Sinc function

In mathematics, physics and engineering, the sinc function (/ˈsɪŋk/ SINK), denoted by sinc(x), is defined as either $${\displaystyle \operatorname {sinc} (x)={\frac {\sin x}{x}}.}$$ or $${\displaystyle \operatorname {sinc} (x)={\frac {\sin \pi x}{\pi x}},}$$

the latter of which is sometimes referred to as the normalized sinc function. The only difference between the two definitions is in the scaling of the independent variable (the x axis) by a factor of π. In both cases, the value of the function at the removable singularity at zero is understood to be the limit value 1. The sinc function is then analytic everywhere and hence an entire function.

The normalized sinc function is the Fourier transform of the rectangular function with no scaling. It is used in the concept of reconstructing a continuous bandlimited signal from uniformly spaced samples of that signal. The sinc filter is used in signal processing.

The function itself was first mathematically derived in this form by Lord Rayleigh in his expression (Rayleigh's formula) for the zeroth-order spherical Bessel function of the first kind.

The sinc function is also called the cardinal sine function.

The sinc function has two forms, normalized and unnormalized.

In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by $${\displaystyle \operatorname {sinc} (x)={\frac {\sin x}{x}}.}$$

Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x).