线性正则变换的离散化研究进展

孙艳楠, 李炳照, 陶然. 线性正则变换的离散化研究进展[J]. 光电工程, 2018, 45(6): 170738. doi: 10.12086/oee.2018.170738
引用本文: 孙艳楠, 李炳照, 陶然. 线性正则变换的离散化研究进展[J]. 光电工程, 2018, 45(6): 170738. doi: 10.12086/oee.2018.170738
Sun Yannan, Li Bingzhao, Tao Ran. Research progress on discretization of linear canonical transform[J]. Opto-Electronic Engineering, 2018, 45(6): 170738. doi: 10.12086/oee.2018.170738
Citation: Sun Yannan, Li Bingzhao, Tao Ran. Research progress on discretization of linear canonical transform[J]. Opto-Electronic Engineering, 2018, 45(6): 170738. doi: 10.12086/oee.2018.170738

线性正则变换的离散化研究进展

  • 基金项目:
    国家自然科学基金资助项目(61671063);国家自然科学基金创新研究群体基金资助项目(61421001)
详细信息
    作者简介:
    通讯作者: 李炳照(1975-),男,博士,教授,主要从事小波变换、分数阶Fourier变换、线性正则变换的基本理论及其在非平稳信号分析与处理中应用的研究。E-mail:li_bingzhao@bit.edu.cn
  • 中图分类号: O436.3

Research progress on discretization of linear canonical transform

  • Fund Project: Supported by National Natural Science Foundation of China (61671063) and Foundation for Innovative Research Groups of the National Natural Science Foundation of China (61421001)
More Information
  • 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。

  • Overview:Linear canonical transformation (LCT) is a generalized form of linear integral transform, which is a three-parameter linear integral transform. The LCT unifies a variety of transforms from the well-known Fourier transform (FT), fractional Fourier transform (FRFT) and Fresnel transform (also known as chirp convolution (CC)) to simple operations such as scaling and chirp multiplication (CM). The LCT is an important tool in optics because a broad class of optical systems including thin lenses, sections of free space in the Fresnel approximation, sections of quadratic graded-index media, and arbitrary concatenations of any number of these, sometimes referred to as first-order optical systems the paraxial light propagation can be modeled by the LCT. Besides, as a generalization of the transforms mentioned above, the LCT could be more useful and attractive in many signal processing applications, such as filter design, radar system, time-frequency analysis, phase reconstructions, and so on. On the other hand, the LCT can also be used in the fields of application mathematics, such as solution of differential equations. Therefore, the LCT has attracted a considerable amount of attention in many areas. In order to promote the applications of LCT, the discretization becomes the key vital issue of the LCT. Since the discretization of LCT cannot be obtained by directly sampling in time domain and LCT domain, it has been investigated recently. After the continuous LCT has been introduced, the definition and implementation of the discrete LCT (DLCT) have been widely considered by many researchers. Based on the development history of LCT discretization, a review of important research progress and current situation of discretization methods in the last nearly two decades is presented in this paper. The discretization algorithms include, discrete-time LCT, linear canonical series, discrete linear canonical transform. In this paper, the existing discretization methods are divided into three categories, directly discrete LCT, which appeared to have been first undertaken by Pei and Ding; operator decomposition LCT, which decomposed into products of these special operator combinations through the benefit of the additive property of the LCT; base decomposition fast discrete LCT, which was also first utilized by Hennelly and Sheridan to fast-implement DLCT; and other discrete LCT. Meanwhile, the connection among different discretization algorithms and the future development direction are given. It provides important reference value for researcher in the related fields, and can further promote its engineering application. With the deepening of research, LCT will be more and more widely used in practical applications.

  • 加载中
  • 图 1  一些变换对信号WVD影响。(a)原信号的WVD;(b)尺度变换之后;(c) Fourier变换之后;(d) Chirp乘积变换之后;(e) Chirp卷积变换之后;(f)分数阶Fourier变换之后

    Figure 1.  WVD of (a) original signal, (b) scale transform, (c) FT, (d) CM, (e) CC, and (f) FRFT

    图 2  任意信号LCT的WVD的影响。(a)变换之前;(b) LCT变换之后

    Figure 2.  WVD of (a) original signal and (b) LCT

    表 1  不同类型信号的线性正则分析

    Table 1.  The linear canonical analysis of different types of signals

    变换类型 时域 LCT域 经典频域
    LCT 连续 连续 连续(FT)
    LCS 连续Chirp周期 离散非Chirp周期 离散非周期(FS)
    DTLCT 离散非周期 连续Chirp周期 连续周期(DTFT)
    DLCT 离散Chirp周期 离散Chirp周期 离散周期(DFT)
    下载: 导出CSV

    表 2  3种主要类型DLCT的比较

    Table 2.  Comparison of 3 main types of DLCT

    性质 直接离散LCT 基分解的快速DLCT 算子分解DLCT
    近似连续变换
    是否具有闭合形式
    计算复杂度 O(N2) O(N·logN) O(N·logkN)
    下载: 导出CSV
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收稿日期:  2017-12-13
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