# Il teorema di Parseval

Parseval's theorem In mathematics, Il teorema di Parseval[1] usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh.[2] Although the term "Il teorema di Parseval" is often used to describe the unitarity of any Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem.[3] Contenuti 1 Statement of Parseval's theorem 2 Notation used in engineering 3 Guarda anche 4 Appunti 5 Riferimenti 6 External links Statement of Parseval's theorem Suppose that {stile di visualizzazione A(X)} e {stile di visualizzazione B(X)} are two complex-valued functions on {displaystyle mathbb {R} } of period {displaystyle 2pi } that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series {stile di visualizzazione A(X)=somma _{n=-infty }^{infty }un_{n}e^{inx}} e {stile di visualizzazione B(X)=somma _{n=-infty }^{infty }b_{n}e^{inx}} rispettivamente. Quindi {somma dello stile di visualizzazione _{n=-infty }^{infty }un_{n}{sopra {b_{n}}}={frac {1}{2pi }}int _{-pi }^{pi }UN(X){sopra {B(X)}},matematica {d} X,} (Eq.1) dove {stile di visualizzazione i} is the imaginary unit and horizontal bars indicate complex conjugation. Substituting {stile di visualizzazione A(X)} e {stile di visualizzazione {sopra {B(X)}}} : {stile di visualizzazione {inizio{allineato}somma _{n=-infty }^{infty }un_{n}{sopra {b_{n}}}&={frac {1}{2pi }}int _{-pi }^{pi }sinistra(somma _{n=-infty }^{infty }un_{n}e^{inx}Giusto)sinistra(somma _{n=-infty }^{infty }{sopra {b_{n}}}e^{-inx}Giusto),matematica {d} x\[6pt]&={frac {1}{2pi }}int _{-pi }^{pi }sinistra(un_{1}e^{i1x}+un_{2}e^{i2x}+cdot a destra)sinistra({sopra {b_{1}}}e^{-i1x}+{sopra {b_{2}}}e^{-i2x}+cdot a destra)matematica {d} x\[6pt]&={frac {1}{2pi }}int _{-pi }^{pi }sinistra(un_{1}e^{i1x}{sopra {b_{1}}}e^{-i1x}+un_{1}e^{i1x}{sopra {b_{2}}}e^{-i2x}+un_{2}e^{i2x}{sopra {b_{1}}}e^{-i1x}+un_{2}e^{i2x}{sopra {b_{2}}}e^{-i2x}+cdot a destra)matematica {d} x\[6pt]&={frac {1}{2pi }}int _{-pi }^{pi }sinistra(un_{1}{sopra {b_{1}}}+un_{1}{sopra {b_{2}}}e^{-ix}+un_{2}{sopra {b_{1}}}e^{ix}+un_{2}{sopra {b_{2}}}+cdot a destra)matematica {d} xend{allineato}}} As is the case with the middle terms in this example, many terms will integrate to {stile di visualizzazione 0} over a full period of length {displaystyle 2pi } (see harmonics): {stile di visualizzazione {inizio{allineato}somma _{n=-infty }^{infty }un_{n}{sopra {b_{n}}}&={frac {1}{2pi }}sinistra[un_{1}{sopra {b_{1}}}x+ia_{1}{sopra {b_{2}}}e^{-ix}-ia_{2}{sopra {b_{1}}}e^{ix}+un_{2}{sopra {b_{2}}}x+cdots right]_{-pi }^{+pi }\[6pt]&={frac {1}{2pi }}sinistra(2pi a_{1}{sopra {b_{1}}}+0+0+2pi a_{2}{sopra {b_{2}}}+cdot a destra)\[6pt]&=a_{1}{sopra {b_{1}}}+un_{2}{sopra {b_{2}}}+cdot \[6pt]fine{allineato}}} Più generalmente, given an abelian locally compact group G with Pontryagin dual G^, Parseval's theorem says the Pontryagin–Fourier transform is a unitary operator between Hilbert spaces L2(G) and L2(G^) (with integration being against the appropriately scaled Haar measures on the two groups.) When G is the unit circle T, G^ is the integers and this is the case discussed above. When G is the real line {displaystyle mathbb {R} } , G^ is also {displaystyle mathbb {R} } and the unitary transform is the Fourier transform on the real line. When G is the cyclic group Zn, again it is self-dual and the Pontryagin–Fourier transform is what is called discrete Fourier transform in applied contexts.

