Sinc function


In mathematics, physics and engineering, the sinc function, denoted by, is defined as either
or
the latter of which is sometimes referred to as the normalized sinc function. The only difference between the two definitions is in the scaling of the independent variable 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 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 sinc filter is used in signal processing.
The function itself was first mathematically derived in this form by Lord Rayleigh in his expression for the zeroth-order spherical Bessel function of the first kind.
The sinc function is also called the cardinal sine function.

Definitions

The sinc function 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, indicated as Sa.
In digital signal processing and information theory, 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 of the function over the real numbers to equal 1. As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of.

Etymology

The function has also been called the cardinal sine or sine cardinal function. The term "sinc" is a contraction of the function's full Latin name, the sinus cardinalis and was introduced by Philip M. Woodward and I.L Davies in their 1952 article "Information theory and inverse probability in telecommunication", saying "This function occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own". It is also used in Woodward's 1953 book Probability and Information Theory, with Applications to Radar.

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 through Euler's reflection formula:
Euler 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 is :
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 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:
  • It is an interpolating function, i.e.,, and for nonzero integer.
  • The functions form an orthonormal basis for bandlimited functions in the function space, with highest angular frequency .
Other properties of the two sinc functions include:
The normalized sinc function can be used as a nascent delta function, meaning that the following weak limit holds:
This is not an ordinary limit, since the left side does not converge. Rather, it means that
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.
We can also make an immediate connection with the standard Dirac representation of by writing and
which makes clear the recovery of the delta as an infinite bandwidth limit of the integral.

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 of the unnormalized function can be obtained from that of the sine :
The series converges for all. The normalized version follows easily:
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 :, whose Fourier transform is the indicator function of a square in the frequency space. The sinc function for a non-Cartesian 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 using the geometric properties of Brillouin zones and their connection to zonotopes.
For example, a hexagonal lattice can be generated by the linear span 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.

Sinhc

Some authors, by analogy, define the hyperbolic sine cardinal function.