Biorthogonal Wavelet Based Identification of Fast Linear Time-Varying Systems—Part I: System Representations¹²

An analytical framework is developed that permits the input-output representations of discrete-time linear time-varying (LTV) systems in terms of biorthogonal bases on compact time intervals. Using these representations, the companion paper, Part II develops computational procedures for rapid identification of fast nonsmooth LTV systems based on short data records. One of the representations proposed is also used in H. Zhao and J. Bentsman, “Block Diagram Reduction of the Interconnected Linear Time-Varying Systems in the Time Frequency Domain,” accepted for publication by Multidimensional Systems and Signal Processing to form system interconnections, or wavelet networks, and develop subsystem connectibility conditions and reduction rules. Under the assumption that the inputs and the outputs of the plants considered in the present work belong to $lp$ spaces, where $p=2$ or $p=∞,$ their impulse responses are shown to belong to Banach spaces. Further on, by demonstrating that the set of all bounded-input bounded-output (BIBO) stable discrete-time LTV systems is a Banach space, the system representation problem is shown to be reducible to the linear approximation problem in the Banach space setting, with the approximation errors converging to zero as the number of terms in the representation increases. Three types of LTV system representation, based on the input-side, the output-side, and the input-output transformations, are developed and the suitability of each representation for matching a particular type of the LTV system behavior is indicated.