Optimal frequency hopping sequences: auto- and cross-correlation properties

  • Authors:

  • Gennian Ge;Ying Miao;Zhongxiang Yao

  • Affiliations:

  • Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, China;Department of Social Systems and Management, Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba, Ibaraki, Japan;Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, China

  • Venue:
  • IEEE Transactions on Information Theory
  • Year:
  • 2009

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Abstract

Frequency hopping (FH) sequences play a key role in frequency hopping spread spectrum communication systems. In order to evaluate the performance of FH sequences, Lempel and Greenberger (1974) and Peng and Fan (2004) derived lower bounds on their Hamming auto- and cross-correlations. In this paper, we construct families of FH sequences with Hamming correlations meeting those bounds by combinatorial and algebraic techniques. We first construct optimal families consisting of a single FH sequence with maximum Hamming correlation equal to 2 from a combinatorial approach. Then we investigate families consisting of multiple FH sequences. We provide a combinatorial characterization for such families, and present a recursive method to construct them by means of this characterization. We also describe two algebraic constructions for such families of FH sequences, generalizing those of Ding, Moisio, and Yuan (2007). As a consequence, many new optimal families of FH sequences are obtained.