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.