Wavelets and variance reduction in non-parametric transfer function estimation

A variance reduction scheme is presented for nonparametric transfer function estimators based on the use of wavelets as an alternative to the traditional spectral windowing. The latter can be generalized into a variance reduction method based on thresholding (omitting or altering) the coefficients of an orthogonal series expansion of the estimator to be smoothed. Crucial is the choice of threshold level, distinguishing between coefficients related predominantly to estimation errors and those associated with the underlying true function. The standard wavelet threshold operation with a constant or level-dependent threshold can not be applied to wavelet coefficients of spectral denstity functions. The nonstationarity in the statistical properties of these estimators reveals itself in the wavelet domain as significant peaks. An efficient threshold level should follow the standard deviation of each wavelet coefficient. New exact expressions of the standard deviation are presented, using the fact that we are dealing with functions associated with linear time invariant systems. An estimator based on these expressions proves to provide an appropriate threshold level.