Noise reduction for NMR FID signals via Gabor expansion

The parameters in a nuclear magnetic resonance (NMR) free induction decay (FID) signal contain information that is useful in biological and biomedical applications and research. A real time-sampled FID signal is well modeled as a finite mixture of modulated exponential sequences plus noise. We propose to use the generalized Gabor expansion for noise reduction, where the generalized Gabor expansion represents a signal in terms of a collection of time-shifted and frequency-modulated versions of a single sequence (prototype sequence). For FID signal-fitting, we choose the exponential sequence as the prototype function. Using the generalized Gabor expansion and exponential prototype sequences for FID model-fitting, an NMR FID signal can be well represented by the Gabor coefficients distributed in the joint time-frequency domain (JTFD). The Gabor coefficients reflect the weights of modulated exponential sequences in a signal. One of the important features is that the nonzero Gabor coefficients of a modulated exponential sequence will span a very small area in the JTFD, whereas the Gabor coefficients of the noise will not. If the exponent constant of the prototype sequence in the generalized Gabor expansion matches that of a modulated exponential sequence in the signal, then only one of the Gabor coefficients is nonzero in the JTFD. This is a very important property since it can be exploited to separate a signal from noise and to estimate modulated exponential sequence parameters.