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Date: 29-9-2019
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The rectangle function is a function that is 0 outside the interval and unity inside it. It is also called the gate function, pulse function, or window function, and is defined by
(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
(2) |
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(3) |
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(4) |
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(5) |
where is the Heaviside step function. The Fourier transform of the rectangle function is given by
(6) |
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(7) |
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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طقطقة الرقبة.. عواقب خطيرة جدا لا يمكن تخيلها
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"ستارلاينر" تترك رائدين عالقين في الفضاء وتعود للأرض
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في يومها الثاني مركز الثقافة الأسرية يواصل تنظيم ورشة العودة إلى المدرسة
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