零相关区序列集与零相关区序列偶集的构造方法研究
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摘要
最佳离散信号及其设计在现代通信、雷达、声纳、制导、以及电子对抗等系统中,起着越来越重要的作用。经过几十年的努力,人们在最佳离散信号的研究上已取得了大量的重要成果。第三代通信系统(3G)都是基于多址技术,为了消除多用户无线通信系统中的多径干扰与多址干扰,因此对于扩频码的数量以及质量的要求与日俱增,这就使得我们不得不想尽办法来增加性能优良的扩频码的存在空间。本文中主要应用迭代法以及交织技术构造序列及序列偶,主要工作如下:
首先,针对多载波码分多址通信系统提出来的零相关互补序列集,它不仅具有较多的子序列而且与传统的零相关区序列集相比具有更好的相关性,从而具有更好的抗干扰能力。从集合的角度,给出了ZCZ(Zero Correlation Zone)互补序列集以及相关定义。提出了一种基于互补序列和正交矩阵通过交织技术的方法来构造零相关互补序列集的方法,并给出了证明及实例。
其次,利用差集偶构造最佳二元序列偶并产生移位序列,随后与Hadamard矩阵交织构造出迭代初始序列偶,再进行迭代生成长度更长的ZCZ序列偶,并给出了证明、性能分析以及实例。
最后,将正交矩阵偶运用到ZCZ序列偶的构造中,并且给出了利用最佳二元序列偶构造正交矩阵偶的方法,最后正交矩阵偶通过综合应用插值、级联、交织、迭代等方法构造含有更多子序列且相关性良好的ZCZ序列偶集;利用正交矩阵偶对基于差集偶构造ZCZ序列偶的方法进行简化,并给出证明、性能分析以及实例。
The perfect discrete signal and its design method play an important role in the optimizing design in the area of modernistic communication, radar, sonar, navigation and electronically antagonism systems.A number of important results on research of the perfect discrete signal are printed during decades development of the research.The third generation communication system(3G) are based on the multiple access technology,and multipath interference and multiple access interference need to be eliminated in multi-user wireless communication system.So the demand of quantity and quality of spreading codes is increasing, which makes us have to find ways to increase the survived space of spreading code of the pefect performance. The construction of the sequence or the sequence pairs in this article,is mainly based on the iterative method and the interleaving technique, the main work as follow:
First,the concept of ZCZ complement sequence set is presented for multi-carrier code division multiple accesscommunication systems.The set has more subsequences and more correlation than traditional sequence set,then has stronger ability to resist multi-path interference and multiple access interference. From set view,the set of ZCZ complement sequence and related definitions are proposed.In this article,the method of construction of ZCZ complement sequence set which make use of interleaving technique based on complement sequence and orthogonal matrix is proposed,and the poof and examples given.
Secondly, the introduction of the concept and definition of difference set pair, pefect binary sequence pairs and shift sequences are produced by difference set pairs, and then initial iteration sequence pairs which constructed from Hadamard matrix form a longer length the ZCZ sequence pair through iterative technique, at last proof, performance analysis and examples of it given.
Finally, in the construction of ZCZ sequence pairs, the concept of the orthogonal matrix pair is applied. The construction of orthogonal matrix pair is proposed by method which based on pefect binary sequence pairs,and then comprehensive application of the methods of interpolation, concatenation, interleaving,iterative,then more subsequences and pefect correlation ZCZ sequence pairs are constructed from orthogonal matrix pair;the method of the construction of ZCZ sequence pair based on difference set pairs is simplified by orthogonal matrix pair,and at last proof, performance analysis and examples of it given.
