## 3 Adaptive Time-Frequency Analysis
### Empirical Wavelet Transform and Optimization Models
#### 3.3.1 Empirical Wavelet Transform  
Definition  
The empirical wavelet transform (EWT) is an adaptive signal decomposition method that constructs a wavelet filter bank by analyzing the spectral characteristics of the signal. The core formulas are as follows:  
1. Fourier Transform: Perform Fourier transform on the signal \(f(t)\) to obtain the spectrum \(F(\omega)\): \[F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}dt\] (5)
2. Frequency Band Division: Detect the local maxima of \(|F(\omega)|\) to determine the frequency band boundaries \(\{\omega_{n}\}_{n=0}^{N+1}\), where \(\omega_{0}=0\) and \(\omega_{N+1}=\pi\) (normalized frequency). 3. Construction of Empirical Wavelet Functions: For each frequency band \([\omega_{n},\omega_{n+1}]\), construct the empirical wavelet function \(\psi_{n}(\omega)\) and scaling function \(\phi_{1}(\omega)\), satisfying the tight frame condition:\[|\phi_{1}(\omega)|^{2}+\sum_{n=1}^{N}|\psi_{n}(\omega)|^{2}=1,\quad\forall\omega \tag{6}\]  
Scaling Function \(\phi_{1}(\omega)\) (Low-Pass Filter):  
\[\phi_{1}(\omega)= \begin{cases}1,&|\omega|\leq\omega_{1}-\tau_{1},\\ \cos\left(\frac{\pi}{2}\beta\left(\frac{|\omega|-(\omega_{1}-\tau_{1})}{2\tau_ {1}}\right)\right),&\omega_{1}-\tau_{1}\leq|\omega|<\omega_{1}+\tau_{1},\\ 0,&\text{otherwise}.\end{cases}  \tag{7}\]  
Wavelet Function \(\psi_{n}(\omega)\) (Band-Pass Filter):  
\[\psi_{n}(\omega)= \begin{cases}\sin\left(\frac{\pi}{2}\beta\left(\frac{|\omega|- (\omega_{n}-\tau_{n})}{2\tau_{n}}\right)\right),&\omega_{n}-\tau_{n}\leq| \omega|<\omega_{n}+\tau_{n},\\ 1,&\omega_{n}+\tau_{n}\leq|\omega|<\omega_{n+1}-\tau_{n+1},\\ \cos\left(\frac{\pi}{2}\beta\left(\frac{|\omega|-(\omega_{n+1}-\tau_{n+1})}{2 \tau_{n+1}}\right)\right),&\omega_{n+1}-\tau_{n+1}\leq|\omega|<\omega_{n+1}+ \tau_{n+1},\\ 0,&\text{otherwise}.\end{cases}  \tag{8}\]  
Here, \(\beta(x)\) is a smoothing function, for example  
\[\beta(x)=x^{4}(35-84x+70x^{2}-20x^{3}), \tag{9}\]  
which satisfies \(\beta(0)=0\), \(\beta(1)=1\), and has zero derivatives at the endpoints. The transition band width \(\tau_{n}\) is usually taken as a certain proportion of the adjacent boundary spacing, such as \(\tau_{n}=\gamma(\omega_{n}-\omega_{n-1})\), (where \(\gamma\in(0,0.5)\)). 4. Mode Decomposition: Extract the mode components via inverse Fourier transform: Residual (Low-Frequency Part): \[W_{f}(0,t)=\mathcal{F}^{-1}[F(\omega)\phi_{1}(\omega)](t)\] (10)  
\(n\)-th Mode Component:  
\[W_{f}(n,t)=\mathcal{F}^{-1}[F(\omega)\psi_{n}(\omega)](t),\quad n\geq 1 \tag{11}\]
5. Signal Reconstruction: The signal can be reconstructed exactly as follows: \[f(t)=W_{f}(0,t)+\sum_{n=1}^{N}W_{f}(n,t)\] (12)  
Key Points  
* Adaptivity: The frequency band division is based on the local maxima of the signal spectrum, without the need for predefined basis functions. * Compact Support: The wavelet functions have compact support and smooth transitions in the frequency domain, reducing ringing effects in the time domain. * Perfect Reconstruction: The filter bank satisfies the condition that the sum of squares is 1, ensuring lossless signal recovery. By combining signal-driven wavelet design with traditional wavelet theory, EWT achieves effective analysis and processing of non-stationary signals. Application  
First, in combination with neural networks or high-order algorithms, researchers have proposed a novel damage detection method based on empirical wavelet transform (EWT) and artificial neural network (ANN) for detecting and localizing damage in steel truss bridge models [108]. Additionally, there is a time series prediction method based on EWT and echo state network (ESN). This method decomposes the original time series into multivariate subsequences and employs ESN for modeling and prediction. Experimental results demonstrate that this method outperforms other benchmark models on multiple public datasets [109].