Rectangle Function
المؤلف:
Bracewell, R.
المصدر:
"Rectangle Function of Unit Height and Base, Pi(x)." In The Fourier Transform and Its Applications. New York: McGraw-Hill, , 1965.
الجزء والصفحة:
pp. 52-53
25-5-2019
2510
Rectangle Function
The rectangle function 
is a function that is 0 outside the interval ![[-1/2,1/2]](http://mathworld.wolfram.com/images/equations/RectangleFunction/Inline2.gif)
and unity inside it. It is also called the gate function, pulse function, or window function, and is defined by
![Pi(x)=<span style=]() {0 for |x|>1/2; 1/2 for |x|=1/2; 1 for |x|<1/2. " src="http://mathworld.wolfram.com/images/equations/RectangleFunction/NumberedEquation1.gif" style="height:86px; width:142px" /> |
(1)
|
The left figure above plots the function as defined, while the right figure shows how it would appear if traced on an oscilloscope. The generalized function 
has height 
, center 
, and full-width 
.
As noted by Bracewell (1965, p. 53), "It is almost never important to specify the values at 
, that is at the points of discontinuity. Likewise, it is not necessary or desirable to emphasize the values 
in graphs; it is preferable to show graphs which are reminiscent of high-quality oscillograms (which, of course, would never show extra brightening halfway up the discontinuity)."
The piecewise version of the rectangle function is implemented in the Wolfram Language as UnitBox[x], while the generalized function version is implemented as HeavisidePi[x].
Identities satisfied by the rectangle function include
where 
is the Heaviside step function. The Fourier transform of the rectangle function is given by
where 
is the sinc function.
REFERENCES:
Bracewell, R. "Rectangle Function of Unit Height and Base, 
." In The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 52-53, 1965.
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