A desirable property of the error norm minimized by a filter-design
technique is concavity of the
error norm with respect to the filter coefficients. When this holds,
the error surface ``looks like a bowl,'' and the *global minimum*
can be found by iteratively moving the parameters in the ``downhill''
(negative gradient) direction. The advantages of concavity are
evident from the following classical results.

**Theorem. ** If is a vector space, a concave subset
of , and a concave functional on , then any local
minimizer of is a global minimizer of in .

**Theorem. **If is a normed linear space, a concave subset of , and a *strictly* concave functional on , then has
*at most* one minimizer in .

**Theorem. ** Let be a closed and bounded subset of .
If
is *continuous* on , then has
*at least* one minimizer in .

Theorem (2.1) bears directly on the existence of a solution to the general filter design problem in the frequency domain. Replacing ``closed and bounded'' with ``compact'', it becomes true for a functional on an arbitrary metric space (Rudin [6], Thm. 14). (In , ``compact'' is equivalent to ``closed and bounded'' [5].) Theorem (2.1) implies only compactness of and continuity of the error norm on need to be shown to prove existence of a solution to the general frequency-domain filter design problem.

Download gradient.pdf

Download the whole thesis containing this material.

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

[Automatic-links disclaimer]