Editing Sampling (signal processing) (section)

== 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 <math>s(t)</math> be a continuous function (or "signal") to be sampled, and let sampling be performed by measuring the value of the continuous function every <math>T</math> 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><ref name=Moir>{{cite book | title = Rudiments of Signal Processing and Systems | author = Tom J. Moir | publisher = Springer International Publishing AG | year = 2022|pages=459 | isbn = 9783030769475 | url = https://public.ebookcentral.proquest.com/choice/publicfullrecord.aspx?p=6809637|doi=10.1007/978-3-030-76947-5 }}</ref> Then the sampled function is given by the sequence:
: <math>s(nT)</math>, for integer values of <math>n</math>.
{{anchor|Sampling rate}}The '''sampling frequency''' or '''sampling rate''', <math>f_s</math>, is the average number of samples obtained in one second, thus <math>f_s=1/T</math>, with the unit ''samples per second'', sometimes referred to as [[hertz]], for example 48&nbsp;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 <math>(T)</math>, 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 <math>s(t)</math>. 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 <math>s(t)</math> 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 <math>0.5</math> cycle/sample&nbsp;× <math>f_s</math> samples/second = <math>f_s/2</math>, known as the [[Nyquist frequency]] of the sampler. Therefore, <math>s(t)</math> 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>