In
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
and
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, a
linear, time-invariant system is said to be minimum-phase if the system and its
inverse are
causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
and
stable.
The most general
causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
LTI transfer function can be uniquely factored into a series of an all-pass and a minimum phase system. The system function is then the product of the two parts, and in the time domain the response of the system is the
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of the two part responses. The difference between a minimum phase and a general transfer function is that a minimum phase system has all of the poles and zeroes of its transfer function in the left half of the s-plane representation (in discrete time, respectively, inside the unit circle of the z-plane). Since inverting a system function leads to
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 ...
s turning to
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
es and vice versa, and poles on the right side (
s-plane
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the com ...
imaginary line
In general, an imaginary line is usually any sort of geometric line that has only an abstract definition and does not physically exist. In fact, they are used to properly identify places on a map.
Some outside geography do exist, such as th ...
) or outside (
z-plane 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 Eucli ...
) of the
complex plane lead to
unstable
In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...
systems, only the class of minimum phase systems is closed under inversion. Intuitively, the minimum phase part of a general causal system implements its amplitude response with minimum
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, amplifie ...
, while its
all pass
All or ALL may refer to:
Language
* All, an indefinite pronoun in English
* All, one of the English determiners
* Allar language (ISO 639-3 code)
* Allative case (abbreviated ALL)
Music
* All (band), an American punk rock band
* ''All'' (All ...
part corrects its
phase response
In signal processing, phase response is the relationship between the phase of a sinusoidal input and the output signal passing through any device that accepts input and produces an output signal, such as an amplifier or a filter.
Amplifiers, f ...
alone to correspond with the original system function.
The analysis in terms of poles and zeroes is exact only in the case of transfer functions which can be expressed as ratios of polynomials. In the continuous time case, such systems translate into networks of conventional, idealized
LCR networks. In discrete time, they conveniently translate into approximations thereof, using addition, multiplication, and unit delay. It can be shown that in both cases, system functions of rational form with increasing order can be used to efficiently approximate any other system function; thus even system functions lacking a rational form, and so possessing an infinitude of poles and/or zeroes, can in practice be implemented as efficiently as any other.
In the context of causal, stable systems, we would in theory be free to choose whether the zeroes of the system function are outside of the stable range (to the right or outside) if the closure condition wasn't an issue. However,
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
is of great practical importance, just as theoretically perfect factorizations are in their own right. (Cf. the spectral symmetric/antisymmetric decomposition as another important example, leading e.g. to
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...
techniques.) Many physical systems also naturally tend towards minimum phase response, and sometimes have to be inverted using other physical systems obeying the same constraint.
Insight is given below as to why this system is called minimum-phase, and why the basic idea applies even when the system function cannot be cast into a rational form that could be implemented.
Inverse system
A system
is invertible if we can uniquely determine its input from its output. I.e., we can find a system
such that if we apply
followed by
, we obtain the identity system
. (See
Inverse matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
for a finite-dimensional analog). That is,
Suppose that
is input to system
and gives output
.
Applying the inverse system
to
gives the following
So we see that the inverse system
allows us to determine uniquely the input
from the output
.
Discrete-time example
Suppose that the system
is a discrete-time,
linear, time-invariant (LTI) system described by the
impulse response for in . Additionally, suppose
has impulse response
. The cascade of two LTI systems is a
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
. In this case, the above relation is the following:
where
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
or the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
system in the discrete-time case. (Changing the order of
and
is allowed because of commutativity of the convolution operation.) Note that this inverse system
need not be unique.
Minimum phase system
When we impose the constraints of
causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
ity and
stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Linear stability
** Lyapunov stability
** Orbital stability
** Structural sta ...
, the inverse system is unique; and the system
and its inverse
are called minimum-phase. The causality and stability constraints in the discrete-time case are the following (for time-invariant systems where is the system's impulse response):
Causality
and
Stability
and
See the article on
stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Linear stability
** Lyapunov stability
** Orbital stability
** Structural sta ...
for the analogous conditions for the continuous-time case.
Frequency analysis
Discrete-time frequency analysis
Performing frequency analysis for the discrete-time case will provide some insight. The time-domain equation is the following:
Applying 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 ...
gives the following relation in the z-domain
From this relation, we realize that
For simplicity, we consider only the case of a
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
. Causality and stability imply that all
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in C ...
of must be strictly inside the
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 Eucli ...
(See
stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Linear stability
** Lyapunov stability
** Orbital stability
** Structural sta ...
). Suppose
where and are
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
in . Causality and stability imply that the
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in C ...
– the
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
s of – must be strictly inside the
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 Eucli ...
. We also know that
So, causality and stability for
imply that its
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in C ...
– the roots of – must be inside the
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 Eucli ...
. These two constraints imply that both the zeros and the poles of a minimum phase system must be strictly inside the unit circle.
Continuous-time frequency analysis
Analysis for the continuous-time case proceeds in a similar manner except that we use the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
for frequency analysis. The time-domain equation is the following.
where
is the
Dirac delta function. The
Dirac delta function is the identity operator in the continuous-time case because of the sifting property with any signal .
Applying the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
gives the following relation in the
s-plane
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the com ...
.
