# grpdelay

Average filter delay (group delay)

## Syntax

[gd,w] = grpdelay(b,a,n)
[gd,w] = grpdelay(B,A,"ctf",n)
[gd,w] = grpdelay({B,A,g},"ctf",n)
[gd,w] = grpdelay(d,n)
[gd,w] = grpdelay(sos,n)
[gd,w] = grpdelay(___,"whole")
[gd,f] = grpdelay(___,n,fs)
[gd,f] = grpdelay(___,n,"whole",fs)
gd = grpdelay(___,win)
gd = grpdelay(___,fin,fs)
grpdelay(___)

## Description

[gd,w] = grpdelay(b,a,n) returns the group delay response of the specified digital filter. Specify a digital filter with numerator coefficients b and denominator coefficients a. The function returns the n-point group delay response vector in gd and the corresponding angular frequency vector w.
[gd,w] = grpdelay(B,A,"ctf",n) returns the n-point group delay response of the digital filter represented as Cascaded Transfer Functions (CTF) with numerator coefficients B and denominator coefficients A. (since R2024b)

example

[gd,w] = grpdelay({B,A,g},"ctf",n) returns the n-point group delay response of the digital filter in CTF format. Specify the filter with numerator coefficients B, denominator coefficients A, and scaling values g across filter sections. (since R2024b)

example

[gd,w] = grpdelay(d,n) returns the n-point group delay response for the digital filter d.

example

[gd,w] = grpdelay(sos,n) returns the n-point group delay response corresponding to the second-order sections matrix sos.

example

[gd,w] = grpdelay(___,"whole") returns the group delay at n sample points around the entire unit circle.
[gd,f] = grpdelay(___,n,fs) returns the group delay response vector gd and the corresponding physical frequency vector f for a digital filter designed to filter signals sampled at a rate fs.
[gd,f] = grpdelay(___,n,"whole",fs) returns the frequency vector at n points ranging between 0 and fs.
gd = grpdelay(___,win) returns the group delay response vector gd evaluated at the normalized frequencies supplied in win.
gd = grpdelay(___,fin,fs) returns the group delay response vector gd evaluated at the physical frequencies supplied in fin.

example

grpdelay(___) with no output arguments plots the group delay response of the filter.

example

## Examples

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Design a Butterworth filter of order 6 with normalized 3-dB frequency $0.2\pi$ rad/sample. Use grpdelay to display the group delay.

[z,p,k] = butter(6,0.2); sos = zp2sos(z,p,k); grpdelay(sos,128)

Plot both the group delay and the phase delay of the system on the same figure.

gd = grpdelay(sos,512); [h,w] = freqz(sos,512); pd = -unwrap(angle(h))./w; plot(w/pi,gd,w/pi,pd) grid xlabel 'Normalized Frequency (\times\pi rad/sample)' ylabel 'Group and phase delays' legend('Group delay','Phase delay')

Use designfilt to design a sixth-order Butterworth Filter with normalized 3-dB frequency $0.2\pi$ rad/sample. Display its group delay response.

d = designfilt('lowpassiir','FilterOrder',6, ... 'HalfPowerFrequency',0.2,'DesignMethod','butter'); grpdelay(d)

Design an 88th-order FIR filter of arbitrary magnitude response. The filter has two passbands and two stopbands. The lower-frequency passband has twice the gain of the higher-frequency passband. Specify a sample rate of 200 Hz. Visualize the magnitude response and the phase response of the filter from 10 Hz to 78 Hz.

fs = 200; d = designfilt('arbmagfir', ... 'FilterOrder',88, ... 'NumBands',4, ... 'BandFrequencies1',[0 20], ... 'BandFrequencies2',[25 40], ... 'BandFrequencies3',[45 65], ... 'BandFrequencies4',[70 100], ... 'BandAmplitudes1',[2 2], ... 'BandAmplitudes2',[0 0], ... 'BandAmplitudes3',[1 1], ... 'BandAmplitudes4',[0 0], ... 'SampleRate',fs); freqz(d,10:1/fs:78,fs)

Compute and display the group delay response of the filter over the same frequency range. Verify that it is one-half of the filter order.

filtord(d)
ans = 88 
grpdelay(d,10:1/fs:78,fs)

Since R2024b

Design a 40th-order lowpass Chebyshev type II digital filter with a stopband edge frequency of 0.4 and stopband attenuation of 50 dB. Plot the group delay response of the filter using its coefficients in the CTF format.

[B,A] = cheby2(40,50,0.4,"ctf"); grpdelay(B,A,"ctf")

Design a 30th-order bandpass elliptic digital filter with passband edge frequencies of 0.3 and 0.7, passband ripple of 0.1 dB, and stopband attenuation of 50 dB. Plot the group delay response of the filter using its coefficients and gain in the CTF format.

