Sinc function

In mathematics, physics and engineering, the sinc function (Template:IPAc-en Template:Respell), denoted by sinc(x), is defined as either $$\mathrm{sinc}(x)=\frac{\sin x}{x}.$$ or $$\mathrm{sinc}(x)=\frac{\sin \pi x}{\pi x},$$

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 [[Cartesian coordinate system|Template:Mvar axis]]) by a factor of Template:Pi. 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

File:Sinc.wav

The sinc function as audio, at 2000 Hz (±1.5 seconds around zero)

The sinc function has two forms, normalized and unnormalized.

In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by $$\mathrm{sinc}(x)=\frac{\sin x}{x}.$$

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 $$\mathrm{sinc}(x)=\frac{\sin (\pi x)}{\pi x}.$$

In either case, the value at x = 0 is defined to be the limiting value $$\mathrm{sinc}(0):=\underset{x\to 0}{\lim }\frac{\sin (ax)}{ax}=1$$ 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 [[pi|Template:Pi]]). As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of Template:Mvar.

Etymology

The function has also been called the cardinal sine or sine cardinal function. The term "sinc" is a contraction of the function's full Latin name, the 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

File:Si cos.svg

The local maxima and minima (small white dots) of the unnormalized, red sinc function correspond to its intersections with the blue cosine function.

The zero crossings of the unnormalized sinc are at non-zero integer multiples of Template:Pi, 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, Template:Sfrac = cos(ξ) for all points Template:Mvar where the derivative of Template:Sfrac is zero and thus a local extremum is reached. This follows from the derivative of the sinc function: $$\frac{d}{dx}\mathrm{sinc}(x)=\{\begin{array}{ll}\frac{\cos (x)-\mathrm{sinc}(x)}{x},& x\ne 0\\ 0,& x=0\end{array}.$$

The first few terms of the infinite series for the Template:Mvar coordinate of the Template:Mvar-th extremum with positive Template:Mvar coordinate are $$x_n=q-q^{-1}-\frac{2}{3}{q}^{-3}-\frac{13}{15}{q}^{-5}-\frac{146}{105}{q}^{-7}-\cdots ,$$ where $$q=(n+\frac{1}{2})\pi ,$$ and where odd Template:Mvar lead to a local minimum, and even Template:Mvar to a local maximum. Because of symmetry around the Template:Mvar axis, there exist extrema with Template:Mvar 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: $$\frac{\sin (\pi x)}{\pi x}=\prod _{n=1}^{\mathrm{\infty }}(1-\frac{x^2}{n^2})$$

The cardinal sine function sinc(z) plotted in the complex plane from -2-2i to 2+2i

The cardinal sine function sinc(z) plotted in the complex plane from -2-2i to 2+2i

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

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

File:Sinc cplot.svg

Domain coloring plot of sinc z = Template:Sfrac

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

The continuous Fourier transform of the normalized sinc (to ordinary frequency) is rect(f): $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\mathrm{sinc}(t)e^{-i2\pi ft}dt=\mathrm{rect}(f),$$ where the rectangular function is 1 for argument between −Template:Sfrac and Template:Sfrac, 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 $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\sin (\pi x)}{\pi x}dx=\mathrm{rect}(0)=1$$ is an improper integral (see Dirichlet integral) and not a convergent Lebesgue integral, as $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\left|\frac{\sin (\pi x)}{\pi x}\right|dx=+\mathrm{\infty }.$$

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) (Template:Mvar 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 = Template:Sfrac).

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, $$\underset{0}{\overset{x}{\int }}\frac{\sin (\theta )}{\theta }d\theta =\mathrm{Si}(x).$$
• λ sinc(λx) (not normalized) is one of two linearly independent solutions to the linear ordinary differential equation $$x\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+{\lambda }^{2}xy=0.$$ The other is Template:Sfrac, which is not bounded at x = 0, unlike its sinc function counterpart.
• Using normalized sinc, $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{{\sin }^2(\theta )}{{\theta }^2}d\theta =\pi \Rightarrow \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\mathrm{sinc}}^2(x)dx=1,$$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\sin (\theta )}{\theta }d\theta =\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\left(\frac{\sin (\theta )}{\theta }\right)}^{2}d\theta =\pi .$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{{\sin }^3(\theta )}{{\theta }^3}d\theta =\frac{3\pi }{4}.$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{{\sin }^4(\theta )}{{\theta }^4}d\theta =\frac{2\pi }{3}.$
• The following improper integral involves the (not normalized) sinc function: $$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{dx}{x^n+1}=1+2\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}}{(kn{)}^2-1}=\frac{1}{\mathrm{sinc}(\frac{\pi }{n})}.$$

