# Notes on a New Structure of Active Noise Control Systems
## 4 Simulation Case Studies
### Case Study 3
These differences were greater than 0 and thus show that adaptation was faster in the new FeFxLMS control system structure than in the corresponding classical FxLMS control system structure in the simulation time window of 50,000 samples. Figure 16: Differences between expressed in dB estimates of the mean square values of signals at the error microphone in the classical FxLMS and new FeFxLMS control system structures—the mixed spectrum noise with different variances of Gaussian white noise. Figure 15: Identified amplitude frequency responses of the reference signal measurement path (blue line), acoustical disturbance path (red line), secondary path control signal part (green line), and error signal measurement path (magenta line). In the next step the length of simulation time window was changed to \(10^{7}\) samples and the noise reduction obtained after the convergence of adaptation algorithm was estimated. The obtained results of the mixed spectrum noise reduction are summarized in Table 3.  
It is important to note that the obtained noise attenuation in the new FeFxLMS control system structure and classical FxLMS control system structure are the same, provided that the error signal \(e(i)\) is equal to 0 and consequently the error signal \(e_{0}(i)\) at the error microphone is equal to 0 as well. As it was mentioned, it was the only case in which the classical FxLMS control system structure could give an optimal result. For most of the attenuated noises, the results obtained in the new FeFxLMS control system structure were better than what was obtained in the classical FxLMS control system structure, besides the cases with Gaussian white noise variances equal to 1 and 2. Consequently, the power spectral densities estimated for the variance equal to 1 for a single realization of the discrete-time signals at the error microphone are presented in Figure 17. The power spectral density of the mixed spectrum noise at the error microphone was plotted with a red line and the power spectral densities or the discrete-time error signals \(e_{0}(i)\) in the classical FxLMS and new FeFxLMS control system structure are plotted with green lines. The noise components placed around the frequency 175 Hz were amplified by the new FeFxLMS control system structure more than by the classical FxLMS control system structure, which indicated that the parameter \(\mu\) of was too large. The parameter \(\mu\) for the variance of Gaussian white noise equal to 2 was too large as well. Figure 17: 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; variance of Gaussian white noise equal to 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variance of White Noise** & **Classical FxLMS Structure** & **New FeFxLMS Structure** \\ \hline \(0\) & \(179.8\) dB & \(179.9\) dB \\ \(1\times 10^{-4}\) & \(36.9\) dB & \(37.4\) dB \\ \(1\times 10^{-3}\) & \(27.0\) dB & \(28.2\) dB \\ \(1\times 10^{-2}\) & \(17.5\) dB & \(18.9\) dB \\ \(1\times 10^{-1}\) & \(9.2\) dB & \(9.5\) dB \\ \(1\) & \(3.0\) dB & \(1.6\) dB \\ \(2\) & \(2.0\) dB & \(-1.5\) dB \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Noise reduction—mixed spectrum case in a laboratory model. According to the above conclusion, the simulations were repeated with the parameter \(\mu\) decreased 100 times to the value \(5\times 10^{-8}\) for Gaussian white noise variances equal to 1 and 2. In Figure 18, the estimated mean square values of the discrete-time signals \(e_{0}(i)\) for the Gaussian white noise variance equal to 1 are compared; the presented curves averaged for 50 realizations of \(d(i)\). It follows that adaptation in the new FeFxLMS control system structure for the lower \(\mu\) value was significantly faster than in the classical FxLMS control system structure. It results in a greater noise reduction (3.40 dB) obtained with the new FeFxLMS algorithm after convergence of the adaptation process.