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 D n 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.