"Sinc" redirects here. For the designation used in the United Kingdom for areas of wildlife interest, see Site of Importance for Nature Conservation. For the signal processing filter based on this function, see Sinc filter.
In mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized.[1]
Sinc
Part of the normalized sinc (blue) and unnormalized sinc function (red) shown on the same scale
General information
General definition
Motivation of invention
Telecommunication
Date of solution
1952
Fields of application
Signal processing, spectroscopy
Domain, codomain and image
Domain
Image
Basic features
Parity
Even
Specific values
At zero
1
Value at +∞
0
Value at −∞
0
Maxima
1 at
Minima
at
Specific features
Root
Related functions
Reciprocal
Derivative
Antiderivative
Series definition
Taylor series
In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by
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 x ≠ 0 by
In either case, the value at x = 0 is defined to be the limiting value
for all real a ≠ 0 (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 π). As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of x.
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 x axis) by a factor of π. 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/ˈsɪŋk/ 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.
^Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), "Numerical methods", NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248..
^Singh, R. P.; Sapre, S. D. (2008). Communication Systems, 2E (illustrated ed.). Tata McGraw-Hill Education. p. 15. ISBN 978-0-07-063454-1. Extract of page 15
^Weisstein, Eric W. "Sinc Function". mathworld.wolfram.com. Retrieved 2023-06-07.
^Merca, Mircea (2016-03-01). "The cardinal sine function and the Chebyshev–Stirling numbers". Journal of Number Theory. 160: 19–31. doi:10.1016/j.jnt.2015.08.018. ISSN 0022-314X. S2CID 124388262.
^Woodward, P. M.; Davies, I. L. (March 1952). "Information theory and inverse probability in telecommunication" (PDF). Proceedings of the IEE – Part III: Radio and Communication Engineering. 99 (58): 37–44. doi:10.1049/pi-3.1952.0011.
^Poynton, Charles A. (2003). Digital video and HDTV. Morgan Kaufmann Publishers. p. 147. ISBN 978-1-55860-792-7.
^Woodward, Phillip M. (1953). Probability and information theory, with applications to radar. London: Pergamon Press. p. 29. ISBN 978-0-89006-103-9. OCLC 488749777.
engineering, the sincfunction, denoted by sinc(x), has two forms, normalized and unnormalized. In mathematics, the historical unnormalized sincfunction is defined...
In signal processing, a sinc filter can refer to either a sinc-in-time filter whose impulse response is a sincfunction and whose frequency response is...
rectangular wave. The rect function has been introduced by Woodward in as an ideal cutout operator, together with the sincfunction as an ideal interpolation...
time domain, the cause of this type of ringing is the ripples in the sincfunction, which is the impulse response (time domain representation) of a perfect...
{\pi z}{\sin(\pi z)}}={\frac {1}{\operatorname {sinc} (z)}}} where sinc is the normalized sincfunction, while the multiplication theorem takes on the...
copy of the Lanczos kernel, which is a sincfunction windowed by the central lobe of a second, longer, sincfunction. The sum of these translated and scaled...
}{N}}\right)\right)} If we introduce the normalized sincfunctionsinc ( x ) = sin ( π x ) π x {\textstyle \operatorname {sinc} (x)={\frac {\sin(\pi x)}{\pi x}}} for...
product of two sincfunctions vs. an Airy function, respectively. Conventions: w 0 ( x ) {\displaystyle w_{0}(x)} is a zero-phase function (symmetrical...
zeroth spherical Bessel function j0(x) is also known as the (unnormalized) sincfunction. The first few spherical Bessel functions are: j 0 ( x ) = sin ...
wavelet. First, we define the scaling function to be the sincfunction. ϕ (Sha) ( t ) := sin π t π t = sinc ( t ) . {\displaystyle \phi ^{\text{(Sha)}}(t):={\frac...
origin. Other approximations to the identity of this kind include the sincfunction (used widely in electronics and telecommunications) η ε ( x ) = 1 π...
generalization to vertex angles Proofs of trigonometric identities Sincfunction Sine and cosine transforms Sine integral Sine quadrant Sine wave Sine–Gordon...
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the resulting function is regular in a neighbourhood of that point. For instance, the (unnormalized) sincfunction, as defined by sinc ( z ) = sin z...
Paul. "Minimization and maximization refresher". "The Unnormalized SincFunction Archived 2017-02-15 at the Wayback Machine", University of Sydney For...
Gustav Lejeune Dirichlet, one of which is the improper integral of the sincfunction over the positive real line: ∫ 0 ∞ sin x x d x = π 2 . {\displaystyle...
if and only if x − 5 = 0. {\displaystyle x-5=0.} "The Unnormalized SincFunction Archived 2017-02-15 at the Wayback Machine", University of Sydney Rockafellar...
Bessel–Clifford function Kelvin functions Legendre function: From the theory of spherical harmonics. Scorer's functionSincfunction Hermite polynomials...
x ≠ − 2. {\displaystyle x\neq -2.} Since the function sine is continuous on all reals, the sincfunction G ( x ) = sin ( x ) / x , {\displaystyle G(x)=\sin(x)/x...
{\frac {\sin(t)}{t}}} is the sincfunction, and also the zeroth spherical Bessel function. Since sinc is an even entire function (holomorphic over the entire...
Wx}{\lambda z}}}\\&=aW\operatorname {sinc} {\frac {\pi Wx}{\lambda z}}\end{aligned}}} where sinc(p) = sin(p)/p. The sincfunction is sometimes defined as sin(πp)/πp...
A sombrero function (sometimes called besinc function or jinc function) is the 2-dimensional polar coordinate analog of the sincfunction, and is so-called...
x(t)} can be written using the Sincfunction, using the definition sinc x = sin π x π x {\displaystyle \operatorname {sinc} x={\frac {\sin \pi x}{\pi...