sinc function

File:Si cos.svg.]]

The zero crossings of the unnormalized sinc are at non-zero integer multiples of {{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, {{math|1={{sfrac|sin(ξ)|ξ}} = cos(ξ)}} for all points {{mvar|ξ}} where the derivative of {{math|{{sfrac|sin(x)|x}}}} is zero and thus a local extremum is reached. This follows from the derivative of the sinc function:

The first few terms of the infinite series for the {{mvar|x}} coordinate of the {{mvar|n}}-th extremum with positive {{mvar|x}} coordinate are {{Citation needed|date=January 2025}}

$$x\_n\; =\; q\; -\; q^\{-1\}\; -\; \backslash frac\{2\}\{3\}\; q^\{-3\}\; -\; \backslash frac\{13\}\{15\}\; q^\{-5\}\; -\; \backslash frac\{146\}\{105\}\; q^\{-7\}\; -\; \backslash cdots,$$

where

$$q\; =\; \backslash left(n\; +\; \backslash frac\{1\}\{2\}\backslash right)\; \backslash pi,$$

and where odd {{mvar|n}} lead to a local minimum, and even {{mvar|n}} to a local maximum. Because of symmetry around the {{mvar|y}} axis, there exist extrema with {{mvar|x}} coordinates {{math|−x_n}}. In addition, there is an absolute maximum at {{math|1=ξ₀ = (0, 1)}}.

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

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

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

and is related to the gamma function {{math|Γ(x)}} through Euler's reflection formula:

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

Euler discovered{{cite arXiv |last=Euler |first=Leonhard |title=On the sums of series of reciprocals |year=1735 |eprint=math/0506415}} that

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

and because of the product-to-sum identity{{cite journal |author1=Sanjar M. Abrarov |author2=Brendan M. Quine |title=Sampling by incomplete cosine expansion of the sinc function: Application to the Voigt/complex error function |year=2015 |journal=Appl. Math. Comput. |volume=258 |issue= |pages=425–435 |doi=10.1016/j.amc.2015.01.072 |arxiv=1407.0533 |bibcode=|url=https://www.sciencedirect.com/science/article/pii/S0096300315001046 |hdl-access= }}

File:Sinc cplot.svg plot of {{math|1=sinc z = {{sfrac|sin z|z}}}}]]

$$\backslash prod\_\{n=1\}^k\; \backslash cos\backslash left(\backslash frac\{x\}\{2^n\}\backslash right)\; =\; \backslash frac\{1\}\{2^\{k-1\}\}\; \backslash sum\_\{n=1\}^\{2^\{k-1\}\}\; \backslash cos\backslash left(\backslash frac\{n\; -\; 1/2\}\{2^\{k-1\}\}\; x\; \backslash right),\backslash quad\; \backslash forall\; k\; \backslash ge\; 1,$$

Euler's product can be recast as a sum

$$\backslash frac\{\backslash sin(x)\}\{x\}\; =\; \backslash lim\_\{N\backslash to\backslash infty\}\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{n=1\}^N\; \backslash cos\backslash left(\backslash frac\{n\; -\; 1/2\}\{N\}\; x\backslash right).$$

The continuous Fourier transform of the normalized sinc (to ordinary frequency) is {{math|rect(f)}}:

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash operatorname\{sinc\}(t)\; \backslash ,\; e^\{-i\; 2\; \backslash pi\; f\; t\}\backslash ,dt\; =\; \backslash operatorname\{rect\}(f),$$

where the rectangular function is 1 for argument between −{{sfrac|1|2}} and {{sfrac|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

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash frac\{\backslash sin(\backslash pi\; x)\}\{\backslash pi\; x\}\; \backslash ,\; dx\; =\; \backslash operatorname\{rect\}(0)\; =\; 1$$

is an improper integral (see Dirichlet integral) and not a convergent Lebesgue integral, as

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash left|\backslash frac\{\backslash sin(\backslash pi\; x)\}\{\backslash pi\; x\}\; \backslash right|\; \backslash ,dx\; =\; +\backslash infty.$$

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., {{math|1=sinc(0) = 1}}, and {{math|1=sinc(k) = 0}} for nonzero integer {{math|k}}.
• The functions {{math|1=x_k(t) = sinc(t − k)}} ({{mvar|k}} integer) form an orthonormal basis for bandlimited functions in the function space {{math|L²(R)}}, with highest angular frequency {{math|1=ω_H = π}} (that is, highest cycle frequency {{math|1=f_H = {{sfrac|1|2}}}}).

