Complete Teager Energy Operator: A Novel Method for Enhancing Micro-impact Signal

: The output of conventional Teager energy operator (TEO) is approximately equal to the square product of the instantaneous amplitude and the instantaneous frequency ( A 2 Ω 2 ). The original TEO can effectively enhance the transient shock components and suppress the non-impacting elements, and it also changes the frequency distribution of the original shock. In this paper, a complete Teager energy operator is proposed, and its expression is more exact than original method. By keeping the positive and negative distribution of the shock signal ( ) x t , the fundamental frequency energy of the impulses can be effectively enhanced. The incipient fault characteristics of large-scale rotating machinery are typically micro shock pulse, extremely weak and mixed with heavy noise. Preprocessing the fault signal and enhancing the micro shock component are essential means to extract the early fault features. In the experiment part, the applicability of the proposed method is verified by the simulated micro impact signal, the common bearing fault data-sets and the practical measured data of the test bench.


Introduction
In the undamped free vibration system, the general solution is A.cos(ω.t+φ), thus, the output of conventional Teager energy operator is T energy = A 2. Ω 2 , which is an approximation of A 2. sin 2 Ω [1]. When large rotating machinery is working at low speed and heavy load, once bearing pitting or defect is developed, it will produce periodic impact signal due to the impact between parts. Through the analysis of the collected vibration signal, it is found that the impulse caused by the fault presents the sine exponential attenuation property [2]. That is to say, the function characteristic of shock signal satisfies the general solution form of damped free vibration system [3].Base on that, this paper proposes an influence factor  , termed energy T = 2 2 A    , which provides the exact output form of Teager energy operator. In addition, the output value of conventional Teager energy operator is nonnegative.
But in practice, the collected fault signals of large, low-speed and heavy-duty machinery are bipolar.
Teager energy operator can enhance the instantaneous change of micro impulses in fault signal, however, its output is consistently positive, which changes the amplitude and frequency distribution of the micro shock signal. Therefore, by distinguishing the polarity of the original signal, the output of Teager energy operator is 2 2 () A      , that is consistent with the positive and negative characteristics of the raw signal, At the same time, the frequency distribution of original micro shock signal is maintained. This kind of Teager energy operator is named as complete Teager energy operator (CTEO) in this paper. It can effectively enhance the magnitude of transient impact components and maintain the frequency distribution of the signal.
With the development of industrial technology, engineering mechanical equipment is developing towards the tendency of maximization, complexity and automation. Under complex working conditions, the transmission part of equipment is easily damaged. Its early defect may cause unexpected equipment accidents, and even endanger personal safety [4]. Therefore, mechanical equipment condition monitoring and fault diagnosis technology have attracted the attention of the industry. Taking the case of the mechanical equipment running under low speed and heavy load, the early shock fault often reflects the characteristics of low signal-to-noise ratio (SNR) and long period. Affected by strong background noise and attenuation of signal transmission process, the fault characteristics are extremely sparse and weak, and it is difficult to extract fault features directly. Therefore, it is necessary to preprocess the original fault signal through denoising and enhancing useful features to improve the SNR. The most commonly used denoising methods are wavelet [5][6], empirical mode decomposition [7], sparse decomposition [8], etc. In addition, Teager energy operator is often applied to enhance the impulse components in fault signal [9]. Teager energy operator is an energy model of dynamics, which is introduced into the field of signal processing. It is a parameter free method, simple and effective [10], and has high real-time performance. The wavelet method needs to select the appropriate basis function and related parameters in advance [11], its validity for the bearing early fault of large rotating machinery with low-speed needs to be further verified by experiments. Therefore, Teager energy operator is often used to improve the SNR of such fault signals [12].
The remainder of the paper is organized as follows. Section 2 introduces the theoretical principles of Teager energy operator and describes the proposed Teager energy operator framework. In Section 3, the proposed framework is verified by simulation analysis. In Section 4,public experiment data set analysis and engineering data analysis are conducted. Finally, conclusions are presented in Section 5.

The proposed method
In present section, we will introduce a novel identification method of shock signal enhancement factor in detail. Firstly, the model of vibration system is introduced and analyzed. After that, the framework of TEO is deduced. Then, a complete enhancement factor is proposed.

Framework of single-freedom vibration system
A single degree of freedom vibration system usually consists of a mass m of directional vibration, an elastic element k connected between the vibration mass and the foundation and a damping c in motion. The system is shown in Fig.1. According to Newton's second law, the motion differential equation of vibration system can be established as: 0 mx cx kx (1) where x is the position of the vibration mass m , c is the damping coefficient, k denotes the elastic coefficient.
x and x represent the first and second derivatives of x , respectively. Eq. (1) can be analyzed in two cases: (1) Eq. (1) is a single degree of freedom undamped free vibration when c = 0, and its general solution is as follow: where A is the amplitude, n  is the arbitrary phase, n k m  is the natural frequency of the vibration system.
(2) When 0 c  , Eq. (1) presents a single degree of freedom mass-spring-damper system. The expression of general solution is deduced as: where A is the amplitude and r  is the initial phase.

