filternorm

2-norm or infinity-norm of digital filter

Syntax

L = filternorm(b,a)

L = filternorm(___,pnorm)

L = filternorm(___,tol)

Description

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

L = filternorm(___,pnorm) specifies the type of filter norm to compute for the digital filter.

L = filternorm(___,tol) specifies the tolerance allowed in the filter norm computation.

Examples

Compute Filter Norms

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.

[b1,a1] = butter(5,0.5);
L2 = filternorm(b1,a1,2,1e-10)

L2 = 
0.7071

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

b2 = firpm(30,[.1 .9],[1 1],"Hilbert");
Linf = filternorm(b2,1,inf)

Linf = 
1.0028

Input Arguments

b — Filter numerator coefficients

real-valued vector

Filter numerator coefficients, specified as a real-valued vector.

Data Types: double
Complex Number Support: Yes

a — Filter denominator coefficients

real-valued vector

Filter denominator coefficients, specified as a real-valued vector.

Data Types: double
Complex Number Support: Yes

pnorm — Filter norm type

2 (default) | inf

Type of filter norm to compute for the digital filter, specified as one of these:

• 2 — The function computes the 2-norm of the filter.

• inf — The function computes the infinity-norm of the filter.

Data Types: double

tol — Computation tolerance

1e-8 (default) | positive scalar

Tolerance allowed for filter norm estimation, specified as a positive scalar.

Note

You can specify the tol argument only for IIR 2-norm computations, which only applies if you set pnorm to 2.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

L — Filter norm

real positive scalar

Filter norm, returned as a real positive scalar.

Tips

• 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.

Algorithms

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

$${\Vert H(e^{j\omega })\Vert }_p={\left(\frac{1}{2\pi }{\displaystyle {\int }_{-\pi }^{\pi }{\left|H(e^{j\omega })\right|}^p}d\omega \right)}^{1/p}.$$

For the case p → ∞, the L_∞-norm is

$${\Vert H(e^{j\omega })\Vert }_{\infty }=\underset{-\pi \le \omega \le \pi }{\max }\left|H(e^{j\omega })\right|.$$

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

$${\Vert H(e^{j\omega })\Vert }_2={\left(\frac{1}{2\pi }{{\displaystyle {\int }_{-\pi }^{\pi }\left|H(e^{j\omega })\right|}}^{2}d\omega \right)}^{1/2}={\left({\displaystyle \sum _n{\left|h(n)\right|}^2}\right)}^{1/2},$$

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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced before R2006a