## 2 Spatially Fixed Virtual Sensing Algorithms
### The forward difference prediction technique
The equation for the estimate of the pressure at the virtual location using three microphone linear forward difference extrapolation is given by [45]  
\[\tilde{e}_{v}(n)=\frac{(-3x-h)}{6h}e_{p1}(n)+\frac{1}{3}e_{p2}(n)+\frac{(3x+5h) }{6h}e_{p3}(n). \tag{11}\]  
The pressure at a virtual location, \(x\), estimated by a first-order finite difference estimate using \(M_{p}=5\) physical microphones, separated by a distance of \(h/2\), is shown in Fig. 6 (b). The equation for the estimate of the pressure at the virtual location using five microphone linear forward difference extrapolation is given by [45]  
\[\tilde{e}_{v}(n)=\frac{(-2x+3h)}{5h}e_{p1}(n)+\frac{(-x+2h)}{5h}e_{p2}(n)+ \frac{1}{5}e_{p3}(n)+\frac{x}{5h}e_{p4}(n)+\frac{(2x-h)}{5h}e_{p5}(n). \tag{12}\]  
The pressure at a virtual location, \(x\), estimated by a second-order finite difference estimate using \(M_{p}=5\) physical microphones, separated by a distance of \(h/2\), is shown in Fig. 6 (c). The equation for the estimate of the pressure at the virtual location using five microphone quadratic forward difference extrapolation is given by [45]  
\[\tilde{e}_{v}(n) =\frac{(20x^{2}-54 xh+31h^{2})}{35h^{2}}e_{p1}(n)+\frac{(-10x^{2}+3 xh+9h^{2})}{35h^{2}}e_{p2}(n)\] \[+\frac{(-20x^{2}-40 xh-31h^{2})}{35h^{2}}e_{p3}(n)+\frac{(-10x^{2}- 27 xh-5h^{2})}{35h^{2}}e_{p4}(n)\] \[+\frac{(20x^{2}-26 xh+3h^{2})}{35h^{2}}e_{p5}(n). \tag{13}\]  
In experiments, the accuracy of these higher-order forward difference prediction virtual sensors was found to be adversely affected by sensitivity and phase mismatches and relative position errors between microphone elements in the array [45]. Such phase mismatches and position errors are unavoidable when a large number of physical microphones is used. It has also been demonstrated by [PERSON] [12], that the estimation problem is ill-conditioned for higher-order forward difference extrapolation. In an attempt to extend the zone of quiet achieved at the virtual location, [PERSON] [40] and [PERSON] et al. [41, 42, 43] developed virtual energy density sensors using the forward difference prediction technique. An estimate of the energy density at a virtual location, \(x\), using two microphone linear forward difference extrapolation, with the arrangement of physical microphones shown in Fig. 5 (a), is given by [40, 41]  
\[\tilde{E}_{Dv}(n) =\frac{1}{4\rho c^{2}}\left[\left(1+\frac{x}{2h}\right)^{2}e_{p2 }^{2}(n)-\frac{x}{h}\left(1+\frac{x}{2h}\right)e_{p1}(n)e_{p2}(n)+\left(\frac {x}{2h}\right)^{2}e_{p1}^{2}(n)\right.\] \[\tilde{e}_{v}(n)=e_{p2}(n)+\frac{e_{p2}(n)-e_{p1}(n)}{2h}x. \tag{9}\]0  
where \(k\) is the wavenumber. An estimate of the energy density at a virtual location, \(x\), using three microphone quadratic forward difference extrapolation, with the arrangement of physical microphonesshown in Fig. 5 (b), is given by [40; 41]  
\[\tilde{e}_{v}(n)=e_{p2}(n)+\frac{e_{p2}(n)-e_{p1}(n)}{2h}x. \tag{9}\]1 \[\tilde{e}_{v}(n)=e_{p2}(n)+\frac{e_{p2}(n)-e_{p1}(n)}{2h}x. \tag{9}\]2 \[\tilde{e}_{v}(n)=e_{p2}(n)+\frac{e_{p2}(n)-e_{p1}(n)}{2h}x. \tag{9}\]3  
The experimental results presented by [PERSON] et al. [43] on the performance of forward difference prediction virtual energy density sensors were inconclusive and it was later demonstrated by [PERSON] et al. [49] that these results were most likely flawed.