15. Specific Norms

GUIDE: Elementary Digital Filter Theory - Julius O. Smith III. Specific 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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III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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Specific Norms

The norms are defined on the space by

norms are technically pseudo-norms; if functions in are replaced by equivalence classes containing all functions equal almost everywhere, then a norm is obtained.

Since all practical desired frequency responses arising in digital filter design problems are bounded on the unit circle, it follows that forms a Banach space under any norm.

The weighted norms are defined by

where is real, positive, and integrable. Typically, . If for a set of nonzero measure, then a pseudo-norm results.

The case gives the popular root mean square norm, and can be interpreted as the total energy of in many physical contexts.

An advantage of working in is that the norm is provided by an inner product,

The norm of a vector is then given by

As approaches infinity in Eq. (3.1), the error measure is dominated by the largest values of . Accordingly, it is customary to define

and this is often called the Chebyshev oruniform norm.

Suppose the norm of is finite, and let

denote the Fourier coefficients of . When is afilter frequency response, is the corresponding impulseresponse. The filter is said to be causal if for.

The norms for impulse response sequences are defined in a manner exactly analogous with the frequency response norms , viz.,

These time-domain norms are called norms.

The and norms are strictly concave functionals for (see below).

By Parseval's theorem, we have , i.e., the and norms are the same for .

The Frobenious norm of an matrix is defined as

That is, the Frobenious norm is the norm applied to the elements of the matrix. For this norm there exists the following.

Theorem. The unique rank matrix which minimizes is given by , where is a singular value decomposition of , and is formed from by setting to zero all but the largest singular values.

Proof. See Golub and Kahan [Golub and Van Loan 1989].

The induced norm of a matrix is defined in terms of the norm defined for the vectors on which it operates,

For the norm, we have

and this is called the spectral norm of the matrix .

The Hankel matrix corresponding to a time series is defined by , i.e.,

Note that the Hankel matrix involves only causal components of the time series.

The Hankel norm of a filter frequency response is defined as the spectral norm of the Hankel matrix of its impulse response,

The Hankel norm is truly a norm only if , i.e., if it is causal. For noncausal filters, it is a pseudo-norm.

If is strictly stable, then is finite for all, and all norms defined thus far are finite. Also, the Hankel matrix is a bounded linear operator in this case.

The Hankel norm is bounded below by the norm, and bounded above by the norm [Genin 1981],

with equality iff is an allpass filter (i.e., constant).

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