doi: 10.3934/amc.2021037
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Z-complementary pairs with flexible lengths and large zero odd-periodic correlation zones

1. 

School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China

2. 

State Key Laboratory of Cryptology, P. O. Box 5159, Beijing 100878, China

3. 

College of Mathematical sciences, Dezhou University, Dezhou 253023, China

4. 

School of Physics, Southwest Jiaotong University, Chengdu 610031, China

5. 

School of Mathematics and Information, China West Normal University, Nanchong 637002, China

* Corresponding author: Yong Wang

Received  January 2021 Revised  May 2021 Early access August 2021

Z-complementary pairs (ZCPs) have been widely used in different communication systems. In this paper, we first investigate the odd-periodic correlation property of ZCPs, and propose a new class of ZCPs, called ZOC-ZCPs with zero correlation zone (ZCZ) width $ Z $ and zero odd-period correlation zone (ZOCZ) width $ Z_{odd} = Z $ by horizontal concatenation of a certain combination of some known ZCPs. Particularly, based on any known Golay pair, we can generate a class of GCPs of more flexible length whose ZOCZ width is larger than a quarter of the sequence length.

Citation: Liqun Yao, Wenli Ren, Yong Wang, Chunming Tang. Z-complementary pairs with flexible lengths and large zero odd-periodic correlation zones. Advances in Mathematics of Communications, doi: 10.3934/amc.2021037
References:
[1]

A. R. AdhikaryS. MajhiZ. L. Liu and L. G. Yong, New sets of even-length binary Z-complementary pairs with asymptotic ZCZ ratio of $3/4$, IEEE Signal Processing Letters, 25 (2018), 970-973.   Google Scholar

[2]

A. R. AdhikaryZ. ZhouY. Yang and P. Fan, Constructions of cross Z-complementary pairs with new lengths, IEEE Transactions on Signal Processing, 68 (2020), 4700-4712.  doi: 10.1109/TSP.2020.3014613.  Google Scholar

[3]

C.-Y. Chen, A novel construction of Z-ccomplementary pairs based on generalized boolean functions, IEEE Signal Processing Letters, 24 (2017), 987-990.  doi: 10.1109/LSP.2017.2701834.  Google Scholar

[4]

C.-Y. Chen and S.-W. Wu, Golay complementary sequence sets with large zero correlation zones, IEEE Transactions on Communications, 66 (2018), 5097-5204.  doi: 10.1109/TCOMM.2018.2857485.  Google Scholar

[5]

L. ChenZ. LuoX. Cheng and S. Li, Golay sequence based time-domain compensation of frequency-dependent I/Q imbalance, Communications China, 11 (2014), 1-11.  doi: 10.1109/CC.2014.6878998.  Google Scholar

[6]

J. D. Coker and A. H. Tewfik, Simplified ranging systems using discrete wavelet decomposition, IEEE Transactions on Signal Processing, 58 (2010), 575-582.  doi: 10.1109/TSP.2009.2032949.  Google Scholar

[7]

P. Z. FanN. SuehiroN. Kuroyanagi and X. M. Deng, Class of binary sequences with zero correlation zone, Electronics Letters, 35 (1999), 777-779.  doi: 10.1049/el:19990567.  Google Scholar

[8]

P. FanW. Yuan and Y. Tu, Z-complementary binary sequences, IEEE Signal Processing Letters, 14 (2007), 509-512.  doi: 10.1109/LSP.2007.891834.  Google Scholar

[9]

M. J. E. Golay, Complementary series, IRE Transactions on Information Theory, IT-7(1961), 82-87. doi: 10.1109/tit.1961.1057620.  Google Scholar

[10]

G. GongF. Huo and Y. Yang, Large zero correlation zone of Golay pairs and QAM Golay pairs, Information Theory Proceedings (ISIT), 2013 IEEE International Symposium on Information Theory, (2013), 3135-3139.  doi: 10.1109/ISIT.2013.6620803.  Google Scholar

[11]

G. GongF. Huo and Y. Yang, Large zero autocorrelation zones of Golay sequences and their applications, IEEE Transactions on Communications, (2013), 3967-3979.  doi: 10.1109/TCOMM.2013.072813.120928.  Google Scholar

[12]

Z. GuY. Yang and Z. Zhou, New sets of even-length binary Z-complementary pairs, 2019 Ninth International Workshop on Signal Design and its Applications in Communications, (2019), 1-5.  doi: 10.1109/IWSDA46143.2019.8966113.  Google Scholar

