All-pass filter

An all-pass filter is a signal processing filter that passes all frequencies equally in gain, but changes the phase relationship among various frequencies. It does this by varying its phase shift as a function of frequency. Generally, the filter is described by the frequency at which the phase shift crosses 90° (i.e., when the input and output signals go into quadrature – when there is a quarter wavelength of delay between them).

They are generally used to compensate for other undesired phase shifts that arise in the system, or for mixing with an unshifted version of the original to implement a notch comb filter.

They may also be used to convert a mixed phase filter into a minimum phase filter with an equivalent magnitude response or an unstable filter into a stable filter with an equivalent magnitude response.

The operational amplifier circuit shown in Figure 1 implements an active all-pass filter with the transfer function

which has one pole at -1/RC and one zero at 1/RC (i.e., they are reflections of each other across the imaginary axis of the complex plane). The magnitude and phase of H(iω) for some angular frequency ω are

As expected, the filter has unity-gain magnitude for all ω. The filter introduces a different delay at each frequency and reaches input-to-output quadrature at ω=1/RC (i.e., phase shift is 90 degrees).

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