EFFECTS OF BIAS AND CHARACTERISTIC PHASE ON THE CROSS-CORRELATION OF M-SEQUENCES

The paper examines the origin of high crosscorrelation between m-seque nces of the same length by exhaustive examination of all pairs of sequ ences 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 th e sequences and the relative sequence phases. Decimations leading to b iased correlations are predictable. For such bias, one of the sidelobe s occurs for characteristic phase alignment. The small, but finite, di screpancy between the predicted and observed sidelobe maxima is analys ed and is found to be due to the partitioning of sequence terms at cha racteristic phase. Sequences of nonprime length are found to exhibit c onsiderable diversity in their cyclotomic set structure and closed-for m expressions for the crosscorrelation at characteristic-phase alignme nt. By contrast, these are particularly simple for Mersenne-prime leng ths, where it is found that suppression of peak sidelobes by at least 3dB is achieved by avoiding characteristic-phase alignment. The impact of the theory developed in the paper on CDMA is that sequences with b iased correlation, and hence high sidelobes, are predictable and there fore avoidable.