A system has a phase response given by $\phi(\omega)$, where $w$ is the angular frequency. The phase delay and group delay at $\omega=\omega_{0}$ are respectively given by
- $ – \frac{\phi\left(\omega_{n}\right)}{\omega_{n}},-\left.\frac{d \phi(\omega)}{d \omega}\right|_{\omega=\omega_{0}}$
- $\phi\left(\omega_{0}\right),-\left.\frac{d^{2} \phi\left(\omega_{0}\right)}{d \omega^{2}}\right|_{\omega=\omega_{0}}$
- $\frac{\omega_{o}}{\phi\left(\omega_{o}\right)},-\left.\frac{d \phi(\omega))}{d\omega}\right|_{\omega=\omega_{0}}$
- $\omega_{o} \phi\left(\omega_{o}\right)_{1} \int_{-\infty}^{\omega_{\infty}} \phi(\lambda) d \lambda$