Sinc function

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.

A good approximation of the x-coordinate of the nth extremum with positive x-coordinate is

x n ≈ ( n + 1 2 ) π − 1 ( n + 1 2 ) π   ,

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 −x_n. In addition, there is an absolute maximum at ξ₀ = (0,1).

The normalized sinc function has a simple representation as the infinite product

sin ⁡ ( π x ) π x = ∏ n = 1 ∞ ( 1 − x 2 n 2 )

and is related to the gamma function Γ(x) through Euler's reflection formula,

sin ⁡ ( π x ) π x = 1 Γ ( 1 + x ) Γ ( 1 − x )   .

Euler discovered that

sin ⁡ ( x ) x = ∏ n = 1 ∞ cos ⁡ ( x 2 n )   .

The continuous Fourier transform of the normalized sinc (to ordinary frequency) is rect( f ),

∫ − ∞ ∞ sinc ⁡ ( t ) e i 2 π f t d t = rect ⁡ ( f )   ,

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.

This Fourier integral, including the special case

∫ − ∞ ∞ sin ⁡ ( π x ) π x d x = rect ⁡ ( 0 ) = 1

is an improper integral (cf. Dirichlet integral) and not a convergent Lebesgue integral, as

∫ − ∞ ∞ | sin ⁡ ( π x ) π x | d x = + ∞   .

The normalized sinc function has properties that make it ideal in relationship to interpolation of sampled bandlimited functions:

It is an interpolating function, i.e., sinc(0) = 1, and sinc(k) = 0 for nonzero integer k.

The functions x_k(t) = sinc(t − k) (k integer) form an orthonormal basis for bandlimited functions in the function space L²(R), with highest angular frequency ω_H = π (that is, highest cycle frequency f_H = 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).

∫ 0 x sin ⁡ ( θ ) θ d θ = Si ⁡ ( x )

where Si(x) is the sine integral.

λ sinc(λx) (not normalized) is one of two linearly independent solutions to the linear ordinary differential equation

The other is cos(λx)/x, which is not bounded at x = 0, unlike its sinc function counterpart.

∫ − ∞ ∞ sin 2 ⁡ ( θ ) θ 2 d θ = π → ∫ − ∞ ∞ sinc 2 ⁡ ( x ) d x = 1   ,

where the normalized sinc is meant.

∫ − ∞ ∞ sin 3 ⁡ ( θ ) θ 3 d θ = 3 π 4

∫ − ∞ ∞ sin 4 ⁡ ( θ ) θ 4 d θ = 2 π 3   .

Relationship to the Dirac delta distribution

The normalized sinc function can be used as a nascent delta function, meaning that the following weak limit holds,

lim a → 0 sin ⁡ ( π x a ) π x = lim a → 0 1 a sinc ( x a ) = δ ( x )   .

This is not an ordinary limit, since the left side does not converge. Rather, it means that

lim a → 0 ∫ − ∞ ∞ 1 a sinc ( x a ) φ ( x ) d x = φ ( 0 )   ,

for any smooth function φ(x) with compact support.

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.

∑ n = 1 ∞ sinc ⁡ ( n ) = sinc ⁡ ( 1 ) + sinc ⁡ ( 2 ) + sinc ⁡ ( 3 ) + sinc ⁡ ( 4 ) + ⋯ = π − 1 2

The sum of the squares also equals π − 1/2.

∑ n = 1 ∞ sinc 2 ⁡ ( n ) = sinc 2 ⁡ ( 1 ) + sinc 2 ⁡ ( 2 ) + sinc 2 ⁡ ( 3 ) + sinc 2 ⁡ ( 4 ) + ⋯ = π − 1 2

When the signs of the addends alternate and begin with +, the sum equals 1/2.

∑ n = 1 ∞ ( − 1 ) n + 1 sinc ⁡ ( n ) = sinc ⁡ ( 1 ) − sinc ⁡ ( 2 ) + sinc ⁡ ( 3 ) − sinc ⁡ ( 4 ) + ⋯ = 1 2

The alternating sums of the squares and cubes also equal 1/2.

∑ n = 1 ∞ ( − 1 ) n + 1 sinc 2 ⁡ ( n ) = sinc 2 ⁡ ( 1 ) − sinc 2 ⁡ ( 2 ) + sinc 2 ⁡ ( 3 ) − sinc 2 ⁡ ( 4 ) + ⋯ = 1 2 ∑ n = 1 ∞ ( − 1 ) n + 1 sinc 3 ⁡ ( n ) = sinc 3 ⁡ ( 1 ) − sinc 3 ⁡ ( 2 ) + sinc 3 ⁡ ( 3 ) − sinc 3 ⁡ ( 4 ) + ⋯ = 1 2

Series expansion

Unnormalized sinc(x):

sinc ⁡ ( x ) = sin ⁡ ( x ) x = ∑ n = 0 ∞ ( − x 2 ) n ( 2 n + 1 ) !

Higher dimensions

The product of 1-D sinc functions readily provides a multivariate sinc function for the square, Cartesian, grid (lattice): sinc_C(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 using the geometric properties of Brillouin zones and their connection to zonotopes.

For example, a hexagonal lattice can be generated by the (integer) linear span of the vectors

u 1 = [ 1 2 3 2 ] and u 2 = [ 1 2 − 3 2 ] .

Denoting

ξ 1 = 2 3 u 1 , ξ 2 = 2 3 u 2 , ξ 3 = − 2 3 ( u 1 + u 2 ) , x = [ x y ] ,

one can derive the sinc function for this hexagonal lattice as:

sinc H ⁡ ( x ) = 1 3 ( cos ⁡ ( π ξ 1 ⋅ x ) sinc ⁡ ( ξ 2 ⋅ x ) sinc ⁡ ( ξ 3 ⋅ x ) + cos ⁡ ( π ξ 2 ⋅ x ) sinc ⁡ ( ξ 3 ⋅ x ) sinc ⁡ ( ξ 1 ⋅ x ) + cos ⁡ ( π ξ 3 ⋅ x ) sinc ⁡ ( ξ 1 ⋅ x ) sinc ⁡ ( ξ 2 ⋅ x ) )

This construction can be used to design Lanczos window for general multidimensional lattices.