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{{short description|Special mathematical function defined as sin(x)/x}}

In [[Mathematics|mathematics]], [[Physics:Physics|physics]] and [[Engineering:Engineering|engineering]], the '''sinc function''', 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 \\[2px] 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>\left [ -0.217234\ldots, 1 \right]</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{\left( -1 \right)^{k} {x}^{2k}}{\left( 2 k + 1 \right)!}</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|digital signal processing]] and [[Information theory|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|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|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|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}}</ref> The term ''sinc'' {{IPAc-en|ˈ|s|ɪ|ŋ|k}} was introduced by Philip M. 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 (Rayleigh's Formula) for the zeroth-order spherical [[Bessel function]] of the first kind.

== Properties ==

[[File:Si cos.svg|thumb|350px|right|The local maxima and minima (small white dots) of the unnormalized, red sinc function correspond to its intersections with the blue cosine function.]]
The [[Zero crossing|zero crossing]]s of the unnormalized sinc are at non-zero integer multiples of {{pi}}, while zero crossings of the normalized sinc occur at non-zero integers.

The local maxima and minima of the unnormalized sinc correspond to its intersections with the [[Cosine|cosine]] function. That is, {{math|1={{sfrac|sin(''ξ'')|''ξ''}} = cos(''ξ'')}} for all points {{mvar|ξ}} where the derivative of {{math|{{sfrac|sin(''x'')|''x''}}}} is zero and thus a local extremum is reached. This follows from the derivative of the sinc function:
<math display="block">\frac{d}{dx}\operatorname{sinc}(x) = \frac{\cos(x) - \operatorname{sinc}(x)}{x}.</math>

The first few terms of the infinite series for the {{mvar|x}} coordinate of the {{mvar|n}}-th extremum with positive {{mvar|x}} coordinate are:
<math display="block">x_n = q - q^{-1} - \frac{2}{3} q^{-3} - \frac{13}{15} q^{-5} - \frac{146}{105} q^{-7} - \cdots,</math>
where:
<math display="block">q = \left(n + \frac{1}{2}\right) \pi,</math>
and where odd {{mvar|n}} lead to a local minimum, and even {{mvar|n}} to a local maximum. Because of symmetry around the {{mvar|y}} axis, there exist extrema with {{mvar|x}} coordinates {{math|−''x<sub>n</sub>''}}. In addition, there is an absolute maximum at {{math|1=''ξ''<sub>0</sub> = (0, 1)}}.

The normalized sinc function has a simple representation as the [[Infinite product|infinite product]]:
<math display="block">\frac{\sin(\pi x)}{\pi x} = \prod_{n=1}^\infty \left(1 - \frac{x^2}{n^2}\right)</math>
[[File:The cardinal sine function sinc(z) plotted in the complex plane from -2-2i to 2+2i.svg|alt=The cardinal sine function sinc(z) plotted in the complex plane from -2-2i to 2+2i|thumb|The cardinal sine function sinc(z) plotted in the complex plane from -2-2i to 2+2i]]
and is related to the [[Gamma function|gamma function]] {{math|Γ(''x'')}} through Euler's reflection formula:
<math display="block">\frac{\sin(\pi x)}{\pi x} = \frac{1}{\Gamma(1 + x)\Gamma(1 - x)}.</math>

Euler discovered<ref>{{cite arXiv |last=Euler |first=Leonhard |title=On the sums of series of reciprocals |year=1735 |eprint=math/0506415}}</ref> that:
<math display="block">\frac{\sin(x)}{x} = \prod_{n=1}^\infty \cos\left(\frac{x}{2^n}\right),</math>
and because of the product-to-sum identity:<ref>{{cite journal |author1=Luis Ortiz-Gracia |author2=Cornelis W. Oosterlee |title=A highly efficient Shannon wavelet inverse Fourier technique for pricing European options |year=2016 |journal=SIAM J. Sci. Comput. |volume=38 |issue=1 |pages=B118–B143 |doi=10.1137/15M1014164 |url=https://ir.cwi.nl/pub/24377}}</ref>
[[File:Sinc cplot.svg|thumb|[[Domain coloring|Domain coloring]] plot of {{math|1=sinc ''z'' = {{sfrac|sin ''z''|''z''}}}}]]
<math display="block">\prod_{n=1}^k \cos\left(\frac{x}{2^n}\right) = \frac{1}{2^{k-1}} \sum_{n=1}^{2^{k-1}} \cos\left(\frac{n - 1/2}{2^{k-1}} x \right),\quad \forall k \ge 1,</math>
Euler's product can be recast as a sum:
<math display="block">\frac{\sin(x)}{x} = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N \cos\left(\frac{n - 1/2}{N} x\right).</math>

The continuous Fourier transform of the normalized sinc (to ordinary frequency) is {{math|[[Rectangular function|rect]](''f'')}}:
<math display="block">\int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f),</math>
where the [[Rectangular function|rectangular function]] is 1 for argument between −{{sfrac|1|2}} and {{sfrac|1|2}}, and zero otherwise. This corresponds to the fact that the [[Sinc filter|sinc filter]] is the ideal (brick-wall, meaning rectangular frequency response) [[Physics:Low-pass filter|low-pass filter]].

