Fast Parallel Algorithms For Real-valued Discrete Gabor Transforms Based On DCT

In 1946, Gabor constructed time-and frequency-shiftable transform kernel(named basis functions) by multiplying the transform kernel of Fourier transform(named complex exponential functions) with a set of time-shiftable window functions, then he presented the complex-valued Gabor transform based on Fourier transform. Although many people thought the Gabor expansion was useful, the applications were restrained for a long time because of the difficulties in the calculation of the transform coefficients.In order to reduce the computational complexity, we have presented the real-valued discrete Gabor transform(RDGT) based on DCT in our previous work. This approach is similar to the traditional complex-valued discrete Gabor transform(CDGT). Besides, it only involves real operations and can utilize fast DCT and IDCT algorithms for fast computation. As a result, the computational complexity of the Gabor transform is significantly reduced. So in practical application, the real-valued discrete Gabor transform is more easily implemented in both software and hardware.In this paper, firstly, the development of the Gabor transform theory is briefly reviewed; then the fast parallel algorithms for real-valued discrete Gabor transforms based on DCT will be presented. In order to compute the real-valued discrete Gabor transform effectively and fast, block time-recursive algorithms for the efficient and fast computation of the 1-D RDGT coefficients and for the fast reconstruction of the original signal from the RDGT coefficients will be developed in both critical sampling case and over-sampling case, and then unified parallel lattice structures for the implementation of the algorithms will be studied. Because the computational complexity of this algorithm is allocated to parallel cells, so the computation speed will be greatly increased. Moreover, the computational complexity compared with other algorithms is also proved that the parallel algorithm is fast and effective. Finally, an application of the real-valued discrete Gabor transforms based on DCT will be presented, The noised NMR FID signal can be enhanced by the DCT-based real-valued discrete Gabor transform.