離散傅立葉轉換特徵向量的生成矩陣之研究

Research on generating matrices for eigenvectors of the discrete Fourier transform

黃聖淇(Sheng-Chi Huang)

指導教授 ： 許文良

中原大學/電機資訊學院/電機工程研究所/碩士(2020年)

https://doi.org/10.6840/cycu202000543

摘要

離散分數傅立葉轉換(discrete fractional Fourier transform, DFRFT)可用離散傅立葉轉換(discrete Fourier transform, DFT)之特徵值與特徵向量組成，其中特徵向量會根據取得的方式而存在差異。本論文將擴展一次微分有限近似矩陣至無限階，然後用不同階數的一次微分有限近似矩陣組成不同階數的生成矩陣，這些生成矩陣會被用於生成DFT特徵向量，然後這些DFT特徵向量會與過去研究中取得的DFT特徵向量進行比較，以此獲得最為近似於連續Hermite-Gaussian function(HGF)取樣的DFT特徵向量。
本論文除了擴展一次微分有限近似矩陣至無限階以此獲得任意階的生成矩陣外，還提出了三種新的生成矩陣用法，前兩項方法分別稱為混合階生成矩陣與適應階生成矩陣，這兩項方法與過去生成矩陣用法的不同點在於生成DFT特徵向量的過程中使用了兩種階數以上的生成矩陣，第三項方法是DFT交替矩陣與適應階生成矩陣的結合，稱為交替適應階生成矩陣，其與適應階生成矩陣的相異點在於初始向量不再只有零階Hermite-Gaussian-like(HGL)向量一個，而是有更多HGL向量作為初始向量。

關鍵字

特徵向量 ； 一次微分有限近似矩陣 ； 生成矩陣 ； 混合階生成矩陣 ； 適應階生成矩陣 ； 交替適應階生成矩陣

並列摘要

Discrete fractional Fourier transform (DFRFT) can be constructed by eigenvalues and eigenvectors of the discrete Fourier transform (DFT). The eigenvectors are different and dependent on how they are obtained. In this thesis, we will extend the finite approximation matrices of the first derivative to the infinite order, and then use the different-order first derivative finite approximation matrices to form different-order generating matrices. These generating matrices will be used to generate the DFT eigenvectors. The eigenvectors of the generating matrices will be compared with the DFT eigenvectors obtained from the past studies. Our purpose is to get the DFT eigenvectors most similar to the samples of continuous Hermite-Gaussian functions (HGFs).
In addition to extending the order of first derivative finite approximation matrices to infinite order and then using these matrices to get the arbitrary-order generating matrices. We also propose three new generating matrix methods. The first two methods are called hybrid-order generating matrix and adaptive-order generating matrix. These two methods differ from the existing generating matrix method in that more than two kinds of generating matrices are used in the process of generating DFT eigenvectors. The third method is the combination of DFT-commuting matrices and adaptive-order generating matrices, called commuting adaptive-order generating matrix method. The difference between the third and adaptive-order generating matrix methods is that the initial vector is no longer only the 0th-order Hermite-Gaussian-Like (HGL) DFT vector, but there are more HGL DFT vectors used as the initial vectors.

並列關鍵字

eigenvectors ； first derivative finite approximation matrix ； generating matrix ； hybrid-order generating matrix ； adaptive-order generating matrix ； commuting adaptive-order generating matrix

參考文獻

參考文獻

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