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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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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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