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 $H(\ejo ),\,-\pi < \omega \leq \pi$ , corresponding to a causal^I.4 desired frequency-response, find a stable digital filter of the form

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

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

$$J(\hat{\theta}) \isdef \left\Vert\,H(\ejo ) - \hat{H}(\ejo )\,\right\Vert$$

is minimum with respect to the filter coefficients

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

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 $\hat{\Theta}\subset\mathbb{C}^{N}$ .

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.