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 causal ^I.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

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

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 do not appear in $\hat{B}(z)$ , stability implies causality. Consequently, the impulse response of the filter $\hat{h}(n)$ is zero for . If were noncausal, all impulse-response components for would be approximated by zero.

Subsections

Order a Hardcopy of Introduction to Digital Filters

Previous: Examples
Next: Equation Error Formulation

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

Sign in

Search Online Books

Free Online Books

Chapters

Introduction to Digital Filters
Recursive Digital Filter Design
Filter Design by Minimizing the L2 Equation-Error Norm