The sinc function, denoted by $\backslash mathrm\{sinc\}(x)\backslash ,$, has two definitions, sometimes distinguished as the normalized sinc function and unnormalized sinc function. Each is the product of a sine function and a monotonically decreasing function 1/x:

1. In digital signal processing and communication theory, the normalized sinc function is commonly defined by

   $\backslash mathrm\{sinc\}(x)\; =\; \backslash frac\{\backslash sin(\backslash pi\; x)\}\{\backslash pi\; x\}$

2. In mathematics, the historical unnormalized sinc function (for sinus cardinalis), is defined by

   $\backslash mathrm\{sinc\}(x)\; =\; \backslash frac\{\backslash sin(x)\}\{x\}$

In both cases, the value of the function at the removable singularity at zero is sometimes specified explicitly as 1. The sinc function is analytic everywhere.

The unnormalized sinc function is identical to the normalized sinc function above except for a missing scaling factor of π in the argument.

Properties

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

• $\backslash mathrm\{sinc\}(0)\; =\; 1\backslash ,$ and $\backslash mathrm\{sinc\}(k)\; =\; 0\backslash ,$ for $k\backslash ne\; 0\backslash ,$ and $k\backslash in\backslash mathbb\{Z\}\backslash ,$ (integers); that is, it is an interpolating function.
• the functions $x\_k(t)=\backslash mathrm\{sinc\}(t-k)\; \backslash$ form an orthonormal basis for bandlimited functions in the function space $L^2(\backslash R)$, with highest angular frequency $\backslash omega\_\backslash mathrm\{H\}=\backslash pi\backslash ,$ (that is, highest cycle frequency $f\_\backslash mathrm\{H\}=1/2\backslash ,$).

Other properties of the two sinc functions include:

• The local maxima and minima of the unnormalized sinc,   $\backslash begin\{matrix\}\backslash frac\{\backslash sin(x)\}\{x\}\; \backslash end\{matrix\}\backslash ,$   correspond to its intersections with the cosine function. I.e. where the derivative of $\backslash begin\{matrix\}\backslash frac\{\backslash sin(x)\}\{x\}\; \backslash end\{matrix\}\backslash ,$ is zero (local extrema at $x\; =\; a\backslash ,$), then   $\backslash begin\{matrix\}\backslash frac\{\backslash sin(a)\}\{a\}\; \backslash end\{matrix\}\; =\; \backslash cos(a)\; \backslash ,$.

• The unnormalized sinc is the zero^th order spherical Bessel function of the first kind, $j\_0(x)\; =\; \backslash begin\{matrix\}\backslash frac\{\backslash sin(x)\}\{x\}\; \backslash end\{matrix\}\backslash ,$. The normalized sinc is $j\_0(\backslash pi\; x)\backslash ,$.

• The zero-crossings of the unnormalized sinc are at nonzero multiples of $\backslash pi\backslash ,$; zero-crossing of the normalized sinc   $\backslash mathrm\{sinc\}(x)\; =\; \backslash begin\{matrix\}\backslash frac\{\backslash sin(\backslash pi\; x)\}\{\backslash pi\; x\}\; \backslash end\{matrix\}\backslash ,$   occur at nonzero integer values.

• The continuous Fourier transform of the normalized sinc   $\backslash mathrm\{sinc\}(x)\; =\; \backslash begin\{matrix\}\backslash frac\{\backslash sin(\backslash pi\; x)\}\{\backslash pi\; x\}\; \backslash end\{matrix\}\backslash ,$   (to ordinary frequency) is  $\backslash mathrm\{rect\}(f)\backslash ,$.

$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash mathrm\{sinc\}(t)\backslash ,e^\{-2\backslash pi\; i\; f\; t\}dt\; =\; \backslash mathrm\{rect\}(f)$,

where the rectangular function is 1 for argument between –1/2 and 1/2, and zero otherwise.

• The integral

$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash begin\{matrix\}\backslash frac\{\backslash sin(\backslash pi\; x)\}\{\backslash pi\; x\}\; \backslash end\{matrix\}\backslash ,\; dx\; =\; 1$

is an improper integral. It is not a Lebesgue integral because:

$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash left|\backslash begin\{matrix\}\backslash frac\{\backslash sin(\backslash pi\; x)\}\{\backslash pi\; x\}\; \backslash end\{matrix\}\backslash right|\backslash \; dx\; =\; \backslash infty\; \backslash ,$

• $\backslash mathrm\{sinc\}(x)\; =\; \backslash frac\{\backslash sin(\backslash pi\; x)\}\{\backslash pi\; x\}\; =\; \backslash prod\_\{n=1\}^\backslash infty\; \backslash left(1\; -\; \backslash frac\{x^2\}\{n^2\}\backslash right)$

• $\backslash mathrm\{sinc\}(x)\; =\; \backslash frac\{\backslash sin(\backslash pi\; x)\}\{\backslash pi\; x\}\; =\; \backslash frac\{1\}\{\backslash Gamma(1+x)\backslash Gamma(1-x)\}\; =\; \backslash frac\{1\}\{x!\; (-x)!\}$
  

where $\backslash Gamma(x)$ is the gamma function.

Relationship to the Dirac delta distribution

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The normalized sinc function can be used as a nascent delta function (see Dirac delta function), even though it is not a distribution.

The normalized sinc function is related to the delta distribution δ(x) by

$\backslash lim\_\{a\backslash rightarrow\; 0\}\backslash frac\{1\}\{a\}\backslash textrm\{sinc\}(x/a)=\backslash delta(x).$

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

$\backslash lim\_\{a\backslash rightarrow\; 0\}\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash frac\{1\}\{a\}\backslash textrm\{sinc\}(x/a)\backslash varphi(x)\backslash ,dx$

=\int_{-\infty}^\infty\delta(x)\varphi(x)\,dx = \varphi(0),

for any smooth function $\backslash varphi(x)$ 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/(πx), 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

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• Anti-aliasing
• Sinc filter
• Whittaker–Shannon interpolation formula

External links

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Signal processing | Elementary special functions

Sinc-Funktion | Sinc関数 | funkcja sinc