# Research on the Robustness of Active Headrest with Virtual Microphones to Human Head Rotation
## 2 Theory
### Robustness Optimization Method  
The use of active headsets involves a transfer path identification stage and a control stage. The various estimates of the transfer response in Equation (15), \(\mathbf{\hat{S}}_{\text{v}}\), \(\mathbf{\hat{S}}_{\text{p}}\) and \(\mathbf{\hat{G}}_{\text{opt}}\), are derived at the identification stage with the head at the nominal position. As the head is rotated or moved, the system still uses these estimated transfer responses with the head at the nominal position, which deviate from the real responses at the new position, leading to reduced control performance. When the human head is rotated by \(\theta\), the residual noise factor at the measurement microphone is  
\[\beta_{\text{r}}(\theta)=1-\frac{\mathbf{S}_{\text{m}}(\theta)\Big{(}\mathbf{ S}^{\text{H}}(\theta)\mathbf{S}(\theta)\Big{)}^{-1}\mathbf{S}^{\text{H}}( \theta)\mathbf{\hat{G}}_{\text{opt}}\mathbf{d}_{\text{p}}(\theta)}{\mathbf{d} _{\text{m}}(\theta)}, \tag{18}\]  
\[\mathbf{S}(\theta)=\mathbf{\hat{G}}_{\text{opt}}\big{(}\mathbf{S}_{\text{p}}( \theta)-\mathbf{\hat{S}}_{\text{p}}\big{)}+\mathbf{\hat{S}}_{\text{v}}, \tag{19}\]  
where \(\mathbf{S}_{\text{m}}(\theta)\), \(\mathbf{S}_{\text{p}}(\theta)\), \(\mathbf{d}_{\text{p}}(\theta)\) and \(\mathbf{d}_{\text{m}}(\theta)\) are the corresponding transfer response matrices and the primary noise vectors as the head rotated by \(\theta\). Correspondingly, the noise reduction can be written as  
\[NR(\theta)=-20\log_{10}(|\beta_{\text{r}}(\theta)|). \tag{20}\]  
The aim of the study is to maintain good noise reduction at the human ear when the head rotates. The human head can be approximated as a sphere, with the different ear positions lying on a horizontal plane as it rotates, so the problem is equivalent to having better results at all points on the horizontal circle of the human ear when the head is in its nominal position. A straightforward approach is to deploy multiple virtual microphones on the \"ear circle\" and minimizing the virtual error signals from them simultaneously to acquire the optimal control signals. However, this approach includes multiple error signals, which significantly increases the computational effort of the adaptive algorithm. An alternative approach does not add virtual error signals and directly optimizes the transfer response models to average or limit the noise reduction of the human ear at each position to obtain similar performance. The cost function is defined as the maximum residual noise factor, expressed respectively as  
\[J=\|\beta_{\text{r}}(\theta)\|_{\infty}, \tag{21}\]  
For the cost function of Equation (14), the update equation of the vector of control signals when using the adaptive algorithm can be obtained by using the stochastic gradient descent method, expressed as [9]  
\[\mathbf{y}(n+1)=\mathbf{y}(n)-\alpha\mathbf{\hat{S}}_{\text{v}}^{\text{H}} \mathbf{\hat{e}}_{\text{v}}(n), \tag{22}\]where \(\alpha\) is the iteration step size. Substituting Equation (12) into Equation (22) yields  
\[\mathbf{y}(n+1)=\mathbf{y}(n)-\alpha\mathbf{S}_{\mathrm{v}}^{\mathrm{H}}(\hat{ \mathbf{G}}_{\mathrm{opt}}\mathbf{d}_{\mathrm{p}}(n)+\mathbf{S}\mathbf{y}(n)). \tag{23}\]  
When the convergence is achieved, \(\mathbf{y}(n+1)=\mathbf{y}(n)\), and therefore the expression in the parentheses in Equation (23) is zero.