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.

Popular Abstract

This thesis explores and improves the reassignment method, used for analyzing signals that change over time. With our newfound knowledge, a special smoothing can be applied, enhancing its performance.
Signals varying over time, so called non-stationary signals, appear all around us. From the changing temperature of our globe to the response from a complex radar system.
The need for analyzing them arises practically everywhere, and is ever so challenging. A fundamental tool in this field of research, called time-frequency analysis, is the spectrogram. It takes the signal and decomposes it into its frequency content at any given point in time.
A problem with the spectrogram is that it is not very precise. Let’s say we want to analyze some bird chirps. Perhaps we want to know both when the chirp happened,
and what pitch it had. This corresponds to localizing the signal in time and in frequency respectively. Unfortunately, we can not obtain sharp measures of both simultaneously.
Either the time or frequency will, to some extent, appear to be smeared out.
This is what the so called reassignment method seek to compensate. For some signals, we can obtain a perfect concentration of a signal both in time and frequency. In other words, it allows us to clearly see both when and with what pitch a bird has chirped, which is great! However, the reassignment method comes with a drawback – it is quite sensitive to noise and disturbances.
If there is some way to work around this, we would get the best of two worlds. And this is exactly what our thesis project is all about. We present a method that mitigates the effects that noise has on the reassignment. To get there, we had to build up a thorough understanding of the method.
Insomesense, the(Gabor)spectrogram can be seen as normal distributed. However, we
saw that this is actually not the case for the reassigned spectrogram. Actually, it turns out to be of infinite variance (a measure of spread), which might seem counterintuitive.
This is not as bad as it seems, and we suggest that the reassigned spectrogram can be roughly seen as normal distributed as well.
Knowing how the reassigned spectrogram behaves, we had an idea on how to modify
it to make it more robust against noise. By smoothing it corresponding to its theoretical distribution, its behavior becomes much more predictable. Our method is basically a way to sacrifice some concentration for the gain of robustness. The concept is similar to kernel density estimation or kernel smoothing. We call it the smoothed reassigned spectrogram, or SRS for short.

• author: Månsson, Erik. M
• supervisor: Maria Sandsten
• organization: Mathematical Statistics
• course: FMSM01 20191
• year: 2019
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• language: English
• id: 8984673
• date added to LUP: 2019-06-17 11:08:01
• date last changed: 2019-06-17 11:08:01