate (BER) and Outage Probability (OP). The study explored the impact of jamming on the system using different receiver apertures in [25]. In [24], an examination was conducted to analyze the impact of jamming on FSO systems, including both single-input single-output (SISO) setups and multiple-input single-output (MISO) configurations. The study assessed the efficacy of these systems by examining the BER and OP in the presence of negative exponential atmospheric turbulence. In [26], A Buffer-Aided Relaying Approach is employed to mitigate jamming in FSO cooperative networks. In [27], Saxena et al. explore the consequences of jamming instigated by a malicious UAV on the performance of an FSO communication system. They utilize a trustworthy UAV as a relay and incorporate an intelligent reflecting surface to improve signal coverage.

However, in all of these works, research on FSO jamming and mitigation techniques is primarily focused on single-hop systems. Furthermore, while literature has conducted jamming mitigation analysis using MISO for single-hop systems, SIMO is relatively easier to implement compared to MISO. Therefore, this prompts us to delve into examining the ABER of dual-hop FSO systems supported by RIS in the presence of jamming. Additionally, we aim to evaluate the improvement of ABER under different jamming scenarios utilizing SIMO technology. In summary, the key contributions of this study can be outlined as follows:

• The closed-form expression of the PDF of the legitimate channel in the RIS-assisted dual-hop FSO system is derived. Additionally, The PDF of the UAV jamming receiver and the RIS channel is derived. Novel closed-form expressions for the end-to-end ABER in both jamming scenarios have been derived based on the obtained link statistics. These expressions are derived considering the impact of non-Gaussian additive noise, which varies depending on the jammer’s location.
• The closed-form PDF of the legitimate channel in a 1 × N SIMO-FSO system and the channels under various jamming scenarios were derived using Mellin transforms to mitigate the impact of jamming. Analytical expressions for the end-to-end ABER in the two jamming scenarios were subsequently provided.
• A comprehensive system ABER analysis is conducted in terms of atmospheric turbulence strength, N (the number of receiving apertures), the probability of jamming and different RIS positions. Moreover, some useful insights are obtained.

The rest of the paper is organized as follows. The FSO system, along with the jammer and optical channel models, is introduced in Section 2. In Section 3, The closed-form expressions for ABER in the SISO System are derived. In Section 4, ABER analysis is carried for a generalized N receiving apertures with a jammer. The numerical results and discussion are presented in Section 5, and we conclude this paper in Section 6.

2. System and Channel Model

2.1. System Model

RIS-assisted FSO communication is shown in Figure 1. There is no direct LoS between the source (S) and the destination (D) due to the obstruction caused by some building and obstacles. The source is equipped with a LD which transmits symbols employing intensity modulation and direct detection (IM/DD) and on-off shift keying (OOK) modulation schemes. A malicious jammer, which UAVs are equipped with, is randomly present, attempting to disrupt the legitimate transmission link by opportunistically sending an optical signal. Thus, at any given instant, the jammer can jam either the received signal at the RIS or the received signal at the destination. Therefore, we consider two jamming scenarios: Scenario 1 involves a malicious jammer causing jamming at the destination, as depicted in Figure 1a, and Scenario 2 involves a malicious jammer causing jamming at the RIS, as depicted in Figure 1b.

2.1.1. System Model for Scenario 1

The signal from S is transmitted to D through the RIS consisting of N reflecting elements. Each component of the RIS receives the incident light and adjusts the phase and amplitude under program control, and then transmits it to the receiver. It is assumed that the signal transmitted through the RIS is fully radiated and undergoes full phase compensation. The received signal is given as

$$y_1=R\sqrt{P_s}{h}_{i}s+R\sqrt{N_J}{h}_J\mathrm{\Lambda }{s}_J+n$$

(1)

where $R$ is related to the responsivity of the photodetector, whose value is considered as 1; $s$ represents the modulated information symbol; $h_i=h_p\partial e^{j{\Psi }_p}{h}_q$, $h_p$, $h_q$ and $h_J$ are the S-RIS, RIS-D and jammer channel fading coefficient, respectively; $\partial e^{j{\Psi }_p}$ characterizes the IRS element at a position P, with $\partial$ representing the amplitude reflection coefficient and ${\Psi }_p$ denoting the induced phase; $s_J$ is the injected jamming signal, whose value is considered as unity; $P_s$ represents the peak transmit energy of the legitimate transmitter; $n$ is the additive white Gaussian noise (AWGN) with zero mean and variance $N_0$; $\mathrm{\Lambda }$ represents the state of jamming, modeled using a Bernoulli distributed random variable, Thus, the probability distribution of $\mathrm{\Lambda }$ is given by [24]

$$\left\{\begin{array}{l}P\left(\mathrm{\Lambda }=1\right)=\rho \\ P\left(\mathrm{\Lambda }=0\right)=1-\rho \end{array}\right.$$

(2)

where $P\left(\mathrm{\Lambda }=1\right)$ represents the probability of the occurrence of a jamming event; $P\left(\mathrm{\Lambda }=0\right)$ represents the probability of the jamming event being idle. we define $P_J$ as the average power of jamming. When the jamming event is in an active state and starts to jam legitimate information, the energy it emits is denoted as $N_J=P_J/\rho$.

The signal-to-jamming-and-noise ratio (SJNR) can be obtained from (1) when the jamming is active.

$${\gamma }_1=\frac{{\left(h_i\right)}^2{P}_s}{N_J{h}_J^2+N_0}$$

(3)

When the jamming is idle, the signal-to-noise ratio (SNR) can be calculated as follows:

$${\gamma }_2=\frac{{\left(h_i\right)}^2{P}_s}{N_0}={\overline{\gamma }}_D{\left(h_i\right)}^2$$

(4)

where ${\overline{\gamma }}_D=\frac{P_s}{N_0}$ represents the average signal-to-noise ratio (SNR).

2.1.2. System Model for Scenario 2

As shown in Figure 1b, the RIS is jammed by the UAV. Assuming that the jammer can effectively align to the RIS unit of the legitimate signal. Thus, the received signal at the destination is denoted as

$$y_2=R\sqrt{P_s}{h}_{i}s+R\sqrt{N_J}{g}_J\mathrm{\Lambda }{s}_J+n$$

(5)

where $g_J=h_J\partial e^{j{\Psi }_p}{h}_q$.

From (5), it can be derived that the SJNR is obtained when the jamming is active.

$${\gamma }_3=\frac{{\left(h_i\right)}^2{P}_s}{{\left(g_J\right)}^2{N}_J+N_0}$$

(6)

When the jamming is idle, the expression of SNR follows (4).