[B,A,g] = ellip(30,0.1,50,[0.3 0.7],"ctf"); grpdelay({B,A,g},"ctf")

## Input Arguments

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Transfer function coefficients, specified as vectors. Express the transfer function in terms of b and a as

$H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{1}+{b}_{2}{z}^{-1}\cdots +{b}_{n}{z}^{-\left(n-1\right)}+{b}_{n+1}{z}^{-n}}{{a}_{1}+{a}_{2}{z}^{-1}\cdots +{a}_{m}{z}^{-\left(m-1\right)}+{a}_{m+1}{z}^{-m}}$

Example: b = [1 3 3 1]/6 and a = [3 0 1 0]/3 specify a third-order Butterworth filter with normalized 3 dB frequency 0.5π rad/sample.

Data Types: double | single
Complex Number Support: Yes

Number of frequency points over which to evaluate response, specified as a positive integer scalar no less than 2. When n is absent, it defaults to 512. For best results, set n to a value greater than the filter order.

Since R2024b

Cascaded transfer function (CTF) coefficients, specified as scalars, vectors, or matrices. B and A list the numerator and denominator coefficients of the cascaded transfer function, respectively.

B must be of size L-by-(m + 1) and A must be of size L-by-(n + 1), where:

• L represents the number of filter sections.

• m represents the order of the filter numerators.

• n represents the order of the filter denominators.

Note

If any element of A(:,1) is not equal to 1, then grpdelay normalizes the filter coefficients by A(:,1). In this case, A(:,1) must be nonzero.

Data Types: double | single
Complex Number Support: Yes

Since R2024b

Scale values, specified as a real-valued scalar or as a real-valued vector with L + 1 elements, where L is the number of CTF sections. The scale values represent the distribution of the filter gain across sections of the cascaded filter representation.

The grpdelay function applies a gain to the filter sections using the scaleFilterSections function depending on how you specify g:

• Scalar — The function distributes the gain uniformly across all filter sections.

• Vector — The function applies the first L gain values to the corresponding filter sections and distributes the last gain value uniformly across all filter sections.

Data Types: double | single

Digital filter, specified as a digitalFilter object. Use designfilt to generate a digital filter based on frequency-response specifications.

Example: d = designfilt('lowpassiir','FilterOrder',3,'HalfPowerFrequency',0.5) specifies a third-order Butterworth filter with normalized 3 dB frequency 0.5π rad/sample.

Second-order section coefficients, specified as a matrix. sos is a K-by-6 matrix, where the number of sections, K, must be greater than or equal to 2. If the number of sections is less than 2, the function treats the input as a numerator vector. Each row of sos corresponds to the coefficients of a second-order (biquad) filter. The ith row of sos corresponds to [bi(1) bi(2) bi(3) ai(1) ai(2) ai(3)].

Example: s = [2 4 2 6 0 2;3 3 0 6 0 0] specifies a third-order Butterworth filter with normalized 3 dB frequency 0.5π rad/sample.

Data Types: double | single
Complex Number Support: Yes

Sample rate, specified as a positive scalar. When the unit of time is seconds, fs is expressed in hertz.

Data Types: double

Angular frequencies, specified as a vector and expressed in rad/sample. win must have at least two elements, because otherwise the function interprets it as n. win = π corresponds to the Nyquist frequency.

Frequencies, specified as a vector. fin must have at least two elements, because otherwise the function interprets it as n. When the unit of time is seconds, fin is expressed in hertz.

Data Types: double

## Output Arguments

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Group delay response, returned as a vector. If you specify n, then gd has length n. If you do not specify n, or specify n as the empty vector, then gd has length 512.

If the input to grpdelay is single precision, the function computes the group delay using single-precision arithmetic. The output h is single precision.

Angular frequencies, returned as a vector. w has values ranging from 0 to π. If you specify "whole" in your input, the values in w range from 0 to 2π. If you specify n, w has length n. If you do not specify n, or specify n as the empty vector, then w has length 512.

Frequencies, returned as a vector expressed in hertz. f has values ranging from 0 to fs/2 Hz. If you specify "whole" in your input, the values in f range from 0 to fs Hz. If you specify n, f has length n. If you do not specify n, or specify n as the empty vector, then f has length 512.

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### Group Delay

The group delay response of a filter is a measure of the average delay of the filter as a function of frequency. It is the negative first derivative of the phase response of the filter. If the frequency response of a filter is H(e), then the group delay is

${\tau }_{g}\left(\omega \right)=-\frac{d\theta \left(\omega \right)}{d\omega },$

where θ(ω) is the phase, or argument, of H(e).

Partitioning an IIR digital filter into cascaded sections improves its numerical stability and reduces its susceptibility to coefficient quantization errors. The cascaded form of a transfer function H(z) in terms of the L transfer functions H1(z), H2(z), …, HL(z) is

$H\left(z\right)=\prod _{l=1}^{L}{H}_{l}\left(z\right)={H}_{1}\left(z\right)×{H}_{2}\left(z\right)×\cdots ×{H}_{L}\left(z\right).$

### Specify Digital Filters in CTF Format

You can specify digital filters in the CTF format for analysis, visualization, and signal filtering. Specify a filter by listing its coefficients B and A. You can also include the filter scaling gain across sections by specifying a scalar or vector g.

