On Finite Group Presentations and Function Decomposition Based on Linearity of Discrete-Time Signal
S. G. Ngulde
Department of Mathematics and Statistics, University of Maiduguri, Nigeria.
B. A. Madu
Department of Mathematics and Statistics, University of Maiduguri, Nigeria.
D. Samaila *
Department of Mathematics, Adamawa State University, Mubi, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
Based on the concept of group representation theory, new representations can be generated by direct product (or tensor product) of any two representations of a group. In such case, their irreducible representations are also the direct product. But the conditions under which these representations can be chosen and how to decompose them is silent. In this work, a clear and efficient method for generating and decomposing representations is presented. The study is restricted to geometric group Dn of order 2n and its subgroups, where a new homomorphism called a transfer function based on the geometric group is constructed. Due to linearity of discrete-time signal, the generated transformations are used on signal space. Thus, a different approach to signal processing with the choice of a group of transformations is established.
Keywords: Finite group, representation, decomposition, Fourier transform, signal processing