Sinc function explained

In mathematics, physics and engineering, the sinc function, denoted by, has two forms, normalized and unnormalized.^[1]

Sinc
Imagealt:	Part of the normalized and unnormalized sinc function shown on the same scale
General Definition:	\operatorname{sinc}x = \begin{cases} \dfrac{ \sin x } x, & x \ne 0 \ 1 , & x = 0 \end{cases}
Motivation Of Creation:	Telecommunication
Date:	1952
Fields Of Application:	Signal processing, spectroscopy
Domain:	R
Range:	[ -0 . 217234 \ldots, 1 ]
Parity:	Even
Zero:	1
Plusinf:	0
Minusinf:	0
Max:	1 at x = 0
Min:	-0 . 21723 \ldots at x = \pm 4 . 49341 \ldots
Root:	\pi k, k \in Z _{≠}
Reciprocal:	\begin{cases} x \csc x, & x \ne 0 \ 1 , & x = 0 \end{cases}
Derivative:	\operatorname{sinc}'x = \begin{cases} \dfrac{\cos x - \operatorname{sinc} x}{x}, & x \ne 0 \ 0 , & x = 0 \end{cases}
Antiderivative:	\int \operatorname{sinc} xdx = \operatorname{Si}(x) + C
Taylor Series:	\operatorname{sinc}x = \sum ^{inf ty} _{k =0} (( -1 ) ^{k} x ^{2k})/(( 2 k +1 )!)

In mathematics, the historical unnormalized sinc function is defined for by$$\backslash operatornamex\; =\; \backslash frac.$$

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

In digital signal processing and information theory, the normalized sinc function is commonly defined for by$$\backslash operatornamex\; =\; \backslash frac.$$

In either case, the value at is defined to be the limiting value$$\backslash operatorname0\; :=\; \backslash lim\_\backslash frac\; =\; 1$$ for all real (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 ). As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of .

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 axis) by a factor of . 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.^{[3] [4]} 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",^[5] and his 1953 book Probability and Information Theory, with Applications to Radar.^{[6] [7]} 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.

Properties

The zero crossings of the unnormalized sinc are at non-zero integer multiples of, 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, for all points where the derivative of is zero and thus a local extremum is reached. This follows from the derivative of the sinc function:$$\backslash frac\backslash operatorname(x)\; =\; \backslash frac.$$

The first few terms of the infinite series for the coordinate of the -th extremum with positive coordinate are$$x\_n\; =\; q\; -\; q^\; -\; \backslash frac\; q^\; -\; \backslash frac\; q^\; -\; \backslash frac\; q^\; -\; \backslash cdots,$$where$$q\; =\; \backslash left(n\; +\; \backslash frac\backslash right)\; \backslash pi,$$and where odd lead to a local minimum, and even to a local maximum. Because of symmetry around the axis, there exist extrema with coordinates . In addition, there is an absolute maximum at .

The normalized sinc function has a simple representation as the infinite product:$$\backslash frac\; =\; \backslash prod\_^\backslash infty\; \backslash left(1\; -\; \backslash frac\backslash right)$$and is related to the gamma function through Euler's reflection formula:$$\backslash frac\; =\; \backslash frac.$$

Euler discovered^[8] that$$\backslash frac\; =\; \backslash prod\_^\backslash infty\; \backslash cos\backslash left(\backslash frac\backslash right),$$and because of the product-to-sum identity^[9] $$\backslash prod\_^k\; \backslash cos\backslash left(\backslash frac\backslash right)\; =\; \backslash frac\; \backslash sum\_^\; \backslash cos\backslash left(\backslash frac\; x\; \backslash right),\backslash quad\; \backslash forall\; k\; \backslash ge\; 1,$$Euler's product can be recast as a sum$$\backslash frac\; =\; \backslash lim\_\; \backslash frac\; \backslash sum\_^N\; \backslash cos\backslash left(\backslash frac\; x\backslash right).$$

The continuous Fourier transform of the normalized sinc (to ordinary frequency) is :$$\backslash int\_^\backslash infty\; \backslash operatorname(t)\; \backslash ,\; e^\backslash ,dt\; =\; \backslash operatorname(f),$$where the rectangular function is 1 for argument between − and, 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$$\backslash int\_^\backslash infty\; \backslash frac\; \backslash ,\; dx\; =\; \backslash operatorname(0)\; =\; 1$$is an improper integral (see Dirichlet integral) and not a convergent Lebesgue integral, as$$\backslash int\_^\backslash infty\; \backslash left|\backslash frac\; \backslash right|\; \backslash ,dx\; =\; +\backslash infty.$$

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.,, and for nonzero integer .
• The functions (integer) form an orthonormal basis for bandlimited functions in the function space, with highest angular frequency (that is, highest cycle frequency).

Other properties of the two sinc functions include:

• The unnormalized sinc is the zeroth-order spherical Bessel function of the first kind, . The normalized sinc is .
• where is the sine integral, $$\backslash int\_0^x\; \backslash frac\backslash ,d\backslash theta\; =\; \backslash operatorname(x).$$
• (not normalized) is one of two linearly independent solutions to the linear ordinary differential equation $$x\; \backslash frac\; +\; 2\; \backslash frac\; +\; \backslash lambda^2\; x\; y\; =\; 0.$$ The other is, which is not bounded at, unlike its sinc function counterpart.
• Using normalized sinc, $$\backslash int\_^\backslash infty\; \backslash frac\backslash ,d\backslash theta\; =\; \backslash pi\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash int\_^\backslash infty\; \backslash operatorname^2(x)\backslash ,dx\; =\; 1,$$

\int ^{inf ty} _{-inf ty} (\sin(\theta))/(\theta) d\theta = \int ^{inf ty} _{-inf ty} \left( (\sin(\theta))/(\theta) \right) ^{2} d\theta = \pi. \int ^{inf ty} _{-inf ty} (\sin ^{3(\theta)})/(\theta ^{3}) d\theta = (3 \pi)/(4) . \int ^{inf ty} _{-inf ty} (\sin ^{4(\theta)})/(\theta ^{4}) d\theta = (2 \pi)/(3) .

