Design And Analysis Of Complementary Sequences With Zero Correlation Zone And Perfect Sequences Over Small Integers

The correlation (including periodic/aperiodic correlation, and auto-correlation/cross-correlation correlation) function of the spreading sequences which are employed in code-division multiple-access (CDMA) communication systems play a critical role on the capability of reducing multiple access interference (MAI), the multipath interference (MI), and adjacent cell interference (ACI), therefore directly influence the performance and capacity of the systems. Existing research on spreading sequences mainly focuses on three aspects, i.e. the theoretic sequence bounds, the sequence design and the sequence applications respectively. In this thesis, complementary sequences with zero correlation zone (ZCZ) and perfect sequences over small integers are investigated concretely, that is,(1) upper bound on zero correlation zone width of aperiodic binary Z-complementary pairs of odd length,(2) existence of aperiodic Z-complementary pairs of binary and quadriphase sequences,(3) constructions of aperiodic quadriphase and four-level Z-complementary sets and their mates, and (4) perfect sequences over small integers and ternary perfect sequences with a few zero elements.By extending the ZCZ concept to complementary sequences, aperiodic Z-complementary sequences (including binary/quadriphase/four-level) are investigated. It is shown that an upper bound on zero correlation zone width of aperiodic Z-complementary pairs of binary sequences of odd length N is (N+1)/2. Moreover, necessary conditions for aperiodic Z-complementary pairs of binary sequences are given.A new recursive construction of aperiodic Z-complementary pairs is proposed. That is, aperiodic Z-complementary pairs of longer length are constructed from aperiodic Z-complementary pairs of shorter one by using two pairs of sequences that satisfied certain conditions. It is proved that there exist aperiodic Z-complementary pairs of binary sequences of any length with ZCZ widths2,3,4,5and6. It is also proved that there exist aperiodic Z-complementary pairs of quadriphase sequences of any length with ZCZ widths2,3and4.Elementary operations on aperiodic Z-complementary pairs and the representatives of the equivalence class of them are proposed. The set of Z-complementary pairs of some fixed lengths is determined by the set of inequivalent representatives. Based on computer search, a specific representative for each aperiodic quadriphase (and four-level) Z-complementary pair of length N≤9, its maximum ZCZ width, and Summed ACF are given. Binary Z-complementary pairs are compared with quadriphase and four-level Z-complementary ones. It is shown that the aperiodic quadriphase and four-level Z-complementary pairs are normally better than aperiodic binary ones of the same length, in terms of the number of Z-complementary pairs and the maximum ZCZ width. Besides, the aperiodic quadriphase Z-complementary pairs are nearly the same as aperiodic four-level Z-complementary pairs in terms of the maximum ZCZ width.By modifying or improving the original methods of constructing complementary sets and their mates, constructions of aperiodic quadriphase (and four-level) Z-complementary sets and their mates are given. Generally speaking, the shorter ZCZ width, the more mates of an aperiodic quadriphase Z-complementary pair.Perfect sequences are studied by using matrix method. The relation between multilevel cyclic Hadamard matrices and multilevel perfect sequences is investigated. A sequence is a multilevel perfect sequence if and only if the cyclic matrix associated with the sequence is a multilevel cyclic Hadamard matrix. Necessary conditions for perfect sequences over integers are proposed. It is shown that, the sum of squares of all elements of perfect sequence of odd length over integers is a square number. The existence of perfect sequences over the alphabets{±1,±2} and {±1,±3} is also investigated. A sufficient condition for perfect sequences over integers is given, and two-level perfect sequences of length3,4,6and7are obtained.Ternary perfect sequences with a few zero elements are studied, and their existence is also investigated. Necessary conditions for ternary perfect sequences with k zero elements are given. It is proved that there exist no ternary perfect sequences of odd lengths with one zero element or two zero elements. It is also proved that there exist no ternary perfect sequences of any length with two and above adjacent zero elements. An efficient search algorithm for ternary perfect sequences is given. The result concerning the non-existence of ternary perfect sequences of lengths less than49with one zero element are obtained. The non-existence of ternary perfect sequences of lengths less than37with two zero elements is determined, but excluding the case of length6.