Relationship And Fast Algorithm For Window Computation In Real-valued Discrete Gabor Transform For Long Sequences

Gabor analysis and synthesis is an important joint time-frequency analysis method proposed firstly by Dennis Gabor in1946. Using Gabor time-frequency analysis, any signal can be mapped into the joint time-frequency domain. Wexler and Qian et al proposed the discrete version of Gabor expansion and transformation for discrete signal. They also proposed the biorthogonal analysis to compute the analysis window function which is required in the Gabor analysis and synthesis. Since signals are always real-valued in practical applications, Tao Liang et al proposed real-valued discrete Gabor transform (RDGT) based on discrete Hartley transform (DHT) thereafter. And block recursive algorithm for RDGT was also proposed which can be implemented with parallel lattice-structure making it more applicable.In the real-valued discrete Gabor transform, we need to compute the synthesis window function according to analysis window or vice versa. However, computing these window functions are very involved because basic functions in discrete Gabor transform is not orthogonal to each other. In order to compute the Gabor transform coefficients and reconstruct the original signal effectively and fast, a fast algorithm to compute the window function is required at first. In order to tackle the problem that computes the synthesis window based on the analysis window, this thesis proposes two biorthogonal relationship pairs based on the biorthogonal analysis. Since the conventional biorthogonal relationship is computationally complex, a fast algorithm based on the discrete Hartley transform is also proposed to resolve the biorthogonal presented relationship. Based on discrete Hartley transform (DHT) and its inversion version, the biorthogonal relationship is solved by several independent small scale linear equations. In order to evaluate the performance of the proposed algorithm, the matlab simulation was conducted and the experimental results indicate that the proposed dual of the biorthogonal relationship and the fast algorithm are correct and effective.