## 1 Introduction
However, the complex vibration signal is often mixed with a lot of background noise, which makes the diagnosis very difficult. Only by effectively extracting multiple fault shock signals from noise signals and enhancing fault characteristics [32; 33], fault characteristic signals can be further separated accurately. For example, [PERSON] [34] proposed an improved matching pursuit algorithm, which successfully extracted the quantitative characteristics of bearing faults. [PERSON] [35] proposed a sparsity-enabled signal decomposition method based on resonance-based sparse signal decomposition (RSSD). According to the oscillatory behavior of the signal components, the analyzed signal can be decomposed into high and low resonance components by using the RSSD. Thus, high Q factor basis and low Q factor basis are constructed, and non-stationary signals are sparse decomposed. In the References [36; 37], the RSSD method was introduced into the fault diagnosis of gearboxes. The results showed that the sparse decomposition effect was better. Therefore, the proposed RSSD method can be applied to extract multiple impact components in compound fault diagnosis. Another important problem is how to set the input parameters appropriately for the best performance, which contains the filter length \(L\), shift order \(M\) and the deconvolution period \(T\). Among those parameters, which will affect the validation of MCKD, \(L\) and \(M\) should be set reasonably to highlight the superiority of MCKD on the base of satisfying the prime requirement of \(T\). If \(L\) and \(M\) are too large, the deconvoluted signal will be distorted. Therefore, the rigorous requirements for the parameters limit the application of MCKD. To solve this problem, the improved maximum correlation kurtosis deconvolution (IMCKD) is proposed in this paper. It can optimize the input parameters: The shift order \(M\) and the filter length \(L\). In the case of different \(M\) values, calculating the mean of Teager envelope spectrum kurtosis (MTEK) to select the optimal shift order \(M\); then, the Teager envelope spectral kurtosis (TEK) is used as the objective function to optimize the filter length \(L\) by the iterative algorithm. Based on the above discussion, a compound fault diagnosis method based on parallel dual-Q-factor and the improved MCKD is proposed in this paper. The method combines the compound fault characteristics of rolling bearings based on frequency characteristic analysis of composite signals. The parallel dual-Q-factor sparse decomposition algorithm is proposed for the compound faults of the inner and outer race of the bearing. The non-stationary signal is decomposed into the high resonance components and low resonance components (compound fault impact component and small amount of noise). Then, the multiple periodic failures of the compound fault signal can be extracted simultaneously. In order to further isolate and extract the compound faults, the low resonance component is processed by IMCKD, which is separated and extracted according to the different fault characteristics periods. The remainder of this paper is organized as follows. Section 2 briefly describes the theory of the parallel dual-Q-factors base sparse decomposition. Section 3 introduces the improved maximum correlation kurtosis deconvolution (IMCKD) for selecting the input parameters appropriately. Section 4 is devoted to descriptions of the proposed method. Section 5 is dedicated to description of application of the proposed method with the simulation and experiment signals. The results show the effectiveness and reliability on the compound faults decoupling diagnosis for rolling bearings.