A method of synthesis and decomposition of autonomous linear sequential circuits

We describe one of the practical methods to synthesize and decompose the autonomous linear sequential circuit (ALSC). The method for decomposition includes two steps. First, we discuss the behavior of the linear sequential circuit (LSC) with a feedback which has a unit time delay. Next, we derive the relation between the number of cycles (which correspond to a periodic output sequence) and that of states.In a forward I-S permutation LSC the number p of cycles is given as 1 ⩽ p ⩽ n(n + 1)/2, where n is the number of states. While, the largest length of a cycle on a I-S permutation LSC may be n2. The necessary and sufficient condition that the state transition diagram of a forward I-S permutation LSC A consists of only cycles is that A is a reverse I-S permutation LSC. Moreover, the decomposition theorem of the resultant LSC is obtained from Hartmanis' theorem