# A New Effective Narrowband Active Noise Control System for Accommodating Frequency Mismatch
## 4 Simulation Results and Discussion
### Simulation 1: The MSE of All NANC System  
This simulation was carried out to demonstrate the MSE and convergence performance of the above NANC algorithms under different FMs. The step sizes were set as: \(\mu_{i}=[0.1\ 0.1\ 0.1],\ (i=1,\ 2,\ 3)\), \(\mu_{c_{i}}=[0.00025\ 0.00025\ 0.00025],\ (i=1,\ 2,\ 3)\) in the conventional NANC system and the NANC system with MLMS; \(\mu_{i}=[0.1\ 0.1\ 0.1],\ (i=1,\ 2,\ 3)\), \(\mu_{c_{i}}=[0.00025\ 0.00025\ 0.00025],\ (i=1,\ 2,\ 3)\) in the proposed NANC system. The momentum factor \(\lambda=10\) was set in the NANC system with MLMS. The step sizes of the delay model were set as \(\mu_{k_{ai}}=\mu_{k_{bi}}=[0.05\ 0.05\ 0.05],\ (i=1,\ 2,\ 3)\) and the delay unit \(k\) was set to \(14\) in the proposed NANC system. Figures 5-8 investigate the steady-state MSE in the different NANC systems, where the FM varied between \(1\%\) and \(20\%\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline \(f_{s}\) & \(1\ \mathrm{kHz}\) \\ \(\delta_{p}^{2}\) & \(0.1\) \\ \(\omega_{p}=\left[\omega_{p,1}\ \omega_{p,2}\ \omega_{p,3}\right]^{T}\) & \([0.1\pi\ 0.2\pi\ 0.3\pi]^{T}\) \\ \(a_{p}=\left[a_{p,1}\ a_{p,2}\ a_{p,3}\right]^{T}\) & \([2\ 1\ 1]^{T}\) \\ \(b_{p}=\left[b_{p,1}\ b_{p,2}\ b_{p,3}\right]^{T}\) & \([1\ 1\ 1]^{T}\) \\ \(a_{i}=b_{i}=a_{k_{i}}=b_{k_{i}}\) & \([1\ 1\ 1]^{T}\) \\ FM C\(\%\) (C is a random constant) & \(\omega_{p,i}(1+C\%)\) \\ \(S(z)\) & \(M=11,cut\_off\) _frequency_\(0.4\pi\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The basic conditions. It is clear that the system eventually converges to \(-10\)dB when the FM varies between 5% and 15%. Moreover, the MSE and convergence rate of the conventional NANC system are similar to the system with MLMS. In Figure 5, given the same 5% FM, it appears that the proposed NANC system can achieve the same steady state and the same convergence rate as the others, but the curve of the new proposed NANC system has the fastest downward trend. Clearly, in Figure 6, the proposed system converges fastest and significantly outperforms its counterpart in the 10% FM condition. In Figure 7, a great superiority in the convergence was shown in the proposed system. However, at 20% FM, it is clear that a serious performance deterioration occurs in the NANC system with MLMS, as shown in Figure 8a. Obviously, in Figure 8b, even if it is unstable at the beginning, the proposed system achieves good noise cancellation after  
Figure 5: Under 5%FM (**a**) the conventional NANC system (**b**) the system with MLMS (**c**) the proposed system. Figure 6: Under 10%FM (**a**) conventional NANC system (**b**) the system with MLMS (**c**) proposed system. Figure 7: Under 15%FM (**a**) the conventional NANC system (**b**) the system with MLMS (**c**) proposed system. multiple iterations. Although the latter takes a long time to reach a stable state, it provides the possibility for accommodating a larger FM. In conclusion, all of the results further reveal that the proposed system converges faster than the others without damaging the noise reduction performance of the system, while both the NANC system and the NANC system with the MLMS algorithm fail to suppress the noise well with a large FM. Certainly, by comparing the two systems, it is confirmed that the method of the MLMS algorithm has little contribution to the noise reduction in the article [27]. Second, the improvement in the proposed NANC system benefits from the new adjustment strategy for the reference signal. When the reference signal is updated, the LMP is used to optimize the AR coefficient. The convergence of the AR coefficients in the NANC system is depicted in Figure 9. The red, pink, and blue lines draw the MSE of \(c_{i}\) in the conventional NANC system, the ones with MLMS and the proposed system, respectively. As it can be seen from (a), the proposed method converges faster than the others.