首先,针对多载波码分多址通信系统提出来的零相关互补序列集,它不仅具有较多的子序列而且与传统的零相关区序列集相比具有更好的相关性,从而具有更好的抗干扰能力。从集合的角度,给出了ZCZ(Zero Correlation Zone)互补序列集以及相关定义。提出了一种基于互补序列和正交矩阵通过交织技术的方法来构造零相关互补序列集的方法,并给出了证明及实例。
其次,利用差集偶构造最佳二元序列偶并产生移位序列,随后与Hadamard矩阵交织构造出迭代初始序列偶,再进行迭代生成长度更长的ZCZ序列偶,并给出了证明、性能分析以及实例。
最后,将正交矩阵偶运用到ZCZ序列偶的构造中,并且给出了利用最佳二元序列偶构造正交矩阵偶的方法,最后正交矩阵偶通过综合应用插值、级联、交织、迭代等方法构造含有更多子序列且相关性良好的ZCZ序列偶集;利用正交矩阵偶对基于差集偶构造ZCZ序列偶的方法进行简化,并给出证明、性能分析以及实例。
The perfect discrete signal and its design method play an important role in the optimizing design in the area of modernistic communication, radar, sonar, navigation and electronically antagonism systems.A number of important results on research of the perfect discrete signal are printed during decades development of the research.The third generation communication system(3G) are based on the multiple access technology,and multipath interference and multiple access interference need to be eliminated in multi-user wireless communication system.So the demand of quantity and quality of spreading codes is increasing, which makes us have to find ways to increase the survived space of spreading code of the pefect performance. The construction of the sequence or the sequence pairs in this article,is mainly based on the iterative method and the interleaving technique, the main work as follow:
First,the concept of ZCZ complement sequence set is presented for multi-carrier code division multiple accesscommunication systems.The set has more subsequences and more correlation than traditional sequence set,then has stronger ability to resist multi-path interference and multiple access interference. From set view,the set of ZCZ complement sequence and related definitions are proposed.In this article,the method of construction of ZCZ complement sequence set which make use of interleaving technique based on complement sequence and orthogonal matrix is proposed,and the poof and examples given.
Secondly, the introduction of the concept and definition of difference set pair, pefect binary sequence pairs and shift sequences are produced by difference set pairs, and then initial iteration sequence pairs which constructed from Hadamard matrix form a longer length the ZCZ sequence pair through iterative technique, at last proof, performance analysis and examples of it given.
Finally, in the construction of ZCZ sequence pairs, the concept of the orthogonal matrix pair is applied. The construction of orthogonal matrix pair is proposed by method which based on pefect binary sequence pairs,and then comprehensive application of the methods of interpolation, concatenation, interleaving,iterative,then more subsequences and pefect correlation ZCZ sequence pairs are constructed from orthogonal matrix pair;the method of the construction of ZCZ sequence pair based on difference set pairs is simplified by orthogonal matrix pair,and at last proof, performance analysis and examples of it given.
引文
1赵晓群,贾志成.最佳信号相关理论及其发展.河北工业大学学报,2000,29(2):1-6
2曾凡鑫,葛利嘉.无线通信中的序列设计原理.北京,国防工业出版社, 2007:163-166
3 P.Z.Fan,M.Darnell.Sequence Design for Communications Applications,Research Studies Press(RSP).John Wiley & Sons,London,UK,1996:1-400
4 Pingzhi Fan. New Direction in Spreading Sequence Design and The Related Theoretical Bounds. IEEE 2002 International Conference on Communications, Circuits and Systems and West Sino Expositions,2002,1: xliii - xlviii
