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Filbir, F. ; Mhaskar, N.* ; Prestin, J.*

On the problem of parameter estimation in exponential sums.

Constr. Approx. 35, 323-343 (2012)
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Let I >= 1 be an integer, omega(0) = 0 < omega(1) < ... omega(I) <= pi, and for j = 0,..., I, a(j) is an element of C, a-j = (aj) over bar = omega-.j, and a(j) not equal 0 if j not equal 0. We consider the following problem: Given finitely many noisy samples of an exponential sum of the form (x)over bar (k) = Sigma(j=-1) (I) ajexp(-iw(j)k) + epsilon(k), k = -2N, ... , 2N, where epsilon (k) are random variables with mean zero, each in the range [- epsilon, epsilon] for some epsilon 0, determine approximately the frequencies.j. We combine the features of several recent works to use the available information to construct the moments (y)over bar(N)(k) yN(k) of a positive measure on the unit circle. In the absence of noise, the support of this measure is exactly {exp(- i.j) : aj = 0}. This support can be recovered as the zeros of the monic orthogonal polynomial of an appropriate degree on the unit circle with respect to this measure. In the presence of noise, this orthogonal polynomial structure allows us to provide error bounds in terms of epsilon and N. It is not our intention to propose a new algorithm. Instead, we prove that a preprocessing of the raw moments (x)over bar(k) to obtain (y)over bar (N) k) enables us to obtain rigorous performance guarantees for existing algorithms. We demonstrate also that the proposed preprocessing enhances the performance of existing algorithms.
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Publikationstyp Artikel: Journalartikel
Dokumenttyp Wissenschaftlicher Artikel
Schlagwörter Frequency analysis; Spectral analysis problem; Parameter estimation; Exponential sums; Prony method; MULTIVARIATE INTEGRATION; APPROXIMATION; TRACTABILITY
ISSN (print) / ISBN 0176-4276
e-ISSN 1432-0940
Quellenangaben Band: 35, Heft: 3, Seiten: 323-343 Artikelnummer: , Supplement: ,
Verlag Springer
Begutachtungsstatus Peer reviewed