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Summary: This module will look at the relationships between the z-transform and the complex plane. Specifically, the creation of pole/zero plots and some of their useful properties are discussed.
It is quite difficult to qualitatively analyze the Laplace transform and Z-transform, since mappings of their magnitude and phase or real part and imaginary part result in multiple mappings of 2-dimensional surfaces in 3-dimensional space. For this reason, it is very common to examine a plot of a transfer function's poles and zeros to try to gain a qualitative idea of what a system does.
Once the Z-transform of a system has been determined, one can use the information contained in function's polynomials to graphically represent the function and easily observe many defining characteristics. The Z-transform will have the below structure, based on Rational Functions:
The two polynomials,
Below is a simple transfer function with the poles and zeros
shown below it.
The zeros are:
The poles are:
Once the poles and zeros have been found for a given
Z-Transform, they can be plotted onto the Z-Plane. The
Z-plane is a complex plane with an imaginary and real axis
referring to the complex-valued variable
| Z-Plane |
|---|
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This section lists several examples of finding the poles and zeros of a transfer function and then plotting them onto the Z-Plane.
The zeros are:
The poles are:
| Pole/Zero Plot |
|---|
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The zeros are:
The poles are:
| Pole/Zero Plot |
|---|
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An easy mistake to make with regards to poles and zeros is to
think that a function like
MATLAB - If access to MATLAB is readily available, then you can use its functions to easily create pole/zero plots. Below is a short program that plots the poles and zeros from the above example onto the Z-Plane.
% Set up vector for zeros
z = [j ; -j];
% Set up vector for poles
p = [-1 ; .5+.5j ; .5-.5j];
figure(1);
zplane(z,p);
title('Pole/Zero Plot for Complex Pole/Zero Plot Example');
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Now that we have found and plotted the poles and zeros, we must ask what it is that this plot gives us. Basically what we can gather from this is that the magnitude of the transfer function will be larger when it is closer to the poles and smaller when it is closer to the zeros. This provides us with a qualitative understanding of what the system does at various frequencies and is crucial to the discussion of stability.
The region of convergence (ROC) for
| Region of Convergence for the Pole/Zero Plot |
|---|
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The reason it is helpful to understand and create these pole/zero plots is due to their ability to help us easily design a filter. Based on the location of the poles and zeros, the magnitude response of the filter can be quickly understood. Also, by starting with the pole/zero plot, one can design a filter and obtain its transfer function very easily.
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This work is licensed by Michael Haag under a Creative Commons Attribution License (CC-BY 3.0), and is an Open Educational Resource.
Last edited by Dan Calderon on Sep 29, 2011 4:43 pm GMT-5.
"My introduction to signal processing course at Rice University."