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On Prime Root-of-Unity Sequences with Perfect Periodic Correlation
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Automatic control. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Systems and Control.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Automatic control. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Systems and Control.
2014 (English)In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 62, no 20, 5458-5470 p.Article in journal (Refereed) Published
##### Abstract [en]

In this paper, Perfect Root-of-Unity Sequences (PRUS) with entries in $\alpha_p = \{ x \in \complexC ~ |~ x^p =1\}$ (where $p$ is a prime) are studied. A lower bound on the number of distinct phases that are used in PRUS over $\alpha_p$ is derived. We show that PRUS of length $L \geq p(p-1)$ must use all phases in $\alpha_p$. Certain conditions on the lengths of PRUS are derived. Showing that the phase values of PRUS must follow a given difference multiset property, we derive a set of equations (which we call the principal equations) that give possible lengths of a PRUS over $\alpha_p$ together with their phase distributions. The usefulness of the principal equations is discussed, and guidelines for efficient construction of PRUS are provided. Through numerical results, also contributions are made to the current state-of-knowledge regarding the existence of PRUS. In particular, a combination of the developed ideas allowed us to numerically settle the problem of existence of PRUS with $(L,p)=(28,7)$ within about two weeks--- a problem whose solution (without using the ideas in this paper) would likely take more than three million years on a standard PC.

##### Place, publisher, year, edition, pages
2014. Vol. 62, no 20, 5458-5470 p.
##### National Category
Signal Processing
##### Identifiers
ISI: 000341982900021OAI: oai:DiVA.org:uu-229414DiVA: diva2:736575
Available from: 2014-08-07 Created: 2014-08-07 Last updated: 2014-10-23Bibliographically approved

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Soltanalian, MojtabaStoica, Petre
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Automatic controlDivision of Systems and Control
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IEEE Transactions on Signal Processing
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Signal Processing