Content deleted Content added
Tags: Mobile edit Mobile web edit |
IznoRepeat (talk | contribs) m add WP:TEMPLATECAT to remove from template; genfixes |
||
Line 1:
{{
The '''Nyquist–Shannon sampling theorem''' is an essential principle for [[digital signal processing]] linking the [[frequency range]] of a signal and the [[sample rate]] required to avoid a type of [[distortion]] called [[aliasing]]. The theorem states that the sample rate must be at least twice the [[Bandwidth (signal processing)|bandwidth]] of the signal to avoid aliasing. In practice, it is used to select [[band-limiting]] filters to keep aliasing below an acceptable amount when an analog signal is sampled or when sample rates are changed within a digital signal processing function.
Line 9:
Strictly speaking, the theorem only applies to a class of [[mathematical function]]s having a [[continuous Fourier transform|Fourier transform]] that is zero outside of a finite region of frequencies. Intuitively we expect that when one reduces a continuous function to a discrete sequence and [[interpolates]] back to a continuous function, the fidelity of the result depends on the density (or [[Sampling (signal processing)|sample rate]]) of the original samples. The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are [[bandlimiting|band-limited]] to a given bandwidth, such that no actual information is lost in the sampling process. It expresses the sufficient sample rate in terms of the bandwidth for the class of functions. The theorem also leads to a formula for perfectly reconstructing the original continuous-time function from the samples.
Perfect reconstruction may still be possible when the sample-rate criterion is not satisfied, provided other constraints on the signal are known (see {{
The name ''Nyquist–Shannon sampling theorem'' honours [[Harry Nyquist]] and [[Claude Shannon]], but the theorem was also previously discovered by [[E. T. Whittaker]] (published in 1915), and Shannon cited Whittaker's paper in his work. The theorem is thus also known by the names ''Whittaker–Shannon sampling theorem'', ''Whittaker–Shannon'', and ''Whittaker–Nyquist–Shannon'', and may also be referred to as the ''cardinal theorem of interpolation''.
Line 26:
The symbol <math>T \triangleq 1/f_s</math> is customarily used to represent the interval between samples and is called the ''sample period'' or ''sampling interval''. The samples of function <math>x(t)</math> are commonly denoted by <math>x[n] \triangleq T\cdot x(nT)</math><ref>
{{cite book |last1=Ahmed |first1=N. |url=https://books.google.com/books?id=F-nvCAAAQBAJ |title=Orthogonal Transforms for Digital Signal Processing |last2=Rao |first2=K.R. |date=July 10, 1975 |publisher=Springer-Verlag |isbn=9783540065562 |edition=1 |___location=Berlin Heidelberg New York |language=English |doi=10.1007/978-3-642-45450-9}}</ref> (alternatively <math>x_n</math> in older signal processing literature), for all integer values of <math>n.</math> The <math>T</math> multiplier is a result of the transition from continuous time to discrete time (see [[Discrete-
A mathematically ideal way to interpolate the sequence involves the use of [[sinc function]]s. Each sample in the sequence is replaced by a sinc function, centered on the time axis at the original ___location of the sample <math>nT,</math> with the amplitude of the sinc function scaled to the sample value, <math>x(nT).</math> Subsequently, the sinc functions are summed into a continuous function. A mathematically equivalent method uses the [[Dirac comb#Sampling and aliasing|Dirac comb]] and proceeds by [[Convolution|convolving]] one sinc function with a series of [[Dirac delta]] pulses, weighted by the sample values. Neither method is numerically practical. Instead, some type of approximation of the sinc functions, finite in length, is used. The imperfections attributable to the approximation are known as ''interpolation error''.
Line 40:
:<math>X(f)\ \triangleq\ \int_{-\infty}^{\infty} x(t) \ e^{- i 2 \pi f t} \ {\rm d}t,</math>
Then the samples, <math>x[n],</math> of <math>x(t)</math> are sufficient to create a [[periodic summation]] of <math>X(f).</math> (see [[Discrete-
{{Equation box 1|title=
Line 115:
Shannon's proof of the theorem is complete at that point, but he goes on to discuss reconstruction via [[sinc function]]s, what we now call the [[Whittaker–Shannon interpolation formula]] as discussed above. He does not derive or prove the properties of the sinc function, as the Fourier pair relationship between the [[rectangular function|rect]] (the rectangular function) and sinc functions was well known by that time.<ref>{{cite book |last1=Campbell |first1=George |last2=Foster |first2=Ronald |title=Fourier Integrals for Practical Applications |date=1942 |publisher=Bell Telephone System Laboratories |___location=New York}}</ref>
{{
Let <math>x_n</math> be the <math>n^{th}</math> sample. Then the function <math>x(t)</math> is represented by:
Line 145:
[[File:CriticalFrequencyAliasing.svg|thumb|right|A family of sinusoids at the critical frequency, all having the same sample sequences of alternating +1 and –1. That is, they all are aliases of each other, even though their frequency is not above half the sample rate.]]
