# Kernel-FastICA-Based Nonlinear Blind Source Separation for Anti-Jamming Satellite Communications
## 5 Conclusions  
To address the challenge of separating nonlinearly mixed signals caused by the soft-clipping effects of power amplifiers in satellite communications, this paper proposes a fast independent component analysis algorithm optimized via the kernel method (Kernel-FastICA). By constructing a post-nonlinear (PNL) mixing model to accurately characterize clipping distortion, the algorithm utilizes a polynomial kernel function to map the observed signals into a high-dimensional reproducing kernel Hilbert space (RKHS), transforming the nonlinear blind source separation (BSS) problem into a linear independent component analysis (ICA) task. Combined with regularized pre-whitening and symmetric fixed-point iteration optimization, it significantly suppresses the computational complexity and numerical instability induced by the kernel space dimensionality expansion. The simulation results demonstrate that, compared with the conventional FastICA algorithm, the proposed method achieves a communication signal correlation coefficient of 0.872-0.981, a peak signal-to-interference ratio (SIR) of 30.27 dB, and an optimized interference-to-signal ratio (ISR) down to \(-\)19.83 dB within an interference range of 3-12 dB ISR, verifying its robustness and generalization capability in nonlinear interference environments.  
Although Kernel-FastICA performs excellently in simulations, the fixed form of its nonlinear mixing function may not fully capture the diversity of distortion variations. Future work will employ neural networks (e.g., multilayer perceptrons or convolutional neural networks) to adaptively fit the nonlinear mixing process. Furthermore, the cubic polynomial kernel mapping expands the input dimensionality from 2 to 10, increasing the computational complexity to \(O(d^{3})\) (where d is the expanded dimensionality). The regularized pre-whitening step requires eigenvalue decomposition of the covariance matrix, with complexity \(O(n^{3})\) at dimension \(n=10\); however, regularization (Equation (7)) optimizes ill-conditioning. The per-iteration complexity of FastICA is \(O(mnt)\) (\(m\): number of sources, \(n\): dimensionality, \(t\): sample size). Combined with its cubic convergence property (Equation (16) in Section 3), the total complexity is controlled at \(O(kmnt)\). Setting the termination  
Figure 11: The variation of the correlation coefficient between the separated signal and the original signal with the number of iterations.  
condition \(T_{max}=1000\) as a practical upper limit makes it suitable for resource-constrained onboard, real-time systems. While this study focused on validating the nonlinear separation mechanism under controlled interference, subsequent work will evaluate the algorithm's robustness in dynamic environments using low-Earth-orbit (LEO) channel models (e.g., Doppler spread, Rician fading). The current model does not consider spatial correlation (e.g., multi-antenna array effects); future work will develop a spatiotemporal BSS framework to suppress interference in dynamic scenarios and leverage array signal processing to enhance multipath robustness. This work provides a solution for satellite communication anti-jamming technology that balances theoretical innovation with engineering feasibility. It overcomes the limitations of traditional linear BSS methods in nonlinear scenarios and offers new avenues for signal processing in complex electromagnetic environments.  
Conceptualization, X.S. and J.L.; methodology, X.S.; software, X.S.; writing--original draft preparation, X.S.; writing--review and editing, X.S. and J.L.; project administration, Q.S.; funding acquisition, C.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant number 62001516.  
Not applicable.  
The data presented in this study are available on request from the first author.  
The authors declare no conflicts of interest.