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

Authors: Haikun Yang; Hong-Xia Pan
Journal: Shock and Vibration
 (2018)

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## 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.


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## Body

## 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(1)mt=e−t2/2ejω0t,ω0≥5.The Fourier transform of (1) is(2)m^ω=∫−∞+∞ejω0te−t2/2ejωtdt=2πe−ω−ω02/2.Given a waveletψt∈L1∩L2, the permissibility condition of the wavelet transform is(3)Cψ=∫−∞+∞ψ^ω2ωdω<+∞.We knowψ^0=0 by (3), whereas m^0≠0; thus, the Morlet wavelet does not satisfy the permissibility condition of the wavelet transform. Following Ji and Yan [17], let ψt=1/2πe−t2/2+j2πt, and the modified Morlet wavelet cluster is then obtained by stretching and translating ψt. Their improved Morlet wavelet cluster is(4)ψf,τt=f2πe−f2t−τ2/2+j2πft−τ,f∈R,τ∈R.Assuming signalht∈L2, the expression for the MJT is [17](5)Jf,τ=ht,ψf,τt=∫−∞+∞htψ¯f,τtdt=∫−∞+∞htf2πe−f2t−τ2/2−j2πft−τdt=∫−∞+∞htf2πe−f2τ−t2/2+j2πfτ−tdt=hτ∗ψf,0τ.According to (5), the Fourier transformation of Jf,τ with respect to τ can be obtained:(6)J^f=∫−∞+∞Jf,τe−j2πfτdτ=∫−∞+∞∫−∞+∞htf2πe−f2τ−t2/2+j2πfτ−tdte−j2πfτdτ=∫−∞+∞∫−∞+∞e−f2τ−t2/2+j2πfτ−te−j2πfτdτhtf2πdt=∫−∞+∞hte−j2πftdt=Hf.Therefore, according to (6), the inverse transformation of ht can be obtained:(7)ht=∫−∞+∞Hfej2πftdf=∫−∞+∞∫−∞+∞Jf,τe−j2πfτdτej2πftdf.Formulas (5) and (7) are, respectively, the positive transformation and inverse transformation formulas of the MJT. According to (5), the signal ht can be transformed into the f–τ 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 φf,δt, is expressed as (8). Formula (9) is the Fourier transform of (8):(8)φf,δt=f+δ2πe−f+δ2t2/2+j2πft,(9)φ^f,δγ=∫−∞+∞φf,δte−j2πγtdt=∫−∞+∞f+δ2πe−f+δ2t2/2ej2πf−γtdt=sgnf+δπ∫−∞+∞e−t2ej22π/f+δf−γtdt=sgnf+δe−2π2f−γ2/f+δ2.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. φf,0t=ψf,0t can be obtained by using (4) and (8). We will now analyse the properties of the time and frequency windows for φf,δt, the centre of which in the time domain is denoted as t∗, which can be obtained by the following equation:(10)t∗=∫−∞+∞tφf,δt2dt∫−∞+∞φf,δt2dt=∫−∞+∞tf+δ/2πe−f+δ2t2/2+j2πft2dt∫−∞+∞f+δ/2πe−f+δ2t2/2+j2πft2dt=0.In the process of calculating the definite integral in (10), we obtain ∫−∞+∞tφf,δt2dt=0 and ∫−∞+∞φf,δt2dt=f+δ/2πsgnf+δ 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 t∗ is independent of f and τ via (10); therefore, the centre of the time domain for ψf,0t is also 0. Furthermore, the radius of the time window, which is denoted as Δt, can be obtained by the following equation:(11)Δt=∫−∞+∞t−t∗2φf,δt2dt∫−∞+∞φf,δt2dt1/2=∫−∞+∞t2f+δ2/2πe−f+δ2t2dt∫−∞+∞φf,δt2dt1/2=sgnf+δ2πf+δΓ3/2∫−∞+∞φf,δt2dt1/2=sgnf+δ2f+δ.Obviously, the radius of the time window forψf,0t is 1/2f, and f together with δ jointly affects the radius of the time window for φf,δt. When the absolute value of f is large, the radii of the time windows for ψf,0t and φf,δt decrease, but the latter can be adjusted by using the parameter δ. Similarly, the centre γ∗ and radius Δγ of the frequency window for φf,δt in the frequency domain are obtained as(12)γ∗=∫−∞+∞γφ^f,δγ2dγ∫−∞+∞φ^f,δγ2dγ=f,(13)Δγ=∫−∞+∞γ−γ∗2φ^f,δγ2dγ∫−∞+∞φ^f,δγ2dγ1/2=sgnf+δf+δ22π.We can concludeΔt×Δγ=1/4π via (11) and (13), which as a constant cannot be affected by δ, but it can be separately adjusted by δ, and the width of the frequency window can be linearly adjusted by f and δ; what is more, f determines the centre of the frequency window.


