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:

where
is the
vector of unknown filter coefficients. Then the problem is to minimize
this norm with respect to
. What makes the equation-error so easy to
minimize is that it is linear in the parameters. In the time-domain
form, it is clear that the equation error is linear in the unknowns
. When the error is linear in the parameters, the sum of
squared errors is a quadratic form which can be minimized using one
iteration of Newton's method. In other words, minimizing the norm of
any error which is linear in the parameters results in a set of linear
equations to solve. In the case of the equation-error minimization at
hand, we will obtain
linear equations in as many unknowns.

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
[45,75] 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.