Correlation and unbalance properties of spectrally sifted collimated sequences

The construction of collections of pseudorandom binary sequences whose elements are chosen from the set (1, -1) and which have “noiselike” properties remains an important issue in spread spectrum communications systems and the search for such sequences may be methodically attacked armed with a spectral approach. Along these lines a reasonable conjecture is that the flatness of the sequence spectrum should be a statistical indication of the superior noiselike quality of the codes, as indicated by the autocorrelation function sidelobe structure. This conjecture is verified here. While one can conceive of a diversity of definitions for a measure of spectral flatness, we adopted as our metric the ratio of maximum to average spectral energy. Insight into the probability density function (PDF) of this normalised maximum spectral energy as connected to spectral flatness was obtained empirically as a prelude to focusing our work in this direction. In the process of gathering results we were startled by the appearance of apparently anomalous δ-function-like discontinuities in the empirical distribution which persisted even for a rather large number of trials. These unexpected discontinuities are explained here in terms of the random walk problem and their existence suggested to us an examination of sequences affiliated with them for possible special properties. The sequences associated with the δ-functions could be appropriately described by the name “aligned sequences”, however this designation has wide connotations in the field of molecular biology in its wide sense, and therefore we denote them instead by the term collimated sequences. When the characteristics of collimated sequences are statistically examined, they are shown to be distinct in terms of their correlation and unbalance properties, setting them apart from the spectral neighbourhood in which they are found. Our results showed that the collimated sequences have clearly higher mean unbalance, a possible explanation for the higher crosscorrelation between them which our investigation showed. This higher mean unbalance, however, challenges their superior mean autocorrelation which we discovered with surprise in the course of our study. There is apparently some special noiselike character pertaining to the collimated sequences which is not linked to unbalance and yet to be uncovered