Teager energy operator
For continuous signals x(t), a kind of nonlinear energy tracking difference operator, named Teager energy operator(TEO) [13] is used to track and capture the instantaneous change of narrowband signals.
By simple mathematical analysis, the nonlinear energy tracing operator for continuous signal is defined as Eq.( 4), referred to as ѱ c : where () xt and () xt denote the first and second derivatives of the signal () xt versus time t, respectively.
The function form of shock signal excitated by large rotating machinery early fault conforms to the general solution form of damped free vibration system. As shown in Fig.1, For discrete signals () xn , Kaiser [14] proposed the differential Teager energy operator, which can quickly track the energy changes of the signal with three close samples. The discrete TEO is defined as Eq. (5): According to Eq. (3), it can be concluded: In Eq. (6), A is the amplitude,  is the initial phase, = 2    is the frequency, and  denotes attenuation coefficient. Apparently, According to the formula Hence, sin  is approximately equal to  for small values of  , that is, sin  . When  =0, the expression of Teager energy operator is obtained as follows: where T is an approximation of Teager energy operator output, which is approximately equal to the square product of the instantaneous amplitude and the instantaneous frequency of a vibration signal.
Due to the high frequency variation of transient impact, this method can effectively enhance the transient impact components.

The proposed CTEO model
In this paper, a complete factor  is introduced and the Eq. (7) is modified as follows: , Eq. (9) can be expressed as: In order to keep the output of Teager energy operator as the product of instantaneous amplitude and instantaneous frequency square, the above formula is modified as follows: where T is the complete Teager energy operator proposed in this paper, which is the exact output of T . Notably, , so the output of energy operator T is greater than that of T . That is, T is an enhanced expression of T . The instantaneous angular frequency  is irrelevant to the attenuation coefficient  and the initial amplitude A of the signal. Considering calculation simplicity, we assume n e  =1 and A =1, and Eq. (7) According to Section 2.2, Teager energy operator can enhance the energy of micro impulses in fault signal. However, its output is consistently positive, which changes the amplitude and frequency distribution of the micro shock signal. Therefore, in order to keep the characteristic of the original signal, the demodulated signal of the complete Teager energy operator is

Simulation and validation
When the bearing of large-scale mechanical equipment has pitting corrosion or defect fault, periodic impact signal will be generated due to the impact between parts [15]. By analyzing the vibration signal captured by sensor, it can be found that the shock signal caused by the fault presents exponential attenuation sine property. In order to verify the validity of the proposed complete TEO method, in this section, a typical simulated fault excitation signal of rolling bearing is employed for analysis. The simulated signal () xt is composed of periodic impulse signal () st and additive white Gaussian noise () nt .  To improve the SNR of shock characterization, Teager energy operator is introduced to enhance the impact components in the signal. In order to evaluate the advantages of the Teager energy operator method, to start with, we use existing Teager energy operator to process the pure shock signal in Fig.2, and the result is illustrated in Fig. 4. The original signal and the energy signal produce the curves shown in red and blue in Fig. 4 (a), respectively. It can be found that the amplitude of preprocessed energy signal has been improved. Then, the envelope spectrum of Teager energy signal is processed, and the results are shown in Fig. 4 (b). It can be observed in Fig. 4 (b) that the spectral amplitudes of fundamental frequency (100Hz) and its multiplications representing fault characteristics are slightly larger than those in Fig. 3 (a). After that, the proposed CTEO is applied to analyze the raw impulse signal in Fig. 2, and the result is shown in Fig. 5 (a). By comparison, the amplitude of the energy signal preprocessed by complete Teager energy operator is improved remarkably, and the energy signal has positive and negative properties. Fig. 5 (b) shows the Hilbert envelope spectrum of CTEO signal. The spectral amplitudes of fundamental frequency and harmonic frequencies are obviously larger than those in Fig. 3 (a) and Fig. 4 (a). However, in application of available engineering, the impact signal is mixed with abundant strong background noise, as shown in Fig.2. To verify the validation of signal enhancement, the conventional TEO and the complete TEO are applied to process the mixed signal, and the results are demonstrated in Fig. 6(a) and Fig. 7(a) respectively. As shown in Fig. 6 (a), the amplitude of the energy waveform of the conventional TEO method is always positive, which is introduced in Eq. (8). From the perspective of time domain, that is, the preprocessed signal has great change with the original signal, and its frequency distribution will also change accordingly. Fig. 7 (a) describes the complete Teager energy waveform. The information is consistent with the positive and negative directions of the original signal. In other words, only the amplitude of the impact component of the original signal is enhanced without changing its directivity, and the frequency distribution is consistent with the original signal. In order to quantitatively explain the effectiveness of CTEO, the envelope spectrum of Fig. 6 (a) and Fig. 7 (a) are shown in Fig. 6 (b) and Fig. 7 (b) respectively. In Fig. 3 (b), the fault characteristic frequency i f (100Hz) is identified hardly and buried in the noise. Although the dual frequency component 2 i f can be barely observed, the 3 i f and 4 i f (300Hz and 400Hz) cannot be clearly identified. As shown in Fig. 6 (b), the energy of i f and its frequency harmonics are significantly improved compared with those in Fig. 3 (b). However, the amplitude of uncorrelated interference items also increase. In Fig. 7   The proposed CTEO method demonstrates its superiorities through these comparison results in impulse signal enhancement. It achieves much better effect for weak signal preprocessing than the conventional TEO method. To exclude the possibility of obtaining the above results due to specific simulation signal, the common platform bearing vibration signal analysis would be conducted in the next subsection.

