In
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, a transfer function (also known as system function or network function) of a system, sub-system, or component is a
mathematical function
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
that
theoretically models the system's output for each possible input.
They are widely used in
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
and
control systems. In some simple cases, this function is a two-dimensional
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
of an independent
scalar input versus the dependent scalar output, called a transfer curve or characteristic curve. Transfer functions for components are used to design and analyze systems assembled from components, particularly using the
block diagram
A block diagram is a diagram of a system in which the principal parts or functions are represented by blocks connected by lines that show the relationships of the blocks. technique, in electronics and
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 ...
.
The dimensions and units of the transfer function model the output response of the device for a range of possible inputs. For example, the transfer function of a
two-port electronic circuit like an
amplifier
An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the magnitude of a signal (a time-varying voltage or current). It may increase the power significantly, or its main effect may be to boost t ...
might be a two-dimensional graph of the scalar voltage at the output as a function of the scalar voltage applied to the input; the transfer function of an electromechanical
actuator might be the mechanical displacement of the movable arm as a function of electrical current applied to the device; the transfer function of a
photodetector
Photodetectors, also called photosensors, are sensors of light or other electromagnetic radiation. There is a wide variety of photodetectors which may be classified by mechanism of detection, such as photoelectric or photochemical effects, or ...
might be the output voltage as a function of the
luminous intensity
In photometry, luminous intensity is a measure of the wavelength-weighted power emitted by a light source in a particular direction per unit solid angle, based on the luminosity function, a standardized model of the sensitivity of the human e ...
of incident light of a given
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, t ...
.
The term "transfer function" is also used in the
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
analysis of systems using transform methods such as 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 ...
; here it means the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of the output as a function of the
frequency
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 ...
of the input signal. For example, the transfer function of an
electronic filter is the voltage amplitude at the output as a function of the frequency of a constant amplitude
sine wave applied to the input. For optical imaging devices, the
optical transfer function
The optical transfer function (OTF) of an optical system such as a camera, microscope, human eye, or projector specifies how different spatial frequencies are captured or transmitted. It is used by optical engineers to describe how the optics pro ...
is the
Fourier transform of the
point spread function (hence a function of
spatial frequency
In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components (as determined by the Fourier tra ...
).
Linear time-invariant systems
Transfer functions are commonly used in the analysis of systems such as
single-input single-output
In control engineering, a single-input and single-output (SISO) system is a simple single variable control system with one input and one output. In radio it is the use of only one antenna both in the transmitter and receiver.
Details
SISO sys ...
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
s in the fields of
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, ...
,
communication theory, and
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 ...
. The term is often used exclusively to refer to
linear time-invariant
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defin ...
(LTI) systems. Most real systems have
non-linear input/output characteristics, but many systems, when operated within nominal parameters (not "over-driven") have behavior close enough to linear that
LTI system theory
LTI can refer to:
* '' LTI – Lingua Tertii Imperii'', a book by Victor Klemperer
* Language Technologies Institute, a division of Carnegie Mellon University
* Linear time-invariant system, an engineering theory that investigates the response o ...
is an acceptable representation of the input/output behavior.
The descriptions below are given in terms of a complex variable,
, which bears a brief explanation. In many applications, it is sufficient to define
(thus
), which reduces 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 ...
s with complex arguments to
Fourier transforms with real argument ω. The applications where this is common are ones where there is interest only in the steady-state response of an LTI system, not the fleeting turn-on and turn-off behaviors or stability issues. That is usually the case for
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, ...
and
communication theory.
Thus, for
continuous-time input signal
and output
, the transfer function
is the linear mapping of the Laplace transform of the input,
, to the Laplace transform of the output
:
:
or
:
In
discrete-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
systems, the relation between an input signal
and output
is dealt with using 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 ...
, and then the transfer function is similarly written as
and this is often referred to as the pulse-transfer function.
Direct derivation from differential equations
Consider a
linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
with constant coefficients
:
where ''u'' and ''r'' are suitably smooth functions of ''t'', and ''L'' is the operator defined on the relevant function space, that transforms ''u'' into ''r''. That kind of equation can be used to constrain the output function ''u'' in terms of the ''forcing'' function ''r''. The transfer function can be used to define an operator
that serves as a right inverse of ''L'', meaning that
.
Solutions_of_the_''homogeneous'',_Linear_differential_equation#Homogeneous_equations_with_constant_coefficients.html" ;"title=".html" ;"title="
[r_=_r.
Solutions_of_the_''homogeneous'',_Linear_differential_equation#Homogeneous_equations_with_constant_coefficients">constant-coefficient_differential_equation_
_can_be_found_by_trying_
._That_substitution_yields_the_Characteristic_equation_(calculus).html" ;"title="">[r = r.
Solutions of the ''homogeneous'', Linear differential equation#Homogeneous equations with constant coefficients">constant-coefficient differential equation
can be found by trying
. That substitution yields the Characteristic equation (calculus)">characteristic polynomial
:
The inhomogeneous case can be easily solved if the input function ''r'' is also of the form
. In that case, by substituting
one finds that
if we define
:
Taking that as the definition of the transfer function requires careful disambiguation between complex vs. real values, which is traditionally influenced by the interpretation of abs(''H''(''s'')) as the
gain and −atan(''H''(''s'')) as the
phase lag
In the aerodynamics of rotorcraft like helicopters, phase lag refers to the angular difference between the point at which a control input to a rotor blade occurs and the point of maximum displacement of the blade in response to that control input ...
