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Phase Response



Definition: The phase response of a filter is defined as the phase of its frequency response:

$\displaystyle \Theta(k) \isdef \angle{H(\omega_k)}
$

From the convolution theorem, we can see that the phase response $ \Theta(k)$ is the phase-shift added by the filter to an input sinusoidal component at frequency $ \omega_k$ , since

$\displaystyle \angle{Y(\omega_k)} = \angle{\left[H(\omega_k)X(\omega_k)\right]}
= \angle{H(\omega_k)} + \angle{X(\omega_k)}
= \Theta(k) + \angle{X(\omega_k)}.
$

The topics touched upon in this section are developed more fully in the next book [71] in the music signal processing series mentioned in the preface.


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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 © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA