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 [46,30,78].
The problem of fitting a digital filter to a given spectrum may be formulated as follows:
Given a continuous complex function , corresponding to a causalI.2 desired frequency-response, find a stable digital filter of the form
with given, such that some norm of the error
is minimum with respect to the filter coefficients
which are constrained to lie in a subset , where . When explicitly stated, the filter coefficients may be complex, in which case .
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.