## 3 Signal Demodulation
### PLS-Based Detection Method
#### 3.2.1 Training Process of PLS Regression  
Let \(\mathbf{x}\) be a random example for training and \(\mathbf{y}\) its response. Either \(\mathbf{x}\) or \(\mathbf{y}\) is zero-mean column vectors. In order to assess their relation with covariance, we project them onto two separate directions  
Figure 4: Correlation between the original chaos phase modulation signal and its underwater sound channel output at \(SNR=-10\) dB. (**a**) Autocorrelation; (**b**) Cross-correlation. specified by unit vectors \(\mathbf{w}_{x}\) and \(\mathbf{w}_{y}\) in order to obtain two random variables \(\mathbf{w}_{x}^{T}\mathbf{x}\) and \(\mathbf{w}_{y}^{T}\mathbf{y}\). Now we present two matrices \(\mathbf{X}\) and \(\mathbf{Y}\) whose \(i\)-th rows are the feature vectors of corresponding examples \(\mathbf{x}_{i}\) and \(\mathbf{y}_{i}\). According to the nonlinear iterative PLS algorithm, PLS searches for the directions \(\mathbf{w}_{x}\) and \(\mathbf{w}_{y}\) such that [45; 51]  
\[\max_{\mathbf{w}_{x},\mathbf{w}_{y}:||\mathbf{w}_{x}||=||\mathbf{w}_{y}||=1}\mathbb{C}(\mathbf{w}_ {x},\mathbf{w}_{y})=\mathbf{w}_{x}^{T}\mathbf{C}_{xy}\mathbf{w}_{y}=\frac{1}{m}\mathbf{w}_{x}^{T} \mathbf{X}^{T}\mathbf{Y}\mathbf{w}_{y} \tag{11}\]  
where \(\mathbf{C}_{xy}=\frac{1}{m}\mathbf{X}^{T}\mathbf{Y}\) is the covariance matrix of \(\mathbf{X}\) and \(\mathbf{Y}\), and \(m\) is the example number. The direction that solve the maximal covariance optimization are the first singular vectors \(\mathbf{w}_{x}=\mathbf{U}_{1}\) and \(\mathbf{w}_{y}=\mathbf{V}_{1}\) of the singular value decomposition of \(\mathbf{C}_{xy}\):  
\[\mathbf{C}_{xy}=\mathbf{U}\Sigma\mathbf{V}^{T} \tag{12}\]  
where the value of the covariance is given by the corresponding singular value \(\sigma_{1}\). In this paper, more than one projection direction are wanted. To do this, the same strategy of data projecting is applied by deflation. This creates the new data matrix  
\[\mathbf{X}^{\prime}=\mathbf{X}(\mathbf{I}-\mathbf{w}_{x}\mathbf{w}_{x}^{T}) \tag{13}\]  
Let \(\mathbf{P}_{j}\) be \(\frac{\mathbf{x}_{j}^{T}\mathbf{x}_{j}\mathbf{u}_{j}}{\mathbf{x}_{j}^{T}\mathbf{u}_{j}^{T}\mathbf{u}_{ j}\mathbf{x}_{j}}\) with \(j=1,2,\ldots,k\), where \(k\) is the projection direction number, the deflation of \(\mathbf{X}=\mathbf{X}_{1}\) is obtained as follows:  
\[\mathbf{X}_{2}=\mathbf{X}_{1}(\mathbf{I}-\mathbf{U}_{1}\mathbf{P}_{1}^{T})\,\,\text{with}\ \mathbf{X}_{1}=\mathbf{X} \tag{14}\]  
Considering a test point with the feature vector \(\varphi(\mathbf{x})\), we define \(\widetilde{\varphi}(\mathbf{x})\) as the feature vector needed for the regression, whose columns are \(\varphi_{j}(\mathbf{x})^{T}\mathbf{U}_{j}\). The new feature vector can be expressed as follows:  
\[\widetilde{\varphi}(\mathbf{x})^{T}=\varphi(\mathbf{x})^{T}\widetilde{\mathbf{U}}(\mathbf{P}^ {T}\widetilde{\mathbf{U}})^{-1} \tag{15}\]  
where \(\widetilde{\mathbf{U}}=[\mathbf{U}_{1},\mathbf{U}_{2},\mathbf{U}_{3},\ldots,\mathbf{U}_{j}]\). Now, we can start to compute the regression coefficients vector \(\mathbf{W}\), which performs the regression of the variables \(\mathbf{Y}\) in terms of \(\mathbf{X}\widetilde{\mathbf{U}}\). We seek a coefficient matrix \(\mathbf{B}\) that solves the optimization  
\[\min_{\mathbf{B}}||\mathbf{X}\widetilde{\mathbf{U}}\mathbf{B}-\mathbf{Y}||^{2}=\min_{\mathbf{B}}\langle \mathbf{X}\widetilde{\mathbf{U}}\mathbf{B}-\mathbf{Y},\mathbf{X}\widetilde{\mathbf{U}}\mathbf{B}-\mathbf{Y}\rangle \tag{16}\]  
The final regression coefficients \(\mathbf{W}\) are given by \(\widetilde{\mathbf{U}}\mathbf{B}\). We seek the minimum by computing the gradient with respect to \(\mathbf{B}\) and setting it to zero. The overall regression coefficients can be computed as follows:  
\[\mathbf{W}=\widetilde{\mathbf{U}}(\mathbf{P}^{T}\widetilde{\mathbf{U}})^{-1}\mathbf{C}^{T} \tag{17}\]  
where \(\mathbf{C}\) is the matrix with columns \(c_{j}=\frac{\mathbf{\gamma}^{T}\mathbf{x}_{j}\mathbf{u}_{j}}{\mathbf{u}_{j}\mathbf{x}_{j}^{T}\mathbf{ x}_{j}\mathbf{u}_{j}}\).