University of Washington - Applied Physics Laboratory
Current methods for producing a power spectrum estimate by wavelet thresholding apply thresholding to the empirical wavelet coefficients derived from the log periodogram. Unfortunately, the periodogram is a very poor spectrum estimation method when the true spectrum has a high dynamic range and/or is rapidly varying (as commonly occurs in the physical sciences). Also, because the distribution of the log periodogram is not well approximated by a Gaussian distribution, complicated wavelet-dependent thresholding schemes are needed. To bypass these difficulties, we start with a multitaper spectrum estimator. The logarithm of this estimator is close to Gaussian distributed provided five or more tapers are used, and this enables the computation of the correlation of the log spectrum estimator. For scale-independent "ideal" thresholding the correlation acts to strongly suppress "noise spikes" while leaving informative coarse-scale coefficients relatively unattenuated. This apparently rather crude approach is seen to work very well in practice. Additionally, the progression of the variance of the wavelet coefficients with scale can be accurately calculated so that it is also possible to compute scale-dependent "ideal" thresholds; in fact these thresholds do not seem to lead to superior spectrum estimates, probably due to the finite sample size sensitivity of the various finely-tuned asymptotic "ideal" thresholds.