mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 boxcar function can be expressed in terms of the

uniform distribution Uniform distribution may refer to: * Continuous uniform distribution * Discrete uniform distribution * Uniform distribution (ecology) * Equidistributed sequence See also * * Homogeneous distribution In mathematics, a homogeneous distribution ...

\operatorname(x)= (b-a)A\,f(a,b;x) = A(H(x-a) - H(x-b)),

where is the uniform distribution of ''x'' for the interval and

H(x)

is the

Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...

. 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 moving average filter. The function is named after its graph's resemblance to a

boxcar A boxcar is the North American (AAR) term for a railroad car that is enclosed and generally used to carry freight. The boxcar, while not the simplest freight car design, is considered one of the most versatile since it can carry most ...

, a type of railroad car.

References

{{mathanalysis-stub Special functions