[13]

T. HuY. Yang and Z. Zhou, Golay complementary sets with large zero odd-periodic correlation zones, Advances in Mathematics of Communications, 15 (2019), 23-33.  doi: 10.3934/amc.2020040.  Google Scholar

[14]

K. M. Z. IslamT. Y. Al-Naffouri and N. Al-Dhahir, On optimum pilot design for comb-type OFDM transmission over doubly-selective channels, IEEE Transactions on Communications, 59 (2011), 930-935.  doi: 10.1109/TCOMM.2011.020411.100151.  Google Scholar

[15]

J. Jedwab and M. G. Parker, Golay complementary array pairs, Designs Codes Cryptography, 44 (2007), 209-216.  doi: 10.1007/s10623-007-9088-z.  Google Scholar

[16]

H. L. JinG. D. LiangC. Q. Xu and J. B. Zhang, The necessary condition of families of odd-periodic perfect complementary sequence pairs, Second International Workshop on Education Technology Computer Science, 2 (2009), 303-307.   Google Scholar

[17]

L. Kai, L. Hao and S. Yan, Construction of 8-QAM+ odd-periodic complementary sequences and sequence sets with zero correlation zone, Electronics Communications and Networks Ⅳ, 2015. Google Scholar

[18]

X. LiP. FanX. Tang and Y. Tu, Existence of binary Z-complementary pairs, IEEE Signal Processing Letters, 18 (2010), 63-66.  doi: 10.1109/LSP.2010.2095459.  Google Scholar

[19]

Z. Liu, Y. Guan and U. Parampalli, On optimal binary Z-complementary pair of odd period, IEEE International Symposium on Information Theory, (2013), 3125–3129. doi: 10.1109/ISIT.2013.6620801.  Google Scholar

[20]

Z. LiuU. Parampalli and Y. L. Guan, On even-period binary Z-complementary pairs with large ZCZs, IEEE Signal Processing Letters, 21 (2014), 284-287.  doi: 10.1109/LSP.2014.2300163.  Google Scholar

[21]

Z. LiuU. Parampalli and Y. L. Guan, Optimal odd-length binary Z-complementary pairs, IEEE Transactions on Information Theory, 60 (2014), 5768-5781.  doi: 10.1109/TIT.2014.2335731.  Google Scholar

[22]

H. D. Lüke and H. D. Schotten, Odd-perfect, almost binary correlation sequences, IEEE Transactions on Aerospace and Electronic Systems, 31 (1995), 495-498.  doi: 10.1109/7.366335.  Google Scholar

[23]

H. D. Lüke, Binary odd-periodic complementary sequences, IEEE Transactions on Information Theory, 43 (1997), 365-367.  doi: 10.1109/18.567768.  Google Scholar

[24]

V. Nee and D. J. R, OFDM codes for peak-to-average power reduction and error correction, Global Telecommunications Conference, (1996), 740-744.   Google Scholar

[25]

K. G. Paterson, Generalized Reed-Muller codes and power control in OFDM modulation, IEEE Transactions on Information Theory, 46 (2000), 104-120.  doi: 10.1109/18.817512.  Google Scholar

[26]

A. PezeshkiA. R. CalderbankW. Moran and S. D. Howard, Doppler resilient Golay complementary waveforms, IEEE Transactions on Information Theory, 54 (2008), 4254-4266.  doi: 10.1109/TIT.2008.928292.  Google Scholar

[27]

M. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication–Part Ⅰ: System analysis, IEEE Transactions Communications, 25 (1977), 795-799.  doi: 10.1109/TCOM.1977.1093915.  Google Scholar

[28]

M. B. Pursley and P. Hall, Introduction to Digital Communications: United States Edition, Pearson Schwz Ag, 2004. Google Scholar

[29]

P. Spasojevic and C. N. Georghiades, Complementary sequences for ISI channel estimation, IEEE Transactions on Information Theory, 47 (2011), 1145-1152.  doi: 10.1109/18.915670.  Google Scholar

[30]

E. Spano and O. Ghebrebrhan, Complementary sequences with high sidelobe suppression factors for ST/MST radar applications, IEEE Transactions on Geoence Remote Sensing, 34 (1996), 317-329.  doi: 10.1109/36.485110.  Google Scholar

[31]

D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences, Proceedings of the IEEE, 68 (1980), 593-619.  doi: 10.1109/PROC.1980.11697.  Google Scholar

[32]

