## 2 Model of Fault Diagnosis System  
In the condition monitoring and fault diagnosis of rotating machinery, the signal measured by the sensor is usually a linear mixture of multiple original signals. The mixing process is shown in Figure 1.  
Suppose the signal vector observed from \(m\) sensors is \(x(t)=\left[x_{1}(t)x_{2}(t)\cdots x_{m}(t)\right]^{T}\), and the vibration signal vector of \(n\) signal sources is \(s(t)=\left[s_{1}(t)s_{2}(t)\cdots s_{n}(t)\right]^{T}\), where \(x_{i}(t)\) represents the signal value observed by the \(i(i=1,2,\cdots,m)\) sensors at time \(t\), while \(s_{i}(t)\) represents the vibration signal value of the \(i(i=1,2,\cdots,n)\) signal sources at time \(t\). The output signal of each sensor can be expressed as follows [41]:  
\[x_{i}=\sum_{j=1}^{n}a_{ij}s_{j}(t)+v_{i}(t);i=1,2,3,\cdots,m \tag{1}\]  
where \(a_{ij}\) is the mixing coefficient, and \(v_{i}(t)\) is the noise signal.  
Its matrix form can be expressed as follows:  
\[\left(\begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\\ \vdots\\ x_{m}\end{array}\right)=\left(\begin{array}{ccc}a_{11}&\cdots&a_{1n}\\ \vdots&\vdots&\vdots\\ a_{m1}&\cdots&a_{mm}\\ \end{array}\right)\left(\begin{array}{c}s_{1}\\ s_{2}\\ s_{3}\\ \vdots\\ s_{n}\end{array}\right)+\left(\begin{array}{c}v_{1}\\ v_{2}\\ v_{3}\\ \vdots\\ v_{n}\end{array}\right) \tag{2}\]  
which is  
\[x(t)=As(t)+v(t), \tag{3}\]  
where \(A\in R^{m\times n}\) is an unknown full-rank mixed matrix; \(s(t)\) is an \(n\)-dimensional source vector; \(v(t)\) is an additive noise vector, and its statistics are independent.  
The purpose of blind source separation is to find a separation matrix \(W\), so that the result of Equation (4) is the optimal estimate of \(s(t)\):  
\[y(t)=Wx(t). \tag{4}\]  
To separate the source signal is to find the separation matrix \(U\) and make it meet the following formula:  
\[\widehat{S}(t)=UX(t)=UAS\overline{S}(t), \tag{5}\]  
if  
\[UA=I. \tag{6}\]  
If \(I\) is a unit matrix, then \(\widehat{S}(t)\) is an effective separation of \(\widetilde{S}(t)\).  
However, the structure of key parts of the rotating machinery is complex: there are many sources of vibration excitation, the actual propagation path of the vibration is complicated, the equipment interferes with each other, and the environmental interference is noisy. The measured signal is often  
Figure 1: Diagram of multisource multi-sensor testing procedure.  
a result of multiple vibration sources overlapping with each other. In a linear convolutional hybrid model, the observation vector can be expressed as follows:  
\[x(t)=A(t)*s(t)+v(t). \tag{7}\]  
The convolution in the time domain is equal to the product in the frequency domain, and the convolution mixture in the time domain is converted into an instantaneous mixture in the frequency domain. Its model can be expressed as follows:  
\[x(f)=A(f)s(f)+v(f). \tag{8}\]  
The two are unified in form. Therefore, no matter how the signals are mixed, they can be processed in the same way.  
In this article, unless otherwise stated, the following assumptions are made:  
1. The mixed matrix \(A\in R^{m\times n}\) is full rank;
2. The source signal is a zero-mean, spatially uncorrelated, but time-correlated signal;
3. The source signal is a stationary process, i.e., \[E\Big{[}s(t+\tau)s^{T}(t)\Big{]}=diag[\rho_{1}(\tau),\cdots,\rho_{n}(\tau)].\] (9)