# Kernel-FastICA-Based Nonlinear Blind Source Separation for Anti-Jamming Satellite Communications
## 3 Kernel-FastICA-Based Nonlinear Blind Source Separation Algorithm  
Building on the post-nonlinear (PNL) mixing model established in Section 2, the core objective of blind source separation (BSS) is to isolate communication signals from interference signals using only observed signals, without prior information, thereby recovering clean communication signals and achieving interference suppression. Traditional linear BSS methods fail in scenarios involving nonlinear distortions caused by components such as limiters during signal transmission. Thus, constructing a BSS framework suitable for nonlinear mixing is imperative. To address the dual challenges of channel distortion and interference coupling in satellite communication, this paper proposes a kernel-method-enhanced FastICA algorithm (Kernel-FastICA), which adopts a two-stage \"nonlinear mapping-linear separation\" architecture, as illustrated in Figure 4.  
The kernel method centers on nonlinear mapping, aiming to transform the nonlinear mixing structures in raw observed signals into linearly separable forms through feature space transformation [27]. Specifically, for an observed signal \(x(t)\in\mathcal{X}\), a mapping function \(\phi:\mathcal{X}\rightarrow\Gamma\) projects it into a high-dimensional reproducing kernel Hilbert space (RKHS) \(\Gamma\). Within this space, complex nonlinear mixing relationships can be approximated as linear superposition models, providing theoretical feasibility for subsequent FastICA implementation. The kernel function is defined as the inner product of data points in the feature space:  
\[\mathrm{K}(x,y)=\langle\phi(x),\phi(y)\rangle, \tag{5}\]  
where \(x,y\in\mathcal{X}\) are samples in the original input space, and \(\phi(\cdot)\) denotes the implicit mapping function. According to [PERSON]'s theorem, the positive definiteness of the kernel function implicitly constructs a high-dimensional feature space, circumventing the curse of dimensionality while preserving statistical independence. To enhance nonlinear separation  
Figure 3: Block diagram of the nonlinear blind source separation signal processing system. capability, this work employs a third-order polynomial kernel to explicitly construct the feature space [28]:  
\[\phi(x)=[1,x_{1},x_{2},x_{1}^{2},x_{1}x_{2},x_{2}^{2},\ldots,x_{2}^{4}]^{T}, \tag{6}\]  
The dimensionality of the input signal \(\mathbf{x}(t)=[x_{1}(t),x_{2}(t)]^{T}\) is expanded from 2D to 10D via the cubic polynomial kernel. Comparative experiments demonstrate that the cubic kernel significantly enhances the modeling capability for nonlinear effects (e.g., power amplifier clipping distortion) compared with the quadratic polynomial kernel, while its compact feature space effectively mitigates the risk of dimensionality explosion relative to quartic and higher-order polynomials. Furthermore, the polynomial kernel requires no parameter tuning, making it better suited than the Gaussian kernel [29] for the deterministic modulation schemes of satellite communication systems. This optimizes real-time performance while ensuring algorithmic robustness. To satisfy the preconditions of the FastICA algorithm and enhance numerical stability, the kernel-mapped signal \(\phi(\mathbf{x}(t))\) undergoes pre-whitening. This process renders the data components uncorrelated with unit variance in the feature space. First, a regularization term \(\lambda=10^{-4}\) is introduced to estimate the covariance matrix:  
\[\mathbf{C}=\frac{1}{T}\sum_{t=1}^{T}\phi(\mathbf{x}(t))\phi(\mathbf{x}(t))^{T }+10^{-4}\mathbf{I} \tag{7}\]  
where \(T\) is the number of sampling points; \(\mathbf{I}\) is the identity matrix; and the regularization term suppresses perturbations from small eigenvalues while ensuring matrix invertibility. Specifically, the \(10^{-4}\mathbf{I}\) term in Equation (7) boosts small eigenvalues, ensuring numerical stability and alleviating ill-conditioning in the high-dimensional kernel space.