S. Wang and A. Abdi, MIMO ISI channel estimation using uncorrelated Golay complementary sets of polyphase sequences, IEEE Transactions on Vehicular Technology, 56 (2007), 3024-3039.  doi: 10.1109/TVT.2007.899947.  Google Scholar

[33]

J.-B. Wang, X.-X. Xie, Y. Jiao, X. Song, X. Zhao, M. Gu and M. Sheng, Optimal odd-periodic complementary sequences for diffuse wireless optical communications, Optical Engineering, 51 (2012), 095002. doi: 10.1117/1.OE.51.9.095002.  Google Scholar

[34]

H. WenF. Hu and F. Jin, Design of odd-periodic complementary binary signal set, Computers and Communications, 2004. Proceedings. ISCC 2004. Ninth International Symposium, 2 (2004), 590-593.   Google Scholar

[35]

K. K. Wong and T. O'Farrell, Application of complementary sequences in indoor wireless infrared communications, IEE Proceedings Optoelectronics, 150 (2003), 453-464.  doi: 10.1049/ip-opt:20030928.  Google Scholar

[36]

D. Wulich, Reduction of peak to mean ratio of multicarrier modulation using cyclic coding, Electronics Letters, 32 (1996), 432-432.  doi: 10.1049/el:19960286.  Google Scholar

[37]

C. Xie and Y. Sun, Constructions of even-period binary Z-complementary pairs with large ZCZs, IEEE Signal Processing Letters, 25 (2018), 1141-1145.  doi: 10.1109/LSP.2018.2848102.  Google Scholar

[38]

J. D. YangX. JinK. Y. SongJ. S. No and D. J. Shin, Multicode MIMO systems with quaternary LCZ and ZCZ sequences, IEEE Transactions on Vehicular Technology, 57 (2008), 234-2341.   Google Scholar

show all references

References:
[1]

A. R. AdhikaryS. MajhiZ. L. Liu and L. G. Yong, New sets of even-length binary Z-complementary pairs with asymptotic ZCZ ratio of $3/4$, IEEE Signal Processing Letters, 25 (2018), 970-973.   Google Scholar

[2]

A. R. AdhikaryZ. ZhouY. Yang and P. Fan, Constructions of cross Z-complementary pairs with new lengths, IEEE Transactions on Signal Processing, 68 (2020), 4700-4712.  doi: 10.1109/TSP.2020.3014613.  Google Scholar

[3]

C.-Y. Chen, A novel construction of Z-ccomplementary pairs based on generalized boolean functions, IEEE Signal Processing Letters, 24 (2017), 987-990.  doi: 10.1109/LSP.2017.2701834.  Google Scholar

[4]

C.-Y. Chen and S.-W. Wu, Golay complementary sequence sets with large zero correlation zones, IEEE Transactions on Communications, 66 (2018), 5097-5204.  doi: 10.1109/TCOMM.2018.2857485.  Google Scholar

[5]

L. ChenZ. LuoX. Cheng and S. Li, Golay sequence based time-domain compensation of frequency-dependent I/Q imbalance, Communications China, 11 (2014), 1-11.  doi: 10.1109/CC.2014.6878998.  Google Scholar

[6]

J. D. Coker and A. H. Tewfik, Simplified ranging systems using discrete wavelet decomposition, IEEE Transactions on Signal Processing, 58 (2010), 575-582.  doi: 10.1109/TSP.2009.2032949.  Google Scholar

[7]

P. Z. FanN. SuehiroN. Kuroyanagi and X. M. Deng, Class of binary sequences with zero correlation zone, Electronics Letters, 35 (1999), 777-779.  doi: 10.1049/el:19990567.  Google Scholar

[8]

P. FanW. Yuan and Y. Tu, Z-complementary binary sequences, IEEE Signal Processing Letters, 14 (2007), 509-512.  doi: 10.1109/LSP.2007.891834.  Google Scholar

[9]

M. J. E. Golay, Complementary series, IRE Transactions on Information Theory, IT-7(1961), 82-87. doi: 10.1109/tit.1961.1057620.  Google Scholar

[10]

G. GongF. Huo and Y. Yang, Large zero correlation zone of Golay pairs and QAM Golay pairs, Information Theory Proceedings (ISIT), 2013 IEEE International Symposium on Information Theory, (2013), 3135-3139.  doi: 10.1109/ISIT.2013.6620803.  Google Scholar

[11]

G. GongF. Huo and Y. Yang, Large zero autocorrelation zones of Golay sequences and their applications, IEEE Transactions on Communications, (2013), 3967-3979.  doi: 10.1109/TCOMM.2013.072813.120928.  Google Scholar

