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### Equation Error Formulation

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

 (I.11)

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.

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.

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