In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, the sinc function, denoted by , has two forms, normalized and unnormalized.
[.]
In mathematics, the historical unnormalized sinc function is defined for by
Alternatively, the unnormalized sinc function is often called the
sampling function
In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula
\operatorname_(t) \ := \sum_^ \delta(t - k T)
for some given period T. Here ''t'' is a real variable and th ...
, indicated as Sa(''x'').
In
digital signal processing
Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
and
information theory
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...
, the normalized sinc function is commonly defined for by
In either case, the value at is defined to be the limiting value
for all real .
The
normalization causes the
definite integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
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
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
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
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or dema ...
(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
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic or analytical can also have the following meanings:
Chemistry
* ...
everywhere and hence an
entire function.
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", and his 1953 book ''Probability and Information Theory, with Applications to Radar''.
The function itself was first mathematically derived in this form by
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. A ...
in his expression (
Rayleigh's Formula
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
) for the zeroth-order spherical
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
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:
The first few terms of the infinite series for the coordinate of the -th extremum with positive coordinate are
where
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:
and is related to the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
through
Euler's reflection formula:
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
discovered that
and because of the product-to-sum identity
Euler's product can be recast as a sum
The
continuous Fourier transform of the normalized sinc (to ordinary frequency) is :
where the
rectangular function is 1 for argument between − and , and zero otherwise. This corresponds to the fact that the
sinc filter
In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given cutoff frequency, without affecting lower frequencies, and has linear phase response. The filter's impulse response is a sinc func ...
is the ideal (
brick-wall, meaning rectangular frequency response)
low-pass filter.
This Fourier integral, including the special case
is an
improper integral (see
Dirichlet integral
In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line:
: \int_0^ ...
) and not a convergent
Lebesgue integral, as
The normalized sinc function has properties that make it ideal in relationship to
interpolation of
sampled bandlimited
Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency.
A band-limited signal is one whose Fourier transform or spectral density has bounded support.
A bandli ...
functions:
* It is an interpolating function, i.e., , and for nonzero
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
.
* The functions ( integer) form an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
for
bandlimited
Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency.
A band-limited signal is one whose Fourier transform or spectral density has bounded support.
A bandli ...
functions in the
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
, 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
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
of the first kind, . The normalized sinc is .
* where is the
sine integral
In mathematics, trigonometric integrals are a family of integrals involving trigonometric functions.
Sine integral
The different sine integral definitions are
\operatorname(x) = \int_0^x\frac\,dt
\operatorname(x) = -\int_x^\infty\f ...
,
* (not normalized) is one of two linearly independent solutions to the linear
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
The other is , which is not bounded at , unlike its sinc function counterpart.
* Using normalized sinc,
*
*
*
* The following improper integral involves the (not normalized) sinc function:
Relationship to the Dirac delta distribution
The normalized sinc function can be used as a ''
nascent delta function'', meaning that the following
weak limit
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
holds:
This is not an ordinary limit, since the left side does not converge. Rather, it means that
for every
Schwartz function
In mathematics, Schwartz space \mathcal is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables o ...
, as can be seen from the
Fourier inversion theorem In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information ...
.
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 :
The sum of the squares also equals :
When the signs of the
addends alternate and begin with +, the sum equals :
The alternating sums of the squares and cubes also equal :
Series expansion
The
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of the unnormalized function can be obtained from that of the sine:
The series converges for all . The normalized version follows easily:
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
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
Multivariate may refer to:
In mathematics
* Multivariable calculus
* Multivariate function
* Multivariate polynomial
In computing
* Multivariate cryptography
* Multivariate division algorithm
* Multivariate interpolation
* Multivariate optical c ...
sinc function for the square Cartesian grid (
lattice): , whose
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
is the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
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
The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120� ...
) is a function whose
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
is the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
of the
Brillouin zone of that lattice. For example, the sinc function for the hexagonal lattice is a function whose
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
is the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
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
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A '' regular hexagon'' has ...
,
body-centered cubic,
face-centered cubic and other higher-dimensional lattices can be explicitly derived
using the geometric properties of Brillouin zones and their connection to
zonotopes.
For example, a
hexagonal lattice
The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120� ...
can be generated by the (integer)
linear span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
of the vectors
Denoting
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)
*
Sinhc function In mathematics, the sinhc function appears frequently in papers about optical scattering, Heisenberg spacetime and hyperbolic geometry. For z \neq 0, it is defined as
\operatorname(z)=\frac
The sinhc function is the hyperbolic analogue of the sinc ...
References
External links
* {{MathWorld, title=Sinc Function, urlname=SincFunction
Signal processing
Elementary special functions