Poles and Zeros and the Unit Circle

Recall the amplitude response

$$G(\omega) = \vert g\vert\frac{\vert e^{j\omega T}-q_1\vert\cdot\v... ...-p_1\vert\cdot\vert e^{j\omega T}-p_2\vert\cdots\vert e^{j\omega T}-p_N\vert}.$$

Figure 12: Measurement of amplitude response from a pole-zero diagram (a bi-quad section).

Thus the term $e^{j\omega T} - q_i$ may be drawn as an arrow from the $i$th zero to the point $e^{j\omega T}$ on the unit circle, and $e^{j\omega T} - p_i$ is an arrow from the $i$th pole.

The amplitude response at frequency $\omega$ is given by

$$G(\omega) = \frac{d_1d_2}{d_3d_4}$$