Poisson's empirical formula

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 ${\displaystyle f\in {\mathcal {S}}(\mathbb {R} )}$

${\displaystyle {\hat {f}}(\omega )={\mathcal {F}}(f)(\omega )=\int _{-\infty }^{\infty }f(t)\,e^{-2\pi i\omega \cdot t}\,dt}$

the continuous Fourier transform of in . Then it says the Poisson's sum formula ${\displaystyle f}$ ${\displaystyle {\mathcal {S}}}$

${\displaystyle \sum _{n\in \mathbb {Z} }f(n)=\sum _{k\in \mathbb {Z} }{\hat {f}}(k).}$

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. ${\displaystyle f}$ ${\displaystyle (1+t^{2})\,(|f(t)|+|f''(t)|)}$

Using the elementary properties of the Fourier transformation, the more general formula with additional parameters results ${\displaystyle t,\nu \in \mathbb {R} }$

${\displaystyle {\begin{aligned}\sum _{n\in \mathbb {Z} }f(t+nT)e^{-2\pi i\nu nT}&=\sum _{k\in \mathbb {Z} }{\mathcal {F}}(f(t+\cdot T)e^{-2\pi i\nu \cdot T})(k)\\&={\frac {1}{T}}\sum _{k\in \mathbb {Z} }{\mathcal {F}}(f(t+\cdot )e^{-2\pi i\nu \cdot })\left({\frac {k}{T}}\right)\\&={\frac {1}{T}}\sum _{k\in \mathbb {Z} }{\mathcal {F}}(f(t+\cdot ))\left({\frac {k}{T}}+\nu \right)\\&={\frac {1}{T}}\sum _{k\in \mathbb {Z} }e^{2\pi i(k/T+\nu )t}{\mathcal {F}}(f)\left({\frac {k}{T}}+\nu \right).\end{aligned}}}$

If one sets in the more general form , ${\displaystyle t=0}$

${\displaystyle \sum _{n\in \mathbb {Z} }f(nT)e^{-2\pi i\nu nT}={\frac {1}{T}}\sum _{k\in \mathbb {Z} }{\mathcal {F}}(f)\left({\frac {k}{T}}+\nu \right),}$

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. ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle L^{2}(\mathbb {R} )}$

Formulation using a Dirac comb

The Dirac comb of the interval length is the distribution ${\displaystyle T\in \mathbb {R} }$

${\displaystyle {\text{III}}_{T}=\sum _{n\in \mathbb {Z} }\delta _{nT}.}$

The Fourier transform of a tempered distribution is defined by ${\displaystyle {\mathcal {F}}A\in {\mathcal {S}}'(\mathbb {R} )}$ ${\displaystyle A\in {\mathcal {S}}'(\mathbb {R} )}$

${\displaystyle \langle {\mathcal {F}}A,\,\phi \rangle =\langle A,\,{\mathcal {F}}\phi \rangle \quad (\phi \in {\mathcal {S}}(\mathbb {R} )),}$

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

${\displaystyle {\mathcal {F}}{\text{III}}_{T}={\frac {1}{T}}{\text{III}}_{1/T}}$

is. This can also be seen in the form

${\displaystyle {\text{III}}_{T}={\frac {1}{T}}\sum _{k\in \mathbb {Z} }e^{i(2\pi k/T)t}}$

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

${\displaystyle g(t):=\sum _{n\in \mathbb {Z} }f(t+n)}$

is continuous, bounded, differentiable and periodic with period 1. This can therefore be expanded into a point-wise convergent Fourier series ,

${\displaystyle g(t)=\sum _{k\in \mathbb {Z} }c_{k}\cdot e^{2\pi ikt}.}$

Their Fourier coefficients are determined according to the formula

${\displaystyle c_{k}=\int _{0}^{1}g(t)\cdot e^{-2\pi ikt}\,dt=\int _{0}^{1}\sum _{n\in \mathbb {Z} }f(t+n)\cdot e^{-2\pi ik(t+n)}\,dt.}$

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

${\displaystyle c_{k}=\sum _{n\in \mathbb {Z} }\int _{n}^{n+1}f(s)\cdot e^{-2\pi iks}\,ds=\int _{-\infty }^{\infty }f(s)\cdot e^{-2\pi iks}\,ds={\mathcal {F}}f(k).}$

In summary:

${\displaystyle \sum _{n\in \mathbb {Z} }f(t+n)=\sum _{k\in \mathbb {Z} }{\mathcal {F}}f(k)e^{2\pi ikt},}$

from which the claim arises. ${\displaystyle t=0}$

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 ${\displaystyle \operatorname {supp} {\hat {x}}\subset [-W,W]}$ ${\displaystyle |WT|\leq \pi ,}$ ${\displaystyle \omega :=-2\pi \nu \in [-W,W]}$

${\displaystyle {\sqrt {2\pi }}{\hat {x}}(\omega )e^{i\omega t}=T\sum _{n\in \mathbb {Z} }x(nT)e^{i\omega (t-nT)}.}$

After multiplication with the indicator function of the interval [-W, W] and subsequently the inverse Fourier transformation, this results

${\displaystyle x(t)={\frac {1}{\sqrt {2\pi }}}\int _{-W}^{W}{\hat {x}}(\omega )e^{i\omega t}\,d\omega =T\sum _{n\in \mathbb {Z} }x(nT){\frac {\sin(W(t-nT))}{\pi (t-nT)}}.}$

In the borderline case this is the reconstruction formula of the Nyquist-Shannon sampling theorem ${\displaystyle WT=\pi }$

${\displaystyle x(t)=\sum _{n\in \mathbb {Z} }x(nT)\operatorname {sinc} (t/T-n),}$

where the sinc function is with . ${\displaystyle \operatorname {sinc} }$ ${\displaystyle \operatorname {sinc} (t):={\tfrac {\sin(\pi t)}{\pi t}}}$

Applications in number theory

With the help of Poisson's sum formula one can show that the theta function

${\displaystyle \theta (t)=\sum _{n\in \mathbb {Z} }e^{-n^{2}\pi t}}$

the transformation formula

${\displaystyle \theta (t)={\frac {1}{\sqrt {t}}}\theta \left({\frac {1}{t}}\right)}$

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 .