Sinc function
Encyclopedia
In mathematics
, the sinc function, denoted by sinc(x) has two nearly equivalent definitions. In digital signal processing
and information theory
, the normalized sinc function is commonly defined by
It is qualified as normalized
because its integral over all x is 1. All of the zeros of the normalized sinc function are integer values of x. The Fourier transform
of the normalized sinc function is the rectangular function with no scaling. This function is fundamental in the concept of reconstructing
the original continuous bandlimited signal from uniformly spaced samples
of that signal.
In mathematics, the historical unnormalized sinc function is defined by
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 analytic
everywhere.
The term "sinc" (ˈsɪŋk) is a contraction of the function's full Latin name, the sinus cardinalis (cardinal sine), first introduced by Phillip M. Woodward in 1953.
s of the unnormalized sinc are at nonzero multiples of π, while zero crossings of the normalized sinc occur at nonzero integer values.
The local maxima and minima of the unnormalized sinc correspond to its intersections with the cosine function. That is, sin(ξ)/ξ = cos(ξ) for all points ξ where the derivative of sin(x)/x is zero and thus a local extremum is reached.
A good approximation of the x-coordinate of the n-th extremum with positive x-coordinate is
where odd n lead to a local minimum and even n to a local maximum. Besides the extrema at xn, the curve has an absolute maximum at ξ0 = (0,1) and because of its symmetry to the y-axis extrema with x-coordinates −xn.
The normalized sinc function has a simple representation as the infinite product
and is related to the gamma function
by Euler's reflection formula:
Euler discovered that
The continuous Fourier transform
of the normalized sinc (to ordinary frequency) is rect(f),
where the rectangular function is 1 for argument between −1/2 and 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
is an improper integral
and not a convergent Lebesgue integral, as
The normalized sinc function has properties that make it ideal in relationship to interpolation
of sampled
bandlimited
functions:
Other properties of the two sinc functions include:
holds:
This is not an ordinary limit, since the left side does not converge. Rather, it means that
for any smooth function
with compact support.
In the above expression, as a approaches zero, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±1/(π a x), and approaches zero for any nonzero value of x. This complicates the informal picture of δ(x) as being zero for all x except at the point 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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the sinc function, denoted by sinc(x) has two nearly equivalent definitions. In digital signal processing
Digital signal processing
Digital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing...
and information theory
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
, the normalized sinc function is commonly defined by
It is qualified as normalized
Normalizing constant
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics.-Definition and examples:In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g.,...
because its integral over all x is 1. All of the zeros of the normalized sinc function are integer values of x. The Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of the normalized sinc function is the rectangular function with no scaling. This function is fundamental in the concept of reconstructing
Whittaker–Shannon interpolation formula
The Whittaker–Shannon interpolation formula or sinc interpolation is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples.-Definition:...
the original continuous bandlimited signal from uniformly spaced samples
Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...
of that signal.
In mathematics, the historical unnormalized sinc function is defined by
The only difference between the two definitions is in the scaling of the independent variable
Independent variable
The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...
(the x-axis) by a factor of π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
. 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 analytic
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
everywhere.
The term "sinc" (ˈsɪŋk) is a contraction of the function's full Latin name, the sinus cardinalis (cardinal sine), first introduced by Phillip M. Woodward in 1953.
Properties
The zero crossingZero crossing
Zero-crossing is a commonly used term in electronics, mathematics, and image processing. In mathematical terms, a "zero-crossing" is a point where the sign of a function changes Zero-crossing is a commonly used term in electronics, mathematics, and image processing. In mathematical terms, a...
s of the unnormalized sinc are at nonzero multiples of π, while zero crossings of the normalized sinc occur at nonzero integer values.
The local maxima and minima of the unnormalized sinc correspond to its intersections with the cosine function. That is, sin(ξ)/ξ = cos(ξ) for all points ξ where the derivative of sin(x)/x is zero and thus a local extremum is reached.
