Even periodic and odd periodic complementary sequence pairs from generalized Boolean functions

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May 2013, 7(2): 113-125. doi: 10.3934/amc.2013.7.113

Even periodic and odd periodic complementary sequence pairs from generalized Boolean functions

Yang Yang ^1, , Xiaohu Tang ^1, and Guang Gong ^2,

1.	Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu, Sicuan 610031, China, China
2.	Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1

Received April 2012 Published May 2013

Abstract
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A pair of two sequences is called the even periodic (odd periodic) complementary sequence pair if the sum of their even periodic (odd periodic) correlation function is a delta function. The well-known Golay aperiodic complementary sequence pair (Golay pair) is a special case of even periodic (odd periodic) complementary sequence pair. In this paper, we presented several classes of even periodic and odd periodic complementary pairs based on the generalized Boolean functions, but which do not form Gloay pairs. The proposed sequences could be used to design signal sets, which have been applied in direct sequence code division multiple (DS-CDMA) cellular communication systems.

Keywords: Even periodic complementary sequence pair, Golay complementary pair, odd periodic complementary sequence pair, Golay sequences, generalized Boolean function..

Mathematics Subject Classification: Primary: 94A05, 60G3.

Citation: Yang Yang, Xiaohu Tang, Guang Gong. Even periodic and odd periodic complementary sequence pairs from generalized Boolean functions. Advances in Mathematics of Communications, 2013, 7 (2) : 113-125. doi: 10.3934/amc.2013.7.113

References:

[1]	L. Bömer and M. Antweiler, Periodic complementary binary sequences,, IEEE. Trans. Inf. Theory, 35 (1990), 1487. Google Scholar
[2]	J. A. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences and Reed-Muller codes,, IEEE Trans. Inf. Theory, 45 (1999), 2397. doi: 10.1109/18.796380. Google Scholar
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[4]	K. Q. Feng, P. J.-S. Shiue and Q. Xiang, On aperiodic and periodic complementary binary sequences,, IEEE Trans. Inf. Theory, 45 (1999), 296. doi: 10.1109/18.746823. Google Scholar
[5]	H. Ganapathy, D. A. Pados and G. N. Karystinos, New bounds and optimal binary signature sets-Part I: Periodic total squared correlation,, IEEE Trans. Inf. Theory, 59 (2011), 1123. Google Scholar
[6]	M. J. E. Golay, Multislit spectroscopy,, J. Opt. Soc. Amer., 39 (1949), 437. doi: 10.1364/JOSA.39.000437. Google Scholar
[7]	M. J. E. Golay, Complementary series,, IRE Trans., 7 (1961), 82. Google Scholar
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[10]	H. L. Jin, G. D. Liang, Z. H. Liu and C. Q. Xu, The necessary condition of families of odd periodic perfect complementary sequence pairs,, in, (2009), 303. doi: 10.1109/CIS.2009.227. Google Scholar
[11]	G. N. Karystinos and D. A. Pados, New bounds on the total squared correlation and optimal design of DS-CDMA binary signature sets,, IEEE Trans. Commun., 51 (2003), 48. doi: 10.1109/TCOMM.2002.807628. Google Scholar
[12]	N. Levanon, "Radar Principles,'', Wiley Interscience, (1988). Google Scholar
[13]	H. D. Lüke, Binary odd periodic complementary sequences,, IEEE Trans. Inf. Theory, 43 (1997), 365. doi: 10.1109/18.567768. Google Scholar
[14]	H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences,, IEEE Trans. Aerosp. Electron. Syst., 31 (1995), 495. doi: 10.1109/7.366335. Google Scholar
[15]	M. G. Parker, K. G. Paterson and C. Tellambura, Golay complementary sequences,, in, (2002). Google Scholar
[16]	K. G. Paterson, Generalized Reed-Muller codes and power control for OFDM modulation,, IEEE. Trans. Inf. Theory, 46 (2000), 104. doi: 10.1109/18.817512. Google Scholar
[17]	M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication--Part I: System analysis,, IEEE Trans. Inf. Theory, 25 (1977), 795. Google Scholar
[18]	M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication--Part II: Code sequence analysis,, IEEE Trans. Inf. Theory, 25 (1977), 800. Google Scholar
[19]	M. B. Pursley, "An Introduction to Digital Communications,'', Pearson Prentice Hall, (2005). Google Scholar
[20]	D. V. Sarwate, Meeting the Welch bound with equality,, in, (1999), 79. Google Scholar
[21]	D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences,, Proc. IEEE, 68 (1980), 593. doi: 10.1109/PROC.1980.11697. Google Scholar
[22]	H. D. Schotten, New optimum ternary complementary sets and almost quadriphase, perfect sequences,, in, (1995), 1106. Google Scholar
[23]	R. Sivaswamy, Self-clutter cancellation and ambiguity properties of subcomplementary sequences,, IEEE Trans. Aerosp. Electron. Sysr., AES-18 (1982), 163. doi: 10.1109/TAES.1982.309223. Google Scholar
[24]	C. C. Tseng and C. L. Liu, Complementary sets of sequences,, IEEE Trans. Inf. Theory, 18 (1972), 644. doi: 10.1109/TIT.1972.1054860. Google Scholar
[25]	H. Wen, F. Hu and F. Jin, Design of odd periodic complementary binary signal set,, in, 2 (2004), 590. Google Scholar

