Editing Nyquist–Shannon sampling theorem (section)

===Why Nyquist?===
Exactly how, when, or why [[Harry Nyquist]] had his name attached to the sampling theorem remains obscure. The term ''Nyquist Sampling Theorem'' (capitalized thus) appeared as early as 1959 in a book from his former employer, [[Bell Labs]],<ref>{{cite book | title = Transmission Systems for Communications | author = Members of the Technical Staff of Bell Telephone Lababoratories | year = 1959 | publisher = AT&T | page = 26-4 |volume=2}}</ref> and appeared again in 1963,<ref>{{cite book | title = Theory of Linear Physical Systems | publisher = Wiley | year = 1963 | url = https://books.google.com/books?id=jtI-AAAAIAAJ |first=Ernst Adolph |last=Guillemin| isbn = 9780471330707 }}</ref> and not capitalized in 1965.<ref>{{cite book |first1=Richard A. |last1=Roberts |first2=Ben F. |last2=Barton |title=Theory of Signal Detectability: Composite Deferred Decision Theory |year=1965 }}</ref> It had been called the ''Shannon Sampling Theorem'' as early as 1954,<ref>{{cite journal |first=Truman S. |last=Gray |title=Applied Electronics: A First Course in Electronics, Electron Tubes, and Associated Circuits |journal=Physics Today |year=1954 |volume=7 |issue=11 |page=17 |doi=10.1063/1.3061438 |bibcode=1954PhT.....7k..17G |hdl=2027/mdp.39015002049487 |hdl-access=free }}</ref> but also just ''the sampling theorem'' by several other books in the early 1950s.

In 1958, [[R. B. Blackman|Blackman]] and [[J. W. Tukey|Tukey]] cited Nyquist's 1928 article as a reference for ''the sampling theorem of information theory'',<ref>{{cite journal
 | last1 = Blackman | first1 = R. B. | author1-link = R. B. Blackman
 | last2 = Tukey | first2 = J. W. | author2-link = J. W. Tukey
 | doi = 10.1002/j.1538-7305.1958.tb03874.x
 | journal = [[The Bell System Technical Journal]]
 | mr = 102897
 | pages = 185–282
 | title = The measurement of power spectra from the point of view of communications engineering. I
 | volume = 37
 | 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:

{{blockquote|
{{glossary}}
{{term|Sampling theorem (of information theory)}}
{{defn|Nyquist's result that equi-spaced data, with two or more points per cycle of highest frequency, allows reconstruction of band-limited functions. (See ''Cardinal theorem''.)}}
{{term|Cardinal theorem (of interpolation theory)}}
{{defn|A precise statement of the conditions under which values given at a doubly infinite set of equally spaced points can be interpolated to yield a continuous band-limited function with the aid of the function <math display="block">\frac{\sin (x - x_i)}{x - x_i}.</math>}}
{{glossary end}}}}

Exactly what "Nyquist's result" they are referring to remains mysterious.

When Shannon stated and proved the sampling theorem in his 1949 article, according to Meijering,<ref name="EM" /> "he referred to the critical sampling interval <math>T = \frac 1 {2W}</math> as the ''Nyquist interval'' corresponding to the band <math>W,</math> in recognition of Nyquist's discovery of the fundamental importance of this interval in connection with telegraphy". This explains Nyquist's name on the critical interval, but not on the theorem.

Similarly, Nyquist's name was attached to ''[[Nyquist rate]]'' in 1953 by [[Harold Stephen Black|Harold S. Black]]:

{{blockquote|If the essential frequency range is limited to <math>B</math> cycles per second, <math>2B</math> was given by Nyquist as the maximum number of code elements per second that could be unambiguously resolved, assuming the peak interference is less than half a quantum step. This rate is generally referred to as '''signaling at the Nyquist rate''' and <math>\frac 1 {2B}</math> has been termed a ''Nyquist interval''.|Harold Black, ''Modulation Theory''<ref>{{cite book |first=Harold S. |last=Black |title=Modulation Theory |year=1953 }}</ref> (bold added for emphasis; italics as in the original)}}

According to the ''[[Oxford English Dictionary]]'', this may be the origin of the term ''Nyquist rate''. In Black's usage, it is not a sampling rate, but a signaling rate.