Researches on Generalized Legendre Sequence and Generalized Walsh-Fourier Transform

本篇論文包含兩個主題:第一個主題是在討論廣義滿秩勒讓德序列(Complete Generalized Legendre Sequence)的性質及應用以及第二個主題是在討論廣義的沃爾許傅立葉轉換(Walsh-Fourier Transform)的性質及應用。 廣義滿秩勒讓德序列一開始是被定義用來解決傅立葉特徵向量的問題。我們提出的傅立葉特徵向量基於廣義滿秩勒讓德序列具有解析解、滿秩、正交、適用於任何長度的離散傅立葉轉換以及具有快速演算法。所以廣義滿秩勒讓德序列可以適合用來推導新的快速傅立葉轉換架構。奠基於實數域的研究，我們可以將研究的成果推廣到有限域當中，我們可以用有限域廣義滿秩勒讓德序列來解決數論轉換的特徵向量問題。更進一步的我們可以用廣義滿秩勒讓德序列來定義分數數論轉換以及推廣到可切換性的洗牌轉換。 沃爾許以及傅立葉轉換對於信號處理來說非常重要。我們的目的是希望用簡單的參數將上述二轉換統整起來。如此我們可以得到具有兩者優點的新轉換以及可很彈性調整的優點。我們首先將先定義離散正交轉換的特定產生方式，共軛離散正交轉換的特定產生方式以及在有限域的快速轉換推導。基於前述的成果，我們可以更進一步的去定義具有頻率順序性的廣義沃爾許傅立葉轉換以及共軛沃爾許傅立葉轉換。我們將上述定義的轉換利用在碼分多址序列設計、頻譜分析以及轉換編碼等應用。The thesis contains two research topics: The first one is the discussion about the properties and applications of the complete generalized Legendre sequence (CGLS) and the second one is about the generalization between the Walsh-Hadamard transform (WHT) and the DFT and their properties. The CGLS is first defined to solve the DFT eigenvector problem. The proposed CGLS based DFT eigenvectors have the advantages of closed-form solutions, completeness, orthogonality, being well defined for arbitrary N, and fast DFT expansion so that the CGLS is helpful for developing DFT fast algorithms. Based on the CGLS researches, we can extend our results to the finite field operation. That means we can also use the CGLS over finite field (CGLSF) to solve the number theoretic transform (NTT) eigenvector problem. Mean while, we can apply the CGLS and CGLSF to constructing fast DFT(NTT) algorithm, fractional number theoretic transform (FNTT) definition and the switchable perfect shuffle transform (PST) system. The WHT and the DFT are two of the most important transforms for signal processing applications. Our purpose is to generalize these two transform by a single parameter so that the generalized transforms can not only have the advantages of the WHT and DFT but also have flexibility to some applications. We will first define the discrete orthogonal transform (DOT), conjugate symmetric discrete orthogonal transform (CS-DOT) and the fast finite field orthogonal transform (FFFOT). From the above transform, we can furthermore define the sequency ordered generalized Walsh-Fourier transform (SGWFT) and conjugate symmetric sequency ordered generalized Walsh-Fourier transform (CS-SGWFT) and show their properties and applications in CDMA sequence design, spectrum estimation and transform coding.口試委員會審定書 # 誌謝 i 中文摘要 ii ABSTRACT iii CONTENTS v LIST OF FIGURES x LIST OF TABLES xiii Chapter 1 Introduction 1 1.1 Complete Generalized Legendre Sequence (CGLS) 1 1.1.1 CGLS and DFT Eigenvector 1 1.1.2 CGLSF and NTT Eigenvector 1 1.1.3 Fractional Number Theoretic Transform (FNTT) 2 1.1.4 CGLS and Perfect Shuffle Transform 2 1.2 Sequency Ordered Walsh-Fourier Transform (SGWFT) 4 1.2.1 Discrete Orthogonal Transform (DOT) and SGWFT 4 1.2.2 CS-DOT and CS-SGWFT 6 1.2.3 Fast Finite Field Orthogonal Transform Without Length Constraint 7 1.3 Organization of the Dissertation 8 Chapter 2 Complete Generalized Legendre Sequence 10 2.1 Original Generalized Legendre Sequence (GLS) 10 2.1.1 Original Generalized Legendre Sequence (GLS) 10 2.1.2 Properties of GLS 11 2.2 Complete Form of