Impulse-train Sampling
One type of sampling that satisfies the Sampling Theorem is called impulse-train sampling. This type of sampling is achieved by the use of a periodic impulse train multiplied by a continuous time signal, $ x(t)\! $. The periodic impulse train, $ p(t)\! $ is referred to as the sampling function, the period, $ T\! $, is referred to as the sampling period, and the fundamental frequency of $ p(t)\! $, $ \omega_s = \frac{2\pi}{T}\! $, is the sampling frequency. We define $ x_p(t)\! $ by the equation,
Graphically, this equation looks as follows,
$ x(t)\! $ ----------> x --------> $ x_p(t)\! $ ^ | | $ p(t) = \sum^{\infty}_{n = -\infty} \delta(t - nT)\! $
By using linearity and the sifting property, $ x_p(t)\! $ can be represented as follows,
$ x_p(t) = x(t)p(t)\! $
$ = x(t)\sum^{\infty}_{n = -\infty} \delta(t - nT)\! $
$ =\sum^{\infty}_{n = -\infty}x(t)\delta(t - nT)\! $
$ =\sum^{\infty}_{n = -\infty}x(nT)\delta(t - nT)\! $
