Nyquist-Shannon sampling theorem

The Nyquist-Shannon sampling theorem is not specific to music, but is fundamental to any digital sampling of a signal.

The Nyquist-Shannon sampling theorem states that the frequency content of a signal is fully represented by sampling at a certain frequency if the signal does not contains frequencies higher than one-half of the sampling rate.

In other words, to properly represent a frequency one must sample it with a sampling rate that is at least twice that frequency. To represent the frequency 1000 Hz properly, we must use at least 2000 samples per second. If we sample with 2000 samples per second, we can "catch" only frequencies up to 1000 Hz.

For example, audio CDs store digital audio at the 44.1 KHz sampling rate. It is accepted that the human ear can perceive frequencies up to 22 KHz and hence a sampling rate of at least 2 * 22 KHz = 44 KHz is necessary to represent the full frequency spectrum of what is humanly audible.

The Nyquist-Shannon sampling theorem is sometimes described as the Shannon-Kotelnikov theorem or the Nyquist-Kotelnikov theorem. The theorem has many variations and the names Nyquist, Shannon, and Kotelnikov are used in various combinations. The following, for example, is the Nyquist-Kotelnikov theorem: "Every signal, which can be integrated in time and has a finite frequency spectrum, can be sampled at intervals of time that are smaller or equal to 1 / (2 f_s), where f_s is the maximum of the frequency spectrum."

A good example of how the Nyquist-Shannon sampling theorem works is to consider sampling at the sampling frequency f_s of a simple wave with zero phase and frequency f_s. At every sample k, the value of this simple wave would be

$$\cos(\frac{2 \pi k f_s }{ f_s}) = \cos(2 \pi k) = 1$$

This simple wave cannot be sampled properly with the given sampling frequency. The sampled data can be confused with the frequency 0 (with DC gain of 1).

In fact, if the sampling rate is fs, any frequency above f_s / 2 will be confused with a frequency below f_s / 2. An example is provided in the topic Aliasing.