A continoustime 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 nth impulse given by
x(nT_{s} ). An ideal lowpass
filter with cutoff frequency w_{c}
rad/sec. is used to obtain the reconstructed signal
x_{r}(t).
Suppose the highestfrequency component in x(t) is at frequency w_{m}. Then the Sampling Theorem states that for w_{s} > 2w_{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.
Applet by Steve Crutchfield.
