## 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.