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| In [[mathematics]], [[physics]] and [[engineering]], the '''cardinal sine function''' or '''sinc function''', denoted by sinc(''x''), has two slightly different definitions.<ref name="dlmf">{{dlmf|title=Numerical methods|id=3.3}}</ref>
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| In mathematics, the historical '''unnormalized sinc function''' is defined by
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| :<math>\mathrm{sinc}(x) = \frac{\sin(x)}{x}.\,\!</math>
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| In [[digital signal processing]] and [[information theory]], the '''normalized sinc function''' is commonly defined by
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| [[File:Si sinc.svg|thumb|350px|right|The normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale.]]
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| :<math>\mathrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x}.\,\!</math>
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| In either case, the value at <math>x = 0</math> is defined to be the limiting value: <math>\mathrm{sinc}(0) = 1</math>.
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| 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|π]]). As a further useful property, all of the zeros of the normalized sinc function are integer values of <math>x</math>. The normalized sinc function is the [[Fourier transform]] of the [[rectangular function]] with no scaling. This function is fundamental in the concept of [[Whittaker–Shannon interpolation formula|reconstructing]] the original continuous bandlimited signal from uniformly spaced [[Nyquist–Shannon sampling theorem|samples]] of that signal.
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| The only difference between the two definitions is in the scaling of the [[independent variable]] (the [[x-axis]]) by a factor of [[pi|π]]. In both cases, the value of the function at the [[removable singularity]] at zero is understood to be the limit value 1.
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| The sinc function is [[Analytic function|analytic]] everywhere.
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| The term "sinc" {{IPAc-en|ˈ|s|ɪ|ŋ|k}} is a contraction of the function's full Latin name, the ''sinus cardinalis'' (cardinal sine).<ref name=Poynton/> It was introduced by Phillip M. Woodward in his 1952 paper "Information theory and inverse probability in telecommunication" in which he said 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|page=147|publisher= Morgan Kaufmann Publishers|year= 2003| isbn =1-55860-792-7}}</ref><ref>{{cite book|first=Phillip M. |last=Woodward|title=Probability and information theory, with applications to radar|page=29|location=London|publisher= Pergamon Press|year= 1953|oclc=488749777|isbn=0-89006-103-3}}</ref>
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| == Properties ==
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| [[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]].]]
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| The [[zero crossing]]s of the unnormalized sinc are at non-zero multiples of π, while zero crossings of the normalized sinc occur at non-zero integers.
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| The local maxima and minima of the unnormalized sinc correspond to its intersections with the cosine function. That is, sin(ξ)/ξ = cos(ξ) for all points ξ where the derivative of sin(''x'')/''x'' is zero and thus a local extremum is reached.
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| A good approximation of the ''x''-coordinate of the ''n''-th extremum with positive x-coordinate is
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| :<math>
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| x_n \approx (n+\tfrac12)\pi - \frac1{(n+\frac12)\pi}
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| </math> | |
| where odd ''n'' lead to a local minimum and even ''n'' to a local maximum. Besides the extrema at ''x<sub>n</sub>'', the curve has an absolute maximum at ξ<sub>0</sub> = (0,1) and because of its symmetry to the ''y''-axis extrema with ''x''-coordinates −''x<sub>n</sub>''.
