All-pass filter

A common application in electronic music production is in the design of an effects unit known as a "phaser", where a number of all-pass filters are connected in sequence and the output mixed with the raw signal.

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.

The filter's transfer function is given by: 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 filter introduces a different delay at each frequency and reaches input-to-output quadrature at ω=1/RC (i.e., phase shift is 90°).

[2] This implementation uses a low-pass filter at the non-inverting input to generate the phase shift and negative feedback.

The filter's transfer function is given by: 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 filter introduces a different delay at each frequency and reaches input-to-output quadrature at ω=1/RC (i.e., phase lead is 90°).

This implementation uses a high-pass filter at the non-inverting input to generate the phase shift and negative feedback.

In electronic music, a phaser typically consists of two, four or six of these phase-shifting sections connected in tandem and summed with the original.

A low-frequency oscillator (LFO) ramps the control voltage to produce the characteristic swooshing sound.

The benefit to implementing all-pass filters with active components like operational amplifiers is that they do not require inductors, which are bulky and costly in integrated circuit designs.

In other applications where inductors are readily available, all-pass filters can be implemented entirely without active components.

While the circuit diagram may look like a low pass filter it is different in that the two inductor branches are mutually coupled.

This results in transformer action between the two inductors and an all-pass response even at high frequency.

This application requires that the filter has a linear phase response with frequency (i.e., constant group delay) over a wide bandwidth and is the reason for choosing this topology.

The pole and zero sit at the same angle but have reciprocal magnitudes (i.e., they are reflections of each other across the boundary of the complex unit circle).

can be rotated in the complex plane by any angle and retain its all-pass magnitude characteristic.

Complex pole-zero pairs in all-pass filters help control the frequency where phase shifts occur.