Sinc function

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The sinc function is an entire function defined as sinc(0) = 0 and $\operatorname {sinc} (x)=\sin(x)/x$ for nonzero x. The function is continuous at x = 0 since $\scriptstyle \lim _{x\to 0}{\frac {\sin x}{x}}=1$.

The term "sinc" (IPA-en: ˈsɪŋk) is a contraction of the function's full Latin name, the sinus cardinalis (cardinal sine). First introduced by Phillip M. Woodward in 1953.

Normalized sinc function

In digital signal processing and information theory, the normalized sinc function $\operatorname {sinc} _{\pi }(x)=\operatorname {sinc} (\pi x)$ is common.

The only difference between sinc and the normalized sinc function is the scaling of the independent variable (the x-axis) by a factor of π. It is called normalized because the integral over all $\scriptstyle x\,$ is 1. All of the zeros of the normalized sinc function are at nonzero integer values of x. The Fourier transform of the normalized sinc function is the rectangular function with no scaling. This function is fundamental in the concept of reconstructing the original continuous bandlimited signal from uniformly spaced samples of that signal.

Reciprocal Gamma function

The domain can be extended to the complex plane via the reciprocal Gamma function $\scriptstyle f(z)\,:=\,{\frac {1}{\Gamma (z)}}\,$

$\operatorname {sinc} (z)=f(1+{\tfrac {z}{\pi }})\,f(1-{\tfrac {z}{\pi }}),\quad z\in \mathbb {C} .\,$

$\operatorname {sinc} _{\pi }(z)=f(1+z)\,f(1-z),\quad z\in \mathbb {C} .\,$

Properties

The sinc function crosses zero at nonzero multiples of π; zero crossings of the normalized sinc occur at nonzero integer values.

The local maxima and minima of sinc correspond to its intersections with the cosine function. That is, $\scriptstyle {\frac {\sin(\xi )}{\xi }}=\cos(\xi )$ for all points $\scriptstyle \xi$ where the derivative of $\scriptstyle {\frac {\sin(x)}{x}}$ is zero (and thus a local extremum is reached.)

Maclaurin series

${\frac {\sin \pi x}{\pi x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}\,\pi ^{2n}x^{2n}}{(2n+1)!}}=1-{\frac {\pi ^{2}x^{2}}{6}}+{\frac {\pi ^{4}x^{4}}{120}}-{\frac {\pi ^{6}x^{6}}{5040}}+{\frac {\pi ^{8}x^{8}}{362880}}-{\frac {\pi ^{10}x^{10}}{39916800}}+{\frac {\pi ^{12}x^{12}}{6227020800}}-{\frac {\pi ^{14}x^{14}}{1307674368000}}+{\frac {\pi ^{16}x^{16}}{355687428096000}}-\cdots \,$

${\frac {\sin x}{x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}\,x^{2n}}{(2n+1)!}}=1-{\frac {x^{2}}{6}}+{\frac {x^{4}}{120}}-{\frac {x^{6}}{5040}}+{\frac {x^{8}}{362880}}-{\frac {x^{10}}{39916800}}+{\frac {x^{12}}{6227020800}}-{\frac {x^{14}}{1307674368000}}+{\frac {x^{16}}{355687428096000}}-\cdots \,$

where the numerators $\scriptstyle (-1)^{n},\,n\,\geq \,0,\,$ give the sequence (Cf. A033999)

{1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ...}

and the denominators $\scriptstyle (2n+1)!,\,n\,\geq \,0,\,$ give the sequence (Cf. A009445)

{1, 6, 120, 5040, 362880, 39916800, 6227020800, 1307674368000, 355687428096000, 121645100408832000, 51090942171709440000, 25852016738884976640000, ...}

Product representations

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

${\frac {\sin \pi x}{\pi x}}=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}}}\right)\,$

and is related to the gamma function $\scriptstyle \Gamma (x)\,$ by Euler's reflection formula

${\frac {\sin \pi x}{\pi x}}={\frac {1}{\Gamma (1+x)\Gamma (1-x)}}.\,$

Euler discovered that

${\frac {\sin x}{x}}=\prod _{n=1}^{\infty }\cos \left({\frac {x}{2^{n}}}\right).\,$

Integral representations

The continuous Fourier transform of the normalized sinc (to cycle frequency) is the rectangular function $\scriptstyle \mathrm {rect} (f)\,$

$\int _{-\infty }^{\infty }e^{-i2\pi ft}\,\mathrm {sinc} _{\pi }\,t\,dt=\mathrm {rect} (f),\,$

where the rectangular function is 1 for argument between $\scriptstyle -{\frac {1}{2}}\,$ and $\scriptstyle {\frac {1}{2}}\,$, 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

$\int _{-\infty }^{\infty }\mathrm {sinc} _{\pi }\,x\,dx=\mathrm {rect} (0)=1,\,$

is an improper integral and not a convergent Lebesgue integral, as

$\int _{-\infty }^{\infty }\left|\mathrm {sinc} _{\pi }\,x\right|\,dx=+\infty .\,$

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. $\scriptstyle \mathrm {sinc} _{\pi }(0)\,=\,1\,$, and $\scriptstyle \mathrm {sinc} _{\pi }(k)\,=\,0\,$ for nonzero integer $\scriptstyle k\,$.

