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研究生:劉英鴻
研究生(外文):Ying-Hung Liu
論文名稱:三值互相關函數值展頻序列之研究
論文名稱(外文):A Study of Spreading Sequences with 3_Valued Crosscorrelation Functions
指導教授:賴辰彥賴辰彥引用關係
指導教授(外文):Chen-Yan Lai
學位類別:碩士
校院名稱:逢甲大學
系所名稱:電子工程所
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:中文
論文頁數:176
中文關鍵詞:展頻碼虛擬雜訊序列
外文關鍵詞:Spreading Codepseudo-noisePNsequences
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針對長度N=2n-1,其中n=mk, M=2m-1和T=N/M的pseudo-noise(PN)sequences可以Width Analyses的方式將其映射成M´T的二維陣列,且此陣列之行向量(子序列)顯現一些重要特性。例如No序列其子序列則為較短的m-序列;unknown type for miscellaneous cyclic Hadamard和理想相關函數值序列,其子序列則為較短的Legendre sequences;Gold序列其子序列則為較短的Gold序列,在其它miscellaneous cyclic Hadamard和Bent序列其子序列重量為balance或三值以上分佈。更短的Gold序列(M=2n/4-1)也可能為Bent序列的子序列。如此可推展到所有不同長度(包含7, 15, 31, 127)的子序列,利用general model for shift register sequence generators依據不同的次取樣(σ)及循環位移(γ)的關係,對應到不同的本原域接法及初始值,來產生所有的子序列,並將其組合成較長週期序列。當長度N=2n-1 m-序列以Width Analyses的方式,其中M=p¹2m-1, p=±1mod4,則可得到另一重要結果。其陣列中行向量顯現長度M的二位元循環碼集特性,此碼集乃由M次單位根之最小多項式生成。利用這些行向量任一向量,以Continued Fraction Algorithm可輕易將cyclotomic polynomial Q(p)(x)分解成二個(p-1)/2階的irreducible polynomials。更進一步,我們將針對IS-95 CDMA,以五個長度215-1的m-序列實施Width Analyses亦可得到重要的QR(151, 75)平方剩餘循環碼生成多項式。
Take a length of the form N=2n-1, n=mk, such that M=2m-1 and T=N/M are relatively prime, width analyses of some pseudo-noise(PN)sequences of length N=2n-1 result in two-dimensional M´T arrays, which reveal significant structure that each column shows the behavior of a subsequence of smaller m-sequence for No family sequences, smaller Legendre sequences of Mersenne prime period, Hall’s sextic residue sequences, and miscellaneous sequences of unknown type for some miscellaneous cyclic Hadamard and optimal sequences, smaller Gold sequences for Gold family sequences, and smaller balance or several-valued weight spectrum sequences for some other miscellaneous cyclic Hadamard and Bent function sequences. It is possible that even smaller (M=2n/4-1) Gold sequences can be subsequences of those arrays for Bent function sequences. There is no possible width analysis for 31 and 127. However, shift registers for generating sequences of length 7, 15, 31, 127 provide a building block for generating many of more sequences of longer period. All the subsequences can be generated by utilizing a general model for shift register sequence generators associated with parameters, s and g, which represent primitive connections and initial loadings of the shift registers respectively. When width analyses of m-sequences of length N=2n-1, M=p¹2m-1, p=±1mod4are compared the result is even more significant. Each column of the arrays shows the behavior of the subsequences of cyclic binary code of block length M generated by the minimal polynomial of a primitive Mth root of unity. By using continued fraction algorithm on any column, the cyclotomic polynomial Q(N)(x) can then be easily factored into two binary irreducible polynomials, each of degree (p-1)/2. Furthermore, width analyses of five m-sequences of length (215-1), which are used in IS-95 CDMA system, give generator polynomial of important Quadratic Residue (151, 75) code.
第一章緒論.………………………………………………………...……1
1.1 研究背景…………………………….……………………...…..1
1.2 研究動機……….. ………………………………………...……3
1.3 各章介紹………………………………………………………..8

第二章理論基礎..…………………………………………………...……9
2.1 Width Analyses………………………….……………………..10
2.1.1 次取樣陣列………..……………..……………………10
2.1.2 Trace Function………………..………………………12
2.1.3 m-及GMW之Width Analyse………..…..……………13
2.2 基底向量…………………………….……………………...15
2.2.1 Walsh-Hadamard轉換…….…………..………………15
2.2.2 布林函數…………….………………...………………17
2.2.1 Bent(almost bent)函數………………..………………20
2.3 Shift Register Sequences………..……………………………21
2.3.1 General Model for Shift Register Sequence Generators..22
2.3.2 Continued Fractions Algorithm………………………..24

第三章三值互相關函數值序列………………………………...……..29
3.1m-序列…..………………………………..……………………..29
3.1.1 線性迴授移位暫存器…………………...……………29
3.1.2 以Trace function產生m-序列………...………………30
3.1.3 m-序列之特性………………..….………..……………31
3.1.4 雙值自相關函數值序列與Hadamard矩陣的關係..32
3.1.5 m-序列間互相關函數值之探討……………………32
3.2 Gold序列………………………………..…………...……….34
3.2.1 Gold序列的產生………………………………………34
3.2.2 Gold序列與Hadamard矩陣關係………………………35
3.2.3 m-序列與Gold 序列最大互相關函數值之比較……36
3.3 Kasami序列…….…….………………………….……...36
3.4 No序列…………………………………………………………38
3.5 Bent序列…………….…………………………………………39
3.5.1 Bent序列之產生………………………………………..40
3.6.建構理想相關函數值序列……………………………………51
3.6.1 雙值自相關函數值序列…….…………………………52
3.6.2 建構三值互相關函數值序列…….……………………52
3.7 Quasi-Orthogonal sequences………….………………56
3.7.1 互相關值範圍……………..…….……………………56
3.7.2 建構 Quasi-Orthogonal Sequence.……………………58

第四章Pseudo-Noise Sequences之Width Analyses……………………61
4.1 具ideal和balance特性之參考子序列………………………61
4.1.1 No序列之Width Analyses……...………….………….62
4.1.2 miscellaneous cyclic Hadamard sequence(M255-1, 2 M1023-1, 2, 3, 4)之Width Analyses………………67
4.1.3三值互相關值序列之Width Analyses….…………72
4.2具Gold序列和非線性特性之參考子序列…………..…..78
4.2.1 Gold序列之Width Analyses..…….……………………..79
4.2.2 Bent序列之Width Analyses….…………………………89
4.2.3 M511序列之Width Analyses………………………92

第五章一些二位元循環碼集的建構…………….…………...………..96
5.1以單一m-序列建構二位元線性循環碼集………….………96
5.1.1 建構長度11的二位元線性循環碼集.………………97
5.1.2 建構長度17的二位元線性循環碼集.………….……99
5.1.3 建構長度23的二位元線性循環碼集.………….…101
5.1.4 建構長度41的二位元線性循環碼集.………….…104
5.1.5 建構長度47的二位元線性循環碼集.…………….106
5.2以多重m-序列建構二位元線性循環碼集列………………108
5.2.1 建構長度73的二位元線性循環碼集.………….…109
5.2.2 建構長度89的二位元線性循環碼集.………….…111
5.2.3 建構長度151的二位元線性循環碼集……………112
5.2.4 建構長度257的二位元線性循環碼集……………116
5.3有限域下xp+1的因式分解..…………………………118

結論及未來研究...………………..………...…………………...121
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