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| The normalized sinc function has a simple representation as the [[infinite product]]
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| :<math>\frac{\sin(\pi x)}{\pi x} = \prod_{n=1}^\infty \left(1 - \frac{x^2}{n^2}\right)\,\!</math>
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| and is related to the [[gamma function]] <math>\Gamma(x)</math> by [[Euler's reflection formula]]:
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| :<math>\frac{\sin(\pi x)}{\pi x} = \frac{1}{\Gamma(1+x)\Gamma(1-x)}.\,\!</math>
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| [[Euler]] discovered<ref>{{cite journal|last=Euler|first=Leonhard|title=On the sums of series of reciprocals|year=1735|url=http://arxiv.org/abs/math/0506415}}</ref> that
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| :<math>\frac{\sin(x)}{x} = \prod_{n=1}^\infty \cos\left(\frac{x}{2^n}\right).</math>
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| The [[continuous Fourier transform]] of the normalized sinc (to ordinary frequency) is [[rectangular function|rect]](''f''),
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| :<math>\int_{-\infty}^\infty \mathrm{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \mathrm{rect}(f),\,\!</math>
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| where the [[rectangular function]] is 1 for argument between −1/2 and 1/2, and zero otherwise. This corresponds to the fact that the [[sinc filter]] is the ideal ([[brick-wall filter|brick-wall]], meaning rectangular frequency response) [[low-pass filter]]. This Fourier integral, including the special case
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| :<math>\int_{-\infty}^\infty \frac{\sin(\pi x)}{\pi x} \, dx = \mathrm{rect}(0) = 1\,\!</math>
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| is an [[improper integral]] and not a convergent [[Lebesgue integral]], as
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| :<math>\int_{-\infty}^\infty \left|\frac{\sin(\pi x)}{\pi x} \right|\, dx = +\infty.</math>
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| The normalized sinc function has properties that make it ideal in relationship to [[interpolation]] of [[sampling (signal processing)|sampled]] [[bandlimited]] functions:
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| * It is an interpolating function, i.e., sinc(0) = 1, and sinc(''k'') = 0 for nonzero [[Number#Integers|integer]] ''k''.
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| * The functions ''x<sub>k</sub>''(''t'') = sinc(''t−k'') (''k'' integer) form an [[orthonormal basis]] for [[bandlimited]] functions in the [[Lp space|function space]] '''''L'''''<sup>2</sup>('''R'''), with highest angular frequency ''ω<sub>''H</sub> = ''π'' (that is, highest cycle frequency ''ƒ''<sub>H</sub> = 1/2).
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| Other properties of the two sinc functions include:
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| * The unnormalized sinc is the zero<sup>th</sup> order spherical [[Bessel function]] of the first kind, <math>\scriptstyle j_0(x)</math>. The normalized sinc is <math>\scriptstyle j_0(\pi x)</math>
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| * <math> \int_0^x \frac{\sin(\theta)}{\theta}\,d\theta = \mathrm{Si}(x) \,\!</math>
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| :where Si(''x'') is the [[sine integral]].
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| * λ sinc(λ ''x'') (not normalized) is one of two linearly independent solutions to the linear [[ordinary differential equation]]
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| ::<math>x \frac{d^2 y}{d x^2} + 2 \frac{d y}{d x} + \lambda^2 x y = 0.\,\!</math>
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| :The other is cos(λ ''x'')/''x'', which is not bounded at ''x'' = 0, unlike its sinc function counterpart.
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| * <math> \int_{-\infty}^\infty \frac{\sin^2(\theta)}{\theta^2}\,d\theta = \pi \,\! \rightarrow \int_{-\infty}^\infty \mathrm{sinc}^2(x)\,dx = 1. </math>
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| :where the normalized sinc is meant.
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| * <math> \int_{-\infty}^\infty \frac{\sin^3(\theta)}{\theta^3}\,d\theta = \frac{3\pi}{4} \,\!</math>
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| * <math> \int_{-\infty}^\infty \frac{\sin^4(\theta)}{\theta^4}\,d\theta = \frac{2\pi}{3} \,\!</math>
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| == Relationship to the Dirac delta distribution ==
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| 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 topology|weak limit]] holds:
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| :<math>\lim_{a\rightarrow 0}\frac{1}{a}\textrm{sinc}(x/a)=\delta(x).</math>
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| This is not an ordinary limit, since the left side does not converge. Rather, it means that
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| :<math>\lim_{a\rightarrow 0}\int_{-\infty}^\infty \frac{1}{a}\textrm{sinc}(x/a)\varphi(x)\,dx
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| = \varphi(0),
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| </math>
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| for any [[smooth function]] <math>\scriptstyle \varphi(x)</math> with [[compact support]].