This Fourier integral, including the special case:
<math display="block">\int_{-\infty}^\infty \frac{\sin(\pi x)}{\pi x} \, dx = \operatorname{rect}(0) = 1</math>
is an [[Improper integral|improper integral]] (see [[Dirichlet integral]]) and not a convergent Lebesgue integral, as:
<math display="block">\int_{-\infty}^\infty \left|\frac{\sin(\pi x)}{\pi x} \right| \,dx = +\infty.</math>

The normalized sinc function has properties that make it ideal in relationship to [[Interpolation|interpolation]] of [[Sampling (signal processing)|sampled]] bandlimited functions:
* It is an interpolating function, i.e., {{math|1=sinc(0) = 1}}, and {{math|1=sinc(''k'') = 0}} for nonzero [[Number#Integers|integer]] {{math|''k''}}.
* The functions {{math|1=''x<sub>k</sub>''(''t'') = sinc(''t'' − ''k'')}} ({{mvar|k}} integer) form an [[Orthonormal basis|orthonormal basis]] for bandlimited functions in the [[Lp space|function space]] {{math|'''''L'''''<sup>2</sup>('''R''')}}, with highest angular frequency {{math|1=''ω''<sub>H</sub> = π}} (that is, highest cycle frequency {{math|1=''f''<sub>H</sub> = {{sfrac|1|2}}}}).

Other properties of the two sinc functions include:
* The unnormalized sinc is the zeroth-order spherical [[Bessel function]] of the first kind, {{math|''j''<sub>0</sub>(''x'')}}. The normalized sinc is {{math|''j''<sub>0</sub>(π''x'')}}.
* where {{math|Si(''x'')}} is the sine integral:<math display="block">\int_0^x \frac{\sin(\theta)}{\theta}\,d\theta = \operatorname{Si}(x).</math>
* {{math|''λ'' sinc(''λx'')}} (not normalized) is one of two linearly independent solutions to the linear [[Ordinary differential equation|ordinary differential equation]]:<math display="block">x \frac{d^2 y}{d x^2} + 2 \frac{d y}{d x} + \lambda^2 x y = 0.</math> The other is {{math|{{sfrac|cos(''λx'')|''x''}}}}, which is not bounded at {{math|1=''x'' = 0}}, unlike its sinc function counterpart.
* Using normalized sinc:<math display="block">\int_{-\infty}^\infty \frac{\sin^2(\theta)}{\theta^2}\,d\theta = \pi \quad \Rightarrow \quad \int_{-\infty}^\infty \operatorname{sinc}^2(x)\,dx = 1,</math>
* <math>\int_{-\infty}^\infty \frac{\sin(\theta)}{\theta}\,d\theta = \int_{-\infty}^\infty \left( \frac{\sin(\theta)}{\theta} \right)^2 \,d\theta = \pi.</math>
* <math>\int_{-\infty}^\infty \frac{\sin^3(\theta)}{\theta^3}\,d\theta = \frac{3\pi}{4}.</math>
* <math>\int_{-\infty}^\infty \frac{\sin^4(\theta)}{\theta^4}\,d\theta = \frac{2\pi}{3}.</math>
* The following improper integral involves the (not normalized) sinc function: <math display="block">\int_0^\infty \frac{dx}{x^n + 1} = 1 + 2\sum_{k=1}^\infty \frac{(-1)^{k+1}}{(kn)^2 - 1} = \frac{1}{\operatorname{sinc}(\frac{\pi}{n})}.</math>

== Relationship to the Dirac delta distribution ==

The normalized sinc function can be used as a ''[[Dirac delta function#Representations of the delta function|nascent delta function]]'', meaning that the following [[Weak convergence (Hilbert space)|weak limit]] holds:

<math display="block">\lim_{a \to 0} \frac{\sin\left(\frac{\pi x}{a}\right)}{\pi x} = \lim_{a \to 0}\frac{1}{a} \operatorname{sinc}\left(\frac{x}{a}\right) = \delta(x).</math>

This is not an ordinary limit, since the left side does not converge. Rather, it means that:

<math display="block">\lim_{a \to 0}\int_{-\infty}^\infty \frac{1}{a} \operatorname{sinc}\left(\frac{x}{a}\right) \varphi(x) \,dx = \varphi(0)</math>

for every [[Schwartz space|Schwartz function]], as can be seen from the [[Fourier inversion theorem]].
In the above expression, as {{math|''a'' → 0}}, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of {{math|±{{sfrac|1|π''x''}}}}, regardless of the value of {{mvar|a}}.

