Sinc function

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: $${\displaystyle {\frac {d}{dx}}\operatorname {sinc} ($$x)={\frac {\cos(x)-\operatorname {sinc} (x)}{x}}.}  

The first few terms of the infinite series for the x coordinate of the n-th extremum with positive x coordinate are $${\displaystyle x_{n}=q-q^{-1}-{\frac {2}{3}}q^{-3}-{\frac {13}{15}}q^{-5}-{\frac {146}{105}}q^{-7}-\cdots ,}$$  where $${\displaystyle q=\left(n+{\frac {1}{2}}\right)\pi ,}$$  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 −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: $${\displaystyle {\frac {\sin(\pi x)}{\pi x}}=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}}}\right)}$$ 

and is related to the gamma function Γ(x) through Euler's reflection formula: $${\displaystyle {\frac {\sin(\pi x)}{\pi x}}={\frac {1}{\Gamma (1+x)\Gamma (1-x)}}.}$$ 

Euler discovered^[8] that $${\displaystyle {\frac {\sin($$x)}{x}}=\prod _{n=1}^{\infty }\cos \left({\frac {x}{2^{n}}}\right),}   and because of the product-to-sum identity^[9]

$${\displaystyle \prod _{n=1}^{k}\cos \left({\frac {x}{2^{n}}}\right)={\frac {1}{2^{k-1}}}\sum _{n=1}^{2^{k-1}}\cos \left({\frac {n-1/2}{2^{k-1}}}x\right),\quad \forall k\geq 1,}$$  Euler's product can be recast as a sum $${\displaystyle {\frac {\sin($$x)}{x}}=\lim _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}\cos \left({\frac {n-1/2}{N}}x\right).}  

The continuous Fourier transform of the normalized sinc (to ordinary frequency) is rect(f): $${\displaystyle \int _{-\infty }^{\infty }\operatorname {sinc} ($$t)\,e^{-i2\pi ft}\,dt=\operatorname {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 $${\displaystyle \int _{-\infty }^{\infty }{\frac {\sin(\pi x)}{\pi x}}\,dx=\operatorname {rect} (0)=1}$$  is an improper integral (see Dirichlet integral) and not a convergent Lebesgue integral, as $${\displaystyle \int _{-\infty }^{\infty }\left|{\frac {\sin(\pi x)}{\pi x}}\right|\,dx=+\infty .}$$ 

The normalized sinc function has properties that make it ideal in relationship to interpolationofsampled 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).
• where Si(x) is the sine integral, $${\displaystyle \int _{0}^{x}{\frac {\sin(\theta )}{\theta }}\,d\theta =\operatorname {Si} ($$x).}  
• λ sinc(λx) (not normalized) is one of two linearly independent solutions to the linear ordinary differential equation $${\displaystyle x{\frac {d^{2}y}{dx^{2}}}+2{\frac {dy}{dx}}+\lambda ^{2}xy=0.}$$  The other is cos(λx)/x, which is not bounded at x = 0, unlike its sinc function counterpart.
• Using normalized sinc, $${\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{2}(\theta )}{\theta ^{2}}}\,d\theta =\pi \quad \Rightarrow \quad \int _{-\infty }^{\infty }\operatorname {sinc} ^{2}($$x)\,dx=1,}  
• ${\displaystyle \int _{-\infty }^{\infty }{\frac {\sin(\theta )}{\theta }}\,d\theta =\int _{-\infty }^{\infty }\left({\frac {\sin(\theta )}{\theta }}\right)^{2}\,d\theta =\pi .}$ 
• ${\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{3}(\theta )}{\theta ^{3}}}\,d\theta ={\frac {3\pi }{4}}.}$ 
• ${\displaystyle \int _{-\infty }^{\infty }{\frac {\sin ^{4}(\theta )}{\theta ^{4}}}\,d\theta ={\frac {2\pi }{3}}.}$ 
• The following improper integral involves the (not normalized) sinc function: $${\displaystyle \int _{0}^{\infty }{\frac {dx}{x^{n}+1}}=1+2\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{($$kn)^{2}-1}}={\frac {1}{\operatorname {sinc} ({\frac {\pi }{n}})}}.}  

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

$${\displaystyle \lim _{a\to 0}{\frac {\sin \left({\frac {\pi x}{a}}\right)}{\pi x}}=\lim _{a\to 0}{\frac {1}{a}}\operatorname {sinc} \left({\frac {x}{a}}\right)=\delta ($$x).}  

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

$${\displaystyle \lim _{a\to 0}\int _{-\infty }^{\infty }{\frac {1}{a}}\operatorname {sinc} \left({\frac {x}{a}}\right)\varphi ($$x)\,dx=\varphi (0)}  

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.

