In
mathematics, a boxcar function is any
function which is zero over the entire
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
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 In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
as
where is the uniform distribution of ''x'' for the interval and
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 argume ...
.
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
In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
of a
filter, the result is a
moving average
In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a moving mean (MM) or rolling mean and is ...
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
A railroad car, railcar ( American and Canadian English), railway wagon, railway carriage, railway truck, railwagon, railcarriage or railtruck (British English and UIC), also called a train car, train wagon, train carriage or train truck, is ...
.
See also
*
Boxcar averager
A boxcar averager (alternative names are gated integrator and boxcar integrator) is an electronic test instrument that integrates the signal input voltage after a defined waiting time (trigger delay) over a specified period of time (gate width) a ...
*
Rectangular function
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as
\operatorname(t) = \Pi(t) =
\left\{\begin{array}{r ...
*
Step function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having on ...
*
Top-hat filter
References
{{mathanalysis-stub
Special functions