5赵晓群,贾世楼,王仲文.序列偶及其应用.遥测遥控,1998,19(3):31-35
6 Solomon W.Golomb,Guang Gong.Signal Design for Good Correlation For Wireless Communication,Cryptography,and Radar.New York:Cambrige university press, 2005: 1-401
7万哲先.代数与编码.第2版.北京:科学出版社,1980:1-100
8石生明.近世代数初步.第二版.北京:高等教育出版社,2006:1-102
9肖国镇,梁传甲,王育民.伪随机序列及其应用.北京:国防工业出版社,1985:47-61
10钟义信.伪噪声编码通信.北京:人民邮电出版社,1979:123-321
11蒋挺,李琦.计算机搜索准最佳二进阵列偶的算法研究.河北工业大学学报,2003,32(2):1-6
12蒋挺,候蓝田.一种构造最佳二进阵列偶的新方法.通信学报,2003,24(4):116-122
13刘凯.二值自相关序列偶的理论研究.[燕山大学博士学位论文].2009:1-103
14 Honggang Hu,Guang Gong. New sets of Zero or Low Correlation Zone Sequences via Interleaving Techniques. IEEE Transactions on Information Theory,2010,56(4):1702- 1713
15 Hideyuki Torii,Makoto Nakamura,Naoki Suehiro.A New Class of Polyphase Sequence Sets with Optimal Zero-correlation Zones. IEICE Trans.Fundamentals,2005, E88-A(7):1987-1993
16 Hideyuki Torii,Makoto Nakamura,Naoki Suehiro. A New Class of Zero-correlation Zone Sequences. IEEE Transactions on Information Theory,2004,50(3):559-565
17 Hideyuki Torii,Makoto Nakamura. Enhancement of ZCZ Sequence Set Construction Procedure. IEICE Trans.Fundamentals,2007, E90-A(2):535-538
18 Kenji Takatsukasa,Shinya Matsufuji,Yoshiaki Watanabeetc.Ternary ZCZ Sequence Sets for Cellular CDMA Systems.IEICE Trans.Fundamentals,2002,E85-A(9):2135-2140
19 S.Matsufuji,N.Suehiro,N.Kuroyanagi,et al..Two types of polyphase sequence sets for approximately synchronized CDMA systems.IEICE Trans.Fundamentals,2001, E86-A(1):229-234
20 J.S.Cha. Class of Ternary Spreading Sequences with Zero Correlation Duration. Electronics Letters,2001,37(10):636-637
21 Q.K.Trinin, P.Fan, E.M.Gabidulin. Multilevel Hadamard matrices and Zero Correlation Zone Sequences. Electronics Letters,2006,42(13):748-750
22 A.Rathinakumar,A.K.Chaturvedi. Mutually Orthogonal Sets of ZCZ Sequences. Electronics Letters,2004,40(18):1133–1134
23 Di Wu, Predrag Spasojevi’c, Ivan Seskar. Ternary Zero Correlation Zone Sequences for Multiple Code UWB. Proceedings of the 38th Conference on Information Sciences and Systems (CISS’04), Princeton, NJ, 2004:939-943
24 Daiyuan Peng,Pingzhi Fan,Naoki Suehiro. Construction of Sequences with Large Zero Correlation Zone.IEICE Trans.Fundamentals,2005,E88-A(11):3254-3259
25 Hideyuki Torii,Makoto Nakamura. New ZCZ Sequence Sets Composed of Two Subsets. Proceedings of the 6th WSEAS Int. Conference on Telecommunications and Informatics,Dallas,Texas,USA,2007:1-6
26佟鑫,温巧燕.新的三元零相关区二维阵列集.通信学报,2008,29(3):123-127
27 Takafumi Hayashi.Zero-Correlation Zone Sequence Set Constructed from a Perfect Sequence and a Complementary Sequence Pair.IEICE Trans.Fundamentals,2008, E91-A(7):1676-1681
28 Takafumi Hayashi. Binary Zero-Correlation Zone Sequence Set Constructed from an M-Sequence. IEICE Trans.Fundamentals,2006,E89-A(2):633-638
29 Takafumi Hayashi. Binary Zero-Correlation Zone Sequence Set Construction Using a Cyclic Hadamard Sequence. IEICE Trans.Fundamentals,2006,E89-A(10):2649-2655
30 Takafumi Hayashi. A Novel Class of Zero-Correlation Zone Sequence Sets Constructed from a Perfect Sequence. IEICE Trans.Fundamentals,2008,E91-A(4):1233-1237
