Computation of the response-error-gradient of linear discrete filters

Iterative time domain design of recursive or nonrecursive digital filters requires the minimization of an error criterion E. If E is a differentiable functional, the most effective methods (like steepest descent, Newton's method, or conjugate gradient method) use first derivatives with respect to the parameters of the filter. Considering an arbitrary filter structure, it is shown in this paper that the gradient of any differentiable error criterion can be computed in two steps. The first step depends only on the choice of E, whereas the second is based exclusively on the knowledge of the filter structure. We show that the gradient computer is also a linear filter-of order less than (N + 1)
²
and of similar structure-and that it requires no additional coefficient memory, a fact allowing a simple hardware or software implementation.