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Embedded SystemsFPGAElectronics

Mathematics of the DFT
    Example Applications of the DFT
       Filters and Convolution
          Phase Response

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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 [68] in the music signal processing series mentioned in the preface.

Previous: Amplitude Response
Next: Correlation Analysis

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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