Boxcar function

{{Short description|Mathematical function resembling a boxcar}}

Image:Boxcar function.svg

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.{{cite web| last=Weisstein|first=Eric W.|title=Boxcar Function|url=http://mathworld.wolfram.com/BoxcarFunction.html| publisher=MathWorld| accessdate=13 September 2013}} 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

$$\backslash operatorname\{boxcar\}(x)=\; (b-a)A\backslash ,f(a,b;x)\; =\; A(H(x-a)\; -\; H(x-b)),$$

where {{math|f(a,b;x)}} is the uniform distribution of x for the interval {{closed-closed|a, b}} and $H(x)$ 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 simple moving average filter, whose frequency response is a sinc-in-frequency, a type of low-pass filter.