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Sinc function

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Special mathematical function defined as sin(x)/x

"Sinc" redirects here; not to be confused withSite of Importance for Nature Conservation.

Sinc
Part of the normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale
General information
General definition	$\operatorname {sinc} x={\begin{cases}{\dfrac {\sin x}{x}},&x\neq 0\\1,&x=0\end{cases}}$
Fields of application	Signal processing, spectroscopy
Domain, codomain and image
Domain	$\mathbb {R}$
Image	$[-0.217234\ldots ,1]$
Basic features
Parity	Even
Specific values
At zero	1
Value at +∞	0
Value at −∞	0
Maxima	1 at $x=0$
Minima	$-0.21723\ldots$ at $x=\pm 4.49341\ldots$
Specific features
Root	$\pi k,k\in \mathbb {Z} _{\neq 0}$
Related functions
Reciprocal	${\begin{cases}x\csc x,&x\neq 0\\1,&x=0\end{cases}}$
Derivative	$\operatorname {sinc} 'x={\begin{cases}{\dfrac {\cos x-\operatorname {sinc} x}{x}},&x\neq 0\\0,&x=0\end{cases}}$
Antiderivative	$\int \operatorname {sinc} x\,dx=\operatorname {Si} (x)+C$
Series definition
Taylor series	$\operatorname {sinc} x=\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k+1)!}}$

Inmathematics,physics andengineering, thesinc function (/ˈsɪŋk/SINK), denoted bysinc(x), is defined as either $\operatorname {sinc} (x)={\frac {\sin x}{x}}.$ or $\operatorname {sinc} (x)={\frac {\sin \pi x}{\pi x}},$

the latter of which is sometimes referred to as thenormalized sinc function. The only difference between the two definitions is in the scaling of theindependent variable (thex axis) by a factor ofπ. In both cases, the value of the function at theremovable singularity at zero is understood to be the limit value 1. The sinc function is thenanalytic everywhere and hence anentire function.

The normalized sinc function is theFourier transform of therectangular function with no scaling. It is used in the concept ofreconstructing a continuous bandlimited signal from uniformly spacedsamples of that signal. Thesinc filter is used in signal processing.

The function itself was first mathematically derived in this form byLord Rayleigh in his expression (Rayleigh's formula) for the zeroth-order sphericalBessel function of the first kind.

Thesinc function is also called thecardinal sine function.

Definitions

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

The sinc function has two forms, normalized and unnormalized.[1]

In mathematics, the historicalunnormalized sinc function is defined forx ≠ 0 by $\operatorname {sinc} (x)={\frac {\sin x}{x}}.$

Alternatively, the unnormalized sinc function is often called thesampling function, indicated as Sa(x).^[2]

Indigital signal processing andinformation theory, thenormalized sinc function is commonly defined forx ≠ 0 by $\operatorname {sinc} (x)={\frac {\sin(\pi x)}{\pi x}}.$

In either case, the value atx = 0 is defined to be the limiting value $\operatorname {sinc} (0):=\lim _{x\to 0}{\frac {\sin(ax)}{ax}}=1$ for all reala ≠ 0 (the limit can be proven using thesqueeze theorem).

Thenormalization causes thedefinite integral of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value ofπ). As a further useful property, the zeros of the normalized sinc function are the nonzero integer values ofx.

Etymology

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The function has also been called thecardinal sine orsine cardinal function.^[3]^[4] The term "sinc" is a contraction of the function's full Latin name, thesinus cardinalis^[5] and was introduced byPhilip M. Woodward and I.L Davies in their 1952 article "Information theory andinverse 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".^[6] It is also used in Woodward's 1953 bookProbability and Information Theory, with Applications to Radar.^[5]^[7]

Properties

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

Thezero crossings of the unnormalized sinc are at non-zero integer multiples ofπ, 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 thecosine function. That is,⁠sin(ξ)/ξ⁠ = cos(ξ) for all pointsξ where the derivative of⁠sin(x)/x⁠ is zero and thus a local extremum is reached. This follows from the derivative of the sinc function: ${\frac {d}{dx}}\operatorname {sinc} (x)={\begin{cases}{\dfrac {\cos(x)-\operatorname {sinc} (x)}{x}},&x\neq 0\\0,&x=0\end{cases}}.$

The first few terms of the infinite series for thex coordinate of then-th extremum with positivex coordinate are^{[citation needed]} $x_{n}=q-q^{-1}-{\frac {2}{3}}q^{-3}-{\frac {13}{15}}q^{-5}-{\frac {146}{105}}q^{-7}-\cdots ,$ where $q=\left(n+{\frac {1}{2}}\right)\pi ,$ and where oddn lead to a local minimum, and evenn to a local maximum. Because of symmetry around they axis, there exist extrema withx coordinates−x_n. In addition, there is an absolute maximum atξ₀ = (0, 1).

The normalized sinc function has a simple representation as theinfinite product: ${\frac {\sin(\pi x)}{\pi x}}=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}}}\right)$

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

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

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

Thecontinuous Fourier transform of the normalized sinc (to ordinary frequency) isrect(f): $\int _{-\infty }^{\infty }\operatorname {sinc} (t)\,e^{-i2\pi ft}\,dt=\operatorname {rect} (f),$ where therectangular function is 1 for argument between −⁠1/2⁠ and⁠1/2⁠, and zero otherwise. This corresponds to the fact that thesinc filter is the ideal (brick-wall, meaning rectangularfrequency response)low-pass filter.

