實數離散分數傅立葉轉換之研究

A Research on Real Discrete Fractional Fourier Transform

張惟晴(Wei-Ching Chang)

指導教授 ： 許文良

中原大學/電機資訊學院/通訊工程碩士學位學程/碩士(2013年)

https://doi.org/10.6840/cycu201300837

摘要

兩個新的實數轉換以特徵分解的形式被建構:實數離散分數傅立葉轉換(real discrete fractional Fourier transform, RDFRFT)、實數離散分數哈特利轉換(real discrete fractional Hartley transform, RDFRHT)。此兩個轉換的特徵向量均為隨機，且每個特徵值均為1或-1。此兩轉換的特徵向量由隨機的DFT交替矩陣來求得。我們也探討RDFRFT與RDFRHT的關係。我們還定義了另一種基於近似對角矩陣的概念的RDFRHT。類似地，我們提出了實數廣義離散分數傅立葉轉換(real generalized discrete fractional Fourier transform, RGDFRFT)與實數廣義離散分數哈特利轉換(real generalized discrete fractional Hartley transform, RGDFRHT)，和定義了另一種基於近似對角矩陣的概念的RGDFRHT。所有新提出的轉換都具有分數轉換須俱備的性質。最後把這些新提出的轉換應用在影像加密，並探討其強健性與敏感度。

關鍵字

特徵向量 ； 離散分數傅立葉轉換 ； 離散傅立葉轉換 ； 廣義離散傅立葉轉換 ； 廣義離散哈特利轉換 ； 離散哈特利轉換 ； 特徵值

並列摘要

Two new real fractional transforms with many parameters are constructed. They are the real discrete fractional Fourier transform (RDFRFT) and the real discrete fractional Hartley transform (RDFRHT). The eigenvectors of these two new transforms are all random, and they both have only two distinct eigenvalues: 1 or -1. Eigenvectors of both two transforms are constructed from random DFT-commuting matrices. Besides, relationship between the RDFRFT and the RDFRHT is discussed. We also propose an alternative definition of RDFRHT based on a diagonal-like matrix. Similarly, we propose the real generalized discrete fractional Fourier transform (RGDFRFT) and the real generalized discrete fractional Hartley transform (RGDFRHT) and an alternative definition of RGDFRHT based on a diagonal-like matrix. All of the proposed new transforms have required good properties to be fractional transforms. Finally, since outputs of proposed new transforms are random, they can be applied in image encryptions. In addition, we discuss the robustness and the sensitivity of these transforms for image encryption applications.

並列關鍵字

DFT ； generalized discrete Hartley transform ； eigenvector ； eigenvalue ； generalized discrete Fourier transform ； discrete Hartley transform ； discrete fractional Fourier transform

參考文獻

[1] L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Transactions on Signal Processing, vol. 42, pp. 3084–3091, Nov. 1994.

[2] S. C. Pei and M. H. Yeh, “Improved discrete fractional Fourier transform,” Optics Letters, vol. 22, pp. 1047–1049, Jul. 1997.

[3] J. H. McClellan and T. W. Parks, “Eigenvalue and eigenvector decomposition of the discrete Fourier transform,” IEEE Transactions on Audio and Electroacoustics, vol. AU-20, pp. 66–74, 1972.

[5] S. C. Pei and W. L. Hsue, “The multiple-parameter discrete fractional Fourier transform,” IEEE Signal Processing Letters, vol. 13, no. 6, pp. 329–332, Jun. 2006.

[6] S.C. Pei, and W. L. Hsue, "Random discrete fractional Fourier transform," IEEE Signal Processing Letters, vol. 16, no. 12, pp. 1015-1018, Dec. 2009.