A Computational Theory of Laurent Polynomial Rings and Multidimensional FIR Systems by HYUNG-JU PARK Doctor of Philosophy in Mathematics University of California at Berkeley A Laurent polynomial ring is a natural ground ring for the study of FIR (Finite Impulse Response) systems in signal processing since an FIR lter bank can be seen as a matrix over this ring, and the notion of perfect reconst...

We begin with some basic definitions. Definition 1.1. Let f, g ∈ F [x]. We say that f divides g, written f ∣∣g, if there exists an h ∈ F [x] such that g = fh, i.e. g is a multiple of f . Thus, for example, every f ∈ F [x] divides the zero polynomial 0, but g is divisible by 0 ⇐⇒ g = 0. By definition, f is a unit ⇐⇒ f ∣∣1. Recall also that the group of units (F [x])∗ of the ring F [x] is F ∗, th...