Editing Dirac comb (section)

{{Short description|Periodic distribution ("function") of "point-mass" Dirac delta sampling}}
{{use dmy dates|date=August 2024}}
[[Image:Dirac comb.svg|thumb|300px|The graph of the Dirac comb function is an infinite series of [[Dirac delta function]]s spaced at intervals of ''T'']]

In [[mathematics]], a '''Dirac comb''' (also known as '''sha function''', '''impulse train''' or '''sampling function''') is a [[periodic function|periodic]] [[Function (mathematics)|function]] with the formula
<math display="block">\operatorname{\text{Ш}}_{\ T}(t) \ :=  \sum_{k=-\infty}^{\infty} \delta(t - k T) </math>
for some given period <math>T</math>.<ref name=":0">{{Cite web |title=The Dirac Comb and its Fourier Transform |url=https://dspillustrations.com/pages/posts/misc/the-dirac-comb-and-its-fourier-transform.html#:~:text=The%20Dirac%20Comb%20function&text=CT(t)=T,(t%E2%88%92nT).&text=As%20shown,%20the%20Dirac%20comb,distinct%20impulses%20are%20T%20apart.&text=or%20equivalently%20as%20a%20sum,%CF%80nt/T). |access-date=2022-06-28 |website=dspillustrations.com}}</ref> Here ''t'' is a real variable and the sum extends over all [[integer]]s ''k.'' The [[Dirac delta function]] <math>\delta</math> and the Dirac comb are [[Distribution_(mathematics)#Tempered_distributions_and_Fourier_transform|tempered distributions]].<ref name="Schwartz 1951">{{cite book|last=Schwartz |first=L. |title=Théorie des distributions |volume=I-II |year=1951 |publisher=Hermann |location=Paris |authorlink=Laurent Schwartz}}</ref><ref name="Strichartz 1994">{{cite book |last=Strichartz |first=R. |title=A Guide to Distribution Theory and Fourier Transforms |year=1994 |publisher=CRC Press |isbn=0-8493-8273-4}}</ref> The graph of the function resembles a [[comb]] (with the <math>\delta</math>s as the comb's ''teeth''), hence its name and the use of the comb-like [[Cyrillic script|Cyrillic]] letter [[Sha (Cyrillic)|sha]] (Ш) to denote the function.

The symbol <math>\operatorname{\text{Ш}}\,\,(t)</math>, where the period is omitted, represents a Dirac comb of unit period. This implies<ref name=":0" />
<math display="block">\operatorname{\text{Ш}}_{\ T}(t) \ = \frac{1}{T}\operatorname{\text{Ш}}\ \!\!\!\left(\frac{t}{T}\right).</math>

Because the Dirac comb function is periodic, it can be represented as a [[Fourier series]] based on the [[Dirichlet kernel]]:<ref name=":0" />
<math display="block">\operatorname{\text{Ш}}_{\ T}(t) = \frac{1}{T}\sum_{n=-\infty}^{\infty} e^{i  2 \pi n \frac{t}{T}}.</math>

The Dirac comb function allows one to represent both [[Continuous function|continuous]] and [[Discrete mathematics|discrete]] phenomena, such as [[sampling (signal processing)|sampling]] and [[aliasing]], in a single framework of [[Fourier transform|continuous Fourier analysis]] on tempered distributions, without any reference to Fourier series.  The [[Fourier transform]] of a Dirac comb is another Dirac comb. Owing to the [[Convolution_theorem#Convolution theorem for tempered distributions|Convolution Theorem]] on tempered distributions which turns out to be the [[Poisson summation formula]], in [[signal processing]], the Dirac comb allows modelling sampling by ''[[multiplication]]'' with it, but it also allows modelling periodization by ''[[convolution]]'' with it.<ref name="Bracewell 1986">{{cite book |last1=Bracewell|first1=R. N.|title=The Fourier Transform and Its Applications|publisher=McGraw-Hill|edition=revised| year=1986 |orig-year=1st ed. 1965, 2nd ed. 1978}}</ref>