Sampling Theorem
The Nyquist-Shannon theorem: a signal can be perfectly reconstructed from samples taken at twice its highest frequency — and what goes wrong (aliasing) if you sample slower.
To store or process a continuous signal (like sound) digitally, you must sample it — record its value at regular instants in time. The Nyquist-Shannon sampling theorem says: a signal can be perfectly reconstructed from its samples if and only if it was sampled at a rate of at least twice its highest frequency component.
That minimum rate, , is called the Nyquist rate. Sample any slower, and information is permanently lost — different signals become indistinguishable from their samples, a phenomenon called aliasing.
Human hearing tops out around 20,000 Hz. To safely capture all audible frequencies, CD audio samples at 44,100 Hz — comfortably above the Nyquist rate of Hz.
If the highest frequency in a signal is 5,000 Hz, what is the minimum sampling rate needed to reconstruct it without aliasing?
Solution
10,000 Hz (10 kHz) — twice the highest frequency, per the Nyquist rate.