[ H(z) = \fraca_2 + a_1 z^-1 + z^-21 + a_1 z^-1 + a_2 z^-2, \quad |a_2| < 1 ]

[ \phi(\omega) = -\omega - 2 \arctan\left( \fraca \sin \omega1 + a \cos \omega \right) ] allpassphase

where ( \omega ) is normalized frequency (0 to ( \pi )). [ H(z) = \fraca_2 + a_1 z^-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: Introduction In signal processing, most filters are designed

| Frequency (Hz) | Phase (degrees) | Group Delay (samples) | |----------------|----------------|----------------------| | 0 | 0 | ≈0.28 | | 500 | -22 | 0.31 | | 2000 | -95 | 0.55 | | 5000 | -162 | 0.21 | | 10000 | -176 | 0.06 |

1. Introduction In signal processing, most filters are designed to modify the magnitude of a signal’s frequency components — boosting bass, cutting treble, or removing noise. But there exists a special class of filters that leaves the magnitude spectrum completely untouched while selectively shifting the phase of different frequencies. These are called all-pass filters .

Allpassphase | ((exclusive))

[ H(z) = \fraca_2 + a_1 z^-1 + z^-21 + a_1 z^-1 + a_2 z^-2, \quad |a_2| < 1 ]

[ \phi(\omega) = -\omega - 2 \arctan\left( \fraca \sin \omega1 + a \cos \omega \right) ]

where ( \omega ) is normalized frequency (0 to ( \pi )).

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:

| Frequency (Hz) | Phase (degrees) | Group Delay (samples) | |----------------|----------------|----------------------| | 0 | 0 | ≈0.28 | | 500 | -22 | 0.31 | | 2000 | -95 | 0.55 | | 5000 | -162 | 0.21 | | 10000 | -176 | 0.06 |

1. Introduction In signal processing, most filters are designed to modify the magnitude of a signal’s frequency components — boosting bass, cutting treble, or removing noise. But there exists a special class of filters that leaves the magnitude spectrum completely untouched while selectively shifting the phase of different frequencies. These are called all-pass filters .