## 3 Simulation and Validation of the Proposed Algorithm
### Validation of the Proposed Algorithm  
When the speed of the engine changes, the engine noise inside the cabin of a car is comprised of the sum of amplitude- and frequency-modulated order signals. Section 3.1 presents the details for when two synthesized signals were simulated. In this section, details for when the synthesized signals were used for the validation of the control performance of the OW-FxLMS algorithm are presented. These results were compared with the conventional method, which is the LMS-algorithm-based adaptive notch filter. The conventional method cannot control the amplitude of the time-varying engine order sound because the frequency of the engine order sound is changed quickly when the rotating speed of the crankshaft increases or decreases. The secondary path was not used in this validation in order to exclude its influence. **Signal 1**. Frequency-modulated signal (single order signal)  
A frequency-modulated signal with a single order is applied to the ANC system with different algorithms. Figure 3 illustrates the time history of error signals controlled by the NLMS algorithm-based adaptive notch filter with two weights (\(L=2\)). Different step sizes are used to observe the resulting effect of convergence. In Figure 3, the letter \"u\" denotes step size. Figure 3a-c illustrate the time history of errors controlled by the NLMS-algorithm-based adaptive notch filter with different step sizes of \(\overline{\mu}=0.5\), \(\overline{\mu}=2\), and \(\overline{\mu}=3\), respectively, but the leakage factor is 0.5. Figure 3d shows the comparison of power spectrum density (PSD). From these results, it can be observed that the convergence of the error signal is fast initially as the step size increases, but that the amplitude of the error signal diverges eventually. The adaptive notch filter cannot trace and control the time-varying signal with fast frequency change. Because the frequency bandwidth of the notch filter is very narrow it is not suitable when the frequency of the signal to be controlled is changed. Therefore, it is finally diverged. Subsequently, in order to observe the effect of leakage, different leakage factors are applied to the leaky NLMS algorithm in Equation (11) with the same step size (\(\overline{\mu}=2\)). Figure 4a-c illustrates the time history of the error signal controlled by the NLMS-algorithm-based adaptive notch filter with different leakage factors of \(\gamma\) = 0.5, \(\gamma\) = 0.7, and \(\gamma\) = 1.0, respectively, and the step size is 0.5. In Figure 4, the letter \"g\" denotes leakage factor. According to these results, the amplitude of the error signal diverges eventually. Even with the selection of any leakage factor, the controller cannot work stably. Subsequently, in order to observe the effect of frequency variation, reference signals with five different sweep rates are used. Figure 5 shows the variation of instantaneous frequency (\(f_{i}\) = rpm/60) of the five references. The sweep time is 5 s for all the references. The selected sweep rates, which are calculated using \(\Delta f_{i}/\Delta time\), are 4, 6, 8, 16, and 36 s.  
Figure 6 illustrates the time history and PSDs of the error signals 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. These results demonstrate that the error signals do not converge and are not stable for all the cases with different sweep rates. Finally, we attempted to determine the optimal step size and leakage factor for signal 1. Using the previous simulation results, we first used the step size of \(\overline{\mu}\) = 2 and leakage factor of \(\gamma\) = 1 in this simulation. Figure 3: Time history and PSD of signal 1 controlled by the normalized least mean squares filter (NLMS) algorithm-based adaptive notch filter with two weights (\(L\) = 2) and leakage factor (\(\gamma\) = 0.5).