# Combined Improved CEEMDAN and Wavelet Transform Sea Wave Interference Suppression
## 3 Proposed Algorithm
### Improved CEEDMAN Algorithm  
Empirical mode decomposition (EMD) is a novel method for handling nonstationary data series, which was proposed by [PERSON] [24] in 1998. Signals are decomposed one by one, and a data series containing different characteristic scales is obtained, which is called the intrinsic mode function (IMF) [24].  
The data collected by radar include sea wave and WSAW data, which are preprocessed to obtain one-dimensional time-distance data. Since the sea waves and WSAWs contain multiple frequency components, they can be separated based on the frequency. The EMD algorithm can decompose a signal into multiple frequency components, but it is prone to modal aliasing. By adding Gaussian white noise in the decomposition process, the modal aliasing phenomenon can be effectively suppressed. The improved CEEMDAN algorithm is as follows.  
The Gaussian white noise decomposed by EMD is added to the improved CEEMDAN's first decomposition instead of the original Gaussian white noise. In contrast, the improved CEEMDAN adds Gaussian white noise with the same multiple decompositions.  
The first decomposition of the improved CEEMDAN obtains the IMF component by calculating the mean value of the IMF component, and each decomposition after the first decomposition obtains the IMF component by calculating the mean value of the residual component and subtracting the mean value of the residual component from the signal.  
Definition: \(E_{k}(\cdot)\) represents the \(kth\)-order IMF component obtained by the EMD of the signal; \(M(\cdot)\) is the average calculation of the local envelope of the signal; and \(n^{i}(t)\) is the zero-mean Gaussian white noise. The phase \(\phi(t)\) is converted from Equation (8) into the distance \(x(t)\). The improved CEEMDAD algorithm flow is as follows.  
White noise is added to \(x(t)\), and the superimposed signal is decomposed to obtain the first-order IMF by EMD:  
\[x^{i}(t)=x(t)+\xi_{0}E_{1}\Big{(}n^{i}(t)\Big{)},i=1,2,\cdots,L \tag{9}\]  
where \(\xi_{k-1}\) is the amplitude coefficient of the noise added to the \(k\)\(th\) decomposition, generally set to 0.1-0.3; \(L\) is the number of times white noise is added, which is usually set between 100 and 1000. The local envelope averaging is solve; then, the ensemble averaging is performed to obtain the first-order residual components:  
\[r_{1}=\frac{1}{L}\sum_{i=1}^{L}M\Big{(}x^{i}(t)\Big{)} \tag{10}\]  
The first-order IMF is calculated:  
\[IMF_{1}=x(t)-r_{1} \tag{11}\]  
The white noise is added to the first residual error, \(r_{1}\), and the superimposed signal is decomposed to obtain the second-order IMF by EMD. The second-order residual error is obtained by \(M(\cdot)\) and ensemble averaging. The second-order residual error is obtained from the first-order residual error to obtain the second-order IMF:  
\[IMF_{2}=r_{1}-r_{2}=\frac{1}{L}\sum_{i=1}^{L}M\Big{(}r_{1}+\xi_{1}E_{2}\Big{(} n^{i}(t)\Big{)}\Big{)} \tag{12}\]Equations (11) and (12) are repeated to calculate the \(k\)\(th\) IMF; then, the \(k\)\(th\)-order residual error, \(r_{k}\), can be expressed as:  
\[r_{k}=\frac{1}{L}\sum_{i=1}^{L}M\Big{(}r_{k-1}+\xi_{k-1}E_{k}\Big{(}n^{i}(t) \Big{)}\Big{)} \tag{13}\]  
The \(k\)\(th\)-order IMF is calculated:  
\[IMF_{k}=r_{k-1}-r_{k} \tag{14}\]  
Whether the above conditions satisfy the termination condition is determined (i.e., the residual error does not exceed two extreme values) and, if so, the decomposition is stopped.