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Transfer function

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From Wikipedia, the free encyclopedia

Function specifying the behavior of a component in an electronic or control system

Not to be confused withTransformation (function).

Inengineering, atransfer function (also known assystem function^[1] ornetwork function) of a system, sub-system, or component is amathematical function thatmodels the system's output for each possible input.^[2]^[3]^[4] It is widely used inelectronic engineering tools likecircuit simulators andcontrol systems. In simple cases, this function can be represented as a two-dimensionalgraph of an independentscalar input versus the dependent scalar output (known as atransfer curve orcharacteristic curve). Transfer functions for components are used to design and analyze systems assembled from components, particularly using theblock diagram technique, in electronics andcontrol 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 atwo-port electronic circuit, such as anamplifier, 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 electromechanicalactuator might be the mechanical displacement of the movable arm as a function of electric current applied to the device; the transfer function of aphotodetector might be the output voltage as a function of theluminous intensity of incident light of a givenwavelength.

The termtransfer function is also used in thefrequency domain analysis of systems using transform methods, such as theLaplace transform; it is theamplitude of the output as a function of thefrequency of the input signal. The transfer function of anelectronic filter is the amplitude at the output as a function of the frequency of a constant amplitudesine wave applied to the input. For optical imaging devices, theoptical transfer function is theFourier transform of thepoint spread function (a function ofspatial frequency).

Linear time-invariant systems

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Transfer functions are commonly used in the analysis of systems such assingle-input single-output filters insignal processing,communication theory, andcontrol theory. The term is often used exclusively to refer tolinear time-invariant (LTI) systems. Most real systems havenon-linear input–output characteristics, but many systems operated within nominal parameters (not over-driven) have behavior close enough to linear thatLTI system theory is an acceptable representation of their input–output behavior.

Continuous-time

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Descriptions are given in terms of acomplex variable, $s=\sigma +j\cdot \omega$ . In many applications it is sufficient to set $\sigma =0$ (thus $s=j\cdot \omega$ ), which reduces theLaplace transforms with complex arguments toFourier transforms with the real argument ω. This is common in applications primarily interested in the LTI system's steady-state response (often the case insignal processing andcommunication theory), not the fleeting turn-on and turn-offtransient response or stability issues.

Forcontinuous-time input signal $x(t)$ and output $y(t)$ , dividing the Laplace transform of the output, $Y(s)={\mathcal {L}}\left\{y(t)\right\}$ , by the Laplace transform of the input, $X(s)={\mathcal {L}}\left\{x(t)\right\}$ , yields the system's transfer function $H(s)$ :

H(s)={\frac {Y(s)}{X(s)}}={\frac {{\mathcal {L}}\left\{y(t)\right\}}{{\mathcal {L}}\left\{x(t)\right\}}}

which can be rearranged as:

Y(s)=H(s)\;X(s)\,.

Discrete-time

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Discrete-time signals may be notated as arrays indexed by aninteger $n {\displaystyle n}$ (e.g. $x[n]$ for input and $y[n]$ for output). Instead of using the Laplace transform (which is better for continuous-time signals), discrete-time signals are dealt with using thez-transform (notated with a corresponding capital letter, like $X(z)$ and $Y(z)$ ), so a discrete-time system's transfer function can be written as:

$H(z)={\frac {Y(z)}{X(z)}}={\frac {{\mathcal {Z}}\{y[n]\}}{{\mathcal {Z}}\{x[n]\}}}.$

Direct derivation from differential equations

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Alinear differential equation with constant coefficients

L[u]={\frac {d^{n}u}{dt^{n}}}+a_{1}{\frac {d^{n-1}u}{dt^{n-1}}}+\dotsb +a_{n-1}{\frac {du}{dt}}+a_{n}u=r(t)

whereu andr are suitably smooth functions oft, hasL as the operator defined on the relevant function space that transformsu intor. That kind of equation can be used to constrain the output functionu in terms of theforcing functionr. The transfer function can be used to define an operator $F[r]=u$ that serves as a right inverse ofL, meaning that $L[F[r]]=r$ .

Solutions of the homogeneousconstant-coefficient differential equation $L[u]=0$ can be found by trying $u=e^{\lambda t}$ . That substitution yields thecharacteristic polynomial

p_{L}(\lambda )=\lambda ^{n}+a_{1}\lambda ^{n-1}+\dotsb +a_{n-1}\lambda +a_{n}\,

The inhomogeneous case can be easily solved if the input functionr is also of the form $r(t)=e^{st}$ . By substituting $u=H(s)e^{st}$ , $L[H(s)e^{st}]=e^{st}$ if we define

H(s)={\frac {1}{p_{L}(s)}}\qquad {\text{wherever }}\quad p_{L}(s)\neq 0.

Other definitions of the transfer function are used, for example $1/p_{L}(ik).$ ^[5]

Gain, transient behavior and stability

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A general sinusoidal input to a system of frequency $\omega _{0}/(2\pi )$ may be written $\exp(j\omega _{0}t)$ . The response of a system to a sinusoidal input beginning at time $t=0$ 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 thedifferential equation. The transfer function for an LTI system may be written as the product:

H(s)=\prod _{i=1}^{N}{\frac {1}{s-s_{P_{i}}}}

wheres_{P_i} are theN roots of the characteristic polynomial and will be thepoles of the transfer function. In a transfer function with a single pole $H(s)={\frac {1}{s-s_{P}}}$ where $s_{P}=\sigma _{P}+j\omega _{P}$ , the Laplace transform of a general sinusoid of unit amplitude will be ${\frac {1}{s-j\omega _{0}}}$ . The Laplace transform of the output will be ${\frac {H(s)}{s-j\omega _{0}}}$ , and the temporal output will be the inverse Laplace transform of that function:

g(t)={\frac {e^{j\,\omega _{0}\,t}-e^{(\sigma _{P}+j\,\omega _{P})t}}{-\sigma _{P}+j(\omega _{0}-\omega _{P})}}

The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity ifσ_P 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:

g(\infty )={\frac {e^{j\,\omega _{0}\,t}}{-\sigma _{P}+j(\omega _{0}-\omega _{P})}}

Thefrequency 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:

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}}}},

which is the absolute value of the transfer function $H(s)$ evaluated at $j\omega _{i}$ . This result is valid for any number of transfer-function poles.

