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Digital Filter Design Overview

This section (adapted from [432]), summarizes some of the more commonly used methods for digital filter design aimed at matching a nonparametric frequency response, such as typically obtained from input/output measurements. This problem should be distinguished from more classical problems with their own specialized methods, such as designing lowpass, highpass, and bandpass filters [346,365], or peak/shelf equalizers [561,452], and other utility filters designed from a priori mathematical specifications.

The problem of fitting a digital filter to a prescribed frequency response may be formulated as follows. To simplify, we set $ T=1$ .

Given a continuous complex function $ H(e^{j\omega}),\,-\pi < \omega \le \pi$ , corresponding to a causal desired frequency response,9.8 find a stable digital filter of the form

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

where
$\displaystyle {\hat B}(z)$ $\displaystyle \isdef$ $\displaystyle {\hat b}_0 + {\hat b}_1 z^{-1} + \cdots + {\hat b}_{\hat{N}_b}z^{-{\hat{N}_b}}$ (9.15)
$\displaystyle {\hat A}(z)$ $\displaystyle \isdef$ $\displaystyle 1 + {\hat a}_1 z^{-1} + \cdots + {\hat a}_{\hat{N}_a}z^{-{\hat{N}_a}} ,$ (9.16)

with $ {\hat{N}_b},{\hat{N}_a}$ given, such that some norm of the error

$\displaystyle J(\hat{\theta}) \isdefs \left\Vert\,H(e^{j\omega}) - {\hat H}(e^{j\omega})\,\right\Vert \protect$ (9.17)

is minimum with respect to the filter coefficients

$\displaystyle \hat{\theta}^K\isdefs [{\hat b}_0,{\hat b}_1,\ldots\,,{\hat b}_{\hat{N}_b},{\hat a}_1,{\hat a}_2,\ldots\,,{\hat a}_{\hat{N}_a}].
$

The filter coefficients are constrained to lie in some subset $ \hat{\Theta}\subset\Re ^{{\hat N}}$ , where $ {\hat N}\isdef {\hat{N}_a}+{\hat{N}_b}+1$ . The filter coefficients may also be complex, in which case $ \hat{\Theta}\subset{\bf C}^{{\hat N}}$ .

The approximate filter $ {\hat H}$ is typically constrained to be stable, and since $ {\hat B}(z)$ is causal (no positive powers of $ z$ ), stability implies causality. Consequently, the impulse response of the model $ {\hat h}(n)$ is zero for $ n<0$ .

The filter-design problem is then to find a (strictly) stable $ {\hat{N}_a}$ -pole, $ {\hat{N}_b}$ -zero digital filter which minimizes some norm of the error in the frequency-response. This is fundamentally rational approximation of a complex function of a real (frequency) variable, with constraints on the poles.

While the filter-design problem has been formulated quite naturally, it is difficult to solve in practice. The strict stability assumption yields a compact space of filter coefficients $ \hat{\Theta}$ , leading to the conclusion that a best approximation $ \hat{H}^\ast $ exists over this domain.9.9Unfortunately, the norm of the error $ J(\hat{\theta})$ typically is not a convex9.10function of the filter coefficients on $ \hat{\Theta}$ . This means that algorithms based on gradient descent may fail to find an optimum filter due to their premature termination at a suboptimal local minimum of $ J(\hat{\theta})$ .

Fortunately, there is at least one norm whose global minimization may be accomplished in a straightforward fashion without need for initial guesses or ad hoc modifications of the complex (phase-sensitive) IIR filter-design problem--the Hankel norm [156,432,178,36]. Hankel norm methods for digital filter design deliver a spontaneously stable filter of any desired order without imposing coefficient constraints in the algorithm.

An alternative to Hankel-norm approximation is to reformulate the problem, replacing Eq.$ \,$ (8.17) with a modified error criterion so that the resulting problem can be solved by linear least-squares or convex optimization techniques. Examples include

In addition to these modifications, sometimes it is necessary to reformulate the problem in order to achieve a different goal. For example, in some audio applications, it is desirable to minimize the log-magnitude frequency-response error. This is due to the way we hear spectral distortions in many circumstances. A technique which accomplishes this objective to the first order in the $ L-infinity$ norm is described in [432].

Sometimes the most important spectral structure is confined to an interval of the frequency domain. A question arises as to how this structure can be accurately modeled while obtaining a cruder fit elsewhere. The usual technique is a weighting function versus frequency. An alternative, however, is to frequency-warp the problem using a first-order conformal map. It turns out a first-order conformal map can be made to approximate very well frequency-resolution scales of human hearing such as the Bark scale or ERB scale [461]. Frequency-warping is especially valuable for providing an effective weighting function connection for filter-design methods, such as the Hankel-norm method, that intrinsically do not offer choice of a weighted norm for the frequency-response error.

There are several methods which produce $ {\hat H}(z){\hat H}(z^{-1})$ instead of $ {\hat H}(z)$ directly. A fast spectral factorization technique is useful in conjunction with methods of this category [432]. Roughly speaking, a size $ 2{\hat{N}_a}$ polynomial factorization is replaced by an FFT and a size $ {\hat{N}_a}$ system of linear equations.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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