## 3 Adaptive Time-Frequency Analysis
### Empirical Wavelet Transform and Optimization Models
#### 3.3.2 Ensemble Empirical Wavelet Transform  
Definition  
Ensemble empirical wavelet transform iiiLEWThijL combines the advantages of EEMD and EWT, aiming to improve the stability and accuracy of signal decomposition. It performs empirical wavelet transform multiple times and averages the results to reduce noise and uncertainty during the decomposition process.  
1. Signal and Noise Superposition: The signal and noise superposition is given by  
\[x^{\prime}(t)=x(t)+n_{i}(t) \tag{13}\]  
where \(x(t)\) is the original signal, \(n_{i}(t)\) is the \(i\)-th added Gaussian white noise, and \(x^{\prime}(t)\) is the signal after noise superposition.
2. EMD Decomposition: Perform EMD decomposition on \(x^{\prime}(t)\) to obtain a series of intrinsic mode functions (IMFs):\[x^{\prime}(t)=\sum_{j=1}^{N}\text{IMF}_{i,j}(t)+r_{i}(t) \tag{14}\]  
where \(\text{IMF}_{i,j}(t)\) is the \(j\)-th IMF obtained from the \(i\)-th decomposition, and \(r_{i}(t)\) is the residual term.
3. Repetition and Averaging: Repeat the above process \(M\) times, adding different white noise each time, and average all the obtained IMFs: \[\text{IMF}_{j}(t)=\frac{1}{M}\sum_{i=1}^{M}\text{IMF}_{i,j}(t)\] (15) The final \(\text{IMF}_{j}(t)\) obtained is the result of the EEMD decomposition.  
Strength and Weakness
* Strong noise resistance and high resolution time frequency analysis
* High computational complexity and endpoint effect  
First, regarding adaptive frequency band partitioning and ensemble processing, EEMT dynamically adjusts the decomposition process based on the spectral characteristics of the signal through adaptive frequency band partitioning, thereby effectively capturing local features of the signal. Compared to the traditional EWT, EWT introduces an ensemble processing mechanism, which significantly enhances the stability and robustness of the decomposition through multiple decompositions and averaging. For instance, during the analysis of bridge vibration signals, EEMT can effectively separate modal components of different frequencies, providing more accurate data support for structural damage identification [123].  
Second, EEMT exhibits strong noise resistance. By utilizing the ensemble processing mechanism, EEMT can effectively suppress noise interference and improve the signal-to-noise ratio. In environments with high noise levels, the decomposition results of EEMT are more stable, allowing for the accurate extraction of effective modal components.  
Third, EEMT offers high-resolution time-frequency analysis through the dilation and translation of wavelet basis functions. This enables EEMT to simultaneously provide both the temporal and frequency information of the signal, making it particularly suitable for analyzing transient signals and complex vibration patterns. For example, in mechanical fault diagnosis, EEMT can accurately capture the transient features of fault signals, thereby improving the accuracy of fault detection.  
Next, the disadvantages of EEMT are discussed:  
First, EEMT has a relatively high computational complexity. Although EEMT performs excellently in terms of decomposition accuracy and noise resistance, its computational complexity is relatively high, especially when processing large-scale or high-dimensional signals, leading to a significant increase in computation time. As a result, in real-time structural health monitoring, EEMT may not meet the requirements for real-time processing.  
Second, parameter selection remains somewhat complex. The performance of EEMT depends on the choice of parameters, such as the thresholds for frequency band partitioning, the number of ensemble iterations, and the selection of wavelet basis functions. Inappropriate parameter settings may lead to modal aliasing or modal loss, affecting the decomposition results.  
Like many signal decomposition methods, EEMT has an endpoint effect, distorting decomposition results at signal boundaries, which can cause error accumulation in real-time structural health monitoring, reducing the monitoring system's reliability.