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) $
^
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$ 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)
^
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p(t)