A good approximation of the x-coordinate of the n-th extremum with positive x-coordinate is
where odd n lead to a local minimum and even n to a local maximum. Besides the extrema at xn, the curve has an absolute maximum at ξ0 = (0,1) and because of its symmetry to the y-axis extrema with x-coordinates −xn.
The normalized sinc function has a simple representation as the infinite product
and is related to the gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
by Euler's reflection formula:Euler discovered that
The continuous Fourier transform
Continuous Fourier transform
The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum. For instance, the transform of a musical chord made up of pure notes is a mathematical representation of the amplitudes of the individual notes that make...
of the normalized sinc (to ordinary frequency) is rect(f),
where the rectangular function is 1 for argument between −1/2 and 1/2, and zero otherwise. This corresponds to the fact that the sinc filter
Sinc filter
In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given bandwidth, leaves the low frequencies alone, and has linear phase...
is the ideal (brick-wall, meaning rectangular frequency response) low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
. This Fourier integral, including the special case
is an improper integral
Improper integral
In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits....
and not a convergent Lebesgue integral, as
The normalized sinc function has properties that make it ideal in relationship to interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
of sampled
Sampling (signal processing)
In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave to a sequence of samples ....
bandlimited
Bandlimited
Bandlimiting is the limiting of a deterministic or stochastic signal's Fourier transform or power spectral density to zero above a certain finite frequency...
functions:
- It is an interpolating function, i.e., sinc(0) = 1, and sinc(k) = 0 for nonzero integer k.
- The functions xk(t) = sinc(t−k) (k integer) form an orthonormal basisOrthonormal basisIn mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
for bandlimitedBandlimitedBandlimiting is the limiting of a deterministic or stochastic signal's Fourier transform or power spectral density to zero above a certain finite frequency...
functions in the function spaceLp spaceIn mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
L2(R), with highest angular frequency ωH = π (that is, highest cycle frequency ƒH = 1/2).
Other properties of the two sinc functions include:
- The unnormalized sinc is the zeroth order spherical Bessel functionBessel functionIn mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
of the first kind,
. The normalized sinc is 
- where Si(x) is the sine integral.
- λ sinc(λ x) (not normalized) is one of two linearly independent solutions to the linear ordinary differential equationOrdinary differential equationIn mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
-
- The other is cos(λ x)/x, which is not bounded at x = 0, unlike its sinc function counterpart.
- where the normalized sinc is meant.
Relationship to the Dirac delta distribution
The normalized sinc function can be used as a nascent delta function, meaning that the following weak limitWeak topology
In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
holds:
This is not an ordinary limit, since the left side does not converge. Rather, it means that
for any smooth function
with compact support.In the above expression, as a approaches zero, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the expression always oscillates inside an envelope of ±1/(π a x), and approaches zero for any nonzero value of x. This complicates the informal picture of δ(x) as being zero for all x except at the point 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
Gibbs phenomenon
In mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity: the nth partial sum of the Fourier series has large...
.
See also
- Borwein integralBorwein integralIn mathematics, a Borwein integral is an integral studied by involving products of sinc, where the sinc function is given by sinc = sin/x. These integrals are notorious for exhibiting apparent patterns that eventually break down...
- Dirichlet integralDirichlet integralIn mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet.One of those isThis can be derived from attempts to evaluate a double improper integral two different ways...
- Anti-aliasingAnti-aliasingIn digital signal processing, spatial anti-aliasing is the technique of minimizing the distortion artifacts known as aliasing when representing a high-resolution image at a lower resolution...
- Sinc filterSinc filterIn signal processing, a sinc filter is an idealized filter that removes all frequency components above a given bandwidth, leaves the low frequencies alone, and has linear phase...
- Lanczos resamplingLanczos resamplingLanczos resampling is an interpolation method used to compute new values for sampled data. It is often used in multivariate interpolation, for example for image scaling , but can be used for any other digital signal...
- Whittaker–Shannon interpolation formulaWhittaker–Shannon interpolation formulaThe Whittaker–Shannon interpolation formula or sinc interpolation is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples.-Definition:...