## 1 Introduction  
Rolling bearings are widely used mechanical parts which are often seen in high-speed, high-load, and other complex conditions [1]. They are also one of the most fault-prone parts. The operation of rolling bearings is easily disturbed by the environment. The collected signal characteristic components can easily be drowned out by strong background noise, especially in the early stage of failure. The fault diagnosis method based on vibration signal analysis is one of the most widely used and effective means to detect rolling bearing faults [2]. Employing an effective signal processing algorithm to extract the fault characteristics of the signal from the original signal with strong interference is key in the effective fault diagnosis of rolling bearings. In modern signal processing techniques, the most commonly used rotating machinery fault diagnosis techniques include empirical mode decomposition [3], wavelet transform [4], singular value decomposition [5], stochastic resonance [6], Principal Component Analysis (PCA) [7], Relative Principal Component Analysis [8], and advanced machine learning models [9, 10]. However, they are limited in some respects; for instance, they lack adaptive signal processing. An adaptive time-frequency analysis method called intrinsic time decomposition (ITD) was proposed by [PERSON] et al. [11], and is now widely utilized in the fault diagnosis of rotating machinery. ITD does not require envelope fitting iterations compared to EMD and its extension algorithm, and only linear interpolation is performed, so the operation is faster, and the modal mixing is improved compared to EMD and its extension algorithm [12]. In recent years, researchers have made many improvements to the ITD algorithm. A diesel engine fault identification method based on fully integrated intrinsic time-scale decomposition was proposed by [PERSON] et al. [13]. [PERSON] et al. investigated the decomposition of gearbox vibration signal using the ITD method [14]. [PERSON] et al. proposed a new approach, the time-scale decomposition of set cost sign, improving the envelope fitting method of ITD, which could prevent signal distortion [15]. [PERSON] et al. utilized intrinsic time-scale decomposition to analyze diesel engine signals [16]. [PERSON] et al. proposed the complete ensemble improved intrinsic time-scale decomposition (CEIITD) and bi-spectrum algorithms [17]. A new approach called the sparse coding shrinkage method was proposed by [PERSON] et al. [18]. This method uses intrinsic time-scale decomposition as a sparse representation to extract pulse components from bearing vibration signals. An improved intrinsic time-scale decomposition (IITD) was proposed by [PERSON] et al. [19]. [PERSON] et al. proposed a new method to extract ship-radiated noise features using the ITD method [20]. [PERSON] et al. proposed a method based on wavelet transform and added it to the iterative process of intrinsic time-scale decomposition to improve the signal-to-noise ratio of the decomposed signal [21]. At present, the envelope method of ITD is fixed throughout the decomposition process. However, the waveform and fluctuation trend of the maximum and minimum mode components obtained by each decomposition may be completely different from that of the original signal. It is, therefore, best to allow the utilization of different envelope interpolation algorithms during the generation of each rotating component. An envelope fitting method that could obtain better decomposition results for all signals is still needed. Given these factors, a novel decomposition method called improved intrinsic time-scale decomposition (IITD) was designed and applied to the complete ensemble intrinsic time-scale with adaptive noise (CEITDAN) framework. This method combines the advantages of several common envelope fitting methods, and is different from the established method using a fixed envelope interpolation algorithm.