31 Takafumi Hayashi. Zero-Correlation Zone Sequence Set Constructed from a Perfect Sequence. IEICE Trans.Fundamentals,2007,E90-A(5):1107-1111
32 Xiaohu Tang, Wai Ho Mow. A New Systematic Construction of Zero Correlation Zone Sequences Based on Interleaved Perfect Sequences. IEEE Transactions on Information Theory,2008,54(12):5729-5734
33 Zhengchun Zhou, Xiaohu Tang, Guang Gong. A New Class of Sequences with Zero or Low Correlation Zone Based on Interleaving Technique. IEEE Transactions on Information Theory,2008,54(9):4267-4273
34曾祥勇,程池,胡磊,等.一类相互正交的零相关区序列集的构造.电子与信息学报,2006,28(12):2347-2350
35 Zhengchun Zhou,Zhen Pan,Xiaohu Tang. A New Family of Optimal Zero Correlation Zone Sequences from Perfect Sequences Based on Interleaved Technique. Proceedings of IWSDA’07,2007:195-199
36王业龙.基于交织方法的零相关区序列设计研究.[西南交通大学硕士学位论文].2005:1-42
37王业龙,唐小虎.基于交织方法的ZCZ序列设计.西南交通大学学报,2005,40(3):422-425
38王业龙,曾晓莉,李勇峰.研究新型二维二元ZCZ信号集的交织构造.计算机科学,2009,36(4B):217-218
39任蕊.基于交织技术的零相关序列偶的研究. [燕山大学硕士学位论文].2008:1-17
40 Guang Gong. New Designs for Signal Sets With Low Cross Correlation,Balance Property,and Large Linear Span:GF(P) Case. IEEE Transactions on Information Theory,2002,48(11):2847-2867
41 Takafumi Hayashi,Yuichi Takahashi, Shigeru Kanemoto. An Novel Optimal Zero-Correlation Zone Sequence Set and Its Application to Ultrasonic Imaging. SICEAnnual Conference 2008,The University Electro-Communications,Japan,2008:592- 595
42 Fanxin Zeng, Lijia Ge. Some Novel Results on 1-D and 2-D Sequences with Zero Correlation Zone.ISSSTA2004,Sydney,Australia,2004:934-938
43 Wang Jinsong, Qi Wenfeng. A Large Class of Binary ZCZ Sequence Families Constructed By Period Doubling. Journal of Electronics(China),2007,24(3):301-304
44王劲松,戚文峰.一类由交织方式构造的二元ZCZ序列簇.电子与信息学报,2007,29(7):1573-1575
45 Xin Tong, Qiaoyan Wen. New Constructions of ZCZ Sequence Set with Large Family Size. Proceedings of IWSDA’07,2007:99-103
46左会娟,佟鑫,温巧燕.由完备序列构造零相关区序列集的方法.北京邮电大学学报,2008,31(4):122-125
47许成谦.差集偶与最佳二进阵列偶的组合研究方法.电子学报,2001,29(1):87-89
48郑娟.广义相对差集偶与特征序列偶的研究.[燕山大学硕士学位论文].2004:17-36
49何文才,赵晓群,贾世楼,等.最佳二进阵列偶的复合构造法.电子学报,1999,27(4):51-54
50贾彦国.几类最佳离散信号设计.[燕山大学博士学位论文].2006:19-25
51王扬志.零相关区序列与零相关区序列偶理论的研究.[燕山大学博士学位论文].2009:1-83
52李琦,高军萍.基于交织方法的ZCZ序列偶集构造.微计算机信息(管控一体化),2007,23(8-3):111-112
53 Takafumi Hayashi. Zero-Correlation Zone Sequence Set Construction Using an Even-Perfect Sequence and an Odd-Perfect Sequence. IEICE Trans.Fundamentals, 2007,E90-A(9):1871-1875
54 Takafumi Hayashi. Ternary Sequence Sets Having Periodic and Aperiodic Zero-Correlation Zone. IEICE Trans.Fundamentals,2006,E89-A(6):1825-1831
55 Takafumi Hayashi.Binary Zero-Correlation Zone Sequence Set Construction Using a Primitive Linear Recuision. IEICE Trans.Fundamentals,2005,E88-A(7):2034-2038
56 Takafumi Hayashi. A Class of Ternary Sequence Sets having a Zero-Correlation Zonefor Even and Odd Correlation Functions. ISIT 2003,Yokohama,Japan,2003:434
57 Takafumi Hayashi. Binary Sequences with Orthogonal Subsequences and a Zero-Correlation Zone: Pair-Preserving Shuffled Sequences. IEICE Trans.Fundamentals,2002,E85-A(6):1420-1425
58 Takafumi Hayashi. Optimal Zero-Correlation Zone Sequence Set Constructed from a Perfect Sequence. 7th IEEE International Conference on Computer and Information Technology (CIT 2007), Aizu-Wakamatsu City, Fukushima, Japan,2007:475-479
