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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 × <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>
== Practical considerations==
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