# Kernel-FastICA-Based Nonlinear Blind Source Separation for Anti-Jamming Satellite Communications
## 2 System Modeling  
Nonlinear distortion constitutes one of the critical factors limiting signal separation and interference suppression performance in satellite communication systems. Such distortion primarily arises from the limiter in the RF front-end, which protects high-sensitivity components like power amplifiers (PAs) and analog-to-digital converters (ADCs). As illustrated in Figure 1, to prevent high-power interference signals from damaging critical components, the limiter is typically placed at the front-end of the signal chain to constrain input signal amplitudes within a predefined threshold range [25]. However, when the received signal power exceeds the linear operating region of the limiter, its nonlinear compression effect significantly disrupts the linear superposition characteristics of the signals. This distortion not only interferes with subsequent signal processing but also degrades the effectiveness of traditional blind source separation (BSS) methods, such as FastICA.  
To address the nonlinear distortion introduced by the limiter, this paper employs a post-nonlinear (PNL) mixing model [26] to characterize the signal mixing process (Figure 2). This model divides the signal propagation into two stages: first, source signals undergo linear mixing via a mixing matrix; subsequently, the mixed signals pass through a nonlinear function simulating the limiter's distortion effect. Unlike conventional pre-nonlinear models, the PNL model aligns more closely with the physical characteristics of satellite communication systems, where the limiter's nonlinearity acts after the linear mixing stage. This modeling approach accurately captures the signal distortion mechanism under amplitude compression conditions.  
Consider two source signals \(\mathbf{s}(t)=[s_{1}(t),s_{2}(t)]^{T}\), which are mixed via a linear mixing matrix (i.e., the channel transmission matrix) \(\mathbf{H}\in\mathbb{R}^{2\times 2}\), generating the antenna-received signal \(\mathbf{r}(t)\):  
\[\mathbf{r}(t)=\mathbf{Hs}(t)+\mathbf{n}(t), \tag{1}\]  
where \(s_{1}(t)\) and \(s_{2}(t)\) represent the communication signal and interference signal, respectively, and \(\mathbf{n}(t)\) denotes the additive noise at the receiving antenna.  
The mixed signal then undergoes a nonlinear transformation induced by the device's nonlinear distortion characteristics (e.g., the limiter), yielding the observed signal \(\mathbf{x}(t)\):  
\[\mathbf{x}(t)=\mathbf{f}(\mathbf{r}(t)), \tag{2}\]  
Here, the function \(\mathbf{f}(\cdot)\) models the nonlinear distortion introduced by the limiter. In this work, the hyperbolic tangent function \(\tanh(\cdot)\) is selected as the nonlinear approximation function:  
\[\mathbf{f}(\mathbf{x})=\tanh\mathbf{h}(\mathbf{x}), \tag{3}\]  
Figure 1: Structure of the uplink signal receiver in an anti-jamming satellite communication system.  
Figure 2: Post-nonlinear mixing model.  
This function exhibits smooth, continuous, and differentiable asymptotic saturation properties. It maintains linear gain in low-power regions while simulating the nonlinear response of a soft limiter through gradual amplitude compression in overload regions. Additionally, its characteristics effectively suppress out-of-band radiation and preserve signal envelope information, thereby satisfying the dual requirements of modern satellite communication systems for limiter-induced distortion mitigation and spectral efficiency optimization.  
The core principle of the proposed anti-jamming method based on nonlinear blind source separation lies in designing a nonlinear demixing system (Figure 3) to process the distorted signal \(\mathbf{x}(t)\), producing an estimated output \(\mathbf{z}(t)\) of the source signal:  
\[\mathbf{z}(t)=\mathbf{g}(\mathbf{x}(t)), \tag{4}\]  
The anti-jamming objective is to accurately estimate the communication signal. The block diagram of the nonlinear blind source separation signal processing system is depicted in Figure 3.