Documentation

This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English verison of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

filternorm

2-norm or infinity-norm of digital filter

Syntax

L = filternorm(b,a)
L = filternorm(b,a,pnorm)
L = filternorm(b,a,2,tol)

Description

A typical use for filter norms is in digital filter scaling to reduce quantization effects. Scaling often improves the signal-to-noise ratio of the filter without resulting in data overflow. You also can use the 2-norm to compute the energy of the impulse response of a filter.

L = filternorm(b,a) computes the 2-norm of the digital filter defined by the numerator coefficients in b and denominator coefficients in a.

L = filternorm(b,a,pnorm) computes the 2- or infinity-norm (inf-norm) of the digital filter, where pnorm is either 2 or inf.

L = filternorm(b,a,2,tol) computes the 2-norm of an IIR filter with the specified tolerance, tol. The tolerance can be specified only for IIR 2-norm computations. pnorm in this case must be 2. If tol is not specified, it defaults to 10–8.

Examples

collapse all

Compute the 2-norm of a Butterworth IIR filter with tolerance $10^{-10}$. Specify a normalized cutoff frequency of $0.5\pi$ rad/s and a filter order of 5.

[b,a] = butter(5,0.5);
L2 = filternorm(b,a,2,1e-10)
L2 =

    0.7071

Compute the infinity-norm of an FIR Hilbert transformer of order 30 and normalized transition width $0.2\pi$ rad/s.

b = firpm(30,[.1 .9],[1 1],'Hilbert');
Linf = filternorm(b,1,inf)
Linf =

    1.0028

More About

collapse all

Algorithms

Given a filter with frequency response H(e), the Lp-norm for 1 ≤ p < ∞ is given by

H(ejω)p=(12πππ|H(ejω)|pdω)1/p.

For the case p → ∞, the L-norm is

H(ejω)=maxπωπ|H(ejω)|.

For the case p = 2, Parseval's theorem states that

H(ejω)2=(12πππ|H(ejω)|2dω)1/2=(n|h(n)|2)1/2,

where h(n) is the impulse response of the filter. The energy of the impulse response is the squared L2-norm.

References

[1] Jackson, L. B. Digital Filters and Signal Processing: with MATLAB Exercises. 3rd Ed. Hingham, MA: Kluwer Academic Publishers, 1996, Chapter 11.

See Also

|

Introduced before R2006a

Was this topic helpful?