## 2 Spatially Fixed Virtual Sensing Algorithms
### The forward difference prediction technique  
The forward difference prediction technique, as proposed by [PERSON] [6], fits a polynomial to the signals from a number of physical microphones in an array. The pressure at the virtual location is estimated by extrapolating this polynomial to the virtual location. This virtual sensing algorithm is suitable for use in low frequency sound fields, when the virtual distance and the spacing between the physical microphones is much less than a wavelength. At low frequencies, the spatial rate of change of the sound pressure between the microphones is small and extrapolation can therefore be applied [6]. Figure 4: Block diagram of the remote microphone technique. Fig. 5 (a) shows the pressure at a virtual location, \(x\), estimated by a first-order finite difference estimate. Using \(M_{p}=2\) physical microphones, separated by a distance of \(2h\), the equation for the estimate of the pressure at the virtual location using two microphone linear forward difference extrapolation is given by [6]  
\[\tilde{e}_{v}(n)=e_{p2}(n)+\frac{e_{p2}(n)-e_{p1}(n)}{2h}x. \tag{9}\]  
The pressure at a virtual location, \(x\), can also be estimated by a second-order finite difference estimate, as shown in Fig. 5 (b). Using \(M_{p}=3\) physical microphones, each separated by a distance of \(h\), the equation for the estimate of the pressure at the virtual location using three microphone quadratic forward difference extrapolation is given by [6]  
\[\tilde{e}_{v}(n)=\frac{x(x+h)}{2h^{2}}e_{p1}(n)+\frac{x(x+2h)}{h^{2}}e_{p2}(n) +\frac{(x+2h)(x+h)}{2h^{2}}e_{p3}(n). \tag{10}\]  
The forward difference prediction technique has several advantages over other virtual sensing algorithms. Firstly, the assumption of equal primary sound pressure at the physical and virtual locations does not have to be made, but also preliminary identification is not required, nor are FIR filters or similar to model the complex transfer functions between the sensors and the sources. Furthermore, this is a fixed gain prediction technique that is robust to physical system changes that may alter the complex transfer functions between the error sensors and the control sources. The performance of forward difference prediction virtual sensors has been evaluated in a long narrow duct and in a free field, both numerically and experimentally, by a number of authors [40; 41; 42; 43; 44; 45; 46; 47]. Using either linear or quadratic prediction techniques, these virtual sensors outperform the physical microphones in terms of the level of attenuation achieved at the virtual location. While the second-order estimate is theoretically more accurate than the first-order estimate, real-time feedforward experiments in a narrow duct demonstrated that quadratic prediction techniques are adversely affected by short wavelength extraneous noise. It was also shown by [PERSON] [12], that the estimation problem is ill-conditioned for the three sensor arrangement, explaining the difference between numerical and experimental results. In an attempt to improve the prediction accuracy of the forward difference algorithm, higher-order forward difference prediction virtual sensors which act to spatially filter out the extraneous noise were  
Figure 5: Diagram of (a) two microphone linear forward difference extrapolation; and (b) three microphone quadratic forward difference extrapolation. The black curved line represents the actual pressure field and the dashed line represents the pressure estimate. developed [45, 48]. Additional physical microphones were added to the array resulting in a greater number of microphones than system order. The microphone weights for this over constrained system were then calculated using a least squares approximation. The pressure at a virtual location, \(x\), estimated by a first-order finite difference estimate using \(M_{p}=3\) physical microphones, each separated by a distance of \(h\), is shown in Fig. 6 (a).