Editing Transfer function

{{Short description|Function specifying the behavior of a component in an electronic or control system}}
{{distinguish|Transformation (function)}}

In [[engineering]], a '''transfer function''' (also known as '''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''') of a system, sub-system, or component is a [[function (mathematics)|mathematical function]] that [[mathematical model|models]] the system's output for each possible input.<ref name="LaughtonWarne2002">{{cite book|author1=M. A. Laughton|author2=D.F. Warne|title=Electrical Engineer's Reference Book|date=27 September 2002|publisher=Newnes|isbn=978-0-08-052354-5|pages=14/9–14/10|edition=16}}</ref><ref name="Parr1993">{{cite book|author=E. A. Parr|title=Logic Designer's Handbook: Circuits and Systems|year=1993|publisher=Newness|isbn=978-1-4832-9280-9|pages=65–66|edition=2nd}}</ref><ref name="SinclairDunton2007">{{cite book|author1=Ian Sinclair|author2=John Dunton|title=Electronic and Electrical Servicing: Consumer and Commercial Electronics|year=2007|publisher=Routledge|isbn=978-0-7506-6988-7|page=172}}</ref> It is widely used in [[electronic engineering]] tools like [[Electronic circuit simulation|circuit simulators]] and [[control system]]s. In simple cases, this function can be represented as a two-dimensional [[graph (function)|graph]] of an independent [[scalar (mathematics)|scalar]] input versus the dependent scalar output (known as 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]] technique, in electronics and [[control theory]].

Dimensions and units of the transfer function model the output response of the device for a range of possible inputs. The transfer function of a [[two-port]] electronic circuit, such as an [[amplifier]], 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 electric 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]]; it is the [[amplitude]] of the output as a function of the [[frequency]] of the input signal. The transfer function of an [[electronic filter]] is the 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]] (a function of [[spatial frequency]]).

== Linear time-invariant systems ==
Transfer functions are commonly used in the analysis of systems such as [[single-input single-output]] [[Filter (signal processing)|filter]]s in [[signal processing]], [[communication theory]], and [[control theory]]. The term is often used exclusively to refer to [[linear time-invariant]] (LTI) systems. Most real systems have [[non-linear]] input-output characteristics, but many systems operated within nominal parameters (not over-driven) have behavior close enough to linear that [[LTI system theory]] is an acceptable representation of their input-output behavior.

=== Continuous-time ===
Descriptions are given in terms of a [[complex variable]], <math>s = \sigma + j \cdot \omega</math>. In many applications it is sufficient to set <math>\sigma=0</math> (thus <math>s = j \cdot \omega</math>), which reduces the [[Laplace transform]]s with complex arguments to [[Fourier transform]]s with the real argument ω. This is common in applications primarily interested in the LTI system's steady-state response (often the case in [[signal processing]] and [[communication theory]]), not the fleeting turn-on and turn-off [[transient response]] or stability issues.

For [[continuous-time]] input signal <math>x(t)</math> and output <math>y(t)</math>, dividing the Laplace transform of the output, <math>Y(s) = \mathcal{L}\left\{y(t)\right\}</math>, by the Laplace transform of the input, <math>X(s) = \mathcal{L}\left\{x(t)\right\}</math>, yields the system's transfer function <math>H(s)</math>:
:<math> H(s) = \frac{Y(s)}{X(s)} = \frac{ \mathcal{L}\left\{y(t)\right\} }{ \mathcal{L}\left\{x(t)\right\} } </math>
which can be rearranged as:
:<math> Y(s) = H(s)\;X(s) \, . </math>

=== Discrete-time ===
{{See also|Z-transform#Linear constant-coefficient difference equation}}
[[Discrete-time]] signals may be notated as arrays indexed by an [[integer]] <math>n</math> (e.g. <math>x[n]</math> for input and <math>y[n]</math> for output). Instead of using the Laplace transform (which is better for continuous-time signals), discrete-time signals are dealt with using the [[z-transform]] (notated with a corresponding capital letter, like <math>X(z)</math> and <math>Y(z)</math>), so a discrete-time system's transfer function can be written as:

<math display="block">H(z) = \frac{Y(z)}{X(z)} = \frac{\mathcal{Z}\{y[n]\}}{\mathcal{Z}\{x[n]\}}.</math>

=== Direct derivation from differential equations ===
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'', has ''L'' as 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 <math>F[r] = u </math> that serves as a right inverse of ''L'', meaning that  <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 equation (calculus)|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>. By substituting <math>u = H(s)e^{s t}</math>, <math>L[H(s) e^{s t}] = e^{s t}</math> if we define

:<math>H(s) = \frac{1}{p_L(s)} \qquad\text{wherever }\quad p_L(s) \neq 0.</math>

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|author2=Rota, Gian-Carlo |year=1978|publisher=John Wiley & Sons |location= New York|isbn= 978-0-471-05224-1}}{{page needed|date=April 2013}}</ref>

=== Gain, transient behavior and stability ===

A general sinusoidal input to a system of frequency <math> \omega_0 / (2\pi)</math> may be written <math>\exp( j \omega_0 t )</math>. The response of a system to a sinusoidal input beginning at time <math>t=0</math> 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 [[differential equation]]. The transfer function for an LTI system may be written as the product:

:<math>
H(s)=\prod_{i=1}^N \frac{1}{s-s_{P_i}}
</math>

where ''s<sub>P<sub>i</sub></sub>'' are the ''N'' roots of the characteristic polynomial and will be the [[Pole (complex analysis)|poles]] of the transfer function. In a transfer function with a single pole <math>H(s)=\frac{1}{s-s_P}</math> where <math>s_P = \sigma_P+j \omega_P</math>, the Laplace transform of a general sinusoid of unit amplitude will be <math>\frac{1}{s-j\omega_i}</math>. The Laplace transform of the output will be <math>\frac{H (s)}{s-j \omega_0}</math>, and the temporal output will be the inverse Laplace transform of that function:

