"Sinc" redirects here. For the designation used in the United Kingdom for areas of wildlife interest, see
Site of Importance for Nature Conservation. For the signal processing filter based on this function, see
Sinc filter.
In either case, the value at x = 0 is defined to be the limiting value
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 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.
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 maxima and minima 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:
The first few terms of the infinite series for the x coordinate of the n-th extremum with positive x coordinate are
where
and where odd n lead to a local minimum, and even n to a local maximum. Because of symmetry around the y axis, there exist extrema with x coordinates −xn. In addition, there is an absolute maximum at ξ0 = (0, 1).
The normalized sinc function has a simple representation as the
infinite product:
where the
rectangular function is 1 for argument between −1/2 and 1/2, and zero otherwise. This corresponds to the fact that the
sinc filter is the ideal (
brick-wall, meaning rectangular frequency response)
low-pass filter.
The normalized sinc function has properties that make it ideal in relationship to
interpolation of
sampledbandlimited functions:
It is an interpolating function, i.e., sinc(0) = 1, and sinc(k) = 0 for nonzero
integerk.
The functions xk(t) = sinc(t − k) (k integer) form an
orthonormal basis for
bandlimited functions in the
function spaceL2(R), with highest angular frequency ωH = π (that is, highest cycle frequency fH = 1/2).
Other properties of the two sinc functions include:
The unnormalized sinc is the zeroth-order spherical
Bessel function of the first kind, j0(x). The normalized sinc is j0(πx).
This is not an ordinary limit, since the left side does not converge. Rather, it means that
for every
Schwartz function, as can be seen from the
Fourier inversion theorem.
In the above expression, as a → 0, 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 complicates 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.
Summation
All sums in this section refer to the unnormalized sinc function.
The sum of sinc(n) over integer n from 1 to ∞ equals π − 1/2:
The sum of the squares also equals π − 1/2:[10][11]
When the signs of the
addends alternate and begin with +, the sum equals 1/2:
The alternating sums of the squares and cubes also equal 1/2:[12]
Series expansion
The
Taylor series of the unnormalized sinc function can be obtained from that of the sine (which also yields its value of 1 at x = 0):
The series converges for all x. The normalized version follows easily:
Euler famously compared this series to the expansion of the infinite product form to solve the
Basel problem.
Higher dimensions
The product of 1-D sinc functions readily provides a
multivariate sinc function for the square Cartesian grid (
lattice): sincC(x, y) = sinc(x) sinc(y), whose
Fourier transform is the
indicator function of a square in the frequency space (i.e., the brick wall defined in 2-D space). The sinc function for a non-Cartesian
lattice (e.g.,
hexagonal lattice) is a function whose
Fourier transform is the
indicator function of the
Brillouin zone of that lattice. For example, the sinc function for the hexagonal lattice is a function whose
Fourier transform is the
indicator function of the unit hexagon in the frequency space. For a non-Cartesian lattice this function can not be obtained by a simple tensor product. However, the explicit formula for the sinc function for the
hexagonal,
body-centered cubic,
face-centered cubic and other higher-dimensional lattices can be explicitly derived[13] using the geometric properties of Brillouin zones and their connection to
zonotopes.