Boxcar function (original) (raw)

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Mathematical function resembling a boxcar

A graphical representation of a boxcar function

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 boxcar ⁡ ( x ) = ( b − a ) A f ( a , b ; x ) = A ( H ( x − a ) − H ( x − b ) ) , {\displaystyle \operatorname {boxcar} (x)=(b-a)A\,f(a,b;x)=A(H(x-a)-H(x-b)),} where f(a,b;x) is the uniform distribution of x for the interval [a, _b_] and H ( x ) {\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. 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.

• Boxcar averager
• Rectangular function
• Step function
• Top-hat filter

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