American Institute of Mathematical Sciences

August  2013, 7(3): 293-310. doi: 10.3934/amc.2013.7.293

New classes of optimal frequency hopping sequences with low hit zone

 1 School of Mathematics and Computer Engineering, The Key Laboratory of Network Intelligent Information Processing, Xihua University, Chengdu, Sichuan 610039, China 2 School of Information Science and Technology, Southwest Jiaotong University, Chengdu, Sichuan 610031, China 3 School of Mathematics, Southwest Jiaotong University, Chengdu, 610031

Received  July 2012 Revised  February 2013 Published  July 2013

In this paper, a new design of frequency hopping sequences (FHSs) sets with low hit zone (LHZ) is presented based on interleaving technique. The key idea of the new design is to use short FHSs with good Hamming correlation together with certain appropriate shift sequences to construct a set of long FHSs with LHZ. By the new design, new sets of FHSs meeting the Peng-Fan-Lee bound are obtained. It is shown that all the sequences in the proposed FHS sets are shift distinct. The proposed FHS sets are suitable for quasi-synchronous frequency hopping code division multiple access systems to eliminate multiple-access interference.
Citation: Xianhua Niu, Daiyuan Peng, Zhengchun Zhou. New classes of optimal frequency hopping sequences with low hit zone. Advances in Mathematics of Communications, 2013, 7 (3) : 293-310. doi: 10.3934/amc.2013.7.293
References:
 [1] W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inf. Theory, 51 (2005), 1139-1141. doi: 10.1109/TIT.2004.842708. [2] J. H. Chung, Y. K. Han and K. Yang, New classes of optimal frequency-hopping sequences by interleaving technique, IEEE Trans. Inf. Theory, 55 (2009), 5783-5791. doi: 10.1109/TIT.2009.2032742. [3] J. H. Chung and K. Yang, Optimal frequency-hopping sequences with new parameters, IEEE Trans. Inf. Theory, 56 (2010), 1685-1693. doi: 10.1109/TIT.2010.2040888. [4] C. Ding, R. Fuji-Hara, Y. Fujiwara, et al., Sets of frequency hopping sequences: Bounds and optimal constructions, IEEE Trans. Inf. Theory, 55 (2009), 3297-3304. doi: 10.1109/TIT.2009.2021366. [5] C. Ding, T. Helleseth and H. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Trans. Inf. Theory, 47 (2001), 428-433. doi: 10.1109/18.904555. [6] C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency hopping sequences, IEEE Trans. Inf. Theory, 53 (2007), 2606-2610. doi: 10.1109/TIT.2007.899545. [7] C. Ding and J. Yin, Sets of optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 54 (2008), 3741-3745. doi: 10.1109/TIT.2008.926410. [8] P. Z. Fan and M. Darnell, "Sequence Design for Communications Applications,'' RSP-John Wiley Sons Inc., London, 1996. [9] P. Z. Fan, M. H. Lee and D. Y. Peng, New family of hopping sequences for time/frequency hopping CDMA systems, IEEE Trans. Wir. Commun., 4 (2005), 2836-2842. [10] R. Fuji-Hara, Y. Miao and M. Mishima, Optimal frequency hopping sequences: A combinatorial approach, IEEE Trans. Inf. Theory, 50 (2004), 2408-2420. doi: 10.1109/TIT.2004.834783. [11] G. Ge, R. Fuji-Hara and Y. Miao, Further combinatorial constructions for optimal frequency hopping sequences, J. Combin. Theory Ser. A, 113 (2006), 1699-1718. doi: 10.1016/j.jcta.2006.03.019. [12] G. Ge, Y. Miao and Z. Yao, Optimal frequency hopping sequences: auto- and crosscorrelation properties, IEEE Trans. Inf. Theory, 55 (2009), 867-879. doi: 10.1109/TIT.2008.2009856. [13] G. Gong, Theory and applications of q-ary interleaved sequences, IEEE Trans. Inf. Theory, 41 (1995), 400-411. doi: 10.1109/18.370141. [14] G. Gong, New designs for signal sets with low cross