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 nth 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 > 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.

Fine Print.

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Applet by Steve Crutchfield.