Sinc function

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The sinc function, denoted by <math>\mathrm{sinc}(x)\,</math>, has two definitions, sometimes distinguished as the normalized sinc function and unnormalized sinc function:

In digital signal processing and communication theory, the normalized sinc function is commonly defined by
<math>\mathrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x}</math>
In mathematics, the historical unnormalized sinc function (for sinus cardinalis), is defined by
<math>\mathrm{sinc}(x) = \frac{\sin(x)}{x}</math>

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.

1 Table and plot
2 Properties
3 Relationship to the Dirac delta distribution
4 See also
5 External links

[edit] Table and plot

Image:Sinc function (both).svg

Position and values of the extrema:
<math>x</math>	<math>x/\pi </math>	<math>\mbox{sinc}(x)</math>	<math>\mbox{sinc}^2(x)</math>	<math>20\log_{10} \mbox{sinc}(x)</math>
0.000000	0.000000	1.000000	1.000000	0.000000
4.493409	1.430297	-0.217234	0.047190	-13.261459
7.725252	2.459024	0.128375	0.016480	-17.830421
10.904122	3.470890	-0.091325	0.008340	-20.788187
14.066194	4.477409	0.070913	0.005029	-22.985427
17.220755	5.481537	-0.057972	0.003361	-24.735664
20.371303	6.484387	0.049030	0.002404	-26.190829
23.519452	7.486474	-0.042480	0.001805	-27.436388
26.666054	8.488069	0.037475	0.001404	-28.525278
29.811599	9.489327	-0.033525	0.001124	-29.492589
32.956389	10.490344	0.030329	0.000920	-30.362789
36.100622	11.491185	-0.027690	0.000767	-31.153625
39.244432	12.491891	0.025473	0.000649	-31.878380
42.387914	13.492492	-0.023585	0.000556	-32.547257

[edit] Properties

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

<math>\mathrm{sinc}(0) = 1\,</math> and <math>\mathrm{sinc}(k) = 0\,</math> for <math>k\ne 0\,</math> and <math>k\in\mathbb{Z}\,</math> (integers); that is, it is an interpolating function.
the functions <math>x_k(t)=\mathrm{sinc}(t-k) \ </math> form an orthonormal basis for bandlimited functions in the function space <math>L^2(\R)</math>, with highest angular frequency <math>\omega_\mathrm{H}=\pi\,</math> (that is, highest cycle frequency <math>f_\mathrm{H}=1/2\,</math>).

Other properties of the two sinc functions include:

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

The unnormalized sinc is the zero^th order spherical Bessel function of the first kind, <math>j_0(x) = \begin{matrix}\frac{\sin(x)}{x} \end{matrix}\,</math>. The normalized sinc is <math>j_0(\pi x)\,</math>.

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

The continuous Fourier transform of the normalized sinc <math>\mathrm{sinc}(x) = \begin{matrix}\frac{\sin(\pi x)}{\pi x} \end{matrix}\,</math> (to ordinary frequency) is <math>\mathrm{rect}(f)\,</math>.

<math>\int_{-\infty}^\infty \mathrm{sinc}(t)\,e^{-2\pi i f t}dt = \mathrm{rect}(f)</math>,

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

The integral

<math>\int_{-\infty}^\infty \begin{matrix}\frac{\sin(\pi x)}{\pi x} \end{matrix}\, dx = 1</math>

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

<math>\int_{-\infty}^\infty \left|\begin{matrix}\frac{\sin(\pi x)}{\pi x} \end{matrix}\right|\ dx = \infty \,</math>

<math> \mathrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x} = \prod_{n=1}^\infty \left(1 - \frac{x^2}{n^2}\right)</math>

<math> \mathrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x} = \frac{1}{\Gamma(1+x)\Gamma(1-x)} = \frac{1}{x! (-x)!}</math>

where <math>\Gamma(x)</math> is the gamma function.

[edit] 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

<math>\lim_{a\rightarrow 0}\frac{1}{a}\textrm{sinc}(x/a)=\delta(x).</math>

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

<math>\lim_{a\rightarrow 0}\int_{-\infty}^\infty \frac{1}{a}\textrm{sinc}(x/a)\varphi(x)\,dx

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

</math>

for any smooth function <math>\varphi(x)</math> 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.