• Source: 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




boxcar
⁡
(
x
)
=
(
b
−
a
)
A

f
(
a
,
b
;
x
)
=
A
(
H
(
x
−
a
)
−
H
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x
−
b
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)
,


{\displaystyle \operatorname {boxcar} (x)=(b-a)A\,f(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
)


{\displaystyle H(x)}

is the Heaviside step function. 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 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




References

Kata Kunci Pencarian:

• Boxcar function
• Rectangular function
• Bessel function
• Boxcar averager
• Wigner distribution function
• Step function
• Piecewise linear function
• Moving average
• Boxcar (disambiguation)
• Kernel (statistics)