2.2. Channel Model

FSO communication links are affected by three main sources of signal attenuation: path loss, PE, and AT. These sources of attenuation affect the transmitted optical signal in various ways. The PDF of the channel attenuation factor $h_x$ is as [27]

$$f_{h_x}\left(h_x\right)=\frac{{\alpha }_x{\beta }_x{\xi }_x^2}{A_{0_x}\mathrm{\Gamma }\left({\alpha }_x\right)\mathrm{\Gamma }\left({\beta }_x\right)}{G}_{1,3}^{3,0}\left(\frac{{\alpha }_x{\beta }_x{h}_x}{A_{0_x}}\left|\begin{array}{c}{\xi }_x^2\\ {\xi }_x^2-1,{\alpha }_x-1,{\beta }_x-1\end{array}\right.\right),x\in \left\{p,q\right\}$$

(7)

Since the jamming event is caused by UAV, the PDF of the channel attenuation factor $h_J$, considering the fluctuations of AT, PE, and AoA, is as [27]

$$\begin{array}{l}{f}_{h_J}(h_J)=\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\delta \left(h_J\right)+\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\times \frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}{G}_{1,3}^{3,0}\left(\frac{{\alpha }_J{\beta }_J{h}_J}{A_{0_J}}\left|\begin{array}{c}{\xi }_J^2\\ {\xi }_J^2-1,{\alpha }_J-1,{\beta }_J-1\end{array}\right.\right)\end{array}$$

(8)

Here ${\phi }_{AoA}$ is AoA [27], ${\sigma }_{AoA}$ is the standard variation of UAV’s orientation [28]; ${\alpha }_x,{\beta }_x$ ($x\in \left\{p,q,J\right\}$) is the parameters for G-G turbulence for xth link, i.e., they attain the values $\left\{4,1.9\right\}$ and $\left\{4,1.4\right\}$ for moderate and strong AT regimes, respectively [28]. ${\xi }_x$ characterizes the PE faced by legitimate or jammer signal: higher the value of ${\xi }_x$, lower the PE [24,27]. $G_{p,q}^{m,n}\left(\cdot \left|\right.\right)$ is the Meijer-G function [29]. $A_{0_x}$($x\in \left\{p,q,J\right\}$) is the fraction of collected power at the center of beam footprint [27].

According to [30], it is known that $h_i=h_p\partial e^{j{\Psi }_p}{h}_q$. Thus, we can obtain

$$f_{h_i}\left(h_i\right)={\displaystyle {\int }_0^{\infty }{f}_{h_p}\left(h_p\right)}{f}_{h_q}\left(\frac{h_i}{h_p}\right)\frac{1}{h_p}d{h}_p$$

(9)

By substituting (7) into (9) and integrating using 07.34.16.0002.01 [29] and 07.34.16.0002.01 [29], we obtain

$$\begin{array}{l}{f}_{h_i}\left(h_i\right)=\frac{{\alpha }_p{\beta }_p{\xi }_p^2}{A_{0_p}\mathrm{\Gamma }\left({\alpha }_p\right)\mathrm{\Gamma }\left({\beta }_p\right)}\frac{{\alpha }_q{\beta }_q{\xi }_q^2}{A_{0_q}\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times G_{2,6}^{6,0}\left(\frac{{\alpha }_p{\beta }_p{\alpha }_q{\beta }_q{h}_i}{A_{0_p}{A}_{0_q}}\left|\begin{array}{c}{\xi }_p^2,{\xi }_q^2\\ {\xi }_p^2-1,{\alpha }_p-1,{\beta }_p-1,{\xi }_q^2-1,{\alpha }_q-1,{\beta }_q-1\end{array}\right.\right)\end{array}$$

(10)

Since $g_J=h_J\partial e^{j{\Psi }_p}{h}_q$, we can obtain

$$f_{g_J}\left(g_J\right)={\displaystyle {\int }_0^{\infty }{f}_{h_q}\left(h_q\right)}{f}_{h_J}\left(\frac{g_J}{h_q}\right)\frac{1}{h_q}d{h}_q=I_1+I_2$$

(11)

By substituting (7) and (8) into (11), integrating using 07.34.16.0002.01 [29], we obtain

$$I_1=\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\delta \left(g_J\right)$$

(12)

$$\begin{array}{l}{I}_2=\frac{{\alpha }_q{\beta }_q{\xi }_q^2}{A_{0_q}\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}\\ \hspace{1em}\times G_{2,6}^{6,0}\left(\frac{{\alpha }_J{\beta }_J{\alpha }_q{\beta }_q{g}_J}{A_{0_J}{A}_{0_q}}\left|\begin{array}{c}{\xi }_J^2,{\xi }_q^2\\ {\xi }_J^2-1,{\alpha }_J-1,{\beta }_J-1,{\xi }_q^2-1,{\alpha }_q-1,{\beta }_q-1\end{array}\right.\right)\end{array}$$

(13)

3. Analysis of BER for SISO System

3.1. Scenario 1

In this section, the BER performance is evaluated for the SISO FSO system under the presence of jamming or AWGN.

3.1.1. BER during Jamming Active

Let us define a random variable, ${\rm T}\triangleq \sqrt{N_J}{h}_J=\sqrt{P_j/\rho }{h}_J$. Using (8), the PDF of $\mathrm{{\rm T}}$ is given as

$$\begin{array}{l}{f}_T(T)=\frac{1}{\sqrt{p_J/\rho }}\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\delta \left(\frac{T}{\sqrt{p_J/\rho }}\right)+\frac{1}{\sqrt{p_J/\rho }}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\times \frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}{G}_{1,3}^{3,0}\left(\frac{{\alpha }_J{\beta }_{J}T}{A_{0_J}\sqrt{p_J/\rho }}\left|\begin{array}{c}{\xi }_J^2\\ {\zeta }_J^2-1,{\alpha }_J-1,{\beta }_J-1\end{array}\right.\right)\end{array}$$

(14)

When jamming is present, the energy of the jamming significantly outweighs that of Gaussian white noise, which allows us to disregard the Gaussian white noise. Therefore, we can derive that $T=y_1-\sqrt{P_s}{h}_{i}s$. By using (14), we can further deduce that

$$\begin{array}{l}{f}_{y_1}(y_1)=\frac{1}{\sqrt{p_J/\rho }}\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\delta \left(\frac{y_1-\sqrt{P_s}{h}_{i}s}{\sqrt{p_J/\rho }}\right)+\frac{1}{\sqrt{p_J/\rho }}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\times \frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}{G}_{1,3}^{3,0}\left(\frac{{\alpha }_J{\beta }_J\left(y_1-\sqrt{P_s}{h}_{i}s\right)}{A_{0_J}\sqrt{p_J/\rho }}\left|\begin{array}{c}{\xi }_J^2\\ {\zeta }_J^2-1,{\alpha }_J-1,{\beta }_J-1\end{array}\right.\right)\end{array}$$

