低碰撞区跳频序列部分汉明相关特性及其应用研究

The design for frequency hopping sequences with good Hamming correlation properties is always one of the important research topics in frequency hopping/time hopping spread spectrum communications. There is a large amount of previous literature related with the period Hamming correlation function of frequency hopping sequences, while few research results are on the partial Hamming correlation function because of the difficulty. For practical applications, the partial Hamming correlation properties of frequency hopping sequences play a major role in the performance of frequency hopping spread spectrum communication system. Low hit zone frequency hopping sequences is a new research direction in recent years, which has attracted many attentions of the scholars in the world. In this project, we will study the partial Hamming correlation properties and the applications of frequency hopping sequences with low hit zone, which is a new topic. The research topics include: (1) the theoretic bound on the partial Hamming correlation of frequency hopping sequences with low hit zone; (2) the design for low hit zone frequency hopping sequences with optimal partial Hamming correlation over a polynomial residue class ring; (3) the design for low hit zone frequency hopping sequences with optimal partial Hamming correlation based on the cyclotomy over finite fields; (4) the design for low hit zone frequency hopping sequences with optimal partial Hamming correlation based on the interleaving technology; (5) the design for low hit zone frequency hopping sequences with optimal partial Hamming correlation based on the special functions over finite field; (6) the applications of new low hit zone frequency hopping sequences in the quasi-synchronous frequency hopping/time hopping spread spectrum communication system.

设计具有优异特性的跳频序列一直是跳频（时）扩频通信领域重要研究课题之一。有关跳频序列周期汉明相关函数的研究成果较多，而对跳频序列部分汉明相关函数的研究成果较少，因为后者的研究难度更大。对于实际应用，跳频序列的部分汉明相关特性直接影响跳频扩频通信系统性能。低碰撞区跳频序列是近年新出现的一个研究方向，现已引起国内外许多学者的重视。本课题对低碰撞区跳频序列的部分汉明相关特性及其应用进行研究，这是一个新课题。主要研究内容包括：(1)低碰撞区跳频序列部分汉明相关函数理论界；(2)多项式剩余类环上最优部分汉明相关的低碰撞区跳频序列设计；(3)基于有限域上分圆理论的最优部分汉明相关的低碰撞区跳频序列设计；(4)基于交织技术的具有最优部分汉明相关的低碰撞区跳频序列集设计；(5)利用有限域上的特殊函数构造最优部分汉明相关的低碰撞区跳频序列；(6)新型低碰撞区跳频序列在准同步跳频（时）扩频通信系统中的应用。

本项研究主要对低碰撞区跳频序列的部分汉明相关特性进行研究，取得丰富的研究成果。（1）获得了一系列低碰撞区跳频序列集的理论界，包括：几个常规跳频序列集的新理论界；跳频序列非周期汉明相关函数的新下界；几个低碰撞区跳频序列集非周期汉明相关函数的新下界；跳频序列集的非周期汉明相关的上界；跳频序列集平均周期部分汉明相关函数的理论下界；跳频序列集周期部分汉明相关函数理论界；低碰撞区跳频序列集最大周期汉明相关函数理论界；新的跳频序列集的Singleton界；无碰撞区跳频序列理论界。（2）基于中国剩余定理构造了几个最优低碰撞区跳频序列集。（3）基于分圆理论构造了几个最优低碰撞区跳频序列集。（4）基于交织技术构造了一系列具有最优部分汉明相关的低碰撞区跳频序列集。（5）基于有限域上多项式理论构造了几个最优跳频序列集。（6）基于笛卡尔积构造了几个最优低碰撞区跳频序列集。（7）分别基于m-序列、GM序列、GGMW序列、Reed-Solomon码，构造了几个最优低碰撞区跳频序列集。（8）研究了低碰撞区跳频序列的最大部分汉明相关、平均汉明相关对准同步跳频扩频通信系统性能的影响。本课题对低碰撞区跳频序列的部分汉明相关特性的许多研究结果，填补了国内外空白。该项研究不但具有重要的理论意义，而且对于推动准同步跳频扩频通信系统的实际应用，具有关键作用。