A cascade of all-pass filters approximates a pure delay (useful in reverberation, phasers, flangers).
A Hilbert transformer is a special case of an allpass filter that shifts phase by -90 degrees for all positive frequencies. By combining a signal with its Hilbert transform, you generate the analytic signal (a complex representation with real and imaginary parts). This is the cornerstone of IQ modulation in 4G/5G radios, radar systems, and even electrocardiogram (ECG) analysis.
An all-pass filter is a signal processing block that passes all frequencies with unity magnitude gain (0 dB). Its only effect is to change the phase of the input signal as a function of frequency. allpassphase
Transfer function (analog, 1st order): [ H(s) = \fracs - \omega_0s + \omega_0 ]
Digital (1st order): [ H(z) = \fraca + z^-11 + a z^-1, \quad |a| < 1 ] A cascade of all-pass filters approximates a pure
Magnitude response: [ |H(j\omega)| = 1 \quad \textfor all \omega ]
The phase shift ( \phi(\omega) ) for the first-order analog all-pass is: [ \phi(\omega) = -2 \arctan\left(\frac\omega\omega_0\right) ] Higher-order all-pass filters are cascaded to achieve more
Higher-order all-pass filters are cascaded to achieve more complex phase shaping.