• The functions $\scriptstyle x_{k}(t)\,=\,\mathrm {sinc} _{\pi }(t-k),\,k\,\in \,\mathbb {Z} ,\,$ form an orthonormal basis for bandlimited functions in the function space $\scriptstyle L_{2}(\mathbb {R} )\,$, with highest angular frequency $\scriptstyle \omega _{H}\,=\,\pi \,$ (that is, highest cycle frequency $\scriptstyle f_{H}\,=\,{\frac {1}{2}}\,$.)

Other properties of the two sinc functions

Derivatives of the sinc functions

${\frac {d\,\mathrm {sinc} \,x}{dx}}={\frac {1}{x}}\,(\cos x-\mathrm {sinc} \,x)\,$

${\frac {d\,\mathrm {sinc} _{\pi }\,x}{dx}}={\frac {1}{x}}\,(\cos \pi \,x-\mathrm {sinc} \,\pi \,x)\,$

The unnormalized sinc is the zero th order spherical Bessel function of the first kind, $\scriptstyle j_{0}(x)\,$. The normalized sinc is $\scriptstyle j_{0}(\pi x)\,$.

Integrals of the two sinc functions

Indefinite integrals of the two sinc functions

$\int _{0}^{x}{\frac {\sin t}{t}}\,dt=\mathrm {Si} (x),\,$

$\int _{0}^{x}{\frac {\sin \pi t}{\pi t}}\,dt={\frac {\mathrm {Si} (\pi x)}{\pi }},\,$

where $\scriptstyle Si(x)\,$ is the sine integral.

Improper integrals of the two sinc functions

$\int _{-\infty }^{\infty }\mathrm {sinc} _{\pi }^{1}\,x\,dx=1,\quad \int _{-\infty }^{\infty }{\frac {\sin ^{1}x}{x^{1}}}\,dx=\pi .\,$

$\int _{-\infty }^{\infty }\mathrm {sinc} _{\pi }^{2}\,x\,dx=1,\quad \int _{-\infty }^{\infty }{\frac {\sin ^{2}x}{x^{2}}}\,dx=\pi .\,$

$\int _{-\infty }^{\infty }\mathrm {sinc} _{\pi }^{3}\,x\,dx={\frac {3}{4}},\quad \int _{-\infty }^{\infty }{\frac {\sin ^{3}x}{x^{3}}}\,dx={\frac {3\pi }{4}}.\,$

$\int _{-\infty }^{\infty }\mathrm {sinc} _{\pi }^{4}\,x\,dx={\frac {2}{3}},\quad \int _{-\infty }^{\infty }{\frac {\sin ^{4}x}{x^{4}}}\,dx={\frac {2\pi }{3}}.\,$

Ordinary differential equation for the sinc function

$\scriptstyle \lambda \,\mathrm {sinc} \,\lambda \,x\,$ (unnormalized sync function) is one of two linearly independent solutions to the linear ordinary differential equation

$x{\frac {d^{2}y}{dx^{2}}}+2{\frac {dy}{dx}}+{\lambda }^{2}xy=0.\,$

The other is $\scriptstyle \lambda \,{\frac {\cos \lambda \,x}{\lambda \,x}}={\frac {\cos \lambda \,x}{x}}\,$, which is not bounded at $\scriptstyle x\,=\,0\,$, unlike its sinc function counterpart.

Relationship to the Dirac delta distribution

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

$\lim _{a\rightarrow 0}{\frac {1}{a}}\,\mathrm {sinc} _{\pi }\left({\frac {x}{a}}\right)=\delta (x).\,$

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

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

for any smooth function $\scriptstyle \varphi (x)\,$ with compact support.

In the above expression, as $\scriptstyle a\,$ approaches zero, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of $\scriptstyle {\frac {\pm 1}{\pi ax}}\,$, and approaches zero for any nonzero value of $\scriptstyle x\,$. This complicates the informal picture of $\scriptstyle \delta (x)\,$ as being zero for all $\scriptstyle x\,$ except at the point $\scriptstyle x\,=\,0\,$ 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.

Generalized sinc functions

Since the sinc function is related to the Fourier transform of a quantity distributed uniformly over a finite interval (i.e. a 1-dimensional sphere,) the sinc function may be generalized by considering the Fourier transform of a quantity distributed uniformly over an $\scriptstyle n\,$-dimensional sphere.

Tanc function

By analogy with the sinc function, the tanc function is defined as

$\mathrm {tanc} (x):={\begin{cases}1&{\text{if }}x=0,\\{\frac {\tan x}{x}}&{\text{otherwise}},\end{cases}}\,$

where we have continuity at the removable singularity at $\scriptstyle x\,=\,0\,$, since $\scriptstyle \lim _{x\to 0}{\frac {\tan x}{x}}\,=\,1\,$.

See also

• 1 - ((sin pi x)/(pi x))^2
• 1 - ((sin x)/x)^2
• 1 - sinc^2 x

• {{sinc}} (mathematical function template)

External links

• Weisstein, Eric W., Sinc Function, from MathWorld—A Wolfram Web Resource.