How it works
The sampling theorem states that a band-limited signal with highest frequency component f_max can be perfectly reconstructed from its samples if the sampling frequency f_s ≥ 2f_max. The Nyquist rate is exactly 2f_max; the Nyquist interval (maximum allowable sampling period) is T_s = 1/(2f_max). In the frequency domain, sampling x(t) at rate f_s produces a spectrum X_s(f) that is a sum of shifted copies of X(f) at integer multiples of f_s. If f_s > 2f_max, these copies do not overlap and x(t) is recoverable by ideal low-pass filtering with cutoff f_s/2.
Key points to remember
Aliasing occurs when f_s < 2f_max, causing spectral copies to overlap and making reconstruction impossible without distortion. An anti-aliasing filter (low-pass, cutoff at f_s/2) must be applied before sampling to prevent this. The ideal reconstruction filter is a low-pass filter with cutoff f_s/2 and gain T_s, implemented approximately by a sample-and-hold circuit followed by a smoothing filter. Practical ADCs like the ADC0804 have a built-in sample-and-hold; its aperture time of about 1 µs limits the maximum signal frequency for distortion-free sampling. Oversampling by a factor of 4× relaxes anti-aliasing filter requirements significantly.
Exam tip
The examiner always asks you to state the sampling theorem, define aliasing, and draw the spectrum of a sampled signal showing overlapping copies — sketch both the aliasing case and the correctly sampled case on the same diagram.