## 3 Results
### Pipeline Defect Simulation Signal Analysis  
A set of simulation signals comprising normal and noisy signals was designed. In this study, two sinusoidal signals of the same frequency with an amplitude of 2.5 were used, and the result of summing is still the sinusoidal signal of the original frequency. The normal signal is expressed as follows:  
\[X=2.5\sin(10\pi t)+\sin(80\pi t)\]  
Three groups of noise signals were generated as follows:  
\[\eta_{1}=\sin(100\pi t)+n(t)\]  
\[\eta_{2}=\sin(100\pi t)+\sin(200\pi t)+n(t)\]  
\[\eta_{3}=\sin(100\pi t)+\sin(200\pi t)+\sin(1000\pi t)+n(t)\]  
where \(\eta(t)\) is Gaussian white noise with a mean of 0 and a variance of 1; \(n(t)\) is Gaussian white noise. To facilitate the calculation, we set the sampling frequency to f\({}_{8}\) = 1000 Hz and used N = 1000 sampling points.  
Taking the first type of noise as an example, the synthesized signal (denoted as \(x_{1}\)) was used for noise reduction. ICEEMDAN was performed on the signals to obtain the corresponding IMF components. Noise \(\eta_{1}\) was also decomposed by ICEEMDAN to obtain the corresponding IMF component as a reference signal. Noise signals were filtered by the improved LMS algorithm for each group of IMF components, and the filtered IMF components were summed to obtain the defect signal \(\eta_{11}\) of the pipeline after noise reduction.  
As the time-domain comparison diagrams in Figures 2 and 3 show, the noise of the signal after noise reduction is substantially lower, and the waveform of the signal can be clearly seen. A comparison of the frequency domain diagrams reveals that the main frequency of the pipeline defect signal before noise reduction is concentrated below 100 Hz, and the amplitude of the sinusoidal signal at 100 Hz is reduced from 3.05 to 0.53. In addition, the peak value of the pipeline defect signal is mostly lost in the noise before noise reduction, but the peak value of the effective signal becomes evident after noise removal. As shown in the spectrum diagram of the pipeline defect signal in Figure 3, the visible noise in the spectrum diagram before noise removal is substantially reduced after noise removal. Therefore, when ICEEMDAN is combined with the improved LMS method, it yields a significant reduction in the noise of pipeline defect signals.  
Figure 2: Time–domain and frequency–domain of the pipeline defect signal before noise removal.  
As can be seen from Table 1, ICEEMDAN-LMS obtains good noise reduction results on pipeline defect signals. As the number of superimposed noise signals increases, the root-mean-squared error (RMSE) after noise reduction gradually increases. Adjusting the step size M and order \(\mu\) in the LMS algorithm enables ICEEMDAN-LMS to have a significant effect on the data processing of pipeline defect signals when the SNR is greater than \(-20\) dB. However, when the SNR is below \(-20\) dB, the active components of the signal are filtered out. As Figure 4 shows, for added noise over \(-20\) dB, the SNR after denoising increases from \(-20\) dB to \(-8.5\) dB.  
After denoising the non-stationary signal of the pipeline defects, it was found that the SNR was improved when the improved LMS method was used (see Table 2).  
It can be seen from the above analysis that the ICEEMDAN-LMS method is suitable not only for the noise reduction analysis of stationary pipeline signals, but also for non-stationary signals, and the reduction in noise is substantial. The simulated signal noise reduction results are shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c c} \hline  & **s/s1** & **s/s22** & **s/s33** \\ \hline SNR/dB & 10.66 & 10.25 & 9.47 \\ MSE & 0.24 & 0.30 & 0.38 \\ \hline \end{tabular}
\end{table}
Table 1: Noise removal results.  
Figure 4: Noise reduction effect when the SNR is \(-20\) dB.  
Figure 3: Time-domain and frequency–domain of the pipeline defect signal after noise removal.