2.1. The virtual sensing problem
The virtual sensing problem and notation used throughout this paper are introduced in this section. It is assumed here that there are
physical microphones,
spatially fixed virtual microphones and
L secondary sources. The vector of the total pressures at the
physical microphones,
, is defined as
The total pressures at the
physical microphones,
, is the sum of the sound fields produced by the primary and secondary sound sources at the physical microphone locations, and may be written as
where
is a vector of the primary pressures at the
physical microphones,
is a vector of the secondary pressures at the
physical microphones,
is a matrix of size
whose elements are the transfer functions between the secondary sources and the physical microphones,
is a vector of the secondary source strengths and
n is the time step.
Similarly, the vector of the total pressures at the
spatially fixed virtual locations,
, is defined as
The total pressures at the
virtual microphones,
, is the sum of the sound fields produced by the primary and secondary sources at the
virtual locations and may be written as
where
is the vector of the primary pressures at the
virtual locations,
is the vector of secondary pressures at the
virtual locations and
is a matrix of size
whose elements are the transfer functions between the secondary sources and the virtual locations.
Using the physical error signals, , a virtual sensing algorithm is used to estimate the pressures, , at the spatially fixed virtual locations. Instead of minimising the physical error signals, the estimated pressures are minimised with the active noise control system to generate zones of quiet at the virtual locations.
2.2. The virtual microphone arrangement
The virtual microphone arrangement, proposed by Elliott and David [
4], 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 Kuo et al. [
15] and Pawelczyk [
16,
17]. 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
[
12]. The microphones are located in
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
. 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,
and
, 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,
, of the total error signals at the virtual locations are calculated using
The performance of the virtual microphone arrangement has been thoroughly investigated in both tonal and broadband sound fields by a number of authors [
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32]. 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 [
18,
19]. 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].
Figure 2.
Block diagram of the virtual microphone arrangement.
Figure 2.
Block diagram of the virtual microphone arrangement.
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. Garcia-Bonito 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.
Figure 3.
Local active headrest [
23].
Figure 3.
Local active headrest [
23].
The performance of a local active headrest system in attenuating a broadband disturbance with a 100 - 400 Hz frequency range was investigated by Rafaely 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 microphone arrangement. This is compared to 19 dB being obtained at the physical microphone by directly minimising the measured pressure signal. Differences in the attenuation achieved by minimising the physical and virtual microphone signals were partly attributed to the physical microphone being significantly closer to the secondary loudspeaker than the virtual microphone. This results in a longer delay in the virtual plant, which has a negative effect on the performance of the feedback control system.
As the performance of the active headrest will be affected by the presence of the passenger’s head, Garcia-Bonito and Elliott [
20] and Garcia-Bonito et al. [
19,
21] theoretically investigated the performance of the virtual microphone in generating a zone of quiet near the surface of a reflecting sphere. The presence of the reflecting sphere was seen to increase the size of the zone of quiet when using the virtual microphone arrangement, especially at high frequencies. This is due to the imposed zero pressure gradient on the reflecting surfaces.
2.3. The remote microphone technique
The remote microphone technique developed by Roure and Albarrazin [
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 Popovich [
33]. Versions of the remote microphone technique have also been suggested by Hashimoto et al. [
34], Friot et al. [
35] and Yuan [
36].
Like the virtual microphone arrangement, the remote microphone technique requires a preliminary identification stage in which the secondary transfer matrices and are modelled as matrices of FIR or IIR filters. The sized matrix of primary transfer functions between the virtual locations and the physical locations, , is also estimated as a matrix of FIR or IIR filters during this preliminary identification stage. The secondary transfer function matrix is identified using the secondary sources and the physical microphones while microphones temporarily placed at the virtual locations are used to obtain matrices and .
A block diagram of the remote microphone technique is given in
Fig. 4. As shown in
Fig. 4, estimates of the primary disturbances,
, at the physical error sensors are first calculated using
Next, estimates of the primary disturbances,
, at the virtual locations are obtained using
Finally, estimates,
, of the total virtual error signals are calculated as
Radcliffe and Gogate [
37] demonstrated that theoretically, a perfect estimate of the tonal disturbance at the virtual location can be achieved with this virtual sensing algorithm provided accurate models of the tonal transfer functions are obtained in the preliminary identification stage. Using a three-dimensional 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.
Figure 4.
Block diagram of the remote microphone technique.
Figure 4.
Block diagram of the remote microphone technique.
Roure and Albarrazin [
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.
2.4. The forward difference prediction technique
The forward difference prediction technique, as proposed by Cazzolato [
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].
Fig. 5 (a) shows the pressure at a virtual location,
x, estimated by a first-order finite difference estimate. Using
physical microphones, separated by a distance of
, the equation for the estimate of the pressure at the virtual location using two microphone linear forward difference extrapolation is given by [
6]
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
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]
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.
