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As depicted, copies of <math>X(f)</math> are shifted by multiples of the sampling rate <math>f_s = 1/T</math> and combined by addition. For a band-limited function <math>(X(f) = 0, \text{ for all } |f| \ge B)</math> and sufficiently large <math>f_s,</math> it is possible for the copies to remain distinct from each other. But if the Nyquist criterion is not satisfied, adjacent copies overlap, and it is not possible in general to discern an unambiguous <math>X(f).</math> Any frequency component above <math>f_s/2</math> is indistinguishable from a lower-frequency component, called an ''alias'', associated with one of the copies. In such cases, the customary interpolation techniques produce the alias, rather than the original component. When the sample-rate is pre-determined by other considerations (such as an industry standard), <math>x(t)</math> is usually filtered to reduce its high frequencies to acceptable levels before it is sampled. The type of filter required is a [[lowpass filter]], and in this application it is called an [[anti-aliasing filter]].
[[File:ReconstructFilter.
[[File:Nyquist sampling.gif|upright=1.8|thumb|right|The figure on the left shows a function (in gray/black) being sampled and reconstructed (in gold) at steadily increasing sample-densities, while the figure on the right shows the frequency spectrum of the gray/black function, which does not change. The highest frequency in the spectrum is half the width of the entire spectrum. The width of the steadily-increasing pink shading is equal to the sample-rate. When it encompasses the entire frequency spectrum it is twice as large as the highest frequency, and that is when the reconstructed waveform matches the sampled one.]]
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===Notes===
<!---Bug report: The group=proof tag attracts the intended footnote, but it also attracts one of the
{{notelist|group=proof}}
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That is, a sufficient no-loss condition for sampling [[signal (information theory)|signal]]s that do not have [[baseband]] components exists that involves the ''width'' of the non-zero frequency interval as opposed to its highest frequency component. See ''[[Sampling (signal processing)|sampling]]'' for more details and examples.
For example, in order to sample [[FM broadcasting|FM radio]] signals in the frequency range of 100–102 [[megahertz|MHz]], it is not necessary to sample at 204 MHz (twice the upper frequency), but rather it is sufficient to sample at 4 MHz (twice the width of the frequency interval). (Reconstruction is not usually the goal with sampled IF or RF signals. Rather, the sample sequence can be treated as ordinary samples of the signal frequency-shifted to near baseband, and digital demodulation can proceed on that basis.)
:<math>\left(\frac{N}2 f_\mathrm{s}, \frac{N+1}2 f_\mathrm{s}\right),</math>
for some nonnegative integer <math>N</math> and some sampling frequency <math>f_\mathrm{s}</math>, it is possible to find an interpolation that reproduces the signal. Note that there may be several combinations of <math>N</math> and <math>f_\mathrm{s}</math> that work, including the normal baseband condition as the case <math>N=0.</math> The corresponding interpolation
▲The corresponding interpolation function is the impulse response of an ideal brick-wall [[bandpass filter]] (as opposed to the ideal [[brick-wall filter|brick-wall]] [[lowpass filter]] used above) with cutoffs at the upper and lower edges of the specified band, which is the difference between a pair of lowpass impulse responses:
<math display="block">(N+1)\,\operatorname{sinc} \left(\frac{(N+1)t}T\right) - N\,\operatorname{sinc}\left( \frac{Nt}T \right).</math>
This function is 1 at <math>t=0</math> and zero at any other multiple of <math>T</math> (as well as at other times if <math>N>0</math>).
Other generalizations, for example to signals occupying multiple non-contiguous bands, are possible as well. Even the most generalized form of the sampling theorem does not have a provably true converse. That is, one cannot conclude that information is necessarily lost just because the conditions of the sampling theorem are not satisfied; from an engineering perspective, however, it is generally safe to assume that if the sampling theorem is not satisfied then information will most likely be lost.
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The sampling theorem was implied by the work of [[Harry Nyquist]] in 1928,<ref>{{cite journal | last=Nyquist |first=Harry | author-link =Harry Nyquist | title =Certain topics in telegraph transmission theory | journal =Transactions of the AIEE | volume =47 | issue =2 | pages =617–644 | date =April 1928 | doi=10.1109/t-aiee.1928.5055024| bibcode =1928TAIEE..47..617N }} [https://web.archive.org/web/20130926031230/http://www.ieee.org/publications_standards/publications/proceedings/nyquist.pdf Reprint as classic paper] in: ''Proceedings of the IEEE'', Vol. 90, No. 2, February 2002.</ref> in which he showed that up to <math>2B</math> independent pulse samples could be sent through a system of bandwidth <math>B</math>; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, [[Karl Küpfmüller]] showed a similar result<ref>{{cite journal |first=Karl |last=Küpfmüller |title=Über die Dynamik der selbsttätigen Verstärkungsregler |journal=Elektrische Nachrichtentechnik |volume=5 |issue=11 |pages=459–467 |year=1928 |language=de}} [http://ict.open.ac.uk/classics/2.pdf (English translation 2005)] {{Webarchive|url=https://web.archive.org/web/20190521021624/http://ict.open.ac.uk/classics/2.pdf |date=2019-05-21 }}.</ref> and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step-response [[sine integral]]; this bandlimiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a ''Küpfmüller filter'' (but seldom so in English).