(15)

(15) can be expanded based on the value of the symbol $s$.

$$f_{y_1}(y_1)=\left\{\begin{array}{l}\frac{1}{\sqrt{p_J/\rho }}\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\delta \left(\frac{y_1}{\sqrt{p_J/\rho }}\right)+\frac{1}{\sqrt{p_J/\rho }}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\\ \hspace{1em}\hspace{1em} \times \frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}{G}_{1,3}^{3,0}\left(\frac{{\alpha }_J{\beta }_J\left(y_1\right)}{A_{0_J}\sqrt{p_J/\rho }}\left|\begin{array}{c}{\xi }_J^2\\ {\zeta }_J^2-1,{\alpha }_J-1,{\beta }_J-1\end{array}\right.\right),\hspace{1em}\hspace{1em} \mathrm{for} s=0\\ \frac{1}{\sqrt{p_J/\rho }}\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\delta \left(\frac{y_1-\sqrt{P_s}{h}_i}{\sqrt{p_J/\rho }}\right)+\frac{1}{\sqrt{p_J/\rho }}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\\ \hspace{1em}\hspace{1em} \times \frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}{G}_{1,3}^{3,0}\left(\frac{{\alpha }_J{\beta }_J\left(y_1-\sqrt{P_s}{h}_i\right)}{A_{0_J}\sqrt{p_J/\rho }}\left|\begin{array}{c}{\xi }_J^2\\ {\zeta }_J^2-1,{\alpha }_J-1,{\beta }_J-1\end{array}\right.\right),\hspace{1em}\hspace{1em} \mathrm{for} s=1\end{array}\right.$$

(16)

ABER can be expressed based on the occurrence of jamming states.

$$P_{e1}=P\left(\mathrm{\Lambda }=1\right)\left[P\left(x=1\right)P\left(\left.\nu \right|x=1\right)+P\left(x=0\right)P\left(\left.\nu \right|x=0\right)\right]+P\left(\mathrm{\Lambda }=0\right)P_{ei}$$

(17)

where $P_{ei}$ represents BER during jamming idle states and $\nu$ denotes the occurrence of a bit error event. Here, we assume $P\left(x=1\right)=P\left(x=0\right)=0.5$, indicating that the symbols ‘1’ and ‘0’ are transmitted with equal probabilities.

When jamming is active and the jamming noise significantly exceeds the additive white Gaussian noise, the additive white Gaussian noise can be disregarded when calculating BER. As a result, the expression for calculating the BER under jamming active, derived from (17), is as follows

$$P_{e1a}=\rho \left[0.5{\displaystyle {\int }_{th}^{\infty }{f}_y\left(y\left|s=0\right.\right)dy}+0.5{\displaystyle {\int }_0^{th}{f}_y\left(y\left|s=1\right.\right)dy}\right]$$

(18)

According to [25], $th=\sqrt{P_s}{h}_i$. Substituting (19) into (21) and integrating using 07.34.21.0084.01 and 07.34.21.0085.01 is as

$$\begin{array}{l}{P}_{e1a}\left(h_i\right)=\frac{\rho }{2}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2\sqrt{\rho {\overline{\gamma }}_{JD}}{h}_i}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times G_{2,4}^{4,0}\left(\frac{{\alpha }_J{\beta }_J{h}_i\sqrt{\rho {\overline{\gamma }}_{JD}}}{A_{0_J}}\left|\begin{array}{c}{\xi }_J^2,0\\ -1,{\xi }_J^2-1,{\alpha }_J-1,{\beta }_J-1\end{array}\right.\right)\end{array}$$

(19)

According to [25], in the presence of jamming, we can derive the expression for ABER from (19) as follows

$$P_{e1a}\left({\overline{\gamma }}_{JD}\right)={\displaystyle {\int }_0^{\infty }{P}_{e1a}\left(h_i\right)f_{h_i}\left(h_i\right)d{h}_i}$$

(20)

By substituting (10) and (19) into (20) and integrating using 07.34.21.0011.01 [29], we can obtain

$$\begin{array}{l}{P}_{e1a}\left({\overline{\gamma }}_{JD}\right)=\frac{{\xi }_p^2}{\mathrm{\Gamma }\left({\alpha }_p\right)\mathrm{\Gamma }\left({\beta }_p\right)}\frac{{\xi }_q^2}{\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\frac{\rho }{2}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2\sqrt{\rho {\overline{\gamma }}_{JD}}}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}\\ \times \frac{A_{0_p}{A}_{0_q}}{{\alpha }_p{\beta }_p{\alpha }_q{\beta }_q}{G}_{8,6}^{4,6}\left(\frac{{\alpha }_J{\beta }_J{A}_{0_p}{A}_{0_q}\sqrt{\rho {\overline{\gamma }}_{JD}}}{{\alpha }_p{\beta }_p{\alpha }_q{\beta }_q{A}_{0_J}}\left|\begin{array}{l} -{\xi }_p^2,-{\alpha }_p,-{\beta }_p,-{\xi }_q^2,-{\alpha }_p,-{\beta }_q,{\xi }_J^2,0\\ -1,{\xi }_J^2-1,{\alpha }_J-1,{\beta }_J-1,-1-{\xi }_p^2,-1-{\xi }_q^2\end{array}\right.\right)\end{array}$$

(21)

where ${\overline{\gamma }}_{JD}=\frac{P_s}{P_J}$ is average signal jamming ratio (SJR).

3.1.2. BER during Jamming Idle

In the absence of jamming, when the system is only subjected to AWGN, we can derive the following expression from (4) and (10)

$$\begin{array}{l}{f}_{{\gamma }_2}\left({\gamma }_2\right)=\frac{{\alpha }_p{\beta }_p{\xi }_p^2}{2\sqrt{{\gamma }_2{\overline{\gamma }}_D}{A}_{0_p}\mathrm{\Gamma }\left({\alpha }_p\right)\mathrm{\Gamma }\left({\beta }_p\right)}\frac{{\alpha }_q{\beta }_q{\xi }_q^2}{A_{0_q}\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times G_{2,6}^{6,0}\left(\frac{{\alpha }_p{\beta }_p{\alpha }_q{\beta }_q}{A_{0_p}{A}_{0_q}}\sqrt{\frac{{\gamma }_2}{{\overline{\gamma }}_D}}\left|\begin{array}{c}{\xi }_q^2,{\xi }_p^2\\ {\xi }_p^2-1,{\alpha }_p-1,{\beta }_p-1,{\xi }_q^2-1,{\alpha }_q-1,{\beta }_q-1\end{array}\right.\right)\end{array}$$

(22)