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| In the above expression, as ''a'' approaches zero, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±1/(π ''a'' ''x''), and approaches zero for any nonzero value of ''x''. This complicates the informal picture of δ(x) as being zero for all ''x'' except at the point ''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]].
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| ==Summation==
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| Summation of sinc(''x'') over ''x'' from 1 to <math>\infty</math> equals <math>\tfrac{\pi-1}{2}</math>.
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| :<math>\sum_{x=1}^\infty \mathrm{sinc}(x) = \mathrm{sinc}(1) + \mathrm{sinc}(2) + \mathrm{sinc}(3) + \mathrm{sinc}(4) +\cdots = \frac{\pi-1}{2}</math>
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| When the signs of the [[addend]]s alternate and begin with +, the sum equals <math>\tfrac{1}{2}</math>.
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| :<math>\sum_{x=1}^\infty (-1)^{x+1}\,\mathrm{sinc}(x) = \mathrm{sinc}(1) - \mathrm{sinc}(2) + \mathrm{sinc}(3) - \mathrm{sinc}(4) +\cdots = \frac{1}{2}</math>
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| == Multidimensions ==
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| The tensor-product of 1-D sinc functions readily provides a [[multivariable calculus|multivariate]] sinc function for the square, Cartesian, grid ([[Lattice graph|Lattice]]): <math> {\rm sinc}_{\rm C}(x, y) = {\rm sinc}(x) {\rm sinc}(y)</math> whose [[Fourier transform]] is the [[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]]) is a function whose [[Fourier transform]] is the [[indicator function]] of the [[Brillouin zone]] of that lattice. For example, the sinc function for the hexagonal lattice is a function whose [[Fourier transform]] is the [[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. |url=http://dx.doi.org/10.1109/TIP.2011.2162421|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}}</ref> using the geometric properties of [[Brillouin zone]]s and their connection to [[zonohedron|zonotopes]].
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| For example, a [[hexagonal lattice]] can be generated by the (integer) [[Linear span]] of the vectors <math>u_1 = \left[\begin{array}{c}1/2\\ \sqrt{3}/2\end{array}\right] </math> and <math>u_2 = \left[\begin{array}{c}1/2\\ -\sqrt{3}/2\end{array}\right]</math>. Denoting <math>\xi_1 = 2/3 u_1, \xi_2 = 2/3 u_2, \xi_3 = -2/3(u_1 + u_2)</math> and <math>\mathbf{x} = \left[\begin{array}{c}x\\ y\end{array}\right]</math>, one can derive<ref name="multiD" /> the sinc function for this hexagonal lattice as:
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| :<math>{\rm sinc}_{\rm H}(\mathbf{x}) = 1/3\left(\cos(\pi\xi_1\cdot\mathbf{x}){\rm sinc}(\xi_2\cdot\mathbf{x}){\rm sinc}(\xi_3\cdot\mathbf{x})+\cos(\pi\xi_2\cdot\mathbf{x}){\rm sinc}(\xi_3\cdot\mathbf{x}){\rm sinc}(\xi_1\cdot\mathbf{x})+\cos(\pi\xi_3\cdot\mathbf{x}){\rm sinc}(\xi_1\cdot\mathbf{x}){\rm sinc}(\xi_2\cdot\mathbf{x})\right)</math>.
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| This construction can be used to design [[Lanczos window]] for general multidimensional lattices.<ref name="multiD" />
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| ==See also==
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| *[[Borwein integral]]
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| *[[Dirichlet integral]]
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| *[[Anti-aliasing filter]]
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| *[[Sinc filter]]
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| *[[Lanczos resampling]]
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| *[[Whittaker–Shannon interpolation formula]]
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| *[[Winkel tripel projection]] (cartography)
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| ==References==
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| {{Reflist}}
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| ==External links==
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| *{{MathWorld|title=Sinc Function|urlname=SincFunction}}
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| [[Category:Signal processing]]
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| [[Category:Elementary special functions]]
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