Sinc function

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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>\operatorname{sinc}(x)

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

which oscillates within an envelope of ±1/x.

It is also sometimes defined as simply sin(x)/x, which has a removable singularity at zero since:

Therefore the latter definition yields a function that is analytic everywhere.

An interesting property of the sinc function is that its local maxima and minima correspond to its intersections with the cosine curve. That is if <math>|\operatorname{sinc}(x)|</math> has a local maximum at <math>x = a\,</math>, then:

<math>\mathrm{sinc}(a)=\textrm{cos}(a). \,</math>

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

1 Normalization and properties
2 Relationship to delta function
3 See also
4 External links

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Normalization and properties

A normalized sinc function is defined as:

<math>\mathrm{sinc}_{\pi}(x) \equiv \textrm{sinc}(\pi x)\,</math>

It is important to note that <math>\mathrm{sinc}_{\pi}(x)\,</math> is also commonly referred to as the sinc function and written <math>\mathrm{sinc}(x)\,</math>. The normalization produces the simplest form of several properties and identities:

1. The zero-crossings occur at integer values of <math>x\,</math>.

2. The Fourier transform [to ordinary frequency] is <math>\mathrm{rect(f)}\,</math>. I.e., row 12 of important Fourier transforms, with <math>a = \pi\,</math>.

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

where the integral is regarded as an improper integral; it cannot be taken to be a Lebesgue integral because:

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

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

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

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

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Relationship to delta function

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 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.

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