correlation, balance property and large linear span: GF(p) case, IEEE Trans. Inf. Theory, 48 (2002), 2847-2867. doi: 10.1109/TIT.2002.804044. [15] S. Hong, C. Seol and K. Cheun, Performance of soft decision decoded synchronous FHSS multiple access networks using MFSK modulation under rayleigh fading, IEEE Trans. Commun., 59 (2011), 1066-1077. [16] H. D. Jia, D. Yuan, D. Y. Peng, et al., On a general class of quadratic hopping sequences, Sci. China Ser. F, 12 (2008), 2101-2114. doi: 10.1007/s11432-008-0136-8. [17] N. R. Lanka, S. A. Patnaik and R. A. Harjani, Frequency-hopped quadrature frequency synthesizer in 0.13-$\mu$m technology, IEEE J. Solid-State Circuits, 46 (2011), 1-12. [18] A. Lempel and H. Greenberger, Families of sequence with optimal Hamming correlation properties, IEEE Trans. Inf. Theory, 20 (1974), 90-94. [19] W. P. Ma and S. H. Sun, New designs of frequency hopping sequences with low hit zone, Des. Codes Crypt., 60 (2010), 145-153. doi: 10.1007/s10623-010-9422-8. [20] X. H. Niu, D. Y. Peng and Z. C. Zhou, New classes of optimal LHZ FHS with new parameters, in "The Sixth International Workshop on Signal Design and Its Applications in Communications (IWSDA'11),'' Guilin, China, (2011), 10-14. [21] D. Y. Peng and P. Z. Fan, Lower bounds on the Hamming auto- and cross correlations of frequency hopping sequences, IEEE Trans. Inf. Theory, 50 (2004), 2149-2154. doi: 10.1109/TIT.2004.833362. [22] D. Y. Peng, P. Z. Fan and M. H. Lee, Lower bounds on the periodic Hamming correlations of frequency hopping sequences with low hit zone, Sci. China Ser. F, 49 (2006), 1-11. doi: 10.1007/s11432-006-0208-6. [23] H. Shao and N. Beaulieu, Direct sequence and time-hopping sequence designs for narrow band interference mitigation in impulse radio UWB systems, IEEE Trans. Commun., 59 (2011), 1957-1965. [24] M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, "Spread Spectrum Communications Handbook,'' McGraw-Hill, New York, 1994. [25] P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings, IEEE Trans. Inf. Theory, 44 (1998), 1492-1503. doi: 10.1109/18.681324. [26] P. Udaya and X. Tang, Low correlation zone sequences from interleaved construction, IEICE Trans. Fund., 93-A (2010), 2220-2226. [27] X. N. Wang and P. Z. Fan, A class of frequency hopping sequences with no hit zone, in "Proc. 4th International Conference on Parallel and Distributed Computing, Applications and Technologies,'' (2003), 896-898. [28] W. X. Ye and P. Z. Fan, Two classes of frequency hopping sequences with no-hit zone, in "Proc. 7th Int. Symp. on Communication Theory and Applications,'' Ambleside, UK, (2003), 304-306. [29] W. X. Ye and P. Z. Fan, Construction of frequency hopping sequences with no hit zone, J. Electronics (China), 24 (2007), 305-308. doi: 10.1007/s11767-005-0202-y. [30] W. X. Ye, P. Z. Fan and E. M. Gabidulin, Construction of non-repeating frequency-hopping sequences with no-hit zone, Electronics Letters, 42 (2006), 681-682. doi: 10.1049/el:20060775. [31] Q. Zeng, H. S. Li, Z. H. Zhang, et al., A frequency-hopping based communication infrastructure for wireless metering in smart grid, in "45th Annual Conference on Information Sciences and Systems (CISS),'' (2011), 23-25. [32] Z. C. Zhou, Z. Pan and X. H. Tang, New families of optimal zero correlation zone sequences based on interleaved technique and perfect sequences, IEICE Trans. Fund., 91 (2008), 3691-3697. [33] Z. C. Zhou, X. H. Tang and G. Gong, A new class of sequences with zero or low correlation zone based on interleaving technique, IEEE Trans. Inf. Theory, 54 (2008), 4267-4273. doi: 10.1109/TIT.2008.928256.