## 2.1. MJT

The analytical expression of the Morlet wavelet is(1)mt=e−t2/2ejω0t,ω0≥5.The Fourier transform of (1) is(2)m^ω=∫−∞+∞ejω0te−t2/2ejωtdt=2πe−ω−ω02/2.Given a waveletψt∈L1∩L2, the permissibility condition of the wavelet transform is(3)Cψ=∫−∞+∞ψ^ω2ωdω<+∞.We knowψ^0=0 by (3), whereas m^0≠0; thus, the Morlet wavelet does not satisfy the permissibility condition of the wavelet transform. Following Ji and Yan [17], let ψt=1/2πe−t2/2+j2πt, and the modified Morlet wavelet cluster is then obtained by stretching and translating ψt. Their improved Morlet wavelet cluster is(4)ψf,τt=f2πe−f2t−τ2/2+j2πft−τ,f∈R,τ∈R.Assuming signalht∈L2, the expression for the MJT is [17](5)Jf,τ=ht,ψf,τt=∫−∞+∞htψ¯f,τtdt=∫−∞+∞htf2πe−f2t−τ2/2−j2πft−τdt=∫−∞+∞htf2πe−f2τ−t2/2+j2πfτ−tdt=hτ∗ψf,0τ.According to (5), the Fourier transformation of Jf,τ with respect to τ can be obtained:(6)J^f=∫−∞+∞Jf,τe−j2πfτdτ=∫−∞+∞∫−∞+∞htf2πe−f2τ−t2/2+j2πfτ−tdte−j2πfτdτ=∫−∞+∞∫−∞+∞e−f2τ−t2/2+j2πfτ−te−j2πfτdτhtf2πdt=∫−∞+∞hte−j2πftdt=Hf.Therefore, according to (6), the inverse transformation of ht can be obtained:(7)ht=∫−∞+∞Hfej2πftdf=∫−∞+∞∫−∞+∞Jf,τe−j2πfτdτej2πftdf.Formulas (5) and (7) are, respectively, the positive transformation and inverse transformation formulas of the MJT. According to (5), the signal ht can be transformed into the f–τ 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 φf,δt, is expressed as (8). Formula (9) is the Fourier transform of (8):(8)φf,δt=f+δ2πe−f+δ2t2/2+j2πft,(9)φ^f,δγ=∫−∞+∞φf,δte−j2πγtdt=∫−∞+∞f+δ2πe−f+δ2t2/2ej2πf−γtdt=sgnf+δπ∫−∞+∞e−t2ej22π/f+δf−γtdt=sgnf+δe−2π2f−γ2/f+δ2.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. φf,0t=ψf,0t can be obtained by using (4) and (8). We will now analyse the properties of the time and frequency windows for φf,δt, the centre of which in the time domain is denoted as t∗, which can be obtained by the following equation:(10)t∗=∫−∞+∞tφf,δt2dt∫−∞+∞φf,δt2dt=∫−∞+∞tf+δ/2πe−f+δ2t2/2+j2πft2dt∫−∞+∞f+δ/2πe−f+δ2t2/2+j2πft2dt=0.In the process of calculating the definite integral in (10), we obtain ∫−∞+∞tφf,δt2dt=0 and ∫−∞+∞φf,δt2dt=f+δ/2πsgnf+δ 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 t∗ is independent of f and τ via (10); therefore, the centre of the time domain for ψf,0t is also 0. Furthermore, the radius of the time window, which is denoted as Δt, can be obtained by the following equation:(11)Δt=∫−∞+∞t−t∗2φf,δt2dt∫−∞+∞φf,δt2dt1/2=∫−∞+∞t2f+δ2/2πe−f+δ2t2dt∫−∞+∞φf,δt2dt1/2=sgnf+δ2πf+δΓ3/2∫−∞+∞φf,δt2dt1/2=sgnf+δ2f+δ.Obviously, the radius of the time window forψf,0t is 1/2f, and f together with δ jointly affects the radius of the time window for φf,δt. When the absolute value of f is large, the radii of the time windows for ψf,0t and φf,δt decrease, but the latter can be adjusted by using the parameter δ. Similarly, the centre γ∗ and radius Δγ of the frequency window for φf,δt in the frequency domain are obtained as(12)γ∗=∫−∞+∞γφ^f,δγ2dγ∫−∞+∞φ^f,δγ2dγ=f,(13)Δγ=∫−∞+∞γ−γ∗2φ^f,δγ2dγ∫−∞+∞φ^f,δγ2dγ1/2=sgnf+δf+δ22π.We can concludeΔt×Δγ=1/4π via (11) and (13), which as a constant cannot be affected by δ, but it can be separately adjusted by δ, and the width of the frequency window can be linearly adjusted by f and δ; what is more, f 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 sequencexnn=1,2,…,M, if xn−ml−1<xn−ml=…,xn−1=xn=xn+1,…=xn+mh>xn+mh+1 and ml∈N,mh∈N, we regard xn−ml,…,xn−1,xn,xn+1,…,xn+mh as one maximum aggregate of xnn=1,2,…,M. Similarly, if xn−ml−1>xn−ml=…,xn−1=xn=xn+1,…=xn+mh<xn+mh+1 and ml∈N,mh∈N, we regard xn−ml,…,xn−1,xn,xn+1,…,xn+mh as one minimum aggregate of xnn=1,2,…,M. To go a step further, assuming that the sequence xnn=1,2,…,M contains Ph amounts of maximum aggregates and Pl amounts of minimum aggregates as well, the extremum frequency of xnn=1,2,…,M, denoted as fe, can be defined as(14)fe≜Ph+PlM.


