Adam Frey-- Nyquist sampling theorem - Rhea

Adam Frey Nyquist Sampling Theorem

Sampling Theorem: Let x(t) be a band-limited signal with X(j w) = 0 for |W| >Wm.

Then x(t) is uniquely determined by its samples x(nT) = 0,(+,-)[1,2,3]. . ., if

Ws > 2 * Wm,

where

                                  Ws  =  (2* pi ) / T   

Then if

X D [n] = X(nTs) are a collection of samples, then x(t) can be uniquely recovered from its samples if

Ts < (1/2) (2 pi)/ Wm

For example

 if  X(w) = u(w +2) - u(w-2),  What is the largest Ts you can use to obtain xr(t)from x(t)?

well, Wm = 2, ( X(w) = 0 for |W | > Wm)

and Ws > 2 Wm

and Ts = 2(pi) / Wm

    Ts =  2(pi) / (2 * 2)
    
    Ts =  pi / 2

    So the greatest Ts that can be used with out aliasing would be  pi/ 2 .