Real-valued Discrete Gabor Transform Fast Algorithm And Its Application

The Gabor transform is one of the most important schemes for time-frequency analysis. Since the traditional Gabor transform is complex-valued, it's real-time applications were limited due to the high complexity involved in the computation of the complex-valued transform. To reduce the computation of the complex-valued discrete Gabor transform (CDGT), in this thesis, the fast real-valued discrete Gabor transform (RDGT) are developed, and their applications are also investigated. The main work and contributions of the thesis are summarized as follows:1 -D and 2-D real-valued discrete Gabor transforms are presented. It is shown that, due to the real operations, the computation of the bi-orthogonal analysis window, the transforms coefficients, and the reconstruction of the original signal in RDGT case, can be significantly reduced as compared with the computation of those in the CDGT case. And the similarity between the RDGT and the discrete Hartley transform (DHT) allows the RDGT to utilize the fast DHT algorithms for fast computation. Meanwhile, the RDGT bears a simple relationship with the CDGT such that the CDGT coefficients can be directly computed from the RDGT coefficients. Therefore, the RDGT also offers a faster and more efficient method to compute the CDGT.Two block time-recursive algorithms are developed for the efficient and fast computation of the 1-D RDGT coefficients and for the fast reconstruction of the original signal from the coefficients in both the critical sampling case and the oversampling case. The two algorithms are implemented respectively by a unified parallel lattice structure. And the computational complexity analysis and comparison show that the proposed algorithms provide a more efficient and faster method for the computation of the discrete Gabor transforms. Due to the real operations and the fast DHT algorithms utilized for the fast computation of the RDGT, the parallel lattice structures of the RDGT can more easily be implemented than in the CDGT case. Moreover, the block time-recursive 2-D RDGT algorithms and their unified parallel lattice structure implementation are extended from 1-D RDGT case.Three applications of the RDGTs are investigated: Gabor representation for transient signals via the RDGTs, MNR FID signal enhancement via the oversampled Gabor transforms, representation and approximation of time-varying systems via the RDGTs. The experimental results show the efficiency and advantages of the RDGTs in applications.