[12]

Z. GuY. Yang and Z. Zhou, New sets of even-length binary Z-complementary pairs, 2019 Ninth International Workshop on Signal Design and its Applications in Communications, (2019), 1-5.  doi: 10.1109/IWSDA46143.2019.8966113.  Google Scholar

[13]

T. HuY. Yang and Z. Zhou, Golay complementary sets with large zero odd-periodic correlation zones, Advances in Mathematics of Communications, 15 (2019), 23-33.  doi: 10.3934/amc.2020040.  Google Scholar

[14]

K. M. Z. IslamT. Y. Al-Naffouri and N. Al-Dhahir, On optimum pilot design for comb-type OFDM transmission over doubly-selective channels, IEEE Transactions on Communications, 59 (2011), 930-935.  doi: 10.1109/TCOMM.2011.020411.100151.  Google Scholar

[15]

J. Jedwab and M. G. Parker, Golay complementary array pairs, Designs Codes Cryptography, 44 (2007), 209-216.  doi: 10.1007/s10623-007-9088-z.  Google Scholar

[16]

H. L. JinG. D. LiangC. Q. Xu and J. B. Zhang, The necessary condition of families of odd-periodic perfect complementary sequence pairs, Second International Workshop on Education Technology Computer Science, 2 (2009), 303-307.   Google Scholar

[17]

L. Kai, L. Hao and S. Yan, Construction of 8-QAM+ odd-periodic complementary sequences and sequence sets with zero correlation zone, Electronics Communications and Networks Ⅳ, 2015. Google Scholar

[18]

X. LiP. FanX. Tang and Y. Tu, Existence of binary Z-complementary pairs, IEEE Signal Processing Letters, 18 (2010), 63-66.  doi: 10.1109/LSP.2010.2095459.  Google Scholar

[19]

Z. Liu, Y. Guan and U. Parampalli, On optimal binary Z-complementary pair of odd period, IEEE International Symposium on Information Theory, (2013), 3125–3129. doi: 10.1109/ISIT.2013.6620801.  Google Scholar

[20]

Z. LiuU. Parampalli and Y. L. Guan, On even-period binary Z-complementary pairs with large ZCZs, IEEE Signal Processing Letters, 21 (2014), 284-287.  doi: 10.1109/LSP.2014.2300163.  Google Scholar

[21]

Z. LiuU. Parampalli and Y. L. Guan, Optimal odd-length binary Z-complementary pairs, IEEE Transactions on Information Theory, 60 (2014), 5768-5781.  doi: 10.1109/TIT.2014.2335731.  Google Scholar

[22]

H. D. Lüke and H. D. Schotten, Odd-perfect, almost binary correlation sequences, IEEE Transactions on Aerospace and Electronic Systems, 31 (1995), 495-498.  doi: 10.1109/7.366335.  Google Scholar

[23]

H. D. Lüke, Binary odd-periodic complementary sequences, IEEE Transactions on Information Theory, 43 (1997), 365-367.  doi: 10.1109/18.567768.  Google Scholar

[24]

V. Nee and D. J. R, OFDM codes for peak-to-average power reduction and error correction, Global Telecommunications Conference, (1996), 740-744.   Google Scholar

[25]

K. G. Paterson, Generalized Reed-Muller codes and power control in OFDM modulation, IEEE Transactions on Information Theory, 46 (2000), 104-120.  doi: 10.1109/18.817512.  Google Scholar

[26]

A. PezeshkiA. R. CalderbankW. Moran and S. D. Howard, Doppler resilient Golay complementary waveforms, IEEE Transactions on Information Theory, 54 (2008), 4254-4266.  doi: 10.1109/TIT.2008.928292.  Google Scholar

[27]

M. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication–Part Ⅰ: System analysis, IEEE Transactions Communications, 25 (1977), 795-799.  doi: 10.1109/TCOM.1977.1093915.  Google Scholar

[28]

M. B. Pursley and P. Hall, Introduction to Digital Communications: United States Edition, Pearson Schwz Ag, 2004. Google Scholar

[29]

P. Spasojevic and C. N. Georghiades, Complementary sequences for ISI channel estimation, IEEE Transactions on Information Theory, 47 (2011), 1145-1152.  doi: 10.1109/18.915670.  Google Scholar

[30]

