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In mathematics, physics and engineering, the sinc function (Template:IPAc-en Template:Respell), denoted by Template:Math, has two forms, normalized and unnormalized.<ref name="dlmf">Template:Dlmf.</ref>
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In mathematics, the historical unnormalized sinc function is defined for Template:Math by <math display="block">\operatorname{sinc}(x) = \frac{\sin x}{x}.</math>
Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x).<ref>Template:Cite book Extract of page 15</ref>
In digital signal processing and information theory, the normalized sinc function is commonly defined for Template:Math by <math display="block">\operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x}.</math>
In either case, the value at Template:Math is defined to be the limiting value <math display="block">\operatorname{sinc}(0) := \lim_{x \to 0}\frac{\sin(a x)}{a x} = 1</math> for all real Template:Math (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.
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 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 function has also been called the cardinal sine or sine cardinal function.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite journal</ref> The term sinc was introduced by Philip M. Woodward in his 1952 article "Information theory and inverse probability in telecommunication", in which he said that the function "occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own",<ref>Template:Cite journal</ref> and his 1953 book Probability and Information Theory, with Applications to Radar.<ref name="Poynton">Template:Cite book</ref><ref>Template:Cite book</ref> 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.
PropertiesEdit
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:Math for all points Template:Mvar where the derivative of Template:Math is zero and thus a local extremum is reached. This follows from the derivative of the sinc function: <math display="block">\frac{d}{dx}\operatorname{sinc}(x) = \begin{cases} \dfrac{\cos(x) - \operatorname{sinc}(x)}{x}, & x \ne 0 \\0, & x = 0\end{cases}.</math>
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 Template:Citation needed <math display="block">x_n = q - q^{-1} - \frac{2}{3} q^{-3} - \frac{13}{15} q^{-5} - \frac{146}{105} q^{-7} - \cdots,</math> where <math display="block">q = \left(n + \frac{1}{2}\right) \pi,</math> 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 Template:Math. In addition, there is an absolute maximum at Template:Math.
The normalized sinc function has a simple representation as the infinite product: <math display="block">\frac{\sin(\pi x)}{\pi x} = \prod_{n=1}^\infty \left(1 - \frac{x^2}{n^2}\right)</math>
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and is related to the gamma function Template:Math through Euler's reflection formula: <math display="block">\frac{\sin(\pi x)}{\pi x} = \frac{1}{\Gamma(1 + x)\Gamma(1 - x)}.</math>
Euler discovered<ref>Template:Cite arXiv</ref> that <math display="block">\frac{\sin(x)}{x} = \prod_{n=1}^\infty \cos\left(\frac{x}{2^n}\right),</math> and because of the product-to-sum identity<ref>Template:Cite journal</ref>
<math display="block">\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 \ge 1,</math> Euler's product can be recast as a sum <math display="block">\frac{\sin(x)}{x} = \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N \cos\left(\frac{n - 1/2}{N} x\right).</math>
The continuous Fourier transform of the normalized sinc (to ordinary frequency) is Template:Math: <math display="block">\int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f),</math> 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 <math display="block">\int_{-\infty}^\infty \frac{\sin(\pi x)}{\pi x} \, dx = \operatorname{rect}(0) = 1</math> is an improper integral (see Dirichlet integral) and not a convergent Lebesgue integral, as <math display="block">\int_{-\infty}^\infty \left|\frac{\sin(\pi x)}{\pi x} \right| \,dx = +\infty.</math>
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., Template:Math, and Template:Math for nonzero integer Template:Math.
- The functions Template:Math (Template:Mvar integer) form an orthonormal basis for bandlimited functions in the function space Template:Math, with highest angular frequency Template:Math (that is, highest cycle frequency Template:Math).
Other properties of the two sinc functions include:
- The unnormalized sinc is the zeroth-order spherical Bessel function of the first kind, Template:Math. The normalized sinc is Template:Math.
- where Template:Math is the sine integral, <math display="block">\int_0^x \frac{\sin(\theta)}{\theta}\,d\theta = \operatorname{Si}(x).</math>
- Template:Math (not normalized) is one of two linearly independent solutions to the linear ordinary differential equation <math display="block">x \frac{d^2 y}{d x^2} + 2 \frac{d y}{d x} + \lambda^2 x y = 0.</math> The other is Template:Math, which is not bounded at Template:Math, unlike its sinc function counterpart.
