Sinc function

The sinus cardinalis, also sinc function, cardinal sine or sinc function is a mathematical function. The name cardinal sine goes to Philip M. Woodward back from the year 1953. The nomenclature is not uniformly established in the literature, particularly in the English literature, the term is used for both the normalized sinc as well as for the non-standard variant. In the German literature, a distinction between the two definitions is made and the non- normalized version as:

• Si (x): Non - normalized sinus cardinalis
• Sinc ( x ) = si ( π · x): Normalized sinus cardinalis

Defined. In the information theory, and the digital signal processing, application areas of the sinc function, however, is usually the normalized form with the label application sinc:

The usual in the German language name si for the non-normalized cardinal sine is not the sine integral Si ( x), the antiderivative of the sinc function to be confused.

• 2.1 Signal processing
• 2.2 Diffraction at a slit
• 2.3 The distribution of primes and Nuclear Physics

Properties

General

At the liftable singularity at the functions to be continued by the limit or steady, resulting from the rule of L'Hospital; sometimes the defining equation is also written with case distinction.

Software packages such as Matlab mostly use the normalized sinc function, which can also be expressed as a product or by using the gamma function as:

The Taylor series of the sinc function can be directly derived from the sin function to:

Spherical Bessel function of the first kind is the same to the si - function:

Zeros

Maxima and minima

• Si (x) = sin ( x) / x
• Cos (x)

Xn the extremes of Si (x) with the positive x -coordinate, n ≥ 1, are in a good approximation with

Where for odd n, a minimum is attained and for even n, a maximum. For the first extreme with the positive x -coordinate - the minimum at x1 ≈ 4,49 - is the absolute error of the approximate value already much smaller than 1/100.

Besides these extremes, and the absolute maximum in the curve has 0, because of their symmetry with respect to the y- axis also at extremes - xn.

Fourier transform of the rectangular function

The sinc function is the Fourier transform of the square function

Because it is

From the properties of the Fourier transform follows that the sinc function is analytically and thus any number of times continuously differentiable. From the Plancherel identity of the Fourier transform follows further that it shifts its orthogonal even by integer multiples of, it is

Wherein the Kronecker delta designated.

With a proper normalization, this shifts the sinc function thus form an orthonormal system in function space. The projection on the sinc ( x - kπ ) subspace spanned results as

Due to the interpolation property is true, so

Functions of this subspace are thus uniquely determined by its values ​​at the points.

The rectangular function as a Fourier transform of the sinc function has bounded support, is therefore tied together with the limited linear combinations of their shifts. Conversely, each bandlimited such as linear combination may be represented, and therefore uniquely determined by the function values ​​to said reference points. This is the statement of the WKS sampling theorem.

Derivations

The n -th derivative of

Can be analytically for all x ≠ 0 to determine:

The first two derivatives formed from them are:

Surface

The total area under the integral is:

Application

Signal processing

The sinc function is of great importance in particular in the digital signal processing. It occurs in the so-called sampling number ( or cardinal number, ET Whittaker 1915) on by means of which a continuous band limited signal reconstructed from its samples x or any nodes sequence continues to a continuous signal:

This is the interpolation lowest variation, that is, the frequency spectrum is limited and has the smallest possible maximum (circular ) frequency or frequency. The consequence of these samples is a prerequisite of the band narrowness of the signal x no longer exists, so the output signal components of higher frequencies, it should be a coarse mesh, the high frequency components are converted into additional low-frequency components, that is, it occurs aliasing ( misallocation of frequency components ) on.

Diffraction at a slit

In the diffraction of waves at a gap amplitudes form a diffraction pattern that can be explained by the Fourier transform of a rectangular aperture function. Therefore, the cardinal sine will be also referred to as sinc function. However, in the diffraction of light perceived by the eye brightness distribution is the square of the wave amplitude; therefore it follows sinc2 the squared function.

Distribution of primes and Nuclear Physics

The function term describes the physics pair correlation distribution of energies of the eigenstates of heavy nuclei. In mathematics, it describes the associated with the distribution of prime numbers pair correlation of zeros of the Riemann zeta function. The commonality lies in the two underlying theory of random matrices, whereupon the first physicist Freeman Dyson in 1972 pointed out in an interview with the mathematician Hugh Montgomery.