Blind source separation of rotor vibration signals in high-noise environments

Abstract

During the operation of the engine rotor, the vibration signal measured by the sensor is the mixed signal of each vibration source, and contains strong noise at the same time. In this paper, a new separation method for mixed vibration signals in strong noise environment (such as SNR=-5dB) is proposed. Firstly, the time-delay autocorrelation de-noising method is used to de-noise the mixed signals. Secondly, one common algorithm (the MSNR algorithm is used here) is adopted to separate the mixed vibration signals, which can improve the separation performance. The simulation results verify the validity of the method. The proposed method provides a new idea for health monitoring and fault diagnosis of engine rotor vibration signals.

Keywords: Blind source separation, rotor, vibration signals, auto-correlation de-noising, high-noise environments

1. Introduction

During the operation of rotating machinery, the changes of physical parameters such as vibration and noise will inevitably occur. These changes are often the early fault factors leading to engine failure. The vibration signal measured by the sensor installed on the rotating machinery is a mixture of several vibration signals. How to analyze, process and identify these signals is very important for judging the working state of rotating machinery and fault diagnosis. Various traditional modern signal processing methods, such as Fourier transform, short-time Fourier transform and wavelet transform, have been widely used in vibration signal analysis. However, for mixed vibration signals in rotating machinery, the above analysis methods have obvious shortcomings, and it is difficult to separate or extract source signals independently.

Blind source separation (BSS) technology can separate multiple mixed signals, and the separated output signal will not lose the weak feature information in the source signal. The seminal work on BSS is by Jutten and Heraultin 1985 [1], the problem is to extract the underlying source signals from a set of mixtures, where the mixing matrix is unknown. In other words, BSS seeks to recover original source signals from their mixtures without any prior information on the sources or the parameters of the mixtures. Its research results have been widely applied in many fields, such as speech recognition, wireless communication,biomedicine, image processing, vibration signals separation, and so on [2-5].

There have been many effective and distinctive BSS algorithms, including fast fixed-point algorithm, natural gradient algorithm, EASI algorithm and JADE algorithm. When separating noiseless mixed signals, these algorithms show good separation performance. However, when the signal-to-noise ratio of the noisy signal is very low, the separation performance will become very poor, because these algorithms are derived without considering the noise model. Noise is ubiquitous, its existence not only has a serious impact on the normal work of the system, but also affects the normal measurement of useful signals. In signal processing, in order to retain useful signals, people always try their best to remove background noise. So the research of signal detection, especially the extraction and detection of weak signals submerged in strong noise, is a common problem that many engineering applications face and need to solve urgently.

In the process of machine operation, the vibration signal measured by means of vibration sensor will inevitably contain the noise signal. When the BSS algorithm is used to separate the mixed vibration signals directly, it may cause great errors or draw wrong conclusions. Therefore, the noise reduction is particularly important before blind separation of mechanical vibration signals.

Many scholars have used the combination of wavelet de-noising and BSS to separate mixed signals in noisy environment, and achieved some results. However, the wavelet de-noising method needs to set a threshold value, which may remove the weak signals of useful components in the mixed signals, leading to wrong separation results. Time-delay autocorrelation de-noising method is widely used in the de-noising of rotor vibration signals, and it does not lose useful components in the de-noising process.

Nowadays, there have been lots of BSS algorithms to calculate a de-mixing matrix, so we can make the estimated source signal only by the received signal. In this paper we select and optimize the BSS algorithm based on maximum signal-to-noise ratio (MSNR) [6]. It has very low computational complexity because de-mixing matrix can be achieved without any iteration.

In this paper, the time-delay autocorrelation method is used to de-noise the noisy mixed signal, and then the MSNR algorithm is used to separate the de-noised mixed signal. The separation effect is further improved.

The rest of the paper is organized as follows. In Section 2, we introduce the noisy signal BSS model and principle of the time-delay auto-correlation method, the improved MSNR algorithm is summarized in the end. In Section 3, the simulation experiment that indicates the effectiveness of the method is presented. The final section is a summary of the content of this paper and possible application areas.

2. Methodology

2.1 Noisy signal BSS model

Source signals ${\textstyle {s_{i}(t)}}$ come from different signal sources (assuming that the signal is a continuous signal), so ${\textstyle {s_{i}(t)}}$ can be regarded as mutual statistical independence, ${\textstyle {x_{i}(t)}}$ is the mixed signals or observation signals (Figure 1).

