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Amplitude Response

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

$\displaystyle 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

$\displaystyle \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$.
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