Lyapunov tuning of the leaky LMS algorithm for single-source, single-point noise cancellation

LMS algorithms have performance issues related to insufficient excitation, nonstationary reference inputs, finite-precision arithmetic, quantization noise, and measurement noise. Such factors cause weight drift and potential instability in the conventional LMS algorithm. Here, we analyze the stability and performance of the leaky LMS algorithm, which is widely used to correct weight drift. A Lyapunov tuning method is developed to find an adaptive leakage parameter and step size that provide optimum performance and retain stability in the presence of measurement noise on the reference input. The method accounts for nonpersistent excitation conditions and nonstationary reference inputs and requires no a priori knowledge of the reference input signal other than a lower bound on its magnitude or a minimum signal-to-noise ratio. The tuning method is demonstrated for three candidate adaptive leaky LMS algorithms. Stability and performance tradeoffs of each candidate Lyapunov tuned algorithm are evaluated experimentally in a single source, single-point acoustic noise cancellation system.