参数化时频信号表示研究
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摘要
本文主要研究如何表示信号,取得的成果有:1)证明了不含交叉项干扰且具有Wigner-Ville分布聚集性的时频分布不存在,并且不含交叉项干扰而聚集性充分接近Wigner-Ville分布聚集性的时频分布,亦不存在。2)提出了一种适于刻画信号中线性和非线性时变成分的参数化时频信号表示方法——FM~mlet变换,研究了FM~mlet变换的性质和指令级并行优化算法,揭示了chirplet变换、频散变换、小波变换、chirp-Fourier变换、STFT、Gabor变换、Fourier变换、余弦变换、正弦变换、Hartley变换、Laplace变换、z变换、Mellin变换、Hilbert变换、自相关函数、互相关函数,以及求信号的能量和均值运算,均是FM~mlet变换的特例,提出了适于刻画滑音信号的FM~mlet变换的另一子空间 蝴蝶子空间,以及时间和频率分辨率均达到理论值极限的“伪时频分布”的概念。3)提出了一种适于刻画Doppler信号的参数化时频信号表示方法——Dopplerlet变换,并给出了基于自适应匹配投影塔形分解法的参数化时频信号表示算法,以及Dopplerlet变换在水下航行物测距测速中的应用实例。对Dopplerlet变换的物理机制的分析表明,小波变换、STFT、Gabor变换、Fourier变换、余弦变换、正弦变换、Hartley变换、Laplace变换、z变换、Mellin变换、Hilbert变换、自相关函数、互相关函数,以及求信号的能量和均值运算,均是Dopplerlet变换在其参数取特定值时的特例。4)提出了一种基于子空间分解的多谱线增强方法,一种基于瞬时频率估计的chirp信号增强方法,以及一种基于Radon-STFT变换的chirp信号增强方法,并给出了重构信号的精度与信号采样点数的定量关系。5)定义了酉对称矩阵,导出并证明了酉对称矩阵的奇异值和奇异向量,与母矩阵的奇异值和奇异向量之间的定量关系,揭示了行(列)对称矩阵、置换对称矩阵、正交对称矩阵和酉对称矩阵四者之间的逐级包含关系,以及Givens对称矩阵、Householder对称矩阵与置换对称矩阵之间的并列关系,最后给出了酉对称矩阵奇异值分解的摄动分析,以及酉对称矩阵奇异值分解在信息检
This dissertation focuses on how to represent signals. The main contributions are as follows:1) The nonexistence theorem for cross-term free time-frequency distribution with concentration of Wigner-Ville distribution is proved. In addition, it is discovered that in general, cross-term free joint distributions with their concentrations close to that of Wigner-Ville distribution do not exist either.2) A novel parametric time-frequency representation called FM~mlet transform, which is suitable for delineating the linear and nonlinear time-varying nature of a signal, is proposed. The mathematical properties and instruction level parallel optimization of FM~mlet transform are investigated accordingly. It is shown that some of the existing integral transforms such as chirplet transform, dispersion transform, wavelet transform, chirp-Fourier transform, short-time Fourier transform, Gabor transform, Fourier transform, cosine transform, sine transform, Hartley transform, Laplace transform, z transform, Mellin transform, Hilbert transform, autocorrelation function, cross-correlation function, and the energy and mean of a signal can each be considered as a special case of the FM~mlet transform with specific parameters. Another subspace of FM~mlet transform for compactly characterizing the gliding tones, i.e., the butterfly subspace, is also presented. A "pseudo time-frequency distribution" which resolutes both time and frequency domains to their theoretical limits is advocated for simple and readable visualization of the signal's time-varying and time-invariant structures. Theoretical analysis and experimental results demonstrate that the FM~mlet-based signal representations naturally lead to highly compact solutions to a plethora of stationary and nonstationary scenarios involving especially the whistler or gliding tone-like signals.3) A novel parametric time-frequency representation called Dopplerlet transform, which is inherently well suited for Doppler signal, is proposed. The algorithm for computing parametric time-frequency representation based on matching pursuit is developed, and the application of Dopplerlet transform to estimation of the range and speed
of an underwater vessel is presented. Physical analysis as well as theoretical predictions indicate that some of the existing integral transforms such as wavelet transform, short-time Fourier transform, Gabor transform, Fourier transform, cosine transform, sine transform, Hartley transform, Laplace transform, z transform, Mellin transform, Hilbert transform, autocorrelation function, crosscorrelation function, and the energy and mean of a signal are all special cases of the Dopplerlet transform with specific parameters. Since many interesting natural and artificial processes yield the phenomena of Doppler effect, the Dopplerlet transform may therefore enable more flexible and parsimonious representation for their time-varying nature.4) A multiple line enhancer based on subspace decomposition, and two chirp signal enhancers based on the estimated instantaneous frequency and Radon-STFT transform respectively, are proposed. A quantitative relation between the reconstruction precision and the length of the signal is also presented.5) A special architecture called unitary-symmetric matrix, which embodies orthogonal, Givens, Householder, permutation, and row (or column) symmetric matrices as its special cases, is defined. A precise correspondence of the singular values and singular vectors between the unitary-symmetric matrix and its mother matrix is derived and proved, and the corresponding perturbation analysis is conducted. As an illustration of potential, it is deducted (and the provided application to information retrieval also confirmed) that, for a class of unitary-symmetric matrices, singular value decomposition using mother matrix instead of unitary-symmetric matrix per se, can dramatically save CPU time and computer memory without the loss of numerical precision.6) The relationship between Q and R matrices of unitary-symmetric matrix and its mother matrix, is clarified and justified. Two new algorithms followed by computational costs for fast computing the QR factorization of unitary-symmetric matrix are addressed and examined, along with some qualitative results obtained from a perturbation analysis. Analytical and numerical studies also give evidence that, for a class of unitary-symmetric matrices, the strategy of QR factorization based on the surrogate, viz., the mother matrix, is far cheaper than the naive scheme of direct QR factorizing the unitary-symmetric matrix.
