分数傅里叶变换这个概念,其实最早在公元1929年,N.Wiener就已提出,但是并没有受到太多的瞩目。过了约莫50年,V.Namias 在公元1980年重新提出(称之为重发明)这个概念,但是一直到公元1994年,才有人真正把分数傅里叶变换用在信号处理上,此人为 L. B. Almeida。详细历史:1937年提出分数傅里叶变换的概念雏形; 1980年Namias较明确地提出分数傅里叶变换的数学表达式,并将其用于具有确定边界条件的量子力学薛定谔方程的求解1987年Bride & Kerr 给出严格的数学定义以及性质1993年由德国的学者罗曼,土耳其的Ozaktas和以色列的Mendlovic等人首次将分数傅里叶变换概念引入光学并给出了相应的光学过程; Mendlovic&Ozaktas:渐变折射率GRIN介质中光传播。 A. W. Lohmann: 维格纳分布函数和以及透镜实现,自由空间的光衍射。 1993年Ozaktas,罗曼,Mendlovic等人在光学中全面引入分数傅里叶变换; 1995年Shih提出了另外一种分数傅里叶变换的形式; 1997年刘树田等人根据Shih的定义给出了广义分数傅里叶变换,1999年刘树田等人将分数傅里叶变换应用于图像加密研究中; 2001年Ozaktas等人出版“分数傅里叶变换及其在光学和信号处理中应用”一书。
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V. Namias, "The fractional order Fourier transform and its application to quantum mechanics," J. Inst. Appl. Math.25, 241–265 (1980).
Luís B. Almeida, "The fractional Fourier transform and time-frequency representations," IEEE Trans. Sig. Processing42 (11), 3084–3091 (1994).
Soo-Chang Pei and Jian-Jiun Ding, "Relations between fractional operations and time-frequency distributions, and their applications," IEEE Trans. Sig. Processing49 (8), 1638–1655 (2001).
D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," SIAM Review33, 389-404 (1991). (Note that this article refers to the chirp-z transform variant, not the FRFT.)
Haldun M. Ozaktas, Zeev Zalevsky and M. Alper Kutay. "The Fractional Fourier Transform with Applications in Optics and Signal Processing". John Wiley & Sons (2001). Series in Pure and Applied Optics.
Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2013