On the BIBO stability condition of adaptive recursive FLANN filters with application to nonlinear active noise control

In this paper, a bounded-input bounded-output (BIBO) stability condition for the recursive functional link artificial neural network (FLANN) filter, based on trigonometric expansions, is derived. This filter is considered as a member of the class of causal shift-invariant recursive nonlinear filters whose output depends linearly on the filter coefficients. As for all recursive filters, its stability should be granted or, at least, tested. The relevant conclusion we derive from the stability condition is that the recursive FLANN filter is not affected by instabilities whenever the recursive linear part of the filter is stable. This fact is in contrast with the case of recursive polynomial filters where, in general, specific limitations on the input range are required. The recursive FLANN filter is then studied in the framework of a feedforward scheme for nonlinear active noise control. The novelty of our study is due to the simultaneous consideration of a nonlinear secondary path and an acoustical feedback between the loudspeaker and the reference microphone. An output error nonlinearly Filtered-U normalized LMS adaptation algorithm, derived for the elements of the above-mentioned class of nonlinear filters, is then applied to the recursive FLANN filter. Computer simulations show that the recursive FLANN filter, in contrast to other filters, is able to simultaneously deal with the acoustical feedback and the nonlinearity in the secondary path.