# islinphase

Determine whether filter has linear phase

## Syntax

## Description

returns `flag`

= islinphase(`B,A`

,"ctf")`1`

if the filter specified as Cascaded Transfer Functions (CTF) with numerator coefficients `B`

and denominator coefficients
`A`

is linear phase.* (since R2024b)*

## Examples

### Linear and Nonlinear Phase Filters

Use the window method to design a tenth-order lowpass FIR filter with normalized cutoff frequency 0.55. Verify that the filter has linear phase.

d = designfilt("lowpassfir",DesignMethod="window", ... FilterOrder=10,CutoffFrequency=0.55); flag = islinphase(d)

`flag = `*logical*
1

[phs,w] = phasez(d); plot(w/pi,phs) xlabel("Normalized Frequency (\times\pi rad/sample)") ylabel("Phase (radians)")

IIR filters in general do not have linear phase. Verify the statement by constructing eighth-order Butterworth, Chebyshev, and elliptic filters with similar specifications.

ord = 8; Wcut = 0.35; atten = 20; rippl = 1; [zb,pb,kb] = butter(ord,Wcut); sosB = zp2sos(zb,pb,kb); [zc,pc,kc] = cheby1(ord,rippl,Wcut); sosC1 = zp2sos(zc,pc,kc); [zd,pd,kd] = cheby2(ord,atten,Wcut); sosC2 = zp2sos(zd,pd,kd); [ze,pe,ke] = ellip(ord,rippl,atten,Wcut); sosE = zp2sos(ze,pe,ke);

Plot the phase responses of the filters. Determine whether they have linear phase.

phasez(sosB) hold on phasez(sosC1) phasez(sosC2) phasez(sosE) hold off ylim([-14 2]) legend("Butterworth","Chebyshev I", ... "Chebyshev II","Elliptic",Location="best")

```
phs = [islinphase(sosB) islinphase(sosC1) ...
islinphase(sosC2) islinphase(sosE)]
```

`phs = `*1x4 logical array*
0 0 0 0

### Verify Linear Phase from Cascaded Transfer Functions

*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. Verify that the filter has linear phase using the filter coefficients in the CTF format.

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

`flag = `*logical*
0

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. Verify that the filter has linear phase using the filter coefficients and gain in the CTF format.

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

`flag = `*logical*
0

## Input Arguments

`b,a`

— Transfer function coefficients

vector

Transfer function coefficients, specified as a vector. The values of
`b`

and `a`

represent the numerator and denominator
polynomial coefficients, respectively.

**Example: **`[b,a] = cheby2(5,30,0.7)`

creates a digital 5th-order
Butterworth lowpass filter with coefficients `b`

and
`a`

, having a normalized 3 dB frequency of 0.7*π*
rad/sample and 30 dB attenuation at stopband.

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

`B,A`

— Cascaded transfer function (CTF) coefficients

scalars | vectors | matrices

*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.

For more information about the cascaded transfer function format and coefficient matrices, see Specify Digital Filters in CTF Format.

**Note**

If any element of `A(:,1)`

is not equal to
`1`

, then `islinphase`

normalizes the
filter coefficients by `A(:,1)`

. In this case,
`A(:,1)`

must be nonzero.

**Data Types: **`double`

| `single`

**Complex Number Support: **Yes

`g`

— Scale values

scalar | vector

*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 `islinphase`

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`

`d`

— Digital filter

`digitalFilter`

object

Digital filter, specified as a `digitalFilter`

object. Use
`designfilt`

to generate
`d`

based on frequency-response specifications.

**Example: **`designfilt("lowpassfir",FilterOrder=10,CutoffFrequency=0.55)`

generates a `digitalFilter`

object for a 10th order FIR lowpass filter
with a normalized 3 dB frequency of 0.55*π* rad/sample.

**Data Types: **`digitalFilter`

`sos`

— Second-order section representation

*L*-by-6 matrix

Second-order section representation, specified as an *L*-by-6
matrix, where *L* is the number of second-order sections. The matrix

$$\text{sos}=\left[\begin{array}{cccccc}{b}_{01}& {b}_{11}& {b}_{21}& 1& {a}_{11}& {a}_{21}\\ {b}_{02}& {b}_{12}& {b}_{22}& 1& {a}_{12}& {a}_{22}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {b}_{0L}& {b}_{1L}& {b}_{2L}& 1& {a}_{1L}& {a}_{2L}\end{array}\right]$$

represents the second-order sections of
*H*(*z*):

$$H(z)={\displaystyle \prod _{k=1}^{L}{H}_{k}(z)={\displaystyle \prod _{k=1}^{L}\frac{{b}_{0k}+{b}_{1k}{z}^{-1}+{b}_{2k}{z}^{-2}}{1+{a}_{1k}{z}^{-1}+{a}_{2k}{z}^{-2}}.}}$$

**Example: **`[z,p,k] = butter(3,1/32); sos = zp2sos(z,p,k)`

specifies a
third-order Butterworth filter with a normalized 3 dB frequency of
*π*/32 rad/sample.

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

`tol`

— Tolerance

`eps^(2/3)`

(default) | positive scalar

Tolerance to distinguish between close numbers, specified as a positive scalar. The tolerance value determines when two numbers are close enough to be considered equal.

**Data Types: **`single`

| `double`

## Output Arguments

`flag`

— Linear phase flag

logical

Linear phase flag, returned as a logical scalar. The function returns
`1`

when the input is a minimum phase filter.

## More About

### Cascaded Transfer Functions

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
*H*_{1}(*z*),
*H*_{2}(*z*), …,
*H*_{L}(*z*) is

$$H(z)={\displaystyle \prod _{l=1}^{L}{H}_{l}(z)}={H}_{1}(z)\times {H}_{2}(z)\times \cdots \times {H}_{L}(z).$$

### 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,

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

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

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

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**

**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

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

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

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

## Tips

## References

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

## Version History

**Introduced in R2013a**

### R2024b: Analyze filters using cascaded transfer functions

The `islinphase`

function supports inputs in the cascaded transfer
function (CTF) format.

## See Also

`ctffilt`

| `designfilt`

| `digitalFilter`

| `isallpass`

| `ismaxphase`

| `isminphase`

| `isstable`

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