The equation error is defined (in the frequency domain) as
By comparison, the more natural frequency-domain error is the so-called output error:
The names of these errors make the most sense in the time domain. Let and denote the filter input and output, respectively, at time . Then the equation error is the error in the difference equation:
while the output error is the difference between the ideal and approximate filter outputs:
Denote the norm of the equation error by
Note that (I.11) can be expressed as
Thus, the equation-error can be interpreted as a weighted output error in which the frequency weighting function on the unit circle is given by . Thus, the weighting function is determined by the filter poles, and the error is weighted less near the poles. Since the poles of a good filter-design tend toward regions of high spectral energy, or toward ``irregularities'' in the spectrum, it is evident that the equation-error criterion assigns less importance to the most prominent or structured spectral regions. On the other hand, far away from the roots of , good fits to both phase and magnitude can be expected. The weighting effect can be eliminated through use of the Steiglitz-McBride algorithm [45,78] which iteratively solves the weighted equation-error solution, using the canceling weight function from the previous iteration. When it converges (which is typical in practice), it must converge to the output error minimizer.