NMR FID signal enhancement via the oversampled Gabor transform using the Gaussian synthesis window

An efficient algorithm to reduce the noise from the nuclear magnetic resonance free induction decay (NMR FID) signals is presented, in this paper, via the real-valued and oversampled discrete Gabor transform using the Gaussian synthesis window. An NMR FID signal in the Gabor transform domain (i.e., a joint time-frequency domain) is concentrated in a few number of Gabor transform coefficients while the noise is fairly distributed among all the coefficients. Therefore, the NMR FID signals can be significantly enhanced by performing a thresholding technique on the coefficients in the transform domain. Theoretical and experimental analyses in this paper show that the oversampled Gabor transform using the Gaussian synthesis window is more suitable for the NMR FID signal enhancement than the critically-sampled one using exponential synthesis window, because both the Gaussian synthesis window and its corresponding analysis window in the oversampling case can have better localization in the frequency domain than the exponential synthesis window and its corresponding analysis window in the critically-sampled case. Moreover, to speed up the transform, instead of the common used complex-valued discrete Gabor transform, the real-valued discrete Gabor transform proposed in our previous work is adopted in this paper.