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zplane

Zero-pole plot for discrete-time systems

Description

example

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 matrices, then zplane plots the poles and zeros in the columns of z and p in different colors.

example

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 the numerator coefficients b and the denominator coefficients a.

[hz,hp,ht] = zplane(___) 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.

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.

[vz,vp,vk] = zplane(d) returns the zeros vz, poles vp, and gain vk corresponding to the digital filter d.

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')

Figure contains an axes object. The axes object with title 4th-Order Elliptic Lowpass Digital Filter, xlabel Real Part, ylabel Imaginary Part contains 3 objects of type line. One or more of the lines displays its values using only markers

Create the same filter using designfilt. Use zplane to plot the poles and zeros. Note that this syntax of zplane calls fvtool.

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

Figure Figure 1: Pole-Zero Plot contains an axes object. The axes object with title Pole-Zero Plot, xlabel Real Part, ylabel Imaginary Part contains 3 objects of type line. One or more of the lines displays its values using only markers

Design an 8th-order Chebyshev Type II bandpass filter with a stopband attenuation of 20 dB. Specify the stopband edge frequencies as π/8 rad/sample and 5π/8 rad/sample.

[b,a] = cheby2(8/2,20,[1 5]/8);

Use zplane to plot the poles and zeros of the transfer function.

zplane(b,a)

Figure contains an axes object. The axes object with title Pole-Zero Plot, xlabel Real Part, ylabel Imaginary Part contains 3 objects of type line. One or more of the lines displays its values using only markers

Visualize the zero-phase response of the filter. Overlay the unit circle and the pole and zero locations.

[hw,fw] = zerophase(b,a,1024,"whole");

z = roots(b);
p = roots(a);

plot3(cos(fw),sin(fw),hw)
hold on
plot3(cos(fw),sin(fw),zeros(size(fw)),'--')
plot3(real(z),imag(z),zeros(size(z)),'o')
plot3(real(p),imag(p),zeros(size(p)),'x')
hold off
xlabel("Real")
ylabel("Imaginary")
view(35,40)
grid

Figure contains an axes object. The axes object with xlabel Real, ylabel Imaginary contains 4 objects of type line. One or more of the lines displays its values using only markers

Input Arguments

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Zeros and poles, specified as column vectors or matrices. If z and p are matrices, then zplane plots the poles and zeros in the columns of z and p in different colors.

Data Types: single | double
Complex Number Support: Yes

Transfer function coefficients, specified as row vectors. The transfer function is defined in terms of z–1:

H(z)=B(z)A(z)=b(1)+b(2)z1++b(n+1)zna(1)+a(2)z1++a(m+1)zm

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: single | double
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.

Output Arguments

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Vectors of handles to the zero lines, hz, and the pole lines, hp, of the pole-zero plot. 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, [].

Zeros, poles, and gain of a digital filter, d, returned as column vectors and a scalar.

Tips

  • You can override the automatic scaling of zplane using

    axis([xmin xmax ymin ymax])

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

Version History

Introduced before R2006a

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