# Kernel-FastICA-Based Nonlinear Blind Source Separation for Anti-Jamming Satellite Communications
## 3 Kernel-FastICA-Based Nonlinear Blind Source Separation Algorithm
The whitening matrix \(\mathbf{V}\) is derived through eigenvalue decomposition (Equation (8)) and inversion (Equation (9)), transforming the data into an uncorrelated, unit-variance basis for subsequent FastICA processing. Figure 4: Flowchart of Kernel-FastICA algorithm. To derive the whitening matrix, we first perform eigenvalue decomposition on \(\mathbf{C}\):  
\[\mathbf{C}=\mathbf{E}\mathbf{D}\mathbf{E}^{T} \tag{8}\]  
where \(\mathbf{E}\) is the matrix of eigenvectors and \(\mathbf{D}=diag(d_{1},\ldots,d_{15})\) is the diagonal matrix of eigenvalues. The whitening transformation matrix is defined as follows:  
\[\mathbf{V}=\mathbf{D}^{-1/2}\mathbf{E}^{T} \tag{9}\]  
yielding the following whitened signal:  
\[\mathbf{z}(t)=\mathbf{V}\phi(\mathbf{x}(t)) \tag{10}\]  
Here, \(\mathbf{z}(t)\) satisfies \(\mathbb{E}\left\{\mathbf{z}\mathbf{z}^{T}\right\}=\mathbf{I}\), providing a standardized input for FastICA. The regularization term in Equation (7) bounds the condition number of the covariance matrix, suppressing noise amplification during whitening. Additionally, the cubic polynomial kernel, combined with the tanh-based PNL (post-nonlinear) model, jointly suppresses Gaussian noise by leveraging the distinct higher-order statistical properties of communication signals versus noise. These characteristics endow Kernel-FastICA with superior noise robustness compared with linear ICA. In the high-dimensional kernel feature space, the FastICA algorithm extracts independent source signals \(\overset{\wedge}{\mathbf{s}}(t)\) by maximizing the non-Gaussianity. The theoretical foundation of FastICA lies in the assumption that stronger non-Gaussianity corresponds to higher statistical independence of source signals. This work adopts negentropy as the non-Gaussianity metric:  
\[J(y)=\mathrm{H}(y_{\mathrm{gauss}})-\mathrm{H}(y), \tag{11}\]  
where \(\mathrm{H}(y)\) is the Shannon entropy of signal \(y\), and \(y_{\mathrm{gauss}}\) is a Gaussian-distributed reference variable with the same variance as \(y\). By maximizing negentropy through iterative optimization [30], the algorithm progressively approximates the true independent sources. To approximate negentropy, a Gaussian nonlinear function \(\mathrm{G}(u)=u\cdot e^{-u^{2}/2}\) is used to formulate the optimization objective:  
\[\mathrm{K}(x,y)=\langle\phi(x),\phi(y)\rangle, \tag{5}\]0  
The gradient update rule is derived by differentiating the objective function:  
\[\mathrm{K}(x,y)=\langle\phi(x),\phi(y)\rangle, \tag{5}\]1  
where \(\mathrm{G}^{\prime}(u)=(1-u^{2})e^{-u^{2}/2}\) and \(\beta=\mathbb{E}\big{\{}G^{\prime\prime}(\mathbf{w}^{T}\phi(\mathbf{x}))\big{\}}\). To avoid local convergence in gradient descent, a symmetric fixed-point iteration updates the separation matrix \(\mathbf{W}\):  
\[\mathrm{K}(x,y)=\langle\phi(x),\phi(y)\rangle, \tag{5}\]2  
The iteration termination conditions are set to a maximum iteration count of \(T_{\mathrm{max}}=1000\) or a convergence threshold of \(\epsilon=10^{-7}\) (convergence is declared when \(\parallel\mathbf{W}\mathbf{W}_{\mathrm{old}}^{T}-\mathbf{I}_{F}<\epsilon\parallel\)). The final source signal estimate is given by  
\[\mathrm{K}(x,y)=\langle\phi(x),\phi(y)\rangle, \tag{5}\]3The algorithm flow is illustrated in \[\mathrm{K}(x,y)=\langle\phi(x),\phi(y)\rangle, \tag{5}\]4 To rigorously characterize the robustness of the Kernel-FastICA algorithm, this section establishes mathematical guarantees for convergence behavior, error bounds, and theoretical performance limits. Convergence Analysis: When the nonlinear function \(g(u)=u\exp(-u^{2}/2)\) satisfies local strong convexity, the symmetric fixed-point iteration (Equation (14)) exhibits local cubic convergence. Formally, letting \(T(\mathrm{W})\) denote the update operator, there exists a neighborhood around the optimal separation matrix \(\mathrm{W}^{*}\) such that  
\[\mathrm{K}(x,y)=\langle\phi(x),\phi(y)\rangle, \tag{5}\]5  
where \(\gamma>0\) is a convergence constant.