Novel DCT-Based Real-Valued Discrete Gabor Transform and Its Fast Algorithms

The oversampled Gabor transform is more effective than the critically sampled one in many applications. The biorthogonality relationship between the analysis window and the synthesis window of the Gabor transform represents the completeness condition. However, the traditional discrete cosine transform (DCT)-based real-valued discrete Gabor transform (RGDT) is available only in the critically sampled case and its biorthogonality relationship for the transform has not been unveiled. To bridge these important gaps, this paper proposes a novel DCT-based RDGT, which can be applied in both the critically sampled case and the oversampled case, and their biorthogonality relationships can be derived. The proposed DCT-based RDGT involves only real operations and can utilize fast DCT algorithms for computation, which facilitates computation and implementation by hardware or software as compared to that of the traditional complex-valued discrete Gabor transform. This paper also develops block time-recursive algorithms for the efficient and fast computation of the RDGT and its inverse transform. Unified parallel lattice structures for the implementation of these algorithms are presented. Computational complexity analysis and comparisons have shown that the proposed algorithms provide a more efficient and faster approach for discrete Gabor transforms as compared to those of the existing discrete Gabor transform algorithms. In addition, an application in the noise reduction of the nuclear magnetic resonance free induction decay signals is presented to show the efficiency of the proposed RDGT for time-frequency analysis.