Sinc function: Difference between revisions

Revision as of 11:42, 24 April 2004

Revision as of 04:59, 10 December 2004

Line 1:

The '''sinc function''', also known as the '''interpolation function''' or '''filtering function''', is the product of a [[sine function]] and a [[monotonic function|monotonically decreasing function]]. It is defined by:

:<math>\operatornamemathbf{sinc} \left( x \right) = \frac{\sin \left( x \right)}{x}</math>

<math>

\operatorname{sinc} \left( x \right) = \frac{\sin \left( x \right)}{x}

</math>

or:

:<math>\operatornamemathbf{sinc} \left( x \right) = \frac{\sin \left( \pi x \right)}{\pi x}</math>

<math>

\operatorname{sinc} \left( x \right) = \frac{\sin \left( \pi x \right)}{\pi x}

</math>

It has a [[removable singularity]] at the origin. Sinc<math>\mathbf{sinc}(0)=1,</math> despiteby the division by[[L'Hôpital's zerorule]].

The sinc [[function (mathematics)|function]] [[oscillation|oscillates]] with decreasing [[amplitude]]. Applications of the sinc function are found in [[communication theory]], [[control theory]], and [[optics]].

The [[Fourier transform]] is

:<math>\frac{W}{\pi} \mathbf{sinc}(W t) = \mathbf{rect}\left(\frac{\omega}{2 W}\right)</math>

:''See also'': [[trigonometric function]]

where <math>rect</math> is the [[rectangular function]].

==See also==

:''See also'': *[[trigonometric function]]

*[[L'Hôpital's rule]]

[[Category:Special functions]]

[[Category:Trigonometry]]