Editing Group delay and phase delay (section)

=== 1st order low- or high-pass RC filter example ===
The phase of a 1st-order [[low-pass filter]] formed by a [[RC circuit]] with [[cutoff frequency]] <math> \omega_o {=} \frac{1}{RC} </math> is:<ref>https://www.tedpavlic.com/teaching/osu/ece209/lab3_opamp_FO/lab3_opamp_FO_phase_shift.pdf {{Bare URL PDF|date=August 2024}}</ref>

<math display="block"> \phi(\omega) = -\arctan(\frac{\omega}{\omega_o}) \, . </math>

Similarly, the phase for a 1st-order RC [[high-pass filter]] is:

<math display="block"> \phi(\omega) = \frac{\pi}{2} -\arctan(\frac{\omega}{\omega_o}) \, . </math>

Taking the negative derivative with respect to <math> \omega </math> for either this low-pass or high-pass filter yields the same group delay of:<ref name="aolson"/>

<math display="block"> \begin{align}
\tau_g(\omega) &= \frac{\omega_o}{\omega^2 + \omega_o^2} \, . \\
\end{align} </math>

For frequencies significantly lower than the cutoff frequency, the phase response is approximately linear (arctan for small inputs can be approximated as a line), so the group delay simplifies to a constant value of:

<math display="block"> \begin{align}
\tau_g(\omega \ll \omega_o) &\approx \frac{1}{\omega_o} = RC \, . \\
\end{align} </math>

Similarly, right at the cutoff frequency, <math> \tau_g(\omega {=} \omega_o) = \frac{1}{2 \omega_o} = \frac{RC}{2} \, . </math>

As frequencies get even larger, the group delay decreases with the inverse square of the frequency and approaches zero as frequency approaches infinity.