Minimum phase - AbsoluteAstronomy.com

Control theory

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

and signal processing

Signal processing

Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

, a linear, time-invariant

LTI system theory

Linear time-invariant system theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant...

system is said to be minimum-phase if the system and its inverse

Inverse

Inverse may refer to:* Inverse , a type of immediate inference from a conditional sentence* Inverse , a program for solving inverse and optimization problems...

are causal

Causal system

A causal system is a system where the output depends on past/current inputs but not future inputs i.e...

and stable

BIBO stability

In electrical engineering, specifically signal processing and control theory, BIBO stability is a form of stability for linear signals and systems that take inputs. BIBO stands for Bounded-Input Bounded-Output...

.

For example, a discrete-time system with rational

Rational function

In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

transfer function

Transfer function

A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...

can only satisfy causality and stability

BIBO stability

requirements if all of its poles are inside the unit circle

Unit circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

. However, we are free to choose whether the zero

Zero (complex analysis)

In complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...

s of the system are inside or outside the unit circle

Unit circle

. A system is minimum-phase if all its zeros are also inside the unit circle. Insight is given below as to why this system is called minimum-phase.

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 for a finite-dimensional analog). I.e.,

Suppose that

is input to system

and gives output

Applying the inverse system

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 theory

(LTI) system described by the impulse response

Impulse response

In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...

. Additionally,

has impulse response

. The cascade of two LTI systems is a convolution

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

. In this case, the above relation is the following:

where

is the Kronecker delta or the identity

Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...

system in the discrete-time case. Note that this inverse system

is not unique.

Minimum phase system

When we impose the constraints of causality and stability

BIBO stability

, 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 h is the system's impulse response):

Causality

and

Stability

and

See the article on stability

BIBO stability

for the analogous conditions for the continuous-time case.

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

Z-transform

In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....

gives the following relation in the z-domain.

From this relation, we realize that

For simplicity, we consider only the case of a rational

Rational function

transfer function

Transfer function

H (z). Causality and stability imply that all poles of H (z) must be strictly inside the unit circle

Unit circle

(See stability). Suppose

where A (z) and D (z) are polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

in z. Causality and stability imply that the poles

Zero (complex analysis)

– the roots of D (z) – must be strictly inside the unit circle

Unit circle

. We also know that

So, causality and stability for

imply that its poles – the roots of A (z) – must be inside the unit circle

Unit circle

. 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 for frequency analysis. The time-domain equation is the following.

where

is the Dirac delta function

Dirac delta function

The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

. The Dirac delta function

Dirac delta function

is the identity operator in the continuous-time case because of the sifting property with any signal x (t).

Applying the Laplace transform gives the following relation in the s-plane.

From this relation, we realize that

Again, for simplicity, we consider only the case of a rational

Rational function

transfer function

Transfer function

H(s). Causality and stability imply that all poles of H (s) must be strictly inside the left-half s-plane (See stability). Suppose

where A (s) and D (s) are polynomial

Polynomial

in s. Causality and stability imply that the poles – the roots of D (s) – must be inside the left-half s-plane. We also know that

So, causality and stability for

imply that its poles – the roots of A (s) – must be strictly inside the left-half s-plane. 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.

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 neper

Neper

The neper is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. It has the unit symbol Np. The unit's name is derived from the name of John Napier, the inventor of logarithms...

s which is proportional to dB

Decibel

The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

) is related to the phase angle of the frequency response (measured in radian

Radian

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s) by the Hilbert transform

Hilbert transform

In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...

. That is, in the continuous-time case, let

be the complex frequency response of system H(s). Then, only for a minimum-phase system, the phase response of H(s) is related to the gain by

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 and stable

BIBO stability

systems that have the same magnitude response

Frequency response

Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...

, the minimum phase system has its energy concentrated near the start of the impulse response

Impulse response

. i.e., it minimizes the following function which we can think of as the delay of energy in the impulse response

Impulse response

Minimum phase as minimum group delay

For all causal and stable

BIBO stability

systems that have the same magnitude response

Frequency response

, the minimum phase system has the minimum group delay

Group delay

Group delay is a measure of the time delay of the amplitude envelopes of the various sinusoidal components of a signal through a device under test, and is a function of frequency for each component...

. The following proof illustrates this idea of minimum group delay

Group delay

.

Suppose we consider one zero

Zero (complex analysis)

of the transfer function

Transfer function

. Let's place this zero

Zero (complex analysis)

inside the unit circle

Unit circle

(

) and see how the group delay

Group delay

is affected.

Since the zero

Zero (complex analysis)

contributes the factor

to the transfer function

Transfer function

, the phase contributed by this term is the following.

contributes the following to the group delay

Group delay

The denominator and

are invariant to reflecting the zero

Zero (complex analysis)

outside of the unit circle

Unit circle

, 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

Unit circle

minimizes the group delay

Group delay

contributed by the factor

. We can extend this result to the general case of more than one zero

Zero (complex analysis)

since the phase of the multiplicative factors of the form

is additive. I.e., for a transfer function

Transfer function

with

zero

Zero (complex analysis)

So, a minimum phase system with all zero

Zero (complex analysis)

s inside the unit circle

Unit circle

minimizes the group delay

Group delay

since the group delay

Group delay

of each individual zero

Zero (complex analysis)

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
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

.
The zeros of the continuous-time system are in the right-hand side of the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

.

Such a system is called a maximum-phase system because it has the maximum group delay

Group delay

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 first 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.

Mixed phase

A mixed-phase system has some of its zero

Zero (complex analysis)

s inside the unit circle

Unit circle

and has others outside the unit circle

Unit circle

. Thus, its group delay

Group delay

is neither minimum or maximum but somewhere between the group delay

Group delay

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

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

. Hence, it is a mixed-phase system.

Linear phase

A linear-phase

Linear phase

Linear phase is a property of a filter, where the phase response of the filter is a linear function of frequency, excluding the possibility of wraps at \pm\pi. In a causal system, perfect linear phase can be achieved with a discrete-time FIR filter...

system has constant group delay

Group delay

. Non-trivial linear phase or nearly linear phase systems are also mixed phase.

Inverse system

Discrete-time example

Minimum phase system

Causality

Stability

Discrete-time frequency analysis

Continuous-time frequency analysis

Relationship of magnitude response to phase response

Minimum phase in the time domain

Minimum phase as minimum group delay

Non-minimum phase

Maximum phase

Mixed phase

Linear phase

See also

Further reading