Difference between revisions of "HW81 Jun Hyeong Park ECE301Fall2008mboutin" - Rhea

Revision as of 15:38, 9 November 2008

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 $ \frac{2\pi}{T} $. 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 ---------> $ x_{p}(t) $

       ^
       |
       |

$ p(t) = \sum^{\infty}_{n=-\infty}\delta(t-nT) $

$ x_{p}(t) = x(t)p(t) = x(t)\sum^{\infty}_{n=-\infty}\delta(t-nT) $

$ = \sum^{\infty}_{n=-\infty}x(t)\delta(t-nT) $

$ = \sum^{\infty}_{n=-\infty}x(nT)\delta(t-nT) $

Above diagram is the sampling process.

Here is a diagram for recovering process.

$ x_{p}(t) ---->Filter, H(w) -----> x(t) $

Here is a whole process from sampling to recovering.

x(t) ------> multiply ---------> $ x_{p}(t) $ ---> Filter, H(w) ----> x(t)

       ^
       |
       |
              p(t)