Remarks on Applicability of Spectral Representations on Finite non-Abelian Groups in the Design for Regularity Radomir S. Stankovi´ c, Jaakko Astola 1 , Claudio Moraga 2 Dept. of Computer Science, Faculty of Electronics, 18 000 Niˇ s, Serbia 1 Institute of Signal Processing, Tampere University of Technology, FIN-33101 Tampere, Finland 2 European Centre for Soft Computing; 33600 Mieres, Asturias, Spain, and Dortmund University of Technology; 44221 Dortmund, Germany Abstract In several publications, the application of non-Abelian groups has been suggested as a tool to derive compact rep- resentations of logic functions. The compactness has been measured in the number of product terms in the case of func- tional expressions and the number of nodes, the width, and the interconnections in the case of decision diagrams. In this paper, we discuss Fourier representations on finite non- Abelian groups as a tool useful in synthesis for regularity. The initial domain group for a logic function (binary or multiple-valued) is replaced by a non-Abelian group by en- coding of variables. The function is then decomposed into matrix-valued Fourier coefficients, that are easy to imple- ment by building blocks over a technological platform with regular structure. We point out that spectral representation of non-Abelian groups are capable of capturing regularities in functions and transferring them in the spectral domain. In many cases, week regularities in the original domain are converted into much stronger regularities in the spectral domain due to the regular structure of unitary irreducible group representations upon which the Fourier expressions are based. 1 Introduction Due to ever increasing complexity of systems, the de- creasing clock period, and portability, which demands low- power consumption and dissipation, are contradictory re- quests imposed on contemporary computing and informa- tion systems. The optimization in terms of performances, area, and power, can be viewed as a three dimensional prob- lem, and regularity of the structure of a network is a neces- sary requirement to resolve it. Regularity is usually understood as appearance of iden- tical or similar patterns in a certain order that can be de- scribed formally. This ability of being able to formally ex- press the interrelationships among objects is essential in im- plementations taking advantages from regularity. The im- plementation of large logic functions often assumes decom- position of the function given into subfunctions that will be implemented separately and then connected into a larger network. It is desirable if the decomposition is done such that automatically expresses certain regularity which can be directly mapped to a technology with a regular structure of basic blocks used to realize subfunctions. Field-programmable gate arrays (FPGAs) are an obvious response to the demands for regularity. In spite of advent of FPGAs, the cell-based layout style Application specific in- tegrated circuits (ASIC) are still dominant in terms of all three parameters. The problem is their high cost in terms of large investments of engineering time and also long time- to-market. The cost and performances discrepancy between FPGAs and cell-based ASIC motivated the introduction of structured-ASICs having most of their parts prefabricated and expressing high regularity [1], [20]. Both FPGAs and Structured-ASICS are built as arrays of basic blocks which can be Look-up-table (LUT) based blocks, Programmable logic array (PLA)-based blocks [8], or blocks based on el- ementary logic circuits as inverters and two-input NAND and NOR circuits [12]. Basic blocks can implement any function up to a certain number of variables and can be connected into larger blocks expressing functionality of a semi-universal logic block. In both cases, whatever FPGA or Structured-ASICs are used, the main task is to decompose a given function into subfunctions that will be realized by basic blocks. To meet the regularity required in the underlying technological structure, it is very desirable if the decomposition is per- formed such that ensures regularity in the subfunctions as