The Unified Theory is based on two mathematical fundamentals: Abelian groups and the Haar integral. Abelian groups, developed in this chapter, will serve as the domains of signals, Fourier transforms and of their periodicity. The main problem is the identification of groups on where the Haar integral can be defined. Topology provides a clear answer to this problem: the groups must be locally compact Abelian (LCA). Another problem is the explicit formulation of these groups and of their operations. This is achieved through the technique of basis signature representation, where the basis identifies the geometry of the group and the signature specifies its nature (continuous, discrete or mixed).
The main topic of the chapter is the formulation and illustration of LCA groups, both one-dimensional and multidimensional. Another important topic developed is that of cells which are subsets of a group with specific properties and represent a fundamental tool for an advanced Signal Theory.
KeywordsAbelian Group Quotient Group Multiplicative Group Signal Classis Primitive Group
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