## 3 Performance Analysis
 \tag{17}\]  
The desired signal is then the product of the chaotic terms having the index of the desired user \(d\) and the filter with the desired user's sampling parameter \(\rho_{(d)}\), resulting in the following signal  
\[U_{i}=\sum_{n=1}^{\beta}\sum_{k=-2}^{\beta+2}\sum_{l=1}^{L}s_{i}^{d}{(\alpha_ {i,l}^{d})}^{2}{(x_{i,k-\tau_{l}}^{d})}^{2}\rho_{(d)}^{2}\times sinc^{2}(\rho_ {(d)}(k-n)), \tag{18}\]  
The cross-product of the chaotic terms in (17) results in interferences expressed as  
\[\begin{split} I_{i}&=\sum_{n=1}^{\beta}\sum_{k=-2}^{ \beta+2}\sum_{k^{\prime}=-\infty,k^{\prime}\
eq k}^{\infty}\bigg{(}\sum_{l=1}^ {L}\sum_{l^{\prime}=1}^{L}s_{i}^{d}\alpha_{i,l}^{d}x_{i,k-\tau_{l,l}}^{d}x_{i,k -\tau_{l,l}}^{d}\rho_{(d)}^{2}sinc(\rho_{(d)}(k-n))\bigg{)}\\ &\quad\times sinc(\rho_{(d)}(k^{\prime}-  
The first term in (19) is the ISI, which occurs because of the reduced distances between samples due to the nature of the FTN filter. The remaining part is the inter-user interference (IUI) for different chaos generators and sampling parameters. The noise term is expressed as  
\[\begin{split} N_{i}&=\sum_{n=1}^{\beta}\bigg{(}W_{n -\beta}\bigg{(}\sum_{k=-2}^{\beta+2}\sum_{l=1}^{L}\alpha_{ij}^{d}x_{i,k-\tau_{ ij}}^{d}\rho_{(d)}\times sinc(\rho_{(d)}(k-n))+\sum_{p=1}^{U-1}\sum_{k=-\infty}^{ \infty}\sum_{l=1}^{L}\alpha_{i,l}^{p}x_{i,l}^{p}\bigg{.}\\ &\quad\times\sqrt{\rho_{(p)}\rho_{(d)}}x_{i,k^{\prime}-\tau_{ij}, \tau_{ij}}^{p}\text{sinc}(k\rho_{(p)}-n\rho_{(d)})\bigg{)}\\ &\quad+W_{n}\bigg{(}\sum_{k=-2}^{\beta+2}\sum_{l=1}^{L}\alpha_{i, l}^{d}s_{i,k-\tau_{ij}}^{d}x_{i,k-\tau_{ij}}^{d}\rho_{(d)}sinc(\rho_{(d)}(k-n))+ \sum_{p^{\prime}=1}^{U-1}\sum_{k^{\prime}=-\infty}^{\infty}\sum_{l=1}^{L}x_{i ^{\prime},l}^{p^{\prime}}x_{i^{\prime},l^{\prime}}^{p^{\prime}}x_{i^{\prime}, l^{\prime}-\tau_{ij},\tau_{ij}}^{p^{\prime}} \tag{20}\]  
In this paper, it is assumed that the sum of \(\beta\) chaotic samples is equal to,  
\[E_{b}=2\sum_{k=1}^{\beta}x_{i,k}^{2} \tag{21}\]  
Hence, the mean value of the desired signal is given as  
\[ \begin{split} E=&\frac{1}{2}\rho_{(d)}^{2}\frac{E_ {b}}{\beta}\sum_{n=}^{\beta}\sum_{k=-2}^{\beta+2}\sum_{l=1}^{L}(\alpha_{i,l}^{ d})^{2}sinc^{2}(\rho_{(d)}(k-n)).\end{split}  \tag{22}\]  
Since the chaotic sequences are generated with zero mean (\(E(x_{i,k})=0\)) and using (21), the variance of the individual chaotic chips is given as,  
\[ \begin{split} var(x_{i,k}^{2})=\frac{E_{b}^{2}}{4\beta}\end{split}  \tag{23}\]  
Hence, the variances of the interference terms in (19) can be split as the ISI, with variance \(V_{I_{1}}\), the interference to the desired user, with variance \(V_{I_{2}}\), and the interference among all other users, with variance \(V_{I_{3}}\) as  
\[ \begin{split} V_{I_{1}}=&\frac{E_{b}^{2}}{4\beta^{ 2}}\sum_{n=1}^{\beta}\sum_{k=-2}^{\beta+2}\sum_{k^{\prime}=-\infty}^{\infty} \sum_{l=1}^{L}\sum_{l^{\prime}=1}^{L}\left(\alpha_{i,l}^{d}\alpha_{i,l^{\prime }}^{d}\right)^{2}\\ &\quad\times\rho_{(d)}^{4}sinc^{2}(\rho_{(d)}(k-n))sinc^{2}(\rho_ {(d)}(k^{\prime}-n))\\  V_{I_{2}}=&\frac{E_{b}^{2}}{2\beta^{2}}\sum_{p=1,p \
eq d}^{\text{UI}}\sum_{n=1}^{\beta}\sum_{k=-2}^{\beta+2}\sum_{k^{\prime}=- \infty}^{\infty}\sum_{l=1}^{L}\sum_{l^{\prime}=1}^{L}\left(\alpha_{i,l}^{d} \alpha_{i^{\prime},l^{\prime}}^{p}\right)^{2}\rho_{(p)}\\ &\quad\times\rho_{(d)}^{3}sinc^{2}(\rho_{(d)}(k-n))sinc^{2}(k^{ \prime}\rho_{(p)}-n\rho_{(d)})\\  V_{I_{3}}=&\frac{E_{b}^{2}}{4\beta^{2}}\sum_{p=1}^{ \text{UI}}\sum_{p^{\prime}=1,p^{\prime}\
eq p}^{\text{UI}}\sum_{n=1}^{\beta+2} \sum_{k=-2}^{\beta+2}\sum_{k^{\prime}=-\infty}^{\infty}\sum_{l=1}^{L}\sum_{l^{ \prime}=1}^{L}\left(\alpha_{i,l}^{p}\alpha_{i^{\prime},l^{\prime}}^{p^{\prime} }\right)^{2}\\ &\quad\times\rho_{(p)}\rho_{(p^{\prime})}\rho_{(d)}^{2}sinc^{2}(k \rho_{(p)}-n\rho_{(d)})\times sinc^{2}(k^{\prime}\rho_{(p^{\prime})}-n\rho_{( d)}).\end{split}  \tag{24}\]  
The noise variance is equally affected by the sinc filter at the reception.