This complicates the informal picture of {{math|''δ''(''x'')}} as being zero for all {{mvar|x}} except at the point {{math|1=''x'' = 0}}, and illustrates the problem of thinking of the delta function as a function rather than as a distribution. A similar situation is found in the [[Gibbs phenomenon]].

== Summation ==
All sums in this section refer to the unnormalized sinc function.

The sum of {{math|sinc(''n'')}} over integer {{mvar|n}} from 1 to {{math|∞}} equals {{math|{{sfrac|{{pi}} − 1|2}}}}:

<math display="block">\sum_{n=1}^\infty \operatorname{sinc}(n) = \operatorname{sinc}(1) + \operatorname{sinc}(2) +  \operatorname{sinc}(3) + \operatorname{sinc}(4) +\cdots = \frac{\pi - 1}{2}.</math>

The sum of the squares also equals {{math|{{sfrac|{{pi}} − 1|2}}}}:<ref>{{cite journal | title = Advanced Problem 6241 | journal = American Mathematical Monthly | date = June–July 1980 | volume = 87 | issue = 6 | pages = 496–498 | publisher = Mathematical Association of America | location = Washington, DC | doi = 10.1080/00029890.1980.11995075}}</ref><ref name="BBB">{{cite journal | author1=Robert Baillie |  author2=David Borwein | author3=Jonathan M. Borwein |  title=Surprising Sinc Sums and Integrals | journal=American Mathematical Monthly | date=December 2008 | volume=115 | issue=10 | pages=888–901 | jstor = 27642636 | doi=10.1080/00029890.2008.11920606 | hdl=1959.13/940062 | s2cid=496934 | hdl-access=free}}</ref>

<math display="block">\sum_{n=1}^\infty \operatorname{sinc}^2(n) = \operatorname{sinc}^2(1) + \operatorname{sinc}^2(2) + \operatorname{sinc}^2(3) + \operatorname{sinc}^2(4) + \cdots = \frac{\pi - 1}{2}.</math>

When the signs of the addends alternate and begin with +, the sum equals {{sfrac|1|2}}:
<math display="block">\sum_{n=1}^\infty (-1)^{n+1}\,\operatorname{sinc}(n) = \operatorname{sinc}(1) - \operatorname{sinc}(2) + \operatorname{sinc}(3) - \operatorname{sinc}(4) + \cdots = \frac{1}{2}.</math>

The alternating sums of the squares and cubes also equal {{sfrac|1|2}}:<ref name="FWFS">{{cite arXiv |last=Baillie |first=Robert |eprint=0806.0150v2 |class=math.CA |title=Fun with Fourier series |date=2008}}</ref>
<math display="block">\sum_{n=1}^\infty (-1)^{n+1}\,\operatorname{sinc}^2(n) = \operatorname{sinc}^2(1) - \operatorname{sinc}^2(2) + \operatorname{sinc}^2(3) - \operatorname{sinc}^2(4) + \cdots = \frac{1}{2},</math>

<math display="block">\sum_{n=1}^\infty (-1)^{n+1}\,\operatorname{sinc}^3(n) = \operatorname{sinc}^3(1) - \operatorname{sinc}^3(2) + \operatorname{sinc}^3(3) - \operatorname{sinc}^3(4) + \cdots = \frac{1}{2}.</math>

== Series expansion ==
The [[Taylor series]] of the unnormalized {{math|sinc}} function can be obtained from that of the sine (which also yields its value of 1 at {{math|1=''x'' = 0}}):
<math display="block">\frac{\sin x}{x} = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n+1)!} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots</math>

The series converges for all {{mvar|x}}. The normalized version follows easily:
<math display="block">\frac{\sin \pi x}{\pi x} = 1 - \frac{\pi^2x^2}{3!} + \frac{\pi^4x^4}{5!} - \frac{\pi^6x^6}{7!} + \cdots</math>

[[Biography:Leonhard Euler|Euler]] famously compared this series to the expansion of the infinite product form to solve the [[Basel problem]].

