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.

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