Effects of bias and characteristic phase on the crosscorrelation of m-sequences

The paper examines the origin of high crosscorrelation between m-sequences of the same length by exhaustive examination of all pairs of sequences for n<18. It is shown that, for composite lengths, there exist forced agreements between terms of the sequences. This leads to long-range order or correlation bias for particular decimations relating the sequences and the relative sequence phases. Decimations leading to biased correlations are predictable. For such bias, one of the sidelobes occurs for characteristic phase alignment. The small, but finite, discrepancy between the predicted and observed sidelobe maxima is analysed and is found to be due to the partitioning of sequence terms at characteristic phase. Sequences of nonprime length are found to exhibit considerable diversity in their cyclotomic set structure and closed-form expressions for the crosscorrelation at characteristic-phase alignment. By contrast, these are particularly simple for Mersenne-prime lengths, where it is found that suppression of peak sidelobes by at least 3 dB is achieved by avoiding characteristic-phase alignment. The impact of the theory developed in the paper on CDMA is that sequences with biased correlation, and hence high sidelobes, are predictable and therefore avoidable