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:
For a second-order all-pass filter:
The pole-zero placement (complex conjugate pair) allows tuning of both the center frequency and bandwidth of the phase transition. While phase shift matters, the group delay ( \tau_g(\omega) = -\fracd\phi(\omega)d\omega ) often matters more in practical systems. allpassphase
[ H(z) = \fraca_2 + a_1 z^-1 + z^-21 + a_1 z^-1 + a_2 z^-2, \quad |a_2| < 1 ] The key property: poles and zeros are