Dirac Delta function

Dirac Delta function $δ (x)$ **

Such a function satisified:

when $x = + \infty$ , $δ (x) = + \infty$ , when $x = 0$ , $δ (x) = 0$
$\int_{- \infty}^{+ \infty} δ (x) d x = 1$

Most important things you should know for the Delta function

A Dirac Delta function has "sifting property":

\int f (y) δ (x - y) d y = f (x)

looks quite like convolution, right?

The Dirac Delta function can be given as Fourier Transformation as:

δ (t) = F T {1 (t)} = \int e^{- i 2 π f t} d t

Dirac pulse train: Train of equidistant $δ$ pulses in time domain corresponds to train of equidistant $δ$ pulses in frequency domain

\sum δ (t - n) \circ - ∙ \sum δ (f - n)