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 .
The phase is given by:
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: allpassphase
| 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 |
The name says it all: they pass all frequencies with unity gain (0 dB magnitude response). Their entire purpose lies in their . 2. Mathematical Definition An all-pass filter’s transfer function ( H(z) ) (in the discrete-time domain) has the general form: These are called all-pass filters
For a first-order all-pass:
[ \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) ] 1. Introduction In signal processing
where ( \omega ) is normalized frequency (0 to ( \pi )).
