## 3 Moving Virtual Sensing Algorithms
### The remote moving microphone technique  
The remote moving microphone technique [10] uses the remote microphone technique [5] to obtain estimates of the virtual error signals at the moving virtual locations. In this section it is assumed that there are \(L\) secondary sources, \(M_{p}\) physical sensors and \(M_{v}\) moving virtual sensors. The time-variant locations of the \(M_{v}\) moving virtual microphones are contained in matrix \(\mathbf{x}_{v}(n)\) of size \(3\times M_{v}\), defined as [12]  
\[\mathbf{x}_{v}(n)=\left[ \begin{array}{cccc}\mathbf{x}_{v1}(n)&\mathbf{x}_{v2 }(n)&...&\mathbf{x}_{vM_{v}}(n)\end{array} \right], \tag{60}\]  
where each of the moving virtual locations, \(\mathbf{x}_{vm_{v}}(n)\), are defined by three spatial co-ordinates with respect to a reference frame and are given by  
\[\mathbf{x}_{vm_{v}}(n)=\left[ \begin{array}{cccc}x_{vm_{v}}(n)&y_{vm_{v}}(n)& z_{vm_{v}}(n)\end{array} \right]^{\mathrm{T}}. \tag{61}\]  
It is assumed here that the \(M_{v}\) moving virtual locations, \(\mathbf{x}_{v}(n)\), are known at every time step. In practice, the moving virtual locations could be measured using a 3D head tracking system based on camera vision or on ultrasonic position sensing [12]. The remote moving microphone technique is used to compute estimates of the virtual error signals, \(\tilde{\mathbf{e}}_{v}(n)\), at the moving virtual locations, \(\mathbf{x}_{v}(n)\). A block diagram of the remote moving virtual sensing algorithm is given in Fig. 10. In this moving virtual sensing algorithm, the remote microphone technique is first used to obtain estimates of the virtual error signals, \(\tilde{\mathbf{e}}_{v}(n)\), at \(\bar{M}_{v}\) spatially fixed virtual microphone locations, \(\bar{\mathbf{x}}_{v}\). It is assumed here that the moving virtual locations, \(\mathbf{x}_{v}(n)\), are confined to a three-dimensional region and that the spatially fixed virtual microphone locations, \(\bar{\mathbf{x}}_{v}\), are therefore located within this region. The vector of the \(\bar{M}_{v}\) spatially fixed virtual microphone locations is given by  
\[\bar{\mathbf{x}}_{v}=\left[ \begin{array}{cccc}\bar{\mathbf{x}}_{v1}&\bar{ \mathbf{x}}_{v2}&...&\bar{\mathbf{x}}_{v\bar{M}_{v}}\end{array} \right], \tag{62}\]  
where each of the spatially fixed virtual locations, \(\bar{\mathbf{x}}_{v\bar{m}_{v}}\), are defined by three spatial co-ordinates with respect to a reference frame and are given by  
\[\bar{\mathbf{x}}_{v\bar{m}_{v}}=\left[ \begin{array}{cccc}\bar{x}_{v\bar{m}_ {v}}&\bar{y}_{v\bar{m}_{v}}&\bar{z}_{v\bar{m}_{v}}\end{array} \right]^{\mathrm{ T}}. \tag{63}\]  
The virtual error signals, \(\tilde{\mathbf{e}}_{v}(n)\), at the spatially fixed virtual locations, \(\bar{\mathbf{x}}_{v}\), are calculated using the remote microphone technique as described in Section 2.2. The remote microphone technique requires a preliminary identification stage in which the secondary transfer matrices, \(\tilde{\mathbf{G}}_{pu}\) of size \(M_{p}\times L\) and \(\tilde{\mathbf{G}}_{vu}\) of size \(\bar{M}_{v}\times L\), are modelled as matrices of FIR or IIR filters. The \(\bar{M}_{v}\times M_{p}\) sized matrix of primary transfer functions at the spatially fixed virtual locations from the physical locations, \(\tilde{\mathbf{M}}\), is also estimated as a matrix of FIR or IIR filters during this preliminary identification stage.