Sinc function 1

Sinc functionIn mathematics, the sinc function, denoted by sinc(x) and sometimes as Sa(x), has two nearly equivalent definitions.In digital signal processing and information theory, the normalized sinc function is commonly defined by

It is qualified as normalized because its integral over all x is 1. The Fourier transform of the normalized sinc functionis the rectangular function with no scaling. This function is fundamental in the concept of reconstructing the originalcontinuous bandlimited signal from uniformly spaced samples of that signal.In mathematics, the historical unnormalized sinc function is defined by

The only difference between the two definitions is in the scaling of the independent variable (the x-axis) by a factorof π. 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 analytic everywhere.The term "sinc" (English pronunciation: /ˈsɪŋk/) is a contraction of the function's full Latin name, the sinus cardinalis(cardinal sine).

PropertiesThe zero crossings of the unnormalized sinc are at nonzero multiples of π; zero crossings of the normalized sincoccur at nonzero integer values.The local maxima and minima of the unnormalized sinc correspond to its intersections with the cosine function. Thatis, 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).The normalized sinc function has a simple representation as the infinite product

and is related to the gamma function by Euler's reflection formula:

Euler discovered that

The continuous Fourier transform of the normalized sinc (to ordinary frequency) is rect(f),

where the rectangular function is 1 for argument between −1/2 and 1/2, and zero otherwise. This corresponds to thefact that the sinc filter is the ideal (brick-wall, meaning rectangular frequency response) low-pass filter. This Fourierintegral, including the special case

is an improper integral and not a convergent Lebesgue integral, as

Sinc function 2

The normalized sinc function has properties that make it ideal in relationship to interpolation of sampled bandlimitedfunctions:• It is an interpolating function, i.e., sinc(0) = 1, and sinc(k) = 0 for nonzero integer k.• The functions xk(t) = sinc(t−k) form an orthonormal basis for bandlimited functions in the function space L2(R),

with highest angular frequency ωH = π (that is, highest cycle frequency ƒH = 1/2).Other properties of the two sinc functions include:• The unnormalized sinc is the zeroth order spherical Bessel function of the first kind, . The normalized sinc is

j0(πx).

•

where Si(x) is the sine integral.• λ sinc(λ x) (not normalized) is one of two linearly independent solutions to the linear ordinary differential

equation

The other is cos(λ x)/x, which is not bounded at x = 0, unlike its sinc function counterpart.

•

where the normalized sinc is meant.

•

•

Relationship to the Dirac delta distributionThe normalized sinc function can be used as a nascent delta function, meaning that the following weak limit holds:

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

for any smooth function with compact support.In the above expression, as a  approaches zero, the number of oscillations per unit length of the sinc functionapproaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±1/(π a x), and approacheszero for any nonzero value of x. This complicates the informal picture of δ(x) as being zero for all x except at thepoint x = 0 and illustrates the problem of thinking of the delta function as a function rather than as a distribution. Asimilar situation is found in the Gibbs phenomenon.

Sinc function 3

ReferencesOlver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., eds. (2010), "Numerical methods" [1], NISTHandbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255

External links• Weisstein, Eric W., "Sinc Function [2]" from MathWorld.

References[1] http:/ / dlmf. nist. gov/ 3. 3[2] http:/ / mathworld. wolfram. com/ SincFunction. html

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