线性正则变换相关理论问题研究
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摘要
线性正则变换是近年发展起来的一种新兴的信号处理工具。作为一种统一的多参数的线性积分变换，线性正则变换在处理非平稳信号时有其独特优势。尽管它还不十分出名，但是其特殊形式如Fourier变换、分数阶Fourier变换、Fresnel变换以及尺度算子等已经在各专业领域得到广泛的应用。研究线性正则变换并将其基本特点和性质融会贯通，可以进一步发展新的、更好的有效变换。然而，从已有的文献来看，线性正则变换的基本理论体系还不完善，与信号处理相关的一些理论如时频分析、采样理论等有待进一步建立或加强，所以开展线性正则变换及其相关理论问题的研究具有重要的理论和现实意义。
针对上述问题，本文在研究线性正则变换的特点及性质的基础上，重点针对线性正则变换域的时频分析、信号采样等相关理论问题进行了深入的研究。本文的主要贡献和创新表现在以下几个方面：
1．从线性正则变换与Fourier变换的关系出发，研究了线性正则变换域的基本理论。在Fourier变换的均匀采样定理基础上，推导了线性正则变换域的均匀采样理论与信号重建公式；在传统的卷积定理基础上，推导了线性正则变换域新的卷积定理；在经典Hilbert变换的基础上，给出了线性正则变换域Hilbert变换的定义、性质和特点分析。考虑到在数字信号处理中信号是时域离散的，本文还给出了离散时间信号的线性正则变换定义，证明了其和连续时间信号的线性正则变换相类似的部分性质。上述基本理论的建立对推广与发展线性正则变换在信号处理中的实际应用具有重要的意义。
2．从线性正则变换与传统时频分布的关系出发，研究了基于线性正则变换的信号时频分析理论。首先从线性正则变换与Wigner分布的关系入手，分析了线性正则变换的时频滤波原理，提出了基于线性正则变换的时频滤波方法，并给出了线性正则变换域滤波器参数的详细设计。其次，研究了线性正则变换与短时Fourier变换的关系，并针对chirp信号干扰抑制和多分量时频信号分离问题，提出了基于线性正则变换与短时Fourier变换联合的时频分析方法。仿真实例表明，该方法能避免交叉项干扰，是时频信号分析的有效手段。最后，研究了线性正则变换对时频面上信号模糊函数分布产生的影响，并通过仿真进行了验证。以上研究结果为进一步开展线性正则变换在时频分析领域的应用研究奠定了理论基础。
3．研究了线性正则变换域带通信号的采样理论。首先，给出了线性正则变换域带限信号的定义和特点；其次，导出了线性正则变换域带通信号采样定理和重建公式，并证明了已有的均匀采样定理都是线性正则变换域带通采样定理的特殊形式；最后，通过对chirp类信号的采样分析以及仿真实验，例证了线性正则变换域采样定理的正确性。线性正则变换域带通信号采样定理的得出丰富和完善了线性正则变换的采样理论体系。
4．研究了偏移线性正则变换域信号的卷积与乘性滤波理论。首先定义了信号基于偏移线性正则变换意义的新卷积运算，在此基础上利用偏移线性正则变换的性质导出了偏移线性正则变换域的卷积定理；其次基于该定理建立了偏移线性正则变换域的乘性滤波理论，给出了偏移线性正则变换域乘性滤波器的一般模型，研究了偏移线性正则变换域乘性滤波器在时域的实现方法，分析了扫频滤波器与偏移线性正则变换域乘性滤波器的关系；最后还利用该卷积定理推导了偏移线性正则变换域带限信号的均匀采样定理。由于偏移线性正则变换是线性正则变换引入时移与频率调制参数后的扩展形式，所以上述研究结果将进一步丰富线性正则变换的理论体系。
5.研究了偏移线性正则变换域带限信号的广义采样理论。广义采样的目的是通过M个不同的线性时不变系统输出信号的均匀采样序列重构原始带限信号，且采样速率可降为奈奎斯特速率的1/M。首先，基于Papoulis的广义采样模型，提出了偏移线性正则变换域带限信号的广义采样定理。其次，利用广义采样定理和偏移线性正则变换的性质，通过构造合适的偏移线性正则变换域滤波器，分别得到了基于信号及其导数的均匀采样序列、信号及其Hilbert变换的均匀采样序列以及信号的周期非均匀采样序列，重构原始偏移线性正则变换域带限信号的表达式。最后通过仿真实验，例证了上述结论的正确性。
The linear canonical transform (LCT) is a new signal processing tool developed inrecent years. As a unified and multi-parameter class of linear integral transform, LCThas its unique advantages in non-stationary signal processing. Although it is not verymuch known, its special cases are widely used in various fields, often under differentnames, such as the Fourier transform, fractional Fourier transform (FRFT), Fresneltransform, time scaling, and others. Therefore, understanding the LCT may help to gainmore insights on its special cases and to carry over knowledge gained from one subjectto others. However, from the existing literature, the basic theoretical system of LCT isnot completed yet, some theories related the signal processing need to be furtherestablished or strengthened, such as the time-frequency analysis and sampling theory.Thus, carrying out the research of the LCT and some related theories is very significantin theory and practice.
