Amplitude Response

Definition: The amplitude response of a filter is defined as the magnitude of the frequency response

$$G(k) \isdef \left\vert H(\omega_k)\right\vert.$$

From the convolution theorem, we can see that the amplitude response $G(k)$ is the gain of the filter at frequency $\omega_k$, since

$$\left\vert Y(\omega_k)\right\vert = \left\vert H(\omega_k)X(\omega_k)\right\vert = G(k)\left\vert X(\omega_k)\right\vert,$$

where $X(\omega_k)$ is the $k$th sample of the DFT of the input signal $x(n)$, and $Y$ is the DFT of the output signal $y$.