On Parametric Time-Frequency Signal Representation

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 speedof 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.