Leaky FXLMS Algorithm: Enhanced Stability for Active Noise Control Systems

The Leaky FXLMS algorithm represents an enhanced adaptive filtering approach in Active Noise Control (ANC) systems, building upon the conventional Filtered-X Least Mean Square (FXLMS) framework by incorporating a leakage factor to address potential system instability issues.

While traditional FXLMS algorithms adjust filter weights to minimize error signals, practical implementations may face numerical instability due to strong input signal correlation or unbounded weight growth. The Leaky FXLMS solution introduces a leakage term (typically a small positive coefficient λ) into the weight update equation, implementing controlled weight decay through a modified update rule: w(n+1) = (1-μλ)w(n) + μe(n)x'(n), where μ represents the step size, e(n) denotes the error signal, and x'(n) is the filtered reference signal.

The algorithm's core design principle involves balancing convergence rate against stability margins. Larger leakage factors (λ) enhance stability at the potential cost of reduced convergence speed, while smaller λ values approach standard FXLMS behavior with increased stability risks. This implementation finds particular relevance in real-time systems requiring robust performance, such as headphone active noise cancellation and pipeline noise elimination applications.

Compared to basic FXLMS, the Leaky variant demonstrates superior performance when handling non-stationary signals or scenarios with modeling inaccuracies. However, optimal performance requires empirical tuning of the λ parameter through systematic experimentation. This foundational improvement has subsequently inspired derivative algorithms including momentum-enhanced Leaky FxLMS implementations that incorporate historical gradient information for accelerated convergence.