# Sinc function

File:SincFunction.png
The sinc function sinc(x) from x = −8π to 8π.

In mathematics, the sinc function (for sinus cardinalis), also known as the interpolation function, filtering function or the first spherical Bessel function $j_{0}(x)$ , is the product of a sine function and a monotonically decreasing function. It is defined by:

${\textrm {sinc}}(x)=\left\{{\begin{matrix}{\frac {\sin(x)}{x}}&:~x\neq 0\\\\1&:~x=0\end{matrix}}\right.$ The sinc function is sometimes defined as simply sin(x)/x. The function sin(x)/x has a removable singularity at zero. Using l'Hôpital's rule it can be seen that:

$\lim _{x\to 0}{\frac {\sin(x)}{x}}=1.\,$ The above definition for the sinc function is preferred since it removes this singularity and yields a function which is analytic everywhere.

The normalized sinc function is defined as:

$\mathrm {sinc} _{N}(x)={\textrm {sinc}}(\pi x)\,$ and, as its name implies, is normalized to unity

$\int _{-\infty }^{\infty }\mathrm {sinc} _{N}(x)\,dx=1.$ This integral must necessarily be regarded as an improper integral; it cannot be taken to be a Lebesgue integral because

$\int _{-\infty }^{\infty }\left|\mathrm {sinc} _{N}(x)\right|\,dx=\infty .$ The normalized sinc function also has the important infinite product

$\mathrm {sinc} _{N}(x)=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}}}\right).$ We also have an expression in terms of the gamma function, as

$\mathrm {sinc} _{N}(x)={\frac {1}{\Gamma (1+x)\Gamma (1-x)}}={\frac {1}{x!(-x)!}}.$ Because of its usefulness, the normalized sinc function is sometimes simply called the sinc function and written sinc(x).

The sinc function oscillates inside an envelope of ±1/x. The Fourier transform of the sinc function can be expressed in terms of the rectangular function:

${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\textrm {sinc}}(x)e^{-ikx}\,dx={\sqrt {\frac {\pi }{2}}}~{\textrm {rect}}(k/2)$ In the language of distributions, the sinc function is related to the delta function δ(x) by

$\lim _{a\rightarrow 0}{\frac {1}{\pi a}}{\textrm {sinc}}(x/a)=\delta (x).$ This is not an ordinary limit, since the left side does not converge. Rather, it means that

$\lim _{a\rightarrow 0}\int _{-\infty }^{\infty }{\frac {1}{\pi a}}{\textrm {sinc}}(x/a)\varphi (x)\,dx=\int _{-\infty }^{\infty }\delta (x)\varphi (x)\,dx=\varphi (0),$ 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 sinc function always oscillates inside an envelope of ±1/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.

An interesting property of the sinc function is that its local maxima correspond to its intersections with the cosine curve. That is, if the sinc function has a local maximum at x then

$\mathrm {sinc} (x)={\textrm {cos}}(x).\,$ Applications of the sinc function are found in digital signal processing, communication theory, control theory, and optics.