Fast RLS adaptive algorithms of quadratic Volterra ADF

It is shown in [6] that quadratic Volterra adaptive filters (ADF) have error surfaces which are always extremely steep on only one particular direction but relatively flat on the other directions. This explains the instability in the learning processing of Volterra ADF and implies unavoidable slow convergence of traditional gradient adaptive algorithms. On the other hand, the RLS algorithm for Volterra ADF costs O(N⁴) multiplications where N is the number of linear terms in Volterra ADF. This paper shows a new algorithm for Gaussian input signals which converges in the same rate as RLS but costs only O(N²) multiplications which is the same as the LMS algorithm. This algorithm is based on a complete analysis of the intrinsic geometry of the error surface for white input signals and whitening operation of any colored input signals. Simulations shown that this algorithm works well even in non-Gaussian input cases.