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References:

[1]	L. Bömer and M. Antweiler, Periodic complementary binary sequences,, IEEE. Trans. Inf. Theory, 35 (1990), 1487. Google Scholar
[2]	J. A. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences and Reed-Muller codes,, IEEE Trans. Inf. Theory, 45 (1999), 2397. doi: 10.1109/18.796380. Google Scholar
[3]	P. Z. Fan and M. Darnell, "Sequence Design for Communications Applications,'', Research Studies Press, (1996). Google Scholar
[4]	K. Q. Feng, P. J.-S. Shiue and Q. Xiang, On aperiodic and periodic complementary binary sequences,, IEEE Trans. Inf. Theory, 45 (1999), 296. doi: 10.1109/18.746823. Google Scholar
[5]	H. Ganapathy, D. A. Pados and G. N. Karystinos, New bounds and optimal binary signature sets-Part I: Periodic total squared correlation,, IEEE Trans. Inf. Theory, 59 (2011), 1123. Google Scholar
[6]	M. J. E. Golay, Multislit spectroscopy,, J. Opt. Soc. Amer., 39 (1949), 437. doi: 10.1364/JOSA.39.000437. Google Scholar
[7]	M. J. E. Golay, Complementary series,, IRE Trans., 7 (1961), 82. Google Scholar
[8]	M. J. E. Golay, Note on complementary series,, Proc. IRE, 50 (1962). Google Scholar
[9]	S. W. Golomb and G. Gong, "Signal Designs with Good Correlation: For Wireless Communication, Cryptography and Radar Applications,'', Cambridge Univeristy Press, (2005). doi: 10.1017/CBO9780511546907. Google Scholar
[10]	H. L. Jin, G. D. Liang, Z. H. Liu and C. Q. Xu, The necessary condition of families of odd periodic perfect complementary sequence pairs,, in, (2009), 303. doi: 10.1109/CIS.2009.227. Google Scholar
[11]	G. N. Karystinos and D. A. Pados, New bounds on the total squared correlation and optimal design of DS-CDMA binary signature sets,, IEEE Trans. Commun., 51 (2003), 48. doi: 10.1109/TCOMM.2002.807628. Google Scholar
[12]	N. Levanon, "Radar Principles,'', Wiley Interscience, (1988). Google Scholar
[13]	H. D. Lüke, Binary odd periodic complementary sequences,, IEEE Trans. Inf. Theory, 43 (1997), 365. doi: 10.1109/18.567768. Google Scholar
[14]	H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences,, IEEE Trans. Aerosp. Electron. Syst., 31 (1995), 495. doi: 10.1109/7.366335. Google Scholar
[15]	M. G. Parker, K. G. Paterson and C. Tellambura, Golay complementary sequences,, in, (2002). Google Scholar
[16]	K. G. Paterson, Generalized Reed-Muller codes and power control for OFDM modulation,, IEEE. Trans. Inf. Theory, 46 (2000), 104. doi: 10.1109/18.817512. Google Scholar
[17]	M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication--Part I: System analysis,, IEEE Trans. Inf. Theory, 25 (1977), 795. Google Scholar
[18]	M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication--Part II: Code sequence analysis,, IEEE Trans. Inf. Theory, 25 (1977), 800. Google Scholar
[19]	M. B. Pursley, "An Introduction to Digital Communications,'', Pearson Prentice Hall, (2005). Google Scholar
[20]	D. V. Sarwate, Meeting the Welch bound with equality,, in, (1999), 79. Google Scholar
[21]	D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences,, Proc. IEEE, 68 (1980), 593. doi: 10.1109/PROC.1980.11697. Google Scholar
[22]	H. D. Schotten, New optimum ternary complementary sets and almost quadriphase, perfect sequences,, in, (1995), 1106. Google Scholar
[23]	R. Sivaswamy, Self-clutter cancellation and ambiguity properties of subcomplementary sequences,, IEEE Trans. Aerosp. Electron. Sysr., AES-18 (1982), 163. doi: 10.1109/TAES.1982.309223. Google Scholar
[24]	C. C. Tseng and C. L. Liu, Complementary sets of sequences,, IEEE Trans. Inf. Theory, 18 (1972), 644. doi: 10.1109/TIT.1972.1054860. Google Scholar
[25]	H. Wen, F. Hu and F. Jin, Design of odd periodic complementary binary signal set,, in, 2 (2004), 590. Google Scholar

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