Based on above issues, this dissertation focuses on the study of time-frequencyanalysis, signal sampling and some related theories of LCT. The main contributions andinnovations of this dissertation are summarized as follows:
1. Beginning with the relationship between the LCT and the Fourier transform,some basic theories associated with the LCT are investigated in this dissertation. Basedon the uniform sampling theorem of Fourier transform, the uniform sampling theoremand signal reconstruction formula associated with the LCT are derived; from thetraditional convolution theorem, the convolution theorem associated with the LCT isalso derived; by analyzing the classic Hilbert transform, the Hilbert transform associatedwith the LCT is proposed.The definition and properties of Hilbert transform of a signalin the LCT domain have been studied. Finally, the definition and properties of LCT fordiscrete time signal are deduced. It’s very important to establish the above basic theoriesassociated with the LCT.
2. Time-frequency analysis theories associated with the LCT are studied in thisdissertation. Starting with the relationship between the linear canonical transform andthe Wigner distribution, the theory of time-frequency filtering about the linear canonical transform has been studied, a time-frequency filtering method based on the linearcanonical transform is proposed and the parameter selection of the filter is discussed indetail. Secondly, the relationship between the linear canonical transform and short-timeFourier transform (STFT) is analyzed. A time-frequency signal analysis method basedon LCT and STFT is proposed, which has no cross-terms problem. It can be used torealize interference suppression of chirp signals and separate components from atime-frequency signal. The simulation results illustrate the validity of the proposedmethod. Finally, the relationship between the linear canonical transform and theambiguity function is analyzed, which is verified by simulations. The above researchresults have laid a good theoretical basis in applied research of further development ofLCT to time-frequency analysis field.
3. The uniform sapling theories for band-pass signal associated with the LCT arededuced in this dissertation. The sampling theories related to LCT have not beencompleted yet, so the sampling theorem needs to be restudied in the LCT domain.Firstly, the definition of band-limited signal in the LCT sense is introduced. Secondly,the band-pass signal sampling theorem and reconstruction formula associated with LCTare deduced. Finally, an example of sampling a chirp signal is provided to demonstratethe application of the sampling theorem. The band-pass signal sampling theoremassociated with the LCT is a generalization of the classical sampling theories and willenrich the theoretical system of the linear canonical transform.
4. The convolution and multiplicative filtering theories associated with the offsetlinear canonical transform (OLCT) are investigated in this dissertation.The OLCT,which is a time-shifted and frequency-modulated version of the linear canonicaltransform, has been shown to be a powerful tool for signal processing and optics.However, some basic results for this transform, such as convolution and correlationtheorems, remain unknown. Based on a new convolution operation, the convolution andcorrelation theorems associated with the OLCT are derived. The sampling theorem forthe bandlimited signal in the OLCT domain is also derived. The formulas of uniformsampling and lowpass reconstruction related to the OLCT are obtained. The designmethod of the multiplicative filter in the OLCT domain is analyzed. Based on the modelof the multiplicative filter in the OLCT domain, a practical method to achievemultiplicative filtering through convolution in the time domain is proposed.These result further enrich the theoretical system of the linear canonical transform.
5. The generalized sampling theories for bandlimited signal associated with theOLCT are deduced in this dissertation.The aim of the generalized sampling is thereconstruction of a bandlimited signal f (t), from the samples of the responses of Mlinear time invariant systems, each sampled by the1Mth Nyquist rate. Firstly, basedon Papoulis’generalized sampling model, the generalized sampling theorem forbandlimited signals in OLCT domains is proposed. Secondly, by designing differentOLCT domain filters, reconstruction formulae for uniform sampling from the signal,from the signal and its first derivative or its generalized Hilbert transform are obtainedbased on the generalized sampling theorem. Since recurrent nonuniform sampling forsignal has valuable applications, reconstruction expression for recurrent nonuniformsamples of signal bandlimited in the offset linear canonical transform domain is alsoobtained by using the generalized sampling theorem and the properties of the offsetlinear canonical transform. Finally, simulations to verify results have been presented.
