## 2 Spatially Fixed Virtual Sensing Algorithms
### The remote microphone technique  
The remote microphone technique developed by [PERSON] and [PERSON] [5] is an extension to the virtual microphone arrangement [4] which uses an additional matrix of filters to compute estimates of the primary disturbances at the virtual sensors from the primary disturbances at the physical sensors. An active acoustic attenuation system designed to attenuate noise at a location that is remote from the physical error sensor using the remote microphone technique was independently patented by [PERSON] [33]. Versions of the remote microphone technique have also been suggested by [PERSON] et al. [34], [PERSON] et al. [35] and [PERSON] [36].  
Like the virtual microphone arrangement, the remote microphone technique requires a preliminary identification stage in which the secondary transfer matrices \(\tilde{\mathbf{G}}_{pu}\) and \(\tilde{\mathbf{G}}_{vu}\) are modelled as matrices of FIR or IIR filters. The \(M_{v}\times M_{p}\) sized matrix of primary transfer functions between the virtual locations and the physical locations, \(\tilde{\mathbf{M}}\), is also estimated as a matrix of FIR or IIR filters during this preliminary identification stage. The secondary transfer function matrix \(\tilde{\mathbf{G}}_{pu}\) is identified using the secondary sources and the physical microphones while microphones temporarily placed at the virtual locations are used to obtain matrices \(\tilde{\mathbf{G}}_{vu}\) and \(\tilde{\mathbf{M}}\).  
A block diagram of the remote microphone technique is given in Fig. 4. As shown in Fig. 4, estimates of the primary disturbances, \(\tilde{\mathbf{d}}_{p}(n)\), at the physical error sensors are first calculated using  
\[\tilde{\mathbf{d}}_{p}(n)=\mathbf{e}_{p}(n)-\tilde{\mathbf{y}}_{p}(  
finite element model of a car cabin, the tonal control achieved at a number of virtual microphones generated with the remote microphone technique was equivalent to that achieved by directly minimising the measured signals at the virtual locations.  
[PERSON] and [PERSON] [5] experimentally investigated the performance of the remote microphone technique in a room simulating an aircraft cabin with periodic noise at 170 Hz. Using twelve virtual microphones, six physical microphones and nine secondary sources, the remote microphone technique achieved an average attenuation of 20 dB at the twelve virtual locations with a feedforward control approach. However, 27 dB of attenuation was obtained by directly minimising the measured pressure at the virtual locations. This disparity was attributed to the sensitivity of the remote microphone technique to errors in the measured transfer functions. The performance of the remote microphone technique has also been investigated in the control of broadband noise in an acoustic enclosure [34], road traffic noise [38] and broadband acoustic duct noise [36].  
The performance of the remote microphone technique has been experimentally compared to that of the virtual microphone arrangement in a broadband primary sound field with 50 - 300 Hz frequency range [39]. Using a feedforward control approach, the two virtual sensing algorithms were used to generate a zone of quiet at a virtual location inside a three-dimensional enclosure using a physical microphone located on the enclosure wall 25 cm from the virtual location. The results demonstrated that greater attenuation is achieved at the virtual location with the remote microphone technique. The inferior performance of the virtual microphone arrangement was again attributed to the invalid assumption of equal primary sound pressure at the physical and virtual microphone locations.