Boxcar function: Difference between revisions
imported>LR.127 Adding local short description: "Mathematical function resembling a boxcar", overriding Wikidata description "type of mathematical function" |
imported>David Eppstein Undid revision 1317204586 by AndyShow1000000 (talk) already sorted into a more specific stub category |
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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 | 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> | <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 | 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, which are usually best chosen depending on the 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]]. | 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]]. | ||
Latest revision as of 00:55, 17 October 2025
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.[1] 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 as where f(a,b;x)Script error: No such module "Check for unknown parameters". is the uniform distribution of x for the interval Template:Closed-closed and is the Heaviside step function. As with most such discontinuous functions, there is a question of the value at the transition points, which are usually best chosen depending on the individual application.
When a boxcar function is selected as the impulse response of a filter, the result is a simple moving average filter, whose frequency response is a sinc-in-frequency, a type of low-pass filter.
See also
References
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- ↑ Script error: No such module "citation/CS1".
Script error: No such module "Check for unknown parameters".