Simulation of FXLMS-based Active Noise Control

The FXLMS (Filtered-X Least Mean Square) algorithm serves as one of the core algorithms in Active Noise Control (ANC) systems. Through adaptive filtering technology, it dynamically adjusts filter coefficients to cancel environmental noise in real-time, widely applied in scenarios such as headphones and automotive cabin noise reduction.

### Core Algorithm Concept A typical FXLMS-ANC system comprises a reference microphone, error microphone, and secondary loudspeaker. The reference microphone captures noise signals, which are processed by an adaptive filter. The secondary loudspeaker then generates anti-phase sound waves to cancel the noise. The error microphone detects residual noise and feeds it back to the algorithm for filter parameter optimization. Code Implementation Insight: The algorithm typically initializes with a W-filter (adaptive filter) and S-filter (secondary path model). The core update equation W(n+1) = W(n) + μ * e(n) * X'(n) requires careful step-size (μ) selection and secondary path modeling for stability.

### Key Simulation Aspects Offline Simulation: Utilizes pre-recorded noise signals for algorithm validation and parameter tuning. MATLAB implementation often involves loading audio data, designing filter structures, and iterating through sample-by-sample processing. Online Simulation: Processes noise signals in real-time, closely resembling practical applications. Requires consideration of computational latency and system stability, often implemented using real-time audio processing frameworks. Secondary Path Modeling: The transfer function from secondary loudspeaker to error microphone must be accurately estimated, typically through white noise excitation methods. System identification toolbox functions like `tfestimate()` can be employed for this modeling.

### Extended Considerations Convergence Speed vs. Steady-State Error: The step-size parameter requires balancing between convergence speed and steady-state error - larger values may cause oscillation while smaller values slow convergence. Multi-Channel Extension: Complex environments (e.g., automotive cabins) require multiple reference microphones and secondary loudspeakers for coordinated noise reduction, expanding to Multiple-Input Multiple-Output (MIMO) systems. Nonlinear Noise Handling: Traditional FXLMS targets linear noise; nonlinear scenarios can be addressed by integrating neural networks or nonlinear filter structures for enhanced adaptability.

Through combined offline and online simulation approaches, comprehensive validation of FXLMS-ANC system's noise reduction performance and real-time capabilities can be achieved, laying the foundation for hardware implementation.