Time-varying filters and filter banks: some basic principles
Journal
IEEE Transactions on Signal Processing
Journal Volume
44
Journal Issue
12
Pages
2971-2987
Date Issued
1996
Author(s)
Vaidyanathan, P.P.
Abstract
In this paper we study the fundamentals of time-varying filter banks (TVFB). Using a polyphase approach to TVFB's 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 will show that we can characterize all TVFB's by characterizing multi-input multi-output (MEVIO) LTV systems. A useful subclass of LTV systems namely the lossless systems will be discussed in detail. All lossless LTV systems are invertible. Moreover the inverse is finite impulse response (FIR) if the original lossless system is FIR. Explicit construction of the inverses is given. However unlike in the LTI case we will 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 will 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 will 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. © 1996 IEEE.
Other Subjects
Electric losses; Electric network analysis; Electric network synthesis; Time varying networks; Linear time variant (LTV) systems; Time varying filter banks (TVFB); Digital filters
Type
journal article