More commonly, for a first-order all-pass filter:
[ \tau_g(\omega) = \frac1 - a^21 + 2a \cos \omega + a^2 ]
The phase is given by:
When she finally returned to Willowdale, the shop was gone, but the lessons she had learned and the connections she had made remained. The townsfolk looked at her differently, as if they saw the phases she had traversed in her eyes.
Allpass phase is a subtle art. It is invisible on a standard frequency analyzer (which only shows magnitude), but it is audible to trained ears. It is the secret behind the "space" in your reverb, the "sweep" of your phaser, and the integrity of your crossover. allpassphase
The phase transitions most rapidly near 2 kHz, where group delay peaks.
:
where ( \omega ) is normalized frequency (0 to ( \pi )).