On the Analysis of Bivariate Non-Stationary Processes

Summary In this paper, we propose a general definition of the evolutionary (time-dependent) cross-spectrum between two non-stationary processes and describe its physical interpretation. We also study the estimation of the evolutionary cross-spectrum at each time instant t from a single realization of a bivariate process. Further, we propose a definition (and a method of estimation) for the coherency (spectrum) between the two components of the bivariate process and show that the notion of residual variance bound first introduced in the analysis of bivariate stationary processes can be extended to that of non-stationary processes. As an application of the evolutionary cross-spectral analysis of bivariate non-stationary stochastic processes, we consider the estimation of the transfer function of a linear open-loop time-dependent system. Numerical illustrations of the estimation of a time-dependent transfer function are included.