Finite and Infinite-Dimensional Models for Oversampled Filter Banks

Thomas Strohmer

A filter bank is a collection of functions $\{\gmn\}$ of the form $\gmn(k) = \gm(k-an)$. Such systems play an important role in digital signal processing and communication. For given analysis functions $\gmn$ we want to find synthesis functions $\gamn(k) = \gam(k-an)$ such that each signal $f$ can be written as $f = \sum \langle f, \gamn \rangle \gmn$. The mathematical theory addresses the questions when such $\gamn$ exist and how to construct them usually in infinite dimensions, whereas numerical methods have to operate with finite-dimensional models. In this chapter we discuss the relation between certain finite-dimensional models used for numerical procedures and infinite-dimensional filter bank theory. We show that the ``synthesis filter bank'' obtained by solving a finite-dimensional problem converges to the synthesis filter bank of the original infinite-dimensional problem in $\ltZ$. We give rates of approximations if the $\gm$ satisfy certain decay properties (such as finite impulse response or polynomial decay) and also address the construction of paraunitary filter banks. Furthermore, we investigate the use of the periodic finite model and validate existing methods that exploit the polyphase representation. Finally for oversampled DFT filter banks a factorization of the frame operator is presented that improves upon the known factorization by means of the Zak transform.

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