Theoretical analysis and fast RLS algorithms of quadratic Volterra adaptive filters

It is shown that the error surface of quadratic Volterra adaptive filters (ADF) is always extremely steep in one particular direction but relatively flat in the other directions. This nontrivial geometry explains the instability and unavoidable slow convergence of gradient adaptive algorithms. On the other hand, the RLS algorithm for Volterra ADF costs O(N⁵) multiplications where N is the number of linear terms in the filter input. The paper presents a new algorithm for Gaussian input signals which converges in the same rate as RLS but costs only O(N²) multiplications, the same order as the LMS algorithm. Simulations shown that this algorithm works well also in non-Gaussian input cases.