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| An '''all-pass filter''' is a [[filter (signal processing)|signal processing filter]] that passes all [[Frequency|frequencies]] equally in gain, but changes the [[Phase (waves)|phase]] relationship between various frequencies. It does this by varying its [[phase (waves)|phase]] shift as a function of frequency. Generally, the filter is described by the frequency at which the [[Phase shifting|phase shift]] crosses 90° (i.e., when the input and output signals go into [[Quadrature phase|quadrature]] — when there is a quarter [[wavelength]] of delay between them).
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| 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]].
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| 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.
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| == Active analog implementation ==
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| [[Image:Active Allpass Filter.svg|thumb|Figure 1: Schematic of an [[op amp]] all-pass filter]]
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| The [[operational amplifier]] circuit shown in Figure 1 implements an [[Passivity (engineering)|active]] all-pass filter with the [[transfer function]]
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| :<math>H(s) = \frac{ sRC - 1 }{ sRC + 1 }, \,</math> | |
| which has one [[pole (complex analysis)|pole]] at -1/RC and one [[zero (complex analysis)|zero]] at 1/RC (i.e., they are ''reflections'' of each other across the [[imaginary number|imaginary]] axis of the [[complex plane]]). The [[complex plane|magnitude and phase]] of H(iω) for some [[angular frequency]] ω are
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| :<math>|H(i\omega)|=1 \quad \text{and} \quad \angle H(i\omega) = 180^{\circ} - 2 \arctan(\omega RC). \,</math>
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| As expected, the filter has [[unity (mathematics)|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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| This implementation uses a [[high-pass filter]] at the [[Operational amplifier#Circuit_notation|non-inverting input]] to generate the phase shift and [[negative feedback]] to compensate for the filter's [[attenuation]].
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| * At high [[frequency|frequencies]], the [[capacitor]] is a [[short circuit]], thereby creating a [[unity (mathematics)|unity]]-[[gain]] [[Operational amplifier applications#Voltage_follower|voltage buffer]] (i.e., no phase shift).
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| * At low frequencies and [[DC offset|DC]], the capacitor is an [[open circuit]]{{dn|date=July 2013}} and the circuit is an [[Operational amplifier applications#Inverting_amplifier|inverting amplifier]] (i.e., 180 degree phase shift) with unity gain.
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| * At the [[corner 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 quarter [[wavelength]]).
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| 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.
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| === Interpretation as a Padé approximation to a pure delay ===
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| The Laplace transform of a pure delay is given by
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| :<math> \exp\{-sT\},</math>
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| where <math>T</math> is the delay (in seconds) and <math>s\in\mathbb{C}</math> is complex frequency. This can be approximated using a [[Padé approximant]], as follows:
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| :<math> \exp\{-sT\} =\frac{\exp\{-sT/2\}}{\exp\{sT/2\} } \approx \frac{1-sT/2}{1+sT/2} ,</math>
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| where the last step was achieved via a first-order [[Taylor series]] expansion of the numerator and denominator. By setting <math>RC = T/2</math> we recover <math>H(s)</math> from above.
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| === Implementation using low-pass filter ===
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| 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 [[Negation (disambiguation)|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.
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| === Voltage controlled implementation ===
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| The resistor can be replaced with a [[field-effect transistor|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 (effect)|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 ([[low-frequency oscillation|LFO]]) ramps the control voltage to produce the characteristic swooshing sound.
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| === General usage ===
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| 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 [[Control theory#Stability|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 [[complex plane|plane]] can be canceled and reflected on the left-hand plane.
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| ==Passive analog implementation==
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| The benefit to implementing all-pass filters with [[Passivity (engineering)|active components]] like [[operational amplifiers]] is that they do not require [[inductor]]s, which are bulky and costly in [[integrated circuit]] designs. In other applications where inductors are readily available,
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| all-pass filters can be implemented entirely without active components. There are a number of circuit [[Topology (electronics)|topologies]] that can be used for this. The following are the most commonly used circuits.
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| ===Lattice filter===
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| [[Image:Lattice filter, low end correction.svg|thumb|200px|An all-pass filter using lattice topology]]
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| {{main|Lattice phase equaliser}}
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| 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 a [[constant-resistance network]] (i.e., its [[image impedance]] is constant over all frequencies).
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| ===T-section filter===
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| 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
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| 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.
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| ===Bridged T-section filter===
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| {{main|Bridged T delay equaliser}}
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| The bridged T topology is used for delay equalisation, particularly the differential delay between two [[landline]]s being used for [[stereophonic sound]] broadcasts. 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.
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| == Digital Implementation ==
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| A [[Z-transform]] implementation of an all-pass filter with a complex pole at <math>z_0</math> is
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| :<math>H(z) = \frac{z^{-1}-\overline{z_0}}{1-z_0z^{-1}} \ </math>
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| which has a zero at <math>1/\overline{z_0}</math>, where <math>\overline{z}</math> denotes the [[complex 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 [[complex plane|complex]] [[unit circle]]). The placement of this pole-zero pair for a given <math>z_0</math> 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.
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| To create an all-pass implementation with real coefficients, the complex all-pass filter can be cascaded with an all-pass that substitutes <math>\overline{z_0}</math> for <math>z_0</math>, leading to the [[Z-transform]] implementation
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| :<math>H(z)
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| =
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| \frac{z^{-1}-\overline{z_0}}{1-z_0z^{-1}} \times
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| \frac{z^{-1}-z_0}{1-\overline{z_0}z^{-1}}
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| =
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| \frac {z^{-2}-2\Re(z_0)z^{-1}+\left|{z_0}\right|^2} {1-2\Re(z_0)z^{-1}+\left|z_0\right|^2z^{-2}}, \ </math>
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| which is equivalent to the [[recurrence relation|difference equation]]
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| :<math>
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| y[k] - 2\Re(z_0) y[k-1] + \left|z_0\right|^2 y[k-2] =
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| x[k-2] - 2\Re(z_0) x[k-1] + \left|z_0\right|^2 x[k], \,</math>
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| where <math>y[k]</math> is the output and <math>x[k]</math> is the input at discrete time step <math>k</math>.
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| Filters such as the above can be cascaded with [[Control theory#Stability|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 <math>z_0</math>, a pole of an unstable system that is outside of the [[unit circle]] can be canceled and reflected inside the unit circle.
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| == See also ==
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| * [[Bridged T delay equaliser]]
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| * [[Lattice phase equaliser]]
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| * [[Minimum phase]]
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| * [[Hilbert transform]]
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| * [[High-pass filter]]
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| * [[Low-pass filter]]
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| * [[Band-stop filter]]
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| * [[Band-pass filter]]
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| == External links ==
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| * [http://ccrma.stanford.edu/~jos/pasp/Allpass_Filters.html JOS@Stanford on all-pass filters]
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| * [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.
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| {{Electronic filters}}
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| [[Category:Linear filters]]
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| [[Category:Filter frequency response]]
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| [[Category:Digital signal processing]]
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