## 5 Simulations and Discussion
### Simulation Results and Discussion  
Figure 4 presents the converged filter weight coefficients of feedforward control structures and feedback control structures for different hybrid sound quality control systems. The converged filter weight coefficients were calculated when the gain factor was set to 0.5. The \"FF structure\" expresses the feedforward structure alone, and the \"FB structure\" represents the feedback structure alone. The \"tra structure\", \"nsim structure\", \"mod structure\", and \"sim structure\" are defined as in Section 4. The filter \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Structure** & **Tra Structure** & **Sim Structure** & **Tsim Structure** & **Nsim** & **Mod Structure** \\  & **[**10**]** & **Sim Structure** & **[**19**]** & **Structure** & **Mod Structure** \\ \hline Time (\(\mathrm{\SIUnitSymbolMicro s}\)) & 15.46 & 15.44 & 17.49 & 17.50 & 15.48 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computing time of the proposed structures compared to the traditional structure. weight vectors of the feedforward or feedback structure alone were regarded as the reference filter weight vector \(\mathbf{w}_{r}\), and the filter weight vectors of other structures were the comparative filter weight vectors \(\mathbf{w}\). The discrepancy \(D\) was calculated as follows [15]:  
\[D=20\log_{10}\frac{\|\mathbf{w}-\mathbf{w}_{r}\|_{2}}{\|\mathbf{w}_{r}\|_{2}}, \tag{24}\]  
where \(\|\cdot\|_{2}\) denotes the Euclidean norm. It can be seen that the discrepancy can validate the effects of the feedback and line-spectrum noise cancellation control filters on feedforward structure. A smaller discrepancy results in a smaller influence of the feedback and line-spectrum noise cancellation control filters on the feedforward structure. The discrepancy between the \"FF structure\" and the \"tra structure\" was \(-11.4\) dB, and that between the \"FF structure\" and the \"nsim structure\" was \(-15.6\) dB. Meanwhile, the discrepancy between the \"FF structure\" and the \"sim structure\" was \(-13.1\) dB, and that between the \"FF structure\" and the \"mod structure\" was \(-14.7\) dB. Furthermore, it should be noted that the two modified hybrid structures had less of a coupling effect compared to the traditional hybrid structure, as depicted in Equations (14) and (19). From Figure 4a, it can also be seen that the two modified hybrid structures had more stable iteration processes in feedforward control filters. At the same time, the discrepancy in Figure 4b was also calculated. The discrepancy between the \"FB structure\" and the \"nsim structure\" was \(-4.8\) dB, while that between the \"FB structure\" and \"tim structure\" was \(-1.1\) dB. However, the discrepancy between the \"FB structure\" and \"tra structure\" was \(-5.5\) dB, and the discrepancy between the \"FB structure\" and \"mod structure\" was \(-3.8\) dB. It is indicated that the discrepancy of the modified hybrid structures did not improve owing to the influence of the line-noise cancellation controller when the gain factor was equal to \(0.5\). As a result, under the synthetic effect of the feedforward and feedback control filters, the three hybrid structures had the same noise reduction. The objective assessments of the different structures are shown by the noise attenuation as a function of frequency [29]. The gain factor can determine the degree of noise control. When the gain factor is zero, the sound quality control systems become ANC systems where the noise to be controlled is minimized. Meanwhile, the noise to be controlled can be reduced, retained, and amplified when the gain factor is not equal to zero. Therefore, only two gain factors of \(0\) and \(0.5\) were selected to validate the performances of the different structures. In addition, the leakage factor of \(0.8\) was selected in the leaky FxLMS algorithm for the simplified structure because this case had better performance. Figure 5 shows the spectra of noise attenuation in different structures when the ANC was turned on and off.