Spectrum estimation by wavelet thresholding of multitaper estimators

Abstract

Current methods for power spectrum estimation by wavelet thresholding use the empirical wavelet coefficients derived from the log periodogram. Unfortunately, the periodogram is a very poor estimate when the true spectrum has a high dynamic range and/or is rapidly varying. In addition, because the distribution of the log periodogram is markedly non-Gaussian, special wavelet-dependent thresholding schemes are needed. These difficulties can be bypassed by starting with a multitaper spectrum estimator. The logarithm of this estimator is close to Gaussian distributed if a moderate number (/spl ges/5) of tapers are used. In contrast to the log periodogram, log multitaper estimates are not approximately pairwise uncorrelated at the Fourier frequencies, but the form of the correlation can be accurately and simply approximated. For scale-independent thresholding, the correlation acts in accordance with the wavelet shrinkage paradigm to suppress small-scale noise spikes while leaving informative coarse scale coefficients relatively unattenuated. This simple approach to spectrum estimation is demonstrated to work very well in practice. Additionally, the progression of the variance of wavelet coefficients with scale can be accurately calculated, allowing the use of scale-dependent thresholds. This more involved approach also works well in practice but is not uniformly preferable to the scale-independent approach.