All-pass_filter

All-pass filter

Schematic of an op amp all-pass filter

An all-pass filter is an electronic filter that passes all frequencies equally, but changes the phase relationship between various frequencies. It does this by varying its propagation delay with frequency. Generally, the filter is described by the frequency at which the phase shift crosses 90°.

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.

Analog Implementation

The following operational amplifier circuit implements an all-pass filter with one pole and one zero. At high frequencies the capacitor is a short circuit, thereby creating a unity-gain voltage buffer. At low frequencies and DC, the capacitor is an open circuit and the circuit is an inverting amplifier with unity gain. Where ωCR=1, the circuit introduces a 90 degree shift.

The resistor has often been replaced with a FET to implement a voltage controlled phase shifter: the voltage on the gate adjusts the phase shift. In electronic music, a phaser consists of typically 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.

These are used as phase shifters and in systems of phase shaping and time delay.

Digital Implementation

A Z-transform implementation of an all-pass filter with a real zero and a real pole is

$H(z) = \frac{1-z_0z^{-1}}{1-z_0^{-1}z^{-1}} \$

The placement of this pole/zero pair for a given $z 0$ 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. A Z-transform expression of an all-pass filter with complex $z 0$ is

$H(z) = \frac{1-z_0z^{-1}}{1-(z_0^*)^{-1}z^{-1}} \$

To create an all-pass implementation with real coefficients, such a complex all-pass filter can be cascaded with an all-pass that substitutes $z_0^*$ for $z 0$ , leading to the Z-transform implementation

$H(z) = \frac {1-2\Re(z_0)z^{-1}+\left|{z_0}\right|^2z^{-2}} {1-\frac{2\Re(z_0)}{\left|z_0\right|^2}z^{-1}+\frac{1}{\left|z_0\right|^2}z^{-2}} \$

Filters such as the above can be cascaded with unstable or mixed phase filters to create a stable or minimum phase filter, without changing the magnitude response of the system. By proper choice of $z 0$ , a pole or zero that is outside of the unit circle can be canceled and reflected inside the unit circle.

Analog Implementation

Digital Implementation

See also