From this relation, we realize that
Again, for simplicity, we consider only the case of a
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
. Causality and stability imply that all
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in C ...
of must be strictly inside the left-half
s-plane
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the com ...
(See
stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Linear stability
** Lyapunov stability
** Orbital stability
** Structural sta ...
). Suppose
where and are
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
in . Causality and stability imply that the
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in C ...
– the
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
s of – must be inside the left-half
s-plane
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the com ...
. We also know that
So, causality and stability for
imply that its
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in C ...
– the roots of – must be strictly inside the left-half
s-plane
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the com ...
. These two constraints imply that both the zeros and the poles of a minimum phase system must be strictly inside the left-half
s-plane
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the com ...
.
Relationship of magnitude response to phase response
A minimum-phase system, whether discrete-time or continuous-time, has an additional useful property that the natural logarithm of the magnitude of the frequency response (the "gain" measured in
nepers which is proportional to
dB) is related to the phase angle of the frequency response (measured in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s) by the
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...
. That is, in the continuous-time case, let
be the complex frequency response of system . Then, only for a minimum-phase system, the phase response of is related to the gain by
where
denotes the Hilbert transform, and, inversely,
Stated more compactly, let
where
and
are real functions of a real variable. Then
and
The Hilbert transform operator is defined to be
An equivalent corresponding relationship is also true for discrete-time minimum-phase systems.
Minimum phase in the time domain
For all
causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
and
stable systems that have the same
magnitude response, the minimum phase system has its energy concentrated near the start of the
impulse response. i.e., it minimizes the following function which we can think of as the delay of energy in the
impulse response.
Minimum phase as minimum group delay
For all
causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
and
stable systems that have the same
magnitude response, the minimum phase system has the minimum
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, amplifie ...
. The following proof illustrates this idea of minimum
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, amplifie ...
.
Suppose we consider one
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
of the
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
. Let's place this
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
inside the
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 Eucli ...
(
) and see how the
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, amplifie ...
is affected.
Since the
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
contributes the factor
to the
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
, the phase contributed by this term is the following.
contributes the following to the
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, amplifie ...
.
The denominator and
are invariant to reflecting the
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
outside of the
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 Eucli ...
, i.e., replacing
with
. However, by reflecting
outside of the unit circle, we increase the magnitude of
in the numerator. Thus, having
inside the
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 Eucli ...
minimizes the
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, amplifie ...
contributed by the factor
. We can extend this result to the general case of more than one
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
since the phase of the multiplicative factors of the form
is additive. I.e., for a
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
with
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
s,
So, a minimum phase system with all
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
s inside the
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 Eucli ...
minimizes the
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, amplifie ...
since the
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, amplifie ...
of each individual
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
is minimized.
Non-minimum phase
Systems that are causal and stable whose inverses are causal and unstable are known as ''non-minimum-phase'' systems. A given non-minimum phase system will have a greater phase contribution than the minimum-phase system with the equivalent magnitude response.
Maximum phase
A ''maximum-phase'' system is the opposite of a minimum phase system. A causal and stable LTI system is a ''maximum-phase'' system if its inverse is causal and unstable. That is,
* The zeros of the discrete-time system are outside the
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 Eucli ...
.
* The zeros of the continuous-time system are in the right-hand side of the
complex plane.
Such a system is called a ''maximum-phase system'' because it has the maximum
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, amplifie ...
of the set of systems that have the same magnitude response. In this set of equal-magnitude-response systems, the maximum phase system will have maximum energy delay.
For example, the two continuous-time LTI systems described by the transfer functions
have equivalent magnitude responses; however, the second system has a much larger contribution to the phase shift. Hence, in this set, the second system is the maximum-phase system and the first system is the minimum-phase system. These systems are also famously known as nonminimum-phase systems that raise many stability concerns in control. One recent solution to these systems is moving the RHP zeros to the LHP using the PFCD method.
Mixed phase
A ''mixed-phase'' system has some of its
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
s inside the
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 Eucli ...
and has others outside the
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 Eucli ...
. Thus, its
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, amplifie ...
is neither minimum or maximum but somewhere between the
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, amplifie ...
of the minimum and maximum phase equivalent system.
For example, the continuous-time LTI system described by transfer function
is stable and causal; however, it has zeros on both the left- and right-hand sides of the
complex plane. Hence, it is a ''mixed-phase'' system. To control the transfer functions that include these systems some methods such as internal model controller (IMC), generalized Smith's predictor (GSP) and parallel feedforward control with derivative (PFCD)
are proposed.
Linear phase
A
linear-phase system has 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, amplifie ...
. Non-trivial linear phase or nearly linear phase systems are also mixed phase.
See also
*
All-pass filter
An all-pass filter is a signal processing filter that passes all frequencies equally in gain, but changes the phase relationship among various frequencies. Most types of filter reduce the amplitude (i.e. the magnitude) of the signal applied to i ...
A special non-minimum-phase case.
*
Kramers–Kronig relation Minimum phase system in physics
References
Further reading
*Dimitris G. Manolakis, Vinay K. Ingle, Stephen M. Kogon : ''Statistical and Adaptive Signal Processing'', pp. 54–56, McGraw-Hill,
*Boaz Porat : ''A Course in Digital Signal Processing'', pp. 261–263, John Wiley and Sons,
{{refend
Digital signal processing
Control theory