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

In this section, we letf+δ≠0 and analyse the real part of φf,δt, which is equal to f+δ/2πe−f+δ2t2/2+j2πft and denoted as φRt:(15)φRt=f+δ2πe−f+δ2t2/2cos2πft,φR′t=f+δ−2πe−f+δ2t2/2f+δ2tcos2πft+2πfsin2πft.Without loss of generality, letf+δ>0. Some properties regarding the maximum of φf,δt are now described as follows:(1)
There is only one pointξn∈−1/4f+n1/2f,1/4f+n1/2f that makes φ′Rξn=0 in the interval −1/4f+n1/2f,1/4f+n1/2f,n∈Z.Proof.(16)φ′Rt=f+δ−2πe−f+δ2t2/2cos2πftf+δ2t+2πftan2πft.Letρt=f+δ2t+2πftan2πft, then φ′Rξn=0⇔ρt=0:(17)ρ′t=f+δ2+2πfsec2πft2>0.Whenf>0,(18)limt→−1/4f+n1/2f+ρt=−∞,limt→1/4f+n1/2f−ρt=+∞.ρ
t is continuous in the interval −1/4f+n1/2f,1/4f+n1/2f, and there is therefore only one point, ξn∈−1/4f+n1/2f,1/4f+n1/2f, that makes ρξn=0, that is, only one point, ξn∈−1/4f+n1/2f,1/4f+n1/2f, that makes φ′Rξn=0.Whenf<0, similarly, there exists only one point ξn∈−1/4f+n1/2f,1/4f+n1/2f that makes φ′Rξn=0.(2)
ξn−1/4f+n1/2f<ξn<1/4f+n1/2f is the only extreme point in the interval −1/4f+n1/2f,1/4f+n1/2f,n∈Z.Proof. When t∈−1/4f+n1/2f,1/4f+n1/2f and f>0,(19)φ′Rt=f+δ−2πe−f+δ2t2/2cos2πftρt.(i)
Letn be an even number. If t∈−1/4f+n1/2f,ξn, then φ′Rt>0; if t∈ξn,1/4f+n1/2f, then φ′Rt<0. That is to say, ξn is the only maximum point in the interval −1/4f+n1/2f,1/4f+n1/2f.(ii)
Letn be an odd number; similarly, ξn is the only minimum point in the interval −1/4f+n1/2f,1/4f+n1/2f.Whent∈−1/4f+n1/2f,1/4f+n1/2f and f<0, ξn is similarly the only extreme point in the interval −1/4f+n1/2f,1/4f+n1/2f.Becauseφ′R−1/4f+n1/2f≠0, φ′R1/4f+n1/2f≠0, 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 asunn=1,2,…,N0, if sampling φRt from the start time ts to the end time te and by sampling interval t0. Figure 1 shows a schematic diagram for sampling φRt.Figure 1
Schematic diagram for samplingφRt.According to the parameters for samplingφRt, there exists the following formula:(20)N0=te−tst0+1.According to the parameters for samplingφRt and the analysis of the extremum for the improved Morlet wavelet cluster in Section 3.2, we can obtain the numbers of the extreme aggregates for unn=1,2,…,N0. Let the numbers be P, which can be expressed as(21)P=2fte−tsorP=2fte−ts±1.Combining (14) and (20) with (21), the extremum frequency of unn=1,2,…,N0, denoted as fem, can be obtained:(22)fem=PN0=t02fte−tste−ts+t0orfem=PN0=t02fte−ts±t0te−ts+t0.Furthermore,(23)limfte−ts→+∞te−ts≫t0fem=2ft0.The effect of analysing a sequence whose extremum frequency isfe may be better when we utilise the improved Morlet wavelet, if we make fem=fe. Because the Morlet wavelet, on the condition of fem=fe, will have better adaptability to the sequence, the following formula is therefore established:(24)f=fe2t0.For a continuous signalht defined in the interval ts,te and containing Pc amounts of extreme points, we can determine that the extremum frequency of the sequence obtained by sampling ht is fec=Pct0/te−ts+t0 if setting the sampling interval equal to t0. When we utilise the improved Morlet wavelet to analyse ht, we set fem=fec. Formula (25) is therefore established:(25)limfte−ts→+∞te−ts≫t0fem=limfte−ts→+∞te−ts≫t0fec.It is equal to(26)f=Pc2te−ts.On the basis of (26), the improved Morlet wavelet, whose parameter f is equal to Pc/2te−ts, will have better adaptability when analysing the signal ht. According to (23)–(25), we obtain f=fe/2t0=Pc/2te−ts on the conditions te−ts→+∞ and te−ts≫t0.


## 3.1. The Extremum Frequency of a Sequence

Assuming a sequencexnn=1,2,…,M, if xn−ml−1<xn−ml=…,xn−1=xn=xn+1,…=xn+mh>xn+mh+1 and ml∈N,mh∈N, we regard xn−ml,…,xn−1,xn,xn+1,…,xn+mh as one maximum aggregate of xnn=1,2,…,M. Similarly, if xn−ml−1>xn−ml=…,xn−1=xn=xn+1,…=xn+mh<xn+mh+1 and ml∈N,mh∈N, we regard xn−ml,…,xn−1,xn,xn+1,…,xn+mh as one minimum aggregate of xnn=1,2,…,M. To go a step further, assuming that the sequence xnn=1,2,…,M contains Ph amounts of maximum aggregates and Pl amounts of minimum aggregates as well, the extremum frequency of xnn=1,2,…,M, denoted as fe, can be defined as(14)fe≜Ph+PlM.


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

In this section, we letf+δ≠0 and analyse the real part of φf,δt, which is equal to f+δ/2πe−f+δ2t2/2+j2πft and denoted as φRt:(15)φRt=f+δ2πe−f+δ2t2/2cos2πft,φR′t=f+δ−2πe−f+δ2t2/2f+δ2tcos2πft+2πfsin2πft.Without loss of generality, letf+δ>0. Some properties regarding the maximum of φf,δt are now described as follows:(1)
There is only one pointξn∈−1/4f+n1/2f,1/4f+n1/2f that makes φ′Rξn=0 in the interval −1/4f+n1/2f,1/4f+n1/2f,n∈Z.Proof.(16)φ′Rt=f+δ−2πe−f+δ2t2/2cos2πftf+δ2t+2πftan2πft.Letρt=f+δ2t+2πftan2πft, then φ′Rξn=0⇔ρt=0:(17)ρ′t=f+δ2+2πfsec2πft2>0.Whenf>0,(18)limt→−1/4f+n1/2f+ρt=−∞,limt→1/4f+n1/2f−ρt=+∞.ρ
t is continuous in the interval −1/4f+n1/2f,1/4f+n1/2f, and there is therefore only one point, ξn∈−1/4f+n1/2f,1/4f+n1/2f, that makes ρξn=0, that is, only one point, ξn∈−1/4f+n1/2f,1/4f+n1/2f, that makes φ′Rξn=0.Whenf<0, similarly, there exists only one point ξn∈−1/4f+n1/2f,1/4f+n1/2f that makes φ′Rξn=0.(2)
ξn−1/4f+n1/2f<ξn<1/4f+n1/2f is the only extreme point in the interval −1/4f+n1/2f,1/4f+n1/2f,n∈Z.Proof. When t∈−1/4f+n1/2f,1/4f+n1/2f and f>0,(19)φ′Rt=f+δ−2πe−f+δ2t2/2cos2πftρt.(i)
Letn be an even number. If t∈−1/4f+n1/2f,ξn, then φ′Rt>0; if t∈ξn,1/4f+n1/2f, then φ′Rt<0. That is to say, ξn is the only maximum point in the interval −1/4f+n1/2f,1/4f+n1/2f.(ii)
Letn be an odd number; similarly, ξn is the only minimum point in the interval −1/4f+n1/2f,1/4f+n1/2f.Whent∈−1/4f+n1/2f,1/4f+n1/2f and f<0, ξn is similarly the only extreme point in the interval −1/4f+n1/2f,1/4f+n1/2f.Becauseφ′R−1/4f+n1/2f≠0, φ′R1/4f+n1/2f≠0, 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 asunn=1,2,…,N0, if sampling φRt from the start time ts to the end time te and by sampling interval t0. Figure 1 shows a schematic diagram for sampling φRt.Figure 1
Schematic diagram for samplingφRt.According to the parameters for samplingφRt, there exists the following formula:(20)N0=te−tst0+1.According to the parameters for samplingφRt and the analysis of the extremum for the improved Morlet wavelet cluster in Section 3.2, we can obtain the numbers of the extreme aggregates for unn=1,2,…,N0. Let the numbers be P, which can be expressed as(21)P=2fte−tsorP=2fte−ts±1.Combining (14) and (20) with (21), the extremum frequency of unn=1,2,…,N0, denoted as fem, can be obtained:(22)fem=PN0=t02fte−tste−ts+t0orfem=PN0=t02fte−ts±t0te−ts+t0.Furthermore,(23)limfte−ts→+∞te−ts≫t0fem=2ft0.The effect of analysing a sequence whose extremum frequency isfe may be better when we utilise the improved Morlet wavelet, if we make fem=fe. Because the Morlet wavelet, on the condition of fem=fe, will have better adaptability to the sequence, the following formula is therefore established:(24)f=fe2t0.For a continuous signalht defined in the interval ts,te and containing Pc amounts of extreme points, we can determine that the extremum frequency of the sequence obtained by sampling ht is fec=Pct0/te−ts+t0 if setting the sampling interval equal to t0. When we utilise the improved Morlet wavelet to analyse ht, we set fem=fec. Formula (25) is therefore established:(25)limfte−ts→+∞te−ts≫t0fem=limfte−ts→+∞te−ts≫t0fec.It is equal to(26)f=Pc2te−ts.On the basis of (26), the improved Morlet wavelet, whose parameter f is equal to Pc/2te−ts, will have better adaptability when analysing the signal ht. According to (23)–(25), we obtain f=fe/2t0=Pc/2te−ts on the conditions te−ts→+∞ and te−ts≫t0.


