# Research on the Robustness of Active Headrest with Virtual Microphones to Human Head Rotation
## 3 Simulations
### Influence of Different Parameters  
There are a few parameters that have an impact on the results of the transfer response model optimization method. Firstly, three types of transfer response model, whose variation with head rotation has been analyzed in Section 3.1, may influence the noise reduction performance, and the model of \(\mathbf{S}_{\mathrm{v}}\) has been chosen for optimization. The optimization results with the model of \(\mathbf{S}_{\mathrm{v}}\) optimized, the model of \(\mathbf{G}\) optimized and the models of \(\mathbf{S}_{\mathrm{v}}\) and \(\mathbf{G}\) optimized simultaneously are presented in Figure 11a and it can be concluded that there is almost no difference between the three results. This can be explained by Equation (17), where the cost function is determined by \(\mathbf{\hat{S}}_{\mathrm{v}}^{-1}\mathbf{\hat{G}}_{\mathrm{opt}}\) when the transfer response \(\mathbf{S}_{\mathrm{p}}\) remains almost constant with head rotation (which is usually the case). Since \(\mathbf{\hat{S}}_{\mathrm{v}}^{-1}\mathbf{\hat{G}}_{\mathrm{opt}}\) can be regarded as an adjustable parameter, optimizing either model of \(\mathbf{S}_{\mathrm{v}}\) or \(\mathbf{G}\) is equivalent to adjusting the entire \(\mathbf{\hat{S}}_{\mathrm{v}}^{-1}\mathbf{\hat{G}}_{\mathrm{opt}}\) to minimize the cost function. The lower threshold of noise reduction at the nominal position can be another factor influencing the optimization result. The noise attenuation with respect to the head rotation angle after optimization for different \(NR_{0}\) constraints is shown in Figure 11b. It is shown that the higher the \(NR_{0}\), the better the control performance at the nominal position, but will slightly reduce the noise attenuation at other rotation angles, especially those where the noise reduction is fairly high, such as \(\pm 15^{\circ}\) and \(\pm 75^{\circ}\). The previous analyses are based on cost function of the maximum residual noise factor. However, there are also some other general cost functions which can be taken into account, such as the sum of squared residual noise factor at each position, denoted as \(J_{2}\), and the corresponding noise attenuation curve after optimization is shown as the dotted line in Figure 11c. Comparing with the red dashed line depicting the optimization result based on \(J_{1}\), i.e., the maximum residual noise factor, the optimized system performance is better when the head rotates from \(-60^{\circ}\) to \(60^{\circ}\), but at larger rotation angles the performance is worse with the minimum noise attenuation still at \(-90^{\circ}\), which is 2.4 dB lower than that based on \(J_{1}\). It can be concluded that, if the minimum reduction at all rotation angles is important, \(J_{1}\) is better, but when only considering the control performance of rotation angles less than \(60^{\circ}\), \(J_{2}\) is more suitable. Moreover, when considering the subjective listening  
Figure 10: Noise attenuation at different frequencies when head rotates: (**a**)125 Hz, (**b**)500 Hz. experience, it is necessary to take into account more factors than the amount of noise reduction in the cost function, such as the uniformity of the noise reduction. This issue is subject to further research. As can be seen from Layout B, using two virtual microphones did not provide more degrees of freedom to significantly improve performance, therefore it is necessary to discuss the case where only one virtual microphone is used in the system with Layout A. A virtual microphone is placed right at the measurement microphone position, the midpoint of the line connecting the two virtual microphones in Figure 4a. Figure 12 shows the noise reduction performances of the system with single virtual microphone before and after optimization at 125 Hz, 250 Hz and 500 Hz and compares them with the results of the system using two virtual microphones.