Site Feedback

# Estimating Frequency Response from Z-Plane

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:

$H⁢w=H⁢z| z , z = ei⁢w =∑ k =0M b k ⁢e−(i⁢w⁢k)∑ k =0N a k ⁢e−(i⁢w⁢k) Hw z w Hz k 0 M b k w k k 0 N a k w k$
1
Next, by factoring the transfer function into poles and zeros and multiplying the numerator and denominator by $ei⁢ww$ we arrive at the following equations:
$H⁢w=| b 0 a 0 |⁢∏ k =1M|ei⁢w− c k |∏ k =1N|ei⁢w− d k | Hw b 0 a 0 k 1 M w c k k 1 N w d k$
2
From Equation 2 we have the frequency response in a form that can be used to interpret physical characteristics about the filter's frequency response. The numerator and denominator contain a product of terms of the form $|ei⁢w−h| w h$, where $hh$ is either a zero, denoted by $c k c k$ or a pole, denoted by $d k d k$. Vectors are commonly used to represent the term and its parts on the complex plane. The pole or zero, $hh$, is a vector from the origin to its location anywhere on the complex plane and $ei⁢w w$ is a vector from the origin to its location on the unit circle. The vector connecting these two points, $|ei⁢w−h| w h$, connects the pole or zero location to a place on the unit circle dependent on the value of $ww$. From this, we can begin to understand how the magnitude of the frequency response is a ratio of the distances to the poles and zero present in the z-plane as $ww$ goes from zero to pi. These characteristics allow us to interpret $|H⁢w|Hw$ as follows:
$|H⁢w|=| b 0 a 0 |⁢∏⁢"distances from zeros"∏⁢"distances from poles" Hw b 0 a 0 ∏ "distances from zeros" ∏ "distances from poles"$
3

# Drawing Frequency Response from Pole/Zero Plot

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.

Example 1

In this first example we will take a look at the very simple z-transform shown below: $H⁢z=z+1=1+z-1 Hz z 1 1 z -1$ $H⁢w=1+e−(i⁢w) Hw 1 w$ For this example, some of the vectors represented by $|ei⁢w−h| w h$, for random values of $ww$, are explicitly drawn onto the complex plane shown in the figure below. These vectors show how the amplitude of the frequency response changes as $ww$ goes from $00$ to $2⁢π2$, and also show the physical meaning of the terms in Equation 2 above. One can see that when $w=0w0$, the vector is the longest and thus the frequency response will have its largest amplitude here. As $ww$ approaches $π$, the length of the vectors decrease as does the amplitude of $|H⁢w|Hw$. Since there are no poles in the transform, there is only this one vector term rather than a ratio as seen in Equation 2.

Example 2

For this example, a more complex transfer function is analyzed in order to represent the system's frequency response. $H⁢z=zz−12=11−12⁢z-1 Hz z z 1 2 1 1 1 2 z -1$ $H⁢w=11−12⁢e−(i⁢w) Hw 1 1 1 2 w$

Below we can see the two figures described by the above equations. The Link represents the basic pole/zero plot of the z-transform, $H⁢wHw$. Link shows the magnitude of the frequency response. From the formulas and statements in the previous section, we can see that when $w=0w0$ the frequency will peak since it is at this value of $ww$ 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 $ww$ moves from $00$ to $π$, we see how the zero begins to mask the effects of the pole and thus force the frequency response closer to $00$.

# Conclusion

In conclusion, using the distances from the unit circle to the poles and zeros, we can plot the frequency response of the system. As $ww$ goes from $00$ to $2⁢π 2$, the following two properties, taken from the above equations, specify how one should draw $|H⁢w| Hw$.

While moving around the unit circle...
1. if close to a zero, then the magnitude is small. If a zero is on the unit circle, then the frequency response is zero at that point.
2. if close to a pole, then the magnitude is large. If a pole is on the unit circle, then the frequency response goes to infinity at that point.