针对上述问题，本文在研究线性正则变换的特点及性质的基础上，重点针对线性正则变换域的时频分析、信号采样等相关理论问题进行了深入的研究。本文的主要贡献和创新表现在以下几个方面：
1．从线性正则变换与Fourier变换的关系出发，研究了线性正则变换域的基本理论。在Fourier变换的均匀采样定理基础上，推导了线性正则变换域的均匀采样理论与信号重建公式；在传统的卷积定理基础上，推导了线性正则变换域新的卷积定理；在经典Hilbert变换的基础上，给出了线性正则变换域Hilbert变换的定义、性质和特点分析。考虑到在数字信号处理中信号是时域离散的，本文还给出了离散时间信号的线性正则变换定义，证明了其和连续时间信号的线性正则变换相类似的部分性质。上述基本理论的建立对推广与发展线性正则变换在信号处理中的实际应用具有重要的意义。
2．从线性正则变换与传统时频分布的关系出发，研究了基于线性正则变换的信号时频分析理论。首先从线性正则变换与Wigner分布的关系入手，分析了线性正则变换的时频滤波原理，提出了基于线性正则变换的时频滤波方法，并给出了线性正则变换域滤波器参数的详细设计。其次，研究了线性正则变换与短时Fourier变换的关系，并针对chirp信号干扰抑制和多分量时频信号分离问题，提出了基于线性正则变换与短时Fourier变换联合的时频分析方法。仿真实例表明，该方法能避免交叉项干扰，是时频信号分析的有效手段。最后，研究了线性正则变换对时频面上信号模糊函数分布产生的影响，并通过仿真进行了验证。以上研究结果为进一步开展线性正则变换在时频分析领域的应用研究奠定了理论基础。
3．研究了线性正则变换域带通信号的采样理论。首先，给出了线性正则变换域带限信号的定义和特点；其次，导出了线性正则变换域带通信号采样定理和重建公式，并证明了已有的均匀采样定理都是线性正则变换域带通采样定理的特殊形式；最后，通过对chirp类信号的采样分析以及仿真实验，例证了线性正则变换域采样定理的正确性。线性正则变换域带通信号采样定理的得出丰富和完善了线性正则变换的采样理论体系。
4．研究了偏移线性正则变换域信号的卷积与乘性滤波理论。首先定义了信号基于偏移线性正则变换意义的新卷积运算，在此基础上利用偏移线性正则变换的性质导出了偏移线性正则变换域的卷积定理；其次基于该定理建立了偏移线性正则变换域的乘性滤波理论，给出了偏移线性正则变换域乘性滤波器的一般模型，研究了偏移线性正则变换域乘性滤波器在时域的实现方法，分析了扫频滤波器与偏移线性正则变换域乘性滤波器的关系；最后还利用该卷积定理推导了偏移线性正则变换域带限信号的均匀采样定理。由于偏移线性正则变换是线性正则变换引入时移与频率调制参数后的扩展形式，所以上述研究结果将进一步丰富线性正则变换的理论体系。
5.研究了偏移线性正则变换域带限信号的广义采样理论。广义采样的目的是通过M个不同的线性时不变系统输出信号的均匀采样序列重构原始带限信号，且采样速率可降为奈奎斯特速率的1/M。首先，基于Papoulis的广义采样模型，提出了偏移线性正则变换域带限信号的广义采样定理。其次，利用广义采样定理和偏移线性正则变换的性质，通过构造合适的偏移线性正则变换域滤波器，分别得到了基于信号及其导数的均匀采样序列、信号及其Hilbert变换的均匀采样序列以及信号的周期非均匀采样序列，重构原始偏移线性正则变换域带限信号的表达式。最后通过仿真实验，例证了上述结论的正确性。
The linear canonical transform (LCT) is a new signal processing tool developed inrecent years. As a unified and multi-parameter class of linear integral transform, LCThas its unique advantages in non-stationary signal processing. Although it is not verymuch known, its special cases are widely used in various fields, often under differentnames, such as the Fourier transform, fractional Fourier transform (FRFT), Fresneltransform, time scaling, and others. Therefore, understanding the LCT may help to gainmore insights on its special cases and to carry over knowledge gained from one subjectto others. However, from the existing literature, the basic theoretical system of LCT isnot completed yet, some theories related the signal processing need to be furtherestablished or strengthened, such as the time-frequency analysis and sampling theory.Thus, carrying out the research of the LCT and some related theories is very significantin theory and practice.
