Filter Types from Poles and Zeros

In general, the poles and zeros relate to the frequency response in that the angle (in radians) around the unit circle (from $0$ to $2 \pi$) corresponds to angular frequency and the radius (from $0$ to $\infty$) corresponds to the strength of the pole or zero at that frequency.

As a zero approaches the unit circle, the magnitude response approaches zero. The magnitude response of a zero on the unit circle is zero at the corresponding frequency (i.e., angle).

As a pole approaches the unit circle, the magnitude response approaches infinity. The magnitude response of a pole on the unit circle is infinity at the corresponding frequency (i.e., angle).

We build different filters by placing poles and zeros throughout the real-imaginary plane. Poles will pull the magnitude response higher around the corresponding frequency. Zeros will push the magnitude response lower around the corresponding frequency. Hence, a pole in the real-imaginary plane at an angle of 0 and a zero in the real-imaginary plane at an angle of $\pi$ will effectively correspond to a low-pass filter.

In the following sections, we demonstrate various examples of filters, their pole-zero plots, the magnitudes of the frequency responses, the phases of the frequency responses, and the impulse responses.