## 1 Introduction
Although improved to a certain extent, the envelope construction issue is still one of the three main factors affecting the decomposition results. In response to the above problems, scholars have carried out a series of research work to improve the performance of LMD through the proper improvements in the boundary condition, envelope estimation function, and sifting-stopping criterion. To allow LMD to accomplish a self-adaptive halt for each sifting operation, [PERSON] [18] created a soft sifting-stopping criterion. [PERSON] [19] created a brand-new signal waveform extension technique to eliminate LMD's aftereffects. To reduce the mode mixing of LMD, [PERSON] [20] created the differential rational spline-based LMD approach. To lessen the mode mixing of LMD, [PERSON] [21] introduced the ensemble LMD (ELMD). In the iterative process of LMD, the accuracy of the envelope estimation function is significantly affected by the precision of its construction method. In the original LMD, this function is gained by the moving average method, while this method may cause phase error of the envelope estimation function [22]. The authors of [23], who were inspired by EMD, computed the envelope estimation function using the cubic spline interpolation (CSI) envelopes of local maximum points and local minimum points, even though these envelopes could lead to overshoot and undershoot issues for strong nonstationary signals. The envelope estimate function was calculated using monotonic piecewise cubic Hermite interpolation (MPCHI) in [24], which has good flexibility and can address the overshoot and undershoot problems brought on by CSI. The authors of [2] also presented a compound interpolation envelope algorithm to match the local properties of the signal. Several other interpolation methods, such as the B-spline, Lagrange, and Newton polynomials, have also been further proposed to improve the performance of the envelope function [25]. Based on the investigation and research of typical single-component signals with physical significance at instantaneous frequency, [PERSON] proposed a relatively novel time-frequency analysis method, local characteristic scale decomposition (LCD), in 2012. Compared with EMD, the iterative process of LCD is calculated according to the eigenscale parameters of the signal itself, which has the characteristics of high efficiency and avoidsthe problems of over-envelope and under-envelope in the process of EMD decomposition. However, the method still has the problem of mode aliasing. In general, the envelopes constructed by different interpolation methods have their unique advantages. For example, CSI is more suitable for the construction of stationary signal envelopes, and MPCHI is appropriate for the construction of strongly nonstationary signal envelopes. The actual vibration signal generally has the property of multi-component coupling, and it may be composed of components with stationary and nonstationary properties; hence, LMD decomposition needs to process multiple PF components with different characteristics [26]. Applying one single interpolation method to construct the envelopes for all PF components in the iteration process is not optimal. Therefore, this paper proposes a double interpolation-based LMD. By setting reasonable judgment conditions, the optimal interpolation method is selected for the envelope of each PF component independently. In the sifting iterations of each order PF component, the stability of the envelope estimation function is also significantly affected by the precision of its envelope construction method. Different from the iterative subtraction process of the intrinsic mode function (IMF) in EMD, the decomposition of the PF component is an iterative division process in LMD, in which the residue signal is the numerator and the envelope estimation function is the denominator. When the numerator is small enough, a tiny error of the denominator may cause a local mutation phenomenon of this demodulation result, which then leads to the non-convergence of the whole decomposition process.