## 4 Simulation
### Periodic Sinusoids: \(\omega_{2,3}=n\omega_{1}\)  
For periodic sinusoids, we used \(A_{1}=0.02\), \(A_{2}=0.05\), \(A_{3}=0.03\), \(\omega_{1}=10\) Hz, \(\omega_{2}=30\) Hz, and \(\omega_{3}=40\) Hz. It was assumed that there was no phase difference in each term. Figure 4 is a graph comparing the signal tracking performance between the LMS algorithm and the proposed MSA-LMS algorithm. It can be seen that the MSA-LMS algorithm with step-size changing for each sampling time according to exponential and autocorrelation functions was superior to the LMS algorithm with fixed step size over the entire simulation range of time. To compare the tracking performance between the LMS algorithm and the proposed MSA-LMS algorithm in detail, the interval from 3.5 to 4 s was enlarged, as shown in Figure 5. In the case of the LMS algorithm, it can be observed that the tracking performance for the desired signal was significantly lowered from the second-high peak to the low peak near 3.55 s. Furthermore, the direction was completely reversed in the middle of a vibration signal, which may have caused a spillover phenomenon with a higher frequency component than the original signal. On the other hand, in  
Figure 4: Control of “Periodic Sinusoids” in the time domain, \(\omega_{2,3}=n\omega_{1}\). (**a**) Comparison of desired signal and filter output controlled with LMS. (**b**) Comparison of desired signal and filter output controlled with MSA-LMS. Key:  
the case of the MSA-LMS algorithm using an energy autocorrelation function of instantaneous error, the misadjustment was found to be properly adjusted and the tracking performance was excellent. Figure 6 compares the estimation error of the active vibration control between the two algorithms. It can be observed that both algorithms exhibited almost the same values of error up to 2 s. However, between 2 and 3 s, the maximum and minimum values of the control error were 0.1963 and \(-\)0.1915, respectively, for the LMS algorithm, while 0.1176 and \(-\)0.1456 for the MSA-LMS algorithm. Similarly, between 3 and 3.5 s, the respective values were 0.1941 and \(-\)0.1917 for LMS, while 0.0928 and \(-\)0.0622 for MSA-LMS. Finally, between 3.5 and 4 s, the respective values were 0.1930 and 0.1946 for LMS, while 0.0558 and 0.0393 for MSA-LMS. Based on the above results, the peak-to-peak value of the proposed MSA-LMS algorithm was found to be 32% less than that of the LMS algorithm with fixed step size between 2 and 3 s, 60% less between 3 and 3.5 s, and 75% less between 3.5 and 4 s. However, it may be noted that the peak-to-peak value of the error did not reduce over time for the LMS algorithm, while it gradually reduced for the MSA-LMS algorithm, owing to its adaptability associated with the multi-stage concept. The above results are summarized in Table 1 below. Figure 5: Comparison of desired signal with filter outputs controlled with LMS and MSA-LMS algorithms in the case of the control of “Periodic Sinusoids” in the time domain, \(\omega_{2,3}=n\omega_{1}\). Key:, desired signal;, controlled with LMS;, controlled with MSA-LMS. Figure 6: Predicted estimation error with LMS and MSA-LMS algorithms. Figure 7 is a graph comparing the error between the LMS algorithm and the proposed MSA-LMS algorithm. Since the MSA-LMS algorithm had an energy autocorrelation function of exponential function and instantaneous error, it can be seen that the error gradually decreased from 2 s onwards, while no such reduction was observed in the case of the LMS algorithm. Figure 8 is a graph comparing the frequency spectra of the proposed MSA-LMS algorithm with the LMS algorithm. The main peaks at \(w_{1}=10\) Hz, \(w_{2}=30\) Hz, and \(w_{3}=40\) Hz for the MSA-LMS algorithm were reduced by 5 to 9 dB, compared with the LMS algorithm. However, in the case of the MSA-LMS algorithm, the spillover phenomenon in the broadband was slightly increased.