A continous-time signal *x(t)* is sampled at a frequency of
*w*_{s} *rad/sec.* to
produce a sampled signal *x*_{s}(t).
We model *x*_{s}(t) as an impulse
train with the area of the *n*th impulse given by
*x(nT*_{s}). An ideal low-pass
filter with cutoff frequency *w*_{c}
*rad/sec.* is used to obtain the reconstructed signal
*x*_{r}(t).

Suppose the highest-frequency component in *x(t)* is at frequency
*w*_{m}.
Then the Sampling Theorem
states that for *w*_{s}
> 2*w*_{m} there is no loss
of information in sampling. In this case, choosing
*w*_{c}
in the range
*w*_{m}
<*w*_{c}
<*w*_{s} -
*w*_{m}
gives *x*_{r}(t) = *x(t)*.
These results can be understood by examining the Fourier transforms
*X(jw)*, *X*_{s}
(jw),
and *X*_{r}(jw). If
*w*_{s}
<2*w*_{m} and/or
*w*_{c}
is chosen poorly,
then *x*_{r}(t) might not resemble
*x(t)*.

To explore sampling and reconstruction, select a signal or
use the mouse to draw a signal *x(t)* in the window below.
After a moment, the magnitude spectrum |*X(jw)*| will appear. Then,
enter a sampling frequency *w*_{s}
and click "Sample" to display the sampled signal and its
magnitude spectrum. Finally, choose a cutoff frequency
*w*_{c} and click "Filter."
The reconstructed signal and its magnitude spectrum will be
shown.

Fine Print.