Optimal Multitaper Spectrograms

Abstract

Multitaper spectrograms have been proposed as a method of improving the spectrogram as a time-frequency representation (TFR). This thesis aimed to investigate both previously used and new methods for combining multitaper spectrograms of a Gaussian signal and a chirp. More specifically, the use of new measures to optimize the weights of a weighted sum of spectrograms was tested and evaluated. The optimization problems were first stated independently of the signal and then solved for a Gaussian and chirp signal. A theorem for the spectrogram of a Hermite function was proved. This allowed an exact solution to be obtained for the least squares optimization between the multitaper spectrogram and the Wigner distribution of a Gaussian signal. The theorem was also used to extend the method used for the Gaussian signal for a chirp based on the use of Hermite expansion. The optimized weights were then evaluated based on concentration and sensitivity to noise.

The results in the case of the Gaussian signal showed that some of the new methods gave improved concentration in comparison to the Wigner distribution and other commonly used methods. The solution derived for the least squares optimization was also seen to be much faster in computation compared to numerical methods. Furthermore, the optimized weights were less sensitive to noise than the Wigner distribution. The results for the chirp signal showed that the new method based on Hermite expansion gave improved results compared to using the weights calculated for the Gaussian signal.

The conclusion was that the new optimization methods were able to not only improve the localization of a time-frequency representation but also give desirable qualities such as non-negativity. This means that one could choose which method to use based on desired properties whether it be concentration or robustness to noise. The conclusion was also that the new method used for the chirp signal gave improved results and could possibly be extended to other signals not covered in the thesis. Further research could also be done pertaining to each of the new optimization methods used in the thesis.

Popular Abstract

Time-frequency analysis comes from the heart, literally as it has been used to study electrical signals from the heart called ECG. Time-frequency analysis as the name suggests is the study of joint representations in both time and frequency of a signal. Rather than constraining the analysis of some signals to only its frequency content, or how the signal behaves over time, one aims to find representations that show how the frequency changes over time. As an example, most music is thankfully not single tones being played constantly throughout but rather different tones of varying lengths appearing at different times: a perfect scenario for time-frequency analysis.

Unfortunately, there is no single time-frequency representation. Rather, there are many possible representations, each with its specific applications and advantages/disadvantages. The most well-known one is the spectrogram which due to its simplicity and computational efficiency is often used. Unfortunately, the spectrogram suffers from quite a few shortcomings such as poor localization and resolution of multiple signals. An alternative that solves some of the issues with the spectrogram is the so-called Wigner distribution. The Wigner distribution however also has its disadvantages like sensitivity to noise and unwanted interference and is therefore not a simple solution.

In reality, the signals that are measured also contain noise and this also leads to the issue of how sensitive the time-frequency representations are to noise. The signal measured is in this case only one realization of some random or stochastic process. There is then an ambiguity in how one should define a time-frequency representation for a random process and which methods previously mentioned are good estimates.

These issues and more are investigated in this thesis. More specifically, methods of combining several spectrograms to improve on the negative aspects of a single spectrogram are proposed. The spectrograms are combined as a weighted sum, where the weights are chosen based on different optimization criteria. The thesis investigates the time-frequency representations using the different optimized weights based on concentration and sensitivity to noise. The result is several methods of choosing weights for two common types of signals that give better concentration than a single spectrogram and are less sensitive to noise than the Wigner distribution. Exact solutions to some explicit cases are also calculated in the thesis, allowing for much faster computations. The thesis, however, leaves plenty of room for future research involving each of the analyzed methods and the extension to other signal types.

• author: Ydreskog, Markus
• supervisor: Maria Sandsten; Oskar Keding
• organization: Mathematical Statistics
• course: FMSM01 20231
• year: 2023
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• keywords: Time-frequency analysis, Multitaper spectrogram, Optimization, Wigner distribution
• publication/series: Master's Theses in Mathematical Sciences
• report number: LUTFMS-3472-2023
• ISSN: 1404-6342
• other publication id: 2023:E21
• language: English
• id: 9116759
• date added to LUP: 2023-05-25 12:33:52
• date last changed: 2023-05-25 12:33:52