## 3 FM-ACSK System's Description
### Chaos Generator and Synchronization  
As a base of ACSK communication system, a fourth-order modified [PERSON]'s circuit chaos generator was used. This circuit can be described as a system of four differential equations and piecewise linear function:  
\[ \begin{cases}\frac{dp_{1}}{dt}=-g(p_{1},p_{3})(p_{1}-p_{3})-p_{2}\\ \frac{dp_{2}}{dt}=p_{1}+\gamma p_{2}\\ \frac{dp_{3}}{dt}=\theta(g(p_{1},p_{3})(p_{1}-p_{3})-p_{4})\\ \frac{dp_{4}}{dt}=\sigma p_{3}\end{cases} , \tag{1}\]\[g(p_{1},p_{3})= \begin{cases}c(p_{1}-p_{3}-d)&(p_{1}-p_{3})>d\\ 0&(p_{1}-p_{3})\leq d\end{cases} , \tag{2}\]  
where \(p_{1},p_{2},p_{3},p_{4}\) are system's state variables, \(\gamma\), \(\theta\), \(\sigma\), \(c\), \(d\) are system coefficients and \(g(p_{1},p_{3})\) is piecewise linear function. System coefficients characterize the chaotic behavior of the system. Master chaos generator's output signal \(R_{\text{out}}\) consists of the weighted sum of the system's state variables \(p_{n}\) and piecewise function \(g\):  
\[R_{\text{out}}=k_{1}p_{1}+k_{2}p_{2}+k_{3}p_{3}+k_{4}p_{4}+g(p_{1},p_{3})\,, \tag{3}\]  
where weight coefficients \(k_{1}\), \(k_{2}\), \(k_{3}\) and \(k_{4}\) define the output signal form variation. Weight coefficients can be potentially used for multiple access implementation, by changing their values for each system. The initial system's state conditions and system coefficient values are shown in Table 1. These values ensure stable chaotic signal generation as provided in [41], and the sample of generated chaotic signal is shown in Figure 1. Equation (3) can be visualized as shown in Figure 2 in the master chaos generator part. Chaotic synchronization recreates the master chaos generator's state variables in the slave chaos generator, which is used to estimate received binary symbol values. Chaotic synchronization is carried out by the linear error feedback method. Both chaos generators have state variables \(p_{n}\), \(p_{n}^{\prime}\) and equal weight coefficients \(k_{n}\). The slave chaos generator's main difference compared to the master chaos generator is in the piecewise linear function that is being restored by subtracting the re-created weighted sum of the state variables \(p_{n}^{\prime}\) from the input signal \(R_{in}\):  
\[g^{\prime}(p_{1},p_{3})=R_{in}-k_{1}p_{1}^{\prime}+k_{2}p_{2}^{\prime}+k_{3}p_{ 3}^{\prime}+k_{4}p_{4}^{\prime}\,, \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(p_{1}\) & \(p_{2}\) & \(p_{3}\) & \(p_{4}\) & \(\gamma\) & \(\theta\) & \(\sigma\) & \(c\) & \(d\) & \(k_{1}\) & \(k_{2}\) & \(k_{3}\) & \(k_{4}\) \\ \hline
0.05 & 0.06 & 0.07 & 0.08 & 0.5 & 10 & 1.5 & 3 & 1 & \(-2.6302\) & \(-0.6054\) & 0.587 & 0.7763 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial system’s state conditions and system coefficients. Figure 1: Master chaos generator’s output signal \(R_{out}\). Figure 2: Block diagram of master and slave chaos generators. where \(R_{in}\) is the input chaotic signal, \(p^{\prime}_{1}\), \(p^{\prime}_{2}\), \(p^{\prime}_{3}\), \(p^{\prime}_{4}\) are state variables of the slave chaos generator and \(g^{\prime}(p_{1},p_{3})\) is the restored piecewise linear function.