According to [31], ABER in a system affected by AWGN is given by

$$P_{e1i}\left({\overline{\gamma }}_D\right)=0.5{\displaystyle {\int }_0^{\infty }\mathit{erfc}\left(\frac{\sqrt{{\gamma }_2}}{2\sqrt{2}}\right)f_{{\gamma }_2}\left({\gamma }_2\right)d{\gamma }_2}$$

(23)

where $\mathit{erfc}$ can be expressed by Meijer-G function [32].

$$\mathit{erfc}\left(\sqrt{y}\right)=\frac{1}{\sqrt{\pi }}{G}_{1,2}^{2,0}\left[y\left|\begin{array}{c}1\\ 0,1/2\end{array}\right.\right]$$

(24)

By substituting (22) and (24) into (23) and integrating using 07.34.21.0013.01, the ABER under jamming idle can be obtained as follows

$$\begin{array}{l}{P}_{e1i}\left({\overline{\gamma }}_D\right)=\frac{1}{\sqrt{\pi }}\frac{2^{\left({\alpha }_p+{\beta }_p+{\alpha }_q+{\beta }_q-7\right)}}{{\left(\pi \right)}^2}\frac{{\xi }_p^2}{\mathrm{\Gamma }\left({\alpha }_p\right)\mathrm{\Gamma }\left({\beta }_p\right)}\frac{{\xi }_q^2}{\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em} \times G_{13,6}^{2,12}\left(32{\overline{\gamma }}_D{\left(\frac{A_{0_p}{A}_{0_q}}{{\alpha }_p{\beta }_p{\alpha }_q{\beta }_q}\right)}^2\left|\begin{array}{c}{\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2,{\mathrm{\Xi }}_3,1\\ 0,\frac{1}{2},\frac{1-{\xi }_p^2}{2},\frac{2-{\xi }_p^2}{2},\frac{1-{\xi }_q^2}{2},\frac{2-{\xi }_q^2}{2}\end{array}\right.\right)\end{array}$$

(25)

where ${\mathrm{\Xi }}_1=\frac{1-{\xi }_p^2}{2},\frac{2-{\xi }_p^2}{2},\frac{1-{\alpha }_p}{2},\frac{2-{\alpha }_p}{2}$, ${\mathrm{\Xi }}_2=\frac{1-{\beta }_p}{2},\frac{2-{\beta }_p}{2},\frac{1-{\xi }_q^2}{2},\frac{2-{\xi }_q^2}{2}$, and ${\mathrm{\Xi }}_3=\frac{1-{\alpha }_q}{2},\frac{2-{\alpha }_q}{2},\frac{1-{\beta }_q}{2},\frac{2-{\beta }_q}{2}$.

The overall ABER of the proposed system for scenario 1 can be expressed as follows

$${\overline{P}}_{e1a}=\rho P_{e1a}\left({\overline{\gamma }}_{JD}\right)+\left(1-\rho \right)P_{e1i}\left({\overline{\gamma }}_D\right)$$

(26)

3.2. Scenario 2

Similar to scenario 1, ABER under jamming for scenario 2 can be obtained as

$$\begin{array}{l}{P}_{e2a}\left({\overline{\gamma }}_{JD}\right)=\frac{A_{0_p}{\xi }_p^2}{{\alpha }_p{\beta }_p\mathrm{\Gamma }\left({\alpha }_p\right)\mathrm{\Gamma }\left({\beta }_p\right)}\frac{{\xi }_q^2}{\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\frac{\rho \sqrt{\rho {\overline{\gamma }}_{JD}}{\xi }_q^2}{2\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em} \times \left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}\\ \hspace{1em}\hspace{1em} \times G_{9,9}^{7,6}\left(\frac{{\alpha }_J{\beta }_J{A}_{0_p}\sqrt{\rho {\overline{\gamma }}_{JD}}}{A_{0_J}{\alpha }_p{\beta }_p}\left|\begin{array}{c}{\mathrm{\Theta }}_1,{\mathrm{\Theta }}_2\\ {\mathrm{\Theta }}_3,{\mathrm{\Theta }}_4\end{array}\right.\right)\end{array}$$

(27)

where ${\mathrm{\Theta }}_1=-{\xi }_p^2,-{\alpha }_p,-{\beta }_p,-{\xi }_q^2$, ${\mathrm{\Theta }}_2=-{\alpha }_q,-{\beta }_q,{\xi }_J^2,{\xi }_q^2,0$, ${\mathrm{\Theta }}_3=-1,{\xi }_q^2-1,{\alpha }_q-1,{\beta }_q-1,{\xi }_J^2-1$, and ${\mathrm{\Theta }}_4={\alpha }_J-1,{\beta }_J-1,-1-{\xi }_p^2,-1-{\xi }_q^2$.

When the jamming is idle, the system and channel models in scenario 1 and scenario 2 are identical. Therefore, the BER between scenario 2 and scenario 1 is the same when the jamming is idle. Consequently, we can obtain the ABER in scenario 2 as follows

$${\overline{P}}_{e2a}=\rho P_{e2a}\left({\overline{\gamma }}_{JD}\right)+\left(1-\rho \right)P_{e1i}\left({\overline{\gamma }}_D\right)$$

(28)

4. Methods for Mitigating Jamming

In this section, the study extends to the evaluation of error performance in the presence of a jammer for a general SIMO FSO communication system. SIMO technology can mitigate jamming factors such as signal attenuation and atmospheric turbulence in the transmission link, thereby enhancing the reliability of communication systems. Even if one receiver encounters jamming, other receivers can still receive the optical signal. They can then perform merging and processing in the signal processing unit to recover the original data. Now, we analyze the impact of jamming on the communication performance of SIMO FSO systems.

4.1. Scenario 1

When the jammer is located at the destination, we can obtain the value of the n-th receiving aperture after photoelectric conversion as follows:

$$y_n=R{a}_n{h}_{in}\sqrt{P_s}s+R\sqrt{N_J}{h}_J\mathrm{\Lambda }{s}_J+n_n$$

(29)

where $h_{in}$ represents the attenuation coefficient of the n-th channel; $a_n$ represents the weighted value of the n-th channel; For the purpose of simplifying calculations, we assume that $a_n=\frac{1}{N}$; $n_n$ is the AWGN with zero mean and variance $N_0$; $\frac{1}{N}$ represents the jamming from N receiving apertures at the receiver, which follows a uniform distribution. Therefore, the probability of jamming for any one of them is $\frac{1}{N}$.