## 4. Analysing a Vibration Signal

### 4.1. Two Adjacent Morlet Wavelets

In this section, we analyse two adjacent Morlet wavelets improved by us. The centres of the frequency windows separately arefk−1 and fk, and the radii are Δγk−1 and Δγk, respectively, where k≥2 and k∈N. We made their radii just connect with each other in the frequency domain, the situations of which are shown in the following two cases and seen in Figures 2 and 3.Figure 2
The first situation of two adjacent Morlet wavelets.Figure 3
The second situation of two adjacent Morlet wavelets.Letf+δ≥0. According to Figures 2 and 3, the following formula must be true if we make the radii of the two adjacent Morlet wavelets just connect with each other in the frequency domain:(27)Δγk=Δγk−1−fk−1−fk.It is equal to(28)δk=fk−1+δk−1−22πfk−1−fk−fk.


### 4.2. YPT

Assume that a signalh0t∈L2R, defined in the interval ts,te and containing Pc amounts of extreme points, is sampled to form a discrete sequence vnn=1,2,…,Nv by setting the sampling interval equal to t0, and the extremum frequency of the sequence is fe. According to Section 3.3, we know we should make f=fe/2t0=Pc/2te−ts, when analysing the signal h0t by utilising the improved Morlet wavelet. We now transform the signal h0t by using the following equations, which are collectively named the “YPT”:(29)Ypkt;fk,δpk=hk−1t∗φfk,δpkt,Ynkt;−fk,δnk=hk−1t∗φ−fk,δnkt,hkt=hk−1t−Ypkt;fk,δpk−Ynkt;−fk,δnk,where k∈N+ in the formulas in (29). On their basis, to go a step further, we take the Fourier transforms of the formulas in (29):(30)Y^pkγ;fk,δpk=FYpkt;fk,δpk,Y^nkγ;−fk,δnk=FYnkt;−fk,δnk,h^kγ=Fhkt.The equations in (29) dispersedly and adaptively transform the signal h0t into the time domain, and on the contrary, the formulas in (30) dispersedly and adaptively transform the signal h0t into the frequency domain. Taken together, both constitute the positive transformation of the YPT. fk and δk are adaptive to hk−1t. We can also obtain the inverse transformation of the YPT according to the formulas in (29) and (30), which can be described by the following equations:(31)h0t=∑i=1kYpit;fi,δpi+Ynit;−fi,δni+hkt,(32)h0t=ℱ−1∑i=1kY^piγ;fi,δpi+Y^niγ;−fi,δni+h^kγ.For the convenience of our expression in this paper,Ypkt;fk,δpk and Ynkt;−fk,δnk are uniformly named Yt;f,δ; similarly, hkt is designated as ht, Y^pkγ;fk,δpk and Y^nkγ;−fk,δnk are designated as Y^γ;f,δ, and h^kγ is designated as h^γ. According to (31), we can directly reconstruct the original signal h0t by adding signals Yt;f,δ and ht, having been analysed and processed, in the time domain. In addition, we can continue to further transform signals Yt;f,δ and ht into signals Y^γ;f,δ and h^γ in the frequency domain and then continue to process signals Y^γ;f,δ and h^γ and reconstruct the original signal h0t through (32). On a similar principle, according to (32), we can reconstruct the original signal h0t through signals Y^γ;f,δ and h^γ, having been analysed and processed, in the frequency domain. In addition, we can continue to transform signals Y^γ;f,δ and h^γ into signals Yt;f,δ and ht in the time domain and then continue to process signals Yt;f,δ and ht and reconstruct the original signal h0t through (31). Therefore, (31) and (32) provide us 4 types of YPT inverse transforms, namely, 2 types of YPT inverse transforms from the frequency domain to the time domain and 2 types of YPT inverse transforms from the time domain to the time domain.


### 4.3. Smoothing Operator

A reconstructed vibration signal can be seriously distorted in some local areas after being analysed and processed. It is usually necessary to properly smooth the signal for the specific transformation, which is helpful to reflect the change law of the original signal. In this section, we propose a smooth operatorS that can smooth the sequence corresponding to the signal. Before introducing the operator S, three operators, A, B, and L, are introduced first. Assuming that there is a finite sequence X=x1,x2,…,xn, the operator A can perform the following operation on the sequence X:(33)AX,k≜x1,…,xk−1,xk−1+xk2,xk+1,…,xn,k=2,3,…,n.In particular,(34)AX,1≜xn+x12,x2,…,xn,AX,0≜X.The operatorB performs the following operation:(35)BX≜AAA…AX,1…,n−1,n.The operatorL performs the following operation:(36)LX,k≜xn−k+1,xn−k+2,…,xn,x1,x2,…,xn−k,k∈Z.Obviously,LX,n=LX,−n=X. On the basis of operators A, B, and L, we can define the operator S:(37)SX,w≜LBB…BX,−w︸w,w∈N+.In particular,SX,0≜X.Through a large number of experiments on sequences obtained by the YPT and the filtering theory that we utilised in this paper, we found that the operatorS can effectively smooth sequences. However, we cannot excessively smooth a signal.


