# Research on the Robustness of Active Headrest with Virtual Microphones to Human Head Rotation
## 2 Theory
### Active Control System with Remote Microphone Technique
Without actual microphones to pick up error signals at the virtual microphone positions, the estimated virtual error signals \(\hat{\mathbf{e}}_{\mathrm{v}}\) can be expressed as  
\[\hat{\mathbf{e}}_{\mathrm{v}}\mathbf{=}\hat{\mathbf{G}}_{\mathrm{opt}}\hat{ \mathbf{d}}_{\mathrm{p}}\mathbf{+}\hat{\mathbf{S}}_{\mathrm{v}}\mathbf{y}, \tag{9}\]  
where \(\hat{\mathbf{S}}_{\mathrm{v}}\) is the corresponding estimated model of the transfer response \(\mathbf{S}_{\mathrm{v}}\), and \(\hat{\mathbf{d}}_{\mathrm{p}}\) can be expressed as  
\[\hat{\mathbf{d}}_{\mathrm{p}}\mathbf{=}\mathbf{e}_{\mathrm{p}}-\hat{\mathbf{ S}}_{\mathrm{p}}\mathbf{y}, \tag{10}\]  
\[\mathbf{e}_{\mathrm{p}}\mathbf{=}\mathbf{d}_{\mathrm{p}}\mathbf{+}\mathbf{S} _{\mathrm{p}}\mathbf{y}, \tag{11}\]  
where \(\mathbf{e}_{\mathrm{p}}\mathbf{=}\left[e_{\mathrm{p}1},e_{\mathrm{p}2},\, \ldots,e_{\mathrm{p}m}\right]^{T}\) is the vector of error signals at the physical microphones, \(\mathbf{S}_{\mathrm{p}}\) is the matrix of physical plant responses, \(\hat{\mathbf{S}}_{\mathrm{p}}\) is the corresponding estimated model. The estimated models of the transfer responses are usually pre-modelled at the nominal head position, which might differ from the real responses with head movement. Substituting Equations (10) and (11) into Equation (9) yields  
\[\hat{\mathbf{e}}_{\mathrm{v}}=\hat{\mathbf{G}}_{\mathrm{opt}}\mathbf{d}_{ \mathrm{p}}+\big{[}\hat{\mathbf{S}}_{\mathrm{v}}+\hat{\mathbf{G}}_{\mathrm{ opt}}\big{(}\mathbf{S}_{\mathrm{p}}-\hat{\mathbf{S}}_{\mathrm{p}}\big{)}\big{]} \mathbf{y}=\hat{\mathbf{G}}_{\mathrm{opt}}\mathbf{d}_{\mathrm{p}}+\mathbf{S} \mathbf{y}, \tag{12}\]  
where \(\mathbf{S}\) is the matrix of effective plant responses between \(\mathbf{y}\) and \(\hat{\mathbf{e}}_{\mathrm{v}}\), which is defined as  
\[\mathbf{S}=\hat{\mathbf{G}}_{\mathrm{opt}}\big{(}\mathbf{S}_{\mathrm{p}}-\hat {\mathbf{S}}_{\mathrm{p}}\big{)}\mathbf{+}\hat{\mathbf{S}}_{\mathrm{v}}. \tag{13}\]  
The cost function of the active control system with remote microphone technique shown in Figure 2 is defined as  
\[J_{\mathrm{y}}=\mathrm{trace}[E[\hat{\mathbf{e}}_{\mathrm{v}}\hat{\mathbf{e} }_{\mathrm{v}}^{\mathrm{H}}]]=\mathrm{trace}[E[\big{(}\hat{\mathbf{G}}_{ \mathrm{opt}}\mathbf{d}_{\mathrm{p}}\mathbf{+}\mathbf{S}\mathbf{y}\big{)} \big{(}\hat{\mathbf{G}}_{\mathrm{opt}}\mathbf{d}_{\mathrm{p}}\mathbf{+} \mathbf{S}\mathbf{y}\big{)}^{\mathrm{H}}]]. \tag{14}\]  
By minimizing the cost function, the optimal control signals can be derived as [9]  
\[\mathbf{d}_{\mathrm{v}}=\mathbf{G}\mathbf{d}_{\mathrm{p}}, \tag{6}\]0  
The performance of the system is evaluated by the noise reduction at an evaluation point close to the human ear. The virtual microphone of a single-channel system may be located at the evaluation point, while the virtual microphone positions of a dual-channel  
\[\mathbf{d}_{\mathrm{v}}=\mathbf{G}\mathbf{d}_{\mathrm{p}}, \tag{6}\]1 Block diagram of the FxLMS algorithm with the remote microphone technique. system may not coincide with the evaluation point. The residual noise signal measured at the evaluation point by a measurement microphone, \(e_{\text{m}}\), can then be expressed as  
\[\mathbf{d}_{\mathrm{v}}=\mathbf{G}\mathbf{d}_{\mathrm{p}}, \tag{6}\]2  
where \(d_{\text{m}}\) is the primary noise signal at the measurement microphone, and \(\mathbf{S}_{\text{m}}\) is the vector of secondary transfer responses from control sources to the measurement microphone. The residual noise factor \(\beta_{\text{r}}\) is defined as the ratio of \(e_{\text{m}}\) to \(d_{\text{m}}\) when the control output is optimized and can be expressed as  
\[\mathbf{d}_{\mathrm{v}}=\mathbf{G}\mathbf{d}_{\mathrm{p}}, \tag{6}\]3  
where the smaller the factor \(\beta_{\text{r}}\), the greater the noise reduction.