Dirac comb

$\delta_T(t) = \sum_{n=-\infty}^{\infty} \delta(t - n T)$

for some given period T. From the orthogonality of the Fourier series, an alternate definition may also be written:

$\delta_T(t) = \frac{1}{T}\sum_{n=-\infty}^{\infty} e^{\pm 2\pi int/T}$

Properties

Many of the properties of the Dirac comb follow from the properties of the Dirac delta function.

$\mathcal{F}(\delta_T)=\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{\infty} e^{-i\omega nT} =\frac{\sqrt{2\pi}}{T}\delta_{2\pi/T}(\omega)$

Multiplication of a continuous signal by a Dirac comb is sometimes called an ideal sampler with sampling rate T. When used as an ideal sampler, it can be used to understand the effects of aliasing and as a proof of the Shannon-Nyquist sampling theorem.

See Shannon-Nyquist sampling theorem for a proof using the Dirac comb.