Tool-Emitted Sound Signal Decomposition Using Wavelet and Empirical Mode Decomposition Techniques—A Comparison

Abstract

Analysis of non-stationary and nonlinear sound signals obtained from dynamical processes is one of the greatest challenges in signal processing. Turning machine operation is a highly dynamic process influenced by many events, such as dynamical responses, chip formations and the operational conditions of machining. Traditional and widely used fast Fourier transformation and spectrogram are not suitable for processing sound signals acquired from dynamical systems as their results have significant deficiencies because of stationary assumptions and having an a priori basis. A relatively new technique, discrete wavelet transform (DWT), which uses Wavelet decomposition (WD), and the recently developed technique, Hilbert–Huang Transform (HHT), which uses empirical mode decomposition (EMD), have notably better properties in the analysis of nonlinear and non-stationary sound signals. The EMD process helps the HHT to locate the signal’s instantaneous frequencies by forming symmetrical envelopes on the signal. The objective of this paper is to present a comparative study on the decomposition of multi-component sound signals using EMD and WD to highlight the suitability of HHT to analyze tool-emitted sound signals received from turning processes. The methodology used to achieve the objective is recording a tool-emitted sound signal by way of conducting an experiment on a turning machine and comparing the results of decomposing the signal by WD and EMD techniques. Apart from the short mathematical and theoretical foundations of the transformations, this paper demonstrates their decomposition strength using an experimental case study of tool flank wear monitoring in turning. It also concludes HHT is more suitable than DWT to analyze tool-emitted sound signals received from turning processes.

1. Introduction

Turning is a highly dynamic process, and its production performance is determined by various parameters, such as cutter geometry, cutting conditions, chip formation, tool and workpiece material, tool wear and so on. According to earlier studies, tool wear is the most important one amongst these parameters that affect the performance of production [1,2]. Thus, tool wear monitoring is vital to attain the surface quality of a workpiece and dimensional accuracy while turning. An indirect way of tool wear monitoring is based on acquiring the values of process variables measured (namely temperature, vibration, cutting force, acoustic emission, spinning-motor current, emitted sound and roughness of the surface) and the relationship among these values and tool wear [1].

Monitoring based on sound is one of the technologies not investigated extensively for monitoring tool wear, although it has been widely used by turning operators in machining workshops for making decisions on tool condition [1,3]. A sound signal may be easily contaminated by noises from motors, adjacent machines, nearby processes and conveyors. These contaminations could be removed by using signal decomposition techniques, namely Wavelet decomposition (WD) and empirical mode decomposition (EMD), which are being used in signal processing techniques, discrete wavelet transform (DWT) and Hilbert–Huang transform (HHT).

Commonly used traditional data analysis methods, for example, Fourier analysis, are all grounded on linear and stationary conventions, i.e., the signal to be processed should be temporarily stationary and linear; if not, the resulting Fourier spectrum will represent only a little physical sense [4]. In addition, it signifies only global properties and not the required local properties of the original signal because of the use of a convolutional integral for decomposing the signal in terms of sine and cosine functions [5]. Short-time Fourier Transform (STFT) is a widely used technique, with limitations due to constant resolution in frequency and time, where the resolution is defined by the width of the windowing function [6]. Conversely, wavelet transform (WT), which is one of the commonly used time–frequency analyzing methods, can produce both frequency and time information of a signal at the same time by matching one-dimensional signals to a multi-dimensional time vs. frequency plane. Nevertheless, wavelet transform still has a few unavoidable insufficiencies, including energy leakage, border distortion and interference terms. These insufficiencies produce numerous, small, unwanted spikes almost all over the frequency scales and make the end results very difficult to infer as well as confusing [4,7]. In order to sense tool condition, it is not sufficient to detect the existence of a particular frequency in the original signal, but it is necessary to localize the frequency with respect to time and its space [8].

DWT is appropriate for localizing the time of a frequency component of a signal and has attracted substantial attention in the area of monitoring machine tool wear [4,7]. HHT is a new technique that is good for time–frequency analysis, which can provide very good resolution in frequency and time simultaneously [5]. HHT has been extensively used in the field of monitoring the condition of roll bearings [9,10]. Moreover, some research has also been carried out in the field of tool condition monitoring using HHT [10,11]. In [12], the authors have proposed an approach to detect flute breakage in end milling using HHT. The use of a marginal Hilbert spectrum from HHT to correlate tool wear in end milling has been explored in [13]. EMD is used to denoise electrocardiogram signals (ECG), which are the most important sources of diagnostic information in healthcare, in [14].

