## 2 Methodology of Time-Frequency Separation Neural Network
### Time-Frequency Analysis of AE Signals Using Smoothed Pseudo Wigner-Ville Distribution  
Due to the non-stationary and time-varying spectral characteristics of crack and abnormal noise signals, classical blind source separation algorithms may not be able to accurately extract the source signal, resulting in aliasing and noticeable degradation in separation performance. For non-stationary signals, the time-frequency analysis (TFA) plays a pivotal role in characterizing signal attributes across time and frequency domains [18]. Using TFA, time-frequency conversion between the mixed signals should be achieved before various separation schemes of the mixed chaotic AE signals.  
Therefore, the first key step of the proposed algorithm is to perform time-frequency conversion on the mixed signal, and the final separation performance will naturally be different depending on the adopted TFA method. Wigner-Ville distribution (WVD) is a quadratic time-frequency conversion method, giving the instantaneous joint energy distribution of a signal in time-frequency domain. Given a non-stationary signal, its instantaneous autocorrelation function is defined as  
\[\mathbf{r}_{x}(t,\tau)=\mathbf{x}(t+\frac{\tau}{2})\mathbf{x}^{*}(t-\frac{\tau }{2}) \tag{4}\]  
where \(t\) is the time variable and \(\tau\) the time shift, and the Wigner-Ville distribution of the signal \(\mathbf{x}(t)\) is the Fourier transform of the instantaneous autocorrelation function \(\mathbf{r}_{x}(t,\tau)\) with respect to the integral variable \(\tau\),  
\[WVD(t,f)=\int_{-\infty}^{\infty}\mathbf{r}_{x}(t,\tau)e^{-\beta\pi f\tau}d\tau =\int_{-\infty}^{\infty}\mathbf{x}\Bigl{(}t+\frac{\tau}{2}\Bigr{)}\mathbf{x}^{ *}\Bigl{(}t-\frac{\tau}{2}\Bigr{)}e^{-\beta\pi f\tau}d\tau \tag{5}\]Different from STFT or CWT, WVD compensates for the shortcomings of linear time-frequency algorithms with high resolution in both domains, and it is unlimited by the uncertainty principle. Additionally, WVD is also widely used due to its excellent properties, such as smearing reduction, energy preservation, time-frequency shift invariance, etc. However, the Wigner-Ville distribution is seriously affected by the cross-term interference, and it might obscure the interpretation of the genuine TF distribution when the analyzed signal is composed of multiple components [25]. The smoothed pseudo Wigner-Ville distribution (SPWVD) is an improved version of the WVD on the basis of a wavelet scaling factor, which is equivalent to adding windows simultaneously and independently in both the time domain and frequency domain to suppress the cross interferences [26]. The expression of the SPWVD for \(\mathbf{x}(t)\) is  
\[SPWVD_{g,h}(t,f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g(u)h(\tau) \mathbf{x}\Big{(}t-u+\frac{\tau}{2}\Big{)}\mathbf{x}^{*}\Big{(}t-u-\frac{\tau }{2}\Big{)}e^{-j2\pi f/\tau}dud\tau \tag{6}\]  
where \(g(u)\) and \(h(\tau)\) denote the window functions in the frequency and time domains, respectively. The general spatial time-frequency distribution of a random signal in SPWVD accurately represents the time-varying spectral characteristics of non-stationary signals, \(STFD(t,f)=SPWVD_{g,h}(t,f)\), making it suitable for analyzing and separating cracks and noises in non-stationary AE signals. Therefore, SPWVD is used in the crack signal extraction method based on time-frequency blind source separation (TFBSS) in the following section. For instance, [PERSON] proposed the most representative TFBSS method to obtain the unitary separation matrix based on the diagonalization of joint Wigner-Ville TF distributions [22]. Additionally, in view of the above reasons, SPWVD is used to replace WVD and obtain a better separation performance in TFBSS as another representative baseline method in the comparison analysis.