### 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

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An FFT-Based Equation-Error Method

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Error Weighting and Frequency Warping