The receiver combines the received electrical signals by using the equal gain combining (EGC) [33] as follows:

$$y_D={\displaystyle \sum _{n=1}^N{y}_n}=R\sqrt{P_s}s{\displaystyle \sum _{n=1}^N{a}_n{h}_{in}}+R\sqrt{N_J}{h}_J\mathrm{\Lambda }{s}_J+{\displaystyle \sum _{n=1}^N{n}_n}$$

(30)

From (30), we can obtain the received SJR as follows

$${\gamma }_{SJR-D}=\frac{P_s{\left({\displaystyle \sum _{n=1}^N{a}_n{h}_{in}}\right)}^2}{N_J{\left(h_J\right)}^2}=\frac{\rho {\overline{\gamma }}_{JD}{\left({\displaystyle \sum _{n=1}^N{a}_n{h}_{in}}\right)}^2}{{\left(h_J\right)}^2}$$

(31)

We can also obtain from (30) the received SNR as follows

$${\gamma }_{SNR}=\frac{P_s{\left({\displaystyle \sum _{n=1}^N{a}_n{h}_{in}}\right)}^2}{N{N}_0}=\frac{{\overline{\gamma }}_D{\left({\displaystyle \sum _{n=1}^N{a}_n{h}_{in}}\right)}^2}{N}$$

(32)

4.2. Scenario 2

When the jammer jams the IRS, we assume that the jamming optical path can effectively align with the legitimate optical path unit of the IRS. As a result, we can obtain the received signal for the n-th path as follows

$$y_{n-IRS}=R{a}_n{h}_{in}\sqrt{P_s}s+R\sqrt{N_J}{b}_n{h}_{Jn}\mathrm{\Lambda }{s}_J+n_n$$

(33)

where $b_n$ represents the weighting value of the nth interfering branch, similarly, we assume $b_n=\frac{1}{N}$.

The receiver combines the received electrical signals by using the EGC as follows

$$y_{I-D}={\displaystyle \sum _{n=1}^N{y}_{n-IRS}}=R\sqrt{P_s}s{\displaystyle \sum _{n=1}^N{a}_n{h}_{in}}+R\sqrt{N_J}{\displaystyle \sum _{n=1}^N{b}_n{g}_{Jn}}\mathrm{\Lambda }{s}_J+{\displaystyle \sum _{n=1}^N{n}_n}$$

(34)

From (34), we can calculate the SJR at the receiver as

$${\gamma }_{SJR-R}=\frac{P_s{\left({\displaystyle \sum _{n=1}^N{a}_n{h}_{in}}\right)}^2}{N_J{\left({\displaystyle \sum _{n=1}^N{b}_n{h}_{Jn}}\right)}^2}=\frac{\rho {\overline{\gamma }}_{JD}{\left({\displaystyle \sum _{n=1}^N{a}_n{h}_{in}}\right)}^2}{{\left({\displaystyle \sum _{n=1}^N{b}_n{h}_{Jn}}\right)}^2}$$

(35)

4.3. ABER Calculation

4.3.1. Analysis of the ABER for SIMO in Scenario 1

The PDF of the received signal can be derived from Equations (8) and (30) as follows

$$f_{y_D}(y_D)=\left\{\begin{array}{l}\frac{1}{\sqrt{p_J/\rho }}\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\delta \left(\frac{y_D}{\sqrt{p_J/\rho }}\right)\\ +\frac{1}{\sqrt{p_J/\rho }}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}\\ \times G_{1,3}^{3,0}\left(\frac{{\alpha }_J{\beta }_J{y}_D}{A_{0_J}\sqrt{p_J/\rho }}\left|\begin{array}{c}{\xi }_J^2\\ {\zeta }_J^2-1,{\alpha }_J-1,{\beta }_J-1\end{array}\right.\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{for} s=0\\ \\ \frac{1}{\sqrt{p_J/\rho }}\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\delta \left(\frac{y_D-\sqrt{P_s}s{\displaystyle \sum _{n=1}^N{a}_n{h}_{in}}}{\sqrt{p_J/\rho }}\right)\\ +\frac{1}{\sqrt{p_J/\rho }}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}\\ \times G_{1,3}^{3,0}\left(\frac{{\alpha }_J{\beta }_J\left(y_D-\sqrt{P_s}s{\displaystyle \sum _{n=1}^N{a}_n{h}_{in}}\right)}{A_{0_J}\sqrt{p_J/\rho }}\left|\begin{array}{c}{\xi }_J^2\\ {\zeta }_J^2-1,{\alpha }_J-1,{\beta }_J-1\end{array}\right.\right),\hspace{1em}\hspace{1em}\mathrm{for} s=1\end{array}\right.$$

(36)

Let $h_{sum}={\displaystyle \sum _{n=1}^N{a}_n{h}_{in}}$, then by substituting (18) and (36), we obtain

$$\begin{array}{l}{P}_{e1a}\left(h_i\right)=\frac{\rho }{2}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2\sqrt{\rho {\overline{\gamma }}_J}{h}_{sum}}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em} \times G_{2,4}^{4,0}\left(\frac{{\alpha }_J{\beta }_J\sqrt{\rho {\overline{\gamma }}_J}{h}_{sum}}{A_{0_J}}\left|\begin{array}{c}{\xi }_J^2,0\\ -1,{\xi }_J^2-1,{\alpha }_J-1,{\beta }_J-1\end{array}\right.\right)\end{array}$$

(37)

Then, we apply the Mellin transform to (43) using 07.34.22.0004.01 [29], which results in

$$\begin{array}{l}{\varphi }_{h_{in}}\left(z\right)=\frac{{\alpha }_q{\beta }_q{\xi }_q^2}{A_{0_q}\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}{\left(\frac{{\alpha }_J{\beta }_J{\alpha }_q{\beta }_q}{A_{0_J}{A}_{0_q}}\right)}^{-z}\\ \hspace{1em}\hspace{1em} \times \frac{\mathrm{\Gamma }\left({\xi }_J^2-1+z\right)\mathrm{\Gamma }\left({\alpha }_J-1+z\right)\mathrm{\Gamma }\left({\beta }_J-1+z\right)\mathrm{\Gamma }\left({\xi }_q^2-1+z\right)\mathrm{\Gamma }\left({\alpha }_q-1+z\right)\mathrm{\Gamma }\left({\beta }_q-1+z\right)}{\mathrm{\Gamma }\left({\xi }_q^2+z\right)\mathrm{\Gamma }\left({\xi }_J^2+z\right)}\end{array}$$

(38)

Considering the independent channels, the Mellin transform of $h_{sum}$ can be expressed as

$${\varphi }_{h_{sum}}\left(z\right)={\displaystyle \prod _{n=1}^N{\varphi }_{h_{in}}\left(z\right)={\left({\varphi }_{h_{in}}\left(z\right)\right)}^N}$$

(39)

