## 3 Performance Analysis  
This section derives the BER expression for the proposed MU-FTN-DCSK system over multi-path Rayleigh fading channels. Following the design presented in Figure 1, the Probability Density Function (PDF) of the received signal follows an exponential distribution expressed as  
\[f(\gamma_{i,l}^{p})=\sum_{l=1}^{L}\frac{\tau_{i,l}^{p}}{\tau_{i,l}^{p}}e^{- \frac{\gamma_{i,l}^{p}}{\gamma_{i,l}^{p}}}, \tag{14}\]  
where \(\gamma_{i,l}^{p}\) is the channel gain, \(\tau_{i,l}^{p}\) is its averaged value, and \(\tau_{i,l}^{p}\) being the weight for the \(i\)th symbol duration over the \(l\)th path and is expressed as  
\[\tau_{i,l}^{p}=\prod_{j=1,\ j\
eq l}^{l}\frac{\gamma_{l}}{\gamma_{l}-\overline {\gamma_{j}}}. \tag{15}\]Hence, considering [28] (Eq. 3.3), the result in (8), and the normalized filters for different users, the received signal \(r_{i,n}^{d}\) post filtering during the first \(\beta\) samples, i.e., the \(n\)th sample of the reference signal, for the desired user with a sampling parameter \(\rho_{(d)}\) is expressed as:  
\[ \begin{split}\mathrm{r}_{i,n}^{d}&=\sum_{k=-2}^{ \beta+2}\sum_{l=1}^{L}\alpha_{i,l}^{d}x_{i,k-\tau_{l}}^{d}\rho_{(d)}sinc(\rho_{ (d)}(k-n))+\sum_{k^{\prime}=-\infty}^{\infty}\sum_{p=1}^{U-1}\sum_{l^{\prime}=1 }^{L}\alpha_{i,p^{\prime},x_{i^{\prime}}^{p}x_{i^{\prime}}^{p}-\tau_{i^{\prime },p^{\prime}}}\\ &\quad\quad\times\sqrt{\rho_{(p)}\rho_{(d)}}sinc(k^{\prime}\rho_ {(p)}-n\rho_{(d)})+W_{n}.\end{split}  \tag{16}\]  
\(W_{n}\) is the noise signal received during the same \(i\)th symbol duration. It is assumed that only two samples prior to and after the desired signal can influence it, and thus the value of \(k=\{-2,\beta+2\}\). Similarly, the value of \(k^{\prime}\in\{-\infty,\infty\}\) is used to express the \(2\beta\) samples transmitted by any other random user asynchronously. The operator \(\lfloor.\rfloor\) in (4) expresses the sampling of \(2\beta\) sample at the moment of detection. Similarly, the data-bearing signal, received during the second \(\beta\) samples is expressed by multiplying the signal of the desired user form (16) by the data bits \(s_{i}^{d}\). Due to the asynchronous nature of the system, both the reference and data-bearing signals only assume the chaos in the expression of other users' signals so that generality is not lost... The assumption does not impact the derivation of the BER expression from a statistical perspective since \(E|e_{i,k}^{p}|=1\). Hence, the decision variable for the desired user's \(i\)th transmitted symbol for the MU-FTN-DCSK system is derived as,  
\[ \begin{split} D_{i}&=\sum_{n=1}^{\beta}r_{i,n}^{d} \ast r_{i,n+\beta}^{d}\\ &=\sum_{n=1}^{\beta}\bigg{(}\sum_{k=-2}^{\beta+2}\sum_{l=1}^{L} \alpha_{i,l}^{d}x_{i,k-\tau_{l}}^{d}\rho_{(d)}sinc(\rho_{(d)}(k-n))+\sum_{p=1 }^{U-1}\sum_{l=1}^{L}\sum_{k^{\prime}=-\infty}^{\infty}\alpha_{i,p^{\prime}_{ l}}^{p}x_{i^{\prime},k^{\prime}-\tau_{i^{\prime},p^{\prime}}}^{p}\sqrt{\rho_{(p)} \rho_{(d)}}\\ &\quad\times sinc(k\rho_{(p)}-n\rho_{(d)})+W_{n}\bigg{)}\times \bigg{(}\sum_{k=-2}^{\beta+2}\sum_{l^{\prime}=1}^{L}\alpha_{i,l^{\prime}_{l} }^{d}s_{i}^{d}x_{i,k-\tau_{l},p^{\prime}}^{d}\rho_{(d)}sinc(\rho_{(d)}(k-n))\\ &\quad+\sum_{p^{\prime}=1}^{U-1}\sum_{l^{\prime}=1}^{L}\sum_{k^{ \prime}=-\infty}^{\infty}\alpha_{i,p^{\prime}_{l,k}}^{p^{\prime}}x_{i^{\prime },k^{\prime}-\tau_{i^{\prime},p^{\prime}}}^{p^{\prime}}\times\sqrt{\rho_{(p^{ \prime})}\rho_{(d)}}sinc(k\rho_{(p^{\prime})}-n\rho_{(d)})+W_{n+\beta}\bigg{)},\end{split}