Step Size Bound of the Sequential Partial Update LMS Algorithm with Periodic Input Signals

Abstract

This paper derives an upper bound for the step size of the sequential partial update (PU) LMS adaptive algorithm when the input signal is a periodic reference consisting of several harmonics. The maximum step size is expressed in terms of the gain in step size of the PU algorithm, defined as the ratio between the upper bounds that ensure convergence in the following two cases: firstly, when only a subset of the weights of the filter is updated during every iteration; and secondly, when the whole filter is updated at every cycle. Thus, this gain in step-size determines the factor by which the step size parameter can be increased in order to compensate the inherently slower convergence rate of the sequential PU adaptive algorithm. The theoretical analysis of the strategy developed in this paper excludes the use of certain frequencies corresponding to notches that appear in the gain in step size. This strategy has been successfully applied in the active control of periodic disturbances consisting of several harmonics, so as to reduce the computational complexity of the control system without either slowing down the convergence rate or increasing the residual error. Simulated and experimental results confirm the expected behavior.