Editing Sinc function (section)

{{short description|Special mathematical function defined as sin(x)/x}}
{{Redirect|Sinc|the designation used in the United Kingdom for areas of wildlife interest|Site of Importance for Nature Conservation|the signal processing filter based on this function|Sinc filter}}
{{Use American English|date = March 2019}}

In [[mathematics]], [[physics]] and [[engineering]], the '''sinc function''' ({{IPAc-en|ˈ|s|ɪ|ŋ|k}} {{respell|SINK}}), denoted by {{math|sinc(''x'')}}, has two forms, normalized and unnormalized.<ref name="dlmf">{{dlmf|title=Numerical methods|id=3.3}}.</ref>

{{Infobox mathematical function
| name = Sinc
| image = Si sinc.svg
| imagesize = 350px
| imagealt = Part of the normalized and unnormalized sinc function shown on the same scale
| caption = Part of the normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale
| general_definition = <math>\operatorname{sinc}x = \begin{cases} \dfrac{ \sin x } x, & x \ne 0 \\ 1, & x = 0\end{cases}</math>
| motivation_of_creation = Telecommunication
| date = 1952
| fields_of_application = Signal processing, spectroscopy
| domain = <math>\mathbb{R}</math>
| range = <math>[-0.217234\ldots, 1]</math>
| parity = Even
| zero = 1
| plusinf = 0
| minusinf = 0
| max = 1 at <math>x = 0</math>
| min = <math>-0.21723\ldots</math> at <math>x = \pm 4.49341\ldots</math>
| root = <math>\pi k, k \in \mathbb{Z}_{\neq 0}</math>
| reciprocal = <math>\begin{cases} x \csc x, & x \ne 0 \\ 1, & x = 0 \end{cases}</math>
| derivative = <math>\operatorname{sinc}'x = \begin{cases} \dfrac{\cos x - \operatorname{sinc} x}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases}</math>
| antiderivative = <math>\int \operatorname{sinc} x\,dx = \operatorname{Si}(x) + C</math>
| taylor_series = <math>\operatorname{sinc}x = \sum_{k=0}^\infty \frac{(-1)^k x^{2k}}{(2k + 1)!}</math>
}}

[[File:Sinc.wav|thumb|The sinc function as audio, at 2000 Hz (±1.5 seconds around zero)]]

In mathematics, the historical '''unnormalized sinc function''' is defined for {{math|''x'' ≠ 0}} by
<math display="block">\operatorname{sinc}(x) = \frac{\sin x}{x}.</math>

Alternatively, the unnormalized sinc function is often called the [[sampling function]], indicated as Sa(''x'').<ref>{{cite book |title=Communication Systems, 2E |edition=illustrated |first1=R. P. |last1=Singh |first2=S. D. |last2=Sapre |publisher=Tata McGraw-Hill Education |year=2008 |isbn=978-0-07-063454-1 |page=15 |url=https://books.google.com/books?id=WkOPPEhK7SYC}} [https://books.google.com/books?id=WkOPPEhK7SYC&pg=PA15 Extract of page 15]</ref>

In [[digital signal processing]] and [[information theory]], the '''normalized sinc function''' is commonly defined for {{math|''x'' ≠ 0}} by
<math display="block">\operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x}.</math>

In either case, the value at {{math|1=''x'' = 0}} is defined to be the limiting value
<math display="block">\operatorname{sinc}(0) := \lim_{x \to 0}\frac{\sin(a x)}{a x} = 1</math> for all real {{math|''a'' ≠ 0}} (the limit can be proven using the [[Squeeze theorem#Second example|squeeze theorem]]).

The [[Normalizing constant|normalization]] causes the [[integral|definite integral]] of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of [[pi|{{pi}}]]). As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of {{mvar|x}}.

The normalized sinc function is the [[Fourier transform]] of the [[rectangular function]] with no scaling. It is used in the concept of [[Whittaker–Shannon interpolation formula|reconstructing]] a continuous bandlimited signal from uniformly spaced [[Nyquist–Shannon sampling theorem|samples]] of that signal.

The only difference between the two definitions is in the scaling of the [[independent variable]] (the [[Cartesian coordinate system|{{mvar|x}} axis]]) by a factor of {{pi}}. In both cases, the value of the function at the [[removable singularity]] at zero is understood to be the limit value 1. The sinc function is then [[Analytic function|analytic]] everywhere and hence an [[entire function]].

The function has also been called the '''cardinal sine''' or '''sine cardinal''' function.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Sinc Function |url=https://mathworld.wolfram.com/ |access-date=2023-06-07 |website=mathworld.wolfram.com |language=en}}</ref><ref>{{Cite journal |last=Merca |first=Mircea |date=2016-03-01 |title=The cardinal sine function and the Chebyshev–Stirling numbers |url=https://www.sciencedirect.com/science/article/pii/S0022314X15002863 |journal=Journal of Number Theory |language=en |volume=160 |pages=19–31 |doi=10.1016/j.jnt.2015.08.018 |s2cid=124388262 |issn=0022-314X|url-access=subscription }}</ref> The term ''sinc'' was introduced by [[Philip Woodward|Philip M.&nbsp;Woodward]] in his 1952 article "Information theory and inverse probability in telecommunication", in which he said that the function "occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own",<ref>{{cite journal |last1=Woodward |first1=P. M. |last2=Davies |first2=I. L. |url=http://www.norbertwiener.umd.edu/crowds/documents/Woodward52.pdf |title=Information theory and inverse probability in telecommunication |journal=Proceedings of the IEE - Part III: Radio and Communication Engineering |volume=99 |issue=58 |pages=37–44 |date= March 1952 |doi=10.1049/pi-3.1952.0011}}</ref> and his 1953 book ''Probability and Information Theory, with Applications to Radar''.<ref name="Poynton">{{Cite book |first=Charles A. |last=Poynton |title=Digital video and HDTV |url=https://archive.org/details/digitalvideohdtv00poyn_079 |url-access=limited |page=[https://archive.org/details/digitalvideohdtv00poyn_079/page/n152 147] |publisher=Morgan Kaufmann Publishers |year=2003 |isbn=978-1-55860-792-7}}</ref><ref>{{cite book |first=Phillip M. |last=Woodward |title=Probability and information theory, with applications to radar |url=https://archive.org/details/probabilityinfor00wood |url-access=limited |page=[https://archive.org/details/probabilityinfor00wood/page/n40 29] |location=London |publisher=Pergamon Press |year=1953 |oclc=488749777 |isbn=978-0-89006-103-9}}</ref>
The function itself was first mathematically derived in this form by [[Lord Rayleigh]] in his expression ([[Bessel functions#Rayleigh's formulas|Rayleigh's formula]]) for the zeroth-order spherical [[Bessel function]] of the first kind.