# A Four-Stage Method for Active Control with Online Feedback Path Modelling Using Control Signal
## 3 Simulations  
In the literature, little research can be found for feedback path modelling using the control signal. The closest related algorithm is [PERSON]'s method [10], which uses random noise for online feedback path modelling, so it is used as a benchmark for comparison with the proposed method. In the simulations, the primary path \(P(z)\), secondary path \(S(z)\) and the feedback path \(F(z)\) are FIR filters of lengths 48, 16 and 32, respectively, which are collected from the data provided in [2], Their spectra are shown in Figure 4. The adaptive filters \(W(z)\), \(\hat{S}(z)\) and \(\hat{F}(z)\) are selected as FIR filters of lengths \(L_{\mathrm{w}}=48\), \(L_{\mathrm{s}}=16\) and \(L_{\mathrm{f}}=32\), respectively. The adaptive de-correlation filter \(A(z)\) is an FIR filter of length \(N=5\)The reference signal \(r(n)\) is generated from an autoregressive process:  
\[r(n)=0.9r(n-1)+v(n), \tag{9}\]  
where \(v(n)\) is a white noise with zero-mean and unit variance. The white noise injected in [PERSON]'s method is a zero-mean with variance 0.05. The sampling frequency used in the simulation is 2 kHz. All the simulation results are averaged over 10 independent realizations. The mean square error (MSE) and the relative modelling error \(\Delta F\) are used as the metrics for comparison, which are defined as:  
\[\text{MSE (dB)}=10\ \log_{10}\left[E\left[e^{2}(n)\right]\right], \tag{10}\]  
\[\Delta F\left(\text{dB}\right)=10\ \log_{10}\left\{\frac{\left\|\vec{f}(n)- \hat{\vec{f}}(n)\right\|^{2}}{\left\|\vec{f}(n)\right\|^{2}}\right\}. \tag{11}\]  
In the first stage, \(\text{K}_{1}\) is turned on and the other switches are turned off for initializing the controller. The control filter is set as \(W(z)=z^{-47}\), representing a delay of \(L_{w}-1\) samples followed by unit gain. The amplitude of the residual error signal \(e(n)\) is higher than that of the undesired noise \(p(n)\) in this initialization, howling occurs. A nominal estimate of the feedback path using Equation (5) is used to maintain the stability of the control system. In the second stage, the feedback path modelling is carried out using the proposed decorrelation filters for 10 s, during which \(\text{K}_{1}\) and \(\text{K}_{3}\) are turned on and the other switches are turned off. The benefit of using the decorrelation filters can be noticed from the modelling error shown in Figure 5.  
Figure 4: Frequency response of (**a**) the primary path and the secondary path and (**b**) feedback path. Figure 5: Feedback path modelling error in the second stage. The improved modelling accuracy is due to the reduction of bias, which is achieved by the whitening property of the decorrelation filter. The steady-state modelling error without using the decorrelation filters fluctuate more as compared to that with the decorrelation filters due to the existence of bias. Furthermore, the decorrelation filters aid in faster convergence of the adaptive filter, e.g., to achieve \(-16\) dB modelling error, the one with the decorrelation filters takes \(0.7\) s compared to \(1.1\) s taken by the one without decorrelation filters. The modelling error obtained with and without decorrelation filters at \(2.0\) seconds are \(-20.3\) dB and \(-17.4\) dB, respectively. After the feedback path modelling, K\({}_{1}\) and K\({}_{2}\) are turned on and the other switches are turned off, the simulation for control operation runs for \(50\) s with the original acoustic paths shown in Figure 4. Then the feedback path changes suddenly, resulting in a situation \(T_{\mathrm{r}}(n)>-10\) dB and \(T_{\mathrm{d}}(n)>-10\) dB, the control update is ceased. The switches K\({}_{1}\), K\({}_{4}\) and K\({}_{5}\) are turned on and the other switches are turned off, the threshold \(T_{\mathrm{s}}(n)\) is checked confirming a feedback path change. The feedback path remodelling is carried out to estimate the new feedback path. After an adequate modelling, the control operation is resumed.