17. Concave Norms

GUIDE: Elementary Digital Filter Theory - Julius O. Smith III. Concave Norms

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Concave Norms

A desirable property of the error norm minimized by a filter-designtechnique 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 minimumcan 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 $X$ is a vector space, $S$ a concave subset of $X$, and $f$ a concave functional on $S$, then any local minimizer of $f$ is a global minimizer of $f$ in $S$.

Theorem. If $X$ is a normed linear space, $S$ a concave subset of $X$, and $f$ a strictly concave functional on $S$, then $f$ hasat most one minimizer in $S$.

Theorem. Let $S$ be a closed and bounded subset of $\. If $f:\ is continuous on $S$, then $f$ hasat least one minimizer in $S$.

Theorem (4.1) 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 [Rudin 1964], Thm. 14). (In $\, ``compact'' is equivalent to ``closed and bounded'' [Royden 1968].) Theorem (4.1) implies only compactness of ${\ and continuity of the error norm $J({\on ${\ need to be shown to prove existence of a solution to the general frequency-domain filter design problem.

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