Rectangular function: Difference between revisions

Revision as of 08:07, 23 December 2004

Revision as of 03:58, 1 February 2005

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[[Image:Rect_function.png|right|Rectangular function]]

The '''rectangular function''' (oralso known as the '''rectangle function''') is definedor as

the normalized '''[[boxcar function]]''') is defined as

:<math>\mathbf{rect}(t) = \left \{ \begin{matrix}

Line 13 ⟶ 14:

:<math>\mathbf{rect}(t) = \mathbf{H}(t + 1/2) - \mathbf{H}(t - 1/2)</math>

The rectangular function is normalized:

The [[Fourier transform]] is

:<math>\mathbf{rect}\left(\fracint_{-\omega}{2 Winfty}^\right) =infty \fractextrm{W}{\pi} \operatorname{sincrect}(W tx)\,dx=1</math>

The [[continuous Fourier transform|Fourier transform]] of the rectangular function is

where "sinc" is the [[sinc function]].

:<math>\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \textrm{rect}(x)e^{-ikx}\,dx

=\sqrt{\frac{\pi}{2}}\,\textrm{sinc}(k/2)</math>

where "sinc" is the normalized [[sinc function]]. Viewed as a [[probability distribution function]], its [[characteristic function]] is therefore written

:<math>\varphi(k)=\pi\,\textrm{sinc}(k/2)\,</math>

and the [[moment generating function]] is:

:<math>M(k)=\frac{2\,\textrm{sinh}(k/2)}{k}\,</math>

==See also==

*[[boxcar function]]

*[[Fourier transform]]

*[[square wave]]

{{math-stub}}

[[Category:Special functions]]