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References:
 [1] W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inf. Theory, 51 (2005), 1139-1141. doi: 10.1109/TIT.2004.842708. [2] J. H. Chung, Y. K. Han and K. Yang, New classes of optimal frequency-hopping sequences by interleaving technique, IEEE Trans. Inf. Theory, 55 (2009), 5783-5791. doi: 10.1109/TIT.2009.2032742. [3] J. H. Chung and K. Yang, Optimal frequency-hopping sequences with new parameters, IEEE Trans. Inf. Theory, 56 (2010), 1685-1693. doi: 10.1109/TIT.2010.2040888. [4] C. Ding, R. Fuji-Hara, Y. Fujiwara, et al., Sets of frequency hopping sequences: Bounds and optimal constructions, IEEE Trans. Inf. Theory, 55 (2009), 3297-3304. doi: 10.1109/TIT.2009.2021366. [5] C. Ding, T. Helleseth and H. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Trans. Inf. Theory, 47 (2001), 428-433. doi: 10.1109/18.904555. [6] C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency hopping sequences, IEEE Trans. Inf. Theory, 53 (2007), 2606-2610. doi: 10.1109/TIT.2007.899545. [7] C. Ding and J. Yin, Sets of optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 54 (2008), 3741-3745. doi: 10.1109/TIT.2008.926410. [8] P. Z. Fan and M. Darnell, "Sequence Design for Communications Applications,'' RSP-John Wiley Sons Inc., London, 1996. [9] P. Z. Fan, M. H. Lee and D. Y. Peng, New family of hopping sequences for time/frequency hopping CDMA systems, IEEE Trans. Wir. Commun., 4 (2005), 2836-2842. [10] R. Fuji-Hara, Y. Miao and M. Mishima, Optimal frequency hopping sequences: A combinatorial approach, IEEE Trans. Inf. Theory, 50 (2004), 2408-2420. doi: 10.1109/TIT.2004.834783. [11] G. Ge, R. Fuji-Hara and Y. Miao, Further combinatorial constructions for optimal frequency hopping sequences, J. Combin. Theory Ser. A, 113 (2006), 1699-1718. doi: 10.1016/j.jcta.2006.03.019. [12] G. Ge, Y. Miao and Z. Yao, Optimal frequency hopping sequences: auto- and crosscorrelation properties, IEEE Trans. Inf. Theory, 55 (2009), 867-879. doi: 10.1109/TIT.2008.2009856. [13] G. Gong, Theory and applications of q-ary interleaved sequences, IEEE Trans. Inf. Theory, 41 (1995), 400-411. doi: 10.1109/18.370141. [14] G. Gong, New designs for signal sets with low cross correlation, balance property and large linear span: GF(p) case, IEEE Trans. Inf. Theory, 48 (2002), 2847-2867. doi: 10.1109/TIT.2002.804044. [15] S. Hong, C. Seol and K. Cheun, Performance of soft decision decoded synchronous FHSS multiple access networks using MFSK modulation under rayleigh fading, IEEE Trans. Commun., 59 (2011), 1066-1077. [16] H. D. Jia, D. Yuan, D. Y. Peng, et al., On a general class of quadratic hopping sequences, Sci. China Ser. F, 12 (2008), 2101-2114. doi: 10.1007/s11432-008-0136-8. [17] N. R. Lanka, S. A. Patnaik and R. A. Harjani, Frequency-hopped quadrature frequency synthesizer in 0.13-$\mu$m technology, IEEE J. Solid-State Circuits, 46 (2011), 1-12. [18] A. Lempel and H. Greenberger, Families of sequence with optimal Hamming correlation properties, IEEE Trans. Inf. Theory, 20 (1974), 90-94. [19] W. P. Ma and S. H. Sun, New designs of frequency hopping sequences with low hit zone, Des. Codes Crypt., 60 (2010), 145-153. doi: 10.1007/s10623-010-9422-8. [20] X. H. Niu, D. Y. Peng and Z. C. Zhou, New classes of optimal LHZ FHS with new parameters, in "The Sixth International Workshop on Signal Design and Its Applications in Communications (IWSDA'11),'' Guilin, China, (2011), 10-14. [21] D. Y. Peng and P. Z. Fan, Lower bounds on the Hamming auto- and cross correlations of frequency hopping sequences, IEEE Trans. Inf. Theory, 50 (2004), 2149-2154. doi: 10.1109/TIT.2004.833362. [22] D. Y. Peng, P. Z. Fan and M. H. Lee, Lower bounds on the periodic Hamming correlations of frequency hopping sequences with low hit zone, Sci. China Ser. F, 49 (2006), 1-11. doi: 10.1007/s11432-006-0208-6. [23] H. Shao and N. Beaulieu, Direct sequence and time-hopping sequence designs for narrow band interference mitigation in impulse radio UWB systems, IEEE Trans. Commun., 59 (2011), 1957-1965. [24] M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, "Spread Spectrum Communications Handbook,'' McGraw-Hill, New York, 1994. [25] P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings, IEEE Trans. Inf. Theory, 44 (1998), 1492-1503. doi: 10.1109/18.681324. [26] P. Udaya and X. Tang, Low correlation zone sequences from interleaved construction, IEICE Trans. Fund., 93-A (2010), 2220-2226. [27] X. N. Wang and P. Z. Fan, A class of frequency hopping sequences with no hit zone, in "Proc. 4th International Conference on Parallel and Distributed Computing, Applications and Technologies,'' (2003), 896-898. [28] W. X. Ye and P. Z. Fan, Two classes of frequency hopping sequences with no-hit zone, in "Proc. 7th Int. Symp. on Communication Theory and Applications,'' Ambleside, UK, (2003), 304-306. [29] W. X. Ye and P. Z. Fan, Construction of frequency hopping sequences with no hit zone, J. Electronics (China), 24 (2007), 305-308. doi: 10.1007/s11767-005-0202-y. [30] W. X. Ye, P. Z. Fan and E. M. Gabidulin, Construction of non-repeating frequency-hopping sequences with no-hit zone, Electronics Letters, 42 (2006), 681-682. doi: 10.1049/el:20060775. [31] Q. Zeng, H. S. Li, Z. H. Zhang, et al., A frequency-hopping based communication infrastructure for wireless metering in smart grid, in "45th Annual Conference on Information Sciences and Systems (CISS),'' (2011), 23-25. [32] Z. C. Zhou, Z. Pan and X. H. Tang, New families of optimal zero correlation zone sequences based on interleaved technique and perfect sequences, IEICE Trans. Fund., 91 (2008), 3691-3697. [33] Z. C. Zhou, X. H. Tang and G. Gong, A new class of sequences with zero or low correlation zone based on interleaving technique, IEEE Trans. Inf. Theory, 54 (2008), 4267-4273. doi: 10.1109/TIT.2008.928256.
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