线性正则变换的离散化研究进展

Research progress on discretization of linear canonical transform

线性正则变换(LCT)是Fourier变换和分数阶Fourier变换的广义形式。近年来研究成果表明,LCT在光学、信号处理及应用数学等领域有广泛的应用,而离散化成为了其得以应用的关键。由于LCT的离散算法不能简单直接地将时域变量和LCT域变量离散化得到,因此LCT的离散算法成为近年来的研究重点。本文依据LCT的离散化发展历史,对其重要研究进展和现状进行了系统归纳和简要评述,并给出不同离散化算法之间的区别和联系,指明了未来发展方向。这对研究者全面了解LCT离散化方法具有很好的参考价值,可以进一步促进其工程应用。

Linear canonical transformation（LCT） is a generalization of the Fourier transform and fractional Fourier transform. The recent studies have shown that LCT is widely used in optics, signal processing and applied mathematics, and the discretization of the LCT becomes vital for the applications of LCT. Since the discretization of LCT cannot be obtained by directly sampling in time domain and LCT domain, the discretization of the LCT becomes the focus of investigation recently. Based on the development history of LCT discretization, a review of important research progress and current situation of discretization of the LCT is presented in this paper. Meanwhile, the connection among different discretization algorithms and the future development direction are given. It is of great reference value for researchers to fully understand the LCT discretization method and can further promote its engineering applications.