## 4 Simulations and Results
### Case 1  
In this case, the primary noise includes three frequencies \((q=3)\): \(w_{1}=0.1\pi\), \(w_{2}=0.2\pi\), and \(w_{3}=0.3\pi\). The corresponding DFCs are set to \(a_{1}=2.0\), \(b_{1}=-1.0\), \(a_{2}=1.0\), \(b_{2}=-0.5\), and \(a_{3}=0.5\), \(b_{3}=0.1\), respectively. \(S(z)\) is the fixed secondary path. The frequency responses of \(S(z)\) are shown in Figure 4a. To compare the scaling strategy of \(|e(n-1)|\) and \(\lambda(n-1)\) more clearly, Method 3, which uses auxiliary noise scheduled based on \(\lambda(n-1)\) but employs the same NANC subsystem as that of Method 2, is included in the comparison. The parameters of the four methods are set as follows. In Method 1, \(\mu_{i}=0.0002\) for all i, \(\mu_{s}=0.004\), and the scaled factor \(k=0.2\). The corresponding parameters in Method 2 are \(\mu_{i}=0.0002\) for all i, \(\mu_{s}=0.02\), and \(k=0.2\). The parameters used in Method 3 and the proposed method are \(\mu_{i}=0.0002\) for all i, \(\mu_{s}=0.02\), \(\gamma=0.9992\), \(\epsilon=0.0012\), \(\lambda_{max}=0.9\), and \(k=1\), and the initial forgetting factor is set as \(\lambda(1)=0.4\). The simulation results for Case 1 are provided in Figures 5-8. Figure 4: Frequency response of the acoustic paths \(S(z)\). (**a**) Original acoustic path; (**b**) Changed acoustic path at \(n=50\),000. Figure 5a-f shows comparisons between the DFC estimation MSEs of Method 1, Method 2, Method 3, and the proposed method. The MSEs \(E[e^{2}(n)]\) are displayed in Figure 6a. To identify the system performance more clearly, each MSE value for the plot was obtained by averaging the square values of 50 past residue errors, as shown in Figure 6b. The modeling error \(\triangle S(z)\) and the mean variable forgetting factor of the proposed method are shown in Figures 7 and 8, respectively. To show the process of adjusting the auxiliary noise carried out in Method 2, the \(e(n-1)\) of Method 2 is shown in Figure 8.  
The following remarks can be made based on above simulation results. (1) Regarding the performance of convergence, the proposed method has a faster rate of convergence, as a result of using the FXWALMS algorithm, than that of Method 1 and Method 2, as shown in Figures 5a-f and 6. (2) Regarding the steady-state performance of the system, the fixed power auxiliary noise used in Method 1 not only contributes to the residual error but also seriously degrades the steady-state performance of the DFC estimation MSEs. However, the proposed method and Method 2 have better steady-state performances than Method 1. Moreover, although Method 2 has a similar steady-state performance to that of the proposed method based on the DFC estimation MSEs, the proposed method has a better steady-state performance for the residual error \(E[e^{2}(n)]\), as shown in Figures 5a-f and 6. The reason for the improved noise-reduction performance is that the proposed auxiliary noise power scheduling strategy results in a smaller auxiliary noise power in the steady state than that of Method 2. Figure 5: Comparison between the discrete Fourier coefficient (DFC) estimation mean square errors (MSEs) of Method 1, Method 2, and the proposed method for the fixed acoustic path \(S(z)\) shown in Figure 4a. (**a**) \(E[\varepsilon_{a_{1}}^{2}(n)]\); (**b**) \(E[\varepsilon_{a_{2}}^{2}(n)]\); (**c**) \(E[\varepsilon_{a_{3}}^{2}(n)]\); (**d**) \(E[\varepsilon_{b_{1}}^{2}(n)]\); (**e**) \(E[\varepsilon_{b_{2}}^{2}(n)]\); (**f**) \(E[\varepsilon_{b_{3}}^{2}(n)]\). Figure 8: Mean variable forgetting factor of the proposed method for the fixed acoustic path \(S(z)\) shown in Figure 4a. Figure 6: Comparison between the residual errors of Method 1, Method 2, and the proposed method for the fixed acoustic path \(S(z)\) shown in Figure 4a. (**a**) Mean square error \(E[e^{2}(n)]\). (**b**) Mean square error \(E[e^{2}(n)]\) obtained by averaging the square values of 50 past residue errors.