Transfer function
transfer function, transfer function examplesIn engineering, a transfer function also known as system function1 or network function and, when plotted as a graph, transfer curve is a mathematical representation for fit23 or to describe inputs and outputs of black box models4
Typically it is a representation in terms of spatial or temporal frequency, of the relation between the input and output of a linear timeinvariant LTI system with zero initial conditions and zeropoint equilibrium5 For optical imaging devices, for example, the optical transfer function is the Fourier transform of the point spread function hence a function of spatial frequency ie, the intensity distribution caused by a point object in the field of viewcitation needed A number of sources however use "transfer function" to mean some inputoutput characteristic in direct physical measures eg, output voltage as a function of the input voltage of a twoport network rather than its transform to the splane678
Contents
 1 Linear timeinvariant systems
 11 Direct derivation from differential equations
 12 Gain, transient behavior and stability
 2 Signal processing
 21 Common transfer function families
 3 Control engineering
 4 Optics
 5 Nonlinear systems
 6 See also
 7 References
 8 External links
Linear timeinvariant systemsedit
Transfer functions are commonly used in the analysis of systems such as singleinput singleoutput filters, typically within the fields of signal processing, communication theory, and control theory The term is often used exclusively to refer to linear timeinvariant LTI systems, as covered in this article Most real systems have nonlinear input/output characteristics, but many systems, when operated within nominal parameters not "overdriven" 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 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 x t and output y t , the transfer function H s is the linear mapping of the Laplace transform of the input, X s = L \left\ , to the Laplace transform of the output Y s = L \left\ :
Y s = H s X sor
H s = Y s X s = L L =\left\\left\In discretetime systems, the relation between an input signal x t and output y t is dealt with using the ztransform, and then the transfer function is similarly written as H z = Y z X z and this is often referred to as the pulsetransfer functioncitation needed
Direct derivation from differential equationsedit
Consider a linear differential equation with constant coefficients
L u = d n u d t n + a 1 d n − 1 u d t n − 1 + ⋯ + a n − 1 d u d t + a n u = r t u+a_u+\dotsb +a_+a_u=rtwhere 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 F r = u that serves as a right inverse of L, meaning that L F r = r
Solutions of the homogeneous, constantcoefficient differential equation L u = 0 can be found by trying u = e λ t That substitution yields the characteristic polynomial
p L λ = λ n + a 1 λ n − 1 + ⋯ + a n − 1 λ + a n \lambda =\lambda ^+a_\lambda ^+\dotsb +a_\lambda +a_\,The inhomogeneous case can be easily solved if the input function r is also of the form r t = e s t In that case, by substituting u = H s e s t one finds that L H s e s t = e s t =e^ if we define
H s = 1 p L s wherever p L s ≠ 0 s\qquad \quad p_s\neq 0Taking that as the definition of the transfer function requires careful disambiguationclarification needed between complex vs real values, which is traditionally influencedclarification needed by the interpretation of absHs as the gain and atanHs as the phase lag Other definitions of the transfer function are used: for example 1 / p L i k ik 9
Gain, transient behavior and stabilityedit
A general sinusoidal input to a system may be written exp j ω i t t The response of a system to a sinusoidal input beginning at time t = 0 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:
H s = ∏ i = 1 N 1 s − s P i ^where sPi 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 H s = 1 s − s P where s P = σ P + j ω P =\sigma _+j\omega _ The Laplace transform of a general sinusoid of unit amplitude will be 1 s − j ω i The Laplace transform of the output will be H s s − j ω i and the temporal output will be the inverse Laplace transform of that function:
g t = e j ω i t − e σ P + j ω P t − σ P + j ω i − ω P \,te^+j\,\omega _t+j\omega _\omega _The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity if σP 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:
g ∞ = e j ω i t − σ P + j ω i − ω P \,t+j\omega _\omega _The frequency response or "gain" G of the system is defined as the absolute value of the ratio of the input amplitude to the steadystate output amplitude:
G ω i =  1 − σ P + j ω i − ω P  = 1 σ P 2 + ω P − ω i 2 =\left+j\omega _\omega _\right=^+\omega _\omega _^which is just the absolute value of the transfer function H s evaluated at j ω i This result can be shown to be valid for any number of transfer function poles
Signal processingedit
Let x t be the input to a general linear timeinvariant system, and y t be the output, and the bilateral Laplace transform of x t and y t be
X s = L = d e f ∫ − ∞ ∞ x t e − s t d t , Y s = L = d e f ∫ − ∞ ∞ y t e − s t d t Xs&=\left\\ \ \int _^xte^\,dt,\\Ys&=\left\\ \ \int _^yte^\,dt\endThen the output is related to the input by the transfer function H s as
Y s = H s X sand the transfer function itself is therefore
