## 3 Bit Error Rate Analysis of RSST-SR-DCSK System  
Without loss of generality, we consider the probability of error for the first symbol sent by the lth user. The input to the threshold detector at the end of the first symbol duration, now denoted with \(y_{l}\), is given using  
If \({}^{\prime}\)+1\({}^{\prime\prime}\) is transmitted for the \(l\)th user, (4) becomes  
\[y_{l}|(b_{l}=+1)=\underbrace{P\sum_{i=0}^{R}x_{i-R,k}^{2}P_{rob}}_{A}+ \underbrace{P\sum_{i=0}^{R}x_{i-R,k}\xi_{i-R,k}}_{B}+\underbrace{P\sum_{i=0}^ {R}x_{i-R,k}\xi_{i,k}}_{C}+\underbrace{\sum_{i=0}^{R}\xi_{i,k}\xi_{i-R,k}}_{D} \tag{11}\]  
If \({}^{\prime\prime}\)\(-\)1\({}^{\prime\prime}\) is transmitted for the \(l\)th user, (4) becomes  
\[y_{l}|(b_{l}=-1)=-\underbrace{P\sum_{i=0}^{R}x_{i-R,k}^{2}}_{A}+\underbrace{P \sum_{i=0}^{R}x_{i-R,k}\xi_{i-R,k}}_{B}+\underbrace{P\sum_{i=0}^{R}x_{i-R,k} \xi_{i,k}}_{C}+\underbrace{\sum_{i=0}^{R}\xi_{i,k}\xi_{i-R,k}}_{D} \tag{12}\]  
We assume that all signals (chaotic or noise) are stationary, i.e., the statistical properties of the signals are not dependent on time \(k\). It is important to note that chaotic sequences from different generators are dependent on one another. As a result, the mean value of \(y_{l}|(b_{l}=+1)\) is equal to Formula (13). \[ \begin{array}{l}E[y_{l}|(b_{l}=+1)]=E[A]+E[B]+E[C]+E[D]\\ E[y_{l}|(b_{l}=-1)]=-E[A]+E[B]-E[C]+E[D]\end{array}  \tag{13}\]  
The variance of \(y_{l}|(b_{l}=+1)\) is found from Formula (14). \[Var[y_{l}|(b_{l}=+1)]=Var[y_{l}|(b_{l}=-1)]\approx Var[A]+Var[B]+Var[C]+Var[D] +Cov(\sim) \tag{14}\]  
The means and variances of \(y_{l}\) given a \"+1\({}^{\prime\prime}\) or \"\(-\)1\({}^{\prime\prime}\) is sent can be evaluated using numerical simulations. Denote the respective means and variances with \(E[y_{l}|(b_{l}=\pm 1)]\) and \(Var[y_{l}|(b_{l}=\pm 1)]\). Since \(y_{l}|(b_{l}=\pm 1)\) is the sum of a large number of random variables, we may assume that it is normally distributed. Hence, the approximate error probability of the \(l\)th transmitted bit is  
\[ \begin{array}{l}BER_{RSST-SR-DCSK}^{(l)}=P_{rob}(b_{l}=+1)\times P_{rob}(y_{l} \leq 0|b_{l}=+1)+P_{rob}(b_{l}=-1)\times P_{rob}(y_{l}>0|b_{l}=-1)\\ =\frac{1}{4}\bigg{[}erfc\bigg{[}\frac{E[y_{l}|(b_{l}=+1)]}{\sqrt{2 Var[y_{l}|(b _{l}=+1)]}}\bigg{]}+erfc\bigg{[}\frac{E[y_{l}|(b_{l}=-1)]}{\sqrt{2 Var[y_{l}|(b _{l}=-1)]}}\bigg{]}\bigg{]}\end{array}  \tag{15}\]  
In order to find the BER, we first obtain the values of the terms (expected values, variances, and covariances) on the right side of the Equations (8) and (9) using numerical simulations. Then, the values of \(E[y_{l}|(b_{l}=\pm 1)]\) and \(Var[y_{l}|(b_{l}=\pm 1)]\) can be computed and substituted into (15) to obtain the BER. Since both numerical simulation and analytical method are involved in obtaining the BER, we refer to this method as the mixed analysis-simulation (MAS) technique. First, we assume that the average transmitting power of the chaotic sequence is stable, meaning that the mean square value of the chaotic signal is constant, and writing the mean power \(P_{s}=E\big{[}x_{k}^{2}\big{]}\), we may simplify (11) and (13) to  
\[E[y_{l}|(b_{l}=+1)]=E[A]+E[B]+E[C]+E[D]=PRE\Big{[}x_{k}^{2}\Big{]}=PRP_{s} \tag{16}\]  
and  
\[\left\{ \begin{array}{l}Var[A]=P^{2}RVar\big{[}x_{k}^{2}\big{]}=P^{2}R\rho\\ Var[B]=P^{2}RP_{S}\frac{N_{0}}{2}\\ Var[C]=PRP_{S}\frac{N_{0}}{2}\\ Var[D]=\frac{PRN_{0}^{2}}{4}\end{array} \right. \tag{17}\]  
where \(\rho=Var\big{[}x_{k}^{2}\big{]}\), it can be readily shown that all the covariance terms are zero. Thus, using the Formula (17), we may simplify (14) to  
\[Var[y_{l}|(b_{l}=+1)]=P^{2}R\rho+P^{2}RP_{s}\frac{N_{0}}{2}+PRP_{s}\frac{N_{0} }{2}+PR\frac{N_{0}^{2}}{4} \tag{18}\]Thus, it can be obtained that  
\[BER^{(l)}_{RSST-SR-DCSK}=\frac{1}{2}erfc\Bigg{[}\left(\frac{2\rho}{RP^{2}_{g}}+ \frac{N_{0}(R+\beta)}{PE_{b}}+\frac{N_{0}(R+\beta)}{PRE_{b}}+\frac{(R+\beta)^{2}N _{0}^{2}}{2 PRE_{b}^{2}}\right)^{-\frac{1}{2}}\Bigg{]} \tag{19}\]  
We redefine \(\psi=\frac{Var[x_{i}^{2}]}{P_{g}^{2}}\).