Filter Coefficients

When you specify the coefficients as L-row matrices,

$B=\left[\begin{array}{cccc}{b}_{11}& {b}_{12}& \cdots & {b}_{1,m+1}\\ {b}_{21}& {b}_{22}& \cdots & {b}_{2,m+1}\\ ⋮& ⋮& \ddots & ⋮\\ {b}_{L1}& {b}_{L2}& \cdots & {b}_{L,m+1}\end{array}\right],\text{ }A=\left[\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1,n+1}\\ {a}_{21}& {a}_{22}& \cdots & {a}_{2,n+1}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{L1}& {a}_{L2}& \cdots & {a}_{L,n+1}\end{array}\right],$

it is assumed that you have specified the filter as a sequence of L cascaded transfer functions, such that the full transfer function of the filter is

$H\left(z\right)=\frac{{b}_{11}+{b}_{12}{z}^{-1}+\cdots +{b}_{1,m+1}{z}^{-m}}{{a}_{11}+{a}_{12}{z}^{-1}+\cdots +{a}_{1,n+1}{z}^{-n}}×\frac{{b}_{21}+{b}_{22}{z}^{-1}+\cdots +{b}_{2,m+1}{z}^{-m}}{{a}_{21}+{a}_{22}{z}^{-1}+\cdots +{a}_{2,n+1}{z}^{-n}}×\cdots ×\frac{{b}_{L1}+{b}_{L2}{z}^{-1}+\cdots +{b}_{L,m+1}{z}^{-m}}{{a}_{L1}+{a}_{L2}{z}^{-1}+\cdots +{a}_{L,n+1}{z}^{-n}},$

where m ≥ 0 is the numerator order of the filter and n ≥ 0 is the denominator order.

• If you specify both B and A as vectors, it is assumed that the underlying system is a one-section IIR filter (L = 1), with B representing the numerator of the transfer function and A representing its denominator.

• If B is scalar, it is assumed that the filter is a cascade of all-pole IIR filters with each section having an overall system gain equal to B.

• If A is scalar, it is assumed that the filter is a cascade of FIR filters with each section having an overall system gain equal to 1/A.

Note

• To convert second-order section matrices to cascaded transfer functions, use the sos2ctf function.

• To convert a zero-pole-gain filter representation to cascaded transfer functions, use the zp2ctf function.

Coefficients and Gain

If you have an overall scaling gain or multiple scaling gains factored out from the coefficient values, you can specify the coefficients and gain as a cell array of the form {B,A,g}. Scaling filter sections is especially important when you work with fixed-point arithmetic to ensure that the output of each filter section has similar amplitude levels, which helps avoid inaccuracies in the filter response due to limited numeric precision.

The gain can be a scalar overall gain or a vector of section gains.

• If the gain is scalar, the value applies uniformly to all the cascade filter sections.

• If the gain is a vector, it must have one more element than the number of filter sections L in the cascade. Each of the first L scale values applies to the corresponding filter section, and the last value applies uniformly to all the cascade filter sections.

If you specify the coefficient matrices and gain vector as

$B=\left[\begin{array}{cccc}{b}_{11}& {b}_{12}& \cdots & {b}_{1,m+1}\\ {b}_{21}& {b}_{22}& \cdots & {b}_{2,m+1}\\ ⋮& ⋮& \ddots & ⋮\\ {b}_{L1}& {b}_{L2}& \cdots & {b}_{L,m+1}\end{array}\right],\text{ }A=\left[\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1,n+1}\\ {a}_{21}& {a}_{22}& \cdots & {a}_{2,n+1}\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{L1}& {a}_{L2}& \cdots & {a}_{L,n+1}\end{array}\right],\text{ }g=\left[\begin{array}{ccccc}{g}_{1}& {g}_{2}& \cdots & {g}_{L}& {g}_{\text{S}}\end{array}\right],$

it is assumed that the transfer function of the filter system is

$H\left(z\right)={g}_{\text{S}}\left({g}_{1}\frac{{b}_{11}+{b}_{12}{z}^{-1}+\cdots +{b}_{1,m+1}{z}^{-m}}{{a}_{11}+{a}_{12}{z}^{-1}+\cdots +{a}_{1,n+1}{z}^{-n}}×{g}_{2}\frac{{b}_{21}+{b}_{22}{z}^{-1}+\cdots +{b}_{2,m+1}{z}^{-m}}{{a}_{21}+{a}_{22}{z}^{-1}+\cdots +{a}_{2,n+1}{z}^{-n}}×\cdots ×{g}_{L}\frac{{b}_{L1}+{b}_{L2}{z}^{-1}+\cdots +{b}_{L,m+1}{z}^{-m}}{{a}_{L1}+{a}_{L2}{z}^{-1}+\cdots +{a}_{L,n+1}{z}^{-n}}\right).$

## References

[1] Lyons, Richard G. Understanding Digital Signal Processing. Upper Saddle River, NJ: Prentice Hall, 2004.

## Version History

Introduced before R2006a

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