sinc function

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sinc function
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{{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}}{{Short description|Special mathematical function defined as sin(x)/x}}In mathematics, physics and engineering, the cardinal sine function or sinc function, denoted by {{math|sinc(x)}}, has two slightly different definitions.{{dlmf|title=Numerical methods|id=3.3}}In mathematics, the historical unnormalized sinc function is defined for {{math|x ≠ 0}} by
operatorname{sinc}(x) = frac{sin(x)}{x}~.
In digital signal processing and information theory, the normalized sinc function is commonly defined for {{math|x ≠ 0}} by(File:Si sinc.svg|thumb|350px|right|The normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale.)
operatorname{sinc}(x) = frac{sin(pi x)}{pi x}~.
In either case, the value at {{math|x {{=}} 0}} is defined to be the limiting value
operatorname{sinc}(0):=lim_{xto 0}frac{sin(a x)}{a x}= 1 for all real {{math|a ≠ 0}}.
The normalization causes the 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}}). 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 reconstructing a continuous bandlimited signal from uniformly spaced samples of that signal.The only difference between the two definitions is in the scaling of the independent variable (the {{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 everywhere and hence an entire function.The term sinc {{IPAc-en|ˈ|s|ɪ|ŋ|k}} is a contraction of the function's full Latin name, the (cardinal sine). It was introduced by Philip 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",JOURNAL, Woodward, P. M., Davies, I. L.,weblink Information theory and inverse probability in telecommunication, Proceedings of the IEE – Part III: Radio and Communication Engineering, 99, 58, 37–44, March 1952, 10.1049/pi-3.1952.0011, and his 1953 book Probability and Information Theory, with Applications to Radar.BOOK, Charles A., Poynton, Digital video and HDTV, 147, Morgan Kaufmann Publishers, 2003, 978-1-55860-792-7, BOOK, Phillip M., Woodward, Probability and information theory, with applications to radar, 29, London, Pergamon Press, 1953, 488749777, 978-0-89006-103-9,


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 functioncosine function(File:Sinc re.svg|thumb|The real part of complex sinc {{math|Re(sinc z) {{=}} Re({{sfrac|sin z|z}})}}.)(File:Sinc im.svg|thumb|The imaginary part of complex sinc {{math|Im(sinc z) {{=}} Im({{sfrac|sin z|z}})}}.)(File:Sinc abs.svg|thumb|The absolute value {{math|{{abs|sinc z}} {{=}} {{abs|{{sfrac|sin z|z}}}}}}.)The zero crossings 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 function. That is, {{math|{{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,
frac{doperatorname{sinc}(x)}{dx} = frac{cos(x) - operatorname{sinc}(x)}{x}
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
x_n = q - q^{-1} - frac{2}{3} q^{-3} - frac{13}{15} q^{-5} - frac{146}{105} q^{-7} -cdots
q = left(n+frac{1}{2}right)pi
and where odd {{math|n}} lead to a local minimum and even {{math|n}} to a local maximum. Because of symmetry around the {{math|y}}-axis, there exist extrema with {{math|x}}-coordinates {{math|−xn}}. In addition, there is an absolute maximum at {{math|ξ0 {{=}} (0,1)}}.The normalized sinc function has a simple representation as the infinite product
frac{sin(pi x)}{pi x} = prod_{n=1}^infty left(1 - frac{x^2}{n^2}right)
and is related to the gamma function {{math|Γ(x)}} through Euler's reflection formula,
frac{sin(pi x)}{pi x} = frac{1}{Gamma(1+x)Gamma(1-x)}~.
Euler discoveredARXIV, Euler, Leonhard, On the sums of series of reciprocals, 1735, math/0506415, that
frac{sin(x)}{x} = prod_{n=1}^infty cosleft(frac{x}{2^n}right)
and because of the product-to-sum identityJOURNAL, Luis Ortiz-Gracia, Cornelis W. Oosterlee, A highly efficient Shannon wavelet inverse Fourier technique for pricing Europe, 2016, SIAM J. Sci. Comput., 38, 1, B118–B143, 10.1137/15M1014164,
prod_{n=1}^kcosleft(frac{x}{2^n}right)=frac{1}{2^{k-1}}sum_{n=1}^{2^{k-1}}cosleft(frac{n-1/2}{2^{k-1}}x right),,qquad qquad forall kge 1,,
the Euler's product can be recast as a sum
frac{sin(x)}{x} = lim_{Ntoinfty}frac{1}{N}sum_{n=1}^Ncosleft(frac{n-1/2}{N}x right),.
The continuous Fourier transform of the normalized sinc (to ordinary frequency) is {{math|rect( f )}},
int_{-infty}^infty operatorname{sinc}(t) , e^{-i 2 pi f t},dt = operatorname{rect}(f)~,
where the 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 is the ideal (brick-wall, meaning rectangular frequency response) low-pass filter.This Fourier integral, including the special case
int_{-infty}^infty frac{sin(pi x)}{pi x} , dx = operatorname{rect}(0) = 1,!
is an improper integral (cf. Dirichlet integral) and not a convergent Lebesgue integral, as
int_{-infty}^infty left|frac{sin(pi x)}{pi x} right|, dx = +infty ~.
The normalized sinc function has properties that make it ideal in relationship to interpolation of sampled bandlimited functions:
  • It is an interpolating function, i.e., {{math|sinc(0) {{=}} 1}}, and {{math|sinc(k) {{=}} 0}} for nonzero integer {{math|k}}.
  • The functions {{math|xk(t) {{=}} sinc(t − k)}} ({{math|k}} integer) form an orthonormal basis for bandlimited functions in the function space {{math|L2(R)}}, with highest angular frequency {{math|ωH {{=}} Ï€}} (that is, highest cycle frequency {{math|fH {{=}} {{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|j0(x)}}. The normalized sinc is {{math| j0(Ï€x)}}.
  • int_0^x frac{sin(theta)}{theta},dtheta = operatorname{Si}(x) ,!

