%0 Journal Article
%T Construction of Optimal Sets of Frequency Hopping Sequences
%A Bin Wen
%J ISRN Combinatorics
%D 2013
%R 10.1155/2013/479408
%X Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are needed in FH-CDMA systems. In this paper, a construction of optimal sets of frequency hopping sequences is presented. The construction is based on the set-theoretic characterization of an optimal set of FH sequences. 1. Introduction Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are an integral part of spread-spectrum communication systems such as FH-CDMA systems (for a description of such systems, see [1]). In modern radar and communication systems, frequency-hopping (FH) spread-spectrum techniques have become popular (see [2], for example). Assume that is a set of available frequencies, called an alphabet. Let be the set of all sequences of length over . Any element of is called a frequency hopping sequence (FHS) of length over . Given two FH sequences, and , we define their Hamming correlation to be where if and if , and where and all operations among position indices are performed modulo . If , is the Hamming autocorrelation. If , is the Hamming cross correlation. 2. Lower Bounds on the Correlations of FHSs FH sequences for FH-CDMA systems are required to have good Hamming correlations and large linear span [3]; the linear span is defined to be the length of the shortest linear feedback shift register that can produce the sequence. FH sequences¡¯ design normally involves six parameters: the size of the frequency library , the sequence length , the family size of the subset , the maximum out-of-phase Hamming autocorrelations , the maximum Hamming cross correlations , and the linear span. It is generally desired that the family of FH sequences has the following properties: (1)the maximum out-of-phase Hamming autocorrelations should be as small as possible,(2)the maximum Hamming cross correlations should be as small as possible,(3)the family size for given , , , and should be as large as possible,(4)the linear span should be as large as possible. In order to evaluate the theoretical performance of the FH sequences, it is important to find some theoretical bounds for these parameters. Given , , and of , Lempel and Greenberger [4] and Peng and Fan [5] derived lower bounds on and of FH sequences in . We restate their results in this section, which will be used later as the criteria to determine whether the new FH sequences constructed in this paper are optimal or not. For any single FH sequence , let be the maximum
%U http://www.hindawi.com/journals/isrn.combinatorics/2013/479408/