This dissertation focuses on how to represent signals. The main contributions are as follows:1) The nonexistence theorem for cross-term free time-frequency distribution with concentration of Wigner-Ville distribution is proved. In addition, it is discovered that in general, cross-term free joint distributions with their concentrations close to that of Wigner-Ville distribution do not exist either.2) A novel parametric time-frequency representation called FM~mlet transform, which is suitable for delineating the linear and nonlinear time-varying nature of a signal, is proposed. The mathematical properties and instruction level parallel optimization of FM~mlet transform are investigated accordingly. It is shown that some of the existing integral transforms such as chirplet transform, dispersion transform, wavelet transform, chirp-Fourier transform, short-time Fourier transform, Gabor transform, Fourier transform, cosine transform, sine transform, Hartley transform, Laplace transform, z transform, Mellin transform, Hilbert transform, autocorrelation function, cross-correlation function, and the energy and mean of a signal can each be considered as a special case of the FM~mlet transform with specific parameters. Another subspace of FM~mlet transform for compactly characterizing the gliding tones, i.e., the butterfly subspace, is also presented. A "pseudo time-frequency distribution" which resolutes both time and frequency domains to their theoretical limits is advocated for simple and readable visualization of the signal's time-varying and time-invariant structures. Theoretical analysis and experimental results demonstrate that the FM~mlet-based signal representations naturally lead to highly compact solutions to a plethora of stationary and nonstationary scenarios involving especially the whistler or gliding tone-like signals.3) A novel parametric time-frequency representation called Dopplerlet transform, which is inherently well suited for Doppler signal, is proposed. The algorithm for computing parametric time-frequency representation based on matching pursuit is developed, and the application of Dopplerlet transform to estimation of the range and speed
of an underwater vessel is presented. Physical analysis as well as theoretical predictions indicate that some of the existing integral transforms such as wavelet transform, short-time Fourier transform, Gabor transform, Fourier transform, cosine transform, sine transform, Hartley transform, Laplace transform, z transform, Mellin transform, Hilbert transform, autocorrelation function, crosscorrelation function, and the energy and mean of a signal are all special cases of the Dopplerlet transform with specific parameters. Since many interesting natural and artificial processes yield the phenomena of Doppler effect, the Dopplerlet transform may therefore enable more flexible and parsimonious representation for their time-varying nature.4) A multiple line enhancer based on subspace decomposition, and two chirp signal enhancers based on the estimated instantaneous frequency and Radon-STFT transform respectively, are proposed. A quantitative relation between the reconstruction precision and the length of the signal is also presented.5) A special architecture called unitary-symmetric matrix, which embodies orthogonal, Givens, Householder, permutation, and row (or column) symmetric matrices as its special cases, is defined. A precise correspondence of the singular values and singular vectors between the unitary-symmetric matrix and its mother matrix is derived and proved, and the corresponding perturbation analysis is conducted. As an illustration of potential, it is deducted (and the provided application to information retrieval also confirmed) that, for a class of unitary-symmetric matrices, singular value decomposition using mother matrix instead of unitary-symmetric matrix per se, can dramatically save CPU time and computer memory without the loss of numerical precision.6) The relationship between Q and R matrices of unitary-symmetric matrix and its mother matrix, is clarified and justified. Two new algorithms followed by computational costs for fast computing the QR factorization of unitary-symmetric matrix are addressed and examined, along with some qualitative results obtained from a perturbation analysis. Analytical and numerical studies also give evidence that, for a class of unitary-symmetric matrices, the strategy of QR factorization based on the surrogate, viz., the mother matrix, is far cheaper than the naive scheme of direct QR factorizing the unitary-symmetric matrix.
引文
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