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It was suggested that spectrum estimation can be accomplished by applying wavelet denoising methodology to wavelet packet coefficients derived from the logarithm of a spectrum estimate. The particular algorithm we consider consists of computing the logarithm of the multitaper spectrum estimator, applying an orthonormal transform derived from a wavelet packet tree to the log multitaper spectrum ordinates, thresholding the empirical wavelet packet coefficients, and then inverting the transform. For a small number of tapers, suitable transforms/partitions for the logarithm of the multitaper spectrum estimator are derived using a method matched to statistical thresholding properties. The partitions thus derived starting from different stationary time series are all similar and easily derived, and any differences between the wavelet packet and discrete wavelet transform (DWT) approaches are minimal. For a larger number of tapers, where the chosen parameters satisfy the conditions of a proven theorem, the simple DWT again emerges as appropriate. Hence, using our approach to thresholding and the method of partitioning, we conclude that the DWT approach is a very adequate wavelet-based approach and that the use of wavelet packets is unnecessary.