Sinc function

In mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized.^[1]

Sinc
Part of the normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale
General information
General definition	${\displaystyle \operatorname {sinc} x={\begin{cases}{\dfrac {\sin x}{x}},&x\neq 0\\1,&x=0\end{cases}}}$
Motivation of invention	Telecommunication
Date of solution	1952
Fields of application	Signal processing, spectroscopy
Domain, codomain and image
Domain	${\displaystyle \mathbb {R} }$
Image	${\displaystyle [-0.217234\ldots ,1]}$
Basic features
Parity	Even
Specific values
At zero	1
Value at +∞	0
Value at −∞	0
Maxima	1 at ${\displaystyle x=0}$
Minima	${\displaystyle -0.21723\ldots }$ at ${\displaystyle x=\pm 4.49341\ldots }$
Specific features
Root	${\displaystyle \pi k,k\in \mathbb {Z} _{\neq 0}}$
Related functions
Reciprocal	${\displaystyle {\begin{cases}x\csc x,&x\neq 0\\1,&x=0\end{cases}}}$
Derivative	${\displaystyle \operatorname {sinc} 'x={\begin{cases}{\dfrac {\cos x-\operatorname {sinc} x}{x}},&x\neq 0\\0,&x=0\end{cases}}}$
Antiderivative	${\displaystyle \int \operatorname {sinc} x\,dx=\operatorname {Si} (x)+C}$
Series definition
Taylor series	${\displaystyle \operatorname {sinc} x=\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k+1)!}}}$

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).^[2]

In digital signal processing and information theory, the normalized sinc function is commonly defined for x ≠ 0 by $${\displaystyle \operatorname {sinc} x={\frac {\sin(\pi x)}{\pi x}}.}$$

In either case, the value at x = 0 is defined to be the limiting value $${\displaystyle \operatorname {sinc} 0:=\lim _{x\to 0}{\frac {\sin(ax)}{ax}}=1}$$ for all real a ≠ 0 (the limit can be proven using the squeeze theorem).

The normalization causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of π). As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of x.

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 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 function has also been called the cardinal sine or sine cardinal function.^[3][4] The term sinc /ˈsɪŋk/ was introduced by Philip M. Woodward in his 1952 article "Information theory and inverse probability in telecommunication", in which he said that the function "occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own",^[5] and his 1953 book Probability and Information Theory, with Applications to Radar.^[6][7] 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.