Shock and Vibration

Volume 2018, Article ID 9892713, 13 pages

https://doi.org/10.1155/2018/9892713

## The Adaptive Analysis of Shock Signals on the Basis of Improved Morlet Wavelet Clusters

^{1}School of Mechatronics Engineering, North University of China, Taiyuan, China^{2}School of Mechanical and Power Engineering, North University of China, Taiyuan, China

Correspondence should be addressed to Haikun Yang; moc.liamxof@0991nukiahgnay

Received 3 February 2018; Accepted 27 June 2018; Published 31 July 2018

Academic Editor: Salvatore Russo

Copyright © 2018 Haikun Yang and Hong-Xia Pan. 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

Morlet wavelets do not satisfy the permissibility condition of wavelet analysis, and there are therefore no inverse transformations for Morlet wavelet transforms. In this paper, we put forward the Yang and Pan transform (YPT), which is an adaptive discrete analysis method for shock signals. First, we improved the Morlet wavelet so that the centre and radius of the frequency window can be easily adjusted in the frequency domain. Second, we proposed the extremum frequency concept and analysed the extremum situation of the improved Morlet wavelet. Third, combining the improved Morlet wavelet and extremum frequency, we advanced the theory of the YPT, which does not need to satisfy the permissibility condition. We then continued by using a smoothing operator that can smooth the potentially distorted signal reconstructed after being analysed by the YPT and filtered by using the threshold filtering theory. This operator proved to be simple and efficient. Finally, a noisy signal was reconstructed after being analysed and filtered using the YPT and threshold filtering, respectively, to verify the validity of the theory, and the YPT was compared with the discrete wavelet transform (DWT). As a supplement to the theory in engineering, the shock signals about a gun automatic mechanism were also analysed using the theory in this paper. Good results were obtained, thereby demonstrating that the YPT can be helpful to further extract the features of shock signals in pattern recognition and fault diagnosis.

#### 1. Introduction

Wavelet transforms are a significant method for analysing signals, and Morlet wavelets are often used as the kernel functions of wavelet transforms. In 1982, Morlet first used them to analyse seismic signals, employing a complex wavelet using a Gauss envelope and a special case of Gabor wavelets as well [1]. The continuous wavelet transform of a signal by Morlet wavelets can achieve arbitrary high resolution in the time or frequency domains [2, 3]. There is a 90-degree phase shift between its real and imaginary parts, which makes it easy to obtain the instantaneous frequency and phase of the signal, and it therefore has a wide range of applications. For example, Morlet wavelets have been applied to signal filtering and denoising [4], mechanical fault diagnosis [5–8], the analysis of medical signals [9], research on river runoff in natural environments [10], research on rainwater evaporation [11], problems of polymer pollution [12], the atmospheric system [13], the influence of cosmic rays on organisms [14], and the motion of celestial bodies [15].

However, the Morlet wavelet does not satisfy the permissibility condition of wavelets, and wavelet reconstruction therefore cannot be realised. To obtain the inverse transformation of the Morlet wavelet, many scholars have improved the Morlet wavelet and obtained corresponding reconstruction transforms. Grossmann et al. [16] added proper correction terms to the Morlet wavelet to satisfy the permissibility condition and properly set the parameters for the tuning signal so that the rounding error of the computer had the same order of magnitude as the correction terms so that the correction terms could be omitted [17]. Ji and Yan [17], undertaking research at Northwestern Polytechnical University in China, improved the Morlet wavelet. They fixed some parameters in the Morlet wavelet and put forward the Morlet and Ji transform (MJT) as well. The MJT can reconstruct a signal without needing to meet the admissible condition. It is a continuous transformation of a signal and has a good time-frequency localisation property. Partially referencing the MJT and on the basis of the Morlet wavelet, we improved the Morlet wavelet in another way, further advancing a method for adaptively and discretely analysing signals.

In this paper, we first summarise the MJT and then propose another method to improve the Morlet wavelet cluster. Subsequently, the time and frequency windows of the Morlet wavelet, which were improved by Ji and Yan [17], are analysed, as is the Morlet wavelet improved by us. Third, the concept of extremum frequency is proposed in this paper, and the extremum properties of the Morlet wavelet cluster as improved by us are then analysed. Based on the improved Morlet wavelet cluster and extremum frequency, this paper then presents a new method that adaptively and discretely analyses a vibration signal and simultaneously yields the YPT, which can completely reconstruct the original signal. Additionally, we continue by putting forward a smooth operator that can smooth the potentially distorted signal reconstructed after analysis using the YPT and filtered employing threshold filtering theory. Finally, through the inverse transformation of the YPT, a filter is constructed to verify the correctness and practicability of the YPT by combining a signal with high noise. At the same time, the YPT is compared with the DWT. As a supplement to the theory in engineering, the shock signals about a gun automatic mechanism are also analysed by using the theory in this paper, which can provide help for further extracting the features of shock signals in pattern recognition and fault diagnosis.

