# A Multi-Mode Recognition Method for Broadband Oscillation Based on Compressed Sensing and EEMD
## 4 Analysis and Comparison of the Results
### Analysis and Comparison, Based on a Theoretical Model  
A broadband oscillation signal, consisting of a supersynchronous oscillation and two medium and high-frequency oscillations in addition to the fundamental frequency signal, was constructed. The sampling frequency was 800 Hz, the length of the data was 256, and the theoretical model was constructed as follows.  
\[x(t)=0.85\sin(2\pi\times 50t)+0.2\sin(2\pi\times 75t)+0.5\sin(2\pi\times 200t)+  
We applied the EEMD to decompose the reconstructed signal. The noise standard deviation was set at 0.2, the test count was 100, and the iterations count was set at 1000. The decomposition results are shown in Figure 2. EEMD decomposes the broadband oscillation signal into a set of IMFs. The IMFs contain different frequency bands, from high to low, of the decomposed signal but the number of IMFs is larger than that of the broadband oscillation mode numbers; thus, it is necessary to eliminate irrelevant IMFs.  
Since the energy coefficient of the real mode is larger than that of the false mode, the IMFs containing oscillation information were selected according to each IMF energy coefficient. Table 1 shows the energy coefficient of each IMF; the first three IMFs retain the main information of the oscillation modes, while the rest are irrelevant IMFs that need to be omitted.  
Figure 1: Comparison between the reconstructed signal and original signal.  
Figure 2: EEMD decomposition results of the reconstructed signal.  
Finally, FFT was used to calculate the oscillation information of the first three IMFs. The test results in Figure 3 show that this technique can accurately identify the oscillation frequency.  
To prove the robustness of the abovementioned method to noise, white noise at about 30 dB, 25 dB, 20 dB, and 15 dB was added to the constructed signal. The proposed method was used to compress and reconstruct the broadband oscillation signal with different signal-to-noise ratios (SNRs). The reconstruction error between the reconstructed and original signal, as well as the reconstructed signal SNR, were calculated as shown in Figure 4.  
With the increase in SNR, the noise component in the signal decreased, the RE after compression and reconstruction also gradually decreased, and the reconstructed SNR increased. When the SNR was 15 dB, the RE was 0.19, and the reconstructed SNR was 19.2044 dB. The proposed compressed sensing algorithm, based on SP, not only has good robustness to noise but also has a good de-noising effect.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**IMFi** & **IMF1** & **IMF2** & **IMF3** & **IMF4** & **IMF5** & **IMF6** & **IMF7** & **IMF8** \\ \hline Energy coefficient & 0.3623 & 0.1092 & 0.3951 & \(0.74\times 10^{-4}\) & \(4.02\times 10^{-5}\) & \(4.42\times 10^{-5}\) & \(3.57\times 10^{-4}\) & \(1.56\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The energy coefficient of each IMF.  
Figure 4: The effect of compression and reconstruction for different SNRs.  
Figure 3: FFT results for the first three IMFs.  
Working according to the proposed method, the identification results of the frequency and amplitude of each oscillation mode in signals with different SNRs are shown in Table 2, indicating that the method is feasible in terms of broadband oscillation signal identification.