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{{Other uses|Sampling (disambiguation)}}
[[Image:Signal Sampling.svg|thumb|300px|Signal sampling representation. The continuous signal ''S''(''t'') is represented with a green colored line while the discrete samples are indicated by the blue vertical lines.]]
In [[signal processing]], '''sampling''' is the reduction of a [[continuous-time signal]] to a [[discrete-time signal]]. A common example is the conversion of a [[sound wave]] to a sequence of "samples".
A '''sample''' is a value of the [[signal]] at a point in time and/or space; this definition differs from [[Sampling (statistics)|the term's usage in statistics]], which refers to a set of such values.{{efn-ua|For example, "number of samples" in signal processing is roughly equivalent to "[[sample size]]" in statistics.}}
A '''sampler''' is a subsystem or operation that extracts samples from a [[continuous signal]]. A theoretical '''ideal sampler''' produces samples equivalent to the instantaneous value of the continuous signal at the desired points.
The original signal can be reconstructed from a sequence of samples, up to the [[Nyquist limit]], by passing the sequence of samples through a [[reconstruction filter]].
== Theory ==
{{See also|Nyquist–Shannon sampling theorem}}
Functions of space, time, or any other dimension can be sampled, and similarly in two or more dimensions.
For functions that vary with time, let ''S''(''t'') be a continuous function (or "signal") to be sampled, and let sampling be performed by measuring the value of the continuous function every ''T'' seconds, which is called the '''sampling interval''' or '''sampling period'''.<ref>{{cite book | title = Communications Standard Dictionary | author = Martin H. Weik | publisher = Springer | year = 1996 | isbn = 0412083914 | url = https://books.google.com/books?id=jxXDQgAACAAJ&q=Communications+Standard+Dictionary }}</ref> Then the sampled function is given by the sequence:
: ''S''(''nT''), for integer values of ''n''.
{{anchor|Sampling rate}}The '''sampling frequency''' or '''sampling rate''', ''f''<sub>s</sub>, is the average number of samples obtained in one second, thus {{nowrap|1=''f''<sub>s</sub> = 1/''T''}}, with the unit ''samples per second'', sometimes referred to as [[hertz]], for example e.g. 48 kHz is 48,000 ''samples per second''.
Reconstructing a continuous function from samples is done by interpolation algorithms. The [[Whittaker–Shannon interpolation formula]] is mathematically equivalent to an ideal [[low-pass filter]] whose input is a sequence of [[Dirac delta functions]] that are modulated (multiplied) by the sample values. When the time interval between adjacent samples is a constant (''T''), the sequence of delta functions is called a [[Dirac comb]]. Mathematically, the modulated Dirac comb is equivalent to the product of the comb function with ''s''(''t''). That mathematical abstraction is sometimes referred to as ''impulse sampling''.<ref>{{cite book |title=Signals and Systems |author=Rao, R. |isbn=9788120338593 |url=https://books.google.com/books?id=4z3BrI717sMC |publisher=Prentice-Hall Of India Pvt. Limited|year=2008 }}</ref>
Most sampled signals are not simply stored and reconstructed. The fidelity of a theoretical reconstruction is a common measure of the effectiveness of sampling. That fidelity is reduced when ''s''(''t'') contains frequency components whose cycle length (period) is less than 2 sample intervals (see ''[[Aliasing#Sampling sinusoidal functions|Aliasing]]''). The corresponding frequency limit, in ''cycles per second'' ([[hertz]]), is 0.5 cycle/sample × ''f''<sub>s</sub> samples/second = ''f''<sub>s</sub>/2, known as the [[Nyquist frequency]] of the sampler. Therefore, ''s''(''t'') is usually the output of a [[low-pass filter]], functionally known as an ''anti-aliasing filter''. Without an anti-aliasing filter, frequencies higher than the Nyquist frequency will influence the samples in a way that is misinterpreted by the interpolation process.<ref>[[Claude E. Shannon|C. E. Shannon]], "Communication in the presence of noise", [[Proc. Institute of Radio Engineers]], vol. 37, no.1, pp. 10–21, Jan. 1949. [http://www.stanford.edu/class/ee104/shannonpaper.pdf Reprint as classic paper in: ''Proc. IEEE'', Vol. 86, No. 2, (Feb 1998)] {{webarchive|url=https://web.archive.org/web/20100208112344/http://www.stanford.edu/class/ee104/shannonpaper.pdf |date=2010-02-08 }}</ref>
== Practical considerations==
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