Multi-window Real-valued Discrete Gabor Transform And Its Fast Algorithms

Gabor transform is one of the most important methods for time-frequency analysis, which is widely used in the detection, analysis and processing of non-stationary signals. The traditional Gabor transform with a single window suffers a limitation of the constrained time-frequency localization (or time-frequency resolution) governed by the Heisenberg uncertainty principle. Because the width of the window used in Gabor transform with a single (analysis) window is fixed, the time and frequency resolutions in the resulting time-frequency spectrum are fixed. Both are contradictory and can not reach the high level at the same time. Using the wide window in the Gabor transform will yield the time-frequency spectrum with high frequency resolution but low time resolution. On the contrary, using the narrow window will result in the time-frequency spectrum with high time resolution but low frequency resolution. To overcome the problem, researchers have proposed the multiwindow complex-valued discrete Gabor transform (M-CDGT) based on the frame theory. By combining the time-frequency spectra obtained from the M-CDGT, the time-frequency resolution in the resulting Gabor time-frequency representation can be effectively improved. To develop the multiwindow discrete Gabor transform more deeply and thoroughly, in this dissertation, the multiwindow real-valued discrete Gabor transform (M-RDGT) is presented and explored based on the biorthogonal analysis approach. The main innovation and work are shown as follows:The M-RDGT and its fast algorithms for periodic (finite) sequences are presented. By replacing the complex exponential kernel in the M-CDGT kernel with the cas kernel in discrete Hartley transform (DHT), the M-RDGT for periodic (finite) sequences is presented. Based on the biorthogonal analysis approach, the biorthogonality constraint between analysis windows and synthesis windows in the M-RDGT and its inverse transform (multiwindow real-valued discrete Gabor expansion) is derived and proved to be equivalent to the completeness condition of the M-RDGT. The biorthogonality constraint is utilized for fast computation of window functions. The fast algorithms for M-RDGT and its inverse transform are presented based on DHT. Because the relationship between M-RDGT coefficients and M-CDGT coefficients is similar to the simple algebraic relation between discrete Fourier transform (DFT) coefficients and DHT coefficients, the fast algorithms of M-RDGT also offer an efficient method to compute the M-CDGT.The M-RDGT and its fast algorithms for long-periodic (or even infinite) sequences are presented. Since the analyzed sequence, analysis and synthesis windows in M-RDGT for finite sequences (periodic sequences) must have an equal length, if the period or length of a sequence is very long or infinite, solving its windows requires a huge amount of memory and computation and sometime could lead to numerical instability. To overcome this problem, a modified M-RDGT for long-periodic (or even infinite) sequences is presented based on the previously proposed M-RDGT for periodic (finite) sequences and its biorthogonality constraint between analysis windows and synthesis windows is derived. In the modified M-RDGT, the period of the analysis and synthesis windows are independent of the periods of the analyzed sequences so that one can apply short windows to process long-periodic (or even infinite) sequences.The multirate-based parallel implementation of the M-RDGT is presented. Because the M-RDGT and its inverse transform are respectively similar to the analysis process of an analysis filter bank and the synthesis process of a synthesis filter bank, an analysis convolver bank and a synthesis convolver bank respectively for the multirate-based parallel implementation of the M-RDGT and its inverse transform can be designed by virtue of the similarity. Every parallel channel in the analysis convolver bank or the synthesis convolver bank has a unified structure and can apply the fast DHT algorithm to reduce the computation load. The computational complexity of each parallel channel is independent of the Gabor oversampling rate and the number of windows used. In fact, it is very low and depends only on the length of the input discrete signal and the number of the Gabor frequency sampling points.Finally, in this dissertation, some computational experiments are given, including the computation of the M-RDGT and the Gabor time-frequency spectra of such three signals as a sinusoid with two impulse functions, an exponentially damped sinusoidal transient signal, and an electrocardiogram (ECG) signal from the Apnea-ECG database. The experimental results show that the proposed M-RDGT provides an effective and fast method to analyze the dynamic time-frequency contents of a signal that contains components with multiple and/or time-varying frequencies.