Analysis and fast RLS algorithms of quadratic Volterra ADF

It is shown that adaptive training of quadratic Volterra filters is an ill-conditioned problem, or the error surfaces of the adaptive filters (ADF) are always extremely steep in one particular direction but relatively flat in the rest of the directions. This result is a generalization of a previous report on the special case of when the inputs are delayed values of a single time series of Gaussian distribution. A complete analysis of the correlation matrix of inputs as multiple time series are also obtained for the unrelated case. This paper then presents a fast RLS algorithm for Gaussian input signals costing only O(N/sup 2/) multiplications where N is the number of linear terms in the filter input, the same order as the LMS algorithm, while the RLS algorithm for Volterra ADF costs O(N/sup 5/) multiplications per sample. Simulations shown that this algorithm works well also in non-Gaussian input cases.