Based on above issues, this dissertation focuses on the study of time-frequencyanalysis, signal sampling and some related theories of LCT. The main contributions andinnovations of this dissertation are summarized as follows:
1. Beginning with the relationship between the LCT and the Fourier transform,some basic theories associated with the LCT are investigated in this dissertation. Basedon the uniform sampling theorem of Fourier transform, the uniform sampling theoremand signal reconstruction formula associated with the LCT are derived; from thetraditional convolution theorem, the convolution theorem associated with the LCT isalso derived; by analyzing the classic Hilbert transform, the Hilbert transform associatedwith the LCT is proposed.The definition and properties of Hilbert transform of a signalin the LCT domain have been studied. Finally, the definition and properties of LCT fordiscrete time signal are deduced. It’s very important to establish the above basic theoriesassociated with the LCT.
2. Time-frequency analysis theories associated with the LCT are studied in thisdissertation. Starting with the relationship between the linear canonical transform andthe Wigner distribution, the theory of time-frequency filtering about the linear canonical transform has been studied, a time-frequency filtering method based on the linearcanonical transform is proposed and the parameter selection of the filter is discussed indetail. Secondly, the relationship between the linear canonical transform and short-timeFourier transform (STFT) is analyzed. A time-frequency signal analysis method basedon LCT and STFT is proposed, which has no cross-terms problem. It can be used torealize interference suppression of chirp signals and separate components from atime-frequency signal. The simulation results illustrate the validity of the proposedmethod. Finally, the relationship between the linear canonical transform and theambiguity function is analyzed, which is verified by simulations. The above researchresults have laid a good theoretical basis in applied research of further development ofLCT to time-frequency analysis field.
3. The uniform sapling theories for band-pass signal associated with the LCT arededuced in this dissertation. The sampling theories related to LCT have not beencompleted yet, so the sampling theorem needs to be restudied in the LCT domain.Firstly, the definition of band-limited signal in the LCT sense is introduced. Secondly,the band-pass signal sampling theorem and reconstruction formula associated with LCTare deduced. Finally, an example of sampling a chirp signal is provided to demonstratethe application of the sampling theorem. The band-pass signal sampling theoremassociated with the LCT is a generalization of the classical sampling theories and willenrich the theoretical system of the linear canonical transform.
4. The convolution and multiplicative filtering theories associated with the offsetlinear canonical transform (OLCT) are investigated in this dissertation.The OLCT,which is a time-shifted and frequency-modulated version of the linear canonicaltransform, has been shown to be a powerful tool for signal processing and optics.However, some basic results for this transform, such as convolution and correlationtheorems, remain unknown. Based on a new convolution operation, the convolution andcorrelation theorems associated with the OLCT are derived. The sampling theorem forthe bandlimited signal in the OLCT domain is also derived. The formulas of uniformsampling and lowpass reconstruction related to the OLCT are obtained. The designmethod of the multiplicative filter in the OLCT domain is analyzed. Based on the modelof the multiplicative filter in the OLCT domain, a practical method to achievemultiplicative filtering through convolution in the time domain is proposed.These result further enrich the theoretical system of the linear canonical transform.
5. The generalized sampling theories for bandlimited signal associated with theOLCT are deduced in this dissertation.The aim of the generalized sampling is thereconstruction of a bandlimited signal f (t), from the samples of the responses of Mlinear time invariant systems, each sampled by the1Mth Nyquist rate. Firstly, basedon Papoulis’generalized sampling model, the generalized sampling theorem forbandlimited signals in OLCT domains is proposed. Secondly, by designing differentOLCT domain filters, reconstruction formulae for uniform sampling from the signal,from the signal and its first derivative or its generalized Hilbert transform are obtainedbased on the generalized sampling theorem. Since recurrent nonuniform sampling forsignal has valuable applications, reconstruction expression for recurrent nonuniformsamples of signal bandlimited in the offset linear canonical transform domain is alsoobtained by using the generalized sampling theorem and the properties of the offsetlinear canonical transform. Finally, simulations to verify results have been presented.
引文
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