On Applications of Operators and Group - Theoretic Concepts in Signal Processing and Telecommunication

Abstract

Operator theory concepts such as frame operator, Gramian operator, synthesis operator and analysis operator are useful in signal processing and telecommunication. Finite dimensional Hilbert spaces and normalized (unit) vectors are rich in signals required in application. One of the most significant fields of applications and source of questions of group theory and operator theory is frames. The Mercedes- Benz frames are highly applicable in signal processing. In this thesis; we investigate frames, fourier analysis and wavelets and some of their applications in signal processing and telecommunication. We investigate why we need frames over bases, we study the dual frames, some operators: synthesis, analysis, frame and Gramian and finalize frames by looking at group frames. Here, we shall investigate the relationship between group symmetry and frames, representation theory and frames and also study group matrices and the Gramian of a group frame. We shall investigate properties of fourier analysis. We finalize by looking at the applications of frames and wavelets in signal processing and telecommunication.