Phase Response

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

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

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

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

The topics touched upon in this section are developed more fully in the next book [66] in the music signal processing series mentioned in the preface.