Galois - Software for a Neglected Part of Control Education

Abstract

Arithmetical mathematics is a neglected part of control engineering education, ignoring which inhibits development of those areas of the subject that need an arithmetical approach. One such area is that of pseudo-random perturbation signals for system identification, in which those signals based on maximum-length sequences require an understanding of finite fields. GALOIS is a program through which a user may be introduced to this part of arithmetical mathematics, with software that provides tutorial facilities suitable for the education of control engineers. Sum and product tables are used to implement the arithmetic of both prime and extension fields with up to 128 elements on a common basis, and these may be viewed to show the field mathematics involved. In all fields both primitive polynomials and maximum-length sequences with periods up to 20000 may also be generated and viewed. The arithmetical mathematics underlying all these processes is described. Finally, the user can convert the maximum- length sequences into pseudo-random sequences and signals for system identification. The paper also shows that arithmetical mathematics underlies other sequences suitable for this purpose.