[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2en\/wiki14\/poissons-sum-formula-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2en\/wiki14\/poissons-sum-formula-wikipedia\/","headline":"Poisson’s sum formula – Wikipedia","name":"Poisson’s sum formula – Wikipedia","description":"before-content-x4 The Poisson sum formula is a tool of Fourier analysis and signal processing. Among other things, it serves to","datePublished":"2021-02-28","dateModified":"2021-02-28","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2en\/wiki14\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2en\/wiki14\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3e3a94bbb23b9dcfa137c357ada80c9ac8256a26","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3e3a94bbb23b9dcfa137c357ada80c9ac8256a26","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2en\/wiki14\/poissons-sum-formula-wikipedia\/","wordCount":8503,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4The Poisson sum formula is a tool of Fourier analysis and signal processing. Among other things, it serves to analyze the properties of sampling methods.        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4May be f \u2208 S ( R ) {displaystyle fin {mathcal {S}}(mathbb {R} )} A Schwartz function and be        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4f^( oh ) = F( f ) ( oh ) = \u222b \u2212\u221e\u221ef ( t ) It is \u22122\u03c0i\u03c9\u22c5td t {displaystyle {hat {f}}(omega )={mathcal {F}}(f)(omega )=int _{-infty }^{infty }f(t),e^{-2pi iomega cdot t},dt} The continuous Fourier transformation of f {displaystyle f} in S {displaystyle {mathcal {S}}} .Then the Poisson’s sum formula says        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u2211 n\u2208Zf ( n ) = \u2211 k\u2208Zf^( k ) . mm slaves Euxt or embb\u0254b 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 f {displaystyle f} dilapidated in two ways and the expression ( first + t 2 ) ( | f ( t ) | + | f \u2033 ( t ) | ) {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 t , n \u2208 R {displaystyle t,nu in mathbb {R} }  \u2211n\u2208Zf(t+nT)e\u22122\u03c0i\u03bdnT=\u2211k\u2208ZF(f(t+\u22c5T)e\u22122\u03c0i\u03bd\u22c5T)(k)=1T\u2211k\u2208ZF(f(t+\u22c5)e\u22122\u03c0i\u03bd\u22c5)(kT)=1T\u2211k\u2208ZF(f(t+\u22c5))(kT+\u03bd)=1T\u2211k\u2208Ze2\u03c0i(k\/T+\u03bd)tF(f)(kT+\u03bd).{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 t = 0 {displaystyle t=0} , \u2211 n\u2208Zf ( n T ) It is \u22122\u03c0i\u03bdnT= 1T\u2211 k\u2208ZF( f ) ( kT+\u03bd) , {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 \u200b\u200bof f {displaystyle f} as coefficients on the left and a periodization of the Fourier transformed of f {displaystyle f} can be read on the right. This identity applies with the exception of a quantity of zero if f {displaystyle f} A band-limited function is, i.e. the Fourier transformed a measurable function in L 2 ( R ) {displaystyle L^{2}(mathbb {R} )} with a compact carrier. The Dirac ridge to the interval length T \u2208 R {displaystyle Tin mathbb {R} } Is the distribution \u0428T= \u2211 n\u2208Zd nT. {Displaystyle {text {\u0448}} _ {t} = sum _ {nin mathbb {z}} delta _ {nt}.} The Fourier transformed F A \u2208 S\u2032 ( R ) {displaystyle {mathcal {F}}Ain {mathcal {S}}'(mathbb {R} )} a tempered distribution A \u2208 S\u2032 ( R ) {displaystyle Ain {mathcal {S}}'(mathbb {R} )} is defined by \u27e8 FA , \u03d5 \u27e9 = \u27e8 A , F\u03d5 \u27e9 ( \u03d5 \u2208 S( R ) ) , {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 F\u0428T= 1T\u04281\/T{displaystyle {mathcal {F}}{text{\u0428}}_{T}={frac {1}{T}}{text{\u0428}}_{1\/T}} is. This can also be in the form \u0428T= 1T\u2211 k\u2208ZIt is i(2\u03c0k\/T)t{displaystyle {text{\u0428}}_{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 g ( t ) : = \u2211 n\u2208Zf ( t + n ) {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, g ( t ) = \u2211 k\u2208Zc k\u22c5 It is 2\u03c0ikt. mms supree Peleek)) yumb) mumbbe m m hubeh hy\u00f3i hyom hort hyom hyo hyag hyo hyh hyh hor. Their Fourier coefficients determine according to the formula c k= \u222b 01g ( t ) \u22c5 It is \u22122\u03c0iktd t = \u222b 01\u2211 n\u2208Zf ( t + n ) \u22c5 It is \u22122\u03c0ik(t+n)d t . m sovet Slele State State Skk k\u00ed mofe syo hok m h\u00e9p h\u00e9\u00e9 Like m\u0254 k\u00f3m 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 integralcan be exchanged. Therefore with s=t+n further c k= \u2211 n\u2208Z\u222b nn+1f ( s ) \u22c5 It is \u22122\u03c0iksd s = \u222b \u2212\u221e\u221ef ( s ) \u22c5 It is \u22122\u03c0iksd s = Ff ( k ) . mm slavetle State State State\u2014empum\u2014emmum\u2014Pe m hk hy hym hork 2 kank kmome, supm ) mmber , lame kmalm mmmmm 22-5-5 Wifetyfee () yoky male -2ym is halm m\u00e9p hjoy mjoy. In summary \u2211 n\u2208Zf ( t + n ) = \u2211 k\u2208ZFf ( k ) It is 2\u03c0ikt, mm slaves Euxt or emb\u0254b m Refine) mm) mmmm mm h\u00e9 h Aske mupe) mupe ) malm mmb) What is from t = 0 {displaystyle t=0} the claim results. May be x Ribbon limited with the highest frequency IN , That means supp \u2061 x^\u2282 [ – IN , IN ] {displaystyle operatorname {supp} {hat {x}}subset [-W,W]} . Is then | IN T | \u2264 Pi , {displaystyle |WT|leq pi ,} So only one summand occurs in the right side of the sum formula, with the replacement oh : = – 2 Pi n \u2208 [ – IN , IN ] {Displastyle Omega: =-2pi nu in [-W, in]} , t=0 and multiplication of a factor you get 2\u03c0x^( oh ) It is i\u03c9t= T \u2211 n\u2208Zx ( n T ) It is i\u03c9(t\u2212nT). {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 x ( t ) = 12\u03c0\u222b \u2212WWx^( oh ) It is i\u03c9td oh = T \u2211 n\u2208Zx ( n T ) sin\u2061(W(t\u2212nT))\u03c0(t\u2212nT). {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 IN T = Pi {displaystyle WT=pi } Is this the reconstruction formula of the Nyquist-Shannon sanctuary theorems x ( t ) = \u2211 n\u2208Zx ( n T ) sinc \u2061 ( t \/ T – n ) , {displaystyle x(t)=sum _{nin mathbb {Z} }x(nT)operatorname {sinc} (t\/T-n),} whereby sinc {displaystyle operatorname {sinc} } The SINC function with sinc \u2061 ( t ) : = sin\u2061(\u03c0t)\u03c0t{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 th ( t ) = \u2211 n\u2208ZIt is \u2212n2\u03c0tm 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 th ( t ) = 1tth ( 1t) {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\u00a0 0002-9904 , S. 45\u201389, 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\u00a0 0002-9904 , S. 505\u2013523, online .      (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2en\/wiki14\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2en\/wiki14\/poissons-sum-formula-wikipedia\/#breadcrumbitem","name":"Poisson’s sum formula – Wikipedia"}}]}]