Linear Time-Varying Models for Signal Processing

The solution of a linear time-invariant differential equation can be obtained as the output of a so-called canonical signal processing filter with the right hand side of the differential equation as input. Such a filter is build up with integrators, adders, multipliers and so on. One distinguishes in the literature between a series, a cascade and a parallel realization of the filter.The coefficients of the differential equation appear to be the multipliers in both, the series and parallel representation of the differential equation. The eigenvalues or poles of the differential equation are the multipliers in the cascade realization. On the basis of a number of recently obtained results one may expect that these three types of canonical representations also exist for linear time-varying differential equations. In this paper this problem is addressed. If we start with the series representation, then it is shown that the cascade realization uses the dynamic eigenvalues and that the parallel realization only in special cases equals the solution of the differential equation. Keywords— linear time-varying systems, canonical representations, dynamic eigenvalues.