Partially decoupled Volterra filters: formulation and LMS adaptation

The adaptation of Volterra lters by one particular method, the method of least mean squares (LMS), while easily implemented, is complicated by the fact that upper bounds for the values of step sizes employed by a parallel update LMS scheme are diicult to obtain. In this paper, we propose a modiication of the Volterra lter in which the lter weights of a given order are optimized independently of those weights of higher order. Using this approach, we then solve the MMSE ltering problem as a series of constrained optimization problems , which produce a partially decoupled normal equation for the Volterra lter. From this normal equation, we are able to develop an adaptation routine which uses the principles of partial decoupling which is similar in form to the Volterra LMS algorithm, but with important structural diierences 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 au-tocorrelation matrix. This produces a reliable set of design guidelines which allow more rapid convergence of the lower-order weight sets.