One of the simplest formulations of recursive digital filter design is based on minimizing the equation error. This method allows matching of both spectral phase and magnitude. Equation-error methods can be classified as variations of Prony's method . Equation error minimization is used very often in the field of system identification [46,31,75].
The problem of fitting a digital filter to a given spectrum may be formulated as follows:
Given a continuous complex function , corresponding to a causalG.2 desired frequency-response, find a stable digital filter of the form
with given, such that some norm of the error
The approximate filter is typically constrained to be stable, and since positive powers of do not appear in , stability implies causality. Consequently, the impulse response of the filter is zero for . If were noncausal, all impulse-response components for would be approximated by zero.