#### 2. MJT and Improved Morlet Wavelet Cluster

This section first introduces the content of the MJT, and we then improve the Morlet wavelet cluster in the MJT to obtain our improved Morlet wavelet cluster and analyse the properties of the time and frequency windows for the latter improved wavelet cluster.

##### 2.1. MJT

The analytical expression of the Morlet wavelet is

The Fourier transform of (1) is

Given a wavelet , the permissibility condition of the wavelet transform is

We know by (3), whereas ; thus, the Morlet wavelet does not satisfy the permissibility condition of the wavelet transform. Following Ji and Yan [17], let , and the modified Morlet wavelet cluster is then obtained by stretching and translating . Their improved Morlet wavelet cluster is

Assuming signal , the expression for the MJT is [17]

According to (5), the Fourier transformation of with respect to can be obtained:

Therefore, according to (6), the inverse transformation of can be obtained:

Formulas (5) and (7) are, respectively, the positive transformation and inverse transformation formulas of the MJT. According to (5), the signal can be transformed into the domain for analysis. The MJT is a continuous transformation that has good time-frequency localisation properties [17], and it is a redundant transformation that requires large amounts of calculations and storage space.

##### 2.2. Improved Morlet Wavelet Cluster

In this section, we improve the Morlet wavelet cluster in (4) and analyse the properties of the time and frequency windows for the improved Morlet wavelet cluster. The improved Morlet wavelet cluster, denoted as , is expressed as (8). Formula (9) is the Fourier transform of (8):

Cauchy’s theorem on the two-dimensional complex plane can be used to compute the definite integral in (9), and it can also be obtained directly from the Poisson integral formula. can be obtained by using (4) and (8). We will now analyse the properties of the time and frequency windows for , the centre of which in the time domain is denoted as , which can be obtained by the following equation:

In the process of calculating the definite integral in (10), we obtain and by utilising the properties of the Gamma function. Of course, as far as (10) is concerned, we do not have to calculate the latter in detail if we obtain the former. We can conclude that is independent of and via (10); therefore, the centre of the time domain for is also 0. Furthermore, the radius of the time window, which is denoted as , can be obtained by the following equation:

Obviously, the radius of the time window for is , and together with jointly affects the radius of the time window for . When the absolute value of is large, the radii of the time windows for and decrease, but the latter can be adjusted by using the parameter . Similarly, the centre and radius of the frequency window for in the frequency domain are obtained as

We can conclude via (11) and (13), which as a constant cannot be affected by , but it can be separately adjusted by , and the width of the frequency window can be linearly adjusted by and ; what is more, determines the centre of the frequency window.

#### 3. Extremum Frequency

In this section, the concept of extremum frequency, which reflects the average rate of change for a sequence and provides a method for adaptively analysing signals, is first proposed. Based on this concept, the extremum and extremum frequency of the improved Morlet wavelet cluster are analysed, which paves the way for the adaptive analysis of signals in the next section.

##### 3.1. The Extremum Frequency of a Sequence

Assuming a sequence , if and , we regard as one maximum aggregate of . Similarly, if and , we regard as one minimum aggregate of . To go a step further, assuming that the sequence contains amounts of maximum aggregates and amounts of minimum aggregates as well, the extremum frequency of , denoted as , can be defined as

##### 3.2. The Extremum of the Improved Morlet Wavelet Cluster

In this section, we let and analyse the real part of , which is equal to and denoted as :

Without loss of generality, let . Some properties regarding the maximum of are now described as follows:(1)There is only one point that makes in the interval .

*Proof*.

Let , then :

When ,

is continuous in the interval , and there is therefore only one point, , that makes , that is, only one point, , that makes .

When , similarly, there exists only one point that makes .(2) is the only extreme point in the interval .

*Proof*. When and ,(i)Let be an even number. If , then ; if , then . That is to say, is the only maximum point in the interval .(ii)Let be an odd number; similarly, is the only minimum point in the interval .

When and , is similarly the only extreme point in the interval .

Because , , and with the information given above, the property of (2) has been proved.

##### 3.3. The Discrete Sampling of the Improved Morlet Wavelet Cluster

We can obtain a sequence, which is denoted as , if sampling from the start time to the end time and by sampling interval . Figure 1 shows a schematic diagram for sampling .