The Poisson empirical formula is an aid to Fourier analysis and signal processing . Among other things, it is used to analyze the properties of scanning methods .
statement
Let be a Schwartz function and be
the continuous Fourier transform of in . Then it says the Poisson's sum formula
This identity also applies to certain more general classes of functions. Suitable prerequisites are, for example, that the function is twofold continuously differentiable and the expression is restricted.
Using the elementary properties of the Fourier transformation, the more general formula with additional parameters results
If one sets in the more general form ,
thus the Poisson's sum formula can also be read as the identity of a Fourier series with function values of as coefficients on the left and a periodization of the Fourier transform of on the right. This identity holds with the exception of a set of measure zero when is a band-constrained function, i.e. the Fourier transform is a measurable function in compact support.
Formulation using a Dirac comb
The Dirac comb of the interval length is the distribution
The Fourier transform of a tempered distribution is defined by
in analogy to the Plancherel identity . Since the Fourier transform is a continuous operator on the Schwartz space, this expression actually defines a tempered distribution.
The Dirac comb is a tempered distribution, and the Poisson empirical formula now says that
is. This can also be seen in the form
write. The exponential functions are to be understood as tempered distributions, and the series converges in the sense of distributions, i.e. in the weak - * - sense , to the Dirac comb. Note, however, that it does not converge anywhere in the usual sense.
For proof
Let f be sufficiently smooth and fall fast enough at infinity that the periodization
is continuous, bounded, differentiable and periodic with period 1. This can therefore be expanded into a point-wise convergent Fourier series ,
Their Fourier coefficients are determined according to the formula
It also follows from the rapid decrease in infinity that the sum can be exchanged for the integral. Therefore applies with s = t + n further
In summary:
from which the claim arises.
Application to band-limited functions
Let x be band-limited with the highest frequency W , that is . If then there is only one summand in the right side of the sum formula, with the substitutions , t = 0 and multiplication of a factor, one obtains
After multiplication with the indicator function of the interval [-W, W] and subsequently the inverse Fourier transformation, this results
In the borderline case this is the reconstruction formula of the Nyquist-Shannon sampling theorem
where the sinc function is with
.
Applications in number theory
With the help of Poisson's sum formula one can show that the theta function
the transformation formula
enough. This transformation formula was used by Bernhard Riemann in the proof of the functional equation of the Riemann zeta function .
literature
- Elias M. Stein, Guido Weiss: Introduction to Fourier Analysis on Euclidean Spaces . 1st edition. Princeton University Press, Princeton, NJ 1971, ISBN 978-0-691-08078-9 .
- JR Higgins: Five short stories about the cardinal series. In: Bulletin of the American Mathematical Society. 12, 1, 1985, ISSN 0002-9904 , pp. 45-89, online (PDF; 4.42 MB) .
- John J. Benedetto, Georg Zimmermann: Sampling multipliers and the Poisson summation formula. In: The journal of Fourier analysis and applications. 3, 5, 1997, ISSN 0002-9904 , pp. 505-523, online .