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以及作为电磁和声学成像的孔径函数得到应用。二维最佳二进阵列在二维同步和时频编码中的应用。这些也用于信道编码和密码编码。该研究涉及到对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、雷达和音频编码是应用这些对象的其他学科。研究者将字母表扩展到三元和四元情况：相应的字母表为{0，+1，-1}和{i，-i，+1，-1}。这里i表示单位的复数四次根。所研究的问题涉及最佳二元阵列、最佳三元阵列、最佳四元阵列、几乎最佳阵列和周期互补序列。调查研究这些通过他们的代数公式，即差集，补充差集和可分差集。利用适当的交换群上群环元的特征标和证明了新的结构定理。这些结果将能够产生新的类的序列和阵列与所需的相关性。

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.