Editing Boxcar function

{{Short description|Mathematical function resembling a boxcar}}
[[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''.<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 function is named after its graph's resemblance to a [[boxcar]], a type of [[railroad car]]. The boxcar function can be expressed in terms of the [[Uniform distribution (continuous)|uniform distribution]] as
<math display="block">\operatorname{boxcar}(x)= (b-a)A\,f(a,b;x) = A(H(x-a) - H(x-b)),</math>
where {{math|''f''(''a'',''b'';''x'')}} is the uniform distribution of ''x'' for the interval {{closed-closed|''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 [[simple moving average]] filter, whose [[frequency response]] is a [[Sinc filter#Frequency-domain sinc|sinc-in-frequency]], a type of [[low-pass filter]].

==See also==
* [[Boxcar averager]]
* [[Rectangular function]]
* [[Step function]]
* [[Top-hat filter]]
==References==
{{reflist}}

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[[Category:Special functions]]