Rectangle Function

The rectangle function Pi(x) is a function that is 0 outside the interval [-1/2,1/2] and unity inside it. It is also called the gate function, pulse function, or window function, and is defined by

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 f(x)=hPi((x-c)/b) has height h, center c, and full-width b.

As noted by Bracewell (1965, p. 53), "It is almost never important to specify the values at x=+/-1/2, that is at the points of discontinuity. Likewise, it is not necessary or desirable to emphasize the values Pi(+/-1/2)=1/2 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] (which takes the value 1 at x=+/-1/2), while the generalized function version is implemented as HeavisidePi [x] (which remains unevalutaed at x=+/-1/2).

Identities satisfied by the rectangle function include

where H(x) is the Heaviside step function. The Fourier transform of the rectangle function is given by

where sinc(x) is the sinc function.

See also

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References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Rectangle Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RectangleFunction.html

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