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:

$${\displaystyle \sum _{n=1}^{\infty }\operatorname {sinc} ($$n)=\operatorname {sinc} (1)+\operatorname {sinc} (2)+\operatorname {sinc} (3)+\operatorname {sinc} (4)+\cdots ={\frac {\pi -1}{2}}.}  

The sum of the squares also equals π − 1/2:^[10][11]

$${\displaystyle \sum _{n=1}^{\infty }\operatorname {sinc} ^{2}($$n)=\operatorname {sinc} ^{2}(1)+\operatorname {sinc} ^{2}(2)+\operatorname {sinc} ^{2}(3)+\operatorname {sinc} ^{2}(4)+\cdots ={\frac {\pi -1}{2}}.}  

When the signs of the addends alternate and begin with +, the sum equals 1/2: $${\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}\,\operatorname {sinc} ($$n)=\operatorname {sinc} (1)-\operatorname {sinc} (2)+\operatorname {sinc} (3)-\operatorname {sinc} (4)+\cdots ={\frac {1}{2}}.}  

The alternating sums of the squares and cubes also equal 1/2:^[12] $${\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}\,\operatorname {sinc} ^{2}($$n)=\operatorname {sinc} ^{2}(1)-\operatorname {sinc} ^{2}(2)+\operatorname {sinc} ^{2}(3)-\operatorname {sinc} ^{2}(4)+\cdots ={\frac {1}{2}},}  

$${\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}\,\operatorname {sinc} ^{3}($$n)=\operatorname {sinc} ^{3}(1)-\operatorname {sinc} ^{3}(2)+\operatorname {sinc} ^{3}(3)-\operatorname {sinc} ^{3}(4)+\cdots ={\frac {1}{2}}.}  

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): $${\displaystyle {\frac {\sin x}{x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n+1)!}}=1-{\frac {x^{2}}{3!}}+{\frac {x^{4}}{5!}}-{\frac {x^{6}}{7!}}+\cdots }$$ 

The series converges for all x. The normalized version follows easily: $${\displaystyle {\frac {\sin \pi x}{\pi x}}=1-{\frac {\pi ^{2}x^{2}}{3!}}+{\frac {\pi ^{4}x^{4}}{5!}}-{\frac {\pi ^{6}x^{6}}{7!}}+\cdots }$$ 

Euler famously compared this series to the expansion of the infinite product form to solve the Basel problem.

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^[13] 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 $${\displaystyle \mathbf {u} _{1}={\begin{bmatrix}{\frac {1}{2}}\\{\frac {\sqrt {3}}{2}}\end{bmatrix}}\quad {\text{and}}\quad \mathbf {u} _{2}={\begin{bmatrix}{\frac {1}{2}}\\-{\frac {\sqrt {3}}{2}}\end{bmatrix}}.}$$ 

Denoting $${\displaystyle {\boldsymbol {\xi }}_{1}={\tfrac {2}{3}}\mathbf {u} _{1},\quad {\boldsymbol {\xi }}_{2}={\tfrac {2}{3}}\mathbf {u} _{2},\quad {\boldsymbol {\xi }}_{3}=-{\tfrac {2}{3}}(\mathbf {u} _{1}+\mathbf {u} _{2}),\quad \mathbf {x} ={\begin{bmatrix}x\\y\end{bmatrix}},}$$  one can derive^[13] the sinc function for this hexagonal lattice as $${\displaystyle {\begin{aligned}\operatorname {sinc} _{\text{H}}(\mathbf {x} )={\tfrac {1}{3}}{\big (}&\cos \left(\pi {\boldsymbol {\xi }}_{1}\cdot \mathbf {x} \right)\operatorname {sinc} \left({\boldsymbol {\xi }}_{2}\cdot \mathbf {x} \right)\operatorname {sinc} \left({\boldsymbol {\xi }}_{3}\cdot \mathbf {x} \right)\\&{}+\cos \left(\pi {\boldsymbol {\xi }}_{2}\cdot \mathbf {x} \right)\operatorname {sinc} \left({\boldsymbol {\xi }}_{3}\cdot \mathbf {x} \right)\operatorname {sinc} \left({\boldsymbol {\xi }}_{1}\cdot \mathbf {x} \right)\\&{}+\cos \left(\pi {\boldsymbol {\xi }}_{3}\cdot \mathbf {x} \right)\operatorname {sinc} \left({\boldsymbol {\xi }}_{1}\cdot \mathbf {x} \right)\operatorname {sinc} \left({\boldsymbol {\xi }}_{2}\cdot \mathbf {x} \right){\big )}.\end{aligned}}}$$ 

This construction can be used to design Lanczos window for general multidimensional lattices.^[13]

• Anti-aliasing filter – Mathematical transformation reducing the damage caused by aliasing
• Borwein integral – Type of mathematical integrals
• Dirichlet integral – Integral of sin(x)/x from 0 to infinity.
• Lanczos resampling – Application of a mathematical formula
• List of mathematical functions
• Shannon wavelet
• Sinc filter – Ideal low-pass filter or averaging filter
• Sinc numerical methods
• Trigonometric functions of matrices – Important functions in solving differential equations
• Trigonometric integral – Special function defined by an integral
• Whittaker–Shannon interpolation formula – Signal (re-)construction algorithm
• Winkel tripel projection – Pseudoazimuthal compromise map projection (cartography)

• Weisstein, Eric W. "Sinc Function". MathWorld.