Boxcar function

From Quire, the open reference·Mathematical function resembling a boxcar·1 min read

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 ${\displaystyle A}$ .^[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

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

where ${\displaystyle f(a,b;x)}$ is the uniform distribution of ${\displaystyle x}$ for the interval ${\displaystyle [a,b]}$ and ${\displaystyle H(x)}$ 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.

References#edit

^ Weisstein, Eric W. "Boxcar function". MathWorld. Retrieved 13 September 2013.

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Boxcar function

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