Nonuniform Filter Banks: New Results and Open Problems

A nonuniform filter bank (FB) is one whose channel decimation rates need not all be equal. While the theory and design of uniform FBs is a very well developed subject, there are several interesting open issues in the area of nonuniform FBs. Most nonuniform FB designs either result in approximate or near–perfect reconstruction, or involve cascading uniform FBs in tree structures. This leaves unanswered many important theoretical issues involved in obtaining perfect reconstruction (PR) in nonuniform FBs. The purpose of this paper is to address these issues. We only study FBs with integer decimation rates, as FBs with rational decimators can also be shown to be transformable to them. The central problem of interest is as follows: Let S be a set of positive integers obeying maximal decimation (i.e., with reciprocals summing to unity). Find necessary and sufficient conditions on S for existence of a PRFB belonging to some FB class C and using S as its set of decimators. The class C is defined by some constraint on the filters of its constituent FBs; examples of interest are the class of all rational FBs (FBs with rational filters), FIR FBs, orthonormal FBs, etc. A condition that immediately suggests itself is the one stating that the integers be arrangeable in a tree so that the required PRFB can be built by cascading uniform PRFBs in a tree structure. However, this condition, while clearly sufficient, is not necessary for many classes C of interest. In fact there are sets violating it which can be used to build delay-chain PRFBs (in which all filters are delays). Many of our new results focus on the class of rational FBs. We strengthen considerably the known necessary conditions in this case, and provide new ones. The basic problem remains unresolved — necessary and sufficient conditions are still unknown, however we believe our work is an important step towards a full solution. We conclude by listing all known conditions, studying their inter-relationship, and pointing out several open problems.