# Research on the Robustness of Active Headrest with Virtual Microphones to Human Head Rotation
## 2 Theory
### Active Control System with Remote Microphone Technique  
The active headrest studied in this paper combines the remote microphone technique with the filtered-x least mean square (FxLMS) algorithm for adaptive feedforward control. In this section, the residual noise at any location in the active headrest system is formulated and the robustness optimization method for improving the noise reduction performance against the human head rotation is presented. All signals are formulated in the frequency domain in the following, as their dependence on frequency is excluded for notational convenience. Figure 2 shows the block diagram of the FxLMS algorithm with the remote microphone technique. Assuming there are \(n\) virtual microphones, \(m\) physical microphones and \(l\) secondary sources, the error signals, \(\mathbf{e}_{\mathrm{v}}\), at the virtual microphones can be expressed as  
\[\mathbf{e}_{\mathrm{v}}\mathbf{=}\mathbf{d}_{\mathrm{v}}\mathbf{+}\mathbf{S}_ {\mathrm{v}}\mathbf{=}\mathbf{P}_{\mathrm{v}}\mathbf{x+}\mathbf{S}_{\mathrm{ v}}\mathbf{W}\mathbf{x}, \tag{5}\]  
where \(\mathbf{e}_{\mathrm{v}}=[e_{\mathrm{v}1},e_{\mathrm{v}2},\ldots,e_{\mathrm{v}n }]^{\mathrm{T}}\), \(\mathbf{d}_{\mathrm{v}}=[d_{\mathrm{v}1},d_{\mathrm{v}2},\ldots,d_{\mathrm{v}n }]^{\mathrm{T}}\) is the vector of primary noise signals at the virtual microphones at a given frequency, \(\mathbf{S}_{\mathrm{v}}\) is the matrix of virtual plant responses, \(\mathbf{y}=[y_{1},\ y_{2},\ \ldots,\ y_{1}]^{\mathrm{T}}\) is the vector of control signals, \(\mathbf{x}\) is the vector of reference signals, \(\mathbf{P}_{\mathrm{v}}\) is the matrix of the primary transfer responses between \(\mathbf{x}\) and \(\mathbf{d}_{\mathrm{v}}\), \(\mathbf{W}\) is the matrix of control filters. \(\mathbf{d}_{\mathrm{p}}=[d_{\mathrm{p}1},d_{\mathrm{p}2},\ldots,d_{\mathrm{p} \mathrm{p}n}]^{\mathrm{T}}\) is the vector of primary noise signals at the physical microphones, \(\mathbf{G}\) is the transfer response between \(\mathbf{d}_{\mathrm{v}}\) and \(\mathbf{d}_{\mathrm{p}}\), then \(\mathbf{d}_{\mathrm{v}}\) can be written as  
\[\mathbf{d}_{\mathrm{v}}=\mathbf{G}\mathbf{d}_{\mathrm{p}}, \tag{6}\]  
where \(\mathbf{d}_{\mathrm{p}}=\mathbf{P}_{\mathrm{p}}\mathbf{x}\), \(\mathbf{d}_{\mathrm{v}}=\mathbf{P}_{\mathrm{v}}\mathbf{x}\), and \(\mathbf{P}_{\mathrm{p}}\) is the matrix of the primary transfer responses between \(\mathbf{x}\) and \(\mathbf{d}_{\mathrm{p}}\). The optimized solution of \(\mathbf{G}\) can be estimated by minimizing the cost function shown below  
\[J_{\mathrm{G}}=\mathrm{trace}[E[(\mathbf{d}_{\mathrm{v}}-\hat{\mathbf{G}} \mathbf{d}_{\mathrm{p}})(\mathbf{d}_{\mathrm{v}}-\hat{\mathbf{G}}\mathbf{d}_{ \mathrm{p}})^{\mathrm{H}}+\lambda\hat{\mathbf{G}}\hat{\mathbf{G}}^{\mathrm{H}} ]], \tag{7}\]  
where the superscript H is the Hermitian operator, \(E[]\) is the expectation operator, and \(\lambda\) is the regularization factor. Then \(\hat{\mathbf{G}}_{\mathrm{opt}}\) can be calculated by [9]  
\[\hat{\mathbf{G}}_{\mathrm{opt}}=\mathbf{P}_{\mathrm{v}}\mathbf{Q}\mathbf{P}_{ \mathrm{p}}^{\mathrm{H}}\Big{(}\mathbf{P}_{\mathrm{p}}\mathbf{Q}\mathbf{P}_{ \mathrm{p}}^{\mathrm{H}}\mathbf{+}\lambda\mathbf{I}\Big{)}^{-1}, \tag{8}\]  
where \(\mathbf{Q}=E[\mathbf{x}\mathbf{x}^{\mathrm{H}}]\) is the power spectral density matrix for \(\mathbf{x}\) and \(\mathbf{I}\) is the identity matrix.