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In mathematics, the 'sinc function', denoted by $mathrm\{sinc\}(x),$, has two definitions, sometimes distinguished as the ''normalized'' sinc function and ''unnormalized'' sinc function:
# In digital signal processing and information theory, the 'normalized sinc function' is commonly defined by
#:
# In mathematics, the historical 'unnormalized sinc function' (for ''sinus cardinalis''), is defined by
#:
In both cases, the value of the function at the removable singularity at zero is sometimes specified explicitly as the limit value 1. The sinc function is analytic everywhere.
The term "sinc" is a contraction of the function's full name, the ''sine cardinal''.

Contents

Properties

Relationship to the Dirac delta distribution

See also

External links

Properties

The '''normalized''' sinc function has properties that make it ideal in relationship to interpolation and bandlimited functions:

★ $mathrm\{sinc\}(0)\; =\; 1,$ and $mathrm\{sinc\}(k)\; =\; 0,$ for $k$
e 0, and $kinmathbb\{Z\},$ (integers); that is, it is an interpolating function.

★ the functions $x\_k(t)=mathrm\{sinc\}(t-k)$ form an orthonormal basis for bandlimited functions in the function space $L^2(R)$, with highest angular frequency $omega\_mathrm\{H\}=pi,$ (that is, highest cycle frequency $f\_mathrm\{H\}=1/2,$).
Other properties of the two sinc functions include:

★ The local maxima and minima of the unnormalized sinc,     correspond to its intersections with the cosine function. That is, for all points ''x'' where the derivative of is zero (and thus a local extremum is reached).

★ The unnormalized sinc is the zero^th order spherical Bessel function of the first kind, . The normalized sinc is $j\_0(pi\; x),$.

★ The zero-crossings of the unnormalized sinc are at nonzero multiples of $pi,$; zero-crossing of the normalized sinc     occur at nonzero integer values.

★ The continuous Fourier transform of the normalized sinc     (to ordinary frequency) is  $mathrm\{rect\}(f),$.
::$int\_\{-infty\}^infty\; mathrm\{sinc\}(t),e^\{-2pi\; i\; f\; t\}dt\; =\; mathrm\{rect\}(f)$,
:where the rectangular function is 1 for argument between –1/2 and 1/2, and zero otherwise.

★ The Fourier integral above, including the special case
::
:is an improper integral. It is not a Lebesgue integral because':'

::
ight| dx = infty ,

★
ight)

★

:where $Gamma(x)$ is the gamma function.

★
:where ''Si''(''x'') is the sine integral.

Relationship to the Dirac delta distribution

The normalized sinc function can be used as a ''nascent delta function'', even though it is not a distribution.
The ''normalized'' sinc function is related to the delta distribution δ(''x'') by
:$lim\_\{a$
ightarrow 0}rac{1}{a} extrm{sinc}(x/a)=delta(x).
This is not an ordinary limit, since the left side does not converge. Rather, it means that
:$lim\_\{a$
ightarrow 0}int_{-infty}^infty rac{1}{a} extrm{sinc}(x/a)arphi(x),dx
=int_{-infty}^inftydelta(x)arphi(x),dx = arphi(0),

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 function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±1/(π''x''), regardless of the value of ''a''. This contradicts 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.

See also

★ Anti-aliasing

★ Sinc filter

★ Whittaker–Shannon interpolation formula

External links

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