Dirac Delta Function - Dirac Comb

Dirac Comb

A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the Shah distribution, creates a 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,

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 ƒ is any Schwartz function, then the periodization of ƒ is given by the convolution

In particular,

(f*\Delta)^\wedge = \hat{f}\widehat{\Delta} = \hat{f} \Delta

is precisely the Poisson summation formula.

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