The discrete fractional Fourier transform is defined by Zeev Zalevsky in
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As the meaning of “canonical”, the linear canonical transform (LCT) is a scalable transform that connects to lots of important kernels such as the Fresnel transform, Fraunhofer transform and the fractional Fourier transform.
M. Haldun Ozaktas, Zeev Zalevsky and M. Alper Kutay, “The fractional Fourier transform with applications in optics and signal processing,” JOHN WILEY & SONS, LTD, New York, 2001.