One of the primary motivating factors for utilizing the z-transform and analyzing the pole/zero plots is due to their relationship to the frequency response of a discrete-time system. Based on the position of the poles and zeros, one can quickly determine the frequency response. This is a result of the correspondence between the frequency response and the transfer function evaluated on the unit circle in the pole/zero plots. The frequency response, or DTFT, of the system is defined as:
Let us now look at several examples of determining the magnitude of the frequency response from the pole/zero plot of a z-transform. If you have forgotten or are unfamiliar with pole/zero plots, please refer back to the Pole/Zero Plots module.
In this first example we will take a look at the very simple z-transform shown below: For this example, some of the vectors represented by , for random values of , are explicitly drawn onto the complex plane shown in the figure below. These vectors show how the amplitude of the frequency response changes as goes from to , and also show the physical meaning of the terms in Equation 2 above. One can see that when , the vector is the longest and thus the frequency response will have its largest amplitude here. As approaches , the length of the vectors decrease as does the amplitude of . Since there are no poles in the transform, there is only this one vector term rather than a ratio as seen in Equation 2.
For this example, a more complex transfer function is analyzed in order to represent the system's frequency response.
Below we can see the two figures described by the above equations. The Link represents the basic pole/zero plot of the z-transform, . Link shows the magnitude of the frequency response. From the formulas and statements in the previous section, we can see that when the frequency will peak since it is at this value of that the pole is closest to the unit circle. The ratio from Equation 2 helps us see the mathematics behind this conclusion and the relationship between the distances from the unit circle and the poles and zeros. As moves from to , we see how the zero begins to mask the effects of the pole and thus force the frequency response closer to .
In conclusion, using the distances from the unit circle to the poles and zeros, we can plot the frequency response of the system. As goes from to , the following two properties, taken from the above equations, specify how one should draw .