## 2 Spatially Fixed Virtual Sensing Algorithms
### The stochastically optimal tonal diffuse field virtual sensing method  
The stochastically optimal tonal diffuse field virtual sensing method generates stochastically optimal virtual microphones and virtual energy density sensors specifically for use in pure tone diffuse sound fields [9; 55]. Like the forward difference extrapolation technique, this virtual sensing method does not require a preliminary identification stage nor models of the complex transfer functions between the error sensors and the sources. It is worth noting that the stochastically optimal tonal diffuse field virtual sensing method is analogous to a fixed gain feedforward control problem. In this section, the primary and secondary acoustic fields are considered diffuse and different notation will be adopted for convenience. The pressure at a position \(\mathbf{x}\) in a single diffuse acoustic field is denoted \(p_{i}(\mathbf{x})\), and \(g_{i}(\mathbf{x})\) denotes the x-axis component of the pressure gradient. In this section, the subscript \(i\) refers to a single diffuse acoustic field, whereas a lack of subscript indicates the total acoustic field produced by superposition of the primary and secondary diffuse acoustic fields. For a displacement vector, \(\mathbf{r}=r_{x}\mathbf{i}+r_{y}\mathbf{j}+r_{z}\mathbf{k}\), the following functions are defined:  
\[A(\mathbf{r}) =\mathrm{sinc}(k|\mathbf{r}|), \tag{43}\] \[B(\mathbf{r}) =\frac{\partial A(\mathbf{r})}{\partial r_{x}}\] \[=-k\left(\frac{\mathrm{sinc}(k|\mathbf{r}|)-\cos(k|\mathbf{r}|)} {k|\mathbf{r}|}\right)\left(\frac{r_{x}}{|\mathbf{r}|}\right), \tag{44}\]  
\[C(\mathbf{r}) =\frac{\partial^{2}A(\mathbf{r})}{\partial r_{x}^{2}}\] \[=-k^{2}\left[\mathrm{sinc}(k|\mathbf{r}|)\left(\frac{r_{x}}{| \mathbf{r}|}\right)^{2}+\left(\frac{\mathrm{sinc}(k|\mathbf{r}|)-\cos(k| \mathbf{r}|)}{(k|\mathbf{r}|)^{2}}\right)\left(1-3\left(\frac{r_{x}}{|\mathbf{ r}|}\right)^{2}\right)\right], \tag{46}\]  
pressure gradients measured by the sensors and calculation of the weight matrices \(\mathbf{H}_{p}(\mathbf{x})\) and \(\mathbf{H}_{g}(\mathbf{x})\) using matrices \(\mathbf{L}_{p}(\mathbf{x})\), \(\mathbf{L}_{g}(\mathbf{x})\) and \(\mathbf{M}\) defined in Eqs. (55) - (57). As the distance between the locations of the physical and virtual sensors increases, the estimates of the virtual quantities approach zero. This is because the virtual and measured quantities become uncorrelated as this distance increases. If none of the distances between the virtual location and the physical sensors are small, then the pressure and pressure gradient at the virtual location will be uncorrelated with the measured quantities and the best estimate of the pressure and pressure gradient at the virtual location will be close to zero. In a pure tone diffuse sound field, a perfect estimate of the pressure at the virtual location may be obtained with the deterministic remote microphone technique [5] provided accurate measurement of the transfer functions occurs in the preliminary identification stage. Although greater control can be achieved with the remote microphone technique, the stochastically optimal tonal diffuse field virtual sensing technique is much simpler to implement because it is a fixed scalar weighting method requiring only sensor position information. Unlike the remote microphone technique, this virtual sensing method is independent of the source or sensor locations within the sound field. The weight functions only need to be updated if the geometric arrangement of physical and virtual locations change with respect to each other. The performance of the stochastically optimal tonal diffuse field virtual sensing method in generating a zone of quiet at a virtual sensor a distance of \(0.1\lambda\) from the physical sensor array has been investigated theoretically and using experimentally measured data [9; 55].