Other properties of the two sinc functions include:

• The unnormalized sinc is the zeroth-order spherical Bessel function of the first kind, {{math|j₀(x)}}. The normalized sinc is {{math|j₀(πx)}}.
• where {{math|Si(x)}} is the sine integral, $$\backslash int\_0^x\; \backslash frac\{\backslash sin(\backslash theta)\}\{\backslash theta\}\backslash ,d\backslash theta\; =\; \backslash operatorname\{Si\}(x).$$
• {{math|λ sinc(λx)}} (not normalized) is one of two linearly independent solutions to the linear ordinary differential equation $$x\; \backslash frac\{d^2\; y\}\{d\; x^2\}\; +\; 2\; \backslash frac\{d\; y\}\{d\; x\}\; +\; \backslash lambda^2\; x\; y\; =\; 0.$$ The other is {{math|{{sfrac|cos(λx)|x}}}}, which is not bounded at {{math|1=x = 0}}, unlike its sinc function counterpart.
• Using normalized sinc, $$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash frac\{\backslash sin^2(\backslash theta)\}\{\backslash theta^2\}\backslash ,d\backslash theta\; =\; \backslash pi\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash operatorname\{sinc\}^2(x)\backslash ,dx\; =\; 1,$$
• $\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash frac\{\backslash sin(\backslash theta)\}\{\backslash theta\}\backslash ,d\backslash theta\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash left(\; \backslash frac\{\backslash sin(\backslash theta)\}\{\backslash theta\}\; \backslash right)^2\; \backslash ,d\backslash theta\; =\; \backslash pi.$
• $\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash frac\{\backslash sin^3(\backslash theta)\}\{\backslash theta^3\}\backslash ,d\backslash theta\; =\; \backslash frac\{3\backslash pi\}\{4\}.$
• $\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash frac\{\backslash sin^4(\backslash theta)\}\{\backslash theta^4\}\backslash ,d\backslash theta\; =\; \backslash frac\{2\backslash pi\}\{3\}.$
• The following improper integral involves the (not normalized) sinc function: $$\backslash int\_0^\backslash infty\; \backslash frac\{dx\}\{x^n\; +\; 1\}\; =\; 1\; +\; 2\backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{(-1)^\{k+1\}\}\{(kn)^2\; -\; 1\}\; =\; \backslash frac\{1\}\{\backslash operatorname\{sinc\}(\backslash frac\{\backslash pi\}\{n\})\}.$$

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

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

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

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

for every Schwartz function, as can be seen from the Fourier inversion theorem.

In the above expression, as {{math|a → 0}}, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of {{math|±{{sfrac|1|πx}}}}, regardless of the value of {{mvar|a}}.

This complicates the informal picture of {{math|δ(x)}} as being zero for all {{mvar|x}} except at the point {{math|1=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 $\backslash delta(x)$ by writing $b=1/a$ and

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

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

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

The sum of {{math|sinc(n)}} over integer {{mvar|n}} from 1 to {{math|∞}} equals {{math|{{sfrac|{{pi}} − 1|2}}}}:

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

The sum of the squares also equals {{math|{{sfrac|{{pi}} − 1|2}}}}:{{cite journal | title = Advanced Problem 6241 | journal = American Mathematical Monthly | date = June–July 1980 | volume = 87 | issue = 6 | pages = 496–498 | publisher = Mathematical Association of America | location = Washington, DC | doi = 10.1080/00029890.1980.11995075}}{{cite journal | author1=Robert Baillie | author2-link=David Borwein | author2=David Borwein | author3=Jonathan M. Borwein | author3-link=Jonathan M. Borwein | title=Surprising Sinc Sums and Integrals | journal=American Mathematical Monthly | date=December 2008 | volume=115 | issue=10 | pages=888–901 | jstor = 27642636 | doi=10.1080/00029890.2008.11920606 | hdl=1959.13/940062 | s2cid=496934 | hdl-access=free}}