### 4.4. The Process of Analysing a Vibration Signal

From the analyses in Sections4.1 and 4.2, if we analyse the signal h0t through the YPT, the kernel is determining the extremum frequency of the signal and controlling the radius of the frequency window. When k≥1, fk corresponds to hk−1t. If k≥2, δk must meet the condition δk=fk−1+δk−1−22πfk−1−fk−fk, which has been discussed through (28) in Section 4.1, and we can let δ1 be 0. The radius of the frequency window will be zero if fk+δk=0 in the process of analysing the signal. However, this matter will make the process of the YPT stagnate. We therefore let δk=0 again if fk+δk=0 to continue transforming the signal until the transformation reaches the required number of the transformation. We can denote this required number as K0. The flow chart of the YPT can be divided into two parts, the first of which concerns analysing the signal, and the second concerns reconstructing the signal; both parts are shown in Figures 4 and 5, respectively.Figure 4
The flow chart for analysing a signal.Figure 5
The flow chart for reconstructing a signal.The parameter model determines how to reconstruct a signal. For this paper, we chosemodel=4 to reconstruct a signal.


## 4.1. Two Adjacent Morlet Wavelets

In this section, we analyse two adjacent Morlet wavelets improved by us. The centres of the frequency windows separately arefk−1 and fk, and the radii are Δγk−1 and Δγk, respectively, where k≥2 and k∈N. We made their radii just connect with each other in the frequency domain, the situations of which are shown in the following two cases and seen in Figures 2 and 3.Figure 2
The first situation of two adjacent Morlet wavelets.Figure 3
The second situation of two adjacent Morlet wavelets.Letf+δ≥0. According to Figures 2 and 3, the following formula must be true if we make the radii of the two adjacent Morlet wavelets just connect with each other in the frequency domain:(27)Δγk=Δγk−1−fk−1−fk.It is equal to(28)δk=fk−1+δk−1−22πfk−1−fk−fk.


## 4.2. YPT

Assume that a signalh0t∈L2R, defined in the interval ts,te and containing Pc amounts of extreme points, is sampled to form a discrete sequence vnn=1,2,…,Nv by setting the sampling interval equal to t0, and the extremum frequency of the sequence is fe. According to Section 3.3, we know we should make f=fe/2t0=Pc/2te−ts, when analysing the signal h0t by utilising the improved Morlet wavelet. We now transform the signal h0t by using the following equations, which are collectively named the “YPT”:(29)Ypkt;fk,δpk=hk−1t∗φfk,δpkt,Ynkt;−fk,δnk=hk−1t∗φ−fk,δnkt,hkt=hk−1t−Ypkt;fk,δpk−Ynkt;−fk,δnk,where k∈N+ in the formulas in (29). On their basis, to go a step further, we take the Fourier transforms of the formulas in (29):(30)Y^pkγ;fk,δpk=FYpkt;fk,δpk,Y^nkγ;−fk,δnk=FYnkt;−fk,δnk,h^kγ=Fhkt.The equations in (29) dispersedly and adaptively transform the signal h0t into the time domain, and on the contrary, the formulas in (30) dispersedly and adaptively transform the signal h0t into the frequency domain. Taken together, both constitute the positive transformation of the YPT. fk and δk are adaptive to hk−1t. We can also obtain the inverse transformation of the YPT according to the formulas in (29) and (30), which can be described by the following equations:(31)h0t=∑i=1kYpit;fi,δpi+Ynit;−fi,δni+hkt,(32)h0t=ℱ−1∑i=1kY^piγ;fi,δpi+Y^niγ;−fi,δni+h^kγ.For the convenience of our expression in this paper,Ypkt;fk,δpk and Ynkt;−fk,δnk are uniformly named Yt;f,δ; similarly, hkt is designated as ht, Y^pkγ;fk,δpk and Y^nkγ;−fk,δnk are designated as Y^γ;f,δ, and h^kγ is designated as h^γ. According to (31), we can directly reconstruct the original signal h0t by adding signals Yt;f,δ and ht, having been analysed and processed, in the time domain. In addition, we can continue to further transform signals Yt;f,δ and ht into signals Y^γ;f,δ and h^γ in the frequency domain and then continue to process signals Y^γ;f,δ and h^γ and reconstruct the original signal h0t through (32). On a similar principle, according to (32), we can reconstruct the original signal h0t through signals Y^γ;f,δ and h^γ, having been analysed and processed, in the frequency domain. In addition, we can continue to transform signals Y^γ;f,δ and h^γ into signals Yt;f,δ and ht in the time domain and then continue to process signals Yt;f,δ and ht and reconstruct the original signal h0t through (31). Therefore, (31) and (32) provide us 4 types of YPT inverse transforms, namely, 2 types of YPT inverse transforms from the frequency domain to the time domain and 2 types of YPT inverse transforms from the time domain to the time domain.


## 4.3. Smoothing Operator

A reconstructed vibration signal can be seriously distorted in some local areas after being analysed and processed. It is usually necessary to properly smooth the signal for the specific transformation, which is helpful to reflect the change law of the original signal. In this section, we propose a smooth operatorS that can smooth the sequence corresponding to the signal. Before introducing the operator S, three operators, A, B, and L, are introduced first. Assuming that there is a finite sequence X=x1,x2,…,xn, the operator A can perform the following operation on the sequence X:(33)AX,k≜x1,…,xk−1,xk−1+xk2,xk+1,…,xn,k=2,3,…,n.In particular,(34)AX,1≜xn+x12,x2,…,xn,AX,0≜X.The operatorB performs the following operation:(35)BX≜AAA…AX,1…,n−1,n.The operatorL performs the following operation:(36)LX,k≜xn−k+1,xn−k+2,…,xn,x1,x2,…,xn−k,k∈Z.Obviously,LX,n=LX,−n=X. On the basis of operators A, B, and L, we can define the operator S:(37)SX,w≜LBB…BX,−w︸w,w∈N+.In particular,SX,0≜X.Through a large number of experiments on sequences obtained by the YPT and the filtering theory that we utilised in this paper, we found that the operatorS can effectively smooth sequences. However, we cannot excessively smooth a signal.


