# phasez

Phase response of digital filter

## Description

[phi,w] = phasez(b,a,n) returns the n-point phase response vector phi and the corresponding angular frequency vector w for the digital filter with the transfer function coefficients stored in b and a.

[phi,w] = phasez(sos,n) returns the n-point phase response corresponding to the second-order sections matrix sos.

[phi,w] = phasez(d,n) returns the n-point phase response for the digital filter d.

[phi,w] = phasez(___,n,'whole') returns the phase response at n sample points around the entire unit circle. This syntax can include any combination of input arguments from the previous syntaxes.

[phi,f] = phasez(___,n,fs) returns the frequency vector.

phi = phasez(___,f,fs) returns the phase response vector phi evaluated at the physical frequencies supplied in f. This syntax can include any combination of input arguments from the previous syntaxes.

phi = phasez(___,w) returns the unwrapped phase response in radians at frequencies specified in w.

example

phasez(___) with no output arguments plots the phase response of the filter.

## Examples

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Use designfilt to design an FIR filter of order 54, normalized cutoff frequency $0.3\pi$ rad/s, passband ripple 0.7 dB, and stopband attenuation 42 dB. Use the method of constrained least squares. Display the phase response of the filter.

Nf = 54;
Fc = 0.3;
Ap = 0.7;
As = 42;

d = designfilt('lowpassfir','CutoffFrequency',Fc,'FilterOrder',Nf, ...
'PassbandRipple',Ap,'StopbandAttenuation',As, ...
'DesignMethod','cls');
phasez(d)

Design the same filter using fircls1. Keep in mind that fircls1 uses linear units to measure the ripple and attenuation.

pAp = 10^(Ap/40);
Apl = (pAp-1)/(pAp+1);

pAs = 10^(As/20);
Asl = 1/pAs;

b = fircls1(Nf,Fc,Apl,Asl);
phasez(b)

Design a lowpass equiripple filter with normalized passband frequency $0.45\pi$ rad/s, normalized stopband frequency $0.55\pi$ rad/s, passband ripple 1 dB, and stopband attenuation 60 dB. Display the phase response of the filter.

d = designfilt('lowpassfir', ...
'PassbandFrequency',0.45,'StopbandFrequency',0.55, ...
'PassbandRipple',1,'StopbandAttenuation',60, ...
'DesignMethod','equiripple');
phasez(d)

Design an elliptic lowpass IIR filter with normalized passband frequency $0.4\pi$ rad/s, normalized stopband frequency $0.5\pi$ rad/s, passband ripple 1 dB, and stopband attenuation 60 dB. Display the phase response of the filter.

d = designfilt('lowpassiir', ...
'PassbandFrequency',0.4,'StopbandFrequency',0.5, ...
'PassbandRipple',1,'StopbandAttenuation',60, ...
'DesignMethod','ellip');
phasez(d)

## Input Arguments

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

$H\left({e}^{j\omega }\right)=\frac{B\left({e}^{j\omega }\right)}{A\left({e}^{j\omega }\right)}=\frac{\text{b(1)}+\text{b(2)}\text{\hspace{0.17em}}{e}^{-j\omega }+\text{b(3)}\text{\hspace{0.17em}}{e}^{-j2\omega }+\cdots +\text{b(M)}\text{\hspace{0.17em}}{e}^{-j\left(M-1\right)\omega }}{\text{a(1)}+\text{a(2)}\text{\hspace{0.17em}}{e}^{-j\omega }+\text{a(3)}\text{\hspace{0.17em}}{e}^{-j2\omega }+\cdots +\text{a(N)}\text{\hspace{0.17em}}{e}^{-j\left(N-1\right)\omega }}.$

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 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.

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

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.

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. w must have at least two elements, because otherwise the function interprets it as n. w = π corresponds to the Nyquist frequency.

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

Data Types: double

## Output Arguments

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Phase response, returned as a vector. If you specify n, then phi has length n. If you do not specify n, or specify n as an empty vector, then phi has length 512.

If the input to phasez is single precision, the function computes the phase response using single-precision arithmetic. The output phi 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, 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 an empty vector, f has length 512.

## Version History

Introduced before R2006a