Transfer function Totally Explained

Everything about Transfer Function totally explained

ť For "transfer function" as used in computer graphics, see lookup table.A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a (linear time-invariant) system. With optical imaging devices, for example, it's the Fourier transform (hence a function of spatial frequency) of the point spread function for example the intensity distribution caused by a point object in the field of view.

Explanation

The transfer function is commonly used in the analysis of single-input single-output electronic filters, 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's 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

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)

:
ť

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

or ť

H(s) = frac

.

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 omega

Common transfer function families

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. Typical infinite impulse response filters are designed to implement one of these special transfer functions.
Some common transfer function families and their particular characteristics are:

matched filter -- optimum pulse response for any arbitrary pulse shape
Butterworth filter - maximally flat pass band for the given order
Chebyshev filter(Type I) - no gain ripple in stop band, sharper cutoff than Butterworth
Chebyshev filter(Type II) - no gain ripple in pass band, sharper cutoff than Butterworth
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
Raised-cosine filter

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.

Further Information

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