Editing Dirac comb (section)

== Dirac-comb identity ==

The Dirac comb can be constructed in two ways, either by using the ''comb'' [[Operator (mathematics)|operator]] (performing [[sampling (signal processing)|sampling]]) applied to the function that is constantly <math>1</math>, or, alternatively, by using the ''rep'' operator (performing [[periodic summation|periodization]]) applied to the [[Dirac delta]] <math>\delta</math>. Formally, this yields the following:{{sfn|Woodward|1953}}{{sfn|Brandwood|2003}}
<math display="block">\operatorname{comb}_T \{1\} = \operatorname{\text{Ш}}_T = \operatorname{rep}_T \{\delta \}, </math>
where
<math display="block">
 \operatorname{comb}_T \{f(t)\} \triangleq \sum_{k=-\infty}^\infty \, f(kT) \, \delta(t - kT)
</math> and <math display="block">
 \operatorname{rep}_T \{g(t)\} \triangleq \sum_{k=-\infty}^\infty \, g(t - kT).
</math>

In [[signal processing]], this property on one hand allows [[sampling (signal processing)|sampling]] a function <math>f(t)</math> by ''multiplication'' with <math>\operatorname{\text{Ш}}_{\ T}</math>, and on the other hand it also allows the [[periodic summation|periodization]] of <math>f(t)</math> by ''convolution'' with <math>\operatorname{\text{Ш}}_T</math>.{{sfn|Bracewell|1986}}
The Dirac comb identity is a particular case of the [[Convolution_theorem#Convolution theorem for tempered distributions| Convolution Theorem]] for tempered distributions.