Convergence analysis of a complex LMS algorithm with tonal reference signals

Often one encounters the presence of tonal noise in many active noise control applications. Such noise, usually generated by periodic noise sources like rotating machines, is cancelled by synthesizing the so-called antinoise by a set of adaptive filters which are trained to model the noise generation mechanism. Performance of such noise cancellation schemes depends on, among other things, the convergence characteristics of the adaptive algorithm deployed. In this paper, we consider a multireference complex least mean square (LMS) algorithm that can be used to train a set of adaptive filters to counter an arbitrary number of periodic noise sources. A deterministic convergence analysis of the multireference algorithm is carried out and necessary as well as sufficient conditions for convergence are derived by exploiting the properties of the input correlation matrix and a related product matrix. It is also shown that under convergence condition, the energy of each error sequence is independent of the tonal frequencies. An optimal step size for fastest convergence is then worked out by minimizing the error energy.