Sampling theorem

Given a function f(t), whose Fourier transform exists, its corresponding aliased version is definedby the band limited function
where is called Nyquist frequency. The two functions and are the same if and only if itself is bandlimited, that is when . It can be shown that the following exact result holds:
where can be thought of as a sampling time. In other words, if we digitize a function we loose some information, and the spectum of the sampled signal is aliased. On the other hand, if the function was bandlimited and the sampling time not above the reciprocal of its stopband frequency, the samples alone contain enough information to reconstruct exactly the original function.
