Difference between revisions of "Adam Frey-- Nyquist sampling theorem" - Rhea

Latest revision as of 07:31, 29 July 2009

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

 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 .

@@ Line 1: / Line 1: @@
+== Adam Frey Nyquist Sampling Theorem ==
 Sampling Theorem:
 Let x(t) be a band-limited signal with X(j w) = 0 for  |W| >Wm.
@@ Line 8: / Line 10: @@
 where
-Ws  =  (2* pi ) / T   .
+                                   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
-For example  if  W(x) =
+     Ts =  2(pi) / (2 * 2)
+     Ts =  pi / 2
+     So the greatest Ts that can be used with out aliasing would be  pi/ 2 .