Frequency Selective Filters

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.