Sparse Time-Frequency Representation Based On Discrete Gabor Transform

Discrete Gabor transform is an important time-frequency analysis tool, which has been widely used in digital signal processing, digital image processing and system mod-eling. In the past decade, the sparse transform has been proved to be a new and effective mathematical tool and successfully applied speech processing, image denoising, com-pressive sensing and other engineering fields. In this dissertation, the research focuses on the sparse time-frequency representation based on the discrete Gabor transform. The traditional discrete Gabor transform is normally a redundant transform in the oversam-pling case, containing many nonzero Gabor coefficients. The sparse time-frequency representation of a signal is to represent the original signal by using as few nonzero Ga-bor coefficients as possible. This motivated us to explore the discrete Gabor transform based on the sparse theory instead of the traditional discrete Gabor transform, which can improve the time-frequency resolution and the time-frequency concentration of Gabor spectrum and make it more effective in the nonstationary signal analysis and process-ing. The main work and contributions of the dissertation are summarized as follows:A sparse time-frequency representation method based on discrete Gabor transform is proposed. The width of window function in discrete Gabor transform straight affects the time-frequency resolution and the time-frequency concentration of Gabor spectrum. First of all, based on the entropy of Gabor spectrum, an optimal win-dow width in discrete Gabor transform can be searched adaptively, then the discrete Gabor transform is converted to a convex optimization equation with a ? norm sparsity constraint model; finally, the approximate resolution can be obtained by sparse theory. Because the sparse constraint model based on the ? norm is often unstable, it can lead to an over-sparse solution and destroy the inner structure of the solution, while the res-olution obtained from the l1-l2 mixed norm-based sparse constraint model has better sparsity and stability, which has a practical engineering application value. Experiments also show that the Gabor spectrum based on this sparse constraint model has better time-frequency concentration and better effects on noise reduction.A sparse time-frequency representation method based on multi-window dis- crete Gabor transform is presented. The traditional discrete Gabor transform with single window suffers a limitation of the constrained time-frequency localization gov-erned by the Heisenberg uncertainty principle and thus it is insufficient to analyze the dynamic time-frequency content of a signal that contains components with multiple and/or time-varying frequencies. The time frequency representation method based on multi-window discrete Gabor transform with structured sparsity can effectively decom-pose and analyze the signal. First of all, the multi-window discrete Gabor transform is converted to a convex optimization equation with lp,q norm constrained, then ac-cording to the different lp,q norms the corresponding soft threshold function is adopted; finally, the sparse Gabor coefficients are obtained by using the block coordinate descent method. Experiments show that the proposed method provides the time-frequency spec-trum with higher time-frequency resolution.A fast algorithm for computing Gabor dual windows in discrete Gabor trans-form is proposed based on the matrix decomposition and fast Fourier transform. In the sparse analysis, it is necessary to study the fast algorithm for computing the synthesis window corresponding to the given analysis window so as to reconstruct the signal. Therefore, this motivated us to explore a fast algorithm for computing Gabor dual windows in discrete Gabor transform based on the matrix decomposition and fast Fourier transform. First of all, A new biorthogonal relationship between synthesis win-dow and analysis window is derived according to the completeness of discrete Gabor transform, then the linear equation set corresponding to the biorthogonal relationship is simplified and separated into a certain number of independent linear sub-equation sets, and each of those sub-equation sets can utilize the fast Fourier transform to compute the dual window so that a large amount of computing time can be saved. The experiments indicate that the proposed algorithm is fast and valid.Efficient discrete Gabor transform with weighted linear combination of anal-ysis windows (WLCAW) and iterative algorithm for its weights of the linear com-bination are proposed. In the traditional multi-window discrete Gabor transform, the time-frequency resolution of the combined Gabor spectrum depends not only on the selected analysis windows but also on the weights of the linear combination. Thus, the WLCAW-based discrete Gabor transform is proposed and then converted to a convex optimization equation with l1-l2 norm constrained according to the transform coeffi-cient sparsity. The weights of the analysis windows can be solved based on the sparse transform theory. During the iterative process of solving the weights of analysis win-dows, it is necessary to calculate the synthesis window sequence to reconstruct the original signal, so the proposed fast algorithm for computing the dual windows can be used to reduce the computation time and increase the computing speed. The experi-ments indicate the proposed discrete Gabor transform to be efficient.