## 1 Introduction  
For the diagnosis of mechanical faults, vibration signals always contain a wealth of information as they indicate the operating status of the equipment, with specific physical meanings. There is no doubt that fault diagnosis technology, based on vibration signal processing, is critical for monitoring the health of key structures or equipment [1; 2; 3; 4; 5]. Generally, different sensors are utilized to obtain mechanical vibration information, from which the features of the running state are characterized [6; 7; 8]. In the practical production environment or industrial field, the fault of a certain part in mechanical equipment is always accompanied by other faults--for instance, faults often occur in the bearings and gears simultaneously. As a result, the measured vibration signals are always overwhelmed by signals from multi-fault vibration sources and other measurement noise [9]. Consequently, there should be close attention paid to the theory of multi-fault signal analysis. Blind source separation (BSS) is one of the effective methods for solving the compound-fault problems as mentioned above [10; 11]. It can be used to separate or recover the unknown source signals through the observed signals in cases where the source signals cannot be acquired accurately [12; 13; 14]. In the fault monitoring of mechanical equipment, the obtained vibration signals of gear and roll bearing from the complex transmission system are often produced by cross-interference. In many cases, only one sensor can be installed, due to the high cost of the hardware and the space limitation. Therefore, research on separation methods of single-channel compound-faults would have extensively practical significance. Single-Channel Blind Source Separation (SCBSS) is a special case of blind source separation, which only requires a single sensor to separate multiple source signals [15; 16]. Compared with the classical BSS method, the SCBSS method only satisfies the condition when the number of source signals is more than the number of observed signals [17; 18; 19]. Therefore, a novel method should be developed to achieve the BSS under the specific condition of a single channel. In the theoretic research of compound-fault diagnosis under a single channel, the main focus falls on methods of virtual multi-channels. The space-time method was first proposed by [PERSON] [20], with its key idea relying on delaying the single-channel mixed signal to obtain the virtual multi-channel signal, which was thereafter processed by Independent Component Analysis (ICA) [21; 22; 23; 24; 25; 26; 27; 28]. Similarly, the wavelet decomposition of the original signal was performed to decompose the single-channel signal into multiple sub-band signals [29]. After that, the Ensemble Empirical Mode Decomposition (EEMD) was used to decompose the single-channel mixed signal into a series of Intrinsic Mode Functions (IMFs) [30], followed by the ICA separation for recovering the original signals [31]. To date, EMD algorithm and its modified versions have been widely used in signal processing due to its advantages of orthogonality, completeness and adaptability. As a result, these algorithms demonstrate a great superiority in the analysis of non-stationary signals [32]. The vibration signals can be decomposed into a series of IMFs. Nevertheless, its decomposition is not stable and there is a mode mixing problem (MMP). This MMP results in a certain IMF component containing different scale signals or similar scale signals existing in different IMFs, which makes it difficult to completely achieve the signal adaptive decomposition based on EMD [33; 34]. To solve this, the EEMD was proposed to alleviate this so-called \"mode mixing\" phenomenon with a noise-assisted version [35; 36; 37]. Considering the uniform distribution of white noise power spectral density, different auxiliary white noise was added to the original signal, so that the signal was continuous at different scales.