Editing Dirac delta function (section)

==Dirac comb==
{{Main|Dirac comb}}
[[File:Dirac comb.svg|thumb|A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of {{mvar|T}}]]
A so-called uniform "pulse train" of Dirac delta measures, which is known as a [[Dirac comb]], or as the [[Sha (Cyrillic)|Sha]] distribution, creates a [[sampling (signal processing)|sampling]] function, often used in [[digital signal processing]] (DSP) and discrete time signal analysis.  The Dirac comb is given as the [[infinite sum]], whose limit is understood in the distribution sense,

<math display="block">\operatorname{\text{Ш}}(x) = \sum_{n=-\infty}^\infty \delta(x-n),</math>

which is a sequence of point masses at each of the integers.

Up to an overall normalizing constant, the Dirac comb is equal to its own Fourier transform.  This is significant because if {{mvar|f}} is any [[Schwartz space|Schwartz function]], then the [[Wrapped distribution|periodization]] of {{mvar|f}} is given by the convolution
<math display="block">(f * \operatorname{\text{Ш}})(x) = \sum_{n=-\infty}^\infty f(x-n).</math>
In particular,
<math display="block">(f*\operatorname{\text{Ш}})^\wedge = \widehat{f}\widehat{\operatorname{\text{Ш}}} = \widehat{f}\operatorname{\text{Ш}}</math>
is precisely the [[Poisson summation formula]].{{sfn|Córdoba|1988}}{{sfn|Hörmander|1983|loc=[{{google books |plainurl=y |id=aaLrCAAAQBAJ|page=177}} §7.2]}}
More generally, this formula remains to be true if {{mvar|f}} is a tempered distribution of rapid descent or, equivalently, if <math>\widehat{f}</math> is a slowly growing, ordinary function within the space of tempered distributions.