## 2 Spatially Fixed Virtual Sensing Algorithms
### The Kalman filtering virtual sensing method  
The Kalman filtering virtual sensing method [8] uses Kalman filtering theory to obtain estimates of the error signals at the virtual locations. In this virtual sensing method, the active noise control system is first modelled as a state space system whose outputs are the physical and virtual error signals. A Kalman filter is formulated to compute estimates of the plant states and subsequently estimate the virtual error signals using the physical error signals. The active noise control system plant is described by the following state space model [8; 12]  
\[ \begin{split}&\mathbf{z}(n+1)=\mathbf{A}\mathbf{z}(n)+\mathbf{B}_{u} \mathbf{u}(n)+\mathbf{B}_{s}\mathbf{s}(n),\\ &\mathbf{e}_{p}(n)=\mathbf{C}_{p}\mathbf{z}(n)+\mathbf{D}_{pu} \mathbf{u}(n)+\mathbf{D}_{ps}\mathbf{s}(n)+\mathbf{v}_{p}(n),\\ &\mathbf{e}_{v}(n)=\mathbf{C}_{v}\mathbf{z}(n)+\mathbf{D}_{vu} \mathbf{u}(n)+\mathbf{D}_{vs}\mathbf{s}(n)+\mathbf{v}_{v}(n),\end{split}  \tag{23}\]  
where \(\mathbf{z}(n)\) are the \(N\) plant states, \(\mathbf{v}_{p}(n)\) are the physical measurement noise signals, \(\mathbf{v}_{v}(n)\) are the virtual measurement noise signals and \(\mathbf{s}(n)\) are the \(K\) primary disturbance signals. In the state space model, \(\mathbf{A}\) is the state matrix of size \(N\times N\) in discrete form, \(\mathbf{B}_{u}\) is the discrete secondary input matrix of size \(N\times L\), \(\mathbf{B}_{s}\) is the discrete primary input matrix of size \(N\times K\), \(\mathbf{C}_{p}\) is the discrete physical output matrix of size \(M_{p}\times N\), \(\mathbf{C}_{v}\) is the discrete virtual output matrix of size \(M_{v}\times N\), \(\mathbf{D}_{pu}\) and \(\mathbf{D}_{ps}\) are the discrete physical feedforward matrices of size \(M_{p}\times L\) and \(M_{p}\times K\) respectively and \(\mathbf{D}_{vu}\) and \(\mathbf{D}_{vs}\) are the discrete virtual feedforward matrices of size \(M_{v}\times L\) and \(M_{v}\times K\) respectively. Inclusion of the measurement noise signals, \(\mathbf{v}_{p}(n)\) and \(\mathbf{v}_{v}(n)\), in the state space model account for measurement noise on the microphones at the physical and virtual locations during the preliminary identification stage. Once the preliminary identification stage is complete, the microphones temporarily positioned at the virtual locations are removed. Implementation of the Kalman filtering virtual sensing method is shown in the block diagram in Fig. 9 (a). In this figure, G is the generalised plant of the acoustic system, \(\tilde{\mathrm{G}}\) is an estimate of the generalised plant given by the state space model in Eq. (23) and K are the Kalman filter gains. This is a form of the generalised control configuration with two sets of inputs and two sets of outputs [53], as shown in Fig. 9 (b). The generalised control configuration with two sets of inputs and two sets of outputs [53] can therefore be interpreted as a virtual sensor arrangement. The covariance properties of the stochastic signals \(\mathbf{s}(n)\), \(\mathbf{v}_{p}(n)\) and \(\mathbf{v}_{v}(n)\) are required when using Kalman filtering theory to estimate the error signals at the virtual locations. These covariance properties and the state space model of the active noise control system plant are estimated during a preliminary identification stage with microphones temporarily positioned at the virtual locations. The primary disturbance signals, \(\mathbf{s}(n)\), the physical measurement noise signals, \(\mathbf{v}_{p}(n)\), and the virtual measurement noise signals, \(\mathbf{v}_{v}(n)\), are all assumed to be zero mean white stationary random processes with the following  
Figure 8: Block diagram of the adaptive LMS virtual microphone technique [38]. covariance properties [8; 12]  
\[E\left[\left[\left[ \begin{array}{c}\mathbf{s}(n)\\ \mathbf{v}_{p}(n)\\ \mathbf{v}_{v}(n)\end{array} \right]\left[ \begin{array}{c}\mathbf{s}(k)\\ \mathbf{v}_{p}(k)\\ \mathbf{v}_{v}(k)\\ 1\end{array} \right]\right]=\left[