By substituting (38) into (39) and performing the Mellin inverse transform, the PDF of $h_{sum}$ can be obtained as follows

$$\begin{array}{l}{f}_{h_{sum}}\left(x\right)={\left(\frac{{\alpha }_p{\beta }_p{\xi }_p^2}{A_{0_p}\mathrm{\Gamma }\left({\alpha }_p\right)\mathrm{\Gamma }\left({\beta }_p\right)}\frac{{\alpha }_q{\beta }_q{\xi }_q^2}{A_{0_q}\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\right)}^N\\ \hspace{1em}\hspace{1em}\hspace{1em}\times G_{2N,6N}^{6N,0}\left({\left(\frac{{\alpha }_p{\beta }_p{\alpha }_q{\beta }_q}{A_{0_p}{A}_{0_q}}\right)}^N\times \left|\begin{array}{c}{S}_1\\ S_2,S_3,S_4\end{array}\right.\right)\end{array}$$

(40)

where $S_1=\stackrel{N}{\overbrace{{\xi }_p^2,\cdots ,{\xi }_p^2,}}\stackrel{N}{\overbrace{{\xi }_q^2,\cdots ,{\xi }_q^2}}$, $S_2=\underset{N}{\underbrace{{\xi }_p^2-1,\cdots ,{\xi }_p^2-1}},\underset{N}{\underbrace{{\alpha }_p-1,\cdots ,{\alpha }_p-1}}$, $S_3=\underset{N}{\underbrace{{\beta }_p-1,\cdots ,{\beta }_p-1}},\underset{N}{\underbrace{{\xi }_q^2-1,\cdots ,{\xi }_q^2-1}}$, and $S_4=\underset{N}{\underbrace{{\alpha }_q-1,\cdots ,{\alpha }_q-1}},\underset{N}{\underbrace{{\beta }_q-1,\cdots ,{\beta }_q-1}}$.

Now, using (37), (40), and 07.34.21.0011.01 [29], after some rigorous mathematics, the final closed-form expression of the ABER during jamming active is given by Appendix A (A1).

When the jamming is idle, according to [34], and assuming perfect Channel State Information at the receiver, the expression for BER is as follows:

$$\begin{array}{l}{P}_{e1i-SIMO}\left({\overline{\gamma }}_D\right)={\displaystyle {\int }_0^{\infty }{\displaystyle {\int }_0^{\infty }\cdots {\displaystyle {\int }_0^{\infty }{f}_{{\gamma }_{11}}\left({\gamma }_{11}\right)f_{{\gamma }_{12}}\left({\gamma }_{12}\right)\cdots f_{{\gamma }_{13}}\left({\gamma }_{13}\right)}}}\\ \hspace{1em}\hspace{1em} \times Q\left(\frac{1}{2\sqrt{N}}\sqrt{{\displaystyle \sum _{n=1}^N{\gamma }_{1n}}}\right)d{\gamma }_{11}d{\gamma }_{12}\cdots d{\gamma }_{1N}\end{array}$$

(41)

where, ${\gamma }_{1n}$ represents the SNR from the transmitter to the nth receiver, while N denotes the total number of receivers. The Q-function can be approximated as $Q\left(x\right)\approx \frac{1}{12}{e}^{-x^2/2}+\frac{1}{4}{e}^{-2x^2/3}$, we can obtain

$$\begin{array}{l}{P}_{e1i-SIMO}\left({\overline{\gamma }}_D\right)\approx \frac{1}{12}{\displaystyle \prod _{n=1}^N{\displaystyle {\int }_0^{\infty }{f}_{h_{in}}\left(h_{in}\right)\exp \left(-\frac{{\overline{\gamma }}_D{h}_{in}^2}{4N}\right)d{h}_{in}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{4}{\displaystyle \prod _{n=1}^N{\displaystyle {\int }_0^{\infty }{f}_{h_{in}}\left(h_{in}\right)\exp \left(-\frac{{\overline{\gamma }}_D{h}_{in}^2}{3N}\right)d{h}_{in}}}\\ \hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{12}{\displaystyle \prod _{n=1}^N{\mathrm{F}}_1}+\frac{1}{4}{\displaystyle \prod _{n=1}^N{\mathrm{F}}_2}\\ \end{array}$$

(42)

where

$$\begin{array}{l}{\mathrm{F}}_1=\frac{{\xi }_p^2}{\mathrm{\Gamma }\left({\alpha }_p\right)\mathrm{\Gamma }\left({\beta }_p\right)}\frac{{\xi }_q^2}{\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\frac{2^{\left({\alpha }_p+{\beta }_p+{\alpha }_q+{\beta }_q-7\right)}}{\sqrt{\pi }{\left(\pi \right)}^2}\\ \hspace{1em}\times G_{12,5}^{1,12}\left(\frac{64{\overline{\gamma }}_D}{N}{\left(\frac{{\alpha }_p{\beta }_p{\alpha }_q{\beta }_q}{A_{0_p}{A}_{0_q}}\right)}^{-2}\left|\begin{array}{c}{E}_1,E_2,E_3\\ 0,\frac{-{\xi }_q^2}{2},\frac{1-{\xi }_q^2}{2},\frac{-{\xi }_p^2}{2},\frac{1-{\xi }_p^2}{2}\end{array}\right.\right)\end{array}$$

(43)

$$\begin{array}{l}{\mathrm{F}}_2=\frac{{\xi }_p^2}{\mathrm{\Gamma }\left({\alpha }_p\right)\mathrm{\Gamma }\left({\beta }_p\right)}\frac{{\xi }_q^2}{\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\frac{2^{\left({\alpha }_p+{\beta }_p+{\alpha }_q+{\beta }_q-7\right)}}{\sqrt{\pi }{\left(\pi \right)}^2}\\ \hspace{1em}\times G_{12,5}^{1,12}\left(\frac{256{\overline{\gamma }}_D}{3N}{\left(\frac{{\alpha }_p{\beta }_p{\alpha }_q{\beta }_q}{A_{0_p}{A}_{0_q}}\right)}^{-2}\left|\begin{array}{c}{E}_1,E_2,E_3\\ 0,\frac{-{\xi }_p^2}{2},\frac{1-{\xi }_p^2}{2},\frac{-{\xi }_q^2}{2},\frac{1-{\xi }_q^2}{2}\end{array}\right.\right)\end{array}$$

(44)

where $E_1=\frac{1-{\xi }_p^2}{2},\frac{2-{\xi }_p^2}{2},\frac{1-{\alpha }_p}{2},\frac{2-{\alpha }_p}{2}$, $E_2=\frac{1-{\beta }_p}{2},\frac{2-{\beta }_p}{2},\frac{1-{\xi }_q^2}{2},\frac{2-{\xi }_q^2}{2}$, $E_3=\frac{1-{\alpha }_q}{2},\frac{2-{\alpha }_q}{2},\frac{1-{\beta }_q}{2},\frac{2-{\beta }_q}{2}$.

