**1. Introduction**

58 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

Yuan, Z. (2003). The Weighted Sum of The Line Spectrum Pair for Noisy Speech. M.Sc.

University of Technology.

Thesis, Department of Electrical and Communications Engineering, Helsinki

Removing noise from signals is possible only if some prior information is available. The information is employed by different estimators to recover the signal and reduce noise. Most noises in one-dimensional transient signal follow Gaussian distribution. The Bayes estimator minimizes the expected risk to get the optimal estimation. The minimax estimator uses a simple model for estimation. They are the most popular estimators in noise estimation.

No matter which estimator is used, the risk should be as small as possible. Donoho and Johnstone have made a breakthrough by proving that thresholding estimator has a small risk which is close to the lower bound. Thereafter, threshold estimation was studied extensively and has been improved by more and more researchers. Besides the universal threshold, some other thresholds, for example SURE threshold and minimax threshold, are also widely applied.

In wavelet denoising, the thresholding algorithm is usually used in orthogonal decompositions: multi-resolution analysis and wavelet packet transform. Wavelet thresholding faces some questions in its application, for example, the selection of hard or soft threshold, fixed or level-dependent threshold. Proper selection of those items helps generating a better estimation.

Besides the influence of thresholding, the influence of wavelets is also an important factor. In most applications, the wavelet transform uses a few non-zero coefficients to describe a signal or function. Producing only a few non-zero coefficients is crucial in noise removal and reducing computing complexity. Choosing a wavelet with optimum design to produce more wavelet coefficients close to zero is crucial in some applications.
