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== Sampling theorem== | == 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, .... | ||
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| + | |||
| + | |||
| + | 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. | ||
| + | |||
| + | |||
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| + | Wc has to exist between Wm and Ws-Wm. | ||
| + | |||
| + | Here is a diagram. | ||
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| + | 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. | ||
Latest revision as of 05:31, 16 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) $
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.