• The following improper integral involves the (not normalized) sinc function: $$\backslash int\_0^\backslash infty\; \backslash frac\; =\; 1\; +\; 2\backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash frac.$$

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:

$$\backslash lim\_\; \backslash frac\; =\; \backslash lim\_\backslash frac\; \backslash operatorname\backslash left(\backslash frac\backslash right)\; =\; \backslash delta(x).$$

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

$$\backslash lim\_\backslash int\_^\backslash infty\; \backslash frac\; \backslash operatorname\backslash left(\backslash frac\backslash right)\; \backslash varphi(x)\; \backslash ,dx\; =\; \backslash varphi(0)$$

for every Schwartz function, as can be seen from the Fourier inversion theorem.In the above expression, as, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of, regardless of the value of .

This complicates the informal picture of as being zero for all except at the point, 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.

Summation

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

The sum of over integer from 1 to equals :

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

The sum of the squares also equals :^{[10] [11]}

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

When the signs of the addends alternate and begin with +, the sum equals :$$\backslash sum\_^\backslash infty\; (-1)^\backslash ,\backslash operatorname(n)\; =\; \backslash operatorname(1)\; -\; \backslash operatorname(2)\; +\; \backslash operatorname(3)\; -\; \backslash operatorname(4)\; +\; \backslash cdots\; =\; \backslash frac.$$

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

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

Series expansion

The Taylor series of the unnormalized function can be obtained from that of the sine (which also yields its value of 1 at):$$\backslash frac\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; 1\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots$$

The series converges for all . The normalized version follows easily:$$\backslash frac\; =\; 1\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots$$

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):, 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^[13] 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$$\backslash mathbf\_1\; =\; \backslash begin\; \backslash frac\; \backslash \backslash \; \backslash frac\; \backslash end\; \backslash quad\; \backslash text\; \backslash quad\; \backslash mathbf\_2\; =\; \backslash begin\; \backslash frac\; \backslash \backslash \; -\backslash frac\; \backslash end.$$

Denoting$$\backslash boldsymbol\_1\; =\; \backslash tfrac\; \backslash mathbf\_1,\; \backslash quad\; \backslash boldsymbol\_2\; =\; \backslash tfrac\; \backslash mathbf\_2,\; \backslash quad\; \backslash boldsymbol\_3\; =\; -\backslash tfrac\; (\backslash mathbf\_1\; +\; \backslash mathbf\_2),\; \backslash quad\; \backslash mathbf\; =\; \backslash begin\; x\; \backslash \backslash \; y\backslash end,$$one can derive the sinc function for this hexagonal lattice as

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

See also

• • • • • • • • • • • • (cartography)

Notes and References

1. .
2. Book: Communication Systems, 2E . illustrated . R. P. . Singh . S. D. . Sapre . Tata McGraw-Hill Education . 2008 . 978-0-07-063454-1 . 15 . Extract of page 15
3. Web site: Weisstein . Eric W. . Sinc Function . 2023-06-07 . mathworld.wolfram.com . en.
4. Merca . Mircea . 2016-03-01 . The cardinal sine function and the Chebyshev–Stirling numbers . Journal of Number Theory . en . 160 . 19–31 . 10.1016/j.jnt.2015.08.018 . 124388262 . 0022-314X.
5. Woodward . P. M. . Davies . I. L. . Information theory and inverse probability in telecommunication . Proceedings of the IEE - Part III: Radio and Communication Engineering . 99 . 58 . 37–44 . March 1952 . 10.1049/pi-3.1952.0011.
6. Book: Poynton, Charles A. . Digital video and HDTV . limited . 147 . Morgan Kaufmann Publishers . 2003 . 978-1-55860-792-7.
7. Book: Woodward, Phillip M. . Probability and information theory, with applications to radar . limited . 29 . London . Pergamon Press . 1953 . 488749777 . 978-0-89006-103-9.
8. Euler . Leonhard . On the sums of series of reciprocals . 1735 . math/0506415.
9. Luis Ortiz-Gracia . Cornelis W. Oosterlee . A highly efficient Shannon wavelet inverse Fourier technique for pricing European options . 2016 . SIAM J. Sci. Comput. . 38 . 1 . B118–B143 . 10.1137/15M1014164 . 2016SJSC...38B.118O . 2072/377498 . free .
10. Advanced Problem 6241 . American Mathematical Monthly . June–July 1980 . 87 . 6 . 496–498 . . Washington, DC . 10.1080/00029890.1980.11995075.
11. Robert Baillie . David Borwein . David Borwein . Jonathan M. Borwein . Jonathan M. Borwein . Surprising Sinc Sums and Integrals . American Mathematical Monthly . December 2008 . 115 . 10 . 888–901 . 27642636 . 10.1080/00029890.2008.11920606 . 1959.13/940062 . 496934 . free.
12. Baillie . Robert . 0806.0150v2 . math.CA . Fun with Fourier series . 2008.
13. Ye . W. . Entezari . A. . A Geometric Construction of Multivariate Sinc Functions . IEEE Transactions on Image Processing . 21 . 6 . 2969–2979 . June 2012 . 10.1109/TIP.2011.2162421 . 21775264 . 2012ITIP...21.2969Y. 15313688 .