Perfect Roots-of-Unity Sequences (PRUS) / Circulant Hadamard Matrices of Butson Type / Constant-Amplitude-Zero-Auto-Correlation (CAZAC) Sequences

Numerous constructions of PRUS, CAZAC, or equivalently, circulant complex Hadamard matrices, were discovered (and re-discovered) by many researchers in the last 40 years. Known constructions by other researchers can be classified into the following four inequivalent classes: 1) the generalized Frank sequences of Kumar, Scholtz and Welch; 2)  the generalized chirp-like polyphase sequences of Popovic; 3)  the Milewski sequences; and 4)  the (generalized bent functions equivlalent) PRUS of Chung and Kumar. We derived a sinlge PRUS contruction that unifies these 4 classes. Not only is our unified construction very general, it also has a very simple closed-form expression. An exhaustive search proves that our unified PRUS includes all PRUS with alphabet size A ≤ 15, sequence length N ≤ 20 and NA ≤ 1111. To the best of our knowledge, no new PRUS that is not a special case of our unified construction presented in ISSSTA'1996 has been discovered up to now. 

  1. W.H. Mow,《 A Study of Correlation of Sequences》, PhD Thesis, Dept. of Information Engineering, Chinese University of Hong Kong, May 1993.

  2. W.H. Mow, "On the Decimations of Frank Sequences", IEEE Transactions on Communications, Vol. 45, March 1995, pp. 751-753.

  3. W.H. Mow, "A Unified Construction of Perfect Polyphase Sequences", Proc. 1995 IEEE International Symposium on Information Theory (ISIT '95), Canada, September 1995, pp. 459.

  4. W.H. Mow, "A New Unified Construction of Perfect Root-of-Unity Sequences", Proc. IEEE 4th International Symposium on Spread Spectrum Techniques and Applications (ISSSTA'96), Germany, September 1996, pp. 955-959.

  5. W.H. Mow,《 Sequence Design for Spread Spectrum》, CU Press, ISBN 962-201-710-X, 1997, 110 pages. Ordering Information  

  6. W.H. Mow, "Towards a Unified Theory of Perfect Polyphase Sequences", Sequence Design and Applications for CDMA Systems, edited by Pingzhi Fan, Naoki Suehiro and Mike Darnell, ISBN 7-81-57-609-7, 2001, pp. 11-15.

  7. Pingzhi Fan and Wai Ho Mow, "On Optimal Training Sequence Design for Multiple-Antenna Systems Over Dispersive Fading Channels and Its Extensions", IEEE Transactions on Vehicular Technology, Vol.53, No.5, Sept.  2004, pp.1623-1626.

  8. Xiaohu Tang and Wai Ho Mow, "A New General Construction of Zero-Correlation-Zone Sequences Based on Interleaved Perfect Sequences", IEEE Transactions on Information Theory, vol.54, no.12, December 2008, pp. 5729-5734.

List of relevant IEEE/IEE papers in the literature