In mathematics, the unit doublet is the derivative of the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
. It can be used to differentiate
signals in
electrical engineering: If ''u''
1 is the unit doublet, then
:
where
is the
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''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 quadrant, and the other in the fourth. The length of each rectangle is k, whereas their breadth is 1/k2, where k tends to zero.
References
Generalized functions
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