Relating Amplitude Response to Poles and Zeros

The frequency response of the transfer function (factored form) is given by

$$H(e^{j\omega t}) = g\frac{(1-q_1e^{-j\omega T})(1-q_2e^{-j\omega ... ...a T})} {(1-p_1e^{-j\omega T})(1-p_2e^{-j\omega T})\cdots(1-p_Ne^{-j\omega T})}$$

Consider the amplitude response $G(\omega) \triangleq \vert H(e^{j\omega T})\vert$

$$G(\omega) = \vert g\vert\frac{\vert 1-q_1e^{-j\omega T}\vert\cdot \vert 1-q_2... ...T}\vert\cdot\vert 1-p_2e^{-j\omega T}\vert\cdots\vert 1-p_Ne^{-j\omega T}\vert}$$

$$= \vert g\vert\frac{\vert e^{-jM\omega T}\vert\cdot \vert e^{j\omeg... ...ega T}\vert\cdot\vert e^{j\omega T}-p_1\vert\cdots\vert e^{j\omega T}-p_N\vert}$$

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