## 3 Blind Demodulation of CDS-SS Signals Based on Particle Filters
### Blind Demodulation of CDS-SS Signals under Colored/Non-Gaussian Noises Condition  
Under the condition that the noises is colored or non-Gaussian as described in Equations (15) and (16), the range-differentiating factor \(\beta\) is imported into Equation (16) as shown in Equation (27):  
\[\left\{ \begin{array}{l}\overline{\mathbf{b}}_{k+1}=\begin{bmatrix}\hat{b}_{k} \\ \mathbf{0}\end{bmatrix}+\begin{bmatrix}\widehat{\mathbf{h}}_{k}\\ \widehat{\mathbf{h}}_{2}\end{bmatrix}\widehat{\mathbf{h}}_{k}+\begin{bmatrix} 1\\ \widehat{\mathbf{h}}_{2}\end{bmatrix}\eta_{k}^{(2)}\\ =\begin{bmatrix}\hat{b}_{k}\\ \mathbf{0}\end{bmatrix}+\overline{\mathbf{A}}_{2}\widehat{\mathbf{b}}_{k}+ \overline{\mathbf{B}}_{2}\eta_{k}^{(2)}\\ z_{k+1}=\hat{b}_{k+1}(g(\hat{x}_{k})+\beta)+\
u_{k+1}^{(2)}\end{array} \right. \tag{27}\]  
Then, the proposed algorithm of alternately estimating the message signal \(\{\hat{b}_{k}\}\) and chaotic state \(\{\hat{x}_{k}\}\) according to Equations (15) and (27) by two PFs is illustrated in Figure 2 and described as follows. (1) Initialization  
Begin by generating samples \(\{\overline{\mathbf{x}}_{0}^{j}\}_{j=1}^{M}\) and \(\{\overline{\mathbf{b}}_{0}^{j}\}_{j=1}^{M}\) from their initial probabilities \(p(\overline{\mathbf{x}}_{0})\) and \(p(\overline{\mathbf{b}}_{0})\), respectively. Since the \(p(\overline{\mathbf{x}}_{0})\) and \(p(\overline{\mathbf{b}}_{0})\) are unknown, they can be assumed to be uniform distributions in bounded state spaces by making use of the limited range of the chaotic system. The algorithm starts from a random measure with equal weight on each of the \(M\) samples (\(M\) will always be 1,000 unless otherwise stated). Weight \(\lambda_{\mathbf{z}}^{j}=1/\,M\), \(\lambda_{\mathbf{\xi}}^{j}=1/\,M\) ;  
(2) Estimate \(\hat{b}_{k+1}\) according to Equation (27) and \(\hat{x}_{k}\) (for \(k=0,1,\ldots\) ). The state equation is  
Figure 2: Flow chart of the proposed PF-based algorithm. where the white non-Gaussian noises \(n_{k}^{(1)}\) and \(n_{k}^{(2)}\) are obey the exponential distributions \(p_{{}_{\mu^{(1)}}}(n_{k}^{(1)})\) = 25 exp(-25\(n_{k}^{(1)}\)) and \(p_{{}_{\mu^{(2)}}}(n_{k}^{(2)})\) = 20 exp(-20\(n_{k}^{(2)}\)), respectively. It should be mentioned that the distribution of the noises is obtained by the priori knowledge in practice. Assume that the measurement noises \(v_{k}^{(1)}\) and \(v_{k}^{(2)}\) with zero-mean and arbitrary distribution are both white Gaussian noises for simplicity, the spreading factor \(N\) is 127, and the range-differentiating factor \(\beta\) is 0.9. Figure 3 shows the estimated message \(\hat{b_{k}}\) without the range-differentiating factor at a signal-to-noise ratio (SNR) of 7 dB, _i.e.,_\(\beta\) = 0, from which the message symbols cannot be recovered because the values of \(\hat{b_{k}^{\dagger}}\) and \(\hat{b_{k}^{\text{-1}}}\) vibrate in the same range. **Figure 3. (a)** The estimated message \(\hat{b_{k}}\) when \(\beta\) = 0 at SNR = 7 dB; (**b**) the result of lowpass filtering; (**c**) the original transmitted binary message \(-b_{n}\). [PERSON] and [PERSON] [15] proposed a modified UKF to blindly demodulate the CDS-SS signals accurately under the white Gaussian noises condition. However, in the case of colored noises in Equations (40) and (41), errors appear as shown in Figure 4.  
**Figure 4. (a)** The message \(\hat{b_{k}}\) estimated by UKF when \(\beta\) = 0.9 at SNR = 7 dB; (**b**) the result of lowpass filtering; (**c**) the original transmitted binary message \(-b_{n}\). Figure 4(a) shows the message \(\hat{b}_{k}\) estimated by UKF at SNR = 7 dB; Figure 4(b) shows the result of lowpass filtering; Figure 4(c) shows the original transmitted binary message \(-b_{n}\). In [5], under a condition of cooperative communication (known transmitter structure), several data blocks containing up to 200 bits were transmitted at a SNR of about 7-8 dB, and no error was noticed at the authorized receiver.