Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 3195492, 8 pages

http://dx.doi.org/10.1155/2016/3195492

## A New Wavelet Threshold Function and Denoising Application

^{1}School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China^{2}Faculty of Electricity and Information Engineering, Northeast Petroleum University, Daqing 163318, China

Received 10 December 2015; Revised 19 March 2016; Accepted 13 April 2016

Academic Editor: Huiyu Zhou

Copyright © 2016 Lu Jing-yi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In order to improve the effects of denoising, this paper introduces the basic principles of wavelet threshold denoising and traditional structures threshold functions. Meanwhile, it proposes wavelet threshold function and fixed threshold formula which are both improved here. First, this paper studies the problems existing in the traditional wavelet threshold functions and introduces the adjustment factors to construct the new threshold function basis on soft threshold function. Then, it studies the fixed threshold and introduces the logarithmic function of layer number of wavelet decomposition to design the new fixed threshold formula. Finally, this paper uses hard threshold, soft threshold, Garrote threshold, and improved threshold function to denoise different signals. And the paper also calculates signal-to-noise (SNR) and mean square errors (MSE) of the hard threshold functions, soft thresholding functions, Garrote threshold functions, and the improved threshold function after denoising. Theoretical analysis and experimental results showed that the proposed approach could improve soft threshold functions with constant deviation and hard threshold with discontinuous function problems. The proposed approach could improve the different decomposition scales that adopt the same threshold value to deal with the noise problems, also effectively filter the noise in the signals, and improve the SNR and reduce the MSE of output signals.

#### 1. Introduction

At present, the speech signal processing techniques have been used in many areas. Denoising is the key to speech signal processing technology, which is the process of removing noise to the maximum extent of noise to restore the original signals [1]. It has become an indispensable link in speech signal processing. Among the methods of denoising, the traditional one is usually based on Fourier transform. In contrast, the wavelet transform is much more automatically adaptive to the requirement of time-frequency signal analysis [2] because any detail of the signal can be captured, and the wavelet transform can solve the problems that Fourier transform cannot in dealing with the nonstationary signals. Using wavelet transform in signal processing is the process of the partial transformation of the spatial domain and frequency domain. Thus, we can get useful information accurately from it with noise. Now, denoising methods can be commonly divided into the following kinds: the modulus maxima denoising method, the correlation denoising method, and the wavelet threshold denoising method [3].

Modulus maxima denoising method is based on the different characteristics of view when the signal and noise models at multiple scales of space maxima are different. Remove those magnitude scales which increase with decreasing extreme value point and then reconstruct these extreme points in order to achieve the elimination of noise [4]. The modulus maxima method in addition to the noise effect is better than any other method when mixed with white noise and singular information is significant, but the amount of calculation is really huge [2]. Correlation denoising method achieves its aim by containing noise signals after wavelet transform so that the correlation degree of the wavelet coefficients of the original part and the noise part on each scale is different. The calculation procedure of this method is complex [5].

Wavelet threshold denoising method was proposed by American scholar Donohue. The method is simple to calculate and the noise can be suppressed to a large extent. At the same time, singular information of the original signal can be preserved well, so it is a simple and effective method [6].

Whether wavelet threshold denoising method is good or bad depends on two decisive factors. One is threshold and the other important factor is the selection of the threshold function [7]. As the most basic threshold function, both hard and soft threshold functions have their own defects. The breakpoint problem of the hard threshold function makes it have no continuity [8]. In the process of reconstructing speech signal, proneness to volatility and pseudo-Gibbs effect will make the speech distorted while the hard threshold function can keep singular information of the original signal. And the constant deviation problem which is born with the soft threshold function cannot be overcome by itself, which means that the reconstructed signal is too smooth and there are constant deviations compared with the original signals. All the abovementioned defects cause part of the high frequency signal information to be missing, which influence the final processing results [9].

In order to overcome the defects of hard and soft thresholding function perspective, this paper constructs an improved wavelet threshold function by increasing the adjustment factor. In order to overcome the problem that the same threshold is used to deal with the different decomposition scales, this paper designs a new fixed threshold formula by introducing the logarithmic functions of layer number of wavelet decomposition.

#### 2. The Principle of Wavelet Threshold Denoising

In practical cases, noise signals usually appear as high frequency signals in signal processing, but useful signals appear as either low frequency or more smooth signals. The signals with noise have the features of the above, so when the signals are decomposed by wavelet, the signals with noise in the high frequency wavelet coefficients, through threshold quantization threshold processing high frequency wavelet coefficients to reconstruct the signals, can eliminate the noise with the signals. One dimensional signal denoising process is as follows: the one dimensional signal is decomposed by wavelet decomposition, selecting threshold and threshold function to quantify the high frequency coefficients of wavelet decomposition and reconstruct the one dimensional wavelet. The key factors affecting the quality of denoising are denoising threshold and the selection of threshold function [10].

#### 3. Wavelet Threshold Function

##### 3.1. Classical Threshold Function

The hard threshold function sets the decomposition coefficient to zero which is less than the threshold value under different scale spaces and reserves the decomposition coefficient which is greater than the threshold at the same time [11]. This method does not change the local properties of the signal, but because of the discontinuity, it leads to a certain fluctuation in the reconstruction of the original signal. Consider

The soft threshold function is to select the specified threshold value of the decomposition coefficient to zero. After the algorithm, the decomposition coefficient is coherent, but it loses a part of the high frequency coefficients above the threshold [12]. ConsiderParameter is the representation of estimated wavelet coefficients, parameter is the representation of the wavelet coefficients after decomposition, is the representation of threshold, and is the symbolic piecewise function in the above two formulas [13].

In order to overcome the shortcomings of above two methods, GaoHong Ye proposed Garrote threshold function in which the denoising effect is the best compared with above two kinds of traditional threshold functions and it is better in continuity with the expressions such as

##### 3.2. The Improved Threshold Function

The continuity in the soft threshold function is much better, but it has a constant deviation. So, in order to overcome its shortcomings, the soft and hard threshold algorithms are compromised process by the literature; the semisoft threshold function [14] (4) has been shown in the formula

Among the soft and hard threshold functions, the value of is taken between 0 and 1. Even if the result of semisoft threshold function is between them, the value of is fixed. So, there will still be fixed bias. Consider

The threshold function is proposed in this paper which is expressed in formula (5). The adjustment factor of the new function is different from the semisoft threshold function. It consists of a complex exponential function which has more adaptability; is the normal number which can be adjusted freely and the values of are different with the different signal. When , and when , . Therefore, continuously in place of , the improved threshold function has the characteristics of soft threshold function; when , improved threshold function based on as the asymptotic line; it can be seen that, with the increase of , will gradually be close to ; when becomes infinite, and can be approximated as equal. That will not only reduce the discreteness of the hard threshold function, but can also avoid the constant problem in soft threshold function at the same time. The variables in the formula are also really important, and the change of the value of can affect the noise directly. When , the function will express soft threshold function and when , the function will express hard threshold function.

Take some simple numerical parameters, and then show the soft, hard, Garrote, and new threshold function in the form of graphics intuitive. Here, the parameter threshold function of each take , , and .

As shown in Figure 1, the hard threshold function is not continuous at the threshold point, so that there are fluctuations in the recovery of the original signal. Compared with the hard threshold function, although there are no discontinuity problems in the soft threshold function, there are still some differences between the original signal and the reconstructed signal, because of the constant problems inherent in the soft threshold function.