Role of the stationarity equation in linear least-squares estimation, spectral analysis and wave propagation
Leroy C. Pusey, A. B. Baggeroer · The Journal of the Acoustical Society of America · 1976
We develop the properties of the innovations representation for continuous-time stationary processes and consider applications to spectral estimation and the description of wave propagation in a lossless nonuniform medium. Both applications are based on the Levinson recursion for the causal least-squares estimation filter and the continuous-time counterpart of this recursion, the stationarity equation. In the first application, we develop an unconstrained representation of positive definite extensions by observing that the stationarity equation completely characterizes the least-squares estimation filter for stationary processes. This representation is incorporated in a noval approach to spectral estimation whereby the maximum entropy method is generalized so as to include arbitrary positive definite extensions of the estimated covariance. In the second application, we observe that the stationarity equation is descriptive of wave propagation in a lossless nonuniform medium and use this fact to obtain an interesting time-domain solution of the nonuniform wave equation. The solution is an extension of discrete-time results in seismology concerning acoustical wave propagation in a layered medium.