==Critical frequency==
To illustrate the necessity of <math>f_s>2B,</math> consider the family of sinusoids generated by different values of <math>\theta</math> in this formula:
Line 159 ⟶ 160:
As discussed by Shannon:<ref name="Shannon49"/>
{{
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.
Line 193 ⟶ 194:
==Historical background==
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.
The sampling theorem, essentially a [[duality (mathematics)|dual]] of Nyquist's result, was proved by [[Claude E. Shannon]].<ref name="Shannon49"/> The mathematician [[E. T. Whittaker]] published similar results in 1915,<ref>{{cite journal |last=Whittaker |first=E. T. |author-link=E. T. Whittaker |title=On the Functions Which are Represented by the Expansions of the Interpolation Theory |journal=Proceedings of the Royal Society of Edinburgh |volume=35 |pages=181–194 |date=1915 |doi=10.1017/s0370164600017806|url=https://zenodo.org/record/1428702 }} ({{lang|de|"Theorie der Kardinalfunktionen"}}).</ref> J. M. Whittaker in 1935,<ref>{{cite book | last=Whittaker |first=J. M. | author-link =J. M. Whittaker | title =Interpolatory Function Theory | url=https://archive.org/details/in.ernet.dli.2015.219870 | publisher =Cambridge University Press | date =1935 | ___location =Cambridge, England}}</ref> and [[Dennis Gabor|Gabor]] in 1946 ("Theory of communication").
Line 202 ⟶ 203:
where <math>X_n = f\left(\frac n {2W} \right).</math>
It was not until these articles were published that the theorem known as "Shannon's sampling theorem" became common property among communication engineers, although Shannon himself writes that this is a fact which is common knowledge in the communication art.{{efn-ua|group=bottom|[[#refShannon49|Shannon 1949]], p. 448.}} A few lines further on, however, he adds: "but in spite of its evident importance, [it] seems not to have appeared explicitly in the literature of [[communication theory]]". Despite his sampling theorem being published at the end of the 1940s, Shannon had derived his sampling theorem as early as 1940.<ref>{{Cite conference |last1=Stanković |first1=Raromir S. |last2=Astola |first2=Jaakko T. |last3=Karpovsky |first3=Mark G. |date=September 2006 |title=Some Historic Remarks On Sampling Theorem |url=https://sites.bu.edu/mark/files/2018/02/196.pdf |conference=Proceedings of the 2006 International TICSP Workshop on Spectral Methods and Multirate Signal Processing}}</ref>
===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 H. Raabe, an assistant to Küpfmüller, proved the theorem in his 1939 Ph.D. dissertation; the term ''Raabe condition'' came to be associated with the criterion for unambiguous representation (sampling rate greater than twice the bandwidth). Meijering<ref name="EM">{{cite journal | last =Meijering | first =Erik | title =A Chronology of Interpolation From Ancient Astronomy to Modern Signal and Image Processing | journal =Proceedings of the IEEE | volume =90 | issue =3 | pages =319–342 | date =March 2002 | doi =10.1109/5.993400 | url =http://bigwww.epfl.ch/publications/meijering0201.pdf }}</ref> mentions several other discoverers and names in a paragraph and pair of footnotes:
{{
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 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 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".}}
Line 229 ⟶ 230:
| year = 1958}} See glossary, pp. 269–279. Cardinal theorem is on p. 270 and sampling theorem is on p. 277.</ref> even though that article does not treat sampling and reconstruction of continuous signals as others did. Their glossary of terms includes these entries:
{{
{{glossary}}
{{term|Sampling theorem (of information theory)}}
Line 300 ⟶ 301:
[[Category:Claude Shannon]]
[[Category:Telecommunication theory]]
[[Category:Data compression]]
| |||