# phasez

Phase response of digital filter

## Syntax

``[phi,w] = phasez(b,a,n)``
``[phi,w] = phasez(B,A,"ctf",n)``
``[phi,w] = phasez({B,A,g},"ctf",n)``
``[phi,w] = phasez(d,n)``
``[phi,w] = phasez(sos,n)``
``````[phi,w] = phasez(___,n,"whole")``````
``````[phi,f] = phasez(___,n,fs)``````
``phi = phasez(___,f,fs)``
``phi = phasez(___,w)``
``phasez(___)``

## Description

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

example

````[phi,w] = phasez({B,A,g},"ctf",n)` returns the `n`-point phase 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

````[phi,w] = phasez(d,n)` returns the `n`-point phase response for the digital filter `d`.```
````[phi,w] = phasez(sos,n)` returns the `n`-point phase response corresponding to the second-order sections matrix `sos`.```
``````[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`.```
````phasez(___)` with no output arguments plots the phase response of the filter.```

example

## 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)```

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 phase response of the filter using its coefficients in the CTF format.

```[B,A] = cheby2(40,50,0.4,"ctf"); phasez(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 phase 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"); phasez({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 `phasez` 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 `phasez` 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. `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.

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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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