On the statistics and practical application of the reassignment method for Gabor spectrograms

Abstract: The reassignment method is a technique for improving the concentration of signals in spectrograms and other time-frequency representations (TFR). It achieves this by displacing the points in a TFR according to the reassignment vector for every point. By doing so, the reassignment method gives perfect concentration of infinite constant frequency sinusoids, impulses and linear chirps. A downside to the reassignment method is that it is fairly sensitive to noise. While this is well known, the subject of how noise affects the reassignment method is largely unexplored. Some important groundwork has been laid by Chassande-Mottin et al. In their report from 1996, they derived the density function of the reassignment vector, given that the signal is subjected to additive white Gaussian noise (AWGN). For Gabor (Gaussian windowed) spectrograms, a closed form expression of the density function is given. This thesis largely builds on top of said result,and aims to extend the general knowledge about the statistics of reassigned spectrograms. The focus lies on Gabor spectrograms, and a rather practical approach is taken. First, some statistical properties of the reas-signment vector are explored. From this, a Gaussian approximation is suggested which makes the density function for the reassignment vector feasible to work with. Then, we look at how the reassigned spectrogram behaves as a whole when subjected to AWGN. The signalsexamined are those previously mentioned, all perfectly localized by the reassignment method. It shows that in the context of reassigning the spectrogram, these signals are equivalent. The resulting reassigned spectrogram turns out to be of infinite variance since the distribution is heavy tailed. However, its shape can still be related to the width of a Gaussian. By doing so, a simple formula is proposed which states the ratio of concentration given by the reassigned spectrogram compared to theoriginal spectrogram. Finally, based on the previous findings, an idea for a new method of resampling reas-signed noisy spectrograms is proposed. This method attempts to mitigate the issue that the reassigned spectrogram “deteriorates” when resampled in a naive manner.