Even though HHT can offer brilliant localization of instantaneous frequencies in time, it has not received as much attention as DWT in the monitoring process of turning. One reason could be due to the fact that HHT is a newer technique than well-established DWT [8]. This paper presents a comparative study on the decomposition of multi-component signals using HHT and DWT to highlight the superiority of HHT for the analysis of tool-emitted sound signals obtained from turning processes.

2. Materials and Methods

It is a known fact that any nonlinear system is identified by its intra-wave frequency variations. To define these intra-wave frequency variations in data, we need a procedure that has an a posteriori-defined basis, which is derived from the data itself. Instantaneous frequency is the one that normally reveals the intra-wave frequency modulation of any system. Hence, the most physically meaningful way to describe nonlinear systems is in terms of their instantaneous frequency. Hilbert transform (HT) is the simplest and easiest way to compute this instantaneous frequency [5]. Hilbert–Huang transform is a relatively new approach to signal processing, which works very well for signals that are nonlinear and non-stationary as the definition of the basis is extracted from the original signal.

HHT is a process of decomposing any multi-component signal into many mono-component signals, so-called intrinsic mode functions (IMFs), and then locating its instantaneous frequencies [5]. It is obtained from the principle of empirical mode decomposition (EMD) and HT. First, the captured nonlinear and non-stationary signal is decomposed into a group of IMFs by using EMD. Every IMF is an adaptive and near orthogonal representation of the decomposed signal. Because an IMF is almost a mono-component, the instantaneous frequencies found in each IMF can easily be determined.

2.1. Empirical Mode Decomposition (EMD)

Any signal (data) might have more than one different co-existing mode of oscillations at the same time, based on its complexity. Every oscillatory mode is represented by an IMF with the definitions given below [5].

(a)

Within a whole dataset, the total count of zero-crossings and the total count of extrema must either be the same or differ by a maximum of one.

(b)

At any point on the whole dataset, the mean value of the envelope formed by the local minima and the envelope formed by the local maxima is zero.

A special sifting process is implemented to remove every IMF from the given dataset [5]. The sifting process begins with identifying all the local minima and then connecting all of the local minima by a cubic spline line. Thereby, a lower envelope is formed for the dataset. Then, by repeating the same procedure for the local maxima, an upper envelope is also formed for the dataset. It is important that the lower and upper envelopes must cover all the data points between them. The lower and upper envelopes should cover all the data between them as shown in Figure 1. Their mean is calculated as m₁(t), and the difference between the data and the mean is also calculated as h₁(t), using Equation (1).

$$x(t)-m_1(t)=h_1(t)$$

(1)

This sifting process is repeated several times until the extracted signal becomes an IMF, by this time the upper and the lower envelopes are almost symmetrical with its mean, satisfying the definitions mentioned above. The diagram in Figure 1 shows a typical series of four iterations until the mean becomes almost zero [15]. It clearly shows the mean value of the envelope formed by the local minima and the envelope formed by the local maxima is zero. This difference h₁(t) is treated as data, and the sifting process is again applied on it, as follows:

$$h_1(t)-m_{11}(t)=h_{11}(t)$$

(2)

where m₁₁(t) is the mean of the lower and upper envelopes of h₁(t). As shown in Equation (3), h_1k(t) is obtained by repeating this process up to k times.

$$h_1(k-1)-m_1(t)=h_{1k}(t)$$

(3)

At the end of each step in the processing, a checking should be performed on whether the number of extrema equals the number of zero-crossings. The resultant time series will be the first IMF, and then, it is marked as c₁(t) = h₁k(t).

Actually, the oscillations with a height frequency found inside the original signal x(t) are extracted into the very first IMF component of the signal. And then, this IMF is subtracted from the original signal, and the result is called a residue, r₁(t), as shown in Equation (4).

$$x(t)-c_1(t)=r_1(t)$$

(4)

The residue r₁(t) is considered as if it was the original signal, and the sifting process is again applied on it. The entire process of finding subsequent intrinsic modes c_i(t) is continued until the last intrinsic mode is found. The end residue will be either a constant or a monotonic function. The entire sifting process may be represented using Equation (5).

$$x(t)={\displaystyle \sum _{j=1}^n{c}_j}(t)+r_n(t)$$

(5)

Hence, a multi-component signal can be decomposed into n-empirical intrinsic mode functions (IMFs), plus a residue, r_n(t), with the help of EMD.

