Arrays over Small Phase Alphabet Having Desirable Correlation Properties

时间离散的一维和二维序列和阵列在信号处理中得到应用，并作为电磁和声学成像的孔径函数。二维最佳二进制阵列在二维同步和时频编码中有广泛的应用。它们也用于信道编码和密码编码。本研究涉及完美阵列及其相关数学对象的研究。宽带数字通信、光信号处理、雷达和音频编码是具有这些对象的应用的其他学科。调查员将字母表扩展到三进制和四进制情况：对应的字母表是{0，1，-1}和{i，-i，1，-1}。在这里，我指的是团结的复杂的第四个根源。所研究的问题涉及最佳二值阵列、最佳三值阵列、最佳四值阵列、几乎最佳阵列和周期互补序列。研究者通过它们的代数公式来研究它们，即。差集、补充差集和可除差集。利用适当交换群上群环元素的特征标和，证明了新的结构定理。这些结果将能够产生具有期望的相关特性的新的序列和阵列类别。

Time-discrete one- and two- dimensional sequences and arrays find applications in signal processing and as aperture functions for electromagnetic and acoustic imaging. Applications of two dimensional perfect binary arrays are found in 2-D synchronization and time-frequency coding. These are also usedin channel coding and cryptographic coding. This research involves the study of perfect arrays and their related mathematical objects. Wide-band digital communications, optical signal processing, radar and audio coding are other disciplines that have applications of these objects.The conventional binary case { +1, -1} as the alphabet. The investigator extends the alphabet to the ternary and quaternary cases: the corresponding alphabets being {0, +1, -1} and {i,-i,+1,-1}. Here i denotes the complex fourth roots of unity. The problems studied pertain to perfect binary arrays, perfect ternary arrays, perfect quaternary arrays, almost perfect arrays and periodic complementary sequences. The investigator studies these via their algebraic formulations viz. difference sets, supplementary difference sets and divisible difference sets. New structural theorems will be proved using character sumsof group ring elements over suitable abelian groups. These results would enable to produce new classes of sequences and arrays with desirable correlation properties.