## 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.