H s = Y s X sIn particular, if a complex harmonic signal with a sinusoidal component with amplitude  X  , angular frequency ω and phase arg X , where arg is the argument
x t = X e j ω t =  X  e j ω t + arg X =Xe^ where X =  X  e j arg Xis input to a linear timeinvariant system, then the corresponding component in the output is:
y t = Y e j ω t =  Y  e j ω t + arg Y , Y =  Y  e j arg Y yt&=Ye^=Ye^,\\Y&=Ye^\endNote that, in a linear timeinvariant system, the input frequency ω has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system The frequency response H j ω describes this change for every frequency ω in terms of gain:
G ω =  Y   X  =  H j ω  =Hj\omega \and phase shift:
ϕ ω = arg Y − arg X = arg H j ωThe phase delay ie, the frequencydependent amount of delay introduced to the sinusoid by the transfer function is:
τ ϕ ω = − ϕ ω ω \omega =The group delay ie, 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 ω ,
τ g ω = − d ϕ ω d ω \omega =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 s = j ω
Common transfer function familiesedit
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 engineeringedit
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 systemscitation 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
Opticsedit
This section does not cite any sources Please help improve this section by adding citations to reliable sources Unsourced material may be challenged and removed December 2014 Learn how and when to remove this template message 
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:
M T F f = M i m a g e M s o u r c e f=Where modulation M is computed from the following image or light brightness:
M = L max − L min L max + L min L_+L_Nonlinear systemsedit
Transfer functions do not properly exist for many nonlinear systems For example, they do not exist for relaxation oscillators;10 however, describing functions can sometimes be used to approximate such nonlinear timeinvariant systems
See alsoedit
 Analog computer
 Black box
 Bode plot
 Convolution
 Duhamel's principle
 Frequency 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
Referencesedit
 ^ Bernd Girod, Rudolf Rabenstein, Alexander Stenger, Signals and systems, 2nd ed, Wiley, 2001, ISBN 0471988006 p 50
 ^ Antunes, Ricardo; Gonzales, Vicente; Walsh, Kenneth July 2016 "Quicker reaction, lower variability: The effect of transient time in flow variability of projectdriven production" Proc 24rd Ann Conf of the Int’l Group for Lean Construction, 21–23 July, Boston, MA, Boston, MA 4: 73–82 doi:1013140/RG2110054647 Retrieved 14 August 2016
 ^ Antunes, Ricardo; González, Vicente; Walsh, Kenneth 29 July 2015 "Identification of Repetitive Processes at Steady and Unsteadystate: Transfer Function" Proc 23rd Ann Conf of the Int’l Group for Lean Construction Perth, Australia: 793–802 doi:1013140/RG2141937364 Retrieved 14 August 2016
 ^ Antunes, Ricardo; González, Vicente; Walsh, Kenneth; Rojas, Omar July 2017 "Dynamics of ProjectDriven Production Systems in Construction: Productivity Function" Journal of Computing in Civil Engineering 31: 17 doi:101061/ASCECP194354870000703 – via ASCE
 ^ The Oxford Dictionary of English, 3rd ed, "Transfer function"
 ^ M A Laughton; DF 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 linksedit
 "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
transfer function, transfer function block diagram, transfer function calculator, transfer function examples, transfer function in control system, transfer function matlab, transfer function of rc circuit, transfer function pdf, transfer function to state space, transfer function tutorial
Transfer function Information about

Transfer function beatiful post thanks!
29.10.2014
Transfer function
Transfer function
Transfer function viewing the topic.
There are excerpts from wikipedia on this article and video
Random Posts
Book
A book is a set of written, printed, illustrated, or blank sheets, made of ink, paper, parchment, or...Boston Renegades
Boston Renegades was an American women’s soccer team, founded in 2003 The team was a member of the U...Sa Caleta Phoenician Settlement
Sa Caleta Phoenician Settlement can be found on a rocky headland about 10 kilometers west of Ibiza T...Bodybuilding.com
Bodybuildingcom is an American online retailer based in Boise, Idaho, specializing in dietary supple...Search Engine
Our site has a system which serves search engine function.
You can search all data in our system with above button which written "What did you look for? "
Welcome to our simple, stylish and fast search engine system.
We have prepared this method why you can reach most accurate and most up to date knowladge.
The search engine that developed for you transmits you to the latest and exact information with its basic and quick system.
You can find nearly everything data which found from internet with this system.
Random Posts
Book
A book is a set of written, printed, illustrated, or blank sheets, made of ink, paper, parchment, or...Boston Renegades
Boston Renegades was an American women’s soccer team, founded in 2003 The team was a member of the U...Sa Caleta Phoenician Settlement
Sa Caleta Phoenician Settlement can be found on a rocky headland about 10 kilometers west of Ibiza T...Bodybuilding.com
Bodybuildingcom is an American online retailer based in Boise, Idaho, specializing in dietary supple...© Copyright © 2014. Search Engine