## 1 Introduction  
In the field of mechanical fault diagnosis, vibration signals always contain a wealth of information about equipment operating status. Thus, a powerful signal processing method is necessary to extract the possible faults [1; 2; 3; 4]. Generally, various sensors are used to obtain the vibration signals of the mechanical equipment. Characteristic information, such as fault feature frequencies, can be extracted from the obtained vibration signals [5; 6]. However, one mechanical fault is usually accompanied by other faults. For example, simultaneous gear fault and bearing fault are common in a damageddecelerator. Therefore, the acquired signal is generally coupled by multiple fault signals along with the background noise, which brings out a consequence that the characteristics of the fault component cannot be directly identified. As an effective approach to solve the problem of complex multiple faults, blind source separation (BSS) can be used to separate the linear mixtures of different unknown source signals [7; 8; 9; 10]. Since the limitation by the cost of equipment, installation conditions and others cases, the measurement scheme using a single sensor is generally considered. Consequently, we can only obtain single channel complex multiple faults signals. Therefore, the research on the fault diagnosis method of rotating machinery under single channel condition has a very wide range of engineering applications. The single channel blind source separation (SCBSS) [11], which can separate each source signal from the collected composite signals obtained by single sensor, is a special case in BSS. However, compared with BSS, there is a serious problem that the number of source signals is not less than the number of observation signals in SCBSS. Hence, the signal decomposition is needed to achieve the SCBSS. In the study of complex fault diagnosis under single channel condition, the general solution to this problem is not unique and various approaches have been proposed, ranging from applying independence assumptions to non-negativity and sparsity constraints [12]. Currently, research in this area mainly focuses on the virtual multi-channel method. The space-time method was first proposed by [PERSON] and [PERSON] [13]. After obtaining a virtual multi-channel signals by delaying the single mixed observation signal, the independent component analysis (ICA) [14] algorithm was utilized to separate the source signals from the obtained virtual multi-channel signals. [PERSON] [15] applied wavelet decomposition to the single-channel signal and a virtual multi-channel signals using sub-frequency band signals was obtained, followed by employing the ICA method. [PERSON] et al. [16] proposed the Ensemble Empirical Mode Decomposition (EEMD) [17] to decompose the mixed single channel signal into a plurality of intrinsic mode functions (IMFs). Moreover, [PERSON] et al. [18] proposed a new method to achieve the separation of complex fault signals by combining with the EEMD and the ICA method. [PERSON] et al. [19] also discovered that the EEMD-ICA method can reduce the dimension to solve the single channel separation problem. [PERSON] et al. [20] applied the EMD-ICA method to the simulation research of bearings and gears with mixed faults. The common character of SCBSS method is based on a virtual multi-channel signal, which is constructed as the input data of the separation algorithm, thus we can obtain a better separation effect. However, the constructed multi-channel signal by the above-mentioned method is difficult to maintain the characteristics of the observed signal, and it may be interfered by the noise or other components. Hence, the frequency domain characteristics of separated signals may be distorted, and a good separation effect may not be achieved. The above methods mainly use EMD or the improved EMD to construct virtual multi-channel signals, which can transform underdetermined condition to positive definite condition in BSS.