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:

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

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

$$\underset{a\to 0}{\lim }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{a}\mathrm{sinc}\left(\frac{x}{a}\right)\phi (x)dx=\phi (0)$$

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 ±Template:Sfrac, regardless of the value of Template:Mvar.

This complicates the informal picture of δ(x) as being zero for all Template:Mvar 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.

We can also make an immediate connection with the standard Dirac representation of $\delta (x)$ by writing $b=1/a$ and

$$\underset{b\to \mathrm{\infty }}{\lim }\frac{\sin \left(b\pi x\right)}{\pi x}=\underset{b\to \mathrm{\infty }}{\lim }\frac{1}{2\pi }\underset{-b\pi }{\overset{b\pi }{\int }}{e}^{ikx}dk=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{ikx}dk=\delta (x),$$

which makes clear the recovery of the delta as an infinite bandwidth limit of the integral.

Summation

All sums in this section refer to the unnormalized sinc function.

The sum of sinc(n) over integer Template:Mvar from 1 to ∞ equals Template:Sfrac:

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

The sum of the squares also equals Template:Sfrac:

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

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

The alternating sums of the squares and cubes also equal Template:Sfrac: $$\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n+1}{\mathrm{sinc}}^2(n)={\mathrm{sinc}}^2(1)-{\mathrm{sinc}}^2(2)+{\mathrm{sinc}}^2(3)-{\mathrm{sinc}}^2(4)+\cdots =\frac{1}{2},$$

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

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

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

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

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 $${\mathbf{𝐮}}_1=\left[\begin{array}{c}\frac{1}{2}\\ \frac{\sqrt{3}}{2}\end{array}\right]\text{and}{\mathbf{𝐮}}_2=\left[\begin{array}{c}\frac{1}{2}\\ -\frac{\sqrt{3}}{2}\end{array}\right].$$

Denoting $${\mathit{\xi }}_1=\frac{2}{3}{\mathbf{𝐮}}_1,{\mathit{\xi }}_2=\frac{2}{3}{\mathbf{𝐮}}_2,{\mathit{\xi }}_3=-\frac{2}{3}({\mathbf{𝐮}}_1+{\mathbf{𝐮}}_2),\mathbf{𝐱}=\left[\begin{array}{c}x\\ y\end{array}\right],$$ one can derive the sinc function for this hexagonal lattice as $$\begin{array}{rl}{\mathrm{sinc}}_{\text{H}}(\mathbf{𝐱})=\frac{1}{3}(& \cos (\pi {\mathit{\xi }}_1\cdot \mathbf{𝐱})\mathrm{sinc}({\mathit{\xi }}_2\cdot \mathbf{𝐱})\mathrm{sinc}({\mathit{\xi }}_3\cdot \mathbf{𝐱})\\ & +\cos (\pi {\mathit{\xi }}_2\cdot \mathbf{𝐱})\mathrm{sinc}({\mathit{\xi }}_3\cdot \mathbf{𝐱})\mathrm{sinc}({\mathit{\xi }}_1\cdot \mathbf{𝐱})\\ & +\cos (\pi {\mathit{\xi }}_3\cdot \mathbf{𝐱})\mathrm{sinc}({\mathit{\xi }}_1\cdot \mathbf{𝐱})\mathrm{sinc}({\mathit{\xi }}_2\cdot \mathbf{𝐱})).\end{array}$$

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

Sinhc

Some authors, by analogy, define the hyperbolic sine cardinal function.

$\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\mathrm{c}(x)=\{\begin{array}{ll}\frac{\sinh (x)}{x},& \text{if }x\ne 0\\ 1,& \text{if }x=0\end{array}$

See also

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