Sinc function

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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

$\mathrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$
In mathematics, the historical unnormalized sinc function (for sinus cardinalis), is defined by

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

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.

1 Properties
2 Relationship to the Dirac delta distribution
3 See also
4 External links

[edit] Properties

The normalized sinc(x) (blue) and unnormalized sinc function (red) shown on the same scale from x = −6π to 6π.

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 $k\ne 0\,$ and $k\in\mathbb{Z}\,$ (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, $\begin{matrix}\frac{\sin(x)}{x} \end{matrix}\,$ correspond to its intersections with the cosine function. I.e. where the derivative of $\begin{matrix}\frac{\sin(x)}{x} \end{matrix}\,$ is zero (local extrema at $x = a\,$ ), then $\begin{matrix}\frac{\sin(a)}{a} \end{matrix} = \cos(a) \,$ .

The unnormalized sinc is the zero^th order spherical Bessel function of the first kind, $j_0(x) = \begin{matrix}\frac{\sin(x)}{x} \end{matrix}\,$ . 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 $\mathrm{sinc}(x) = \begin{matrix}\frac{\sin(\pi x)}{\pi x} \end{matrix}\,$ occur at nonzero integer values.

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

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

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

$\int_{-\infty}^\infty \begin{matrix}\frac{\sin(\pi x)}{\pi x} \end{matrix}\, dx = \mathrm{rect}(0) = 1$

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

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

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

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

where

Γ(x)

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

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

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

$\lim_{a\rightarrow 0}\int_{-\infty}^\infty \frac{1}{a}\textrm{sinc}(x/a)\varphi(x)\,dx =\int_{-\infty}^\infty\delta(x)\varphi(x)\,dx = \varphi(0),$

for any smooth function $\varphi(x)$ 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.