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In engineering, a '''transfer function''' (also known as the '''system function'''<ref>Bernd Girod, Rudolf Rabenstein, Alexander Stenger, ''Signals and systems'', 2nd ed., Wiley, 2001, ISBN 0-471-98800-6 p. 50</ref> or '''network function''' and, when plotted as a [[Line chart|graph]], '''transfer curve''') is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a [[LTI system theory|linear]] [[time-invariant system]] with zero initial conditions and zero-point equilibrium.<ref>The Oxford Dictionary of English, 3rd ed., "Transfer function"</ref> With [[optical transfer function|optical imaging devices]], for example, it is the [[Fourier transform]] of the [[point spread function]] (hence a function of [[spatial frequency]]) i.e. the intensity distribution caused by a point object in the field of view.
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== Explanation ==
Transfer functions are commonly used in the analysis of systems such as [[single-input single-output]] [[Filter (signal processing)|filter]]s, typically within the fields of [[signal processing]], [[communication theory]], and [[control theory]].  The term is often used exclusively to refer to [[LTI system|linear, time-invariant systems]] (LTI), as covered in this article. Most real systems have [[non-linear]] input/output characteristics, but many systems, when operated within nominal parameters (not "over-driven") have behavior that is 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, s = σ + j*ω, which bears a brief explanation.  In many applications, it is sufficient to define σ=0 (and s = j*ω), which reduces the [[Laplace transform]]s with complex arguments to [[Fourier transform]]s 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]] and [[communication theory]].
 
Thus, for [[continuous-time]] input signal <math>x(t)</math> and output <math>y(t)</math>, the transfer function <math>H(s)</math> is the linear mapping of the Laplace transform of the input, <math>X(s) = \mathcal{L}\left\{x(t)\right\}</math>, to the Laplace transform of the output <math>Y(s) = \mathcal{L}\left\{y(t)\right\}</math>:
 
:<math> Y(s) = H(s)\;X(s) </math>
or
:<math> H(s) = \frac{Y(s)}{X(s)} = \frac{ \mathcal{L}\left\{y(t)\right\} }{ \mathcal{L}\left\{x(t)\right\} } </math>
 
In [[discrete-time]] systems, the relation between an  input signal <math>x(t)</math> and output <math>y(t)</math> is dealt with using the [[z-transform]], and then the transfer function is similarly written as <math>H(z) = \frac{Y(z)}{X(z)}</math>  and this is often referred to as the pulse-transfer function.{{cn|date=April 2013}}
 
=== Direct derivation from differential equations ===
Consider a [[linear differential equation]] with constant coefficients
 
:<math> L[u] = \frac{d^nu}{dt^n} + a_1\frac{d^{n-1}u}{dt^{n-1}} + \dotsb + a_{n-1}\frac{du}{dt} + a_nu = r(t) </math>
 
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, written as an operator <math>F[r] = u </math>, is the right inverse of ''L'', since <math>L[F[r]] = r</math>.
 
Solutions of the ''homogeneous'', [[Linear differential equation#Homogeneous equations with constant coefficients| constant-coefficient differential equation]] <math>L[u] = 0</math> can be found by trying <math>u = e^{\lambda t}</math>. That substitution yields the ''characteristic polynomial''
 
:<math> p_L(\lambda) = \lambda^n + a_1\lambda^{n-1} + \dotsb + a_{n-1}\lambda + a_n\,</math>
 
The inhomogeneous case can be easily solved if the input function ''r'' is also of the form <math>r(t) = e^{s t}</math>. In that case, by substituting <math>u = H(s)e^{s t}</math> one finds that <math>L[H(s) e^{s t}] = e^{s t}</math> if and only if
 
:<math>H(s) = \frac{1}{p_L(s)}, \qquad p_L(s) \neq 0.</math>
 
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 (waves)#Phase difference|phase lag]]. Other definitions of the transfer function are used: for example <math>1/p_L(ik) .</math><ref>{{cite book |title= Ordinary differential equations|last= Birkhoff |first= Garrett|coauthors= Rota, Gian-Carlo|year=1978|publisher=John Wiley & Sons |location= New York|isbn= 0-471-05224-8}}{{pn|date=April 2013}}</ref>
 
==Signal processing==
 
Let <math> x(t) \ </math> be the input to a general [[LTI system theory|linear time-invariant system]], and <math> y(t) \ </math> be the output, and the [[bilateral Laplace transform]] of <math> x(t) \ </math> and <math> y(t) \ </math> be
 
: <math> X(s)  =  \mathcal{L}\left \{ x(t) \right \} \ \stackrel{\mathrm{def}}{=}\  \int_{-\infty}^{\infty} x(t) e^{-st}\, dt  </math>
 
: <math> Y(s)  =  \mathcal{L}\left \{ y(t) \right \} \ \stackrel{\mathrm{def}}{=}\  \int_{-\infty}^{\infty} y(t) e^{-st}\, dt </math>.
 