## 4.4. The Process of Analysing a Vibration Signal

From the analyses in Sections4.1 and 4.2, if we analyse the signal h0t through the YPT, the kernel is determining the extremum frequency of the signal and controlling the radius of the frequency window. When k≥1, fk corresponds to hk−1t. If k≥2, δk must meet the condition δk=fk−1+δk−1−22πfk−1−fk−fk, which has been discussed through (28) in Section 4.1, and we can let δ1 be 0. The radius of the frequency window will be zero if fk+δk=0 in the process of analysing the signal. However, this matter will make the process of the YPT stagnate. We therefore let δk=0 again if fk+δk=0 to continue transforming the signal until the transformation reaches the required number of the transformation. We can denote this required number as K0. The flow chart of the YPT can be divided into two parts, the first of which concerns analysing the signal, and the second concerns reconstructing the signal; both parts are shown in Figures 4 and 5, respectively.Figure 4
The flow chart for analysing a signal.Figure 5
The flow chart for reconstructing a signal.The parameter model determines how to reconstruct a signal. For this paper, we chosemodel=4 to reconstruct a signal.


## 5. Analysis and Reconstruction of a Signal

In this section, the correctness and practicability of the YPT are examined from two aspects: signal analysis and reconstruction. Based on the above research, we analysed a signal according to the flow chart in Figure4 and reconstructed it according to the flow chart in Figure 5, using model=4. We first analysed a signal containing high amounts of white noise, and we then reconstructed the signal after the process of filtering it. Threshold filtering was adopted as filtering theory to remove the noise in the signal, while the threshold filtering established by us is neither hard threshold filtering nor conventional soft threshold filtering but is rather a theory established on the probability that an interval signal belongs to noise. In summary, first, the probability function that judges the probability that an interval signal belongs to noise is established on the basis of the characteristics that the expectation of the noise is zero and that the noise has a uniform distribution; the threshold is then determined by the probability function, and at last, the interval signals belonging to noise are filtered according to the threshold function. In this paper, this filtering theory is named “probability filtering” and is not discussed in detail.


### 5.1. Analysis of a Signal

Noise was added to the ideal signalht=sint+sin2t+π/3,t∈−10π,10π, by utilising the random function randt in the Matlab numerical calculation software environment. The maximum noise amplitude was 3, and the noisy signal is therefore expressed as ht=sint+sin2t+π/3+3randt. The noisy signal was sampled into 2048 data points, which are plotted in Figure 6.Figure 6
The noisy signal.From Figure6, it is difficult to observe the change regulation of the noisy signal when we set the maximum noise amplitude to 3. After calculation, the signal-to-noise ratio (SNR) of the noisy signal was 3.13 dB. This noise can therefore be regarded as high noise. The signal ht was transformed 6 times using the YPT, and K0 was therefore equal to 6. For the convenience of calculation, we equally divided the transformed signal hK0 into two parts when we stored the transformed signals in memory. There were a total of 14 transformed signals resulting from the YPT. To conveniently observe the transformed signals in the time domain, the real parts of the signals in three-dimensional space are plotted in Figure 7.Figure 7
The real part of the signals in the time domain.Because the noisy signal is real, the two transformed signals whose serial numbers are opposite are conjugates and have the same real part. According to Figure7, the signals with serial numbers 1 and −1 are noise, and on the contrary, the signals with serial numbers 2 and −2 are useful signals and have higher amplitudes when compared with those of the others. Similarly, the signals with serial numbers 3 and −3 are also useful signals, although they have lower amplitudes when compared with those of the former. To observe them clearly, the real parts of the signals on the two-dimensional plane are plotted in Figure 8. The other signals approximate to straight lines in three-dimensional space because of their lower amplitudes, but they are not straight lines. From the YPT results, we can conclude that the YPT can separate noise and useful signals, for which the essential aspect is that noise is different from the useful signal at the extremum frequency.Figure 8
The real part of the serial number 3 or −3 signals.We can obtain the transformed signals in the frequency domain if continuing to transform the transformed signals in the time domain into the frequency domain using the YPT. The real parts of the transformed signals in three-dimensional space are plotted in Figure9. From Figure 9, we can clearly see that the signals with serial numbers 1 and −1 contain large amounts of noise, which accords with the characteristics that the expectation of the noise is zero and is uniformly distributed in the frequency domain. For the signals with serial numbers 2 and −2, each of them has a positive impact amplitude and a lower negative impact amplitude in the frequency domain.Figure 9
The real part of the signals in the frequency domain.The real parts of the signals with serial numbers 3 and −3 are separately plotted on two-dimensional planes in Figures10 and 11. The signals in Figures 10 and 11 and those in Figure 9 have similar properties. For the convenience of expression in this paper, the signals whose serial numbers are opposite to each other are, respectively, named “positive and negative signals.” Their real parts have symmetry if the signal ht is real.Figure 10
The real part of the positive signal in the frequency domain.Figure 11
The real part of the negative signal in the frequency domain.


