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

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Filter Norms

Open Live Script

Compute the 2-norm of a Butterworth IIR filter with tolerance $1 0^{- 10}$ . Specify a normalized cutoff frequency of $0.5 π$ 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 $0.2 π$ rad/s.

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

Linf = 1.0028

Algorithms

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

${‖ H (e^{j ω}) ‖}_{p} = {(\frac{1}{2 π} \int_{- π}^{π} {| H (e^{j ω}) |}^{p} d ω)}^{1 / p} .$

For the case p → ∞, the L_∞-norm is

${‖ H (e^{j ω}) ‖}_{\infty} = \max_{- π \leq ω \leq π} | H (e^{j ω}) | .$

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

${‖ H (e^{j ω}) ‖}_{2} = {(\frac{1}{2 π} {\int_{- π}^{π} | H (e^{j ω}) |}^{2} d ω)}^{1 / 2} = {(\sum_{n} {| h (n) |}^{2})}^{1 / 2},$

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.

Extended Capabilities

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

Version History

Introduced before R2006a