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Stability of Equation Error Designs

A problem with equation-error methods is that stability of the filter design is not guaranteed. When an unstable design is encountered, one common remedy is to reflect unstable poles inside the unit circle, leaving the magnitude response unchanged while modifying the phase of the approximation in an ad hoc manner. This requires polynomial factorization of $ \hat{A}(z)$ to find the filter poles, which is typically more work than the filter design itself.

A better way to address the instability problem is to repeat the filter design employing a bulk delay. This amounts to replacing $ H(\ejo )$ by

$\displaystyle H_\tau(\ejo ) \isdef e^{-\omega \tau} H(\ejo ),\quad\tau>0,

and minimizing $ \vert\vert\,\hat{A}(\ejo )H_\tau(\ejo ) - \hat{B}(\ejo )\,\vert\vert _2$ . This effectively delays the desired impulse response, i.e., $ h_\tau(n)=h(n-\tau)$ . As the bulk delay is increased, the likelihood of obtaining an unstable design decreases, for reasons discussed in the next paragraph.

Unstable equation-error designs are especially likely when $ H(\ejo )$ is noncausal. Since there are no constraints on where the poles of $ \hat{H}$ can be, one can expect unstable designs for desired frequency-response functions having a linear phase trend with positive slope.

In the other direction, experience has shown that best results are obtained when $ H(z)$ is minimum phase, i.e., when all the zeros of $ H(z)$ are inside the unit circle. For a given magnitude, $ \vert H(\ejo )\vert$ , minimum phase gives the maximum concentration of impulse-response energy near the time origin $ n = 0$ . Consequently, the impulse-response tends to start large and decay immediately. For non-minimum phase $ H$ , the impulse-response $ h(n)$ may be small for the first $ {{n}_b}+1$ samples, and the equation error method can yield very poor filters in these cases. To see why this is so, consider a desired impulse-response $ h(n)$ which is zero for $ n\leq{{n}_b}$ , and arbitrary thereafter. Transforming $ J_E^2$ into the time domain yields

J_E^2(\hat{\theta}) &=& \left\Vert\,\hat{a}\ast h(n) - \hat{b}(n)\,\right\Vert^2_2 \\
&=& \sum_{n=0}^\infty\left(\hat{a}\ast h(n) - \hat{b}(n)\right)^2 \\
&=& \sum_{n=0}^{{n}_b}\hat{b}_n^2 + \sum_{n={{n}_b}+1}^\infty
\left(\hat{a}\ast h(n)\right)^2,\\

where ``$ \ast $ '' denotes convolution, and the additive decomposition is due the fact that $ \hat{a}\ast h(n)=0$ for $ n\leq{{n}_b}$ . In this case the minimum occurs for $ \hat{B}(z)=0\,\,\Rightarrow\,\,\hat{H}(z)\equiv 0$ ! Clearly this is not a particularly good fit. Thus, the introduction of bulk-delay to guard against unstable designs is limited by this phenomenon.

It should be emphasized that for minimum-phase $ H(\ejo )$ , equation-error methods are very effective. It is simple to convert a desired magnitude response into a minimum-phase frequency-response by use of cepstral techniques [22,60] (see also the appendix below), and this is highly recommended when minimizing equation error. Finally, the error weighting by $ \vert\hat{A}(e^{j\omega_k})\vert$ can usually be removed by a few iterations of the Steiglitz-McBride algorithm.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-04-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University