## 2 Spatially Fixed Virtual Sensing Algorithms
### The adaptive LMS virtual microphone technique  
The adaptive LMS virtual microphone technique developed by [PERSON] [7] employs the adaptive LMS algorithm [13] to adapt the weights of physical microphones in an array so that the weighted summation of these signals minimises the mean square difference between the predicted pressure and that measured by a microphone temporarily placed at the virtual location. For the case of \(M_{v}=1\) virtual microphones, an estimate of the total disturbance at the virtual microphone location, \(\tilde{e}_{v}(n)\), is calculated as the sum of the weighted physical sensor signals at \(M_{p}\) physical  
Figure 6: Diagram of (a) three microphone linear forward difference extrapolation; (b) five microphone linear forward difference extrapolation; and (c) five microphone quadratic forward difference extrapolation. The black curved line represents the actual pressure field and the dashed line represents the pressure estimate. sensors in an array and this is given by  
\[\tilde{e}_{v}(n)=\sum_{i=1}^{M_{p}}w_{i}e_{pi}(n)=\mathbf{w}^{\mathrm{T}}\mathbf{e }_{p}(n), \tag{16}\]  
where \(\mathbf{w}\) is a vector containing the \(M_{p}\) physical error sensor weights,  
\[\mathbf{w}=\left[ \begin{array}{cccc}w_{1}&w_{2}&\cdots&w_{M_{p}} \end{array} \right]^{\mathrm{T}}. \tag{17}\]  
The weights, \(\mathbf{w}\), are calculated in a preliminary identification stage by switching the primary source off and exciting the secondary source with band-limited white noise [12]. A modified version of the adaptive LMS algorithm is used to adapt the microphone weights. This algorithm can be used to find the optimal solution for the weights that minimises the mean square difference between the predicted sensor quantity, \(\tilde{y}_{v}(n)\), and that measured by a physical sensor temporarily placed at the virtual location, \(y_{v}(n)\). A block diagram of the adaptive LMS virtual microphone technique used to estimate the physical error sensor weights is shown in Fig. 7. As only a single temporal tap is used, the real valued weights correspond to pure gain and are calculated using  
\[\mathbf{w}(n+1)=\mathbf{w}(n)+2\mu\mathbf{y}_{p}(n)\epsilon(n), \tag{18}\]  
where \(\mu\) is the convergence coefficient and \(\epsilon(n)\) is the error term. This error term, \(\epsilon(n)\), is defined as the difference between the actual virtual secondary disturbance and the estimated virtual secondary disturbance, given by  
\[\epsilon(n)=y_{v}(n)-\tilde{y}_{v}(n), \tag{19}\]  
where the estimated virtual secondary disturbance is given by  
\[\tilde{y}_{v}(n)=\mathbf{w}^{\mathrm{T}}\mathbf{y}_{p}(n). \tag{20}\]  
Once the weights have converged, they are fixed and the temporary microphone is removed from the virtual location. Figure 7: Block diagram of the adaptive LMS algorithm used to calculate the physical sensor weights. A virtual sensing algorithm similar to the adaptive LMS virtual microphone technique was also proposed by [PERSON] and [PERSON] [50]. The performance of the adaptive LMS virtual microphone technique has been investigated for tonal duct noise, both numerically and experimentally [7; 45; 51; 52]. The adaptive virtual sensors were found to be unaffected by sensitivity mismatches and relative position errors adversely affecting the forward difference prediction technique. The adaptive sensors were seen to predict the pressure at the virtual location more accurately than the equivalent forward difference prediction virtual sensor. [PERSON] [12] investigated the performance of the adaptive LMS virtual microphone technique in a broadband sound field with a frequency range of \(50-500\) Hz, in a long narrow duct. For an array of \(M_{p}=2,3\) and \(5\) physical sensors, the overall estimation performance decreased with an increasing distance between the physical sensor array and the virtual location, for all three configurations of physical sensors.