The Theory of Autonomous Linear Sequential Networks

Analysis and synthesis techniques for a class of sequential discrete-state networks are discussed. These networks, made up of arbitrary interconnections of unit-delay elements (or of trigger flip-flops), modulo-p adders, and scalar multipliers (modulo, prime p), are of importance in unconventional radar and communication systems, in automatic error-correction circuits, and in the control circuits of digital computers. In addition, these networks are of theoretical significance to the study of more general sequential networks. The basic problem with which this paper is concerned is that of finding economical realizations of such networks for prescribed autonomous (excitation-free) behavior. To this end, an analytical-algebraic model is described which permits the investigation of the relation between network logical structure and state-sequential behavior. This relation is studied in detail for nonsingular networks (those with purely cyclic behavior). Among the results of this investigation is the establishment of relations between the state diagram of the network and a characteristic polynomial derived from its logical structure, An operation of multiplication of state diagrams is shown to correspond to multiplication of the corresponding polynomials. A criterion is established for the realizability of prescribed cyclic behavior by means of linear autonomous sequential networks. An effective procedure for the economical realization of such networks is described, and it is shown that linear feedback shift registers constitute a canonical class of realizations. Examples are given of the realization procedure. The problem of synthesis with only one-cycle length specified is also discussed. A partial solution is obtained to this "don't care" problem. Some special families of feedback shift registers are investigated in detail, and the state-diagram structures are obtained for an arbitrary number of stages and an arbitrary (prime) modulus. Mathematical appendixes are included which summarize the pertinent results in Galois field theory and in the factorization of cyclotomic polynomials into irreducible factors over a modular field. The relation of the theory developed in this paper to Huffman's description of linear sequence transducers in terms of the D operator is discussed, as well as unsolved problems and directions for further generalization.