# zerophase

Zero-phase response of digital filter

## Syntax

`[Hr,w] = zerophase(b,a)`

[Hr,w] = zerophase(sos)

[Hr,w] = zerophase(d)

[Hr,w] = zerophase(...,nfft)

[Hr,w] = zerophase(...,nfft,'whole')

[Hr,w] = zerophase(...,w)

[Hr,f] = zerophase(...,f,fs)

[Hr,w,phi] = zerophase(...)

zerophase(...)

## Description

`[Hr,w] = zerophase(b,a)`

returns
the zero-phase response `Hr`

, and the frequency vector `w`

(in
radians/sample) at which `Hr`

is computed, given
a filter defined by numerator `b`

and denominator `a`

.
For FIR filters where `a=1`

, you can omit the value `a`

from
the command. The zero-phase response is evaluated at `512`

equally
spaced points on the upper half of the unit circle.

The zero-phase response, *H _{r}*(

*ω*), is related to the frequency response,

*H*(

*e*), by

^{jω}$$H({e}^{j\omega})={H}_{r}(\omega ){e}^{j\phi (\omega )},$$

where *φ*(*ω*) is
the continuous phase.

**Note**

The zero-phase response is always real, but it is not the equivalent of the magnitude response. The former can be negative while the latter cannot be negative.

`[Hr,w] = zerophase(sos)`

returns the zero-phase
response for the second order sections matrix, `sos`

. `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, `zerophase`

considers the input to be the
numerator vector, `b`

. Each row of `sos`

corresponds
to the coefficients of a second order (biquad) filter. The *i*th
row of the `sos`

matrix corresponds to ```
[bi(1)
bi(2) bi(3) ai(1) ai(2) ai(3)]
```

.

`[Hr,w] = zerophase(d)`

returns the zero-phase
response for the digital filter, `d`

. Use `designfilt`

to generate `d`

based
on frequency-response specifications.

`[Hr,w] = zerophase(...,nfft)`

returns
the zero-phase response `Hr`

and frequency vector `w`

(radians/sample),
using `nfft`

frequency points on the upper half of
the unit circle. For best results, set `nfft`

to
a value greater than the filter order.

`[Hr,w] = zerophase(...,nfft,'whole')`

returns
the zero-phase response `Hr`

and frequency vector `w`

(radians/sample),
using `nfft`

frequency points around the whole unit
circle.

`[Hr,w] = zerophase(...,w)`

returns
the zero-phase response `Hr`

and frequency vector `w`

(radians/sample)
at frequencies in vector `w`

. The vector `w`

must
have at least two elements.

`[Hr,f] = zerophase(...,f,fs)`

returns
the zero-phase response `Hr`

and frequency vector `f`

(Hz),
using the sampling frequency `fs`

(in Hz), to determine
the frequency vector `f`

(in Hz) at which `Hr`

is
computed. The vector `f`

must have at least two elements.

`[Hr,w,phi] = zerophase(...)`

returns
the zero-phase response `Hr`

, frequency vector `w`

(rad/sample),
and the continuous phase component, `phi`

. (Note
that this quantity is not equivalent to the phase response of the
filter when the zero-phase response is negative.)

`zerophase(...)`

plots the zero-phase response versus
frequency. If you input the filter coefficients or second order sections matrix, the
current figure window is used. If you input a `digitalFilter`

, the step response is displayed in **FVTool**.

**Note**

If the input to `zerophase`

is single precision,
the zero-phase response is calculated using single-precision arithmetic.
The output, `Hr`

, is single precision.

## Examples

## References

[1] Antoniou, Andreas.
*Digital Filters*. New York: McGraw-Hill, Inc.,
1993.

## See Also

`designfilt`

| `digitalFilter`

| `freqs`

| `freqz`

| FVTool | `grpdelay`

| `invfreqz`

| `phasedelay`

| `phasez `

**Introduced before R2006a**