# A Four-Stage Method for Active Control with Online Feedback Path Modelling Using Control Signal
## 2 Proposed Method
A higher value of the feedback to reference signal ratio can have shorter time, whereas a smaller value requires a longer time. Here, the modelling time duration is chosen by trial and error and can be started with the same time that is used to model the secondary path. In the third stage, the switches K\({}_{1}\) and K\({}_{2}\) are turned on and the other switches are turned off, the control operation is carried out with the filtered-x least mean square (FxLMS) algorithm. The control weights are updated as:  
\[\mathbf{w}(n+1)=\mathbf{w}(n)-\mu_{w}\mathbf{e}(n)\mathbf{x}_{\mathrm{s}}(n), \tag{8}\]  
where \(\mu_{\mathrm{w}}\) is the step size with initial control weights \(\mathbf{w}(0)=\mathbf{0}\), a null vector of length \(L_{\mathrm{w}}\), \(\mathbf{x}_{\mathrm{s}}(n)=[x_{\mathrm{s}}(n)\), \(x_{\mathrm{s}}(n\!-\!1)\),..., \(x_{\mathrm{s}}\) (\(n\!-\!L_{\mathrm{w}}\!+\!1\))]\({}^{\mathrm{T}}\) with \(L_{\mathrm{w}}\) denoting the tap-weight length of the control filter, and \(x_{\mathrm{s}}(n)\) is the filtered reference signal, which is obtained by filtering \(x(n)\) with the secondary path estimate \(\hat{\mathbf{\mathrm{s}}}(n)\). The residual noise is reduced continuously towards normal noise reduction \(T_{\mathrm{r}0}\). If the noise reduction level increases suddenly, it triggers the system into the fourth stage. The fourth stage involves the feedback path changing detection. Assume that the normal residual noise level is \(T_{\mathrm{r}0}=-10\) dB in the operation. If suddenly it becomes greater than \(-10\) dB, i.e., \(T_{\mathrm{r}}\) (\(n\)) \(>\)\(T_{\mathrm{r}0}\), there is a path change. It is to be noted that \(T_{\mathrm{r}}(n)\) is calculated for each instance of time. At the same time, \(\sigma_{\mathrm{x}}^{2}(n)\), \(\sigma_{l}^{2}(n)\) and \(T_{\mathrm{d}}(n)\) are also estimated in a side branch of the system (detector in the schematic diagram shown in Figure 1). If \(T_{\mathrm{d}}\) (\(n\)) \(<\)\(T_{\mathrm{d}0}\) (\(T_{\mathrm{d}0}=-10\) dB for example), the system is stable and the change is caused by the primary path and the control operation is resumed. If \(T_{\mathrm{d}}(n)\)\(>\)\(T_{\mathrm{d}0}\), the rise in \(T_{\mathrm{r}}(n)\) may be due to the secondary path change or the feedback path change, and the control filter update is ceased by turning off K\({}_{2}\). In practice, the secondary path and the feedback path usually change simultaneously when the secondary source or sound propagation paths change. To check whether the change is due to the secondary path, K\({}_{1}\), K\({}_{4}\) and K\({}_{5}\) are turned on and the other switches are turned off. At this time, there is controller output but without controller update, \(\sigma_{\mathrm{\varepsilon}}^{2}(n)\) and \(T_{\mathrm{s}}(n)\) are estimated. If \(T_{\mathrm{s}}(n)\)\(>\)\(T_{\mathrm{s}0}\) (\(T_{\mathrm{s}0}=-15\) dB for example), the change is in the secondary path. In this case, the secondary path is remodelled and then the control operation is resumed with the new model. Here, \(T_{\mathrm{s}0}=-15\) dB is chosen based on an adequate secondary path model, which can be obtained with the extended filtering method [21; 22]. Although the proposed method focuses on the feedback path modelling, the secondary path modelling is highlighted here for the sake of completeness of the whole ANC system. If \(T_{\mathrm{s}}(n)\)\(<\)\(T_{\mathrm{s}0}\) (when K\({}_{1}\), K\({}_{4}\) and K\({}_{5}\) are turned on and the other switches are turned off), the rise in \(e(n)\) is due to the feedback path change. The feedback path is modelled by turning on switches K\({}_{1}\) and K\({}_{3}\) and turning off the other switches with the previously obtained \(W(z)\). After obtaining a new \(\hat{F}(z)\), K\({}_{1}\) and K\({}_{2}\) are turned on, the other switches are turned off, the active control operation is resumed by updating the control coefficients as in Equation (8), and the system runs in the third stage.