An Alternative Approach to Obtain a New Gain in Step-Size of LMS Filters Dealing with Periodic Signals

Abstract

Partial updates (PU) of adaptive filters have been successfully applied in different contexts to lower the computational costs of many control systems. In a PU adaptive algorithm, only a fraction of the coefficients is updated per iteration. Particularly, this idea has been proved as a valid strategy in the active control of periodic noise consisting of a sum of harmonics. The convergence analysis carried out here is based on the periodic nature of the input signal, which makes it possible to formulate the adaptive process with a matrix-based approach, the periodic least-mean-square (P-LMS) algorithm In this paper, we obtain the upper bound that limits the step-size parameter of the sequential PU P-LMS algorithm and compare it to the bound of the full-update P-LMS algorithm. Thus, the limiting value for the step-size parameter is expressed in terms of the step-size gain of the PU algorithm. This gain in step-size is the quotient between the upper bounds ensuring convergence in the following two scenarios: first, when PU are carried out and, second, when every coefficient is updated during every cycle. This step-size gain gives the factor by which the step-size can be multiplied so as to compensate for the convergence speed reduction of the sequential PU algorithm, which is an inherently slower strategy. Results are compared with previous results based on the standard sequential PU LMS formulation. Frequency-dependent notches in the step-size gain are not present with the matrix-based formulation of the P-LMS. Simulated results confirm the expected behavior.