## 4 Simulation Results  
The real-time and precision identification performance of the proposed Adaline-LMM algorithm is initially tested by simulation using MATLAB/SIMULINK. The simulation input is:  
\[ \begin{array}{ll}a(k)&=3\sin(\omega k+1.2)+0.6\sin(2\omega k+0.9)+0.5\sin(3 \omega k+0.7)+0.4\sin(4\omega k+0.5)\\ &+0.3\sin(5\omega k+0.4)+0.2\sin(6\omega k+0.3)\\ \end{array}  \tag{24}\]  
Apparently, there are 6 harmonics whose frequencies are the integral multiple of the fundamental frequency. Its sampling frequency is 1000 Hz. The fundamental frequency is 3 Hz. The optimum values of constant parameters such as \(C_{1}\), \(C_{2}\), are 0.995, 2.157 respectively. \(L_{w}\) is set as 12 because there are 6 harmonics existing in the simulation input. The initial values of both \(\omega_{ai}(k)\) and \(\omega_{bi}(k)\) are chosen as 0.018. From the error plot in Figure 6, it is noted that the estimation error is converging to nearly zero within 1s, in spite of large fluctuation existing in the initial stage of harmonic identification. Figure 7 contains two different kinds of signals, both the actual signal denoted by dashed line and the identified signal denoted by red line. Although the estimation error is quite large originally, it is noted that the identified signal is well converged to the actual signal within 0.7 s which means both the amplitude and the phase of each harmonic are precisely estimated. Figure 5: Flow chart of the least mean M-estimate algorithm. The estimation amplitude and phase of each identified harmonic are separately shown in Figures 8 and 9, and the result of contrast between the estimated values and the set values is showed in Table 3. At the initial stage of the harmonic identification, there are relatively large fluctuation. But both amplitudes and phases of all the harmonics are converging to their specific values which are exactly the same as what the simulation input signal contains originally, i.e., the amplitude and phase of each harmonic are precisely estimated by using Adaline-LMM algorithm. Besides, harmonics can be directly identified and its results are shown in Figure 10. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Set Values**} & \multicolumn{2}{c}{**Estimated Values (LMM)**} \\ \cline{2-5}
**Harmonic Order** & **Amplitude (m/s\({}^{2}\))** & **Phase (rad)** & **Amplitude (m/s\({}^{2}\))** & **Phase (rad)** \\ \hline Fundamental response & 3 & 1.2 & 2.999999 & 1.199999 \\ Second harmonic & 0.6 & 0.9 & 0.599999 & 0.89999 \\ Third harmonic & 0.5 & 0.7 & 0.500000 & 0.700000 \\ Fourth harmonic & 0.4 & 0.5 & 0.399999 & 0.500000 \\ Fifth harmonic & 0.3 & 0.4 & 0.300000 & 0.40000 \\ Sixth harmonic & 0.2 & 0.3 & 0.199999 & 0.299999 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The result of contrast between the estimated values and the given values. Figure 6: The estimation error. Figure 7: The identification acceleration. To validate the impulse noise suppression ability of the Adaline-LMM algorithm in harmonic identification, impulse noises (SNR: 15 dB) are independently added into the simulation input signal and the appearing time of these are fixed at 2 s, 3 s and 4 s during the simulation. The amplitudes are  
Figure 8: Estimation amplitudes of simulation results. Figure 9: Estimation phases of simulation results. set as 0.8891. For confirming the advantage of LMM on impulse noise suppression objectively, as a contrast, LMS algorithm is also used in harmonic identification when the input signal is contaminated by the impulse noise. Other parameters are set as the same with the above. According to the results which is shown in Figure 11, the estimation error has large fluctuation at the time when impulse noise is acted in the input signal and there still has small oscillation after 2.5 s, 3.5 s and 4.5 s (the action time of the impulse noise are 2 s, 3 s and 4 s), but the estimation error is converged to zero and the error oscillation does emerge during the whole identification time.