## 3 Simulation Results
### Case I  
In this experiment, the stability of the system will be studied in the presence of a strong variation in the primary and secondary paths. At stage I, in the first 50,000 simulation samples, the impulse response of the primary path and the secondary path is \(\mathbf{p}_{1}(n)\), \(\mathbf{s}_{1}(n)\), respectively. At stage II, from n = 50,000 to n = 100,000, the impulse response of the primary path and the secondary path is \(\mathbf{p}_{2}(n)\), \(\mathbf{s}_{2}(n)\), respectively. The impulse response of the primary path and secondary path is shown in Figure 2. \(\mathbf{p}_{1}(n)\) and \(\mathbf{s}_{1}(n)\) are set as:  
\[\mathbf{p}_{1}(n)=[p_{1}(0),p_{1}(1),\dots,p_{1}(L_{p})]^{T}, \tag{67}\]  
\[\mathbf{s}_{1}(n)=[s_{1}(0),s_{1}(1),\dots,s_{1}(L)]^{T}, \tag{68}\]  
where \(L_{p}=48\) is the order of primary path impulse response. \(\mathbf{p}_{2}(n)\) and \(\mathbf{s}_{2}(n)\) are set as:  
\[\mathbf{p}_{2}(n)= \begin{array}{l}[-p_{1}(0),\frac{1}{3}p_{1}(0)-p_{1}(1),\dots,\\ \frac{1}{3}p_{1}(L_{p}-1)-p_{1}(L_{p})]^{T}\end{array}, \tag{69}\] \[\mathbf{s}_{2}(n)= \begin{array}{l}[-s_{1}(0),\frac{1}{3}s_{1}(0)-s_{1}(1),\dots, \\ \frac{1}{3}s_{1}(L_{s}-1)-s_{1}(L)]^{T}\end{array}. \tag{70}\]According to the definition of \(\mu_{w\max}\) in (44), \(\mu_{w}\) is set to \(\mu_{w\max}/K\), where \(K=3,6,10\) and \(\mu_{s}=0.002\). \(P_{x}(0)\) and \(P_{f}(0)\) are initialized as 1, \(\lambda=0.99\). The upper bound of R is set to 1. The simulation results are shown in Figure 3.  
The system can remain stable with two different input signals. As can be seen from Figure 3a,c,d,f, the larger the \(\mu_{w}\), the faster the ANC filter converges, but the steady-state relative error is also larger. Second, the system can remain stable when there is a sudden change in the secondary and the primary paths. The smaller the value of \(\mu_{w}\), the smaller the fluctuation of the ANC filter. It seems surprising that the smaller step size has a faster convergence speed in stage II. When the relative error of the ANC filter reaches a minimum, there is a rising stage. This can be explained by Figure 3c,f. As the system converges, the value of \(\mu_{w\max}\) gradually increases, which causes the steady-state relative error of the ANC filter to increase.  
Figure 2: The impulse responses of primary path and secondary path. (**a**) The impulse response of the primary path \(\mathbf{p}_{1}(n)\); (**b**) the impulse response of the primary path \(\mathbf{p}_{2}(n)\); (**c**) the impulse response of the secondary path \(\mathbf{s}_{1}(n)\); (**d**) the impulse response of the secondary path \(\mathbf{s}_{2}(n)\).  
Figure 3: The white Gaussian input: (**a**) the relative modeling error \(\Delta w(n)(dB)\); (**b**) the relative modeling error \(\Delta s(n)(\mathrm{dB})\); (**c**) the time-varying step-size upper bound parameter \(\mu_{\mathrm{zmax}}(n)\). The first-order AR process input: (**d**) the relative modeling error \(\Delta w(n)(dB)\); (**e**) the relative modeling error \(\Delta s(n)(\mathrm{dB})\); (**f**) the time-varying step-size upper bound parameter \(\mu_{w\mathrm{max}}(n)\).  
It can be seen from Figure 3b,e that the convergence speed and steady-state error of the SPM filter are almost the same in the three cases because of the same value of \(\mu_{s}\). Figure 3b,c shows that \(\mu_{w}\) is controlled by the modeling accuracy of the secondary path. The value of \(\mu_{w\text{max}}\) was smaller at the initial stage of the system and when the secondary path underwent a sudden change. As the SPM filter converges, the value of \(\mu_{w\text{max}}\) gradually increases and tends to be stable. Figure 3d suggests that K = 3 results in a more significant steady-state relative error of ANC filter, which leads to a larger steady-state relative error for the SPM filter.