## 2 Spatially Fixed Virtual Sensing Algorithms
### The virtual microphone arrangement  
The virtual microphone arrangement, proposed by [PERSON] and David [PERSON] and [PERSON] (1994), was the first virtual sensing algorithm suggested for active noise control. This virtual sensing algorithm uses the assumption of equal primary sound pressure at the physical and virtual microphone locations. Virtual sensing algorithms similar to the virtual microphone arrangement have also been proposed by [PERSON] et al. [PERSON] et al. (2005) and [PERSON] (2006, 2007). A block diagram of the virtual microphone arrangement is shown in Fig. 2. The virtual microphone arrangement is most easily implemented with equal numbers of physical and virtual sensors, so \(M_{v}=M_{p}\)[PERSON] et al. (2008). The microphones are located in \(M_{v}\) pairs, each consisting of one physical microphone and one virtual microphone. In this virtual sensing algorithm the primary sound pressure is assumed to be equal at the physical and virtual microphones in each pair, i.e. that \(\mathbf{d}_{v}(n)=\mathbf{d}_{p}(n)\). This assumption holds if the primary sound field does not change significantly between the physical and virtual microphones in each pair. A preliminary identification stage is required in this virtual sensing algorithm in which the matrices of transfer functions, \(\tilde{\mathbf{G}}_{pu}\) and \(\tilde{\mathbf{G}}_{vu}\), are modelled as matrices of FIR or IIR filters. Once this preliminary identification stage is complete, the microphones temporarily placed at the virtual locations are removed. As shown in Fig. 2, estimates, \(\tilde{\mathbf{e}}_{v}(n)\), of the total error signals at the virtual locations are calculated using  
\[\tilde{\mathbf{e}}_{v}(n)=\mathbf{e}_{p}(n)-(\tilde{\mathbf{G}}_{pu}-\tilde{ \mathbf{G}}_{vu})\mathbf{u}(n). \tag{5}\]  
The performance of the virtual microphone arrangement has been thoroughly investigated in both tonal and broadband sound fields by a number of authors [PERSON] (2006, 2007, 2008, 2009, 2010, 2011); [PERSON] et al. (2011); [PERSON] et al. (2012); [PERSON] et al. (2013); [PERSON] et al. (2014). Theoretical analysis in a pure tone diffuse sound field demonstrated that at low frequencies, the zone of quiet generated at a virtual microphone with the virtual microphone arrangement is comparable to that achieved by directly minimising the measured pressure of a physical microphone at the virtual location [PERSON] et al. (2011); [PERSON] et al. (2013). At higher frequencies however, those above 500 Hz, the 10 dB zone of quiet is substantially smaller when using a virtual microphone compared to a physical microphone at the same location. This is due to the assumption of equal primary pressure at the physical and virtual microphone locations being less valid as the wavelength decreases [18; 19]. The performance of a local active headrest system implementing the virtual microphone arrangement has been experimentally investigated by a number of authors [18; 19; 22; 23; 25; 27; 32]. An example of a local active headrest system is shown in Fig. 3. [PERSON] et al. [18; 19] investigated the performance of a local active headrest system in minimising a tonal primary disturbance at virtual microphones located 2 cm from the ears of a manikin and 10 cm from the physical microphones. Below 500 Hz, the 10 dB zone of quiet generated at the virtual microphone extends approximately 8 cm forward and 10 cm side to side. At higher frequencies however, the assumption relating to the similarity of the primary field at the physical and virtual microphones is no longer valid and limited attenuation is achieved at the virtual location. The performance of a local active headrest system in attenuating a broadband disturbance with a 100 - 400 Hz frequency range was investigated by [PERSON] et al. [22; 23] using feedback control. An overall attenuation of 9.5 dB was obtained at a virtual microphone located at the ear of a manikin with the virtual  
Figure 3: Local active headrest [23]. Figure 2: Block diagram of the virtual microphone arrangement. microphone arrangement.