Time-varying filters and filter banks: some basic principles

Abstract

We study the fundamentals of time-varying filter banks (TVFB). Using a polyphase approach to TVFBs, we are able to show some unusual properties that are not exhibited by the conventional LTI filter banks. For example, we can show that for a perfect reconstruction (PR) TVFB, the losslessness of analysis bank does not always imply that of the synthesis bank, and replacing the delay z-1 in an implementation of a lossless linear time-variant (LTV) system with z-L for integer L in general will result in a nonlossless system. Moreover, we show that interchanging the analysis and synthesis filters of a PR TVFB will usually destroy the PR property, and a PR TVFB in general will not generate a discrete-time basis for l2. Furthermore, we show that we can characterize all TVFBs by characterizing multi-input multi-output (MIMO) LTV systems. A useful subclass of LTV systems, namely the lossless systems, is discussed in detail. All lossless LTV systems are invertible. Moreover, the inverse is a finite impulse response (FIR) if the original lossless system is an FIR. Explicit construction of the inverses is given. However, unlike in the LTI case, we show that the inverse system is not necessarily unique or invertible. In fact, the inverse of a lossless LTV system is not necessarily lossless. Depending on the invertibility of their inverses, the lossless systems are divided into two groups: (i) invertible inverse lossless (IIL) systems and (ii) noninvertible inverse lossless (NIL) systems. We show that an NIL PR TVFB will only generate a discrete-time tight frame with unity frame bound. However if the PR FB is IIL, we have an orthonormal basis for ℓ2. In a companion paper, some of these results are used to derive deeper properties of lossless TVFB including factorization theorems.