where {{math|Si(x)}} is the sine integral.

x frac{d^2 y}{d x^2} + 2 frac{d y}{d x} + lambda^2 x y = 0.,!
The other is {{math|{{sfrac|cos(λx)|x}}}}, which is not bounded at {{math|x {{=}} 0}}, unlike its sinc function counterpart.
  • int_{-infty}^infty frac{sin^2(theta)}{theta^2},dtheta = pi ,! rightarrow int_{-infty}^infty operatorname{sinc}^2(x),dx = 1~,

where the normalized sinc is meant.
  • int_{-infty}^infty frac{sin(theta)}{theta},dtheta = int_{-infty}^infty left ( frac{sin(theta)}{theta} right )^2 ,dtheta = pi ,!
  • int_{-infty}^infty frac{sin^3(theta)}{theta^3},dtheta = frac{3pi}{4} ,!
  • int_{-infty}^infty frac{sin^4(theta)}{theta^4},dtheta = frac{2pi}{3} ~.
  • The following improper integral involves the (not normalized) sinc function:
  • int_{0}^{infty} frac{dx}{x^n+1} = 1+2sum_{k=1}^{infty} frac{(-1)^{k+1}}{(kn)^2-1} = frac{1}{operatorname{sinc}(frac{pi}{n})}

Relationship to the Dirac delta distribution

The normalized sinc function can be used as a nascent delta function, meaning that the following weak limit holds,
lim_{arightarrow 0}frac{sinleft(frac{pi x}{a}right)}{pi x} = lim_{arightarrow 0}frac{1}{a}operatorname{sinc}left(frac{x}{a}right)=delta(x)~.
This is not an ordinary limit, since the left side does not converge. Rather, it means that
lim_{arightarrow 0}int_{-infty}^infty frac{1}{a} operatorname{sinc}left(frac{x}{a}right)varphi(x),dx
= varphi(0)~,
for every 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|πx}}}}, 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|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.


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}}}}.
sum_{n=1}^infty operatorname{sinc}(n) = operatorname{sinc}(1) + operatorname{sinc}(2) + operatorname{sinc}(3) + operatorname{sinc}(4) +cdots = frac{pi-1}{2}
The sum of the squares also equals {{math|{{sfrac|{{pi}} − 1|2}}}}."MEMBERWIDE">AUTHOR2-LINK=DAVID BORWEINAUTHOR3-LINK=JONATHAN M. BORWEIN, Jonathan M. Borwein, Surprising Sinc Sums and Integrals, American Mathematical Monthlyvolume=115pages=888–901, 27642636, 10.1080/00029890.2008.11920606,
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}
When the signs of the addends alternate and begin with +, the sum equals {{sfrac|1|2}}.
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}
The alternating sums of the squares and cubes also equal {{sfrac|1|2}}.ARXIV, Baillie, Robert, 0806.0150v2, math.CA, Fun with Fourier series, 2008,
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}
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}

Series expansion

Unnormalized {{math|sinc(x)}}:
operatorname{sinc}(x) = frac{sin(x)}{x} = sum_{n=0}^infty frac{left( -x^2 right)^n}{(2n+1)!}

Higher dimensions

The product of 1-D sinc functions readily provides a multivariate sinc function for the square, Cartesian, grid (lattice): {{math|sincC(x, y) {{=}} sinc(x)sinc(y)}} 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 (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, body centered cubic, face centered cubic and other higher-dimensional lattices can be explicitly derivedJOURNAL, Ye, W., Entezari, A., A Geometric Construction of Multivariate Sinc Functions, IEEE Transactions on Image Processing, 21, 6, 2969–2979, June 2012, 10.1109/TIP.2011.2162421, 21775264, 2012ITIP...21.2969Y, using the geometric properties of Brillouin zones and their connection to zonotopes.For example, a hexagonal lattice can be generated by the (integer) linear span of the vectors
mathbf{u}_1 = begin{bmatrix} frac{1}{2} [4pt] frac{sqrt{3}}{2} end{bmatrix} quad text{and} quad
mathbf{u}_2 = begin{bmatrix} frac{1}{2} [4pt] -frac{sqrt{3}}{2} end{bmatrix}.
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 yend{bmatrix},
one can derive the sinc function for this hexagonal lattice as:
operatorname{sinc}_text{H}(mathbf{x}) = tfrac{1}{3} big(
& cosleft(piboldsymbol{xi}_1cdotmathbf{x}right) operatorname{sinc}left(boldsymbol{xi}_2cdotmathbf{x}right) operatorname{sinc}left(boldsymbol{xi}_3cdotmathbf{x}right)
& {} + cosleft(piboldsymbol{xi}_2cdotmathbf{x}right) operatorname{sinc}left(boldsymbol{xi}_3cdotmathbf{x}right) operatorname{sinc}left(boldsymbol{xi}_1cdotmathbf{x}right)
& {} + cosleft(piboldsymbol{xi}_3cdotmathbf{x}right) operatorname{sinc}left(boldsymbol{xi}_1cdotmathbf{x}right) operatorname{sinc}left(boldsymbol{xi}_2cdotmathbf{x}right)
end{align}This construction can be used to design Lanczos window for general multidimensional lattices.

See also



External links

  • {{MathWorld|title=Sinc Function|urlname=SincFunction}}

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