Thresholding functions in the noise suppression methods based on the wavelet expansion of the signal

When transmitted over communication channels, signals are usually contaminated with noise. Noise suppression methods based on thresholding of wavelet expansion coefficients have become popular due to their simplicity, speed, and ability to adapt to nonstationary signals. The analysis of the errors of these methods is an important practical task, since it makes it possible to assess the quality of both the methods themselves and the equipment used for processing. The most popular types of thresholding are hard and soft thresholding but each has its own drawbacks. In an attempt to address these shortcomings, various alternative thresholding methods have been proposed in recent years. The paper considers a model of a signal contaminated with additive Gaussian noise and discusses the general formulation of the thresholding problem with a thresholding function belonging to a certain class. An algorithm for calculating the threshold that minimizes the unbiased risk estimate is described. Conditions are also given under which this risk estimate is asymptotically normal and strongly consistent.