## 2 The Theory of the Proposed Method
### Spectral Subtraction Method  
The spectral subtraction method is widely employed in signal processing for speech enhancement in order to remove acoustic noise [31; 32; 33]. Spectral subtraction is usually used as a denoising method for fault detection. The proposed technique is appropriate for the fault diagnosis of rotating machinery in steady state at constant speed. It is only applied to stationary signals, which means the frequency content is independent of time. In this context, a new denoising scheme is employed to enhance the useful information of the vibration signals measured from a faulty gearbox. It is assumed that the vibration signal is disturbed by random noise, which can be removed by subtracting of an estimation of the mean of the noise spectrum from the noisy signal spectrum. The measured signal model in the time domain is given by:  
\[z(m)=s(m)+n(m) \tag{1}\]  
Figure 1: Flow chart of the fault diagnosis method.  
where \(z(m)\) is the measured mixed-signal (including the vibration signal and noise signal), \(s(m)\) is the vibration signal of the gear, \(n(m)\) is the additive noise, and \(m\) is the discrete-time index. The frequency-domain representation of the relation is given by:  
\[Z(k)=\mathcal{F}[z(m)]=S(k)+N(k) \tag{2}\]  
where \(Z(k)\), \(S(k)\), and \(N(k)\) are the discrete Fourier transforms of \(z(m)\), \(s(m)\), and \(n(m)\), respectively. \(k\) is the discrete-frequency index.  
In the spectral subtraction method, the estimation of the original signal power spectrum can be obtained with the variation of parameters. It includes an over-subtracting estimate of the noise power spectrum and a lower limiting value for the resulting spectrum. The form of the enhanced signal power spectrum is:  
\[\left|\overset{\wedge}{S}(k)\right|_{2}=\max\left\{\left|Z(k)\right|_{2}-a \left|\overset{\wedge}{N}(k)\right|_{2},\beta\left|\overset{\wedge}{N}(k) \right|_{2}\right\} \tag{3}\]  
where \(\alpha\) is the over-subtracting multiplication factor, which has a value greater than or equal to 1 and determines the degree of distortion of the fault signal. \(\beta\) is the spectral flooring parameter, which has a value between 0 and 1, and controls the effect of residual noise. The multiplication factor \(\alpha\) in Equation (3) is defined by the function of the segmental signal-noise ratio (SNR), which is estimated in the frame and defined by:  
\[\alpha=\left\{\begin{matrix}5&SNR<-5\\ 4-\frac{3}{20}SNR&-5\leq SNR\leq 20\\ 1&SNR>20\end{matrix}\right. \tag{4}\]  
\[SNR=10\log 10\left(\begin{matrix}\overset{L-1}{\underset{k=0}{\overset{L-1}{ \sum}}}\left|Z(k)\right|^{2}\\ \overset{L-1}{\underset{k=0}{\overset{L-1}{\sum}}}\left|N(k)\right|^{2}\end{matrix}\right) \tag{5}\]  
Finally, the enhanced signal is obtained by the inverse discrete Fourier transform (IDFT):  
\[\overset{\wedge}{S}(m)=\mathcal{F}^{-1}\left\{\left|\overset{\wedge}{S}(k) \right|e^{i\phi(k)}\right\} \tag{6}\]  
where \(\phi(k)\) is the phase function of the DFT of the input signal.  
According to the basic theory of spectral subtraction, the vibration noise is estimated over the whole spectrum by adjusting the multiplication factor \(\alpha\)[34]. By using the spectral subtract technique, the faulty signal is enhanced. However, it cannot extract the mono-component information. To identify the side-band frequency feature, EWT is used to analyze the faulty signal.