Recursive digital filter synthesis via gradient based algorithms

The three gradient-based algorithms of 1) steepest descent, 2) Newton´s method, and 3) the linearization algorithm are applied to the problem of synthesizing linear recursive filters in the time domain. It is shown that each of these algorithms requires knowledge of the associated recursive filter´s first-order sensitivity vectors, and, in the case of the Newton method, second-order sensitivity vectors as well. Systematic procedures for generating these sensitivity vectors by computing the response of a companion filter structure are then presented. Using the ideal low-pass filter as a design objective, it is then demonstrated that the linearization algorithm is particularly well suited for recursive filter design. On the other hand, the steepest descent and Newton methods are found to work rather poorly for this class of problems. Reasons for these empirical observed results are postulated.