Partially decoupled Volterra filters: formulation and LMS adaptation

The adaptation of Volterra filters by one particular method-the method of least mean squares (LMS)-while easily implemented, is complicated by the fact that upper hounds for the values of step sizes employed by a parallel update LMS scheme are difficult to obtain. In this paper, we propose a modification of the Volterra filter in which the filter weights of a given order are optimized independently of those weights of higher order. Using this approach, we then solve the minimum mean square error (MMSE) filtering problem as a series of constrained optimization problems, which produce a partially decoupled normal equation for the Volterra filter. From this normal equation, we are able to develop an adaptation routine that uses the principles of partial decoupling that is similar in form to the Volterra LMS algorithm but with important structural differences that allow a straightforward derivation of bounds on the algorithm's step sizes; these bounds can be shown to depend on the respective diagonal blocks of the Volterra autocorrelation matrix. This produces a reliable set of design guidelines that allow more rapid convergence of the lower order weight sets.