## 1 Introduction
These four methods all have been used in the separation of the deterministic signal and PIS signal successfully. Except for these methods, a new deterministic component cancellation method was proposed, recently [24]. It is based on an iterative calculation of the signal envelope when the DC offset is cancelled before envelope calculation. The only parameter that needs to be pre-determined is the iteration number for DC offset cancellation. In the paper, a threshold of cross-correlation coefficient interval of [0.9, 0.95] was selected to judge the proper number of envelope calculations. The selected correlation coefficient interval was efficient for the examples in the paper, but this threshold method is not a proper way to acquire the most powerful PIS. Sometimes, a greater iteration number will distort the waveform of PIS and lead to the misdiagnosis of faults. In order to resolve this problem, correlated kurtosis (CK) is proposed to be used in determining when to stop the iteration for envelope calculation [25]. This indicator can detect the PIS properly. However, CK is efficient only with strong enough PIS. When the PIS is very weak, CK is inefficient due to the inability to correctly detect the PIS. Therefore, in the case of very weak PIS, it is very difficult to determine the appropriate iteration number for envelope calculation. Time frequency analysis methods, such as short time Fourier transform, the Wigner-Ville distribution and wavelet transform, were widely used in rotating machinery fault diagnosis. However, they all have some demerits. Compared to these time frequency analysis methods, the S transform can greatly avoid the existing deficiencies of the above three methods [26]. Therefore, it can be used to display the time-frequency characteristics of the separation results of the deterministic component and the random component. In addition, fault features can be extracted from the resulting signal of the S transform to track the degradation of the component of interest. From the time-frequency contour produced by the S transform, one can distinguish the different faults more easily; or it can be used as the input for the deep learning methods to intelligent fault classification. The above discussions show the need to use CK to optimize the iteration number of the envelope calculation with strong enough PIS. This can suppress the deterministic component effectively and properly. The S transform advantages inspire us to diagnose faults of the gearbox component by separation of the results of the deterministic component and the random component. The method is validated using the experimental datasets collected from a lab AC motor faulty bearing; a lab fixed-axis gearbox faulty bearing and gear and a lab planetary gearbox faulty sun, planet and ring gear. Experimental case studies will benefit the application of this novel method and will also provide a detailed guide for engineers in the industrial application. In addition, validation has been conducted using both intentionally-implanted bearing and gear damage data and naturally-developed distributed gear and bearing wear data under constant motor speed and various loads.