### 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
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
by

*delays*the desired impulse response,

*i.e.*, . 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
is
*noncausal*. Since there are no constraints on where the poles of
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 is *minimum phase*, *i.e.*, when all the zeros of are
inside the unit circle. For a given magnitude,
,
minimum phase gives the maximum concentration of impulse-response energy
near the time origin . Consequently, the impulse-response tends to start
large and decay immediately. For non-minimum phase , the
impulse-response may be small for the first 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 which is zero
for
, and arbitrary thereafter. Transforming into the
time domain yields

where ``'' denotes convolution, and the additive decomposition is due the fact that for . In this case the minimum occurs for ! 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 , 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 can usually be removed by a few iterations of the Steiglitz-McBride algorithm.

**Next Section:**

An FFT-Based Equation-Error Method

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Error Weighting and Frequency Warping