[ \phi(\omega) = -2\omega - 2 \arctan\left( \fraca_1 \sin \omega + a_2 \sin 2\omega1 + a_1 \cos \omega + a_2 \cos 2\omega \right) ]
[ H(z) = \fraca_2 + a_1 z^-1 + z^-21 + a_1 z^-1 + a_2 z^-2, \quad |a_2| < 1 ] allpassphase
where ( \omega ) is normalized frequency (0 to ( \pi )). [ \phi(\omega) = -2\omega - 2 \arctan\left( \fraca_1
The key property: poles and zeros are . If a pole is at ( z = p ), a zero is at ( z = 1/p^* ). This reciprocal relationship ensures unity magnitude response for all frequencies. 3. Phase Response Characteristics First-Order All-Pass The phase response ( \phi(\omega) ) for a first-order all-pass is: \quad |a_2| <