### 5.2. Reconstruction of a Signal

For this section, we adoptedmodel=4 to reconstruct the signal ht from Section 5.1. First, the signals in the frequency domain were filtered employing probability filtering to remove a part of the noise, and we then transformed the filtered signals in the frequency domain into the time domain. After that, we continued to again filter the signals in the time domain using probability filtering to remove a part of the noise in the different space. We finally reconstructed the original signal ht using the filtered signals in the time domain. The process of reconstructing the original signal employed the YPT inverse transform, the filtering theory, and the smoothing operator. In Figure 12, the real parts of the new filtered signals in the frequency domain are shown in three-dimensional space.Figure 12
The real part of the new filtered signals in the frequency domain.Compared with those shown in Figure9, the signals with serial numbers 1 and −1 in Figure 12 significantly reduced the noise and retained a part of the useful signals, as was the case for the signals with serial numbers 2 and −2. However, for the serial number 2 signal, the positive impact amplitude was reduced slightly because the positive impact amplitude was affected by the noise and the different filtering order when the threshold function readjusted the values of the interval signals. To compare with the signals in Figures 10 and 11 and to also observe the effects of the YPT and filtering theory, the real parts of the new filtered signals with serial numbers 3 and −3 are separately plotted on two-dimensional planes in Figures 13 and 14. From Figures 13 and 14, they have properties similar to those of the new filtered signals with serial numbers 2 and −2.Figure 13
The real part of the new positive filtered signal in the frequency domain.Figure 14
The real part of the new negative filtered signal in the frequency domain.What follows is to transform the new filtered signals in the frequency domain into the time domain, and we can then continue to filter them in the time domain by probability filtering. The new filtered signals in three-dimensional space are plotted in the time domain in Figure15. Compared with the signals in Figure 7, each of the signals in Figure 15 generally has less noise. On the contrary, the new filtered signal with serial number 2 lost a large amount of energy in the filtering process, which exposes the lack of this filtering theory but does not influence the validity and practicability of the YPT.Figure 15
The real part of the new filtered signals in the time domain.Finally, according to (31), we reconstructed the original signal ht through the new filtered signals in Figure 15. We then properly smoothed the signal using the smoothing operator SX,w, and the parameter w was 5. The new signal processed by the YPT and ideal signal without noise correspond to the B curve and the C curve in Figure 16, respectively. From Figure 16, we can determine that the amplitude variation of the new signal was not as high as the ideal signal. Combining Figures 6 and 16, we can conclude that the effect of reconstructing a signal is ideal when utilising the YPT, probability filtering, and the smoothing operator. Compared with the noisy signal in Figure 6, the noise in the new signal was very low, and it clearly reflected its own law of change.Figure 16
The new signals and ideal signal.To further verify the validity of the YPT, it is compared with the DWT. We also decomposed the noisy signal into 4 layers by the DWT, and the wavelets for the DWT are 3 order Daubechies wavelets. For the noisy signal decomposed by the DWT, after removing the first three layers of detail signals, we can then obtain the best reconstructed signal by the inverse discrete wavelet transform (IDWT), which corresponds to the A curve in Figure16. After calculation, the SNR of the noisy signal, A curve, and B curve is 3.13 dB, 10.61 dB, and 16.39 dB, respectively, and as far as denoising and the noisy signal in this paper are concerned, the YPT is therefore better than the DWT.


## 5.1. Analysis of a Signal

Noise was added to the ideal signalht=sint+sin2t+π/3,t∈−10π,10π, by utilising the random function randt in the Matlab numerical calculation software environment. The maximum noise amplitude was 3, and the noisy signal is therefore expressed as ht=sint+sin2t+π/3+3randt. The noisy signal was sampled into 2048 data points, which are plotted in Figure 6.Figure 6
The noisy signal.From Figure6, it is difficult to observe the change regulation of the noisy signal when we set the maximum noise amplitude to 3. After calculation, the signal-to-noise ratio (SNR) of the noisy signal was 3.13 dB. This noise can therefore be regarded as high noise. The signal ht was transformed 6 times using the YPT, and K0 was therefore equal to 6. For the convenience of calculation, we equally divided the transformed signal hK0 into two parts when we stored the transformed signals in memory. There were a total of 14 transformed signals resulting from the YPT. To conveniently observe the transformed signals in the time domain, the real parts of the signals in three-dimensional space are plotted in Figure 7.Figure 7
The real part of the signals in the time domain.Because the noisy signal is real, the two transformed signals whose serial numbers are opposite are conjugates and have the same real part. According to Figure7, the signals with serial numbers 1 and −1 are noise, and on the contrary, the signals with serial numbers 2 and −2 are useful signals and have higher amplitudes when compared with those of the others. Similarly, the signals with serial numbers 3 and −3 are also useful signals, although they have lower amplitudes when compared with those of the former. To observe them clearly, the real parts of the signals on the two-dimensional plane are plotted in Figure 8. The other signals approximate to straight lines in three-dimensional space because of their lower amplitudes, but they are not straight lines. From the YPT results, we can conclude that the YPT can separate noise and useful signals, for which the essential aspect is that noise is different from the useful signal at the extremum frequency.Figure 8
The real part of the serial number 3 or −3 signals.We can obtain the transformed signals in the frequency domain if continuing to transform the transformed signals in the time domain into the frequency domain using the YPT. The real parts of the transformed signals in three-dimensional space are plotted in Figure9. From Figure 9, we can clearly see that the signals with serial numbers 1 and −1 contain large amounts of noise, which accords with the characteristics that the expectation of the noise is zero and is uniformly distributed in the frequency domain. For the signals with serial numbers 2 and −2, each of them has a positive impact amplitude and a lower negative impact amplitude in the frequency domain.Figure 9
The real part of the signals in the frequency domain.The real parts of the signals with serial numbers 3 and −3 are separately plotted on two-dimensional planes in Figures10 and 11. The signals in Figures 10 and 11 and those in Figure 9 have similar properties. For the convenience of expression in this paper, the signals whose serial numbers are opposite to each other are, respectively, named “positive and negative signals.” Their real parts have symmetry if the signal ht is real.Figure 10
The real part of the positive signal in the frequency domain.Figure 11
The real part of the negative signal in the frequency domain.


