Transfer function
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In engineering, a transfer function (also known as system function^{[1]} or network function) of an electronic or control system component gives the device's output for each possible input.^{[2]}^{[3]}^{[4]} It is often represented as a graph, called a transfer curve or characteristic curve. The transfer function is used in the mathematical analysis of systems, particularly using the block diagram technique, in electronics and control theory.
The units of the transfer function depend on the device. For example, the transfer function of a twoport electronic circuit like an amplifier might be a graph of the voltage at the output as a function of the voltage applied to the input; the transfer function of an electromechanical actuator might be the displacement of the moveable arm as a function of current applied to the device; the transfer function of a photodetector might be the output voltage as a function of the luminous intensity of incident light of a given wavelength.
The term "transfer function" is also used in the frequency domain analysis of systems using transform methods such as the Laplace transform; here it means the amplitude of the output as a function of the frequency 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 is the Fourier transform of the point spread function (hence a function of spatial frequency).
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Linear timeinvariant systems
Transfer functions are commonly used in the analysis of systems such as singleinput singleoutput filters in the fields of signal processing, communication theory, and control theory. The term is often used exclusively to refer to linear timeinvariant (LTI) systems. Most real systems have nonlinear input/output characteristics, but many systems, when operated within nominal parameters (not "overdriven") have behavior close enough to linear that LTI system theory 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 (and ), which reduces the Laplace transforms 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 steadystate response of an LTI system, not the fleeting turnon and turnoff behaviors or stability issues. That is usually the case for signal processing and communication theory.
Thus, for continuoustime 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 discretetime systems, the relation between an input signal and output is dealt with using the ztransform, and then the transfer function is similarly written as and this is often referred to as the pulsetransfer function.^{[citation needed]}
Direct derivation from differential equations
Consider a linear differential equation 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, constantcoefficient differential equation can be found by trying . That substitution yields the 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^{[clarification needed]} between complex vs. real values, which is traditionally influenced^{[clarification needed]} by the interpretation of abs(H(s)) as the gain and atan(H(s)) as the phase lag. Other definitions of the transfer function are used: for example ^{[5]}
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 steadystate response and a transient response. The steadystate 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 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 steadystate output will be:
The frequency response (or "gain") G of the system is defined as the absolute value of the ratio of the output amplitude to the steadystate 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 timeinvariant 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 harmonic signal with a sinusoidal component with amplitude , angular frequency and phase , where arg is the argument
 where
is input to a linear timeinvariant system, then the corresponding component in the output is:
Note that, in a linear timeinvariant 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 describes this change for every frequency in terms of gain:
and phase shift:
The phase delay (i.e., the frequencydependent amount of delay introduced to the sinusoid by the transfer function) is:
The group delay (i.e., the frequencydependent 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 – sharpest cutoff (narrowest transition between pass band and stop band) for the given order
 Optimum "L" filter
 Gaussian filter – minimum group delay; gives no overshoot to a step function
 Hourglass filter
 Raisedcosine filter
Control engineering
In control engineering and control theory the transfer function is derived using the Laplace transform.
The transfer function was the primary tool used in classical control engineering. However, it has proven to be unwieldy for the analysis of multipleinput multipleoutput (MIMO) systems, and has been largely supplanted by state space representations for such systems.^{[citation needed]} In spite of this, a transfer matrix can be always 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 and transfer function methods was proposed by Howard H. Rosenbrock and is referred to as Rosenbrock system matrix.
Optics
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In optics, modulation transfer function indicates the capability of optical contrast transmission.
For example, when observing a series of blackwhitelight 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:
Nonlinear systems
Transfer functions do not properly exist for many nonlinear systems. For example, they do not exist for relaxation oscillators;^{[6]} however, describing functions can sometimes be used to approximate such nonlinear timeinvariant systems.
See also
 Analog computer
 Black box
 Bode plot
 Convolution
 Duhamel's principle
 Frequency response
 Impulse response
 Laplace transform
 LTI system theory
 Nyquist plot
 Operational amplifier
 Optical transfer function
 Proper transfer function
 Rosenbrock system matrix
 Semilog graph
 Signalflow graph
 Signal transfer function
References
 ^ Bernd Girod, Rudolf Rabenstein, Alexander Stenger, Signals and systems, 2nd ed., Wiley, 2001, ISBN 0471988006 p. 50
 ^ M. A. Laughton; D.F. Warne. Electrical Engineer's Reference Book (16 ed.). Newnes. pp. 14/9–14/10. ISBN 9780080523545.
 ^ E. A. Parr (1993). Logic Designer's Handbook: Circuits and Systems (2nd ed.). Newness. pp. 65–66. ISBN 9781483292809.
 ^ Ian Sinclair; John Dunton (2007). Electronic and Electrical Servicing: Consumer and Commercial Electronics. Routledge. p. 172. ISBN 9780750669887.
 ^ Birkhoff, Garrett; Rota, GianCarlo (1978). Ordinary differential equations. New York: John Wiley & Sons. ISBN 0471052248.^{[page needed]}
 ^ Valentijn De Smedt, Georges Gielen and Wim Dehaene (2015). Temperature and Supply VoltageIndependent Time References for Wireless Sensor Networks. Springer. p. 47. ISBN 9783319090030.
External links
 "Transfer function". PlanetMath.
 ECE 209: Review of Circuits as LTI Systems — Short primer on the mathematical analysis of (electrical) LTI systems.
 ECE 209: Sources of Phase Shift — Gives an intuitive explanation of the source of phase shift in two simple LTI systems. Also verifies simple transfer functions by using trigonometric identities.
 Transfer function model in Mathematica
 [1] for resonant frequency
 [2] frequency scaling op amp