## 3 Simulation and Validation of the Proposed Algorithm
### Validation of the Proposed Algorithm
(**a**) Time history (\(\overline{\mu}\) = 0.5); (**b**) Time history (\(\overline{\mu}\) = 2); and (**c**) Time history (\(\overline{\mu}\) = 3); and (**d**) PSD. Figure 4: Time history and PSD of signal 1 controlled by the NLMS algorithm-based adaptive notch filter with two weights (\(L=2\)) and leakage factor (\(\gamma=0.5\)). (**a**) Time history (\(\gamma=0.5\)); (**b**) Time history (\(\gamma=0.7\)); and (**c**) Time history (\(\gamma=1\)); and (**d**) PSD. Figure 5: Variation of instantaneous frequency (\(f_{i}=\mathrm{rpm}/60\)) of five references. Figure 6: _Cont._  
The time history of the error signal is shown in Figure 7a. The error signal still diverges. In order to converge the error signal, we decreased the step size from \(\overline{\mu}=2\) to \(\overline{\mu}=1.75\) and the leakage factor from \(\gamma=1\) to \(\gamma=0.75\) corresponding to the rule of leaky LMS algorithm. The error signal converges initially as shown in Figure 7b but eventually it diverges. For faster convergence at the start, we increased the step size from \(\overline{\mu}=2\) to \(\overline{\mu}=2.75\). In this case, a jump in the error signal appears initially as shown in Figure 7c and eventually the error diverges. In any of the cases, the PSD of the error signal is not attenuated as shown in Figure 7d. It can be concluded that the leaky NLMS-algorithm-based adaptive notch filter is not suitable for active cancellation of the rapid frequency varying disturbances. Figure 6: Time history and PSD of the signal \(1\) with different sweep rate controlled by the Leaky NLMS algorithm-based adaptive notch filter with two weights (\(L=2\)), leakage factor (\(\gamma=1\)), and step size \(\overline{\mu}=2\). Figure 7: Time history and PSD of signal \(1\) controlled by the NLMS algorithm-based adaptive notch filter with two weights (\(L=2\)) (**a**) Time history (\(\overline{\mu}=2\), \(\gamma=1\)); (**b**) Time history (\(\overline{\mu}=1.75\), \(\gamma=0.75\)); and (**c**) Time history (\(\overline{\mu}=2.75\), \(\gamma=0.75\)) and (**d**) PSD. Figure 8a shows a comparison between the time history of error signals controlled by the LMS-algorithm-based adaptive notch filter and the OW-LMS algorithm. The error signal controlled by the adaptive notch filter shows incomplete convergence as the frequency increases according to the time increment. However, the OW-LMS algorithm with a short length (\(L=30\)) of weight vectors is sufficient to attenuate this frequency modulated signal. The results show fast and stable convergence. The PSD function of the error signals is compared, and the results are plotted as shown in Figure 8c. According to these results, the OW-LMS algorithm exhibits superior control performance compared to the conventional method for the frequency-modulated signal. Figure 8e shows an image map of the power spectrum of the weight vector of the adaptive filter used in the OW-LMS algorithm. This map shows the adaptive performance of the filter. Since the length of the weight vector is 30, the frequency bandwidth of the filter is 200 Hz as shown in Figure 8g. In the case of signal 1, according to Equation (15), the optimal length (\(L\)) of the weight vector is changed from 300 (at 600 rpm) to 60 (at 6000 rpm) because \(\Delta k\) is 1 (single frequency) and fs is 3000 Hz. If we use the adaptive filter with the largest optimal length (\(L=300\)) of weight vector, the convergence is faster and more stable as compared to other algorithms as shown in Figure 8b. The attenuation level of the error signal is the largest in Figure 8d. The frequency bandwidth of the adaptive filter is narrow as shown in Figure 8f,h. Figure 8: \(Cont.\)  
In conclusion, in the case of a sinusoidal signal with single frequency modulation (i.e., single order signal), the OW-LMS algorithm with the optimal length of weight vector can be used for sufficient control of the error signal. Its convergence performance is better than that of the conventional algorithm. **Signal 2**.