Stability of non-Wiener solutions of the filtered LMS algorithm

This paper analyzes the stability of Non-Wiener solutions to the Filtered LMS algorithm. The algorithm inputs are sinusoidal and the loop filters are modelled as pure delays. A pair of time-invariant coupled first order difference equations are derived which define the behavior of the time varying algorithm weights. The state-transition matrix for the time-invariant difference equations is studied using root-locus techniques. The stability of the Filtered LMS algorithm is determined as a function of the algorithm step size μ, the number of taps N and the reference signal power. The theory is applied to the algorithm design for active acoustic noise cancellation. A design procedure is presented which allows the theory to be extended to frequency dependent delays in the adaptation loop. Simulation results an in shown to be in close agreement with the theory