自反且自补二元周期序列的性质与构造方法

本文研究了自反、自补、自反且自补的二元周期序列的性质,给出了周期为n的自反且自补序列的数目计算公式以及构造方法。

Properties and Construction Method of Self Converse and Complementary Dyadic Periodic Sequence

In primitive dyadic periodic sequences, some properties are particular to the self converse and complementary periodic sequence. Shapes of switching piece which correspond to the above sequence are of optimum.Therefore, to study the properties and construction method of this kind of dyadic periodic sequences is practically meaningful for the design of switching pieces in electromechanically follow-up control system.Summary, In this paper, four difinitions are given together with six theorems, a formula and an algorithm. They can be summarized as follows.Letbe a dyadic periodic sequence, where n is the period and n0 the number of zero elements.Def. If dividing a circle into n equal arcs and putting every element of sequence (1) on the points of division in successive order, such a graph is called a circular graph of sequence (1).Thm..1 Sequence (1) is self converse if and only if there exists at least a symmetrical axis in the circular graph of (1).Thm.2 If sequence ( 1 ) is self complementary, then n - 2n0,Thm.3 Let sequence ( 1 ) be self converse and complementary, then(a) If HO is odd number, the symmetric axis of the circular graph (1) must pass through a pair of complementary diagonal elements;(b) If n0 is even number, the symmetric axis of the circular graph of ( 1 ) will either pass through a pair of equal diagonal elements or not pass through any element at all.Thm.4 If sequence ( 1 ) is self complementary, all the pairs of diagonal elements of the circular graph of ( 1 ) are either complementary or equal. Form. The number of self converse and complementary sequence, in which all the periods are n, is computed according to the following formula:Finally, an algolithm for constructing all the self converse and complementary sequence is given.