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 [46,31,75].

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^G.2 desired frequency-response, find a stable digital filter of the form

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

where

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

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

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

is minimum with respect to the filter coefficients

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

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.