2.2. Wavelet Decomposition (WD)

The idea of wavelet-based analysis was invented by Jean Morlet, who was an engineer from France, in 1982. The primary aim of wavelet transform (WT) is to get a more accurate illustration of both the time and frequency content of a signal. Analyzing according to scale is the essential principle behind wavelets. Adopting a mathematically defined wavelet prototype function termed mother wavelet is the core process in wavelet analysis [16].

The concept is almost similar to short-time Fourier transform (STFT), except the windowing function width is based on the central frequency. Therefore, for any given signal, the best bargains between time and frequency resolution can be attained automatically. Actually, “a wavelet is a kind of kernel function to be used in an integral transformation” [16].

If x(t) is a continuous signal, its wavelet transform, CWT, can be obtained using Equation (6).

$$CWT\{x\left(t\right);a,b\}=\int x\left(t\right){\psi }_{a,b}^{*}\left(t\right)dt$$

(6)

where ‘*’ denotes the complex conjugation.

Equation (7) represents the wavelet function defined by transforming and expanding a “mother” function.

$${\psi }_{a,b}={\displaystyle \frac{1}{\sqrt{\left|a\right|}}}\psi \left({\displaystyle \frac{t-b}{a}}\right) a,b ϵ R, a\ne 0$$

(7)

where ψ(t) is the basic or mother wavelet with the translation parameter ‘b’ and the dilation factor ‘a’.

The parameters a and b are frequently discretized for practical applications, which results in the so-called discrete wavelet transform (DWT). A wavelet function can be defined by discretizing a = 2^j and b = k2^j (j, k ∈ Z), as shown in Equation (8).

$${\psi }_{j,k}=2^{{\displaystyle \frac{-i}{2}}}\psi (2^{-j}t-k)$$

(8)

DWT can then be defined mathematically using Equation (9).

$$C_{j,k}=\int x\left(t\right){\psi }_{j,k}^{*}\left(t\right)dt$$

(9)

where C_j,k is given the name wavelet coefficient, and it can be viewed, in general, as a time vs. frequency map of the original continuous signal x(t).

Even though DWT is able to give a complete and orthogonal representation of any signal, it is still considered as an a priori approach because of the adaptation of a mother wavelet in the decomposition process [17,18].

2.3. General Experimental Setup

The experimental setup is shown in the form of a graphic diagram in Figure 2. The aim of this experiment was to collect tool-emitted sound signals from fresh, slightly and severely worn tools to compare the performance of EMD and WD signal-decomposing techniques. This experiment was conducted on a normal lathe EME turning machine. Some carbide tool inserts of type NM6 were used to machine unalloyed mild steel AISI 1040. To obtain the tool sound, produced because of the friction between the tool-bit tip and the workpiece surface, a PCB130D20 microphone was placed near the tool insert tip. A specifically designed signal conditioner was used to provide the appropriate voltage level for the PCP microphone. Hence, the microphone was linked to a computer through this signal conditioner. A special software, GoldWave (GoldWave v5.57), was used to save the recorded sound by the microphone at a sampling rate of 44.1 KHz. Many experiments (machining) were conducted.

The experiment began with recording a free-run sound of the spindle rotating at a speed of 570 rev/min while keeping the tool insert not in contact with the workpiece. Then, the emitted sound while machining using a fresh tool (without any flank wear) was recorded, maintaining the same speed (570 rev/min) for a 1 mm depth of cut. This process of recording was repeated distinctly for a slightly worn tool with a flank wear of 0.2 mm and a severely worn tool with a flank wear of 0.4 mm. Throughout the experiment, a constant feed rate set at 0.5 mm/rotation was uniformly maintained. The recording was carried out for 12 s. Each 12 s long signal was split into twelve one-second-long signals to be used in the following signal decomposition process by WD and EMD. Each one-second-duration sound signal was digitized using the Matlab waveread function.

3. Results and Discussion: Comparison of EMD and WD

Figure 3a shows the result of the Wavelet decomposition applied on the sound signal captured when the tool insert was not in contact with the workpiece, i.e., free-running sound. Figure 3b shows the result of the Wavelet decomposition applied on the sound signal captured while machining using the fresh tool (without flank wear). Similarly, Figure 4a shows the result of the Wavelet decomposition of the sound signal captured while machining using the slightly worn tool with a flank wear of 0.2 mm. And Figure 4b shows the result of the Wavelet decomposition of the sound signal captured while machining using the severely worn tool with a flank wear of 0.4 mm.