Steady state behavior for sinusoidal excitation

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The steady state behavior of a linear system

\sum _{i=0}^{n}a_{i}y^{(i)}+\sum _{j=0}^{m}b_{j}u^{(j)}=0

for sinusoidal excitation $u(t)=\sin(\omega t)$ can be expressed in terms of its transfer function

g(s)={\frac {b_{m}s^{m}+...+b_{0}}{a_{n}s^{n}+...+a_{0}}},

evaluated at $s=j\omega$ , i.e. with real part $\sigma =0$ :

y(t)=|g(j\omega )|\sin(\omega t+\arg(g(j\omega ))).

To show this, use the ansatz function

y(t)=ce^{j\omega t},

plug it into the above given differential equation, solve for $c {\displaystyle c}$ , and note that $c=g(j\omega )$ .

From the complex identity $z=|z|e^{j\arg(z)}$ , the argument follows.

Signal processing

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If $x(t)$ is the input to a generallinear time-invariant system, and $y(t)$ is the output, and thebilateral Laplace transform of $x(t)$ and $y(t)$ is

{\begin{aligned}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{aligned}}

The output is related to the input by the transfer function $H(s)$ as

Y(s)=H(s)X(s)

and the transfer function itself is

H(s)={\frac {Y(s)}{X(s)}}.

If acomplex harmonic signal with asinusoidal component withamplitude $|X|$ ,angular frequency $\omega$ andphase $\arg(X)$ , where arg is theargument

x(t)=Xe^{j\omega t}=|X|e^{j(\omega t+\arg(X))}

where

X=|X|e^{j\arg(X)}

is input to alinear time-invariant system, the corresponding component in the output is:

{\begin{aligned}y(t)&=Ye^{j\omega t}=|Y|e^{j(\omega t+\arg(Y))},\\Y&=|Y|e^{j\arg(Y)}.\end{aligned}}

In a linear time-invariant system, the input frequency $\omega$ has not changed; only the amplitude and phase angle of the sinusoid have been changed by the system. Thefrequency response $H(j\omega )$ describes this change for every frequency $\omega$ in terms of gain

G(\omega )={\frac {|Y|}{|X|}}=|H(j\omega )|

and phase shift

\phi (\omega )=\arg(Y)-\arg(X)=\arg(H(j\omega )).

Thephase delay (the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is

\tau _{\phi }(\omega )=-{\frac {\phi (\omega )}{\omega }}.

Thegroup 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 $\omega$ ,

\tau _{g}(\omega )=-{\frac {d\phi (\omega )}{d\omega }}.

The transfer function can also be shown using theFourier transform, a special case ofbilateral Laplace transform where $s=j\omega$ .

Common transfer-function families

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Although any LTI system can be described by some transfer function, "families" of special transfer functions are commonly used:

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 – maximally constantgroup delay for a given order
Elliptic filter – sharpest cutoff (narrowest transition between passband and stopband) for the given order
Optimum "L" filter
Gaussian filter – minimum group delay; gives no overshoot to a step function
Raised-cosine filter

Control engineering

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Incontrol engineering andcontrol theory, the transfer function is derived with theLaplace transform. The transfer function was the primary tool used in classical control engineering. Atransfer 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 bridgingstate space and transfer function methods was proposed byHoward H. Rosenbrock, and is known as theRosenbrock system matrix.

Imaging

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Main article:Transfer functions in imaging

Inimaging, transfer functions are used to describe the relationship between the scene light, the image signal and the displayed light.

Non-linear systems

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Transfer functions do not exist for manynon-linear systems, such asrelaxation oscillators;^[6] however,describing functions can sometimes be used to approximate such nonlinear time-invariant systems.

References

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^Bernd Girod, Rudolf Rabenstein, Alexander Stenger,Signals and systems, 2nd ed., Wiley, 2001,ISBN 0-471-98800-6 p. 50
^M. A. Laughton; D.F. Warne (27 September 2002).Electrical Engineer's Reference Book (16 ed.). Newnes. pp. 14/9–14/10.ISBN 978-0-08-052354-5.
^E. A. Parr (1993).Logic Designer's Handbook: Circuits and Systems (2nd ed.). Newness. pp. 65–66.ISBN 978-1-4832-9280-9.
^Ian Sinclair; John Dunton (2007).Electronic and Electrical Servicing: Consumer and Commercial Electronics. Routledge. p. 172.ISBN 978-0-7506-6988-7.
^Birkhoff, Garrett; Rota, Gian-Carlo (1978).Ordinary differential equations. New York: John Wiley & Sons.ISBN 978-0-471-05224-1.^{[page needed]}
^Valentijn De Smedt, Georges Gielen and Wim Dehaene (2015).Temperature- and Supply Voltage-Independent Time References for Wireless Sensor Networks. Springer. p. 47.ISBN 978-3-319-09003-0.