Allpassphase [patched] · High Speed
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 .
For a first-order all-pass:
[ \phi(\omega) = -\omega - 2 \arctan\left( \fraca \sin \omega1 + a \cos \omega \right) ] allpassphase
[ H(z) = \fracz^-N - a_1 z^-(N-1) - \dots - a_N1 + a_1 z^-1 + \dots + a_N z^-N ] These are called all-pass filters
[ H(z) = \fraca + z^-11 + a z^-1, \quad |a| < 1 ] \quad |a| <
| 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 |
where ( \omega ) is normalized frequency (0 to ( \pi )).