Bounds on the Cross-Correlation Functions of State m-Sequences

The performance of the periodic Hamming correlation function of the sequences of states of -sequence simple linear feedback shift register generators is evaluated. We call these sequences state sequences. Theoretical lower and upper bounds on the peaks of the Hamming cross-correlation functions of those sequences, which are often used in frequency-hopped spread-spectrum systems, are derived. Numerical results in support of the theory are presented, and it is shown that the worst-case shift of the cross-correlation for relatively short sequences is of burst nature. The state position mapped (SPM) sequences of the state -sequences are introduced. Their suitability for OR-channel code division multiplexing is numerically examined. The behavior of the periodic crosscorrelation function of these SPM sequences is used as the basis for judging their performance. It is explained also that the lower bound on the peak of the periodic Hamming cross-correlation function of the state -sequences is naturally a lower bound on the peak of the periodic crosscorrelation function of SPM state -sequences. This is due to the general relationship (a decimation relationship) between the two types of correlation functions. This relationship is explained and illustrated by an example. Tables of the peaks of both cross-correlation functions are presented for all sets of sequences for sequence generators of 5, 6, 7, and 8 stages.