Now, we can calculate the ABER for Scenario 1 of SIMO-FSO as follows

$$P_{e1-SMIO}=\rho P_{e1a-SIMO}\left({\overline{\gamma }}_J\right)+\left(1-\rho \right)P_{e1i-SIMO}\left({\gamma }_D\right)$$

(45)

4.3.2. Analysis of the ABER for SIMO in Scenario 2

Since Scenario 2 has the same system model as Scenario 1 when jamming is idle, we only need to calculate the BER when jamming is active in Scenario 2.

Let $g_{Jsum}={\displaystyle \sum _{n=1}^N{g}_{Jn}}$, then applying the Mellin transform to (11) with the transformation function 07.34.22.0004.01, we can obtain

$$\begin{array}{l}{\varphi }_{h_{Jn}}\left(z\right)=\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}\\ \hspace{1em}\hspace{1em}\times {\left(\frac{{\alpha }_J{\beta }_J}{A_{0_J}}\right)}^{-z}\frac{\mathrm{\Gamma }\left({\zeta }_J^2-1+z\right)\mathrm{\Gamma }\left({\alpha }_J-1+z\right)\mathrm{\Gamma }\left({\beta }_J-1+z\right)}{\mathrm{\Gamma }\left({\zeta }_J^2+z\right)}\end{array}$$

(46)

Thus we have

$${\varphi }_{g_{Jsum}}\left(z\right)={\displaystyle \prod _{n=1}^N{\varphi }_{g_{Jn}}\left(z\right)={\left({\varphi }_{g_{Jn}}\left(z\right)\right)}^N}$$

(47)

The Mellin inverse transform of (53) can be obtained as:

$$\begin{array}{l}{f}_{g_{Jsum}}\left(x\right)={\left(\frac{{\alpha }_q{\beta }_q{\xi }_q^2}{A_{0_q}\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}\right)}^N\\ \hspace{1em}\hspace{1em}\hspace{1em}\times G_{2N,6N}^{6N,0}\left({\left(\frac{{\alpha }_J{\beta }_J{\alpha }_q{\beta }_q}{A_{0_J}{A}_{0_q}}\right)}^N\times \left|\begin{array}{c}{C}_1\\ C_2,C_3,C_4\end{array}\right.\right)\end{array}$$

(48)

where $C_1=\stackrel{N}{\overbrace{{\xi }_J^2,\cdots ,{\xi }_J^2,}}\stackrel{N}{\overbrace{{\xi }_q^2,\cdots ,{\xi }_q^2}}$, $C_2=\underset{N}{\underbrace{{\xi }_q^2-1,\cdots {\xi }_q^2-1}},\underset{N}{\underbrace{{\alpha }_q-1,\cdots ,{\alpha }_q-1}}$, $C_3=\underset{N}{\underbrace{{\beta }_q-1,\cdots ,{\beta }_q-1}},\underset{N}{\underbrace{{\xi }_J^2-1,\cdots ,{\xi }_J^2-1}}$, and $C_4=\underset{N}{\underbrace{{\alpha }_J-1,\cdots ,{\alpha }_J-1}},\underset{N}{\underbrace{{\beta }_J-1,\cdots ,{\beta }_J-1}}$.

From (48), we can obtain

$$f_{y_{I-D}}\left(y_{I-D}\right)=\left\{\begin{array}{l}\frac{1}{\sqrt{N_J}}{\left(\frac{{\alpha }_q{\beta }_q{\xi }_q^2}{A_{0_q}\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}\right)}^N\\ \times G_{2N,6N}^{6N,0}\left({\left(\frac{{\alpha }_J{\beta }_J{\alpha }_q{\beta }_q}{A_{0_J}{A}_{0_q}}\right)}^N\frac{y_{I-D}}{\sqrt{N_J}}\left|\begin{array}{c}{C}_1\\ C_2,C_3,C_4\end{array}\right.\right),\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{for} s=0\\ \frac{1}{\sqrt{N_J}}{\left(\frac{{\alpha }_q{\beta }_q{\xi }_q^2}{A_{0_q}\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}\right)}^N\\ \times G_{2N,6N}^{6N,0}\left({\left(\frac{{\alpha }_J{\beta }_J{\alpha }_q{\beta }_q}{A_{0_J}{A}_{0_q}}\right)}^N\frac{y_{I-D}-\sqrt{P_s}{\displaystyle \sum _{n=1}^N{h}_{in}}}{\sqrt{N_J}}\left|\begin{array}{c}{C}_1\\ C_2,C_3,C_4\end{array}\right.\right),\hspace{1em}\mathrm{for} s=1\end{array}\right.$$

(49)

From (18) and (49), and $h_{sum}={\displaystyle \sum _{n=1}^N{a}_n{h}_{in}}$, we get:

$$\begin{array}{l}{P}_{e2a-SIMO}\left(h_{isum}\right)=\frac{\rho }{2}\sqrt{\rho {\overline{\gamma }}_{JD}}{h}_{isum}{\left(\frac{{\alpha }_q{\beta }_q{\xi }_q^2}{A_{0_q}\mathrm{\Gamma }\left({\alpha }_q\right)\mathrm{\Gamma }\left({\beta }_q\right)}\left[1-\exp \left(-\frac{{\phi }_{AoA}^2}{2{\sigma }_{AoA}^2}\right)\right]\frac{{\alpha }_J{\beta }_J{\xi }_J^2}{A_{0_J}\mathrm{\Gamma }\left({\alpha }_J\right)\mathrm{\Gamma }\left({\beta }_J\right)}\right)}^N\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\times G_{2N+1,6N+1}^{6N+1,0}\left({\left(\frac{{\alpha }_J{\beta }_J{\alpha }_q{\beta }_q}{A_{0_J}{A}_{0_q}}\right)}^N\sqrt{\rho {\overline{\gamma }}_{JD}}{h}_{isum}\left|\begin{array}{c}{C}_1,0\\ -1,C_2,C_3,C_4\end{array}\right.\right)\end{array}$$

(50)

By using (48) and (50), ABER can be obtained by Appendix A (A2).

Now, we can calculate the ABER for Scenario 2 of SIMO-FSO as follows:

$$P_{e2-SMIO}=\rho P_{e2a-SIMO}\left({\overline{\gamma }}_J\right)+\left(1-\rho \right)P_{e1i-SIMO}\left({\gamma }_D\right)$$

(51)

5. Numerical Results

Unless otherwise stated, ${\xi }_p=1.2528,$ ${\phi }_{AoA}=8 \mathrm{mrad}$, $d_p=1 \mathrm{mm}$, $d_q=3 \mathrm{mm}$, and ${\phi }_p=0.175 \mathrm{mrad}$. The following simulation results were obtained by using MATLAB 2022.

