Fast Algorithms Research For Solving The Windows In Multi-window Real-valued Discrete Gabor Transform

Traditional complex-valued Gabor expansion and transform based on Fourier transform was first proposed in 1946 by Dennis Gabor, which has now become an important tool for signal time-frequency analysis and has widely been used in one-dimensional or multidimensional signal processing after several decades of development. The main issues surrounding the Gabor transform theory were the problem for calculation of dual window function, Gabor expansion or reconstruction computing speed and Gabor time-frequency resolution and so on. The proposal of the method of real-valued discrete Gabor transform (RDGT) enriched and expanded the Gabor time-frequency analysis theory. The traditional discrete Gabor expansion and transform with single-window suffered a limitation of the constrained time-frequency resolution governed by the Heisenberg uncertainty principle and improving the signal time-frequency resolution is critical for most signal time-frequency analysis. Multi-window complex-valued discrete Gabor transform theory came into being, and it was used in the signal evolutionary spectral analysis, image watermarking, image compression, etc.In order to improve the time-frequency resolution of the Gabor transform and reduce the computational time complexity of the multi-window Gabor transform, this paper further studied fast algorithms for solving the windows based on the biorthogonality condition in multi-window RDGT. The biorthogonality condition between analysis windows and synthesis windows for the multi-window real-valued discrete Gabor expansion and transform for limit sequences was derived and proved to be equivalent to the completeness condition. A fast algorithm for solving the windows was presented by using fast discrete Hartley transform (DHT) and the orthogonality of the Hartley functions for finite sequences multi-window RDGT. To improve practicability of Gabor transform, this paper also extended the multi-window real-valued discrete Gabor transform for finite sequences to the multi-window RDGT for infinite sequences, which was more fit for dealing with long sequence signal. The bi-orthogonal relationship between the analysis window and the synthesis window in the multi-window RDGT for infinite sequences was given and a fast algorithm to computing the analysis window was presented. Finally, the Matlab simulation’s results also indicated the correctness of the proposed dual biorthogonal relationship and the effectiveness of the proposed fast algorithm.