A continoustime signal x(t) is sampled at a frequency of
f_{s} Hz 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 f_{c}
Hz is used to obtain the reconstructed signal
x_{r}(t).
Suppose the highestfrequency component in x(t) is at frequency f_{m}. Then the Sampling Theorem states that for f_{s} > 2f_{m} there is no loss of information in sampling. In this case, choosing f_{c} in the range f_{m} <f_{c} <f_{s}  f_{m} gives x_{r}(t) = x(t). These results can be understood by examining the Fourier transforms X(f), X_{s}(f), and X_{r}(f). If f_{s} < 2f_{m} and/or f_{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(f) will appear. Then, enter a sampling frequency f_{s} and click "Sample" to display the sampled signal and its magnitude spectrum. Finally, choose a cutoff frequency f_{c} and click "Filter." The reconstructed signal and its magnitude spectrum will be shown.
Applet by Steve Crutchfield.