Parseval's theorem can also be expressed as follows: Supponiamo {stile di visualizzazione f(X)} is a square-integrable function over {stile di visualizzazione [-pi ,pi ]} (cioè., {stile di visualizzazione f(X)} e {stile di visualizzazione f^{2}(X)} are integrable on that interval), with the Fourier series {stile di visualizzazione f(X)simeq {frac {un_{0}}{2}}+somma _{n=1}^{infty }(un_{n}cos(nx)+b_{n}peccato(nx)).} Quindi[4][5][6] {stile di visualizzazione {frac {1}{pi }}int _{-pi }^{pi }f^{2}(X),matematica {d} x={frac {un_{0}^{2}}{2}}+somma _{n=1}^{infty }sinistra(un_{n}^{2}+b_{n}^{2}Giusto).} Notation used in engineering In electrical engineering, Parseval's theorem is often written as: {displaystyle int _{-infty }^{infty }|X(t)|^{2},matematica {d} t={frac {1}{2pi }}int _{-infty }^{infty }|X(omega )|^{2},matematica {d} omega =int _{-infty }^{infty }|X(2pi f)|^{2},matematica {d} f} dove {stile di visualizzazione X(omega )={matematico {F}}_{omega }{X(t)}} represents the continuous Fourier transform (in normalized, unitary form) di {stile di visualizzazione x(t)} , e {displaystyle omega =2pi f} is frequency in radians per second.

The interpretation of this form of the theorem is that the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency.

For discrete time signals, the theorem becomes: {somma dello stile di visualizzazione _{n=-infty }^{infty }|X[n]|^{2}={frac {1}{2pi }}int _{-pi }^{pi }|X_{2pi }({fi })|^{2}matematica {d} fi } dove {stile di visualizzazione X_{2pi }} is the discrete-time Fourier transform (DTFT) di {stile di visualizzazione x} e {stile di visualizzazione phi } represents the angular frequency (in radians per sample) di {stile di visualizzazione x} .

In alternativa, for the discrete Fourier transform (DFT), the relation becomes: {somma dello stile di visualizzazione _{n=0}^{N-1}|X[n]|^{2}={frac {1}{N}}somma _{k=0}^{N-1}|X[K]|^{2}} dove {stile di visualizzazione X[K]} is the DFT of {stile di visualizzazione x[n]} , both of length {stile di visualizzazione N} .

We show the DFT case below. For the other cases, the proof is similar. By using the definition of inverse DFT of {stile di visualizzazione X[K]} , we can derive {stile di visualizzazione {frac {1}{N}}somma _{k=0}^{N-1}|X[K]|^{2}={frac {1}{N}}somma _{k=0}^{N-1}X[K]cdot X^{*}[K]={frac {1}{N}}somma _{k=0}^{N-1}sinistra[somma _{n=0}^{N-1}X[n],esp a sinistra(-j{frac {2pi }{N}}K,giusto)Giusto],X^{*}[K]={frac {1}{N}}somma _{n=0}^{N-1}X[n]sinistra[somma _{k=0}^{N-1}X^{*}[K],esp a sinistra(-j{frac {2pi }{N}}K,giusto)Giusto]={frac {1}{N}}somma _{n=0}^{N-1}X[n](Ncdot x^{*}[n])=somma _{n=0}^{N-1}|X[n]|^{2},} dove {stile di visualizzazione *} represents complex conjugate.

See also Parseval's theorem is closely related to other mathematical results involving unitary transformations: Parseval's identity Plancherel's theorem Wiener–Khinchin theorem Bessel's inequality Notes ^ Parseval des Chênes, Marc-Antoine Mémoire sur les séries et sur l'intégration complète d'une équation aux différences partielles linéaire du second ordre, à coefficients constants" presented before the Académie des Sciences (Parigi) Su 5 aprile 1799. This article was published in Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savants, et lus dans ses assemblées. Sciences, mathématiques et physiques. (Savants étrangers.), vol. 1, pages 638–648 (1806). ^ Rayleigh, J.W.S. (1889) "On the character of the complete radiation at a given temperature," Rivista filosofica, vol. 27, pages 460–469. Available on-line here. ^ Plancherel, Michele (1910) "Contribution à l'etude de la representation d'une fonction arbitraire par les integrales définies," Rendiconti del Circolo Matematico di Palermo, vol. 30, pages 298–335. ^ Arthur E. Danese (1965). Advanced Calculus. vol. 1. Boston, MA: Allyn and Bacon, Inc. p. 439. ^ Wilfred Kaplan (1991). Advanced Calculus (4th ed.). Lettura, MA: Addison Wesley. p. 519. ISBN 0-201-57888-3. ^ Georgi P. Tolstov (1962). Fourier Series. Translated by Silverman, Richard. Englewood Cliffs, NJ: Prentice Hall, Inc. p. 119. References Parseval, Archivio di storia della matematica di MacTutor. George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001). Hubert Kennedy, Eight Mathematical Biographies (Peremptory Publications: San Francisco, 2002). Alan V. Oppenheim and Ronald W. Schafer, Discrete-Time Signal Processing 2nd Edition (Sala dell'Apprendista: Upper Saddle River, NJ, 1999) p 60. William McC. Siebert, Circuits, Signals, and Systems (MIT stampa: Cambridge, MA, 1986), pp. 410–411. David W. Kammler, A First Course in Fourier Analysis (Prentice–Hall, Inc., Upper Saddle River, NJ, 2000) p. 74. External links Parseval's Theorem on Mathworld Categories: Theorems in Fourier analysis

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