Rectangular function

Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way From Wikipedia, the free encyclopedia

Rectangular function

The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function,[1] gate function, unit pulse, or the normalized boxcar function) is defined as[2]

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Rectangular function with a = 1

Alternative definitions of the function define to be 0,[3] 1,[4][5] or undefined.

Its periodic version is called a rectangular wave.

History

The rect function has been introduced 1953 by Woodward[6] in "Probability and Information Theory, with Applications to Radar"[7] as an ideal cutout operator, together with the sinc function[8][9] as an ideal interpolation operator, and their counter operations which are sampling (comb operator) and replicating (rep operator), respectively.

Relation to the boxcar function

Summarize
Perspective

The rectangular function is a special case of the more general boxcar function:

where is the Heaviside step function; the function is centered at and has duration , from to

Fourier transform of the rectangular function

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Plot of normalized function (i.e. ) with its spectral frequency components.

The unitary Fourier transforms of the rectangular function are[2] using ordinary frequency f, where is the normalized form[10] of the sinc function and using angular frequency , where is the unnormalized form of the sinc function.

For , its Fourier transform is

Relation to the triangular function

We can define the triangular function as the convolution of two rectangular functions:

Use in probability

Summarize
Perspective

Viewing the rectangular function as a probability density function, it is a special case of the continuous uniform distribution with The characteristic function is

and its moment-generating function is

where is the hyperbolic sine function.

Rational approximation

Summarize
Perspective

The pulse function may also be expressed as a limit of a rational function:

Demonstration of validity

First, we consider the case where Notice that the term is always positive for integer However, and hence approaches zero for large

It follows that:

Second, we consider the case where Notice that the term is always positive for integer However, and hence grows very large for large

It follows that:

Third, we consider the case where We may simply substitute in our equation:

We see that it satisfies the definition of the pulse function. Therefore,

Dirac delta function

The rectangle function can be used to represent the Dirac delta function .[11] Specifically,For a function , its average over the width around 0 in the function domain is calculated as,

To obtain , the following limit is applied,

and this can be written in terms of the Dirac delta function as, The Fourier transform of the Dirac delta function is

where the sinc function here is the normalized sinc function. Because the first zero of the sinc function is at and goes to infinity, the Fourier transform of is

means that the frequency spectrum of the Dirac delta function is infinitely broad. As a pulse is shorten in time, it is larger in spectrum.

See also

References

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