原创 Nyquist–Shannon sampling theorem

The Nyquist–Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling
is the process of converting a signal (for example, a function of
continuous time or space) into a numeric sequence (a function of
discrete time or space). The theorem states, in the original words of
Shannon (where he uses "cps" for "cycles per second" instead of the
modern unit hertz):^[1]

If a function f(t) contains no frequencies higher than W cps, it is completely determined by giving its ordinates at a series of points spaced 1/(2W) seconds apart.

In essence this means that an analog signal that has been digitized can be perfectly reconstructed if the sampling rate was 1/(2W) seconds, where W is the highest frequency in the original signal.

More recent statements of the theorem are sometimes careful to exclude the equality condition; that is, the condition is if f(t) contains no frequencies higher than or equal to W; this condition is equivalent to Shannon's except when the function includes a steady sinusoidal component at exactly frequency W.

The assumptions necessary to prove the theorem form a mathematical
model that is only an idealization of any real-world situation. The
conclusion, that perfect reconstruction is possible, is mathematically
correct for the model, but only an approximation for actual signals and
actual sampling techniques.

The theorem also leads to a formula for reconstruction of the original signal.

http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem