LINEAR ESTIMATION OF STATIONARY STOCHASTIC PROCESSES , VIBRATING STRINGS , AND INVERSE SCATTERING *

Some vibrating string equations are derived for estimating a stationary stochastic process given some observations of this process over a finite interval. These equations are the time-domain counterpart of equations introduced by Krein, and Dym and McKean in the frequency domain. They are obtained by decomposing the observation and signal processes into even and odd processes, and by solving some associated filtering problems. The resulting linear estimation procedure is very efficient and is well adapted to the estimation of random fields, We also show that it is identical to the Gelfand-Levitan inverse scattering method of quantum mechanics.

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