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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

$\mathrm{boxcar}\left(x\right)=(b-a)Af(a,b;x)=A\left(H\right(x-a)-H(x-b\left)\right),$

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

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