Experiment result and discussion
In this section, the feasibility and effectiveness of the CTEO method are evaluated by analyzing the experimental data provided by Case Western Reserve University (CWRU) and the engineering signal. Furthermore, the comparisons are carried out to evaluate the advantages of the proposed method.

Common experimental signal processing
The experimental signals coming from the bearing fault test-bed of Case Western Reserve University (CWRU) are introduced to verify the validities of the proposed method [16][17]. As shown in Fig.8, the bearing fault test rig mainly consists of an induction motor, a torque transducer, a dynamometer, and several units. In the experiment, the vibration signals of rolling bearing (SKF6205) outer ring fault were collected for analysis. The signals are collected form electrical discharge machining single point damage with 0.1734 mm pitting diameter to simulate micro fault of bearing outer ring(including micro pulse component). The sampling frequency is 12 kHz, the sampling points are 4096, and the motor speed is 1725 r/min. According to Ref. [18], the theoretical calculation of bearing outer ring fault frequency is 3.5948 *1725/60 =103.364 Hz. The conventional TEO and the proposed TEO are used to process the raw signal of the bearing out race fault in Fig. 9 (a). The processing results are shown in Fig. 9 (b) and Fig. 9 (c), respectively. Fig. 9 (b) illustrates that the conventional TEO has good enhancement effect on the shock components in the vibration signal, but the amplitudes of preprocessing signal are all positive. As shown in Fig. 9 (c), CTEO outperforms conventional TEO in terms of amplitude, especially the directivity consistent with the original signal. In order to quantitatively describe the enhancement effect of the two Teager energy operators, the envelope spectrums of raw signal and two energy signals are presented in Fig. 10(a)-(c), respectively.
As shown in Fig. 10(

Application
The experimental results of the above available public data show that the method proposed in this paper is effective for the measured test set. To further verify the effectiveness of the proposed method in the actual test-bed, in this section, the weak impulse signal excitated by adjusting the bearing capacity and speed of the fault diagnosis simulator is processed. The experimental bench is shown in Fig. 11. The pitch diameter of the bearing used in the test platform is 39.5mm. The diameter and the number of rolling elements are 7.5mm and 12 respectively .The radial load force is set as 4000N.
Under heavy load, the rotating speed is 150 r / min, and the sampling frequency is 10 kHz. According to the bearing specification and rotating speed, the fault characteristic frequency of the inner race is calculated and shown in Table 1.   Fig. 12(a) and Fig. 13(a) show the time-domain waveform and envelope spectrum of the raw bearing inner race fault vibration signal, respectively. As shown in Fig. 13(a), the amplitude of the rotating frequency r f (2.5Hz) is 0.06793, the amplitude of the fault characteristic frequency i f (12.2Hz) is 0.0766, and its dual multiplication (22.4Hz) amplitude is 0.05015. Although the fault frequency i f and its harmonics can be identified (2 i f is low-amplitude and 3 i f can not be clearly detected), the amplitudes of uncorrelated interference items (such as 22.6 Hz and 30 Hz) are relatively large in the spectrum. The conventional TEO and complete TEO signals are presented in Fig. 12(b)-(c), and the corresponding Hilbert envelope spectrums are demonstrated in Fig. 13(b)-(c). As shown in Fig.   12(b)-(c), the impact components in the vibration signal have received good enhancement under the representation of the two energy operators. In Fig. 13 In summary, this case further demonstrates the effectiveness of the proposed method to improve impulse components of the signal and restrain the other items. The signal energy representation is the accurate value of the squared product of its instantaneous amplitude (A) and the instantaneous frequency (Ω). Complementarily, the method maintains the positive and negative distribution of the shock signal ( [] n x ), effectively enhances the fundamental frequency energy of the shock signal, and provides a new approach to improving the signal-to-noise ratio of the weak shock signal.

Conclusion
In this paper, an impulses enhancement method, named complete Teager energy operator is proposed based on the traditional Teager energy operator. The method focuses on weak feature enhancement of transient signal in low amplitude. The validity and stability of the proposed method are evaluated by simulation signal analysis, common fault signal testing and fault signal of low speed machinery with heavy load. The proposed method demonstrated better performance than the traditional Teager energy operator in enhancing the efficient impulse components of the signal.

Availability of data and material
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.