## 5.2. Reconstruction of a Signal

For this section, we adoptedmodel=4 to reconstruct the signal ht from Section 5.1. First, the signals in the frequency domain were filtered employing probability filtering to remove a part of the noise, and we then transformed the filtered signals in the frequency domain into the time domain. After that, we continued to again filter the signals in the time domain using probability filtering to remove a part of the noise in the different space. We finally reconstructed the original signal ht using the filtered signals in the time domain. The process of reconstructing the original signal employed the YPT inverse transform, the filtering theory, and the smoothing operator. In Figure 12, the real parts of the new filtered signals in the frequency domain are shown in three-dimensional space.Figure 12
The real part of the new filtered signals in the frequency domain.Compared with those shown in Figure9, the signals with serial numbers 1 and −1 in Figure 12 significantly reduced the noise and retained a part of the useful signals, as was the case for the signals with serial numbers 2 and −2. However, for the serial number 2 signal, the positive impact amplitude was reduced slightly because the positive impact amplitude was affected by the noise and the different filtering order when the threshold function readjusted the values of the interval signals. To compare with the signals in Figures 10 and 11 and to also observe the effects of the YPT and filtering theory, the real parts of the new filtered signals with serial numbers 3 and −3 are separately plotted on two-dimensional planes in Figures 13 and 14. From Figures 13 and 14, they have properties similar to those of the new filtered signals with serial numbers 2 and −2.Figure 13
The real part of the new positive filtered signal in the frequency domain.Figure 14
The real part of the new negative filtered signal in the frequency domain.What follows is to transform the new filtered signals in the frequency domain into the time domain, and we can then continue to filter them in the time domain by probability filtering. The new filtered signals in three-dimensional space are plotted in the time domain in Figure15. Compared with the signals in Figure 7, each of the signals in Figure 15 generally has less noise. On the contrary, the new filtered signal with serial number 2 lost a large amount of energy in the filtering process, which exposes the lack of this filtering theory but does not influence the validity and practicability of the YPT.Figure 15
The real part of the new filtered signals in the time domain.Finally, according to (31), we reconstructed the original signal ht through the new filtered signals in Figure 15. We then properly smoothed the signal using the smoothing operator SX,w, and the parameter w was 5. The new signal processed by the YPT and ideal signal without noise correspond to the B curve and the C curve in Figure 16, respectively. From Figure 16, we can determine that the amplitude variation of the new signal was not as high as the ideal signal. Combining Figures 6 and 16, we can conclude that the effect of reconstructing a signal is ideal when utilising the YPT, probability filtering, and the smoothing operator. Compared with the noisy signal in Figure 6, the noise in the new signal was very low, and it clearly reflected its own law of change.Figure 16
The new signals and ideal signal.To further verify the validity of the YPT, it is compared with the DWT. We also decomposed the noisy signal into 4 layers by the DWT, and the wavelets for the DWT are 3 order Daubechies wavelets. For the noisy signal decomposed by the DWT, after removing the first three layers of detail signals, we can then obtain the best reconstructed signal by the inverse discrete wavelet transform (IDWT), which corresponds to the A curve in Figure16. After calculation, the SNR of the noisy signal, A curve, and B curve is 3.13 dB, 10.61 dB, and 16.39 dB, respectively, and as far as denoising and the noisy signal in this paper are concerned, the YPT is therefore better than the DWT.


## 6. The Application of the YPT in Engineering

In this section, two types of serious fault cracks on the latch sheets of a gun automatic mechanism were set by us, and the shock signals that are obtained by sampling the data, respectively, when the gun automatic mechanism works in normal and two fault patterns were analysed by utilising the method in this paper. The firing frequency of the gun automatic mechanism is10Hz, and the main parameters of the sensors that we used in this paper are shown in Table 1. To obtain more details in the time domain, we set the sampling frequency to the maximum sampling frequency (204.8kHz) of the signal collector, and the sampling time is 5s. On the contrary, for a shock signal, the sampling frequency should be as high as possible to utilise the signal near the resonant frequency of the sensor, although the output of the sensor may be distorted at high frequency. The environment of the experiment is shown in Figure 17, and the main measuring point is shown in Figure 17(d).Table 1
The parameters of the sensor.
Performance
Value
Sensitivity
1.0 mV/g (±15%)
Measurement range
±5000 g pk
Frequency range
0.4 to 7500 Hz (±10%)
Electrical filter cutoff frequency
≥7.5 kHz (−10%)
Resonant frequency
≥50 kHz
Broadband resolution
0.02 g·rms (1 to 10 kHz)Figure 17
The environment of the experiment. (a) The positions prone to noise; (b) the place for the experiment; (c) signal collector; (d) the main measuring point; (e) the first kind of cracks; (f) the second kind of cracks.
(a)
(b)
(c)
(d)
(e)
(f)Due to the bad working environment or high overload of the equipment, a lot of noise is usually contained in the shock signals. The noise reduction for the signals therefore needs to be done to eliminate the effect of the noise if we do pattern recognition or fault diagnosis for the equipment. For the gun automatic mechanism, there are two main sources of noise, one of which is at ammunition feeding mechanism that is numbered (1) in Figure17(a) and another is at the base of the equipment, numbered (2) in Figure 17(a). The former is produced by the irregular friction and collision between the bullet chain and the body of the equipment, and the latter is produced by self-excited vibration on the base that is with the complex structure or the random vibration of the base such as a gun automatic mechanism that works on a tank or fighter plane. These two kinds of noise have the property that the frequency of them is lower when compared with that of the efficient signals that are produced by the free vibrations of each part of the gun automatic mechanism.The actual effective time period of original signals was intercepted to be analysed. The original signals and the signals filtered by the YPT are plotted in Figure18. Figures 18(a)–18(c) are the original signals, and Figures 18(d)–18(f) are the signals that correspond to Figures 18(a)–18(c) and are filtered by the method, respectively. To further observe the effects of the method, we transformed them into the frequency domain, which is shown in Figure 19. From Figure 19, we can conclude that contrast to original signals, the amplitude of the part in low frequency is significantly reduced, relatively to the part in high frequency, and the shape of the part in high frequency almost does not change. The results of the analysis show that the noise in shock signals is filtered effectively. The YPT therefore shows good performance in the analysis of the shock signal in engineering, which can provide help for further extracting features for shock signals in pattern recognition and fault diagnosis.Figure 18
The original signals and the signals filtered by the YPT.
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(f)Figure 19
The original signals and the signals filtered by the YPT in the frequency domain.
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## 7. Summary and Outlook

Because the Morlet wavelet does not satisfy the permissibility condition of the wavelet transform, it has no inverse wavelet transform. On the contrary, the Morlet wavelet has good time-frequency characteristics. In this paper, it was easy to adaptively adjust the centre and radius of the frequency window for the Morlet wavelet after its improvement. The essence of the wavelet transform is that it is a type of filtering. For this paper, the essence of transforming a signal into the time and frequency domains is also a filter. The YPT in this paper can separate noise from the signal, the essential aspect of which is that noise and useful signals have different extreme frequencies. On the principle of extreme frequency, the original signal is decomposed into different components using the YPT.In this paper, the noisy signal and the shock signals in engineering could also be complex or real. The extremum frequency is determined by the real part of the signal when analysing a complex signal. There will be two ways for us to conduct research if further analysing a multidimensional signal. For the first, we can analyse each dimension of the multidimensional signal using the YPT alone. For the other, we can extend the dimension of the improved Morlet wavelet so that it has the ability to analyse multidimensional signals. For the latter method, how to determine the centre and radius of the Morlet wavelet in the frequency domain needs to be researched more deeply. Matching a multidimensional signal to the Morlet wavelet in phase is worthy of study.



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*Source: 9892713-2018-07-31.xml*