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{{refimprove|date=August 2012}}
In mathematics, the '''unit doublet''' is the derivative of the [[Dirac delta function]].  It can be used to differentiate [[Signal (electronics)|signals]] in [[electrical engineering]]:<ref>{{cite web|url=http://ocw.mit.edu/NR/rdonlyres/Electrical-Engineering-and-Computer-Science/6-003Fall-2003/9BEA2A94-230F-4F38-ABAE-F552B8E549A9/0/lecture4.pdf |title=Signals and Systems Lecture #4|date=16 September 2003|work=Mit.edu|accessdate=2 September 2009}}
</ref> If ''u''<sub>1</sub> is the unit doublet, then


: <math> (x * u_1)(t) = \frac{dx(t)}{dt} </math>


where * refers to the [[convolution]] operator.
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The function is zero for all values except zero, where its behaviour is interesting. Its integral over any interval enclosing zero is zero. However, the integral of its absolute value over any region enclosing zero goes to infinity. The function can be thought of as the [[limiting case]] of two rectangles, one in the second [[Cartesian_coordinate_system#Quadrants_and_octants|quadrant]], and the other in the fourth. The length of each rectangle is k, whereas their breadth is 1/k<sup>2</sup>, where k tends to zero.
 
==References==
{{reflist}}
 
[[Category:Generalized functions]]
 
 
{{DEFAULTSORT:Unit Doublet}}
 
 
{{mathanalysis-stub}}

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