 The sinc function sinc(x) from x = −8π to 8π. In mathematics, the sinc function (for sinus cardinalis), also known as the interpolation function, filtering function or the first spherical Bessel function j0(x), is the product of a sine function and a monotonically decreasing function. It is defined by: Image File history File links SincFunction. ...
Euclid, detail from The School of Athens by Raphael. ...
In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number α (the order). ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...
 The sinc function is sometimes defined as simply sin(x)/x, which has a removable singularity at zero since In complex analysis, a removable singularity of a function is a point at which the function is not defined (a singularity) but at which the function can be defined without creating any problems. ...
 The above definition for the sinc function is preferred since it removes this singularity and yields a function which is analytic everywhere.
The normalized sinc function is defined as:
 and, as its name suggests, is normalized to unity The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. ...
 This integral must necessarily be regarded as an improper integral; it cannot be taken to be a Lebesgue integral because It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ...
In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ...
 The normalized sinc function also has the important infinite product  We also have an expression in terms of the gamma function, as The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...
 Because of its usefulness,  is commonly referred to as simply "the sinc function" and written  .
The sinc function oscillates inside an envelope of ±1/x. The Fourier transform of the sinc function can be expressed in terms of the rectangular function: Oscillation is the periodic variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ...
The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...
The rectangular function (also known as the rectangle function or the normalized boxcar function) is defined as or in terms of the Heaviside step function The rectangular function is normalized: The Fourier transform of the rectangular function is where sinc is the sinc function. ...
 In the language of distributions, the sinc function is related to the delta function δ(x) by In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...
The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ...
 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 mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...
In the above expression, as a approaches zero, the number of oscillations per unit length of the sinc function approaches infinity. Nevertheless, the sinc function always oscillates inside an envelope of ±1/x, regardless of the value of a. This contradicts 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. Approximation of square wave in 5 steps Approximation of square wave in 25 steps Approximation of square wave in 125 steps In mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, (also known as ringing artifacts) is the peculiar manner in which the Fourier series of a...
An interesting property of the sinc function is that its local maxima correspond to its intersections with the cosine curve. That is, if the sinc function has a local maximum at x then
 Applications of the sinc function are found in digital signal processing, communication theory, control theory, and optics. Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ...
Communication is a slippery concept, and while we may casually use the word with some frequency, it is difficult to arrive at a precise definition that is agreeable to most of those who consider themselves communication scholars. ...
For the application to living systems, see perceptual control theory. ...
See also: List of optical topics Optics (appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ...
See also
In digital signal processing, anti-aliasing is the technique of minimizing aliasing (jagged or blocky patterns) when representing a high-resolution signal at a lower resolution. ...
In signal processing, the sinc filter strips high-frequency data from a signal. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
In calculus, lHôpitals rule (alternately, lHospitals rule) uses derivatives to help compute limits with indeterminate forms. ...
The Nyquist-Shannon interpolation formula or Cardinal series dates back to works of E. Borel in 1898 and was cited from works of J. M. Whittaker in 1935 in the formulation of the Nyquist-Shannon sampling theorem by C. E. Shannon in 1949. ...
External links - Eric W. Weisstein, Sinc Function at MathWorld.
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