Figure 1. Noisy blind source separation model

The problem of basic linear BSS can be expressed algebraically as follows:

$x_{i}(t)=\sum _{1}^{n}a_{ij}(s_{i}(t)+v(t))$

(1)

where ${\textstyle {a_{ij}}}$ is a mixed coefficient. Equation (1) can be written in vector as follow:

$x(t)=A(s(t)+v(t))$

(2)

where ${\textstyle {s(t)=[s_{1}(t)\cdots s_{n}(t)]^{T}}}$ is a column vector of source signals, ${\textstyle {x(t)=[x_{1}(t)\cdots x_{n}(t)]^{T}}}$ is a vector of the mixed signals or observation signals, ${\textstyle {v(t)}}$ is the additive white Gaussian noise, which is a basic noise model used in the information theory to mimic the effect of many random processes that occur in nature. ${\textstyle {n\times n}}$ is a ${\textstyle {n\times n}}$ mixing matrix. The problem of BSS only knows the observation signals and statistical independence property of the source signals. In virtue of the knowledge of probability distribution of source signals we can recover source signals. Assuming ${\textstyle {W}}$ is a ${\textstyle {}^{n\times n}}$ demixing matrix or separating matrix, the problem of BSS can be described as follows:

$y(t)=Wx(t)$

(3)

where ${\textstyle {y(t)}}$ is an estimate of or separated signals. BSS has two steps, firstly, create a cost function ${\textstyle {F(W)}}$ with respect to ${\textstyle {W}}$, if ${\textstyle {W'}}$ can make ${\textstyle {F(W)}}$ reach to maximum, ${\textstyle {W'}}$ is the demixing matrix . Secondly, find a effective iterative algorithm for the solution of ${\textstyle {\partial F/\partial W=0}}$. In this paper, the cost function is the function of signal noise ratio, optimizing the cost function results in a generalized eigenvalue problem, a demixing matrix is achieved by solving the generalized eigenvalue problem without any iteration.

2.2 MSNR Algorithm

The MSNR algorithm belongs to matrix eigenvalue decomposition method. By constructing the signal-to-noise ratio contrast function and estimating the separation matrix by the eigenvalue decomposition or generalized eigenvalue decomposition, the closed-form solution can be found directly without iterative optimization process. Therefore, it has the advantages of simple algorithm and fast running speed, and is convenient for real-time processing and hardware implementation of FPGA. The time continuous signal is sampled and changed into a discrete value. In the following equation, the time mark ${\textstyle {t}}$ becomes ${\textstyle {n}}$.

According to the model of BSS, the error ${\textstyle {e(n)=s(n)-y(n)}}$ between the source signal ${\textstyle {s(n)}}$ and the output signal ${\textstyle {y(n)}}$ is regarded as noise. When the minimum value of ${\textstyle {e(n)}}$ is taken, the estimated value ${\textstyle {y(n)}}$ is the optimal approximation of the source signal ${\textstyle {s(n)}}$, and the effect of BSS is the best. The power ratio of the source signal ${\textstyle {s(n)}}$ to the error ${\textstyle {e(n)}}$ is defined as the signal-to-noise ratio. When ${\textstyle {e(n)}}$ reaches the minimum, the signal-to-noise ratio reaches the maximum. According to the estimation criterion, the signal-to-noise ratio function is constructed as follows [6]:

$F_{SNR}=10\log {\frac {s\cdot s^{T}}{e\cdot e^{T}}}=10\log {\frac {s\cdot s^{T}}{(s-y)\cdot (s-y)^{T}}}$

(4)

Because the source signal ${\textstyle {s(n)}}$ is unknown, the mean value of noise is 0, so we use the moving average of the estimate signals ${\textstyle {{\bar {y}}(n)}}$ instead of the source signals ${\textstyle {s(n)}}$. Equation (4) can be written as follows:

$F_{SNR}=10\log {\frac {s\cdot s^{T}}{e\cdot e^{T}}}=10\log {\frac {{\bar {y}}\cdot {\bar {y}}^{T}}{({\bar {y}}-y)\cdot ({\bar {y}}-y)^{T}}}$

(5)

where ${\textstyle ^{{\bar {y}}_{i}(n)}}$ is the moving average of the estimate signals ${\textstyle {y(n)}}$. We replace ${\textstyle {{\bar {y}}(n)}}$ with ${\textstyle {}^{y(n)}}$ in the molecule of Equation (5) to simplify calculation, so we gained maximum signal noise ratio cost function as follows:

$F_{SNR}*=10\log {\frac {y\cdot y^{T}}{({\bar {y}}-y)\cdot ({\bar {y}}-y)^{T}}}$

(6)

According to Equation (3), we get the Equation (7) as follows.