This Wavelet decomposition was performed using the Matlab function wavedec(X,N,’wname’), where X is the tool sound signal to be decomposed, and N is the level, which was set to 5. The parameter wname represents the mother wavelet, and ‘db3’ (‘Daubechies Extremal Phase Wavelet’ including ‘3 Vanishing Moments’) was selected. It is worth noting that the selection of ‘db3’ as the mother wavelet and setting the decomposition level to 5 make WD an a priori approach.

Figure 5a,b show the result of empirical mode decomposition on the sound signals captured during free-running and machining using a fresh tool, respectively. Similarly, Figure 6a,b show the result of the same decomposition on sound signals captured while machining with slightly and severely worn tools, respectively. The important point here is that the level of decomposition was not set to any initial value. Also, there was no specific basis selected, like mother wavelet in WD, as it was not required, and hence, EMD is an a posteriori approach. Now, it is evident from Figure 5 and Figure 6 that the level of decomposition was not fixed in EMD, as the IMFs generated were different in numbers for free-run (13 IMFs), slightly worn (14 IMFs) and severely worn (15 IMFs) tool sound signals. It is evident that EMD is able to produce a variable number of components because it is a fully data-driven and adaptive method, which can obtain the various oscillatory modes inside an original signal. This enabled EMD to overcome the inherent limitations of signal decomposition using the wavelet approach WD.

Wavelet analysis managed to extract only coefficients throughout the decomposition process. Moreover, interpretation as amplitude and frequency should be performed in a circuitous process [19]. It could be possible by way of using a scaling concept, but it still drives again toward an a priori approach. It is a well-known fact that intra-wave frequency modulation is the signature of any nonlinear system. Hence, a nonlinear system can only be efficiently described by approaches with an a posteriori-defined basis extracted from a signal.

Information found in a signal is not missing during the process of EMD. So, every individual piece of information within a signal is reflected in conceivable different instantaneous frequencies and instantaneous amplitudes. These instantaneous frequencies and amplitudes could easily be extracted by applying Hilbert transform on each and every IMF. Therefore, we can conclude that use of the EMD method is more appropriate and efficient than the Wavelet decomposition method for decomposing tool-emitted sound signals used in our research.

The results produced in this research agree with the comparison found in [20] using a respiratory waveform. The authors of [8] also documented in their conclusion that EMD overcomes the effect of the choice of mother wavelet in wavelet decomposition. They also mentioned that the reason is because of the adaptive sifting procedure used in EMD to identify the basis from the same analyzed signal, which is an a posteriori approach. This is in contrast with the a priori approach used in Wavelet decomposition to choose the mother wavelet but not from the signal being analyzed.

4. Conclusions

A comparison is made here to verify the performance of Wavelet decomposition and empirical mode decomposition methods in the process of decomposing tool sound signals obtained from a turning process. This study enabled us to summarize their performance on various factors in the form of a table (

Table 1. Comparing the performance of EMD and WD.

Feature	EMD	WD
Basic building block	Intrinsic mode function (IMF)	Wavelet family
Approach	A posteriori (Data-driven and adaptive)	A priori assumed
Level of Decomposition	Variable, automatically generated from signal	Constant set by user
Completeness	Yes	Yes
Orthogonality	Almost	Fully orthogonal
Inverse transform	Not possible	Possible
Instantaneous frequency	Possible	Not possible
Computational complexity	Very high	Reasonable

), which will be a useful reference for researchers performing research related to the decomposition of sound signals. EMD offers a complete, adaptive (because IMFs are data-driven) and nearly orthogonal (because there is a residue) signal/data representation. WD also offers a complete and orthogonal (because a signal is decomposed to form atomic functions) signal/data representation; however, it is non-adaptive because the mother wavelet is chosen in advance (a priori approach). The instantaneous frequency can be determined with high precision by applying Hilbert transform on each IMF. On the other hand, the instantaneous frequency cannot be revealed by applying DWT on the atomic function; it can only be used to perceive the relative change in frequency content of the original signal. Hence, HHT is more suitable than DWT to analyze tool-emitted sound signals received from turning processes. However, implementation of HHT is computationally very expensive, O(n*log(n)), because of the amount of time consumed for the sifting process in EMD. Perhaps because of this shortcoming, and the fact that HHT is younger than DWT, the use of HHT in tool condition monitoring has received slightly less attention among the research community. There are two more improved versions of EMD algorithms, namely variational mode decomposition (VMD) and ensemble empirical mode decomposition (EEMD), which are available to reduce the time complexity of EMD. It is hoped that HHT will be an efficient and promising method in the near future, with these improved versions of EMD.