- Using normalized sinc, <math display="block">\int_{-\infty}^\infty \frac{\sin^2(\theta)}{\theta^2}\,d\theta = \pi \quad \Rightarrow \quad \int_{-\infty}^\infty \operatorname{sinc}^2(x)\,dx = 1,</math>
- <math>\int_{-\infty}^\infty \frac{\sin(\theta)}{\theta}\,d\theta = \int_{-\infty}^\infty \left( \frac{\sin(\theta)}{\theta} \right)^2 \,d\theta = \pi.</math>
- <math>\int_{-\infty}^\infty \frac{\sin^3(\theta)}{\theta^3}\,d\theta = \frac{3\pi}{4}.</math>
- <math>\int_{-\infty}^\infty \frac{\sin^4(\theta)}{\theta^4}\,d\theta = \frac{2\pi}{3}.</math>
- The following improper integral involves the (not normalized) sinc function: <math display="block">\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})}.</math>
Relationship to the Dirac delta distributionEdit
The normalized sinc function can be used as a nascent delta function, meaning that the following weak limit holds:
<math display="block">\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).</math>
This is not an ordinary limit, since the left side does not converge. Rather, it means that
<math display="block">\lim_{a \to 0}\int_{-\infty}^\infty \frac{1}{a} \operatorname{sinc}\left(\frac{x}{a}\right) \varphi(x) \,dx = \varphi(0)</math>
for every Schwartz function, as can be seen from the Fourier inversion theorem. In the above expression, as Template:Math, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of Template:Math, regardless of the value of Template:Mvar.
This complicates the informal picture of Template:Math as being zero for all Template:Mvar except at the point Template:Math, 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 <math>\delta(x)</math> by writing <math> b=1/a </math> and
<math display="block">\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^{ik x}dk= \frac{1}{2\pi} \int_{-\infty}^\infty e^{i k x} dk=\delta(x),</math>
which makes clear the recovery of the delta as an infinite bandwidth limit of the integral.
SummationEdit
All sums in this section refer to the unnormalized sinc function.
The sum of Template:Math over integer Template:Mvar from 1 to Template:Math equals Template:Math:
<math display="block">\sum_{n=1}^\infty \operatorname{sinc}(n) = \operatorname{sinc}(1) + \operatorname{sinc}(2) + \operatorname{sinc}(3) + \operatorname{sinc}(4) +\cdots = \frac{\pi - 1}{2}.</math>
The sum of the squares also equals Template:Math:<ref>Template:Cite journal</ref><ref name="BBB">Template:Cite journal</ref>
<math display="block">\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}.</math>
When the signs of the addends alternate and begin with +, the sum equals Template:Sfrac: <math display="block">\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}.</math>
The alternating sums of the squares and cubes also equal Template:Sfrac:<ref name="FWFS">Template:Cite arXiv</ref> <math display="block">\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},</math>
<math display="block">\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}.</math>
Series expansionEdit
The Taylor series of the unnormalized Template:Math function can be obtained from that of the sine (which also yields its value of 1 at Template:Math): <math display="block">\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</math>
The series converges for all Template:Mvar. The normalized version follows easily: <math display="block">\frac{\sin \pi x}{\pi x} = 1 - \frac{\pi^2x^2}{3!} + \frac{\pi^4x^4}{5!} - \frac{\pi^6x^6}{7!} + \cdots</math>
Euler famously compared this series to the expansion of the infinite product form to solve the Basel problem.
Higher dimensionsEdit
The product of 1-D sinc functions readily provides a multivariate sinc function for the square Cartesian grid (lattice): Template:Math, 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<ref name="multiD">Template:Cite journal</ref> 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 <math display="block">
\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}.
</math>
Denoting <math display="block">
\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},
</math> one can derive<ref name="multiD" /> the sinc function for this hexagonal lattice as <math display="block">\begin{align}
\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{align}</math>
This construction can be used to design Lanczos window for general multidimensional lattices.<ref name="multiD" />
SinhcEdit
Some authors, by analogy, define the hyperbolic sine cardinal function.<ref>Template:Cite book</ref><ref>Template:Cite book</ref><ref>Template:Cite book</ref>
- <math>\mathrm{sinhc}(x) = \begin{cases}
{\displaystyle \frac{\sinh(x)}{x},} & \text{if }x \ne 0 \\
{\displaystyle 1,} & \text{if }x = 0
\end{cases}</math>
See alsoEdit
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ReferencesEdit
Further readingEdit
External linksEdit
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:SincFunction%7CSincFunction.html}} |title = Sinc Function |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}