A Unified Construction of Perfect Polyphase Sequences

Polyphase sequences over N-th complex roots of unity are considered. A sequence is perfect if all its out-of-phase periodic autocorrelation equal zero. Numerous constructions of perfect polyphase sequences (PPS) have been proposed due to their importance in various applications such as pulse compression radar, fast-startup equalization and channel estimation, and spread spectrum multiple access systems. We show that all previous PPS constructions, known to us, can be classified into four classes: (i) generalized Frank sequences due to Kumar, Scholtz and Welch (1985), (ii) generalized chirp-like polyphase sequences due to Popovic (see IEEE Trans. Inform. Theory, vol.IT-38, p.1406, 1992), (iii) Milewski (1983) sequences, and (iv) PPS associated with the general construction of the generalized bent function due to Chung and Kumar (1989). The key result is a unified construction of PPS which includes the above four classes as special cases. Only explicit constructions of PPS are considered.