Researches on Generalized Legendre Sequence and Generalized Walsh-Fourier Transform
Date Issued
2014
Date
2014
Author(s)
Wen, Chia-Chang
Abstract
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.
Subjects
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
Type
thesis
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