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 stop-band frequency, the samples alone contain enough information to reconstruct exactly the original function.


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