:<math>
g(t)=\frac{e^{j\,\omega_0\,t}-e^{(\sigma_P+j\,\omega_P)t}}{-\sigma_P+j (\omega_0-\omega_P)}
</math>

The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity if ''σ<sub>P</sub>'' is positive. 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:

:<math>
g(\infty)=\frac{e^{j\, \omega_0\,t}}{-\sigma_P+j (\omega_0-\omega_P)}
</math>

The [[frequency response]] (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:

:<math>
G(\omega_i)=\left|\frac{1}{-\sigma_P+j (\omega_0-\omega_P)}\right|=\frac{1}{\sqrt{\sigma_P^2+(\omega_P-\omega_0)^2}},
</math>

which is the absolute value of the transfer function <math> H(s) </math> evaluated at <math> j\omega_i </math>. This result is valid for any number of transfer-function poles.

==Signal processing==
If <math> x(t) </math> is the input to a general [[LTI system theory|linear time-invariant system]], and <math> y(t) </math> is the output, and the [[bilateral Laplace transform]] of <math> x(t) </math> and <math> y(t) </math> is

: <math>\begin{align}
  X(s) &= \mathcal{L}\left \{ x(t) \right \} \ \stackrel{\mathrm{def}}{=}\  \int_{-\infty}^{\infty} x(t) e^{-st}\, dt, \\
  Y(s) &= \mathcal{L}\left \{ y(t) \right \} \ \stackrel{\mathrm{def}}{=}\  \int_{-\infty}^{\infty} y(t) e^{-st}\, dt.
\end{align}</math>

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 

: <math> H(s) = \frac{Y(s)} {X(s)}. </math>

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>, where arg is the [[Argument (complex analysis)|argument]]

:<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, the corresponding component in the output is:

:<math>\begin{align}
  y(t) &= Ye^{j\omega t} = |Y|e^{j(\omega t + \arg(Y))}, \\
     Y &= |Y|e^{j\arg(Y)}.
\end{align}</math>

In a linear time-invariant system, the input frequency <math> \omega </math> has not changed; only the amplitude and phase angle of the sinusoid have 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]] (the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is

:<math>\tau_{\phi}(\omega) = -\frac{\phi(\omega)}{\omega}.</math>

The [[group delay]] (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) = -\frac{d\phi(\omega)}{d\omega}.</math>

The transfer function can also be shown using the [[Fourier transform]], a special case of [[bilateral Laplace transform]] where <math> s = j \omega </math>.

=== {{anchor|Common transfer function families}}Common transfer-function families ===

Although any LTI system can be described by some transfer function, "families" of special transfer functions are commonly used:
* [[Butterworth filter]]&nbsp;– maximally flat in passband and stopband for the given order
* [[Chebyshev filter|Chebyshev filter (Type I)]]&nbsp;– maximally flat in stopband, sharper cutoff than a Butterworth filter of the same order
* Chebyshev filter (Type II)&nbsp;– maximally flat in passband, sharper cutoff than a Butterworth filter of the same order
* [[Bessel filter]]&nbsp;– maximally constant [[group delay]] for a given order
* [[Elliptic filter]]&nbsp;– sharpest cutoff (narrowest transition between passband and stopband) for the given order
* [[Optimum "L" filter]]
* [[Gaussian filter]]&nbsp;– minimum group delay; gives no overshoot to a step function
* [[Raised-cosine filter]]

==Control engineering==

In [[control engineering]] and [[control theory]], the transfer function is derived with the [[Laplace transform]]. The transfer function was the primary tool used in classical control engineering. A [[transfer function matrix|transfer matrix]] can be obtained for any linear system 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 representation bridging [[state space]] and transfer function methods was proposed by [[Howard Harry Rosenbrock|Howard H. Rosenbrock]], and is known as the [[Rosenbrock system matrix]].

== Imaging ==
{{Main|Transfer functions in imaging}}
In [[imaging]], 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 exist for many [[nonlinear control|non-linear systems]], such as [[relaxation oscillator]]s;<ref name="Dehaene">{{cite book|author=Valentijn De Smedt, Georges Gielen and Wim Dehaene|title=Temperature- and Supply Voltage-Independent Time References for Wireless Sensor Networks|publisher=Springer|isbn=978-3-319-09003-0|page=47|year=2015}}</ref> however, [[describing function]]s can sometimes be used to approximate such nonlinear time-invariant systems.

==See also==
{{div col |colwidth=14em |content=

* [[Analog computer]]
* [[Black box]]
* [[Bode plot]]
* [[Convolution]]
* [[Duhamel's principle]]
* [[Frequency response]]
* [[Impulse response]]
* [[Laplace transform]]
* [[Linear time-invariant system|LTI system theory]]
* [[Nyquist plot]]
* [[Operational amplifier]]
* [[Optical transfer function]]
* [[Proper transfer function]]
* [[Rosenbrock system matrix]]
* [[Semi-log plot]]
* [[Signal-flow graph]]
* [[Signal transfer function]]

}}

== References ==
{{reflist|25em}}

==External links==
* [http://www.tedpavlic.com/teaching/osu/ece209/support/circuits_sys_review.pdf ECE 209: Review of Circuits as LTI Systems] — Short primer on the mathematical analysis of (electrical) LTI systems.

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[[Category:Electrical circuits]]
[[Category:Transfer functions| ]]
[[Category:Frequency-domain analysis]]
[[Category:Types of functions]]