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摘要:
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。
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关键词:
- 分数阶Fourier变换 /
- 线性正则变换 /
- 离散时间线性正则变换 /
- 线性正则级数 /
- 离散线性正则变换
Abstract: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.
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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.
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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) 
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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 是 是 是否具有闭合形式 是 是 否 计算复杂度 O(N2) O(N·logN) O(N·logkN)
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