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Information about Transfer Function

For "transfer function" as used in computer graphics, see lookup table.
A transfer function is a mathematical representation of the relation between the input and output of a (linear time-invariant) system.

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.

In its simplest form for continuous-time input signal and output , the transfer function is the linear mapping of the Laplace transform of the input, , to the output :

or


where is the transfer function of the LTI system.

In discrete-time systems, the function is similarly written as (see Z transform).

Signal processing

Let be the input to a general linear time-invariant system, and be the output, and the Laplace transform of and be



.


Then the output is related to the input by the transfer function as

:


and the transfer function itself is therefore

: .


In particular, if a complex harmonic signal with a sinusoidal component with amplitude , angular frequency and phase



where


is input to a linear time-invariant system, then the corresponding component in the output is:



and .


Note that, in a linear time-invariant system, the input frequency has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system. The frequency response describes this change for every frequency in terms of gain:



and phase shift:

.


The phase delay (i.e., the frequency-dependent amount of delay to the sinusoid introduced by the transfer function) is:

.


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 ,

.


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 .

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 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.

See also

External link

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LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. Though the standard independent variable is time, it could just as easily be space (as in image processing and field theory) or some other coordinate.
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communication theory.

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..... Click the link for more information.
A continuous signal or a continuous-time signal is a varying quantity (a signal) that is expressed as a function of a real-valued domain, usually time. The function of time need not be continuous.
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discrete signal or discrete-time signal is a time series, perhaps a signal that has been sampled from a continuous-time signal. Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous-time argument, but is a sequence of quantities; that is,
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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.

The Z-transform and advanced Z-transform were introduced (under the Z-transform name) by E. I.
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LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. Though the standard independent variable is time, it could just as easily be space (as in image processing and field theory) or some other coordinate.
..... Click the link for more information.
In the branch of mathematics called functional analysis, the Laplace transform, , is a linear operator on a function f(t) (original ) with a real argument t (t ≥ 0) that transforms it to a function F(s) (
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The concept is broad, and hard to define precisely.
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prevew not available
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Frequency response is the measure of any system's response at the output to a signal of varying frequency (but constant amplitude) at its input. In the audible range it usually referred to in connection with Electronic amplifiers, microphones and loudspeakers.
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group delay and phase delay respectively, are as shown below and potentially functions of ω. In a linear phase system (with non-inverting gain), both and are equal to the same constant delay of the system and the phase shift of the system increases linearly with
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group delay and phase delay respectively, are as shown below and potentially functions of ω. In a linear phase system (with non-inverting gain), both and are equal to the same constant delay of the system and the phase shift of the system increases linearly with
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Fourier transform, named in honor of French mathematician Joseph Fourier, is a certain linear operator that maps functions to other functions. Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its frequency components
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In the branch of mathematics called functional analysis, the Laplace transform, , is a linear operator on a function f(t) (original ) with a real argument t (t ≥ 0) that transforms it to a function F(s) (
..... Click the link for more information.
Control engineering is the engineering discipline that focuses on mathematical modelling of systems of a diverse nature, analyzing their dynamic behavior, and using control theory to create a controller that will cause the systems to behave in a desired manner.
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