${\bar {y}}(n)=W{\bar {x}}(n)$

(7)

where ${\textstyle {{\bar {x}}(n)}}$ is a moving average of the mixed signals ${\textstyle {x(n)}}$. The definition uses the moving average algorithm to predict the source signal. We substitute Equations (3) and (7) into Equations (6) and (8) is deduced as follows

${\begin{array}{l}{F_{SNR}*(W)=10\log {\frac {y\cdot y^{T}}{({\bar {y}}-y)\cdot ({\bar {y}}-y)^{T}}}=10\log {\frac {Wx\cdot x^{T}W^{T}}{W({\bar {x}}-x)\cdot ({\bar {x}}-x)^{T}W^{T}}}}\\{=10\log {\frac {WCW^{T}}{W{\bar {C}}W^{T}}}=10\log {\frac {V}{U}}}\end{array}}$

(8)

where ${\textstyle {{\bar {C}}=({\bar {x}}-x)({\bar {x}}-x)^{T}}}$ and ${\textstyle {C=xx^{T}}}$ are correlation matrices, ${\textstyle {U=W{\bar {C}}W^{T}}}$, ${\textstyle {V=WCW^{T}}}$.

2.3 Derivation of Separation Algorithms

According to Equation (8), the derivative of ${\textstyle {F_{SNR}*(W)}}$ is as follows:

${\frac {\partial F_{SNR}*(W)}{\partial W}}={\frac {2W}{V}}C-{\frac {2W}{U}}{\bar {C}}$

(9)

According to the definition, when the maximum value of the function ${\textstyle {F_{SNR}*(W)}}$ is obtained, the gradient is 0. So we get the following equation.

$WC={\frac {V}{U}}W{\bar {C}}$

(10)

We can obtain demixing matrix ${\textstyle {W'}}$ by solving the Equation (10), it has been proved that solution of the Equation (10) is the eigenvector of ${\textstyle {{\bar {C}}\cdot C^{-1}}}$ [7]. All sources can be recovered once: ${\textstyle {y=W'x}}$, where each row of ${\textstyle {y}}$ corresponds to exactly one extracted signal ${\textstyle {y_{i}}}$.

2.4 Autocorrelation De-noising

The autocorrelation function describes the relationship of the same signal at different times [8]. For the signal ${\textstyle {x(t)}}$, its autocorrelation function is defined as follows:

$R_{x}(\tau )=\lim _{T\to \infty }{\frac {1}{T}}\int _{0}^{T}x(t)x(t+\tau )dt$

(11)

where ${\textstyle {\tau }}$ is the time delay of the autocorrelation function, ${\textstyle {T}}$ is the period of the signal. Equation (11) shows that the autocorrelation function of the periodic signal is the same period as that of the original signal. However, the noise signals are generally uncorrelated. When the time delay is zero, the maximum autocorrelation value is obtained and tends to zero with the increase of the time delay. Therefore, the autocorrelation function can be used in the noise reduction of mechanical vibration signal, so as to retain the useful periodic signal in the vibration signal, effectively remove the random aperiodic white Gaussian noise, and achieve remarkable noise reduction effect.

The autocorrelation function values of the white Gaussian noise and rotor vibration signal are shown in Figure 2. When the vibration periodic signal contains Gauss white noise, the autocorrelation value is the largest near this condition ${\textstyle {\tau =0}}$, which is affected by the noise. Therefore, we can remove some autocorrelation data near the condition ${\textstyle {\tau =0}}$ during removing noises.

Figure 2. Auto-correlation function values of white Gaussian noise signal and rotor vibration signals

The improved MSNR algorithm based on autocorrelation de-noising can be summarized as: (1) Finding the autocorrelation function of the noisy mixed signals ${\textstyle {x(t)}}$. (2) Removing the data near the condition ${\textstyle {\tau =0}}$ and using the remaining data ${\textstyle {{\hat {x}}(t)}}$ as the data of blind separation. (3) Blind separation of the de-noised mixed signals ${\textstyle {{\hat {x}}(t)}}$ by means of the MSNR algorithm. The improved MSNR algorithm with four lines of Matlab code are listed in Table 1.