E. Spano and O. Ghebrebrhan, Complementary sequences with high sidelobe suppression factors for ST/MST radar applications, IEEE Transactions on Geoence Remote Sensing, 34 (1996), 317-329.  doi: 10.1109/36.485110.  Google Scholar

[31]

D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences, Proceedings of the IEEE, 68 (1980), 593-619.  doi: 10.1109/PROC.1980.11697.  Google Scholar

[32]

S. Wang and A. Abdi, MIMO ISI channel estimation using uncorrelated Golay complementary sets of polyphase sequences, IEEE Transactions on Vehicular Technology, 56 (2007), 3024-3039.  doi: 10.1109/TVT.2007.899947.  Google Scholar

[33]

J.-B. Wang, X.-X. Xie, Y. Jiao, X. Song, X. Zhao, M. Gu and M. Sheng, Optimal odd-periodic complementary sequences for diffuse wireless optical communications, Optical Engineering, 51 (2012), 095002. doi: 10.1117/1.OE.51.9.095002.  Google Scholar

[34]

H. WenF. Hu and F. Jin, Design of odd-periodic complementary binary signal set, Computers and Communications, 2004. Proceedings. ISCC 2004. Ninth International Symposium, 2 (2004), 590-593.   Google Scholar

[35]

K. K. Wong and T. O'Farrell, Application of complementary sequences in indoor wireless infrared communications, IEE Proceedings Optoelectronics, 150 (2003), 453-464.  doi: 10.1049/ip-opt:20030928.  Google Scholar

[36]

D. Wulich, Reduction of peak to mean ratio of multicarrier modulation using cyclic coding, Electronics Letters, 32 (1996), 432-432.  doi: 10.1049/el:19960286.  Google Scholar

[37]

C. Xie and Y. Sun, Constructions of even-period binary Z-complementary pairs with large ZCZs, IEEE Signal Processing Letters, 25 (2018), 1141-1145.  doi: 10.1109/LSP.2018.2848102.  Google Scholar

[38]

J. D. YangX. JinK. Y. SongJ. S. No and D. J. Shin, Multicode MIMO systems with quaternary LCZ and ZCZ sequences, IEEE Transactions on Vehicular Technology, 57 (2008), 234-2341.   Google Scholar

Figure 1.  The odd-periodic correlation magnitudes of ZCP in Example 1
Figure 2.  The odd-periodic correlation magnitudes of a GCP in Example 2
Table 1.  The Parameters of Some ZCPs by Construction 1
$ Sequence\; Length $ $ ZCZ\; width $ $ ZOCZ\; width $
$ 4(2^{m+1}+2^m) $ $ 2^{m+1} $ $ 2^{m+1} $
$ 4(2^m+1) $ $ 2^{m-1}+1 $ $ 2^{m-1}+1 $
$ 4(2^m-1) $ $ 2^{m-1} $ $ 2^{m-1} $
$ 4(2^{m+3}+2^{m+2}+2^{m+1} $) $ 2^{m+3} $ $ 2^{m+3} $
$ 56\cdot 2^{\alpha}10^{\beta}26^{\gamma} $ $ 12\cdot 2^{\alpha}10^{\beta}26^{\gamma} $ $ 12\cdot 2^{\alpha}10^{\beta}26^{\gamma} $
$ 48 \cdot 2^{\alpha}10^{\beta}26^{\gamma} $ $ 10\cdot 2^{\alpha}10^{\beta}26^{\gamma} $ $ 10\cdot 2^{\alpha}10^{\beta}26^{\gamma} $
$ Sequence\; Length $ $ ZCZ\; width $ $ ZOCZ\; width $
$ 4(2^{m+1}+2^m) $ $ 2^{m+1} $ $ 2^{m+1} $
$ 4(2^m+1) $ $ 2^{m-1}+1 $ $ 2^{m-1}+1 $
$ 4(2^m-1) $ $ 2^{m-1} $ $ 2^{m-1} $
$ 4(2^{m+3}+2^{m+2}+2^{m+1} $) $ 2^{m+3} $ $ 2^{m+3} $
$ 56\cdot 2^{\alpha}10^{\beta}26^{\gamma} $ $ 12\cdot 2^{\alpha}10^{\beta}26^{\gamma} $ $ 12\cdot 2^{\alpha}10^{\beta}26^{\gamma} $
$ 48 \cdot 2^{\alpha}10^{\beta}26^{\gamma} $ $ 10\cdot 2^{\alpha}10^{\beta}26^{\gamma} $ $ 10\cdot 2^{\alpha}10^{\beta}26^{\gamma} $
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