Then the output is related to the input by the transfer function <math> H(s) \ </math> as
 
:: <math> Y(s) = H(s) X(s)  \, </math>
 
and the transfer function itself is therefore
 
:: <math> H(s) = \frac{Y(s)} {X(s)} </math> .
 
In particular, if a [[complex number|complex]] [[harmonic]] [[signal (information theory)|signal]] with a [[sinusoidal]] component with [[amplitude]] <math>|X| \ </math>, [[angular frequency]] <math>\omega \ </math> and [[Phase (waves)|phase]] <math>\arg(X) \ </math>
 
:<math> x(t) = Xe^{j\omega t} = |X|e^{j(\omega t + \arg(X))} </math>
 
:where <math> X = |X|e^{j\arg(X)}  </math>
 
is input to a [[linear]] time-invariant system, then the corresponding component in the output is:
 
:<math> y(t) = Ye^{j\omega t} = |Y|e^{j(\omega t + \arg(Y))} </math>
 
:and <math> Y = |Y|e^{j\arg(Y)}. </math>
 
Note that, in a linear time-invariant system, the input frequency <math> \omega \ </math> has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system.  The [[frequency response]] <math> H(j \omega) \ </math> describes this change for every frequency <math> \omega \ </math> in terms of ''gain'':
 
:<math>G(\omega) = \frac{|Y|}{|X|} = |H(j \omega)| \ </math>
 
and ''phase shift'':
 
:<math>\phi(\omega) =  \arg(Y) -  \arg(X) = \arg( H(j \omega))</math>.
 
The [[phase delay]] (i.e., the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is:
 
:<math>\tau_{\phi}(\omega) = -\begin{matrix}\frac{\phi(\omega)}{\omega}\end{matrix}</math>.
 
The [[group delay]] (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 <math> \omega \ </math>,
 
:<math>\tau_{g}(\omega) = -\begin{matrix}\frac{d\phi(\omega)}{d\omega}\end{matrix}</math>.
 
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 <math> s = j \omega </math>.
 
=== 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.
Typical [[infinite impulse response]] filters are designed to implement one of these special transfer functions.
 
Some common transfer function families and their particular characteristics are:
 
*[[Butterworth filter]]&nbsp;&ndash; maximally flat in passband and stopband for the given order
*[[Chebyshev filter|Chebyshev filter (Type I)]]&nbsp;&ndash; maximally flat in stopband, sharper cutoff than Butterworth of same order
*[[Chebyshev filter|Chebyshev filter (Type II)]]&nbsp;&ndash; maximally flat in passband, sharper cutoff than Butterworth of same order
*[[Bessel filter]]&nbsp;&ndash; best pulse response for a given order because it has no group delay ripple
*[[Elliptic filter]]&nbsp;&ndash; sharpest cutoff (narrowest transition between pass band and stop band) for the given order
*[[Optimum "L" filter]]
*[[Gaussian filter]]&nbsp;&ndash; minimum group delay; gives no overshoot to a step function.
*[[Hourglass filter]]
*[[Raised-cosine 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 multiple-input multiple-output (MIMO) systems, and has been largely supplanted by [[state space (controls)|state space]] representations for such systems. 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.
 
== 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:
 
: <math> \mathrm{MTF}(f) = \frac{M(\mathrm{image})} {M(\mathrm{source})}</math>
 
Where modulation (M) is computed from the following image or light brightness:
 
: <math> M = \frac{(L_\mathrm{max} - L_\mathrm{min} )} {(L_\mathrm{max} + L_\mathrm{min})} </math>
 
==See also==
* [[Proper transfer function]]
* [[Duhamel's principle]]
* [[Bode plot]]
* [[Convolution]]
* [[Laplace transform]]
* [[Frequency response]]
* [[LTI system theory]]
* [[Nyquist plot]]
* [[Semilog graph]]
* [[Signal transfer function]]
* [[Analog computer]]
* [[Operational amplifier]]
* [[Optical transfer function]]
 
== References ==
{{reflist}}
 
==External links==
*{{planetmath reference|id=5394|title=Transfer function}}
* [http://www.tedpavlic.com/teaching/osu/ece209/support/circuits_sys_review.pdf ECE 209: Review of Circuits as LTI Systems] &mdash; Short primer on the mathematical analysis of (electrical) LTI systems.
* [http://www.tedpavlic.com/teaching/osu/ece209/lab3_opamp_FO/lab3_opamp_FO_phase_shift.pdf ECE 209: Sources of Phase Shift] &mdash; Gives an intuitive explanation of the source of phase shift in two simple [[LTI system theory|LTI]] systems. Also verifies simple transfer functions by using trigonometric identities.
* [http://reference.wolfram.com/mathematica/ref/TransferFunctionModel.html Transfer function model in Mathematica]
 
[[Category:Electrical circuits]]
[[Category:Signal processing]]
[[Category:Control theory]]
[[Category:Frequency domain analysis]]
[[Category:Types of functions]]

Latest revision as of 23:58, 2 December 2014

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