## 3 Adaptive Time-Frequency Analysis
### Empirical Wavelet Transform and Optimization Models
#### 3.3.1 Empirical Wavelet Transform
Another fault diagnosis method for rolling bearings combines an improved adaptive parameter-free empirical wavelet transform (IAPEWT) with an adaptive sparse coding shrinkage denoising (ASCSD) algorithm. IAPEWT automatically segments the spectrum and determines the boundaries of the filter bank, and when combined with the ASCSD algorithm, it enhances periodic impulse features, improving fault identification accuracy [110]. Furthermore, technologies that enhance the mathematical logic of EWT have also emerged. A new EWT method based on Fourier-Bessel series expansion (FBSE) has been used for the time-frequency representation of non-stationary signals. By improving EWT through FBSE, experimental results indicate that FBSE-EWT performs better than existing EWT and Hilbert-Huang transform (HHT) methods in terms of representing non-stationary signals with closely adjacent frequency components [111]. A new multichannel signal processing method called quaternion empirical wavelet transform (QEWT) has been used for fault diagnosis in rolling bearings. QEWT can synchronously process multiple sets of vibration signals, achieving spectrum segmentation, fusion, and modal decomposition, thereby improving the accuracy and efficiency of fault diagnosis [112]. Similarly, an EWT method based on Fourier-Bessel series expansion (FBSE-EWT) [113] has also been mentioned. Moreover, in the iterative process of various optimization models, an improved empirical wavelet transform (IEWT) has emerged, leading to subsequent optimizations based on it. A hybrid model based on IEWT and a long short-term memory network (LSTM) has been used to predict significant wave height. By decomposing buoy data using IEWT and combining them with LSTM for time series prediction, results show that the model exhibits higher prediction accuracy over longer prediction windows [114]. Additionally, a scheme based on IEWT, energy feature vector (EFV), and Bayesian optimized convolutional neural network (BOCNN) has been used to extract the hydrodynamic features of hydraulic turbines and intelligently identify their flow states. By decomposing signals using IEWT and combining EFV and BOCNN for feature extraction and classification, experimental results indicate that the average recognition accuracy of this scheme reaches 99.15% [115]. Furthermore, an approach combining IEWT and a sensitive component selection method has been used for bearing fault diagnosis. It enhances EWT performance through cyclic envelope spectrum segmentation and correlation-based component optimization, and it extracts fault information using the sensitive intrinsic mode function evaluation index (SIAI) [116]. Finally, an improved extended empirical wavelet transform (IEEWT) method has been used to detect and characterize multivariate oscillations in industrial control systems. Combined with a noise-reduced power spectrum and detrended fluctuation analysis (DFA) techniques, the IEEWT method demonstrates higher accuracy and robustness when processing multivariate signals with high noise levels [117]. In addition, researchers have professionally implemented EWT, leading to the emergence of innovative methods and studies. An improved demodulation analysis method based on cepstrum-assisted empirical wavelet transform (CEWT) has been used for fault diagnosis in planetary gearboxes. As demonstrated through numerical simulation and experimental validation, this method outperforms existing signal decomposition methods in terms of fault diagnosis [118]. Two other methods excel in noise reduction. A new variable spectrum segmentation empirical wavelet transform (VEWT) improves signal decomposition efficiency and noise resistance through selective spectrum segmentation and modal estimation based on multi-window power spectral density. Experimental results show that VEWT outperforms traditional EWT and extended EWT (EEWT) at processing noisy signals [119]. An EWT method enhanced by synchronous extraction transform (SET) has been used for the identification of time-varying systems.