An all-pass filter is a
signal processing filter
In signal processing, a filter is a device or process that removes some unwanted components or features from a signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspe ...
that passes all
frequencies
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
equally in gain, but changes the
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
relationship among various frequencies. Most types of filter reduce the amplitude (i.e. the magnitude) of the signal applied to it for some values of frequency, whereas the all-pass filter allows all frequencies through without changes in level.
Common applications
A common application in
electronic music
Electronic music is a genre of music that employs electronic musical instruments, digital instruments, or circuitry-based music technology in its creation. It includes both music made using electronic and electromechanical means ( electroac ...
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.
It does this by varying its
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
shift as a function of frequency. Generally, the filter is described by the frequency at which the
phase shift
In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it v ...
crosses 90° (i.e., when the input and output signals go into
quadrature – when there is a quarter
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
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
In signal processing, a comb filter is a filter implemented by adding a delayed version of a signal to itself, causing constructive and destructive interference. The frequency response of a comb filter consists of a series of regularly spaced no ...
.
They may also be used to convert a
mixed phase filter into a
minimum phase
In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.
The most general causal LTI transfer function can be uniquely factored into a series of a ...
filter with an equivalent magnitude response or an unstable filter into a stable filter with an equivalent magnitude response.
Active analog implementation
Implementation using low-pass filter
![Schem All-Pass Filter Producing Lag](https://upload.wikimedia.org/wikipedia/commons/3/3f/Schem_All-Pass_Filter_Producing_Lag.png)
The
operational amplifier
An operational amplifier (often op amp or opamp) is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op amp produces an output potential (relative to c ...
circuit shown in adjacent figure implements a single-pole
active
Active may refer to:
Music
* ''Active'' (album), a 1992 album by Casiopea
* Active Records, a record label
Ships
* ''Active'' (ship), several commercial ships by that name
* HMS ''Active'', the name of various ships of the British Royal ...
all-pass filter that features a
low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
at the non-inverting input of the opamp. The filter's
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
is given by:
:
which has one
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
at -1/RC and one
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
at 1/RC (i.e., they are ''reflections'' of each other across the
imaginary axis of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
). The
magnitude and phase of H(iω) for some
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
ω are
:
The filter has
unity
Unity may refer to:
Buildings
* Unity Building, Oregon, Illinois, US; a historic building
* Unity Building (Chicago), Illinois, US; a skyscraper
* Unity Buildings, Liverpool, UK; two buildings in England
* Unity Chapel, Wyoming, Wisconsin, US; a h ...
-
gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in de ...
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°).
This implementation uses a low-pass filter at the
non-inverting input to generate the phase shift and
negative feedback
Negative feedback (or balancing feedback) occurs when some function (Mathematics), function of the output of a system, process, or mechanism is feedback, fed back in a manner that tends to reduce the fluctuations in the output, whether caused by ...
.
* At high
frequencies
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, the
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
is a
short circuit
A short circuit (sometimes abbreviated to short or s/c) is an electrical circuit that allows a current to travel along an unintended path with no or very low electrical impedance. This results in an excessive current flowing through the circuit ...
, creating an
inverting amplifier
This article illustrates some typical operational amplifier applications. A non-ideal operational amplifier's equivalent circuit has a finite input impedance, a non-zero output impedance, and a finite gain. A real op-amp has a number of non-ideal f ...
(i.e., 180° phase shift) with unity gain.
* At low frequencies and
DC, the capacitor is an
open circuit Open circuit may refer to:
*Open-circuit scuba, a type of SCUBA-diving equipment where the user breathes from the set and then exhales to the surroundings without recycling the exhaled air
*Open-circuit test, a method used in electrical engineering ...
, creating a
unity
Unity may refer to:
Buildings
* Unity Building, Oregon, Illinois, US; a historic building
* Unity Building (Chicago), Illinois, US; a skyscraper
* Unity Buildings, Liverpool, UK; two buildings in England
* Unity Chapel, Wyoming, Wisconsin, US; a h ...
-
gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in de ...
voltage buffer (i.e., no phase shift).
* At the
corner frequency
In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced ( attenuated or reflected) rather than ...
ω=1/RC of the low-pass filter (i.e., when input frequency is 1/(2πRC)), the circuit introduces a 90° shift (i.e., output is in quadrature with input; the output appears to be delayed by a quarter
period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
from the input).
In fact, the phase shift of the all-pass filter is double the phase shift of the low-pass filter at its non-inverting input.
Interpretation as a Padé approximation to a pure delay
The Laplace transform of a pure delay is given by
:
where
is the delay (in seconds) and
is complex frequency. This can be approximated using a
Padé approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is ap ...
, as follows:
:
where the last step was achieved via a first-order
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
expansion of the numerator and denominator. By setting
we recover
from above.