. Other definitions of the transfer function are used: for example
Gain, transient behavior and stability
A general sinusoidal input to a system of frequency
may be written
. The response of a system to a sinusoidal input beginning at time
will consist of the sum of the steady-state response and a transient response. The steady-state response is the output of the system in the limit of infinite time, and the transient response is the difference between the response and the steady state response (it corresponds to the homogeneous solution of the above differential equation). The transfer function for an LTI system may be written as the product:
:
where ''s
Pi'' are the ''N'' roots of the characteristic polynomial and will therefore be 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 ...
of the transfer function. Consider the case of a transfer function with a single pole
where
. The Laplace transform of a general sinusoid of unit amplitude will be
. The Laplace transform of the output will be
and the temporal output will be the inverse Laplace transform of that function:
:
The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity if ''σ
P'' is positive. In order for a system to be stable, its transfer function must have no poles whose real parts are positive. If the transfer function is strictly stable, the real parts of all poles will be negative, and the transient behavior will tend to zero in the limit of infinite time. The steady-state output will be:
:
The
frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...
(or "gain") ''G'' of the system is defined as the absolute value of the ratio of the output amplitude to the steady-state input amplitude:
:
which is just the absolute value of the transfer function
evaluated at
. This result can be shown to be valid for any number of transfer function poles.
Signal processing
Let
be the input to a general
linear time-invariant system, and
be the output, and the
bilateral Laplace transform of
and
be
:
Then the output is related to the input by the transfer function
as
:
and the transfer function itself is therefore
:
In particular, if a
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 ...
harmonic signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
with a
sinusoidal component with
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
,
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 ...
and
phase , where arg is the
argument
:
:where
is input to a
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
time-invariant system, then the corresponding component in the output is:
:
Note that, in a linear time-invariant system, the input frequency
has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system. The
frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...
describes this change for every frequency
in terms of ''gain'':
:
and ''phase shift'':
:
The
phase 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, ...
(i.e., the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is:
:
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 ...
(i.e., the frequency-dependent amount of delay introduced to the envelope of the sinusoid by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency
,
:
The transfer function can also be shown using the
Fourier transform which is only a special case of the
bilateral Laplace transform for the case where
.
Common transfer function families
While any LTI system can be described by some transfer function or another, there are certain "families" of special transfer functions that are commonly used.
Some common transfer function families and their particular characteristics are:
*
Butterworth filter – maximally flat in passband and stopband for the given order
*
Chebyshev filter (Type I) – maximally flat in stopband, sharper cutoff than a Butterworth filter of the same order
*
Chebyshev filter (Type II) – maximally flat in passband, sharper cutoff than a Butterworth filter of the same order
*
Bessel filter – best pulse response for a given order because it has no group delay ripple
*
Elliptic filter
An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The ...
– sharpest cutoff (narrowest transition between pass band and stop band) for the given order
*
Optimum "L" filter
The Optimum "L" filter (also known as a Legendre–Papoulis filter) was proposed by Athanasios Papoulis in 1958. It has the maximum roll off rate for a given filter order while maintaining a monotonic frequency response. It provides a compromise ...
*
Gaussian filter
In electronics and signal processing mainly in digital signal processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response would have infinite impulse respons ...
– minimum group delay; gives no overshoot to a step function
*
Hourglass filter
*
Raised-cosine filter The raised-cosine filter is a filter frequently used for pulse-shaping in digital modulation due to its ability to minimise intersymbol interference (ISI). Its name stems from the fact that the non-zero portion of the frequency spectrum of its simp ...
Control engineering
In
control engineering
Control engineering or control systems engineering is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with desired behaviors in control environments. The discipline of controls o ...
and
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 ...
the transfer function is derived using 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 ...
.
The transfer function was the primary tool used in classical control engineering. However, it has proven to be unwieldy for the analysis of
multiple-input multiple-output (MIMO) systems, and has been largely supplanted by
state space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory.
For instance, the to ...
representations for such systems. In spite of this, a
transfer matrix can always be obtained for any linear system, in order to analyze its dynamics and other properties: each element of a transfer matrix is a transfer function relating a particular input variable to an output variable.
A useful representation bridging
state space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory.
For instance, the to ...
and transfer function methods was proposed by
Howard H. Rosenbrock and is referred to as
Rosenbrock system matrix.
Optics
In optics,
modulation transfer function indicates the capability of optical contrast transmission.
For example, when observing a series of black-white-light fringes drawn with a specific spatial frequency, the image quality may decay. White fringes fade while black ones turn brighter.
The modulation transfer function in a specific spatial frequency is defined by
:
where modulation (M) is computed from the following image or light brightness:
:
Imaging
In
imaging
Imaging is the representation or reproduction of an object's form; especially a visual representation (i.e., the formation of an image).
Imaging technology is the application of materials and methods to create, preserve, or duplicate images.
...
, transfer functions are used to describe the relationship between the scene light, the image signal and the displayed light.
Non-linear systems
Transfer functions do not properly exist for many
non-linear systems
Non-Linear Systems is an electronics manufacturing company based in San Diego, California. Non-Linear Systems was founded in 1952, by Andrew Kay, the inventor of the digital voltmeter in 1954.relaxation oscillator
In electronics a relaxation oscillator is a nonlinear electronic oscillator circuit that produces a nonsinusoidal repetitive output signal, such as a triangle wave or square wave. on Peter Millet'Tubebookswebsite The circuit consists of a feedb ...
s;
however,
describing functions can sometimes be used to approximate such nonlinear time-invariant systems.
See also
References
External links
ECE 209: Review of Circuits as LTI Systems— Short primer on the mathematical analysis of (electrical) LTI systems.
{{Authority control
Electrical circuits
Frequency-domain analysis
Types of functions