allpass

An all-pass filter is a filter that passes all frequencies with equal gain while altering the phase of the signal. It preserves the amplitude spectrum of the input but changes the phase response, which, in turn, affects the signal’s time-domain characteristics such as group delay.

In discrete-time, a common first-order all-pass section has the transfer function H(z) = (a + z^-1) / (1 + a z^-1), where a is a real coefficient with |a| < 1 to ensure stability. In continuous-time, a first-order analog all-pass is H(s) = (s - a) / (s + a) with a > 0. A key property is that |H(e^jω)| = 1 for all frequencies ω, meaning no magnitude distortion.

Because the magnitude is unity, cascading multiple all-pass sections yields a desired phase response without changing the overall amplitude. This makes all-pass filters useful for shaping phase and group delay, such as time alignment of signals, phase correction, or creating phase-based audio effects.

Applications include audio processing (phase equalization, phasers, and compensation for phase distortions in complex filters), multichannel and stereo signal alignment, digital communications for phase equalization, and general signal processing where phase control is required without amplitude alteration. Higher-order all-pass filters can be constructed by cascading several first-order sections or via lattice realizations, offering tunable phase response with numerically robust implementations.

In summary, all-pass filters modify phase while preserving magnitude, enabling precise control of timing and phase relationships in a wide range of systems.