## 2 Novel Pre-Whitening Method and S Transform  
### Novel Pre-Whitening Method  
The vibration signal acquired from a gearbox is usually complex, containing many different signal components. Generally speaking, deterministic components (also called discrete frequency) represent signals produced by gear meshing, shaft bending, shaft misalignment, etc. However, random components, also called PIS, usually represent signals produced by gear or bearing faults. Traditionally, many scholars represent random components as signals of bearing faults. In this sense, when a gear has faults, such as a chipped tooth, a cracked tooth and wear of the tooth face, the meshingwill change. Every time the faulty tooth meshes, it produces impulse signals. If the gear has severe faults, the transmission ratio will vary also. Impulse signals induced by the faulty gear are very different with the signals induced by the normal gear. The PIS produced by the faulty gear also has a random property, such as the faulty bearing, because the gear faults usually produce impulse vibration signals. Therefore, signals produced by the gear fault also belong to the random components. It is now well established that the impact produced by the local gear fault is non-stationary in nature, and conventional signal processing approaches are inappropriate; and local tooth damage (i.e., fatigue crack, pitting, etc.) produces sharp transients in the vibration signature, which can be classified as non-stationary, non-linear and non-Gaussian in nature [20]. In this content, signals collected from the gearbox can be represented as:  
\[x(t)=u(t)+f(t) \tag{1}\]  
where \(u(t)\) denotes the deterministic components and \(f(t)\) denotes random components induced by the gear or bearing fault. This is under the assumption that the rotating speed is constant or can be transformed to be constant in the angular domain by the assistance of the tachometer signal. The envelope signal can be represented as:  
\[y(t)=x(t)+j\hat{x}(t)=u(t)+f(t)+j\Big{[}\hat{u}(t)+\hat{f}(t)\Big{]} \tag{2}\]  
Because the squared envelope has a bigger SNR than the envelope alone, the squared envelope of the hybrid signal can be calculated as:  
\[|y(t)|^{2}=x(t)^{2}+\hat{x}(t)^{2} \tag{3}\]  
\[|y(t)|^{2}=u(t)^{2}+f(t)^{2}+2u(t)f(t)+\hat{u}(t)^{2}+\hat{f}(t)^{2}+2\hat{u}( t)\hat{f}(t) \tag{4}\]  
Then, [PERSON] et al. [24] derived the analytic forms of the squared envelope and the squared envelope spectrum. They found that the primary energy of the multi-component signal was shifted to the direct current (DC) offset of the envelope signal. Both the deterministic component and the random component are reserved in the envelope of the multi-component signal when the DC offset has not been cancelled. If the DC offset is cancelled, the cross-terms of different mono-components would dominate the envelope. However, the cross-terms of the deterministic component have one fewer harmonic than the original ones. In addition, all harmonics of cross-terms between deterministic and random component shift to low band direction with different distances. That is to say, one DC offset elimination operation can subtract one harmonic of the cross-terms of the deterministic component from the envelope signal. If we repeat this process, the deterministic component would be suppressed efficiently. According to this theory, they proposed a novel deterministic component cancellation method. The detail procedure is illustrated in Figure 1.  
Figure 1: Framework for the deterministic component cancellation method. According to [24], the detailed process can be depicted as follows:  
1. [label=(0)]
2. Cancel the DC offset from the original multi-component signal \(x(t)\), and calculate its envelope denoted as \(\mathrm{H}^{k}[x(t)]\); \(k=1,2,\ldots,\mathrm{H}^{k}[x(t)]\) represent the \(k\)-th envelope obtained by Hilbert transform. Specifically, \(\mathrm{H}^{\tau}[x(t)]=x(t)\). 3.