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 [48]. 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.4 desired
frequency-response, find a stable digital filter of the form
where
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
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.