When ${\xi }_q=1.2528,$ ${\xi }_J=1.2528,$ ${\alpha }_p=2.49$, ${\beta }_p=3.9$, ${\alpha }_q=4.0$, ${\beta }_q=1.9$, ${\alpha }_J=4.0$, ${\beta }_J=1.9$, Figure 2 illustrates the variation of ABER with SJR obtained by (21), and Monte Carlo simulation at $\rho =1$. It can be observed from Figure 2 that the concordance between the analytical and simulation plots validates the accuracy of the derived expression. It can be observed from Figure 2 that as the receiving aperture N increases, the ABER shows significant improvement. However, with the increase of SJR, the degree of improvement decreases. This is because the jamming signal only jams one of the receiving apertures, and under high SJR, each receiving aperture can effectively receive signals. Since the additive white Gaussian noise is not considered in Figure 2, when SJR is high, increasing the number of receiving apertures cannot significantly improve the ABER when the SJR is large. When N = 5, the improvement of ABER with SJR increasing is not very obvious. The reason is that increasing the number of receiving apertures can effectively suppress the influence of jamming. At the same time, even though the additive Gaussian white noise is not considered, the FSO system are also affected by path loss, PE, and AT, which are a static variable, so the changing trend of ABER with SJR is not obvious.

Let $k=\frac{P_J}{N_0}=100$. In scenario 1, the relationship between SJR and ABER is illustrated in Figure 3, which is derived from (26) and (45). Similarly, the relationship between SJR and ABER for scenario 2 is depicted in Figure 4, derived from (28) and (51). From Figure 3 and Figure 4, it can be observed that for SISO-FSO systems, as the jamming probability $\rho$ increases, the ABER also increases; For SIMO-FSO systems, when SJR is greater than 20dB, the ABER shows a positive correlation with $\rho$. Conversely, when SJR is less than 20dB, ABER decreases as $\rho$ increases. This is because the use of SIMO technology enables effective signal recovery at high SJR. By comparing Figure 3 and Figure 4, it is observed that jamming at the receiver has a significant impact on the system’s BER at the RIS. For instance, when $\rho =0.01$ and ${\overline{\gamma }}_J=60 \mathrm{dB}$, ABER for Scenario 1 with SISO and SIMO are $3.49\times {10}^{-4}$ and $4.77\times {10}^{-5}$, respectively. For Scenario 2, ABER with SISO and SIMO are $9.08\times {10}^{-5}$ and $1.35\times {10}^{-6}$, respectively. This indicates that jamming at the receiver has a greater impact on the system compared to interference at the RIS.

In Figure 5, we analyze the impact of different atmospheric turbulence on the ABER in two scenarios. The analytical plots closely align with the Monte Carlo simulations conducted, thereby confirming the accuracy of the derived expressions. It can be observed from Figure 5 that the system’s overall performance improves as the turbulence ranges from strong (${\alpha }_p=4.0$, ${\beta }_p=1.9$) to moderate (${\alpha }_p=2.49$, ${\beta }_p=3.9$). Nevertheless, under the same turbulence parameters, the ABER performance for Scenario 1 is notably inferior to that of Scenario 2. This suggests that jamming at the RIS has a smaller impact on the system compared to jamming at the intended receiver of the legitimate signal. One possible reason for this is that the RIS distributes the power of the jamming signal.

Figure 6 and Figure 7 analyze the impact of the RIS location on the system ABER in two scenarios. Let $\eta =L_p/L_q$, where $L_p$ is the link length between S and RIS and $L_q$ is the link length between RIS and the destination [19]. Based on the value of $\eta$, the value of ${\xi }_q^2$ can be calculated according to [20], with other simulation parameters set to ${\xi }_p=4.5856$, ${\alpha }_p={\alpha }_q=5.4$, ${\beta }_p={\beta }_q=3.77$, and $\rho =0.5$. As $\eta$ decreases, it indicates that the RIS is relatively closer to the source node than the destination node. From Figure 6 and Figure 7, it can be observed that a smaller value of $\eta$ leads to better system ABER performance. Therefore, it should be positioned as close to the source node as possible. Furthermore, it is noted that under weak turbulence with jamming present, the adoption of SIMO technology does not effectively enhance the ABER in Scenario 1, as depicted in Figure 6. However, in Scenario 2, SIMO notably improves the ABER, as shown in Figure 7. This phenomenon is attributed to the minimal channel attenuation in weak turbulence conditions, where the introduction of multiple additive Gaussian noises in SIMO results in an increase in the ABER. Therefore, it is not recommended to employ SIMO technology to mitigate jamming in Scenario 1 under weak turbulence conditions.

Computational complexity analysis: From (45), and (51), it can be concluded that the computational complexity of ABER using SIMO is $O\left(N^3\right)$. When N is large, the computational complexity of (45), and (51) is high. When N is small, the complexity is relatively small. Since SIMO is typically employed in FSO with 2 receiving apertures [24], our proposed ABER calculation can be effectively applied to FSO systems.

6. Conclusions

We have conducted a study on the impact of jamming in an RIS-assisted Dual-Hop FSO Communication system in the presence of a malicious jammer. A novel PDF for the S-IRS-D, UAV-IRS-D, and UAV-D links under the combined influence of AT, PE, and AoA fluctuations has been developed. Based on the jamming scenario, closed-form expressions for the end-to-end ABER have been derived for SISO-FSO systems. To mitigate the impact of jamming in an RIS-assisted Dual-Hop FSO Communication system, a SIMO FSO system has been implemented. We have utilized the Mellin transform to derive the PDF of the legitimate channel and the jamming channel in the SIMO FSO system and the end-to-end ABER have been derived for SIMO-FSO systems under different jamming Scenarios. The simulation results are compared with the analytical results to validate the accuracy of the derived expression. It has been observed that as the number of receiving apertures increases, the system shows a significant improvement in bit error performance. For the SISO and 1 × 2 FSO systems, we conducted separate analyses on the impact of different jamming probabilities. In Scenario 1, the jamming occurs at D, while in Scenario 2, the jamming occurs at the RIS, affecting the system’s ABER. In regions with low SJR regions, it was found that ABER is inversely proportional to the jamming probability $\rho$; conversely, in relatively high SJR regions, ABER exhibits a direct proportionality with $\rho$. Simultaneously, the jamming in Scenario 1 has a greater impact on the system’s ABER compared to the interference in Scenario 2. The impact of different positions of the RIS on the overall system performance has also been investigated. The system achieves optimal performance when the RIS is situated in closer proximity to the source.