Implementation using high-pass filter
The
operational amplifier
An operational amplifier (often op amp or opamp) is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op amp produces an output potential (relative to c ...
circuit shown in the adjacent figure implements a single-pole
active
Active may refer to:
Music
* ''Active'' (album), a 1992 album by Casiopea
* Active Records, a record label
Ships
* ''Active'' (ship), several commercial ships by that name
* HMS ''Active'', the name of various ships of the British Royal ...
all-pass filter that features a
high-pass filter
A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency d ...
at the non-inverting input of the opamp. The filter's
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...
is given by:
:
[Williams, A.B.; Taylor, F.J., Electronic Filter Design Handbook'', McGraw-Hill, 1995 , p. 10.7.]
which has one
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
at -1/RC and one
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
at 1/RC (i.e., they are ''reflections'' of each other across the
imaginary axis of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
). The
magnitude and phase of H(iω) for some
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
ω are
:
The filter has
unity
Unity may refer to:
Buildings
* Unity Building, Oregon, Illinois, US; a historic building
* Unity Building (Chicago), Illinois, US; a skyscraper
* Unity Buildings, Liverpool, UK; two buildings in England
* Unity Chapel, Wyoming, Wisconsin, US; a h ...
-
gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in de ...
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 lead is 90°).
This implementation uses a
high-pass filter
A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency d ...
at the
non-inverting input to generate the phase shift and
negative feedback
Negative feedback (or balancing feedback) occurs when some function (Mathematics), function of the output of a system, process, or mechanism is feedback, fed back in a manner that tends to reduce the fluctuations in the output, whether caused by ...
.
* At high
frequencies
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, the
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
is a
short circuit
A short circuit (sometimes abbreviated to short or s/c) is an electrical circuit that allows a current to travel along an unintended path with no or very low electrical impedance. This results in an excessive current flowing through the circuit ...
, thereby creating a
unity
Unity may refer to:
Buildings
* Unity Building, Oregon, Illinois, US; a historic building
* Unity Building (Chicago), Illinois, US; a skyscraper
* Unity Buildings, Liverpool, UK; two buildings in England
* Unity Chapel, Wyoming, Wisconsin, US; a h ...
-
gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in de ...
voltage buffer (i.e., no phase lead).
* At low frequencies and
DC, the capacitor is an
open circuit Open circuit may refer to:
*Open-circuit scuba, a type of SCUBA-diving equipment where the user breathes from the set and then exhales to the surroundings without recycling the exhaled air
*Open-circuit test, a method used in electrical engineering ...
and the circuit is an
inverting amplifier
This article illustrates some typical operational amplifier applications. A non-ideal operational amplifier's equivalent circuit has a finite input impedance, a non-zero output impedance, and a finite gain. A real op-amp has a number of non-ideal f ...
(i.e., 180° phase lead) with unity gain.
* At the
corner frequency
In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced ( attenuated or reflected) rather than ...
ω=1/RC of the high-pass filter (i.e., when input frequency is 1/(2πRC)), the circuit introduces a 90° phase lead (i.e., output is in quadrature with input; the output appears to be advanced by a quarter
period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
from the input).
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.
Voltage controlled implementation
The resistor can be replaced with a
FET
The field-effect transistor (FET) is a type of transistor that uses an electric field to control the flow of current in a semiconductor. FETs (JFETs or MOSFETs) are devices with three terminals: ''source'', ''gate'', and ''drain''. FETs contr ...
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 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.
Passive analog implementation
The benefit to implementing all-pass filters with
active components like
operational amplifiers
An operational amplifier (often op amp or opamp) is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op amp produces an output potential (relative to c ...
is that they do not require
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s, which are bulky and costly in
integrated circuit
An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tiny ...
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
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 a
constant-resistance network A constant-resistance network in electrical engineering is a network whose input resistance does not change with frequency when correctly terminated. Examples of constant resistance networks include:
* Zobel network
* Lattice phase equaliser
* Bo ...
(i.e., its
image impedance Image impedance is a concept used in electronic network design and analysis and most especially in filter design. The term ''image impedance'' applies to the impedance seen looking into a port of a network. Usually a two-port network is implied but ...
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
A landline (land line, land-line, main line, home phone, fixed-line, and wireline) is a telephone connection that uses metal wires or optical fiber telephone line for transmission, as distinguished from a mobile cellular network, which uses ...
s being used for
stereophonic sound
Stereophonic sound, or more commonly stereo, is a method of sound reproduction that recreates a multi-directional, 3-dimensional audible perspective. This is usually achieved by using two independent audio channels through a configuration ...
broadcasts. This application requires that the filter has a
linear phase
In signal processing, linear phase is a property of a filter where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time (usually delayed) by the sam ...
response with frequency (i.e., constant
group delay
In signal processing, group delay and phase delay are delay times experienced by a signal's various frequency components when the signal passes through a system that is linear time-invariant (LTI), such as a microphone, coaxial cable, amplifier, l ...
) over a wide bandwidth and is the reason for choosing this topology.
Digital implementation
A
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-tim ...
implementation of an all-pass filter with a complex pole at
is
:
which has a zero at
, where
denotes the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
. 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
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
). The placement of this pole-zero pair for a given
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
for
, leading to the
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation.
It can be considered as a discrete-tim ...
implementation
:
which is equivalent to the
difference equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
:
where