**ABSTRACT**
### 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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