线性正则变换与信号重构：理论与算法

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"这篇资源是关于线性正则变换在信号处理中的应用，特别是涉及到信号的重构和采样。文章出自IEEE Transactions on Signal Processing期刊2012年11月刊，探讨了如何避免某些问题并采用Airy波let进行信号分析。文中还提到了实值或非分析型波let在处理不连续性时的作用，以及分析振荡现象时通常首选解析波let的原因。文章指出，根据信号特征选择不同类型的波let是重要的，例如高斯和Cauchy波let可能适合建模时间局部化或类似波let的瞬态事件。当信号的振荡变化接近奈奎斯特频率时，可能会选择牺牲波let的对称性以实现高频更快的衰减。作者建议进一步研究广义Morse波let参数空间，以获取更多有关解析波let性质和潜在用途的信息。" 线性正则变换是一种在信号处理中广泛使用的工具，它保持了信号的基本结构，并允许通过变换对信号进行分析、压缩或重构。在该文章中，作者关注的是如何在变换过程中有效地重构和采样信号，以保持信号的重要特性。线性正则变换通常与傅立叶变换、小波变换等经典方法结合，提供了一种理解和表示信号频域特性的途径。文章特别提到了Airy波let，这是一种特殊的波let类型，它在某些情况下可以避免某些问题，比如在处理具有特定频率成分的信号时。Airy波let因其特殊的数学特性而被青睐，尤其是在分析周期性或振荡现象时。此外，文章讨论了实值或非分析型波let在分析信号不连续性时的优势。例如，对于包含突变或边缘的信号，非分析波let能够更好地捕捉这些特征。然而，对于振荡现象，解析波let（如Airy、高斯或Cauchy波let）往往更受欢迎，因为它们能更好地模拟和解析这类信号。文章还提到，信号的特性可能决定最适合的波let类型。例如，如果信号本身与高斯或Cauchy波let家族有相似形式，那么这些波let可能作为有用的模型来描述时间局部化的事件或“脉冲”事件。在接近奈奎斯特频率的振荡变化中，可能需要非对称的波let来提高高频部分的衰减速度，以适应高频细节的捕获。最后，作者指出，对广义Morse波let的参数空间进行深入研究将有助于发现新的解析波let特性，这可能为信号处理带来新的洞察和应用。这一研究领域对于优化信号分析方法和开发新算法具有重要意义。

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 11, NOVEMBER 2012 6041

be avoided and the Airy wavelet used in its place, reinforcing the con-

clusion of [9].

Of course, real-valued or non-analytic wavelets may be useful, par-

ticularly in the analysis of discontinuities [3], but for analyzing oscilla-

tory phenomena analytic wavelets are commonly favored. There may

be occasions when other members of the generalized Morse wavelet

family than the Airy wavelets would be more suitable. One instance

arises when the signal of interest may have a form corresponding to

one of the wavelet families; this is particularly true of the Gaussian

and Cauchy families which may provide useful models for time-local-

ized or “impulsive” events that themselves resemble wavelets. Another

common scenario is that oscillatory variability may exist close to the

Nyquist frequency, in which case one would like to sacriﬁce symmetry

of the wavelet for a more rapid high-frequency decay. Further inves-

tigation of the generalized Morse wavelet parameter space may yield

useful information regarding the properties and possible uses of an-

alytic wavelets, in particular, the relationship between the roles of the

parameters



and



as differentiation and “warping” operators explored

by [9], and the variations of the wavelet form.

PPENDIX

All software associated with this paper is distributed as a part

of a freely available MATLAB toolbox called JLAB, available at

http://www.jmlilly.net. The generalized Morse wavelets are imple-

mented with

morsewave, while various properties are computed in

the functions morseprops, morsefreq, morsederiv, mors-

espace, and morsebox. The wavelet transform wavetrans uses

the generalized Morse wavelets by default. Finally, makefigs_su-

perfamily generates all ﬁgures in this paper.

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Sampling and Reconstruction of Signals in Function Spaces

Associated With the Linear Canonical Transform

Jun Shi, Xiaoping Liu, Xuejun Sha, and Naitong Zhang

Abstract—The linear canonical transform (LCT) has been shown to be

useful and powerful in signal processing, optics, etc. Many results of this

transform are already known, including sampling theory. Most existing

sampling theories of the LCT consider the class of bandlimited signals.

However, in the real world, many analog signals arising in engineering ap-

plications are non-bandlimited.

In this correspondence, we propose a sampling and reconstruction

strategy for a class of function spaces associated with the LCT, which

can provide a suitable and realistic model for real applications. First,

we introduce deﬁnitions of semi- and fully-discrete convolutions for the

LCT. Then, we derive necessary and sufﬁcient conditions pertaining to the

LCT, under which integer shifts of a chirp-modulated function generate a

Riesz basis for the function spaces. By applying the results, we present a

more comprehensive sampling theory for the LCT in the function spaces,

and further, a sampling theorem which recovers a signal from its own

samples in the function spaces is established. Moreover, some sampling

theorems for shift-invariant spaces and some existing sampling theories

for bandlimited signals associated with the Fourier transform (FT), the

fractional FT, or the LCT are noted as special cases of the derived results.

Finally, some potential applications of the derived theory are presented.

Index Terms—Convolution theorem, function spaces, linear canonical

transform, sampling and reconstruction.

I. I

NTRODUCTION

The linear canonical transform (LCT) [1], [2] is a four-parameter

class of linear integral transforms, which includes the classical Fourier

transform (FT), the fractional Fourier transform (FRFT), the Fresnel

transform, as well as certain other transforms among its members.

Serving as a useful and powerful analyzing tool, the LCT has received

much attention in many ﬁelds, including quantum mechanics [1],

optics [3], communications [4], signal and image processing [5]–[18],

etc. For more details of the LCT, see [2], [3].

Manuscript received February 05, 2012; revised June 08, 2012; accepted July

13, 2012. Date of publication July 31, 2012; date of current version October 09,

2012. The associate editor coordinating the review of this manuscript and ap-

proving it for publication was Prof. Namrata Vaswani. This work was completed

in part while J. Shi was the Department of Electrical and Computer Engineering,

University of Delaware. The work was supported in part by the National Nat-

ural Science Foundation of China under Grant 61171110 and the Short-Term

Overseas Visiting Scholar Program of Harbin Institute of Technology.

J. Shi was with the Department of Electrical and Computer Engineering, Uni-

versity of Delaware, Newark, DE 19716 USA. He is now with the Communi-

cation Research Center, Harbin Institute of Technology, Harbin 150001, China

(e-mail: j.shi@hit.edu.cn).

X. Liu, and X. Sha are with the Communication Research Center, Harbin

Institute of Technology, Harbin 150001, China (e-mail: xp.liu@hit.edu.cn;

shaxuejun@hit.edu.cn).

N. Zhang is with the Communication Research Center, Harbin Institute

of Technology, Harbin 150001, China, and also with the Shenzhen Graduate

School, Harbin Institute of Technology, Shenzhen 518055, China (e-mail:

ntzhang@hit.edu.cn).

Digital Object Identiﬁer 10.1109/TSP.2012.2210887

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