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Dipartimento di Fisica

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The Prony method of Frequency Analysis
Speaker Speaker: Prof. Alain Noullez
Affiliato Affiliato: CNRS, Observatoire de la Cote d'Azur
Data Evento Martedì 28 Marzo , alle ore 15.30 - Aula Seminari, Dipartimento di Fisica
Frequency Analysis has proven to be a very powerful technique to characterize the transition to chaos in dissipative and conservative dynamical systems. The
classical standard method of Frequency Analysis is Fourier series, but it is limited to harmonic frequencies and signals that are periodic over the length of the observation period, and it thus needs samples much longer than the slowest frequency present in the signal. More general and accurate frequency determination methods exist, but they generally require very delicate numerical procedures that are also computationaly expensive. In this talk, I will present an alternative method for Frequency Analysis, originally due to Prony, based on the construction of an approximate linear difference equation of high order, exactly or best (in the least-squares sense) representing an observed time series. Resonances are then obtained as the eigenfrequencies of this difference equation, by solving its characteristic polynomial. In the case of conservative dynamics, resonances correspond exactly to pure frequencies, and the method can be seen as another algorithm for Frequency Analysis. But resonances are more general, and can also represesent damped frequencies, such as those observed in dissipative systems, or spectra with broadband features or even secular drifts. This representation is thus more general than Fourier series, and is suited to both periodic, quasi-periodic and chaotic time series. The method is also very efficient numerically and can work with short time series to extract the frequencies. Results obtained on simple dynamical systems or in celestial mechanics will be presented, and some limitations of the method will be discussed.