## 3 Proposed Method  
Here, a new online SPM method with an auxiliary noise power scheduling strategy for the NANC system is proposed by extending the method developed by the authors [28] to improve the performance of the whole system. The block diagram of the proposed method is shown in Figure 3. We see that the structure of the proposed method is similar to that of [PERSON]'s method shown in Figure 2, except for some modifications in the following subsystem. First, let us consider the NANC subsystem shown in Figure 3. To increase the convergence rate and traceability, a modified FXWALMS (MFXWALMS) algorithm with the variable forgetting factor scheme [28] is used in the NANC subsystem. The MFXWALMS algorithm for the DFC estimates of the secondary source can be rewritten as  
\[\hat{a}_{i}(n+1) =\hat{a}_{i}(n)+\mu_{i}e_{p}(n)\hat{x}_{a_{i}}(n)+\lambda(n)(\hat{ a}_{i}(n)-\hat{a}_{i}(n-1)) \tag{27}\] \[\hat{b}_{i}(n+1) =\hat{b}_{i}(n)+\mu_{i}e_{p}(n)\hat{x}_{b_{i}}(n)+\lambda(n)(\hat {b}_{i}(n)-\hat{b}_{i}(n-1)) \tag{28}\]  
where \(\lambda(n)\) is the time-varying forgetting factor, the update equation of which can be expressed as  
\[\lambda(n+1)=\gamma\lambda(n)+\epsilon e(n)e(n-1),0\leq\lambda(n)<1 \tag{29}\]  
In the above equation, \(\gamma\) and \(\epsilon\) are similar to those used in the algorithm of [28]. The upper limit parameter \(\lambda_{max}\) can be used to avoid \(\lambda(n+1)\) values that are too large and to make the system more stable. Here, we substitute the error autocorrelation of the entire system \(e(n)e(n-1)\) for the  
Figure 3: NANC system with online SPM using new auxiliary noise power scheduling strategy (\(i\)th channel). error autocorrelation of the NANC subsystem \(e_{p}(n)e_{p}(n-1)\) to update \(\lambda(n)\), as \(\lambda(n)\) is used in the two subsystems and the residual error of the entire system can better indicate the status of the whole system compared to the error signal \(e_{p}(n)\) of the NANC subsystem. For the online SPM subsystem, according to the characteristics of the range of the variable forgetting factor \(\lambda(n)\), new auxiliary noise power scheduled based on the convergence status of \(\lambda(n)\) can be expressed as  
\[v_{g}(n)=\lambda(n-1)v(n) \tag{30}\]  
Using \(v_{g}(n)\) to replace \(v(n)\) in Equations (15) and (16), one obtains  
\[y_{v}(n)=\mathbf{V_{g}^{T}}(\mathbf{n})\mathbf{\hat{S}}(\mathbf{n})=\sum_{j=0 }^{\hat{M}-1}\hat{s}_{j}(n)v_{g}(n-j)=\sum_{j=0}^{\hat{M}-1}\hat{s}_{j}(n) \lambda(n-1-j)v(n-j) \tag{31}\]  
\[e_{v}(n)=e_{p}(n)=e(n)-y_{v}(n)=e(n)-\sum_{j=0}^{\hat{M}-1}\hat{s}_{j}(n) \lambda(n-1-j)v(n-j) \tag{32}\]  
where \(\mathbf{V_{g}}(\mathbf{n})\) is redefined as  
\[\mathbf{V_{g}}(\mathbf{n})=[\lambda(n-1)v(n)\quad\lambda(n-2)v(n-1)\quad \cdots\quad\lambda(n-\hat{M}-2)v(n-\hat{M}-1)]^{T} \tag{33}\]  
Similarly, the signal \(y(n)\) can be recalculated as  
Figure 3.0  
Substituting the above equation into Equation (5), one has  
Figure 3.1  
The LMS algorithm for the coefficient vector of \(\hat{S}(z)\) is given by  
Figure 3.2  
Using error autocorrelation to update \(\lambda(n)\) not only is a good measure of the proximity to the optimum but also can reduce the effect of the uncorrelated noise. Generally, the error autocorrelation estimate is a large value in the convergence stages or error bursting, which may be caused by the changes in the secondary path or primary noise, resulting in a large \(\lambda(n)\). When the adaptation process approaches the steady state, the value of the error autocorrelation approaches zero, resulting in a small \(\lambda(n)\). These characteristics of the change in \(\lambda(n)\) are very useful for the two subsystems. For the NANC subsystem, the large \(\lambda(n)\) improves the convergence speed and tracing capabilities, and the small \(\lambda(n)\) can ensure a low misadjustment. For the SPM subsystem, the auxiliary noise \(v(n)\) is scaled by \(\lambda(n)\) before it is added to the entire system. In this way, the large \(\lambda(n)\) will yield a large \(v_{g}(n)\), which can improve modeling accuracy.