Optimal phases for a family of quadriphase CDMA sequences

Signature sequences with good even even and odd (or polyphase) correlations are crucial for asynchronous code-division multiple access (CDMA). When the data sequence is random, the even and odd (or polyphase) correlations are equally important. However, for most known signature sequences, only their even correlations were analyzed. It appears that determining the odd (or the polyphase) correlations is generally a very hard problem since the odd (or the polyphase) correlations depend on the phases of the signature sequences. Sole (1989), Boztas, Hammons, and Kumar (1992) found a family of quadriphase sequences that are asymptotically optimal. These sequences gain a factor √2 in terms of their maximum periodic even correlations when compared with the best possible binary phase-shift keying (BPSK) sequences. We find the optimal phases of these sequences. The optimality is in the sense that at these phases, the mean square values of the even, odd, and the polyphase correlations are minimal, and achieve the Welch (1974) bound-equality simultaneously. Furthermore, we show that at these phases, the average user interference of these sequences is always smaller than that of the ideal random signature sequences. Comprehensive analytical and numerical results show that good phase sequences can offer a nonnegligible amount of gain over bad phase sequences at modest and high signal-to-noise ratios