Filter Design by Minimizing the

L2 Equation-Error Norm

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 causal^{I.2} 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 , 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.

- Equation Error Formulation
- Error Weighting and Frequency Warping
- Stability of Equation Error Designs
- An FFT-Based Equation-Error Method
- Prony's Method
- The Padé-Prony Method

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University