# A Four-Stage Method for Active Control with Online Feedback Path Modelling Using Control Signal
## 3 Simulations
The computational complexity is calculated in terms of the number of multiplications (per samples) and number of additions (per sample). All the methods require \(L_{\text{w}}\) multiplications and \(L_{\text{w}}-1\) additions to obtain the controller output; \(L_{\text{s}}\) multiplications and \(L_{\text{s}}-1\) additions to obtain the filtered reference signal; \(L_{\text{f}}\) multiplications and \(L_{\text{f}}-1\) additions to obtain the estimated feedback signal; \(L_{\text{w}}+1\) multiplications and \(L_{\text{w}}\) additions to update the control filter. Unlike [PERSON]'s method a single filter is used for feedback path modelling and neutralization. The proposed method requires \(2N\) multiplications and \(2(N-1)\) additions to obtain the decorrelation filter outputs; \(2L_{\text{f}}+1\) multiplications and \(2L_{\text{f}}-1\) additions for updating the feedback path modelling and neutralization filter; and \(2N+1\) multiplications and \(2N-1\) additions for updating the decorrelation filter. [PERSON]'s method requires \(L_{\text{h}}\) multiplications and \(L_{\text{h}}-1\) additions for obtaining the ADNC filter output; \(2L_{\text{h}}+1\) multiplications and \(2L_{\text{h}}-1\) additions for updating the ADNC filter using the NLMS algorithm. The basic computational load (filtering and filter update) for the related methods are summarized in Table 1, in which \((\times)\) denotes the number of multiplications per sample and \((+)\) denotes the number of additions per sample associated with different operations. For a particular case considered in the simulations, the total number of multiplications per sample and additions per sample are also included in the last row of Table 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Operation**} & \multicolumn{3}{c}{**Proposed (With the**} & \multicolumn{3}{c}{**Proposed (Without the**} & \multicolumn{3}{c}{**Akhtar’s Method**} \\  & **Decorrelation Filters)** & \multicolumn{3}{c}{**Decorrelation Filters)**} & \multicolumn{3}{c}{**Akhtar’s Method**} \\ \cline{2-7}  & \(\times\) & **+** & \(\times\) & **+** & \(\times\) & **+** \\ \hline Controller output & \(L_{\text{w}}\) & \(L_{\text{w}}-1\) & \(L_{\text{w}}\) & \(L_{\text{w}}-1\) & \(L_{\text{w}}\) & \(L_{\text{w}}-1\) \\ Filtered signal & \(L_{\text{s}}\) & \(L_{\text{s}}-1\) & \(L_{\text{s}}\) & \(L_{\text{s}}-1\) & \(L_{\text{s}}\) & \(L_{\text{s}}-1\) \\ Controller update & \(L_{\text{w}}+1\) & \(L_{\text{w}}\) & \(L_{\text{w}}+1\) & \(L_{\text{w}}\) & \(L_{\text{w}}+1\) & \(L_{\text{w}}\) \\ Neutralization filter output & \(L_{\text{f}}\) & \(L_{\text{f}}-1\) & \(L_{\text{f}}\) & \(L_{\text{f}}-1\) & \(L_{\text{f}}\) & \(L_{\text{f}}-1\) \\ Neutralization filter update & \(2L_{\text{f}}+1\) & \(2L_{\text{f}}-1\) & \(2L_{\text{f}}+1\) & \(2L_{\text{f}}-1\) & None & None \\ Decorrelation filter output & \(2N\) & \(2(N-1)\) & None & None & None & None \\ Decorrelation filter update & \(2N+1\) & \(2N-1\) & None & None & None & None \\ ADNC filter output & None & None & None & None & \(L_{\text{h}}+1\) & \(2L_{\text{h}}-1\) \\ ADNC filter update & None & None & None & None & \(2L_{\text{h}}+1\) & \(2L_{\text{h}}-1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational load for different methods. In summary, the proposed four-stage method and the decorrelation filters maintain the control operation with improved performance and system stability when the feedback path changes. In contrast to the existing algorithm, the proposed method detects the change of acoustic paths by monitoring the preset thresholds, and uses five switches to decouple the feedback path modelling from control operation. Therefore, the implementation of the proposed algorithm is quite simple. Although, the proposed method can be useful for broadband signals for feedback path modelling, it is not effective for tonal signals because it is not possible to compensate all the feedback components when the feedback path modelling is carried out using a tonal control signal.