Python zplane function

Christopher Felton December 17, 20112 comments Coded in Python

Python is a popular general purpose programming language with powerful numerical and scientific packages, numpy and scipy.  The Python ecosystem also has a impressive plotting package, matplotlib.  The language and packages (the Python ecosystem) creates an ideal computing platform for Science and Engineering analysis and design. 

But the current distributions does not include a zplane function which many DSP'ers might use because it is availabe in other packages Matlab, Octave, etc.  Below is a function to implement similar behavior to the common used zplane function.

The zplane function takes the numerator and denominator polynomial representation of a transfer function and plots the complex z-plane poles and zeros.


H(z) = \frac{b_0 + b_1z^{-1} + ... + b_Nz^{-N}}{a_0 + a_1z^{-1} + ... + a_Nz^{-N}}


For those unfamiliar with a numerical computing package, polynomials are usually represented least order coefficient to the highest order coefficient.  An array assignment might look like the following, where *N* is the polynomial order.

b\_array = [b_0, b_1, ..., b_{N-2}, b_{N}]

The function below will plot a complex zplane given an array of b and a coefficients, numerator and denominator respectively. 

This code requires the following packages:

  • matplotlib (plotting routines)
  • numpy  (array object and roots)

Example usage:

H(z) = \frac{z^{-1} + z^{-2}}{1 + \frac{1}{4}z^{-1} - \frac{3}{8}z^{-2}}

>>> import numpy as np
# If the code is in a file called
>>> from plot_zplane import zplane
>>> b = np.array([0, 1, 1])
>>> a = np.array([1, 1/4., -3/8.])
>>> zplane(b,a)

Complex Plane



# Copyright (c) 2011 Christopher Felton
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# GNU Lesser General Public License for more details.
# You should have received a copy of the GNU Lesser General Public License
# along with this program.  If not, see <>.

# The following is derived from the slides presented by
# Alexander Kain for CS506/606 "Special Topics: Speech Signal Processing"
# CSLU / OHSU, Spring Term 2011.

import numpy as np
import matplotlib.pyplot as plt
from  matplotlib import patches
from matplotlib.figure import Figure
from matplotlib import rcParams
def zplane(b,a,filename=None):
    """Plot the complex z-plane given a transfer function.

    # get a figure/plot
    ax = plt.subplot(111)

    # create the unit circle
    uc = patches.Circle((0,0), radius=1, fill=False,
                        color='black', ls='dashed')

    # The coefficients are less than 1, normalize the coeficients
    if np.max(b) > 1:
        kn = np.max(b)
        b = b/float(kn)
        kn = 1

    if np.max(a) > 1:
        kd = np.max(a)
        a = a/float(kd)
        kd = 1
    # Get the poles and zeros
    p = np.roots(a)
    z = np.roots(b)
    k = kn/float(kd)
    # Plot the zeros and set marker properties    
    t1 = plt.plot(z.real, z.imag, 'go', ms=10)
    plt.setp( t1, markersize=10.0, markeredgewidth=1.0,
              markeredgecolor='k', markerfacecolor='g')

    # Plot the poles and set marker properties
    t2 = plt.plot(p.real, p.imag, 'rx', ms=10)
    plt.setp( t2, markersize=12.0, markeredgewidth=3.0,
              markeredgecolor='r', markerfacecolor='r')


    # set the ticks
    r = 1.5; plt.axis('scaled'); plt.axis([-r, r, -r, r])
    ticks = [-1, -.5, .5, 1]; plt.xticks(ticks); plt.yticks(ticks)

    if filename is None:

    return z, p, k


Works fine, thank you! I created a little longer example showing also the impulse response and frequency response of the system. For more details see: I hope your code finds its way to the official scipy.signal distribution.
6 years ago
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Thanks for the additional examples! I don't know if a function like this would be added to the scipy package. Functions that rely on plotting are hard to fit into packages like scipy. A more appropriate spot might be the matplotlib.mlab package. Which isn't a perfect fit either since the z-plane and s-plane are specific to signal processing.
6 years ago
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