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

When the signs of the addends alternate and begin with +, the sum equals {{sfrac|1|2}}:

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

The alternating sums of the squares and cubes also equal {{sfrac|1|2}}:{{cite arXiv |last=Baillie |first=Robert |eprint=0806.0150v2 |class=math.CA |title=Fun with Fourier series |date=2008}}

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

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

The Taylor series of the unnormalized {{math|sinc}} function can be obtained from that of the sine (which also yields its value of 1 at {{math|1=x = 0}}):

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

The series converges for all {{mvar|x}}. The normalized version follows easily:

$$\backslash frac\{\backslash sin\; \backslash pi\; x\}\{\backslash pi\; x\}\; =\; 1\; -\; \backslash frac\{\backslash pi^2x^2\}\{3!\}\; +\; \backslash frac\{\backslash pi^4x^4\}\{5!\}\; -\; \backslash frac\{\backslash pi^6x^6\}\{7!\}\; +\; \backslash cdots$$

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): {{math|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{{cite journal |last1=Ye |first1= W. |last2=Entezari |first2= A. |title=A Geometric Construction of Multivariate Sinc Functions |journal=IEEE Transactions on Image Processing |volume=21 |issue=6 |pages=2969–2979 |date=June 2012 |doi=10.1109/TIP.2011.2162421 |pmid=21775264 |bibcode=2012ITIP...21.2969Y|s2cid= 15313688 }} 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 = \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

\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 the sinc function for this hexagonal lattice as

$$\backslash begin\{align\}$$

\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}

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

Some authors, by analogy, define the hyperbolic sine cardinal function.{{cite book |last=Ainslie |first=Michael |date=2010 |title=Principles of Sonar Performance Modelling |publisher=Springer |isbn=9783540876625 |page=636 |url=https://books.google.com/books?id=EqDnP-lAw40C&pg=PA636}}{{cite book |last=Günter |first=Peter |date=2012 |title=Nonlinear Optical Effects and Materials |publisher=Springer |isbn=9783540497134 |page=258 |url=https://books.google.com/books?id=8QTpCAAAQBAJ&pg=PA258}}{{cite book |last=Schächter |first=Levi |date=2013 |title=Beam-Wave Interaction in Periodic and Quasi-Periodic Structures |publisher=Springer |isbn=9783662033982 |page=241 |url=https://books.google.com/books?id=jQb9CAAAQBAJ&pg=PA241}}

:$\backslash mathrm\{sinhc\}(x)\; =\; \backslash begin\{cases\}$

{\displaystyle \frac{\sinh(x)}{x},} & \text{if }x \ne 0 \\

{\displaystyle 1,} & \text{if }x = 0

\end{cases}

• {{annotated link|Anti-aliasing filter}}
• {{annotated link|Borwein integral}}
• {{annotated link|Dirichlet integral}}
• {{annotated link|Lanczos resampling}}
• {{annotated link|List of mathematical functions}}
• {{annotated link|Shannon wavelet}}
• {{annotated link|Sinc filter}}
• {{annotated link|Sinc numerical methods}}
• {{annotated link|Trigonometric functions of matrices}}
• {{annotated link|Trigonometric integral}}
• {{annotated link|Whittaker–Shannon interpolation formula}}
• {{annotated link|Winkel tripel projection}} (cartography)

{{Reflist|30em}}

• {{cite book |last=Stenger |first=Frank |date=1993 |title=Numerical Methods Based on Sinc and Analytic Functions |publisher=Springer-Verlag New York, Inc. |series=Springer Series on Computational Mathematics|volume=20|doi=10.1007/978-1-4612-2706-9|isbn=9781461276371}}

• {{MathWorld|title=Sinc Function|urlname=SincFunction}}

Category:Signal processing

Category:Elementary special functions