A continous-time 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 low-pass filter with cutoff frequency f_c Hz is used to obtain the reconstructed signal x_r(t).

Suppose the highest-frequency 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.

Fine Print.

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Applet by Steve Crutchfield.