Generalized Legendre Sequence 12 2.2.1 Complete Generalized Legendre Sequence (CGLS) Definition 12 2.2.2 CGLS Properties 14 2.2.3 CGLS for Arbitrary Length 16 Chapter 3 Closed Form DFT Eigenvector 19 3.1 Existing DFT Eigenvector Methods 19 3.2 DFT Eigenvectors by CGLS 21 3.2.1 Linear Combination of CGLS 21 3.2.2 Comparison with Other DFT eigenvector Methods 23 3.2.3 AM-FM Transform Relation 24 3.3 DFRFT Based on CGLS-Like DFT Eigenvectors 25 Chapter 4 Complete Generalized Legendre Sequence over Finite Field and Closed Form NTT Eigenvector 29 4.1 Number Theoretic Transform and Eigenvalue Distribution 29 4.2 Fermet Number Transform and Complex Mersenne Number Transform 30 4.3 Complete Generalized Legendre Sequence over Finite Field 32 4.4 Complete and Orthogonal NTT Eigenvectors 36 4.5 CGLSF with Arbitrary Transform Length 39 4.6 Fast Fractional NTT 41 4.6.1 Fractional NTT Based on the CGLSF Eigenvectors 41 4.6.2 Fractional Fermat Number Transform 42 4.6.3 Fractional Complex Mersenne Number Transform 43 4.6.4 Fractional New Mersenne Number Transform 44 4.6.5 Complexity Analysis 46 4.6.6 Summary on the FNTT 46 Chapter 5 DFT and NTT Implementation by CGLS and CGLSF 48 5.1 DFT Implementation by CGLS based DFT Eigenvectors 48 5.2 NTT Implementation By CGLSF based NTT Eigenvectors 53 Chapter 6 Eigenstructure of Perfect Shuffle Invariant System 58 6.1 Introduction 58 6.2 Review on Perfect Shuffle System 62 6.3 Perfect Shuffle Convolution 63 6.3.1 Perfect Shuffle Operator 63 6.3.2 Perfect Shuffle System 65 6.3.3 Perfect Shuffle Convolution Form 67 6.4 Complete Generalized Legendre Sequence Transform (CGLST) 69 6.5 Fast PST Convolution 78 6.6 Applications 79 6.6.1 Scalable Data Shuffling 79 6.6.2 Dirichlet Convolution Computation 82 6.7 Conclusions 85 Chapter 7 Discrete Orthogonal Transform (DOT) and Conjugate Symmetric Discrete Orthogonal Transform (CS-DOT) Generating Method 86 7.1 Discrete Orthogonal Transform (DOT) 88 7.1.1 Orthogonality 88 7.1.2 Radix-2 Fast Implementation Algorithm 90 7.1.3 Split-Radix Fast Implementation Algorithm 92 7.2 Conjugate Symmetric Discrete Orthogonal Transform (CS-DOT) 95 Chapter 8 Sequency Ordered Generalized Walsh-Fourier Transform 99 8.1 Review on Sequency Ordered Complex Hadamard Transform (SCHT) 100 8.2 Sequency Ordered generalized Walsh–Fourier Transform (SGWFT) 101 8.2.1 Closed-Form Expression 104 8.2.2 Properties of SGWFT 107 8.2.3 Complexity of SGWFT 111 8.3 Application of SGWFT 113 8.3.1 Signal Multiplexing 113 8.3.2 Transform Coding 116 Chapter 9 Conjugate Symmetric Sequency Ordered Generalized Walsh-Fourier Transform and Its Application 120 9.1 Review on Conjugate Symmetric Sequency Ordered Complex Hadamard Transform (CS-SCHT) 121 9.2 Conjugate Symmetric Sequency Ordered Generalized Walsh-Fourier Transform (CS-SGWFT) 123 9.3 Applications on CS-SGWFT 134 9.3.1 Spectrum Estimation 134 9.3.2 Image Interference Removal 136 Chapter 10 Fast Finite Field Orthogonal Transform (FFOT) 138 10.1 Number Theoretic Transform with Length Constraint 138 10.2 Finite Field Orthogonal Transform Generating Process 139 10.3 Fast FFOT Implementation 142 Chapter 11 Conclusion 146 REFERENCE 147 Appendix A 158 Appendix B 159 Appendix C 162 Appendix D 163 Publication List 16