In mathematics, physics and engineering, the sinc function (), denoted by sinc(x), is defined as either
sinc ( x ) = sin x x {\displaystyle \operatorname {sinc} (x)={\frac {\sin x}{x}}}
or
sinc ( x ) = sin π x π x , {\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.
Definitions
The sinc function has two forms, normalized and unnormalized. In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by
sinc ( x ) = sin x x . {\displaystyle \operatorname {sinc} (x)={\frac {\sin x}{x}}.}
Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x). In digital signal processing and information theory, the normalized sinc function is commonly defined for x ≠ 0 by
sinc ( x ) = sin ( π x ) π x . {\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
sinc ( 0 ) := lim x → 0 sin ( a x ) a x = 1 {\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.
Etymology The function has also been called the cardinal sine or sine cardinal function, since it returns 1 (a cardinal number) for x = 0 {\displaystyle x=0} and 0 (another cardinal number) for every other integer. The term "sinc" is a contraction of the function's full Latin name, the sinus cardinalis and was introduced by Philip M. Woodward and I.L Davies in their 1952 article "Information theory and inverse probability in telecommunication", saying "This function occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own". It is also used in Woodward's 1953 book Probability and Information Theory, with Applications to Radar.
Properties
The zero crossings of the unnormalized sinc are at non-zero integer multiples of π, while zero crossings of the normalized sinc occur at non-zero integers. The local extrema of the unnormalized sinc correspond to its intersections with the cosine function. That is, sin(ξ)/ξ = cos(ξ) for all points ξ where the derivative of sin(x)/x is zero and thus a local extremum is reached. This follows from the derivative of the sinc function:
d d x sinc ( x ) = { cos ( x ) − sinc ( x ) x , x ≠ 0 0 , x = 0 . {\displaystyle {\frac {d}{dx}}\operatorname {sinc} (x)={\begin{cases}{\dfrac {\cos(x)-\operatorname {sinc} (x)}{x}},&x\neq 0\\0,&x=0\end{cases}}.}
The first few terms of the infinite series for the x coordinate of the nth extremum with positive x coordinate are
… excerpt ends here. Continue reading the full article.
