Difference between revisions of "HW ECE301Fall2008mboutin" - Rhea

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 == Sampling theorem==
-Here is a signal, x(t) with X(w) = 0 when |W| > Wm.
-With sampling period T, samples of x(t),x(nT), can be obtained , where n = 0 +-1, +-2, ....
-The sampling frequency is <math>\frac{2\pi}{T}</math>. It is called Ws.
-If Ws is greater than 2Wm, x(t) can be recovered from its samples.
-Here, 2Wm is called the "Nyquist rate".
-To recover, first we need a filter with amplited T when |W| < Wc.
-Wc has to exist between Wm and Ws-Wm.
-Here is a diagram.
-x(t) ------> multiply ---------> <math>x_{p}(t)</math>
-        ^
-        |
-        |
-<math>    p(t) = \sum^{\infty}_{n=-\infty}\delta(t-nT)</math>
-<math> x_{p}(t) = x(t)p(t) = x(t)\sum^{\infty}_{n=-\infty}\delta(t-nT)</math>
-<math>          = \sum^{\infty}_{n=-\infty}x(t)\delta(t-nT)</math>
-<math>          = \sum^{\infty}_{n=-\infty}x(nT)\delta(t-nT)</math>
-Above diagram is the sampling process.
-Here is a diagram for recovering process.
-<math> x_{p}(t) ---->Filter, H(w) -----> x(t)</math>
-Here is a whole process from sampling to recovering.
-x(t) ------> multiply ---------> <math>x_{p}(t)</math> ---> Filter, H(w) ----> x(t)
-        ^
-        |
-        |
-       <math>p(t)</math>
-Here is an important point.
-If Ws is not greater than 2Wm, the aliasing will occur.
-Then we cannot recover the original signal
-Therefore, the sampling period has to be selected well.

Revision as of 05:31, 16 November 2008