Research On Gaussian Integer Complementary Pairs Based On Cyclotomy

As an important part of modern communication technology,sequence design plays an important role in many communication systems.There is a strong correlation between the performance of communication system and the performance of sequence.Therefore,the study of sequences with excellent performance has important theoretical significance and application value.Gaussian integer sequence is a special kind of complex sequence,and it’s real and imaginary parts are integers.Different from the traditional complex root sequence,the amplitude of Gaussian integer is not constant.In addition,Gauss integer has higher spectrum utilization and transmission efficiency when it is transmitted in communication system.In this paper,we study the complementary pairs and complete sequences on the Gauss integers based on the theory of cyclotomy,and obtain the following results.First of all,based on the cyclotomy theory of order 2,according to the properties and calculation rules of cyclotomy class,the sequences are assigned on the Gaussian integer set,and the necessary and sufficient conditions for the constructed sequences to be Gaussian integer complementary pairs are given.By using this condition,the complementary pairs with degree 3 and 4 can be obtained in length 2 f +1.By using this complementary pair,a class of complete Gaussian integer sequences is constructed,which has high energy efficiency and the efficiency value is infinitely close to 1.Secondly,based on the cyclotomic theory of order 4,the cyclotomy classes of order 4are combined in a certain way to construct a kind of multilevel three-value autocorrelation sequence of integer set,and the autocorrelation function distribution is given.Based on this kind of three-value autocorrelation sequence,the multilevel complementary pair on the integer set is constructed.Finally,a kind of Gauss integer complementary pair with length 4 f +1 and higher degree is constructed by filtering the multilevel complementary pair.Two construction methods of sequence length of extended Gaussian integer complementary pair are proposed.The length of the sequence obtained by these two methods is twice of the base sequence.