The aim of this work is to present several topics in time-frequency
analysis as subjects in abelian group theory. The algebraic point
of view pre dominates as questions of convergence are not
considered. Our approach emphasizes the unifying role played by
group structures on the development of theory and algorithms. This
book consists of two main parts. The first treats Weyl-Heisenberg
representations over finite abelian groups and the second deals
with mul tirate filter structures over free abelian groups of
finite rank. In both, the methods are dimensionless and
coordinate-free and apply to one and multidimensional problems. The
selection of topics is not motivated by mathematical necessity but
rather by simplicity. We could have developed Weyl-Heisenberg
theory over free abelian groups of finite rank or more generally
developed both topics over locally compact abelian groups. However,
except for having to dis cuss conditions for convergence, Haar
measures, and other standard topics from analysis the underlying
structures would essentially be the same. A re cent collection of
papers 17] provides an excellent review of time-frequency analysis
over locally compact abelian groups. A further reason for limiting
the scope of generality is that our results can be immediately
applied to the design of algorithms and codes for time frequency
processing."
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