# How to match zero-pole diagrams to their frequency responses (Discrete Time)

I get confused when there are a lot of zeros/poles in the zero-pole diagram and I find difficulty understanding their frequency response.

I know the following: 1. Complex conjugates cause double peaks at their respective angles. The larger their magnitude, the sharper their peaks.

1. A single pole causes a peak at its respective angle. The closer it is to the unit circle the sharper the peak in the frequency response.

2. A single zero causes a dip at its respective angle. The closer it is to the unit circle the sharper the dip.

I am not sure how to deal with zero pole diagrams when they come in bundles as shown below where it is asked to match each zero-pole diagram to its frequency response.

My trials:

I matched:

Frequency Response 3 to Zero-Pole Diagram 2

Frequency Response 6 to Zero-Pole Diagram 3

As an example, let's consider PZ-diagram $$\#1$$: since the stopband is around DC (i.e., angle zero), and there are no more zeros, just poles, at angles greater than approximately $$\pi/3$$ (and, by symmetry, smaller than approximately $$-\pi/3$$), this must be a high pass filter. The cut-off frequency must be close to $$\pi/3$$, so all frequency responses can be eliminated except for $$\#4$$ and $$\#5$$. However, frequency response $$\#5$$ has a zero at DC, which is not present in the PZ-diagram. Consequently, we can match PZ-diagram $$\#1$$ to frequency response $$\#4$$. I'm sure you can match the other diagrams yourself.
• @HaneenSu: It's not exactly at $z=0$, but at a small negative value. Without that pole the frequency response would have a slight dip at Nyquist. – Matt L. Mar 8 '20 at 21:17