# Unit doublet

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{{Short description|General function in mathematics}}
{{refimprove|date=August 2012}}
alt=Live graphic of a unit doublet of width k as k goes to zero.|thumb|Approximation of a unit doublet with two rectangles of width k as k goes to zero.
In mathematics, the '''unit doublet''' is a [generalized function](/source/generalized_function), the derivative of the [Dirac delta function](/source/Dirac_delta_function).  It can be used to differentiate [signals](/source/Signal_(electronics)) in [electrical engineering](/source/electrical_engineering): If ''u''<sub>1</sub> is the unit doublet, then

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

where <math>*</math> is the [convolution](/source/convolution) operator.<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|url-status=dead|archiveurl=https://web.archive.org/web/20090219052652/http://ocw.mit.edu/NR/rdonlyres/Electrical-Engineering-and-Computer-Science/6-003Fall-2003/9BEA2A94-230F-4F38-ABAE-F552B8E549A9/0/lecture4.pdf|archivedate=19 February 2009}}
</ref>

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](/source/limiting_case_(mathematics)) of two rectangles, one in the second [quadrant](/source/Cartesian_coordinate_system), and the other in the fourth. The width of each rectangle is ''k'', whereas the height of each is 1/''k''<sup>2</sup>, in the limit as ''k'' approaches zero.

==References==
{{reflist}}

{{DEFAULTSORT:Unit Doublet}}
Category:Generalized functions
Category:Digital signal processing

{{mathanalysis-stub}}

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Adapted from the Wikipedia article [Unit doublet](https://en.wikipedia.org/wiki/Unit_doublet) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Unit_doublet?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.