# Notes on a New Structure of Active Noise Control Systems
## 4 Simulation Case Studies
### Case Study 2  
In the second simulation case study, dynamic properties of the disturbance and secondary paths are changed to represent a dynamic system in which the prediction of the noise to be attenuated is more difficult than in the previous simulation case study. The total time delay in the error signal measurement path, controller, and secondary path (14 ms) is longer than that the time delay in the disturbance path (6 ms). The corresponding impulse responses are presented in Figure 10.  
The above change implies that the low correlated noise used in the previous simulation case study could not be effectively attenuated. To obtain a visible noise reduction, the power spectral density of noise to be attenuated was also changed to a power spectral density defining a highly correlated wide-sense stationary Gaussian random noise. Its power spectral density is presented in Figures 11 and 12 with red lines. This noise was attenuated in the classical FxLMS and new FeFxLMS control system structures. Parameters of the adaptive control algorithms were the same as for the previous simulation example except of the parameter \(\mu\) that was equal to 0.03. In Figure 11, the estimated power spectral densities of the discrete-time signals \(e(i)\) and \(e_{0}(i)\) in the classical FxLMS control system structure and corresponding estimates for the new FeFxLMS control system structure calculated using the discrete-time signals \(e(i)\), \(e_{0}(i)\), and \(\delta_{0}(i)\) are presented. Power spectral densities of the discrete-time signals \(e_{0}(i)\) estimated in the classical FxLMS and new FeFxLMS control system structures are compared in Figure 12. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Signal** & **Classical FxLMS Structure** & **New FeFxLMS Structure** \\ \hline \(e_{0}(i)\) & 26.2 dB & 29.0 dB \\ \(e(i)\) & 30.9 dB & 30.9 dB \\ \(\delta_{0}(i)\) & \(----\) & 30.4 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Noise reduction: Continuous spectrum low-correlated case. Figure 10: Impulse responses of the disturbance and secondary paths. Noise reduction obtained with the classical FxLMS and new FeFxLMS control system structures is presented in Table 2. Similarly, as in the previous simulation case study, the new FeFxLMS control system structure gave a greater (approximately 50%) noise reduction in the error microphone (signal \(e_{0}(i)\)). It was also observed that similarly to the previous case study, the discrete-time signal \(e_{0}(i)\) exposed a greater rate of convergence in the new FeFxLMS control system structure than in the classical FxLMS control system structure. Figure 11: Estimated power spectral densities of the discrete-time signals \(d(i)\), \(e(i)\), \(e_{0}(i)\), and \(\hat{e}_{0}(i)\) (\(e_{est}(i)\))—the classical FxLMS (**a**) and new FeFxLMS (**b**) control system structure. Figure 12: Estimated power spectral densities of the discrete-time signals \(d(i)\) and \(e_{0}(i)\) at the error microphone—the classical FxLMS and new FeFxLMS control system structures. In the next step, the rate of convergence of the discrete-time signal \(e_{0}(i)\) is compared in the classical FxLMS and new FeFxLMS control system structures in simulations run to attenuate another noise. It is a wide-sense stationary random process with a mixed spectrum defined at the primary noise source, as a sum of two random sines and a discrete-time Gaussian white noise with a given variance. The random sine components frequencies were equal to 50 and 120 Hz, the amplitudes were equal to 1, and the phase shifts were random, equally distributed in the range \([0,2\pi)\). It should be noted that the Gaussian white noise after processing by dynamics of the reference signal measurement path and acoustical disturbance path was no longer white. Its power spectral density at the error microphone looked similar to the power spectral density of the noise used in the first simulation case study.