## 3 Performance Analysis
The result shown in [28] (Eq. 3.3) is valid for our case since the noise is only affected by the filter of the desired user at detection. As a result, the power of the noise is reduced by a factor of \(G_{k,n}\) for the sampling parameter \(\rho_{(d)}\) as :  
\[ \begin{split} Cov(w_{k},w_{n})=&\sigma^{2}G_{k,n}\\ =&\sigma^{2}\rho_{(d)}since(\rho_{(d)}(k-n)),\end{split}  \tag{25}\]  
where \(\sigma\) is the standard deviation of the noise. Henceforth, for a noise with a double-sided power spectral density \(N_{0}\), the variance of the noise terms is expressed as  
\[ \begin{split} V_{N_{i}}=&\frac{N_{0}E_{b}}{2\beta} \sum_{n=1}^{\beta}\sum_{k=-2}^{\beta+2}\bigg{(}\sum_{k^{\prime}=-\infty}^{ \infty}\sum_{l=1}^{L}(\alpha_{i,l}^{d})^{2}\rho_{(d)}^{3}since(\rho_{(d)}(k-n) )\tilde{s}inc^{2}(\rho_{(d)}(k^{\prime}-n))\\ &+\frac{N_{0}E_{b}}{2\beta}\sum_{p=1}^{U-1}\sum_{n=1}^{\beta}\sum _{k^{\prime}=-\infty}^{\infty}\sum_{l=1}^{L}(\alpha_{i,l}^{p})^{2}\rho_{(p)} \rho_{(d)}^{2}\tilde{s}inc(\rho_{(d)}(k-n))\tilde{s}inc^{2}(k^{\prime}\rho_ {(p)}-n\rho_{(d)})\\ &+\frac{N_{0}^{2}}{4}\sum_{n=1}^{\beta}\rho_{(d)}^{2}\tilde{s}inc ^{2}(\rho_{(d)}(k-n))\bigg{)}.\end{split}  \tag{26}\]  
For non-coherent chaos-based communications in regular Nyquist transmissions, the covariance is often neglected while expressing variances, given the low cross-correlation between chaotic sequences when time delayed or for different chaos generators,  
\[x_{(i,t)}.(x_{(j,t)})^{T}\approx 0\qquad i\
eq j. \tag{27}\]  
However, in sub-Nyquist transmissions with the introduction of the \(sinc\) terms as seen in the derivation of the variances and the expectation, covariances are introduced in the formulation of the BER expression. Here, we explore the different covariances that may occur in this scenario of multi-user chaos based transmission below the Nyquist rate. For the Chebychev map [29], we know that the joint PDF of two chaotic variables \(x_{0}\) and \(x_{1}\), time delayed by \(t\), is expressed as  
\[f(x_{0},x_{1};t)=\pi^{-1}(1-x_{0})^{-\frac{1}{2}}\delta(x_{1}-T_{k^{\prime}}( x_{0})), \tag{28}\]  
where \(T_{k^{\prime}}\) is the Chebychev polynomial for a variable \(x\) expressed as  
\[T_{k^{\prime}}(x)=\cos(k.\arccos(\mathrm{x})), \tag{29}\]  
where \(cos\) is the cosine function and \(\delta\) is the delta function. The variance of a sum of dependent random variables \(X_{i}\) with weights \(a_{i}\) is known as [30],  
\[var(\sum_{i=1}^{n}a_{i}X_{i})=\sum_{i=1}^{n}a_{i}^{2}var(X_{i})+2\sum_{i=1}^{n }\sum_{j=i+1}^{n}a_{i}a_{j}cov(X_{i},X_{j}), \tag{30}\]  
and the covariance is given as  
\[cov(X_{i},X_{j})=E(X_{i}.X_{j})-E(X_{i})E(X_{j}). \tag{31}\]  
Given that the chaotic sequences are generated with zero mean (\(E(X_{i})=0\)), the covariance of two FTN filtered chaotic sequences is derived as  
\[ \begin{split} Cov(X_{i},X_{j})&=E(X_{i}.X_{j})\\ &=E(\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x_{i}.x_{j}.G_{ \mathrm{T}}^{i}(t)G_{\mathrm{T}}^{j}(s)dtds)\\ &=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}E(x_{i}.x_{j}).G_ {\mathrm{T}}^{i}(t)G_{\mathrm{T}}^{j}(s)dtds,\end{split}  \tag{32}\]where \(G_{T}^{i}(t)G_{T}^{j}(s)\) are two sub-Nyquist filters from two random users with different sampling rates. We use the joint PDF of two chaotic variables to compute the expectation \(E(x_{i}.x_{j})\)  
\[ \begin{split} E(x_{i}.x_{j})&=\int_{-\infty}^{ \infty}\int_{-\infty}^{\infty}x_{i}.x_{j}.f_{x_{i}x_{j}}(x_{i},x_{j})dx_{i}.dx_{ j}\\ &=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x_{i}.x_{j}.\pi^ {-1}(1-x_{i}^{2})^{-\frac{1}{2}}\delta(x_{j}-x_{i})dx_{i}.dx_{j}\\ &=(\frac{E_{b}}{\beta})^{2}\int_{-1}^{1}\pi^{-1}(1-x_{i}^{2})^{- \frac{1}{2}}\\ &=\frac{E_{b}^{2}}{2\beta^{2}}\end{split}