Stochastic Decomposition and Application to Probabilistic Dynamics

Abstract

The frequency-domain analysis concerning the response of nested-cascade multiple input/output systems requires computation of the cross-spectral density matrices that involve the input, intermediate, and output vectors. Clearly, as the number of nested systems increases, the order of the cross-spectral density matrix increases, demanding additional computational effort. This feature lessens the computational attractiveness of the frequency-domain analysis. A stochastic decomposition technique is developed that improves the efficiency of conventional frequency-domain analysis by eliminating the intermediate step of estimating cross-spectral density matrices. Central to this technique is the decomposition of a set of correlated random processes into a number of component random processes. Statistically, any two processes decomposed in this manner are either fully coherent or noncoherent. A random subprocess obtained from this decomposition is expressed in terms of a decomposed spectrum. A theoretical basis for this approach and computational procedures for carrying out such decompositions in probabilistic dynamics are presented.