Sinc function

The sinc function, also known as the interpolation function or filtering function, is the product of a sine function and a monotonically decreasing function. It is defined by:

${\displaystyle \mathbf {sinc} (x)={\frac {\sin \left(x\right)}{x}}}$

or

${\displaystyle \mathbf {sinc} (x)={\frac {\sin \left(\pi x\right)}{\pi x}}}$

It has a removable singularity at the origin. ${\displaystyle \mathbf {sinc} (0)=1}$ by L'Hôpital's rule.

The sinc function oscillates with decreasing amplitude. Applications of the sinc function are found in communication theory, control theory, and optics.

The Fourier transform is

${\displaystyle {\frac {W}{\pi }}\mathbf {sinc} (Wt)=\mathbf {rect} \left({\frac {\omega }{2W}}\right)}$

where ${\displaystyle rect}$ is the rectangular function.

See also

• trigonometric function
• L'Hôpital's rule