## 2 Preliminaries  
As shown in Figure 1, we consider an FSO system with \(N\) transmitting apertures (LEDs/LASERs) that transmit an on-off keying (OOK) modulated optical signal, \(x\), an optical receiver, and a jammer that transmits a variable energy jamming signal to disrupt the legitimate communication. The channel is modeled for saturated AT with PE, which deteriorates the authorized and jamming signal likewise. During a transmission interval, the receiver selects the signal from the transmitting aperture having the most robust signal strength.  
At the receiver, a photodiode with responsivity \(R\) is used; we take the value of \(R\) as unity throughout our calculations. The input-output relation can be mathematically written as  
\[y=R\sqrt{E_{x}}\ h_{m}\ x+R\sqrt{N_{j}}\ h_{j}\ J+\eta. \tag{1}\]  
Here, \(E_{x}\) represents the peak electrical transmitted power of the legitimate transmitter. This study has considered a random jammer that alters its radiating power randomly to intercept the licensed transmission. Hence, the net contributing jamming power is \(N_{j}=\frac{E_{j}}{\rho}\), where \(E_{j}\) is the average jamming power. The degree of activity of the jammer is caught  
Figure 1: System model.  
by \(\rho\in[0,1]\). The random jammer has two states: active and idle state. It toggles between them with probability \(\rho\), following [PERSON]'s distribution. Thus, we can write  
\[P(J=1)=\rho;\quad P(J=0)=1-\rho, \tag{2}\]  
where \(P(\cdot)\) is the probability function, and \(J\) is the jammer's state. In this paper, the random jammer is modeled to impact the authentic signal promptly and randomly. Moreover, \(h_{m}\) represents the channel fading of a path between the transmitter and receiver that optimizes the system's performance. In other words, we can say  
\[h_{m}=max\left\{h_{u_{i}}\right\}, \tag{3}\]  
where \(h_{u_{i}}\in\Re,i\in 1,2\ldots N\) is the channel fading coefficient between the \(i\)th transmitter and receiver. Furthermore, \(h_{j}\) denotes the channel fading between the jammer and receiver; it is assumed that \(h_{u_{i}}\)s are independent and identically distributed (i.i.d.) random variables (RVs). In addition, the legitimate and jamming channels are assumed to be 'quasi-static'. \(\eta\in\Re\) represents the additive Gaussian noise with zero mean and \(N_{o}\) variance. It is noted that a jammer takes no account of eye safety rules obligatory in FSO transmission and increases its transmitting power perilously to obstruct the communication of user information. Therefore, the jamming noise takes precedence over the Gaussian noise. Mathematically, we can infer the average signal-to-jamming-plus-noise ratio (SJNR) without conditioning over the fading channels, denoted by \(\gamma\), as  
\[\gamma=\frac{E_{x}}{E_{j}+N_{0}}. \tag{4}\]  
Let \(E_{x}\) be taken to be of unit power. Hence, when the jammer is active (\(J=1\)), we have \(E_{j}\gg N_{0}\); consequently, SJNR reduces to the signal-to-jamming ratio (SJR) as \(\gamma=\frac{1}{E_{j}}\). Conversely, for an idle jammer (\(J=0\)), we know the power level alters to \(N_{0}\gg E_{j}\); thus, we obtain a signal-to-noise ratio (SNR) as \(\gamma=\frac{1}{N_{0}}\). To expedite the mathematical computation, it is assumed that \(N_{0}=\frac{E_{j}}{\kappa}\), where \(\kappa\) is an arbitrary positive real number. Therefore, we can construe that the legitimate information carrying signal is distorted by the jamming signal whenever the jammer is in active state. For an idle jammer, the signal only suffers from the Gaussian noise.