## 3 Blind Demodulation of CDS-SS Signals Based on Particle Filters
### Blind Demodulation of CDS-SS Signals under the White Gaussian Noises Condition  
Firstly, consider the situation that the signal is influenced by the white Gaussian noises, _i.e._, \(w_{k}^{(1)}\) in Equation (4) and \(w_{k}^{(2)}\) in Equation (5) are white Gaussian noises. Since the transmitter's chaotic map is unknown to the intruder, Equations (4) and (5) use another \(g(\cdot)\) to fit the transmitter's chaotic map \(f(\cdot)\). Thus, the difference between \(g(\cdot)\) and \(f(\cdot)\) will inevitably obviously result in a chaos fitting error. Therefore, the best method is to utilize the chaos fitting error. The chaos fitting method [15] employs a range-differentiating factor \(\beta\) in Equation (5) to differentiate the vibration range of the amplitude of \(\hat{b}_{k}\) corresponding to \(-1\) and \(1\). Accordingly, the following state-space equations can be obtained corresponding to Equations (4) and (5), respectively:  
\[ \begin{cases}\hat{x}_{k+1}=g(\hat{x}_{k})+w_{k}^{(1)}\\ z_{k+1}=\text{sgn}(\hat{b}_{k+1})\hat{x}_{k+1}+v_{k+1}^{(1)}\end{cases}  \tag{22}\]  
\[ \begin{cases}\hat{b}_{k+1}=\hat{b}_{k}+w_{k}^{(2)}\\ z_{k+1}=\hat{b}_{k+1}(g(\hat{x}_{k})+\beta)+v_{k+1}^{(2)}\end{cases}  \tag{23}\]  
Equations (22) and (23) are the state-space equations of the PFs, and then the chaotic state \(\{\hat{x}_{k}\}\) and the message signal \(\{\hat{b}_{k}\}\) can be estimated by the PFs. At time step \(k+1\), the transmitted CDS-SS signal is \(s_{k+1}\!=\!b_{k+1}f(x_{k})\), and the chaos fitting is \(z_{k+1}\!=\!\hat{b}_{k+1}(g(\hat{x}_{k})+\beta)+v_{k+1}^{(2)}\). Then, the chaos fitting error is:  
\[ \begin{split} e_{k+1}&=s_{k+1}-z_{k+1}\\ &=b_{k+1}f(x_{k})-\hat{b}_{k+1}(g(\hat{x}_{k})+\beta)-v_{k+1}^{(2 )}\end{split}  \tag{24}\]  
Let \(\dot{e_{k+1}}\!=\!-\!e_{k+1}-v_{k+1}^{(2)}\), then:  
\[\hat{b}_{k+1}\!=\!\frac{b_{k+1}f(x_{k})+\dot{e_{k+1}}}{g(\hat{x}_{k})+\beta} \tag{25}\]  
\(\dot{e_{k+1}}\) is caused by the fitting error and the measurement noise, so it is random, then the \(\dot{e_{k+1}}\) and \(-\dot{e_{k+1}}\) are considered indiscriminating. \(\hat{b}_{k+1}\) is rewritten as \(\hat{b}_{k+1}^{\text{\tiny{$1$}}}\) or \(\hat{b}_{k+1}^{\text{\tiny{$-1$}}}\) corresponding to the true value \(b_{k+1}\!=\!1\) or \(b_{k+1}\!=\!-1\), respectively, then \(\hat{b}_{k+1}^{\text{\tiny{$1$}}}\) and \(\hat{b}_{k+1}^{\text{\tiny{$-1$}}}\) are expressed as follows:  
\[\hat{b}_{k+1}^{\text{\tiny{$1$}}}\!=\!\frac{f(x_{k})+\dot{e_{k+1}}}{g(\hat{x} _{k})+\beta}\qquad\hat{b}_{k+1}^{\text{\tiny{$-1$}}}\!=\!-\!\frac{f(x_{k})+ \dot{e_{k+1}}}{g(\hat{x}_{k})+\beta} \tag{26}\]  
If there is not a range-differentiating factor, _i.e._, \(\beta\!=\!0\), let \(\hat{b}_{k+1}^{\text{\tiny{$1$}}}\!\in\! [-\!c,d]\), then \(\hat{b}_{k+1}^{\text{\tiny{$-1$}}}\!\in\![-\!d,c]\). Because the range of \(f(x_{k})\!+\!\dot{e_{k+1}}\!/g(\hat{x}_{k})\) approximates axial symmetry [27], _i.e._, \(c\!\approx\!d\), the difference between \([-\!c,\!d]\) and \([-\!d,\!c]\) is very small. Thus, \(\hat{b}_{k+1}^{\text{\tiny{$1$}}}\) or \(\hat{b}_{k+1}^{\text{\tiny{$-1$}}}\) cannot be differentiated from \(\hat{b}_{k+1}\), as shown in Figure 1(a). If \(\beta\!\
eq\!0\), it will break the axial symmetry, then \(c\!\
eq\!-\!d\), \(\hat{b}_{k+1}^{\text{\tiny{$1$0}}}\!\in\! [-\!c,d]\) and \(\hat{b}_{k+1}^{\text{\tiny{$1$1}}}\!\in\! [-\!d,\!c]\) will be obviously different as demonstrated in Figure 1(b). Therefore, \(\hat{b}_{k+1}^{\text{\tiny{$1$2}}}\) can be differentiated from \(\hat{b}_{k+1}^{\text{\tiny{$1$3}}}\) by a threshold, and then the message \(b_{k+1}\) can be recovered. The threshold is chosen as \(0\) since the range of the estimated message \(\hat{b}_{k+1}\) is on the \(x\) axis symmetry when \(\beta\!=\!0\). Based on the chaos fitting method, the approach to estimating the message \(\{\hat{b}_{k}\}\) through Equations (22) and (23) by two alternate PFs is summarized as follows:  
Initiate \(\hat{x}_{k}\) and \(\hat{b}_{k}\), then operate circularly starting from \(k\!=\!0\).