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 .^[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

where is the uniform distribution of for the interval and 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