# Documentation

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# zplane

Zero-pole plot

## Syntax

`zplane(z,p)zplane(b,a)zplane(d)[hz,hp,ht] = zplane(z,p)`

## Description

This function displays the poles and zeros of discrete-time systems.

`zplane(z,p)` plots the zeros specified in column vector `z` and the poles specified in column vector `p` in the current figure window. The symbol `'o'` represents a zero and the symbol `'x'` represents a pole. The plot includes the unit circle for reference. If `z` and `p` are arrays, `zplane` plots the poles and zeros in the columns of `z` and `p` in different colors.

`zplane(b,a)` where `b` and `a` are row vectors, first uses `roots` to find the zeros and poles of the transfer function represented by numerator coefficients `b` and denominator coefficients `a`. The transfer function is defined in terms of z–1:

`$H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{b\left(1\right)+b\left(2\right){z}^{-1}+\cdots +b\left(n+1\right){z}^{-n}}{a\left(1\right)+a\left(2\right){z}^{-1}+\cdots +a\left(m+1\right){z}^{-m}}$`

`zplane(d)` finds the zeros and poles of the transfer function represented by the digital filter, `d`. Use `designfilt` to generate `d` based on frequency-response specifications. The pole-zero plot is displayed in `fvtool`.

`[hz,hp,ht] = zplane(z,p)` returns vectors of handles to the zero lines, `hz`, and the pole lines, `hp`. `ht` is a vector of handles to the axes/unit circle line and to text objects, which are present when there are multiple zeros or poles. If there are no zeros or no poles, `hz` or `hp` is the empty matrix `[]`.

## Examples

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For data sampled at 1000 Hz, plot the poles and zeros of a 4th-order elliptic lowpass digital filter with cutoff frequency 200 Hz, 3 dB of ripple in the passband, and 30 dB of attenuation in the stopband.

```[z,p,k] = ellip(4,3,30,200/500); zplane(z,p) grid title('4th-Order Elliptic Lowpass Digital Filter') ```

Create the same filter using `designfilt`. Use `fvtool` to plot its poles and zeros.

```d = designfilt('lowpassiir','FilterOrder',4,'PassbandFrequency',200, ... 'PassbandRipple',3,'StopbandAttenuation',30, ... 'DesignMethod','ellip','SampleRate',1000); zplane(d) ```

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### Tips

• You can override the automatic scaling of `zplane` using

`axis([xmin xmax ymin ymax])`

after calling `zplane`. This is useful in cases where one or more of the zeros or poles have such a large magnitude that the others are grouped tightly around the origin and are hard to distinguish.