grpdelay

Average filter delay (group delay)

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Syntax

[gd,w] = grpdelay(b,a,n)

[gd,w] = grpdelay(sos,n)

[gd,w] = grpdelay(d,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 n-point group delay response vector, gd, and the corresponding angular frequency vector, w, for the digital filter with transfer function coefficients stored in b and a.

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(d,n) returns the n-point group delay response for the digital filter d.

[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.

example

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.

Examples

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Group Delay of a Butterworth Filter

Open Live Script

Design a Butterworth filter of order 6 with normalized 3-dB frequency $0.2 π$ 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')

Group Delay Response of a Butterworth digitalFilter

Open Live Script

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

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

Group Delay Response of Arbitrary Magnitude Response FIR Filter

Open Live Script

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)

Input Arguments

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`b`, `a` — Transfer function coefficients
vectors

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

$H (e^{j ω}) = \frac{B (e^{j ω})}{A (e^{j ω})} = \frac{b(1) + b(2) e^{- j ω} + b(3) e^{- j 2 ω} + \dots + b(M) e^{- j (M - 1) ω}}{a(1) + a(2) e^{- j ω} + a(3) e^{- j 2 ω} + \dots + a(N) e^{- j (N - 1) ω}} .$

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

`n` — Number of evaluation points
512 (default) | positive integer scalar

Number of evaluation points, 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.

Data Types: double

`sos` — Second-order section coefficients
matrix

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

`d` — Digital filter
`digitalFilter` object

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.

`fs` — Sample rate
positive scalar

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

Data Types: double

`win` — Angular frequencies
vector

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.

`fin` — Frequencies
vector

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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`gd` — Group delay response
vector

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.

`w` — Angular frequencies
vector

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.

`f` — Frequencies
vector

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.

More About

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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^jω), then the group delay is

$τ_{g} (ω) = - \frac{d θ (ω)}{d ω},$

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

Documentation

grpdelay

Syntax

Description

Examples

Group Delay of a Butterworth Filter

Group Delay Response of a Butterworth digitalFilter

Group Delay Response of Arbitrary Magnitude Response FIR Filter

Input Arguments

`b`, `a` — Transfer function coefficients
vectors

`n` — Number of evaluation points
512 (default) | positive integer scalar

`sos` — Second-order section coefficients
matrix

`d` — Digital filter
`digitalFilter` object

`fs` — Sample rate
positive scalar

`win` — Angular frequencies
vector

`fin` — Frequencies
vector

Output Arguments

`gd` — Group delay response
vector

`w` — Angular frequencies
vector

`f` — Frequencies
vector

More About

Group Delay

See Also

Signal Processing Toolbox Documentation

Support

Documentation

grpdelay

Syntax

Description

Examples

Group Delay of a Butterworth Filter

Group Delay Response of a Butterworth digitalFilter

Group Delay Response of Arbitrary Magnitude Response FIR Filter

Input Arguments

b, a — Transfer function coefficients vectors

n — Number of evaluation points 512 (default) | positive integer scalar

sos — Second-order section coefficients matrix

d — Digital filter digitalFilter object

fs — Sample rate positive scalar

win — Angular frequencies vector

fin — Frequencies vector

Output Arguments

gd — Group delay response vector

w — Angular frequencies vector

f — Frequencies vector

More About

Group Delay

See Also

Signal Processing Toolbox Documentation

Support

`b`, `a` — Transfer function coefficients
vectors

`n` — Number of evaluation points
512 (default) | positive integer scalar

`sos` — Second-order section coefficients
matrix

`d` — Digital filter
`digitalFilter` object

`fs` — Sample rate
positive scalar

`win` — Angular frequencies
vector

`fin` — Frequencies
vector

`gd` — Group delay response
vector

`w` — Angular frequencies
vector

`f` — Frequencies
vector