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New Vistas to the Signal Processing of Nonstationary Time Series via an Operator Algebraic Way
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Publication Date: 2001/01/01
Print ISSN: 0916-8508
Type of Manuscript: INVITED PAPER (Special Section on the 10th Anniversary of the IEICE Transactions of Fundamentals: "Last Decade and 21st Century")
signal processing, nonstationary time series, time-frequency analysis, time-varying circuits and systems, operator algebra,
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This paper is, in half part, written in review nature, and presents recent theoretical results on linear-filtering and -prediction problems of nonstationary Gaussian processes. First, the basic concepts, signal and noise, are mathematically characterized, and information sources are defined by linear stochastic differential equations. Then, it is shown that the solution to a conventional problem of filtering or prediction of a nonstationary time series is, in principle, reducible to a problem, of which solution is given by Kalman-Bucy's theory, if one can solve a problem of finding the canonical representation of a Gaussian process such that it has the same covariance functions as those of the time series under consideration. However, the problem mentioned above is left open. Further, the problem of time-frequency analysis is discussed, and physical realizability of the evolutionary, i.e., the online, spectral analyzer is shown. Methods for dealing with differential operators are presented and their basic properties are clarified. Finally, some of related open problems are proposed.