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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 $ H(e^{j\omega}),\,-\pi < \omega \leq \pi$ , corresponding to a causalI.2 desired frequency-response, find a stable digital filter of the form

$\displaystyle \hat{H}(z) \isdef \frac{\hat{B}(z)}{\hat{A}(z)},
$

where

\begin{eqnarray*}
\hat{B}(z) &\isdef \hat{b}_0 + \hat{b}_1 z^{-1} + \cdots + \hat{b}_{{n}_b}z^{-{{n}_b}} \\
\hat{A}(z) &\isdef 1 + \hat{a}_1 z^{-1} + \cdots + \hat{a}_{{n}_a}z^{-{{n}_a}} ,\\
\end{eqnarray*}

with $ {{n}_b},{{n}_a}$ given, such that some norm of the error

$\displaystyle J(\hat{\theta}) \isdef \left\Vert\,H(e^{j\omega}) - \hat{H}(e^{j\omega})\,\right\Vert
$

is minimum with respect to the filter coefficients

$\displaystyle \hat{\theta}^T\isdef \left[\hat{b}_0,\hat{b}_1,\ldots\,,\hat{b}_{{n}_b},\hat{a}_1,\hat{a}_2,\ldots\,,\hat{a}_{{n}_a}\right]^T,
$

which are constrained to lie in a subset $ \hat{\Theta}\subset\Re ^{N}$ , where $ N\isdef {{n}_a}+{{n}_b}+1$ . When explicitly stated, the filter coefficients may be complex, in which case $ \hat{\Theta}\subset{\bf C}^{N}$ .

The approximate filter $ \hat{H}$ is typically constrained to be stable, and since positive powers of $ z$ do not appear in $ \hat{B}(z)$ , stability implies causality. Consequently, the impulse response of the filter $ \hat{h}(n)$ is zero for $ n < 0$ . If $ H$ were noncausal, all impulse-response components $ h(n)$ for $ n < 0$ would be approximated by zero.



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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA