Analysis of LMS algorithm with delayed coefficient adaptation for sinusoidal reference

In this paper, we analyse the stability of LMS algorithms with delayed coefficient adaptation, such as FxLMS and DLMS algorithms, when sinusoidal references are used. We demonstrate that in two common situations, that are two in-phase and quadrature sinusoidal components and a tapped delay line with a large number of weights, the adaptive system turns to be linear and time-invariant. Without the need of any independence assumption, stability is then analysed in those two situations and an upper bound for the adaptation step size is derived.