Multitaper Spectrum Estmation and Time-Frequency Analysis

Multitaper Spectrum Estimation

Multiple Window or Multitaper spectral estimation is a technique that allow efficient and robust use of finite length data with respect to bias and variance. The idea, first presented by David Thomson in 1982, was to average windowed periodograms with the aim to reduce the variance of the spectral estimate. The commonly used method today rely on that different data intervals give almost uncorrelated spectral estimates and when the spectral estimates are averaged, the variance of the result is reduced. Thomson suggested that different data windows, multitapers, should be applied to the same data sequence and the properties of the windows should give uncorrelated periodograms for the final average. The Thomson method work well for smooth spectra, ideally a white noise spectrum. We have applied the multitapers by Thomson for coherence estimation in double talk detection, which was a part of a larger project, ``Non-intrusive measurements of the telephone channel", initiated and done in co-operation with Telia. This project has also generated several patents. In spectral analysis of the electrical activity from the brain, the ElectroEncephaloGram (EEG), the Thomson method has shown to be not always appropriate and as the spectra of EEG usually includes sharp peaks instead of being smooth, an idea has been to use a predefined peaked spectrum as a spectrum model. A set of multitapers giving minimum variance is computed where the suppression of the side-lobes outside a predefined frequency band is determined by a parameter. The weighting factors in the average of the multitaper spectra are optimized for low bias, variance as well as mean squared error.

Statistical Time-Frequency Analysis

Multitaper implementation of time-frequency kernels has the advantage of being much more computationally efficient as compared to the common implementation techniques in time-frequency analysis but in the design of these windows, other aspects could also be taken into account, e.g., by using a penalty function that suppresses power outside a certain area of the kernel while the properties of the kernel still is retained. The advantage of this technique is the suppression of cross-terms. Multitapers are currently being applied to non-stationary processes but often properties of stationary processes are used and the different spectrograms are often equally weighted in the average (as suggested by D. Thomson). We have optimized weights for different non-stationary processes and it is shown that the usual equal weighting is insufficient. We have also studied optimal time-frequency kernels and corresponding multitapers for non-stationary processes and especially a class of locally stationary processes (LSP) where the mean squared error optimal kernel in the ambiguity domain has been calculated. For this class of LSPs, the different multitapers are also shown to be dependent only on a scale factor and different sets of weighting factors. These properties are very useful from an implementation aspect. The approximation of these multitapers as Hermite functions, giving a more easy computation of the windows, are also evaluated, and the Hermite functions are shown to be appropriate as an approximation.