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[[Image:Boxcar function.svg|right|thumb|250px|A graphical representation of a boxcar function]] | |||
In [[mathematics]], a '''boxcar function''' is any [[function (mathematics)|function]] which is zero over the entire | |||
[[real line]] except for a single [[interval (mathematics)|interval]] where it is equal to a constant, ''A;'' it is a simple [[step function]].<ref>{{cite web|last=Weisstein|first=Eric W.|title=Boxcar Function|url=http://mathworld.wolfram.com/BoxcarFunction.html|publisher=MathWorld|accessdate=13 September 2013}}</ref> The boxcar function can be expressed in terms of the [[Uniform distribution (continuous)|uniform distribution]] as | |||
:<math>\operatorname{boxcar}(x)= (b-a)A\,f(a,b;x) = H(x-a) - H(x-b),</math> | |||
where ''f(a,b;x)'' is the uniform distribution of ''x'' for the interval [''a'', ''b''] and <math>H(x)</math> is the [[Heaviside step function]]. | |||
As with most such [[continuous function|discontinuous functions]], there is a question of the value at the transition points. These values are probably best chosen for each individual application. | |||
When a boxcar function is selected as the [[impulse response]] of a [[Digital filter|filter]], the result is a [[moving average]] filter. | |||
The function is named after its resemblance to a [[boxcar]], a type of [[railroad car]]. | |||
==See also== | |||
* [[Rectangular function]] | |||
* [[Step function]] | |||
==References== | |||
{{reflist}} | |||
{{mathanalysis-stub}} | |||
[[Category:Special functions]] | |||
Revision as of 08:10, 4 December 2013
In mathematics, a boxcar function is any function which is zero over the entire real line except for a single interval where it is equal to a constant, A; it is a simple step function.[1] The boxcar function can be expressed in terms of the uniform distribution as
where f(a,b;x) is the uniform distribution of x for the interval [a, b] and is the Heaviside step function. As with most such discontinuous functions, there is a question of the value at the transition points. These values are probably best chosen for each individual application.
When a boxcar function is selected as the impulse response of a filter, the result is a moving average filter.
The function is named after its resemblance to a boxcar, a type of railroad car.
See also
References
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