2013 | OriginalPaper | Chapter
Initial Value Problems in the Time-Frequency Domain
Author : Lorenzo Galleani
Published in: Pseudo-Differential Operators, Generalized Functions and Asymptotics
Publisher: Springer Basel
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We transform an initial value problem for a stochastic differential equation to the time-frequency domain. The result is a deterministic time-frequency equation whose forcing term incorporates the given set of initial values. The structure and solution of the time-frequency equation reveal the spectral properties of the nonstationary random process solution to the stochastic differential equation. By applying our method to the Langevin equation, we obtain the exact time-frequency spectrum for an arbitrary initial value.