59 Takafumi Hayashi.A Class of Ternary Sequence Sets with a Zero-Correlation Zone for Periodic,Aperiodic, and Odd Correlation Functions. IEICE Trans.Fundamentals,2003, E86-A(7):1850-1857
60张振宇,陈卫,曾凡鑫.多载波码分多址通信系统中抑制干扰的序列设计.电子与信息学报,2009,31(10):2354-2358
61 Chenggao Han,Takeshi Hashimoto, Naoki Suehiro. Poly Phase Zero-Correlation Zone Sequences based on Complete Complementary Codes and DFT matrix. Proceedings of IWSDA’07,2007:172-175
62 Chenggao Han,Takeshi Hashimoto, Naoki Suehiro. A Novel Construction Method of Zero-correlation Zone Sequences based on Complete Complementary Codes. ISIT 2008,Toronto,Canada,2008:1931-1934
63 Rathinakumar Appuswamy,Ajit Kumar Chaturvedi.A New framework for Constructing Mutually Orthogonal Complementary Sets and ZCZ Sequences.IEEE Trans. Information Theory,2006,52(8):3817-3826
2曾凡鑫,葛利嘉.无线通信中的序列设计原理.北京,国防工业出版社, 2007:163-166
3 P.Z.Fan,M.Darnell.Sequence Design for Communications Applications,Research Studies Press(RSP).John Wiley & Sons,London,UK,1996:1-400
4 Pingzhi Fan. New Direction in Spreading Sequence Design and The Related Theoretical Bounds. IEEE 2002 International Conference on Communications, Circuits and Systems and West Sino Expositions,2002,1: xliii - xlviii
5赵晓群,贾世楼,王仲文.序列偶及其应用.遥测遥控,1998,19(3):31-35
6 Solomon W.Golomb,Guang Gong.Signal Design for Good Correlation For Wireless Communication,Cryptography,and Radar.New York:Cambrige university press, 2005: 1-401
7万哲先.代数与编码.第2版.北京:科学出版社,1980:1-100
8石生明.近世代数初步.第二版.北京:高等教育出版社,2006:1-102
9肖国镇,梁传甲,王育民.伪随机序列及其应用.北京:国防工业出版社,1985:47-61
10钟义信.伪噪声编码通信.北京:人民邮电出版社,1979:123-321
11蒋挺,李琦.计算机搜索准最佳二进阵列偶的算法研究.河北工业大学学报,2003,32(2):1-6
12蒋挺,候蓝田.一种构造最佳二进阵列偶的新方法.通信学报,2003,24(4):116-122
13刘凯.二值自相关序列偶的理论研究.[燕山大学博士学位论文].2009:1-103
14 Honggang Hu,Guang Gong. New sets of Zero or Low Correlation Zone Sequences via Interleaving Techniques. IEEE Transactions on Information Theory,2010,56(4):1702- 1713
15 Hideyuki Torii,Makoto Nakamura,Naoki Suehiro.A New Class of Polyphase Sequence Sets with Optimal Zero-correlation Zones. IEICE Trans.Fundamentals,2005, E88-A(7):1987-1993
16 Hideyuki Torii,Makoto Nakamura,Naoki Suehiro. A New Class of Zero-correlation Zone Sequences. IEEE Transactions on Information Theory,2004,50(3):559-565
17 Hideyuki Torii,Makoto Nakamura. Enhancement of ZCZ Sequence Set Construction Procedure. IEICE Trans.Fundamentals,2007, E90-A(2):535-538
18 Kenji Takatsukasa,Shinya Matsufuji,Yoshiaki Watanabeetc.Ternary ZCZ Sequence Sets for Cellular CDMA Systems.IEICE Trans.Fundamentals,2002,E85-A(9):2135-2140
19 S.Matsufuji,N.Suehiro,N.Kuroyanagi,et al..Two types of polyphase sequence sets for approximately synchronized CDMA systems.IEICE Trans.Fundamentals,2001, E86-A(1):229-234
20 J.S.Cha. Class of Ternary Spreading Sequences with Zero Correlation Duration. Electronics Letters,2001,37(10):636-637
21 Q.K.Trinin, P.Fan, E.M.Gabidulin. Multilevel Hadamard matrices and Zero Correlation Zone Sequences. Electronics Letters,2006,42(13):748-750
22 A.Rathinakumar,A.K.Chaturvedi. Mutually Orthogonal Sets of ZCZ Sequences. Electronics Letters,2004,40(18):1133–1134
23 Di Wu, Predrag Spasojevi’c, Ivan Seskar. Ternary Zero Correlation Zone Sequences for Multiple Code UWB. Proceedings of the 38th Conference on Information Sciences and Systems (CISS’04), Princeton, NJ, 2004:939-943
