Sinc function

From Academic Kids

 The sinc function sinc(x) from x = −8π to 8π.
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 <math>j_0(x)<math>, is the product of a sine function and a monotonically decreasing function. It is defined by:

<math>\textrm{sinc}(x)

= \left\{ \begin{matrix} \frac{\sin(x)}{x}&:~x\ne 0 \\ \\ 1 &:~x=0 \end{matrix} \right. <math>

The sinc function is sometimes defined as simply sin(x)/x. The function sin(x)/x has a removable singularity at zero, so that, by L'H˘pital's rule we have:

<math>\lim_{x\to 0} \frac{\sin(x)}{x}=1.\,<math>

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:

<math>\mathrm{sinc}_N(x) = \textrm{sinc}(\pi x)\,<math>

and, as its name implies, is normalized to unity

<math>\int_{-\infty}^\infty \mathrm{sinc}_N(x)\,dx = 1.<math>

This integral must necessarily be regarded as an improper integral; it cannot be taken to be a Lebesgue integral because

<math>\int_{-\infty}^\infty \left|\mathrm{sinc}_N(x)\right|\,dx = \infty.<math>

The normalized sinc function also has the important infinite product

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

We also have an expression in terms of the gamma function, as

<math>\mathrm{sinc}_N(x) = \frac{1}{\Gamma(1+x)\Gamma(1-x)} = \frac{1}{x! (-x)!}.<math>

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:

<math>\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \textrm{sinc}(x)e^{-ikx}\,dx=

\sqrt{\frac{\pi}{2}}~\textrm{rect}(k/2)<math>

In the language of distributions, the sinc function is related to the delta function δ(x) by

<math>\lim_{a\rightarrow 0}\frac{1}{\pi 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}{\pi 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 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.

Applications of the sinc function are found in digital signal processing, communication theory, control theory, and optics.

See also

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