## 2 Problem Formulation  
Consider a CDS-SS communication system, in which the transmitter's chaotic system is described by:  
\[x_{k+1}=f(x_{k}) \tag{1}\]  
The chaotic spreading sequence \(\{x_{k}\}\) is supposed to be a simple one that is generated by one chaotic map that is doubly symmetrical, _i.e._, the range of its values and its probability distribution are both symmetrical [5; 19]. Each original binary message \(b_{n}\in\left\{-1,1\right\}\) is associated with a string of \(N\) elements \(x_{k}(k=(n-1)N+1,\cdots,nN)\), where \(N\) is the spreading factor. Namely, \(b_{n}\) is encoded as the string:  
\[s_{k}=b_{n}x_{k}\,,\ \ k=(n-1)N+1,\cdots,nN\,\ \ n=1,2,\cdots \tag{2}\]  
Let \(b_{n}=b_{k}\,,k=(n-1)N+1,\cdots,nN\), then Equation (2) is rewritten as \(s_{k}=b_{k}x_{k}\). Since the transmitter's chaotic map is unknown, the intruder uses another chaotic map \(g(\cdot)\) instead of \(f(\cdot)\), and its chaotic system equation is given by:  
\[\hat{x}_{k+1}=g(\hat{x}_{k}) \tag{3}\]  
Because the transmitted CDS-SS signal is \(s_{k+1}=b_{k+1}f(x_{k})\), the CDS-SS signal estimated by the intruder could be \(\hat{s}_{k+1}=\hat{b}_{k+1}g(\hat{x}_{k})\), where \(\hat{b}\) and \(\hat{x}\) are the estimation of \(b\) and \(x\), respectively. Considering the model error or noise, the following system and measurement equations [15] can be obtained:  
\[ \begin{cases}\hat{x}_{k+1}=g(\hat{x}_{k})+w_{k}^{(1)}\\ z_{k+1}=\text{sgn}(\hat{b}_{k+1})\hat{x}_{k+1}+v_{k+1}^{(1)}\end{cases}  \tag{4}\]where \(z_{k+1}\) is the observation, \(v_{k+1}^{(1)}\) is the measurement noise with zero-mean and arbitrary distribution, \(w_{k}^{(1)}\) is the colored or non-Gaussian process noise that drives the dynamic systems through the nonlinear state transition function. On the other hand, since the message \(b_{k}\) in CDS-SS varies more slowly than the chaotic sequence \(\left\{x_{k}\right\},\)\(\hat{b}_{k+1}=\hat{b}_{k}\) approximately. Thus, the system and measurement equations can also be written as:  
\[ \begin{cases}\hat{b}_{k+1}=\hat{b}_{k}+w_{k}^{(2)}\\ z_{k+1}=\hat{b}_{k+1}g(\hat{x}_{k})+v_{k+1}^{(2)}\end{cases}  \tag{5}\]  
where sgn(\(\hat{b}_{k+1}\)) = 1 (if \(\hat{b}_{k+1}>0\) ) or sgn(\(\hat{b}_{k+1}\)) = -1 (if \(\hat{b}_{k+1}<0\) ), \(v_{k+1}^{(2)}\) is the measurement noise with zero-mean and arbitrary distribution, \(w_{k}^{(2)}\) is the colored or non-Gaussian process noise that drive the dynamic systems through the nonlinear state transition function. The colored noises \(w_{k}^{(1)}\) and \(w_{k}^{(2)}\) can be formulated by ARMA models [26] as:  
\[w_{k}^{(1)}+c_{1}^{(1)}w_{k-1}^{(1)}+c_{2}^{(1)}w_{k-2}^{(1)}+\cdots+c_{p_{1}} ^{(1)}w_{k-p_{1}}^{(1)}=n_{k}^{(1)}+d_{1}^{(1)}n_{k-1}^{(1)}  
\[\widetilde{\mathbf{b}}_{k+1}= \begin{bmatrix}-\mathbf{c}_{1}^{(2)}&-\mathbf{c}_{2}^{ (2)}&\cdots&\cdots&-\mathbf{c}_{p_{z}}^{(2)}\\ 1&&&&0\\ &1&&&\vdots\\ &&\ddots&&\vdots\\ &&&1&0\end{bmatrix}\begin{bmatrix}d_{1}^{(2)}&d_{2}^{(2)}&\cdots&d_{q_{z}}^{(2)} \\ \vdots\\ \widetilde{\mathbf{b}}_{k}+\begin{bmatrix}1\\ 0\\ \vdots\\ 0\\ \vdots\\ 0\end{bmatrix}n_{k}^{(2)}\\ \end{bmatrix}  \tag{14}\]  
Then, the state vectors are augmented as \(\overline{\mathbf{x}}_{k}= \begin{bmatrix}\hat{x}_{k}&\widetilde{\mathbf{x}}_{ k}^{T}\end{bmatrix} ^{T}\) and \(\overline{\mathbf{b}}_{k}= \begin{bmatrix}\hat{b}_{k}&\widetilde{\mathbf{b}}_{ k}^{T}\end{bmatrix} ^{T}.\) Therefore, the augmented state-space models corresponding to Equations (4) and (5) can be rewritten as:  
\[ \begin{bmatrix}\overline{\mathbf{x}}_{k+1}=\begin{bmatrix}g(\hat{x}_{k})\\ \mathbf{0}\end{bmatrix}+\begin{bmatrix}\mathbf{T}^{(1)}\\ \widetilde{\mathbf{A}}_{1}\end{bmatrix}\widetilde{\mathbf{x}}_{k}+\begin{bmatrix} 1\\ \widetilde{\mathbf{B}}_{1}\end{bmatrix}n_{k}^{(1)}\\ =\begin{bmatrix}g(\hat{x}_{k})\\ \mathbf{0}\end{bmatrix}+\overline{\mathbf{A}}_{1}\widetilde{\mathbf{x}}_{k} +\overline{\mathbf{B}}_{1}n_{k}^{(1)}\\ z_{k+1}=\text{sgn}(\hat{b}_{k+1})\hat{x}_{k+1}+v_{k+1}^{(1)}\\ \end{bmatrix}  \tag{15}\]  
\[