15. Specific Norms

GUIDE: Elementary Digital Filter Theory. Specific Norms

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Specific Norms

The $Lp$ norms are defined on the space $Lp$ by


$Lp$ norms are technically pseudo-norms; if functions in $Lp$ 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 $Lp$ norm.

The weighted $Lp$ norms are defined by


where $W(e^{j\ is real, positive, and integrable. Typically, $\. If $W(e^{j\ for a set of nonzero measure, then a pseudo-norm results.

The case $p=2$ gives the popular root mean square norm, and $ \ can be interpreted as the total energy of$F$ in many physical contexts.

An advantage of working in $L2$ is that the norm is provided by an inner product,


The norm of a vector $H\ is then given by


As $p$ 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 $L^1$ norm of $F(e^{j\ is finite, and let


denote the Fourier coefficients of $F(e^{j\. When $F(e^{j\ is afilter frequency response, $f(n)$ is the corresponding impulseresponse. The filter $F$ is said to be causal if $f(n)=0$ for$n<0$.

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 $lp$ norms.

The $Lp$ and $lp$ norms are strictly concave functionals for$1<p<\ (see below).

By Parseval's theorem, we have $ \, i.e., the $Lp$and $lp$ norms are the same for $p=2$.

The Frobenious norm of an $m\ matrix $A$ is defined as


That is, the Frobenious norm is the $L2$ norm applied to the elements of the matrix. For this norm there exists the following.

Theorem. The unique $m\ rank $k$ matrix $B$ which minimizes $ \ is given by $U\, where $A=U\is a singular value decomposition of $A$, and $\ is formed from $\ by setting to zero all but the $k$ largest singular values.

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

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


For the $L2$ norm, we have


and this is called the spectral norm of the matrix $A$.

The Hankel matrix corresponding to a time series $f$ 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 $H(z)\, i.e., if it is causal. For noncausal filters, it is a pseudo-norm.

If $F$ 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 $L2$ norm, and bounded above by the $L-infinity$ norm [Genin 1981],


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

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