- All-pass filter
An

**all-pass filter**is anelectronic filter that passes all frequencies equally, but changes the phase relationship between various frequencies. It does this by varying itspropagation delay with 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 intoquadrature — when there is a quarterwavelength 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.**Active analog implementation**The

operational amplifier circuit shown in Figure 1 implements an active all-pass filter with thetransfer function :$H(s)\; riangleq\; frac\{\; sRC\; -\; 1\; \}\{\; sRC\; +\; 1\; \},\; ,$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 thecomplex plane ). The magnitude and phase of H(iω) for someangular frequency ω are:$|H(iomega)|=1\; quad\; ext\{and\}\; quad\; angle\; H(iomega)\; =\; 180^\{circ\}\; -\; 2\; arctan(omega\; RC).\; ,$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).This implementation uses a

high-pass filter at the non-inverting input to generate the phase shift andnegative feedback to compensate for the filter'sattenuation .

* At high frequencies, thecapacitor is ashort circuit , thereby creating a unity-gain voltage buffer (i.e., no phase shift).

* At low frequencies and DC, the capacitor is anopen circuit and the circuit is an inverting amplifier (i.e., 180 degree phase shift) with unity gain.

* At thecorner frequency ω=1/RC of the high-pass filter (i.e., when input frequency is 1/(2πRC)), the circuit introduces a 90 degree shift (i.e., output is in quadrature with input; it is delayed by a quarterwavelength ).In fact, the phase shift of the all-pass filter is double the phase shift of the high-pass filter at its non-inverting input.**Implementation using low-pass filter**A similar all-pass filter can be implemented by interchanging the position of the resistor and capacitor, which turns the high-pass filter into a

low-pass filter . The result is a phase shifter with the same quadrature frequency but a 180 degree shift at high frequencies and no shift at low frequencies. In other words, the transfer function is negated, and so it has the same pole at -1/RC and reflected zero at 1/RC. Again, the phase shift of the all-pass filter is double the phase shift of the first-order filter at its non-inverting input.**Voltage controlled implementation**The resistor can be replaced with a FET in its "ohmic mode" to implement a voltage-controlled phase shifter; the voltage on the gate adjusts the phase shift. In electronic music, a phaser typically consists of 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.

**General usage**These circuits are used as phase shifters and in systems of phase shaping and time delay. 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. For example, by proper choice of pole (and therefore zero), a pole of an unstable system that is in the right-hand plane can be canceled and reflected on the left-hand plane.

**Passive analog implementation**The benefit to implementing all-pass filters with active components like

operational amplifiers is that they do not requireinductor s, which are bulky and costly inintegrated circuit designs. In other applications where inductors are readily available, all-pass filters can be implemented entirely without active components. There are a number of circuit topologies that can be used for this. The following are the most commonly used circuits.**Lattice filter**The

**lattice phase equaliser**, or**filter**, is a filter composed of lattice, or X-sections. With single element branches it can produce a phase shift up to 180°, and with resonant branches it can produce phase shifts up to 360°. The filter is an example of aconstant-resistance network (i.e., itsimage impedance is constant over all frequencies).**T-section filter**The phase equaliser based on T topology is the unbalanced equivalent of the lattice filter and has the same phase response. 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.

**Bridged T-section filter**The bridged T topology is used for delay equalisation, particularly the differential delay between two

landline s being used forstereophonic sound broadcasts. This application requires that the filter has alinear phase response with frequency (i.e., constantgroup delay ) over a wide bandwidth and is the reason for choosing this topology.**Digital Implementation**A

Z-transform implementation of an all-pass filter with a complex pole at $z\_0$ is:$H(z)\; =\; frac\{z^\{-1\}-z\_0\}\{1-z\_0z^\{-1$which has a zero at $1/z\_0^*$, where $^*$ denotes thecomplex conjugate . 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 complexunit circle ). 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.To create an all-pass implementation with real coefficients, the complex all-pass filter can be cascaded with an all-pass that substitutes $z\_0^*$ for $z\_0$, leading to the

Z-transform implementation:$H(z)\; =\; frac\{z^\{-1\}-z\_0\}\{1-z\_0z^\{-1\; imes\; frac\{z^\{-1\}-z\_0^*\}\{1-z\_0^*z^\{-1=frac\; \{z^\{-2\}-2Re(z\_0)z^\{-1\}+left|\{z\_0\}\; ight|^2\}\; \{1-2Re(z\_0)z^\{-1\}+left|z\_0\; ight|^2z^\{-2,$which is equivalent to the difference equation:$y\; [k]\; -\; 2Re(z\_0)\; y\; [k-1]\; +\; left|z\_0\; ight|^2\; y\; [k-2]\; =x\; [k-2]\; -\; 2Re(z\_0)\; x\; [k-1]\; +\; left|z\_0\; ight|^2\; x\; [k]\; ,\; ,$where $y\; [k]$ is the output and $x\; [k]$ is the input at discrete time step $k$.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. For example, by proper choice of $z\_0$, a pole of an unstable system that is outside of the

unit circle can be canceled and reflected inside the unit circle.**See also***

All-pass transform

*Lattice phase equaliser

*Minimum phase

*Hilbert transform

*High-pass filter

*Low-pass filter

*Band-stop filter

*Band-pass filter **External links*** [

*http://ccrma.stanford.edu/~jos/pasp/Allpass_Filters.html JOS@Stanford on all-pass filters*]

* [*http://www.tedpavlic.com/teaching/osu/ece209/lab1_intro/lab1_intro_phase_shifter.pdf ECE 209 Phase-Shifter Circuit*] — Analysis steps for a common analog phase-shifter circuit.

*Wikimedia Foundation.
2010.*