This Fourier integral, including the special case $\int _{-\infty }^{\infty }{\frac {\sin(\pi x)}{\pi x}}\,dx=\operatorname {rect} (0)=1$ is animproper integral (seeDirichlet integral) and not a convergentLebesgue integral, as $\int _{-\infty }^{\infty }\left|{\frac {\sin(\pi x)}{\pi x}}\right|\,dx=+\infty .$

The normalized sinc function has properties that make it ideal in relationship tointerpolation ofsampled bandlimited functions:

It is an interpolating function, i.e.,sinc(0) = 1, andsinc(k) = 0 for nonzerointegerk.
The functionsx_k(t) = sinc(t −k) (k integer) form anorthonormal basis forbandlimited functions in thefunction spaceL²(R), with highest angular frequencyω_H = π (that is, highest cycle frequencyf_H =⁠1/2⁠).

Other properties of the two sinc functions include:

The unnormalized sinc is the zeroth-order sphericalBessel function of the first kind,j₀(x). The normalized sinc isj₀(πx).
whereSi(x) is thesine integral, $\int _{0}^{x}{\frac {\sin(\theta )}{\theta }}\,d\theta =\operatorname {Si} (x).$
λ sinc(λx) (not normalized) is one of two linearly independent solutions to the linearordinary differential equation $x{\frac {d^{2}y}{dx^{2}}}+2{\frac {dy}{dx}}+\lambda ^{2}xy=0.$ The other is⁠cos(λx)/x⁠, which is not bounded atx = 0, unlike its sinc function counterpart.
Using normalized sinc, $\int _{-\infty }^{\infty }{\frac {\sin ^{2}(\theta )}{\theta ^{2}}}\,d\theta =\pi \quad \Rightarrow \quad \int _{-\infty }^{\infty }\operatorname {sinc} ^{2}(x)\,dx=1,$
$\int _{-\infty }^{\infty }{\frac {\sin(\theta )}{\theta }}\,d\theta =\int _{-\infty }^{\infty }\left({\frac {\sin(\theta )}{\theta }}\right)^{2}\,d\theta =\pi .$
$\int _{-\infty }^{\infty }{\frac {\sin ^{3}(\theta )}{\theta ^{3}}}\,d\theta ={\frac {3\pi }{4}}.$
$\int _{-\infty }^{\infty }{\frac {\sin ^{4}(\theta )}{\theta ^{4}}}\,d\theta ={\frac {2\pi }{3}}.$
The following improper integral involves the (not normalized) sinc function: $\int _{0}^{\infty }{\frac {dx}{x^{n}+1}}=1+2\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{(kn)^{2}-1}}={\frac {1}{\operatorname {sinc} ({\frac {\pi }{n}})}}.$

Relationship to the Dirac delta distribution

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The normalized sinc function can be used as anascent delta function, meaning that the followingweak limit holds:

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

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

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

for everySchwartz function, as can be seen from theFourier inversion theorem.In the above expression, asa → 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 ofa.

This complicates the informal picture ofδ(x) as being zero for allx except at the pointx = 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 theGibbs phenomenon.

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

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

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

Summation

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

The sum ofsinc(n) over integern from 1 to∞ equals⁠π − 1/2⁠:

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

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

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

When the signs of theaddends alternate and begin with +, the sum equals⁠1/2⁠: $\sum _{n=1}^{\infty }(-1)^{n+1}\,\operatorname {sinc} (n)=\operatorname {sinc} (1)-\operatorname {sinc} (2)+\operatorname {sinc} (3)-\operatorname {sinc} (4)+\cdots ={\frac {1}{2}}.$

The alternating sums of the squares and cubes also equal⁠1/2⁠:^[12] $\sum _{n=1}^{\infty }(-1)^{n+1}\,\operatorname {sinc} ^{2}(n)=\operatorname {sinc} ^{2}(1)-\operatorname {sinc} ^{2}(2)+\operatorname {sinc} ^{2}(3)-\operatorname {sinc} ^{2}(4)+\cdots ={\frac {1}{2}},$

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

Series expansion

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TheTaylor series of the unnormalizedsinc function can be obtained from that of the sine (which also yields its value of 1 atx = 0): ${\frac {\sin x}{x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n+1)!}}=1-{\frac {x^{2}}{3!}}+{\frac {x^{4}}{5!}}-{\frac {x^{6}}{7!}}+\cdots$

The series converges for allx. 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 theBasel problem.

Higher dimensions

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The product of 1-D sinc functions readily provides amultivariate sinc function for the square Cartesian grid (lattice):sinc_C(x,y) = sinc(x) sinc(y), whoseFourier transform is theindicator function of a square in the frequency space (i.e., the brick wall defined in 2-D space). The sinc function for a non-Cartesianlattice (e.g.,hexagonal lattice) is a function whoseFourier transform is theindicator function of theBrillouin zone of that lattice. For example, the sinc function for the hexagonal lattice is a function whoseFourier transform is theindicator function of the unit hexagon in the frequency space. For a non-Cartesian lattice this function can not be obtained by a simpletensor product. However, the explicit formula for the sinc function for thehexagonal,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 tozonotopes.