## 1 Introduction
Compared with other popular heuristic methods (e.g., genetic algorithm [48], particle swarm optimization [49], differential evolution [50], and gravitational search algorithm [51]), GWO supports the distributed parallel computing and has the merit of fewer control variables, a stronger global search capability, and an excellent multi-parameter optimization performance. These merits have been applied in solving complex parameter optimization problems by many scholars [52]. Given this, we introduce GWO to search the optimal parameters (\(\hat{\alpha}\), \(K\)) in VMD, which ensures the self-adaptability and flexibility of VMD. Moreover, note that mirroring extreme extension in the preparation of VMD is adopted, which is equal to the function of boundary pretreatment. Nevertheless, mirroring extreme extension utilizes merely the extreme value point of both ends of the initial signal to accomplish the extension procedure, which cannot reflect exactly the natural trend of raw data. The waveform matching extension method presented by Hu et al. [53] is effective in suppressing the end effect, and the extension waveform obtained by this method can conform to the changing trend of the raw data as much as possible. Hence, we adopt waveform matching extension to replace mirroring extreme extension in the original VMD, which will ensure that VMD has a good boundary processing performance. As everybody knows, the demodulation of IMF after signal decomposition is the key step of the TFA tool. Hilbert transform (HT) is an effective demodulation method, but it will generate an edge error because of dewindowing and easily make the negative frequency appear. Compared with HT, the Teager energy operator (TEO) with simple computation can overcome the demodulation error of HT [54]. Hence, in this paper, TEO is selected as the assisted tool to estimate the straightly instantaneous amplitude (IA) and instantaneous frequency (IF) of the decomposition contents obtained by the parameter optimization VMD, for the purpose of obtaining a good time-frequency trajectory. Novelties and attractions in the article are that a novel TFA technique, named the improved adaptive variational mode decomposition (IAVMD), is designed for rotor fault diagnosis, which can automatically separate a non-stationary compound signal into several IMF ingredients and achieve directly time-frequency contents. In the preparation phase of the method, we first present the waveform matching extension to obtain the extended signal. After that, GWO is introduced to determine adaptively the inside parameters of VMD. Moreover, we employ the parameter-optimized VMD to divide the extended signal into several IMF ingredients and truncate the extended segment to obtain the final decomposition results. Finally, TEO demodulation for all IMFs is conducted to achieve the corresponding instantaneous frequency (IF), and the IF of all IMFs is painted together to obtain our desired time-frequency representation (TFR). One obvious merit of the proposed IAVMD is that it does not depend on any parameterization, which makes this method more adaptive to actual situations. Crucially, the study of simulated and actual laboratory data validates the effectiveness of our approach. The remainder of this paper is as follows. Section 2 reviews the basic theory of VMD. Section 3 introduces the theory of the waveform matching extension method and shows the parameter optimization process of VMD based on GWO. Meanwhile, the overall realization process of the proposed IAVMD is illustrated in Section 3. Sections 4 and 5 prove the effectiveness of the presented approach through a simulation analysis and experimental examples, respectively. Section 6 draws some conclusions and provides some future works.