Analysis on error surface and fast algorithms of multichannel quadratic Volterra adaptive filters

This paper presents a theoretical analysis on multichannel quadratic Volterra adaptive filters (ADF). It is shown that adaptive training of these filters is an ill-conditioned problem, or the error surfaces are always extremely steep in one particular direction but relatively flat in the rest directions for Gaussian inputs. This result generalizes previous reports in the case of single channel or transversal filters. A complete analysis on eigen-structure of correlation matrix of the Gaussian inputs is also explicitly obtained for the uncorrelated case. A fast Newton-Raphson algorithm is shown for Gaussian input signals costing O(N²) multiplications where N is the number of linear terms in the filter input, the same cost as the NLMS algorithm, while the RLS algorithm for Volterra ADF costs O(N⁵) multiplications per sample. Simulations shown that this algorithm works well also in non-Gaussian input cases.