== Higher dimensions ==
The product of 1-D sinc functions readily provides a [[Multivariable calculus|multivariate]] sinc function for the square Cartesian grid ([[Lattice graph|lattice]]): {{math|sinc<sub>C</sub>(''x'', ''y'') {{=}} sinc(''x'') sinc(''y'')}}, whose [[Fourier transform]] is the [[Indicator function|indicator function]] of a square in the frequency space (i.e., the brick wall defined in 2-D space). The sinc function for a non-Cartesian [[Lattice (group)|lattice]] (e.g., [[Hexagonal lattice|hexagonal lattice]]) is a function whose [[Fourier transform]] is the [[Indicator function|indicator function]] of the [[Physics:Brillouin zone|Brillouin zone]] of that lattice. For example, the sinc function for the hexagonal lattice is a function whose [[Fourier transform]] is the [[Indicator function|indicator function]] of the unit hexagon in the frequency space. For a non-Cartesian lattice this function can not be obtained by a simple tensor product. However, the explicit formula for the sinc function for the [[Hexagonal lattice|hexagonal]], body-centered cubic, face-centered cubic and other higher-dimensional lattices can be explicitly derived<ref name="multiD">{{cite journal |last1=Ye |first1= W. |last2=Entezari |first2= A. |title=A Geometric Construction of Multivariate Sinc Functions |journal=IEEE Transactions on Image Processing |volume=21 |issue=6 |pages=2969–2979 |date=June 2012 |doi=10.1109/TIP.2011.2162421 |pmid=21775264 |bibcode=2012ITIP...21.2969Y|s2cid= 15313688 }}</ref> using the geometric properties of Brillouin zones and their connection to [[Zonohedron|zonotopes]].

For example, a [[Hexagonal lattice|hexagonal lattice]] can be generated by the (integer) [[Linear span|linear span]] of the vectors:
<math display="block">
  \mathbf{u}_1 = \begin{bmatrix} \frac{1}{2} \\  \frac{\sqrt{3}}{2} \end{bmatrix} \quad \text{and} \quad
  \mathbf{u}_2 = \begin{bmatrix} \frac{1}{2} \\ -\frac{\sqrt{3}}{2} \end{bmatrix}.
</math>

Denoting:
<math display="block">
  \boldsymbol{\xi}_1 =  \tfrac{2}{3} \mathbf{u}_1, \quad
  \boldsymbol{\xi}_2 =  \tfrac{2}{3} \mathbf{u}_2, \quad
  \boldsymbol{\xi}_3 = -\tfrac{2}{3} (\mathbf{u}_1 + \mathbf{u}_2), \quad
          \mathbf{x} = \begin{bmatrix} x \\ y\end{bmatrix},
</math>
one can derive<ref name="multiD" /> the sinc function for this hexagonal lattice as:
<math display="block">\begin{align}
  \operatorname{sinc}_\text{H}(\mathbf{x}) = \tfrac{1}{3} \big(
    &      \cos\left(\pi\boldsymbol{\xi}_1\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_2\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_3\cdot\mathbf{x}\right) \\
    & {} + \cos\left(\pi\boldsymbol{\xi}_2\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_3\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_1\cdot\mathbf{x}\right) \\
    & {} + \cos\left(\pi\boldsymbol{\xi}_3\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_1\cdot\mathbf{x}\right) \operatorname{sinc}\left(\boldsymbol{\xi}_2\cdot\mathbf{x}\right)
  \big).
\end{align}</math>

This construction can be used to design Lanczos window for general multidimensional lattices.<ref name="multiD" />

==See also==

* {{annotated link|Anti-aliasing filter}}
* {{annotated link|Borwein integral}}
* {{annotated link|Dirichlet integral}}
* {{annotated link|Lanczos resampling}}
* {{annotated link|List of mathematical functions}}
* {{annotated link|Shannon wavelet}}
* {{annotated link|Sinc filter}}
* {{annotated link|Trigonometric functions of matrices}}
* {{annotated link|Trigonometric integral}}
* {{annotated link|Whittaker–Shannon interpolation formula}}
* {{annotated link|Winkel tripel projection}} (cartography)
* [[Sinhc function]]

== References ==
{{Reflist|30em}}

== External links ==
* {{MathWorld|title=Sinc Function|urlname=SincFunction}}

[[Category:Signal processing]]
[[Category:Elementary special functions]]

{{Sourceattribution|Sinc function}}