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Understanding Pole/Zero Plots on the Z-Plane

Module by: Michael Haag

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.

Introduction to Poles and Zeros of the Z-Transform

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:
Xz=PzQz Xz Pz Qz (1)
The two polynomials, PzPz and QzQz, allow us to find the poles and zeros of the Z-Transform.
Definition 1: zeros
1. The value(s) for zz where Pz=0 Pz 0.
2. The complex frequencies that make the overall gain of the filter transfer function zero.
Definition 2: poles
1. The value(s) for zz where Qz=0 Qz 0.
2. The complex frequencies that make the overall gain of the filter transfer function infinite.
Example 1 
Below is a simple transfer function with the poles and zeros shown below it. Hz=z+1z-12z+34 Hz z 1 z 1 2 z 3 4
The zeros are: -1 1
The poles are: 12-34 1 2 3 4

The Z-Plane

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 zz. The position on the complex plane is given by rⅇⅈθ r θ and the angle from the positive, real axis around the plane is denoted by θθ. When mapping poles and zeros onto the plane, poles are denoted by an "x" and zeros by an "o". The below figure shows the Z-Plane, and examples of plotting zeros and poles onto the plane can be found in the following section.
Z-Plane
zplane.jpg
Figure 1

Examples of Pole/Zero Plots

This section lists several examples of finding the poles and zeros of a transfer function and then plotting them onto the Z-Plane.
Example 2: Simple Pole/Zero Plot 
Hz=zz-12z+34 Hz z z 1 2 z 3 4
The zeros are: 0 0
The poles are: 12-34 1 2 3 4
Pole/Zero Plot
zp_eg1.jpg
Figure 2: Using the zeros and poles found from the transfer function, the one zero is mapped to zero and the two poles are placed at 1212 and -3434
Example 3: Complex Pole/Zero Plot 
Hz=z-ⅈz+ⅈz-12-12ⅈz-12+12ⅈ Hz z z z 1 2 1 2 z 1 2 1 2
The zeros are: ⅈ-ⅈ
The poles are: -112+12ⅈ12-12ⅈ 1 1 2 1 2 1 2 1 2
Pole/Zero Plot
zp_eg2.jpg
Figure 3: Using the zeros and poles found from the transfer function, the zeros are mapped to ±ⅈ± , and the poles are placed at -11, 12+12ⅈ 1 2 1 2 and 12-12ⅈ 1 2 1 2
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');

Pole/Zero Plot and Region of Convergence

The region of convergence (ROC) for XzXz in the complex Z-plane can be determined from the pole/zero plot. Although several regions of convergence may be possible, where each one corresponds to a different impulse response, there are some choices that are more practical. A ROC can be chosen to make the transfer function causal and/or stable depending on the pole/zero plot.
    Filter Properties from ROC
  • If the ROC extends outward from the outermost pole, then the system is causal.
  • If the ROC includes the unit circle, then the system is stable.
Below is a pole/zero plot with a possible ROC of the Z-transform in the Simple Pole/Zero Plot discussed earlier. The shaded region indicates the ROC chosen for the filter. From this figure, we can see that the filter will be both causal and stable since the above listed conditions are both met.
Example 4 
Hz=zz-12z+34 Hz z z 1 2 z 3 4
Region of Convergence for the Pole/Zero Plot
zp_roc.jpg
Figure 4: The shaded area represents the chosen ROC for the transfer function.

Frequency Response and the Z-Plane

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. Refer to this module for information on the relationship between the pole/zero plot and the frequency response.

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