24 Daiyuan Peng,Pingzhi Fan,Naoki Suehiro. Construction of Sequences with Large Zero Correlation Zone.IEICE Trans.Fundamentals,2005,E88-A(11):3254-3259
25 Hideyuki Torii,Makoto Nakamura. New ZCZ Sequence Sets Composed of Two Subsets. Proceedings of the 6th WSEAS Int. Conference on Telecommunications and Informatics,Dallas,Texas,USA,2007:1-6
26佟鑫,温巧燕.新的三元零相关区二维阵列集.通信学报,2008,29(3):123-127
27 Takafumi Hayashi.Zero-Correlation Zone Sequence Set Constructed from a Perfect Sequence and a Complementary Sequence Pair.IEICE Trans.Fundamentals,2008, E91-A(7):1676-1681
28 Takafumi Hayashi. Binary Zero-Correlation Zone Sequence Set Constructed from an M-Sequence. IEICE Trans.Fundamentals,2006,E89-A(2):633-638
29 Takafumi Hayashi. Binary Zero-Correlation Zone Sequence Set Construction Using a Cyclic Hadamard Sequence. IEICE Trans.Fundamentals,2006,E89-A(10):2649-2655
30 Takafumi Hayashi. A Novel Class of Zero-Correlation Zone Sequence Sets Constructed from a Perfect Sequence. IEICE Trans.Fundamentals,2008,E91-A(4):1233-1237
31 Takafumi Hayashi. Zero-Correlation Zone Sequence Set Constructed from a Perfect Sequence. IEICE Trans.Fundamentals,2007,E90-A(5):1107-1111
32 Xiaohu Tang, Wai Ho Mow. A New Systematic Construction of Zero Correlation Zone Sequences Based on Interleaved Perfect Sequences. IEEE Transactions on Information Theory,2008,54(12):5729-5734
33 Zhengchun Zhou, Xiaohu Tang, Guang Gong. A New Class of Sequences with Zero or Low Correlation Zone Based on Interleaving Technique. IEEE Transactions on Information Theory,2008,54(9):4267-4273
34曾祥勇,程池,胡磊,等.一类相互正交的零相关区序列集的构造.电子与信息学报,2006,28(12):2347-2350
35 Zhengchun Zhou,Zhen Pan,Xiaohu Tang. A New Family of Optimal Zero Correlation Zone Sequences from Perfect Sequences Based on Interleaved Technique. Proceedings of IWSDA’07,2007:195-199
36王业龙.基于交织方法的零相关区序列设计研究.[西南交通大学硕士学位论文].2005:1-42
37王业龙,唐小虎.基于交织方法的ZCZ序列设计.西南交通大学学报,2005,40(3):422-425
38王业龙,曾晓莉,李勇峰.研究新型二维二元ZCZ信号集的交织构造.计算机科学,2009,36(4B):217-218
39任蕊.基于交织技术的零相关序列偶的研究. [燕山大学硕士学位论文].2008:1-17
40 Guang Gong. New Designs for Signal Sets With Low Cross Correlation,Balance Property,and Large Linear Span:GF(P) Case. IEEE Transactions on Information Theory,2002,48(11):2847-2867
41 Takafumi Hayashi,Yuichi Takahashi, Shigeru Kanemoto. An Novel Optimal Zero-Correlation Zone Sequence Set and Its Application to Ultrasonic Imaging. SICEAnnual Conference 2008,The University Electro-Communications,Japan,2008:592- 595
42 Fanxin Zeng, Lijia Ge. Some Novel Results on 1-D and 2-D Sequences with Zero Correlation Zone.ISSSTA2004,Sydney,Australia,2004:934-938
43 Wang Jinsong, Qi Wenfeng. A Large Class of Binary ZCZ Sequence Families Constructed By Period Doubling. Journal of Electronics(China),2007,24(3):301-304
44王劲松,戚文峰.一类由交织方式构造的二元ZCZ序列簇.电子与信息学报,2007,29(7):1573-1575
45 Xin Tong, Qiaoyan Wen. New Constructions of ZCZ Sequence Set with Large Family Size. Proceedings of IWSDA’07,2007:99-103
46左会娟,佟鑫,温巧燕.由完备序列构造零相关区序列集的方法.北京邮电大学学报,2008,31(4):122-125
47许成谦.差集偶与最佳二进阵列偶的组合研究方法.电子学报,2001,29(1):87-89
48郑娟.广义相对差集偶与特征序列偶的研究.[燕山大学硕士学位论文].2004:17-36
49何文才,赵晓群,贾世楼,等.最佳二进阵列偶的复合构造法.电子学报,1999,27(4):51-54
50贾彦国.几类最佳离散信号设计.[燕山大学博士学位论文].2006:19-25
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