Research On Some Perfect Discrete Signal

The perfect discrete signal and its design plays an increasingly important role inmodern communications, radar, sonar, navigation, space ranging and controlling, andelectronically countermeasures design optimization of other wired and wireless systems.Well-structured signals can enhance system's properties of anti-interference, anti-noise,anti-doubts, the decline of performance such as resistance to the system, and increase dataconfidentiality. And CDMA communications and synchronous communication andrapture can be achieved. Therefore, an in-depth study of the perfect discrete signals is ofvital importance both in theory and applications. Over the past decades, the research hasmade great achievements and further in-depth study is underway at present.Perfect discrete signal research includes period-related signal, aperiod-related signal,related signal based on pairs and other such aspects. The newly proposed principle basedon all relevant pair-based signals and perfect complementary quaternary arrays isdiscussed in this thesis. This thesis only studied the theoretical issues rather than itsengineering applications.In the research into perfect binary array pairs and difference set pairs, it was provedthat perfect binary array pairs exist when sequence length multiplier for four;question tothe uniqueness of perfect binary array pairs was presented and proved. The method usedin the solution to this problem can be extended to other perfect signal based on pairs.Based on the equivalence relation between difference set pairs and perfect binary arrayspairs, an in-depth research into difference set pairs was done and as a result, sevenconstruction methods of difference set pairs were given. The first and second multipliertheorems based on difference set pairs were given and proved, which could find somedifference set pairs non-existent and could construct new difference set pairs. Computersearch algorithm was presented. Multiplier guess of difference sets was introduced intothe multiplier guess of difference set pairs. Accordingly, search algorithm was improved,resulting in several examples of difference set pairs up to the rang of 100 unknown toanyone up till now and further confirmed the feasibility of multiplier guess.In the research into perfect quaternary array pairs, perfect quaternary array pairs andits related characters were discussed;the equivalence relation between perfect quaternaryarray pairs and its polynomial character was established and the construction methods of alot of perfect quaternary array pairs by using arrays transform, recursion, period product,perfect binary array pairs were proposed.In the research of perfect complementary binary sequence pairs, polynomialcharacter of perfect complementary binary sequence pairs was given;two constructionmethods by using difference set pairs and finite field theory were proposed. The twomethods can construct perfect complementary binary sequence pairs conveniently andquickly, and meanwhile a preliminary attempt to study perfect signals based on pairs byusing the modern mathematical theory was made as well.In the research of perfect complementary quaternary arrays, the equivalent relationsbetween perfect complementary quaternary arrays and family of relative difference setpairs was established, thus providing perfect complementary quaternary arrays studieswith new mathematical tools. The construction of perfect complementary quaternaryarrays by using array transform, orthogonal arrays, perfect quaternary arrays, periodproduct, period product complementary companion arrays, by recursion and othermethods was presented. And finally a special category of perfect complementaryquaternary arrays and two-dimensional canonical perfect complementary quaternaryarrays was put forward. This array can not only be used as constructing methods forperfect complementary quaternary arrays but also be able to construct new perfectcomplementary quaternary arrays.In the research of families of dyadic complementary sequences pairs, a new categoryof perfect signal --families of dyadic complementary sequences pairs were discussed;necessary conditions of families of dyadic complementary sequences pairs and equalweight families of dyadic complementary sequences pairs were provided;a new conceptof combination design --family of dyadic addition set pairs was proposed, and theequivalence relations between families of dyadic complementary sequence pairs andfamily of dyadic addition set pairs were established;the two construction methods byusing sequence transform and dyadic complementary companion sequence pairs werepresented.Constructing methods of perfect signal proposed in this thesis are achieved throughcomputer algorithms and with examples elaborated.