## 3 Adaptive Time-Frequency Analysis
### Empirical Wavelet Transform and Optimization Models
#### 3.3.3 Empirical Fourier Decomposition  
Definition  
Empirical Fourier decomposition (EFD) is a signal processing technique that analyzes the local frequency spectrum of a signal, identifies the main frequency components and their variation characteristics in the signal, and then decomposes the signal into several components based on this. It aims to decompose complex signals into a series of components with clear physical meaning, which are usually expressed in the form of amplitude modulation-frequency modulation (AM-FM). Application  
Many researchers have employed spectral analysis methods to optimize and implement empirical Fourier decomposition (EFD). A spectral envelope-based adaptive empirical Fourier decomposition (SEAEFD) method has been proposed and applied to fault diagnosis in rolling bearings. The core of this method lies in optimizing the EFD process through spectral envelope analysis, thereby enhancing the orthogonality, correlation coefficient, and overall decomposition accuracy. By adaptively adjusting the spectral envelope analysis strategy, SEAEFD can maintain high decomposition accuracy in highly noisy environments and effectively extract the fault features of rolling bearings [124]. Another automatic modal identification method based on free vibration response and enhanced empirical Fourier decomposition (EFD) technology has also been developed. By combining improved spectral segmentation techniques with zero-phase shift filters, this method can more accurately decompose non-stationary signals and closely spaced modal signals. Through the automatic adjustment of the number of spectral segmentations, it can adapt to different signal characteristics, thereby improving the accuracy and robustness of modal identification [125]. Furthermore, a power spectrum envelope adaptive empirical Fourier decomposition (PSEEFD) method has been proposed for processing signals with low signal-to-noise ratios. The PSEEFD method adaptively adjusts the EFD decomposition strategy based on signal characteristics, thereby automatically selecting appropriate decomposition parameters in order to achieve more accurate signal decomposition under low signal-to-noise ratio conditions. Experimental results have verified the effectiveness of this method in processing vibration signals from ship stabilizing platforms [126]. The integration of EFD with algorithms, neural networks, and other classic signal processing techniques is also one of its important applications. First, improved spectral segmentation techniques and the construction of zero-phase shift filters have been utilized to optimize EFD. The improved spectral segmentation techniques can more accurately identify abrupt changes in signals, thereby enhancing decomposition accuracy. The construction of zero-phase shift filters ensures that the decomposed components maintain the correct phase relationships, thus improving the reliability of the decomposition results. Numerical and experimental methods have verified the effectiveness of EFD in processing non-stationary signals and closely spaced modal signals, and comparisons have been made with other decomposition methods [127]. Additionally, a wind speed prediction model that combines empirical Fourier decomposition, a long short-term memory network (LSTM), and grey wolf optimizer (GWO) has been proposed. The literature provides a detailed description of the working principle and implementation steps of this model, and experimental results have verified its effectiveness in wind speed prediction [128]. Finally, due to its superiority, empirical Fourier decomposition has been applied to bridge health monitoring [129]. Strength and Weakness  
* Components with clear physical meanings
* Excellent modal separation for multi-modal vibration identificationSensitive to noise and possible modal aliasing  
Empirical Fourier decomposition (EFD), as a signal processing technique, demonstrates remarkable advantages in fields such as structural health monitoring.