## Sampling

The function x(t) is sampled at times t = kT. The sampled function y(t) can be represented as the product of x(t) and a sampling function p(t).
 1
For impulse sampling,
 2
Combining equations 1 and 2.
 3
Since p(t) is a periodic function it can be represented by the Fourier series shown in equation 4.
 4
where for impulse sampling, ak = 1 / T.
 5
Multiplying equation 5 by x(t).
 6
Multiplying a time function by results in a shift of kws in the frequency domain. That is,
 7
Applying equation 7 to equation 6 results in the following for the frequency spectrum of y(t) the sampled signal.
 8
Consider a signal with the spectrum X(jw), of the baseband x(t).
The spectrum of the sampled function, Y(jw), contains duplicates of the baseband signal shifted to multiples of kws , as shown below.

Aliasing

To avoid the overlap distortion called aliasing, wb must be less than -wb + ws

That is,

ws > 2 wb
This result, that the sampling frequency must be at least twice the largest baseband frequency is called the Sampling Theorem.