# Kernel-FastICA-Based Nonlinear Blind Source Separation for Anti-Jamming Satellite Communications
## 4 Results and Discussion  
To validate the effectiveness of the proposed Kernel-FastICA algorithm in separating nonlinearly mixed signals, this paper designs multiple simulation experiments based on the characteristics of practical receivers. The separation performance is evaluated using the correlation coefficient matrix between the source signals and the separated signals, whose elements are defined as follows:  
\[\rho_{ij}=\frac{\left|\sum\limits_{k=1}^{n}\hat{s}_{i}(k)s_{j}(k)\right|}{ \sqrt{\sum\limits_{k=1}^{n}\hat{s}_{i}^{2}(k)\sum\limits_{k=1}^{n}\hat{s}_{j}^{ 2}(k)}} \tag{20}\]  
where \(\rho_{ij}\in[-1,1]\). An absolute value approaching 1 indicates higher waveform similarity between the separated signal \(\hat{s}_{i}\) and the source signal \(s_{j}\). The correlation coefficient \(\rho_{ij}\) is adopted as the primary metric due to its theoretical foundation in information theory for measuring statistical independence and its practical sensitivity to waveform fidelity. This is crucial for assessing phase-preserving signal recovery in satellite communications [31]. This choice aligns with standard BSS evaluation protocols and ensures the consistent assessment of nonlinear distortion effects. The experimental parameters are listed in Tables 1 and 2. The initial interference-to-signal ratio (ISR) is set to 10 dB, motivated by the nonlinear compression characteristics of satellite receiver chains: When the input ISR reaches 40 dB, the limiter compresses the amplitude of the strong interference signal, stabilizing the residual ISR near 9 dB. This setting accurately simulates typical scenarios with a constrained receiver dynamic range. The experimental results demonstrate that the conventional FastICA algorithm, reliant on linear mixing assumptions, suffers significant performance degradation under nonlinear distortion. As shown in Figure 5, at ISR = 10 dB, the correlation coefficients between separated communication/interference signals are \(\rho_{\text{communication}}=0.412\) and \(\rho_{interference}=0.852\), indicating severe waveform distortion. When the ISR increases to 12 dB (Figure 6), the separation performance further deteriorates to \(\rho_{\text{communication}}=0.357\) and \(\rho_{interference}=0.845\), confirming its inadequacy in nonlinear scenarios. In contrast, the Kernel-FastICA algorithm explicitly characterizes nonlinear features via fourth-order polynomial kernel mapping, achieving signal decoupling in the high-dimensional reproducing kernel Hilbert space (RKHS). As illustrated in Figure 7, at ISR = 10 dB, the algorithm elevates the separated communication signal's signal-to-interference ratio (SIR) to 22.31 dB, optimizes the interference component's ISR to \(-16.18\) dB, and achieves correlation coefficients of \(\rho_{\text{communication}}=0.886\) and \(\rho_{interference}=0.963\). The time-domain waveforms fully retain the amplitude and phase information of the source signals. \begin{table}
\begin{tabular}{c c c} \hline
**Category** & **Parameter** & **Value** \\ \hline \multirow{4}{*}{Communication signal} & Modulation type & BPSK \\  & Roll-off factor & 0.35 \\  & Symbol rate & 256 Mbps \\  & Sampling rate & 4.096 Mbps (16\(\times\) oversampling) \\ \hline \multirow{2}{*}{Interference signal} & Signal type & Multi-tone jamming \\  & Jamming frequencies & 16 kHz, 32 kHz, 64 kHz \\ \multicolumn{2}{c}{ISR} & 10 dB \\ \hline \end{tabular}
\end{table} 
Table 1: Communication and interference signal parameters. \begin{table}
\begin{tabular}{c c c} \hline
**Category** & **Parameter** & **Value** \\ \hline \multirow{3}{*}{Hybrid model} & Mixing type & Non-linear mixing \\  & Linear mixing matrix & [2, \(-1.5\); 2, 1.8] \\  & Non-linear mixing function & tanh(\(\cdot\)) \\ \hline \multirow{2}{*}{Kernel function} & \multirow{2}{*}{Kernel type} & Fourth-order polynomial kernel \\  & & mapping (2D \(\rightarrow\) 15D feature space) \\ \hline \end{tabular}
\end{table} 
Table 2: Hybrid model and kernel function parameters. Figure 5: Signal separation results of FastICA at ISR = 10 dB.