Sampling
The process of sampling can be defined as the process of converting continuous-time signal to discrete-time signal, process the discrete-time signal using a discrete-time system and convert it back to continuous time.
Application
This process is widely used in moving pictures where a sequence of individual frames are used to form continuously changing scene. These individual frames are moved at a sufficiently fast rate to perceive an accurate representation of the original continuously moving scene.
Sampling Theorem
Let $ x(t)\, $ represent a continuous-time signal and X(j$ \omega\, $) be the continuous Fourier transform of that signal. Then,
$ X(f)\ \stackrel{=}\ \int_{-\infty}^{\infty} x(t) \ e^{- j 2 \omega t} \ dt. \ $
The signal $ x(t)\, $ is band-limited with $ X(j\omega) = 0 \quad $ for all $ |\omega| > \omega_M \, $
Then $ x(t)\, $ is uniquely determined by its samples $ x(nT)\, $ n=....,-2,-1,0,1,2,....., if
$ \omega_s > 2\omega_M,\, $
The time interval between successive samples is referred to as the sampling interval
$ T=\ \frac{2\pi}{\omega_s},\, $
or, $ \omega_s=\ \frac{2\pi}{T},\, $
The using the sampling theorem we can reconstruct the original $ x(t)\ $ from the samples and states exactly.
