Sampling Theory and Aliasing

Sampling is the process of converting a continuous-time analog signal into a discrete sequence of numbers by measuring the signal's amplitude at regular time intervals. The rate of measurement is the sample rate (or sampling frequency), measured in samples per second (Hz). An ADC with a 1 MHz sample rate takes one measurement every microsecond, producing a million numbers per second.

The crucial question is: how fast must you sample to faithfully represent a given signal? Intuition says at least fast enough to capture the signal's highest-frequency content. The Nyquist–Shannon theorem makes this mathematically precise: to perfectly reconstruct a signal containing frequencies up to f_max, the sample rate must be at least f_s ≥ 2 × f_max. The minimum rate of 2 × f_max is the Nyquist rate. The corresponding maximum frequency representable at a given sample rate, f_s / 2, is the Nyquist frequency (also called the folding frequency).

When the sampling theorem is satisfied, perfect reconstruction is theoretically possible by passing the discrete samples through an ideal low-pass filter with cutoff at f_s/2. The filter interpolates between sample points, perfectly recovering the original continuous waveform. In practice, imperfect reconstruction filters leave some ripple, which is why oversampling (sampling well above the Nyquist rate) is common in high-quality audio and SDR systems.