2-norm or infinity-norm of digital filter
L = filternorm(b,a)
L = filternorm(b,a,pnorm)
L = filternorm(b,a,2,tol)
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
b and denominator coefficients in
L = filternorm(b,a,pnorm) computes
the 2- or infinity-norm (inf-norm) of the digital filter, where
either 2 or
L = filternorm(b,a,2,tol) computes
the 2-norm of an IIR filter with the specified tolerance,
The tolerance can be specified only for IIR 2-norm computations.
this case must be 2. If
tol is not specified, it
defaults to 10–8.
Compute the 2-norm of a Butterworth IIR filter with tolerance . Specify a normalized cutoff frequency of 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 rad/s.
b = firpm(30,[.1 .9],[1 1],'Hilbert'); Linf = filternorm(b,1,inf)
Linf = 1.0028
Given a filter with frequency response H(ejω), the Lp-norm for 1 ≤ p < ∞ is given by
For the case p → ∞, the L∞-norm is
For the case p = 2, Parseval's theorem states that
where h(n) is the impulse response of the filter. The energy of the impulse response is the squared L2-norm.
 Jackson, L. B. Digital Filters and Signal Processing: with MATLAB Exercises. 3rd Ed. Hingham, MA: Kluwer Academic Publishers, 1996, Chapter 11.