The sampling theorem, essentially a [[duality (mathematics)|dual]] of Nyquist's result, was proved by [[Claude E. Shannon]].<ref name="Shannon49"/>
In 1948 and 1949, Claude E. Shannon published the two revolutionary articles in which he founded [[information theory]].<ref>{{cite journal |ref=refShannon48 |last=Shannon |first=Claude E. |author-link=Claude Shannon |title=A Mathematical Theory of Communication |journal=Bell System Technical Journal |volume=27 |issue=3 |pages=379–423 |date=July 1948 |doi=10.1002/j.1538-7305.1948.tb01338.x|hdl=11858/00-001M-0000-002C-4317-B |hdl-access=free }}</ref><ref>{{cite journal |ref=refShannon48oct |last=Shannon |first=Claude E. |author-link=Claude Shannon |title=A Mathematical Theory of Communication |journal=Bell System Technical Journal |volume=27 |issue=4 |pages=623–666 |date=October 1948 |doi=10.1002/j.1538-7305.1948.tb00917.x|hdl=11858/00-001M-0000-002C-4314-2 |hdl-access=free }}</ref><ref name="Shannon49"/> In Shannon's "[[A Mathematical Theory of Communication]]", the sampling theorem is formulated as "Theorem 13": Let <math>f(t)</math> contain no frequencies over W. Then
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===Other discoverers===
Others who have independently discovered or played roles in the development of the sampling theorem have been discussed in several historical articles, for example, by Jerri<ref>{{cite journal | last=Jerri | first=Abdul | author-link=Abdul Jerri | title=The Shannon Sampling Theorem—Its Various Extensions and Applications: A Tutorial Review | journal=Proceedings of the IEEE | volume=65 | issue=11 | pages=1565–1596 | date=November 1977 | doi=10.1109/proc.1977.10771 | bibcode=1977IEEEP..65.1565J | s2cid=37036141 }} See also {{cite journal | last=Jerri | first=Abdul | title=Correction to 'The Shannon sampling theorem—Its various extensions and applications: A tutorial review' | journal=Proceedings of the IEEE | volume=67 | issue=4 | page=695 | date=April 1979 | doi=10.1109/proc.1979.11307 }}</ref> and by Lüke.<ref>{{cite journal | last=Lüke | first=Hans Dieter | title =The Origins of the Sampling Theorem | journal =IEEE Communications Magazine | pages =106–108 | date =April 1999 | issue=4 | doi =10.1109/35.755459 | volume=37| url=http://www.hit.bme.hu/people/papay/edu/Conv/pdf/origins.pdf | citeseerx=10.1.1.163.2887 }}</ref> For example, Lüke points out that
{{blockquote|
As pointed out by Higgins, the sampling theorem should really be considered in two parts, as done above: the first stating the fact that a bandlimited function is completely determined by its samples, the second describing how to reconstruct the function using its samples. Both parts of the sampling theorem were given in a somewhat different form by [[J. M. Whittaker]] and before him also by Ogura. They were probably not aware of the fact that the first part of the theorem had been stated as early as 1897 by [[Émile Borel|Borel]].{{refn|group= Meijering|Several authors, following Black, have claimed that this first part of the sampling theorem was stated even earlier by Cauchy, in a paper published in 1841. However, the paper of Cauchy does not contain such a statement, as has been pointed out by Higgins.}} As we have seen, Borel also used around that time what became known as the cardinal series. However, he appears not to have made the link. In later years it became known that the sampling theorem had been presented before Shannon to the Russian communication community by [[Vladimir Kotelnikov|Kotel'nikov]]. In more implicit, verbal form, it had also been described in the German literature by [[Herbert Raabe|Raabe]]. Several authors have mentioned that Someya introduced the theorem in the Japanese literature parallel to Shannon. In the English literature, Weston introduced it independently of Shannon around the same time.{{refn|group= Meijering|As a consequence of the discovery of the several independent introductions of the sampling theorem, people started to refer to the theorem by including the names of the aforementioned authors, resulting in such catchphrases as "the Whittaker–Kotel'nikov–Shannon (WKS) sampling theorem" or even "the Whittaker–Kotel'nikov–Raabe–Shannon–Someya sampling theorem". To avoid confusion, perhaps the best thing to do is to refer to it as the sampling theorem, "rather than trying to find a title that does justice to all claimants".}}
{{reflist|group= Meijering}}|Eric Meijering, "A Chronology of Interpolation From Ancient Astronomy to Modern Signal and Image Processing" (citations omitted)
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