all-pass filter
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An all-pass filter is a filter (signal processing) that passes all Frequency equally, but changes the Phase (waves) 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 shifting 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.

Active analog implementation

Image:Active Allpass Filter.svg all-pass filter]] The operational amplifier circuit shown in Figure 1 implements an Passivity (engineering) all-pass filter with the transfer function :$H(s)\; \backslash triangleq\; \backslash frac\; sRC\; -\; 1\; \}\; sRC\; +\; 1\; \},\; \backslash ,$ which has one pole (complex analysis) at -1/RC and one zero (complex analysis) at 1/RC (i.e., they are reflectionsof each other across the imaginary number axis of the complex plane . The complex plane of H(iω) for some angular frequency ω are :$|H(i\backslash omega)|1\; \backslash quad\; \backslash textand\}\; \backslash quad\; \backslash angle\; H(i\backslash omega)\; 180^\backslash circ\}\; -\; 2\; \backslash arctan(\backslash omega\; RC).\; \backslash ,$ As expected, the filter has unity (mathematics) gain magnitude for all ω. The filter introduces a different delay at each frequency and reaches input-to-output [[quadrature]]at ω1/RC (i.e., phase shift is 90 degrees). This implementation uses a high-pass filter at the Operational amplifier#Circuit_notation to generate the phase shift and negative feedback to compensate for the filters attenuation * At high frequency the capacitor is a short circuit thereby creating a unity (mathematics) gain Operational amplifier applications#Voltage_follower (i.e., no phase shift). * At low frequencies and DC offset the capacitor is an open circuit and the circuit is an Operational amplifier applications#Inverting_amplifier (i.e., 180 degree phase shift) with unity gain. * At the corner frequency ω1/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 . 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 Negation (disambiguation) 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 field-effect transistor in its ohmic modeto implement a voltage-controlled phase shifter; the voltage on the gate adjusts the phase shift. In electronic music, a phaser (effect) typically consists of four or six of these phase-shifting sections connected in tandem and summed with the original. A low-frequency oscillator (low-frequency oscillation 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 Control theory#Stability 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 can be canceled and reflected on the left-hand plane.

Passive analog implementation

The benefit to implementing all-pass filters with Passivity (engineering) like operational amplifiers is that they do not require inductor , 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. There are a number of circuit Topology (electronics) that can be used for this. The following are the most commonly used circuits.

Lattice filter

Image:Lattice filter, low end correction.svg 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).

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 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.

Digital Implementation

A Z-transform implementation of an all-pass filter with a complex pole at $z\_0$ is :$H(z)\; \backslash fracz^-1\}-z\_0^*\}1-z\_0z^-1\}\}\; \backslash$ which has a zero at $1/z\_0^*$, where $^*$ denotes the complex conjugate The pole and zero sit at the same angle but have reciprocal magnitudes (i.e., they are reflectionsof each other across the boundary of the complex plane unit 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 :$H(z)\; \backslash fracz^-1\}-z\_0^*\}1-z\_0z^-1\}\}\; \backslash times\; \backslash fracz^-1\}-z\_0\}1-z\_0^*z^-1\}\}\; \backslash frac\; z^-2\}-2\backslash Re(z\_0)z^-1\}+\backslash left|z\_0\}\backslash right|^2\}\; 1-2\backslash Re(z\_0)z^-1\}+\backslash left|z\_0\backslash right|^2z^-2\}\},\; \backslash$ which is equivalent to the recurrence relation :$yk]\; -\; 2\backslash Re(z\_0)\; yk-1]\; +\; \backslash left|z\_0\backslash right|^2\; yk-2]\; xk-2]\; -\; 2\backslash Re(z\_0)\; xk-1]\; +\; \backslash left|z\_0\backslash right|^2\; xk],\; \backslash ,$ where $yk]$ is the output and $xk]$ is the input at discrete time step $k$. Filters such as the above can be cascaded with Control theory#Stability 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

* Bridged T delay equaliser * 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. Category:Linear filters Category:Filter frequency response Category:Digital signal processing de:Allpassfilter it:Filtro passa tutto ru:Фазовый фильтр uk:Фазовий фільтр