NationMaster - Encyclopedia: Transfer function

A transfer function is a mathematical representation of the relation between the input and output of a (linear time-invariant) system. Linear time invarient systems are called as lti systems it should satisify both linearity and time invarient quality. ... System analysis is the branch of electrical engineering that characterizes electrical systems and their properties. ...

1 Explanation
2 Signal processing
3 Control engineering
4 See also
5 External link

Explanation

The transfer function is commonly used in the analysis of single-input single-output analog electronic circuits, for instance. It is mainly used in signal processing, communication theory, and control theory. The term is often used exclusively to refer to 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. It has been suggested that this article or section be merged with Analog electronics. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... There is much discussion in the academic world of communication as to what actually constitutes communication. ... In engineering and mathematics, control theory deals with the behavior of dynamical systems. ... Linear time invarient systems are called as lti systems it should satisify both linearity and time invarient quality. ... To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. ...

In its simplest form for continuous-time input signal $x(t),$ and output $y(t),$ , the transfer function is the linear mapping of the Laplace transform of the input, $X(s),$ , to the output $Y(s),$ : Continuous time occurs when time is sampled continuously. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ...

$Y(s) = H(s) , X(s)$

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

where $H(s),$ is the transfer function of the LTI system.

In discrete-time systems, the function is similarly written as $H(z) = frac{Y(z)}{X(z)}$ (see Z transform). Discrete time is non-continuous time. ... In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...

Signal processing

Let $x(t)$ be the input to a general linear time-invariant system, and $y(t)$ be the output, and the Laplace transform of $x(t)$ and $y(t)$ be In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ...

$X(s) = mathcal{L}left { x(t) right } stackrel{mathrm{def}}{=} int_{-infty}^{infty} x(t) e^{-st}, dt$

$Y(s) = mathcal{L}left { y(t) right } stackrel{mathrm{def}}{=} int_{-infty}^{infty} y(t) e^{-st}, dt$ .

Then 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 therefore

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

In particular, if a complex harmonic signal with a sinusoidal component with amplitude $|X|$ , angular frequency $omega$ and phase $arg(X)$ In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... This article is about the components of sound. ... In information theory, a signal is the sequence of states of a communications channel that encodes a message. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... It has been suggested that pulse amplitude be merged into this article or section. ... It has been suggested that this article or section be merged into Angular velocity. ... This article is about a portion of a periodic process. ...

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

where $X = |X|e^{jarg(X)}$

is input to a linear time-invariant system, then the corresponding component in the output is: The word linear comes from the Latin word linearis, which means created by lines. ...

$y(t) = Ye^{jomega t} = |Y|e^{j(omega t + arg(Y))}$

and $Y = |Y|e^{jarg(Y)}$ .

Note that, in a linear time-invariant system, the input frequency $omega$ has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system. The frequency response $H(j omega)$ describes this change for every frequency $omega$ in terms of gain: Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ...

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

and phase shift:

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

The phase delay (i.e., the frequency-dependent amount of delay to the sinusoid introduced by the transfer function) is: In LTI system theory, control theory, and in digital or analog signal processing, the relationship between the input signal, x(t), to output signal, y(t), of an LTI system is governed by: or where and . Here h(t) is the impulse response of the LTI system and X(s...

$tau_{phi}(omega) = -begin{matrix}frac{phi(omega)}{omega}end{matrix}$ .

The group delay (i.e., the frequency-dependent amount of delay to the envelope of the sinusoid introduced by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency $omega$ , In physics, and in particular in optics, the study of waves and digital signal processing, the term group delay has the following meanings: 1. ...

$tau_{g}(omega) = -begin{matrix}frac{dphi(omega)}{domega}end{matrix}$ .

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 $s = j ω$ . In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ...

Control engineering

In control engineering and control theory the transfer function is derived using the Laplace transform. Control engineering is the engineering discipline that focuses on the mathematical modelling systems of a diverse nature, analysing their dynamic behaviour, and using control theory to make a controller that will cause the systems to behave in a desired manner. ... In engineering and mathematics, control theory deals with the behavior of dynamical systems. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. ...

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 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. In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. ... The transfer matrix is a formulation in terms of a matrix of the two-scale equation, which characterizes refinable functions. ...

External link

transfer function on PlanetMath

Categories: Electrical circuits | Signal processing | Control theory | Cybernetics PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

Results from FactBites:

Nikon MicroscopyU: Modulation Transfer Function (4055 words)
	Therefore, the optical transfer function is a spatial frequency-dependent complex variable whose modulus is the modulation transfer function, and whose phase is described by the phase transfer function.
	The modulation transfer function is also related to the point spread function, which is the image of a point source of light (commonly referred to as the Airy disk) from the specimen projected by the microscope objective onto the intermediate image plane.
	In this case, however, the point spread function is replaced by the time response to a very short electrical impulse, and the optical transfer function is replaced by the imaging system's response to the sinusoidal electrical signal with respect to amplitude and phase.

A Standard Default Color Space for the Internet - sRGB (6019 words)
	The ITU-R BT.709 transfer function in combination with its target monitor is attempting to achieve a viewing gamma of 1.125 by incorrectly assuming a CRT gamma of 2.5 and an LUT gamma of 1.0/2.222 as shown in the equation below.
	Despite the fact that the exponent of the 709 function is 1.0/2.222, the actual 709 encoding transfer function is closer to a CRT gamma of 1.0/1.956 than 1.0/2.222.
	This is due to the large offset of 0.099 in the transfer function equation.

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents

Explanation GA_googleFillSlot("encyclopedia_square");

Signal processing

Control engineering

See also

External link

Explanation