Poisson’s sum formula – Wikipedia

The Poisson sum formula is a tool of Fourier analysis and signal processing. Among other things, it serves to analyze the properties of sampling methods.

May be

${displaystyle fin {mathcal {S}}(mathbb {R} )}$

A Schwartz function and be

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

The continuous Fourier transformation of

${displaystyle f}$

in

${displaystyle {mathcal {S}}}$

.
Then the Poisson’s sum formula says

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$mm slaves Euxt or embbɔb m Refine ) mmmm mm mm hm hmb)$

This identity also applies to certain general classes of functions. Suitable requirements are, for example, that the function

${displaystyle f}$

dilapidated in two ways and the expression

${displaystyle (1+t^{2}),(|f(t)|+|f”(t)|)}$

is limited.

Using the elementary properties of the Fourier transformation, this results in the more general formula with additional parameters

${displaystyle t,nu in mathbb {R} }$

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

You set in the more general form

${displaystyle t=0}$

,

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

In this way, the Poisson sum formula can also be the identity of a Fourier series with functional values ​​of

${displaystyle f}$

as coefficients on the left and a periodization of the Fourier transformed of

${displaystyle f}$

can be read on the right. This identity applies with the exception of a quantity of zero if

${displaystyle f}$

A band-limited function is, i.e. the Fourier transformed a measurable function in

${displaystyle L^{2}(mathbb {R} )}$

with a compact carrier.

The Dirac ridge to the interval length

${displaystyle Tin mathbb {R} }$

Is the distribution

${Displaystyle {text {ш}} _ {t} = sum _ {nin mathbb {z}} delta _ {nt}.}$

The Fourier transformed

${displaystyle {mathcal {F}}Ain {mathcal {S}}'(mathbb {R} )}$

a tempered distribution

${displaystyle Ain {mathcal {S}}'(mathbb {R} )}$

is defined by

${DISPLYSTYLE LONGLE {MATCAL {F} A ,. PHI RANGLE = PHICAL {F} Phi-{S}}))),})$

In analogy to the Plancerel identity.
Since the Fourier transformation is a constant operator on the Schwartz room, this expression actually defines a tempered distribution.

The Dirac comb is a tempered distribution, and the Poisson sum formula now says that

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

is. This can also be in the form

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

write. The exponential functions are to be understood as tempered distributions, and the series converged in the sense of distributions, i.e. in the weak-*sense, against the Dirac comb. However, it is noted that it does not converge anywhere in the ordinary sense.

May be f enough smooth and sufficiently quickly falling in the infinite, so that the periodization

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

is constant, limited, differentiable and periodically with period 1. So this can be developed into a fourier series convergent,

$mms supree Peleek)) yumb) mumbbe m m hubeh hyói hyom hort hyom hyo hyag hyo hyh hyh hor.$

Their Fourier coefficients determine according to the formula

$m sovet Slele State State Skk kí mofe syo hok m hép héé Like mɔ kóm mmb) malm mupm km km km 15- 22-2 (Deut. Ares) skletttttttttttate looking at 2alk 2ek) Repucious.$

Also from the quick waste in the infinite follows that the sum with the integral
can be exchanged. Therefore with s=t+n further

$mm slavetle State State State—empum—emmum—Pe m hk hy hym hork 2 kank kmome, supm ) mmber , lame kmalm mmmmm 22-5-5 Wifetyfee () yoky male -2ym is halm mép hjoy mjoy.$

In summary

$mm slaves Euxt or embɔb m Refine) mm) mmmm mm hé h Aske mupe) mupe ) malm mmb)$

What is from

${displaystyle t=0}$

the claim results.

May be x Ribbon limited with the highest frequency IN , That means

${displaystyle operatorname {supp} {hat {x}}subset [-W,W]}$

. Is then

${displaystyle |WT|leq pi ,}$

So only one summand occurs in the right side of the sum formula, with the replacement

${Displastyle Omega: =-2pi nu in [-W, in]}$

, t=0 and multiplication of a factor you get

${displaystyle {sqrt {2pi }}{hat {x}}(omega )e^{iomega t}=Tsum _{nin mathbb {Z} }x(nT)e^{iomega (t-nT)}.}$

After multiplication with the indicator function of the interval [-W,W] And below the inverse Fourier transformation results

${displaystyle x(t)={frac {1}{sqrt {2pi }}}int _{-W}^{W}{hat {x}}(omega )e^{iomega t},domega =Tsum _{nin mathbb {Z} }x(nT){frac {sin(W(t-nT))}{pi (t-nT)}}.}$

In the borderline case

${displaystyle WT=pi }$

Is this the reconstruction formula of the Nyquist-Shannon sanctuary theorems

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

whereby

${displaystyle operatorname {sinc} }$

The SINC function with

${displaystyle operatorname {sinc} (t):={tfrac {sin(pi t)}{pi t}}}$

is.

With the help of the Poisson sum formula you can show that the theta function

$m tume Flay The Pregux) Yumass embb-Bal mpie mpie al hale hyom hal hal hal hal hal hal hor mjoy mal sal hym hyk.$

there transformation formula

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

enough. This transformation formula was used by Bernhard Riemann to prove the functional equation of the Riemann Zeta function.

• Elias M. Stein, Guido Weiss: Introduction to Fourier Analysis on Euclidean Spaces . 1st edition. Princeton University Press, Princeton, N.J. 1971, ISBN 978-0-691-08078-9.
• J. R. Higgins: Five short stories about the cardinal series. In: Bulletin of the American Mathematical Society. 12, 1, 1985, ISSN  0002-9904 , S. 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 , S. 505–523, online .