## 1 Introduction
It is clear that wavelet denoising methods are widely used in signal processing, and a discrete wavelet transform (DWT) can effectively remove noise and obtain the detailed information of the signal [15; 16]. In the case of VMD, the thresholding operation should be properly adapted to be consistent with the energy characteristics of the decomposition basic modes. Inspired by improved wavelet thresholding [17], in this paper, we propose a novel denoising method that combines variational mode decomposition with the changed thresholding discrete wavelet transform, called the VMD-DWT algorithm, to process experimental signals. Several EMD denoising algorithms have been developed, such as EMD-soft [18], EMD-DFA [19], and EMD-changed thresholding wavelet [20]. EMD [8] is a recursive procedure to analyze non-linear and non-stationary multicomponent signals by deposing them into several amplitude- and frequency-modulated (AM/FM) zero-mean principle mode signals, which are without predefined basis vectors. However, EMD decomposition highly depends on the extremal point findings method. A lack of a mathematical theory and the problem of recursive shifting and mode mixing in the extraction process of IMF by EMD have inspired such solutions as empirical wavelets [21] and recursive variational decomposition [4]. Nevertheless, these attempts have only partially addressed the drawbacks of EMD. Recently, VMD, due to its advantages in noise resistance, the balance of the backward error, and a narrow-band definition of meaningful modes, has been proposed as an alternative to EMD for dividing multi-component signals into different modes based on a clear variational model [22]. VMD combined with a discrete wavelet thresholding algorithm for denoising mixed and real experiment signals under various working conditions presents a superior effect compared with traditional denoising based on EMD methods. The proposed approach has two key advantages over the previous studies. First, the novel changed thresholding VMD-DWT algorithm can be effectivity applied to denoise the FBG signals that are obtained from the fatigue crack propagation detection experiment. Once the crack is initiated, the amplitude fluctuation of the noise spectral signal measured from the experiment would be obviously changed [3]. Therefore, a VMD combined with the changed thresholding wavelet method is proposed to satisfy the future precise denoising demand of the damage signals. Second, although the VMD or the DWT denoising algorithm has been studied in other signals, such as the EEG [11] and the vibration signals [6], the combined denoising algorithm is seldom adapted in fatigue crack experiment FBG signals. Moreover, the energy distribution in each basic mode has been considered in wavelet thresholding. Additionally, compared with other decomposition parameter definition methods, in this paper, it focuses on the physical characteristic of the signal and the real experiment environment. Through analyzing the denoising results of different methods in mixed and real signals, the proposed denoising method shows good performance in FBG signal denoising processing in the SHM field. The remainder of the paper is organized as follows. We present the experimental design in Sections 2 and 3 provides a brief description of VMD and the major concepts of DWT. Meanwhile, it also discusses the possibility of adapting the wavelet thresholding principle in VMD decomposition. Consequently, novel changed thresholding VMD-DWT strategies are presented. Section 4 explores two cases. In Case 1, four denoising algorithms are applied to denoise the mixed signals. The processing results show the effectiveness of the proposed method with respect to FBG signals, and the best decomposition number \(K\) and data-fidelity constraint parameter \(\alpha\) were obtained. In Case 2, real experimental signals used to evaluate the novel denoising techniques are illustrated. This section also includes a discussion of the results. In Section 5, the final conclusions are drawn.