Table 1. The improved MSNR algorithm

Input: The mixed signals ${X}$. Output: The demixing matrix ${W}$ and the separated signal .

${\textstyle Xd=}$ denoise ${X}$; % ${X}$ is denoised based on the auto-correlation de-noising. ${\textstyle XS=}$ smoothdata(${\textstyle Xd=}$,'movmean'); % Smooth ${X}$ by averaging over each window. ${\textstyle [W,d]=eig(cov(Xd-XS),cov(Xd))}$; % Demixing matrix ${W}$ is obtained from Equation (1). ${\textstyle Y=(X*W)'}$;  % Separated signal ${Y}$.

Figure 3 presents the new system model of BSS based on the above mentioned algorithm. The sequence numbers ①, ②, ③ and ④ in Figure 3 represent steps 1, 2, 3 and 4 in Table 1, respectively.

Figure 3. System model based on the improved MSNR algorithm

3. Simulations and results

In order to verify the effectiveness of the improved MSNR algorithm, two sinusoidal periodic signals with different frequencies are used to simulate the mixing of vibration signals caused by different rotors. After the original vibration signal ${\textstyle s(t)}$ is superimposed with Gaussian white noise whose signal-to-noise ratio is -5dB, the source signal completely submerged by a strong noise is more difficult to be restored and identified in the engineering fields [9]. The noisy mixed signal ${\textstyle x(t)}$ is obtained by a random mixing matrix A (such as A =[0.4684 0.1952; 0.7384 0.5483]). The number of samples N=1000. Evaluating the performance of BSS, a correlation coefficient ${\textstyle C}$ is introduced as a performance index [2].

$C(x,y)={\frac {cov(x,y)}{{\sqrt {cov(x,x)}}{\sqrt {cov(y,y)}}}}$

(12)

where $C(x,y)=0$ means that $x$ and $y$ are uncorrelated, and the signals correlation increases as $C(x,y)$ approaches unity, the signals become fully correlated as $C(x,y)$ becomes unity.

In the first simulation, the noisy mixed signals $x(t)$ are separated directly by the original MSNR algorithm, the separation results are shown in Figure 4. After separation, the correlation coefficients between the separated signals and the sources are 0.4978 and 0.4806 respectively, the separation effect is not good and it is very difficult to recognize separated signals correctly.

Figure 4. Separation of noisy mixed signals by the original MSNR algorithm (SNR=-5dB)

In the second simulation, the noisy mixed signals $x(t)$] are separated by the improved MSNR algorithm. The separation results are shown in Figure 5. After separation, the correlation coefficients between the separated signals and the sources are 0.9987 and 0.9988 respectively, the sources are well recovered and the separation effect has been significantly improved.

Figure 5. Separation of de-noised mixed signals by the improved MSNR algorithm (SNR=-5dB)

Many repeated tests can reduce the randomness and improve the reliability of results. Therefore, in order to evaluate the stability of these algorithms, total number of iterations in the present study is set to 50. The two algorithms are compared with each other from the separation accuracy (average correlation coefficient). Table 2 presents obtained values after 50 iterations.

Table 2. Average correlation coefficient for different algorithms after 50 iterations (SNR=-5dB)

Algorithm	Average correlation coefficient
MSNR algorithm [6]	0.4862
Proposed algorithm in this paper	0.9988

By comparing the two experiments, it is fully demonstrated that time-delay correlation de-noising can effectively remove noise and improve the signal-to-noise ratio, which provides the precondition for the accurate realization of BSS of noisy mixed signals.

4. Conclusion

Aiming at blind source separation of rotor vibration signals in high-noise environments, an improved MSNR algorithm is proposed in this paper. Blind separation of mixed signals with strong noise can lead to large errors or even incorrect separation results. The time-delay autocorrelation de-noising method can effectively remove the strong noise signal without losing the useful components of the original signal, which greatly improves the signal-to-noise ratio and provides the precondition for the accurate realization of blind separation. It provides a new method for separating mixed signals in strong noise environment and further expands the applicability of the MSNR algorithm. Due to its simple principle and good transplantation capability, it can be applied to the vibration signals of various mechanical rotors, such as the separation and detection of vibration signals of aero-engine and internal combustion engine.

Acknowledgments

This work was partially supported by the project of industrial-academic-research cooperation of Jiangsu province (No.BY2020241)

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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