Phase Delay and Group Delay

Phase Response:

$$\zbox{\Theta(\omega) \mathrel{\stackrel{\Delta}{=}}\angle H(e^{j\omega T})}$$

Phase Delay:

$$\zbox{P(\omega) \mathrel{\stackrel{\Delta}{=}}- \frac{\Theta(\omega)}{\omega}}$$   (Phase Delay)

Group Delay:

$$\zbox{D(\omega) \mathrel{\stackrel{\Delta}{=}}- \frac{d}{d\omega}\Theta(\omega)}$$   (Group Delay)

• For a slowly modulated sinusoidal input signal $x(n)=A(nT)\cos(\omega nT + \phi)$, the output signal is
  $$\zbox{y(n) \approx G(\omega)A[nT - D(\omega)]\cdot \cos\{\omega[nT-P(\omega)] + \phi\}}$$
  where $G(\omega)\mathrel{\stackrel{\Delta}{=}}\vert H(e^{j\omega T})\vert$ is the amplitude response.

• Unwrap phase response $\Theta(\omega)$ to uniquely define it:
  • $\Theta(0)\mathrel{\stackrel{\Delta}{=}}0$ or $\pm\pi$ for real filters
  • Discontinuities in $\Theta(\omega)$ cannot exceed $\pm\pi$ radians
  • Phase jumps $\pm\pi$ radians are equivalent
  • See Matlab function unwrap