Algorithm for Separating Original Signals from Mixed Signals

In the field of signal processing, recovering original signals from mixed observations is a classic problem known as Blind Source Separation (BSS). Independent Component Analysis (ICA) serves as one of the core techniques for solving such problems, with FastICA being its efficient implementation.

### Core Concept ICA relies on a key assumption: the original components within mixed signals are statistically independent. By maximizing the non-Gaussianity of these components—using measures like kurtosis or negentropy—the algorithm can reverse the mixing process to extract source signals. Due to inherent uncertainties in the problem (such as ambiguity in order and scaling), results often require post-processing adjustments.

### Advantages of FastICA Efficiency: Utilizes fixed-point iteration optimization, achieving faster convergence compared to traditional gradient descent methods. Flexibility: Supports multiple non-Gaussianity measures, allowing adaptation to different signal characteristics. Parameter-free: Unlike algorithms that depend on pre-defined mixing models, FastICA imposes no strong assumptions on the mixing process.

### Implementation Challenges Despite theoretical maturity, practical applications require attention to the following: Sensor noise may violate the independence assumption. When the number of signals exceeds available sensors (underdetermined problem), additional prior knowledge must be incorporated. Open-source implementations are limited, often necessitating custom adjustments using libraries like Python's `scikit-learn` or MATLAB toolboxes.

This algorithm performs remarkably well in EEG signal analysis and speech dereverberation, but parameter optimization remains essential for specific application scenarios.