Near perfect sequences of odd and even lengths.

Abstract

Sequences having zero correlation zones are of vital importance in applications, for example, wireless communications and quasi-synchronous CDMA systems. In this thesis, we construct new near perfect sequences of odd and even lengths, and those of odd lengths are constructed here, for the first time. Near perfect sequences are a special category of zero correlation zone sequences. Near perfect sequences over roots of unity have many potential applications, in cellular communication systems, radar, position sensing and ultrasonic imaging. Our contribution consists of three parts. Part I: First, we provide a method of constructing new near perfect sequences of odd lengths, over the roots of unity. We use a completely orthogonal pair of sequences and a shift sequence obtained by folding a binary M-sequence of length 2^2J-1, row-wise, into a (2^J-1)×(2^J+1) array for J≥2. We provide many examples of near perfect sequences of odd lengths, illustrating our construction. We prove our main result, that near perfect sequences over the m^th roots of unity, m any odd prime, can be constructed for unbounded lengths. These lengths guarantee arbitrarily long zero correlation zones. Global Position Sensing (GPS) uses sequences of lengths approximately equal to 2^40. Our method can be used to construct new near perfect sequences of similar lengths, with very large zero correlation zones, since our construction produces near perfect sequences of unbounded lengths. To examine the universality of our construction, we perform an exhaustive computer search of near perfect sequences of length 15. We found that, of all sequences obtained, one third of the sequences were equivalent to our constructed sequences. We classify these sequences into three types. We also present a variation of our first construction of near perfect sequences. We obtain shift sequences by the method of folding M-sequences of length 2^2J-1, diagonally (rather than row-wise) into arrays with co-prime sizes. Part II: Next, we construct new near perfect sequences of even lengths, by concatenating two distinct near perfect sequences of the same odd length, under some given conditions. Part III: Finally, we examine some cross correlation sequences of some of our near perfect sequences of odd lengths. We find that, for each of our new near perfect sequences, say, s, the cross correlation between s and s^* is also near perfect.