Linear time dependent systems

In studies of linear open-loop systems, the assumption of time invariance is often tacitly made. In this paper we show how a time dependent system can arise quite naturally. The estimation of time dependent transfer function, on the basis of a single realization, when the input/output processes are nonstationary is considered. We also consider the problem of testing a given open-loop system for time dependence. The tests described here make use of the concept of the "evolutionary cross spectra," and rests essentially on testing the "uniformity" of a set of vectors, whose components consist of the "evolutionary gain spectra" and "evolutionary phase spectra." Using a logarithmic transformation on the evolutionary gain spectra, we show that the mechanics of the tests are formally equivalent to a two-factor multivariate analysis of variance (MANOVA) procedure. Numerical illustrations, from the real and simulated data, of the proposed tests are included.