# Combined Improved CEEMDAN and Wavelet Transform Sea Wave Interference Suppression
## 3 Proposed Algorithm
### Reconstruction Scale  
In the previous step, Gaussian white noise was added to suppress mode aliasing. Here, wavelet transform is needed to de-noise each modal component. WT is a function, \(\psi(t)\), that satisfies the condition \(\int_{-\infty}^{+\infty}\psi(t)dt=0\) and is generated by translation and scaling. The WT of signal \(x(t)\) is defined as:  
\[W(a,b)=|a|^{-1/2}\int_{-\infty}^{+\infty}x(t)\psi^{*}\Big{(}\frac{t-b}{a} \Big{)}dt \tag{15}\]  
where \(W(a,b)\) is the wavelet transform coefficient; \(\psi(t)\) is the wavelet basis function; and \(a\) and \(b\) are the stretch and translation factors, respectively. Equation (15) shows that the choices of the wavelet basis function significantly influence each IMF. Through experimental tests, the wavelet basis function \"coif4\" was selected in this manuscript to obtain better results. After the interference signal is removed by wavelet decomposition, the new signal can be reconstructed using the inverse wavelet transform:  
\[x^{{}^{\prime}}(t)=\frac{1}{c_{\psi}}\int_{-\infty}^{+\infty}\frac{1}{a^{2}}W (a,b)\cdot\psi\left(\frac{t-b}{a}\right)\textit{d}adb \tag{16}\]  
where \(C_{\psi}=\int_{-\infty}^{+\infty}|\Psi(\omega)|^{2}/\omega d\omega\) is the wavelet admissible constant, and \(\Psi(\omega)\) is the Fourier transform of the wavelet basis function \(\psi(t)\). The received phase signals include the WSAW signal and sea wave interference signal. The novel sea wave interference suppression algorithm can divide \(x(t)\) into multiple different IMFs based on the frequency. In order to determine the modal function of the WSAW signal, the corresponding reconstructed scale is calculated by correlation to separate the WSAW and sea wave interference. The separated WSAW signal is reconstructed to further restore the transmission information of the underwater speaker. A flowchart of the sea wave interference suppression is shown in Figure 2. The received echo is de-sloped and low-pass filtered. The yellow line indicates the one-dimensional range result of the superposition of the WSAW and the sea wave interference. The maximum value in each range bin is determined, and the corresponding phase is extracted. The main purpose of preprocessing is to estimate the frequency of the transmitted signal. The frequencies will correlate with the results after the wavelet decomposition to determine the reconstructed scale, \(k\). Then, the superimposed signal \(x(t)\) is decomposed into multiple IMFs and the residual error by the improved CEEMDAN, and the frequencies of the IMFs vary from high frequency to low frequency. In the above process, white noise is added to achieve the continuity of frequencies between adjacent scales, which can eliminate the influence of frequency aliasing. The key point shown in Figure 2 is to determine the reconstruction scale, which needs to be determined by the multiscale IMF and the residual error. The product, \(P_{i}\), of the energy density, \(E_{i}\), and the average cycle, \(\overline{T}_{i}\), are calculated, where \(E_{i}\) is the \(i\)th IMF energy density, \(E_{i}=\frac{1}{N}\sum\limits_{j=1}^{N}\left(A_{i}(j)^{2}\right)\). \(\overline{T}_{i}\) is the \(i\)th IMF average cycle, \(\overline{T}_{i}=2N/O_{i}\). Moreover, \(N\) is the length of the \(i\)th IMF, \(A_{i}\) is the amplitude of the \(i\)th IMF, and \(O_{i}\) is the sum of the extreme points of the \(i\)th IMF. The coefficient \(R_{i}\) is calculated as:  
\[R_{i}=P_{i}/P_{i-1},i=1,2,\cdots N \tag{17}\]  
The product, \(P_{i}\), of the energy density, \(E_{i}\), and the average cycle, \(T_{i}\), is a constant [25]. When a point in \(P_{i}\) is a local minimum, it contains useful frequency components. When \(R_{i}\geq 2\), the initial reconstructed scale \(l\) is equal to \(i\). The signals of the initial reconstructed scale \(l\) and the preprocess frequencies are reconstructed. Correlation is performed on the reconstructed signals. When the correlation coefficient is greater than 0.95, the IMFs are considered to be the WSAW signals.