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Title: | 廣義勒讓德序列以及廣義沃爾許傅立葉轉換之研究 Researches on Generalized Legendre Sequence and Generalized Walsh-Fourier Transform |
Authors: | Chia-Chang Wen 溫家昶 |
Advisor: | 貝蘇章 |
Keyword: | 沃爾許轉換,傅立葉轉換,特徵向量,洗牌轉換,勒讓德序列,數論轉換, Legendre Sequence,Walsh-Fourier Transform,DFT Eigenvector,Number Theoretic Transform,Sequency Ordered,Conjugate Sequency Ordered Orthogonal Transform,Perfect Shuffle, |
Publication Year : | 2014 |
Degree: | 博士 |
Abstract: | 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. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/58215 |
Fulltext Rights: | 有償授權 |
Appears in Collections: | 電信工程學研究所 |
Files in This Item:
File | Size | Format | |
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ntu-103-1.pdf Restricted Access | 3.32 MB | Adobe PDF |
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