Dirac comb

In mathematics, a Dirac comb (also known as an impulse train and sampling function in electrical engineering) is a periodic tempered distribution constructed from Dirac delta functions

for some given period T. The symbol ${\displaystyle \operatorname {III} (t)}$, where the period is omitted, represents a Dirac comb of unit period. Some authors, notably Bracewell as well as some textbook authors in electrical engineering and circuit theory, refer to it as the Shah function (possibly because its graph resembles the shape of the Cyrillic letter sha Ш). Because the Dirac comb function is periodic, it can be represented as a Fourier series:

The Dirac comb function allows one to represent both continuous and discrete phenomena, such as sampling and aliasing, in a single framework of continuous Fourier analysis on Schwartz distributions, without any reference to Fourier series. Owing to the Poisson summation formula, in signal processing, the Dirac comb allows modelling sampling via multiplication with it, but it also allows modelling periodization via convolution with it.

The Dirac comb can be constructed in two ways, either by using the comb operator (performing sampling) applied to the function that is constantly ${\displaystyle 1}$, or, alternatively, by using the rep operator (performing periodization) applied to the Dirac delta ${\displaystyle \delta }$. Formally, this yields

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