Factorability of lossless time-varying filters and filter banks

In this paper, we study the factorability of linear time-varying (LTV) lossless filters and filter banks. We give a complete characterization of all degree-one lossless LTV systems and show that all degree-one lossless systems can be decomposed into a time-dependent unitary matrix followed by a lossless dyadicbased LTV system. The lossless dyadic-based system has several properties that make it useful in the factorization of lossless LTV systems. The traditional lapped orthogonal transform (LOT) is also generalized to the LTV case. We identify two classes of TVLOT’s, namely, the invertible inverse lossless (IIL) and noninvertible inverse lossless (NIL) TVLOT’s. The minimum number of delays required to implement a TVLOT is shown to be a nondecreasing function of time, and it is a constant if and only if the TVLOT is IIL. We also show that all IIL TVLOT’s can be factorized uniquely into the proposed degreeone lossless building block. The factorization is minimal in terms of delay elements. For NIL TVLOT’s, there are factorable and unfactorable examples. Both necessary and sufficient conditions for factorability of lossless LTV systems will be given. We also introduce the concept of strong eternal reachability (SER) and strong eternal observability (SEO) of LTV systems. The SER and SEO of an implementation of LTV systems imply the minimality of the structure. Using these concepts, we are able to show that the cascade structure for a factorable IIL LTV system is minimal. That implies that if a IIL LTV system is factorable in terms of the lossless dyadic-based building blocks, the factorization is minimal in terms of delays as well as the number of building blocks. We also prove the BIBO stability of the LTV normalized IIR lattice.