Novel FxLMS convergence condition with deterministic reference

A novel analysis of FxLMS convergence when the reference signal is deterministic is presented in this paper. The simple case of a sinusoidal reference is considered first, to be later extended to any combination of multiple sinusoids. In both cases, we derive an upper bound for the algorithm step size which ensures convergence. In the derivation of this result there is no need of any of the usual approximations, such as independence between reference and weights or slow convergence, which are not suitable for deterministic references. Instead, we consider the common cases where the adaptive system shows linear time-invariant behavior. The upper bound obtained for the step size is in good agreement with empirical measurements.