Boxcar function

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. 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 $$\mathrm{boxcar}(x)=(b-a)Af(a,b;x)=A(H(x-a)-H(x-b)),$$ where f(a,b;x) is the uniform distribution of x for the interval [a, b] and $H(x)$ 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

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