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.
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 Petersen [
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 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
physical microphones, each separated by a distance of
h, is shown in
Fig. 6 (a). The equation for the estimate of the pressure at the virtual location using three microphone linear forward difference extrapolation is given by [
45]
The pressure at a virtual location,
x, estimated by a first-order finite difference estimate using
physical microphones, separated by a distance of
, 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]
The pressure at a virtual location,
x, estimated by a second-order finite difference estimate using
physical microphones, separated by a distance of
, 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]
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 Petersen [
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, Kestell [
40] and Kestell 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]
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 microphones shown in
Fig. 5 (b), is given by [
40,
41]
Figure 6.
Diagram of (a) three microphone linear forward difference extrapolation; (b) five microphone linear forward difference extrapolation; and (c) five microphone quadratic forward difference extrapolation. The black curved line represents the actual pressure field and the dashed line represents the pressure estimate.
Figure 6.
Diagram of (a) three microphone linear forward difference extrapolation; (b) five microphone linear forward difference extrapolation; and (c) five microphone quadratic forward difference extrapolation. The black curved line represents the actual pressure field and the dashed line represents the pressure estimate.
The experimental results presented by Kestell et al. [
43] on the performance of forward difference prediction virtual energy density sensors were inconclusive and it was later demonstrated by Cazzolato et al. [
49] that these results were most likely flawed.
2.5. The adaptive LMS virtual microphone technique
The adaptive LMS virtual microphone technique developed by Cazzolato [
7] employs the adaptive LMS algorithm [
13] to adapt the weights of physical microphones in an array so that the weighted summation of these signals minimises the mean square difference between the predicted pressure and that measured by a microphone temporarily placed at the virtual location.
For the case of
virtual microphones, an estimate of the total disturbance at the virtual microphone location,
, is calculated as the sum of the weighted physical sensor signals at
physical sensors in an array and this is given by
where
is a vector containing the
physical error sensor weights,
The weights,
, are calculated in a preliminary identification stage by switching the primary source off and exciting the secondary source with band-limited white noise [
12]. A modified version of the adaptive LMS algorithm is used to adapt the microphone weights. This algorithm can be used to find the optimal solution for the weights that minimises the mean square difference between the predicted sensor quantity,
, and that measured by a physical sensor temporarily placed at the virtual location,
. A block diagram of the adaptive LMS virtual microphone technique used to estimate the physical error sensor weights is shown in
Fig. 7. As only a single temporal tap is used, the real valued weights correspond to pure gain and are calculated using
where
μ is the convergence coefficient and
is the error term. This error term,
, is defined as the difference between the actual virtual secondary disturbance and the estimated virtual secondary disturbance, given by
where the estimated virtual secondary disturbance is given by
Once the weights have converged, they are fixed and the temporary microphone is removed from the virtual location.
Figure 7.
Block diagram of the adaptive LMS algorithm used to calculate the physical sensor weights.
Figure 7.
Block diagram of the adaptive LMS algorithm used to calculate the physical sensor weights.
A virtual sensing algorithm similar to the adaptive LMS virtual microphone technique was also proposed by Gawron and Schaaf [
50]. The performance of the adaptive LMS virtual microphone technique has been investigated for tonal duct noise, both numerically and experimentally [
7,
45,
51,
52]. The adaptive virtual sensors were found to be unaffected by sensitivity mismatches and relative position errors adversely affecting the forward difference prediction technique. The adaptive sensors were seen to predict the pressure at the virtual location more accurately than the equivalent forward difference prediction virtual sensor.
Petersen [
12] investigated the performance of the adaptive LMS virtual microphone technique in a broadband sound field with a frequency range of
Hz, in a long narrow duct. For an array of
and 5 physical sensors, the overall estimation performance decreased with an increasing distance between the physical sensor array and the virtual location, for all three configurations of physical sensors. The best estimation performance is theoretically achieved with an array of five physical sensors, however, this configuration was found to be ill-conditioned in experiments and a similar estimation performance was achieved with all three physical sensor configurations.
Despite being calculated by exciting the secondary source only, the weights,
, in Eq. (
18), are applied to both the primary and secondary disturbances. It has thus been assumed that the weights are optimal in the estimation of both disturbances. This however, may not always be true, especially in the near field of the secondary source where the spatial properties of the primary and secondary sound fields are very different [
38]. As a result, Petersen [
12] suggested that the optimal weights for the estimation of both the primary and secondary disturbances should be found separately, with the adaptive LMS virtual microphone technique being implemented as shown in
Fig. 8.
As shown in
Fig. 8, the virtual sensing algorithm separates the physical error signals into their primary and secondary components using the vector of the physical secondary transfer functions
. This vector of FIR or IIR filters is estimated in the preliminary identification stage. The primary component of the physical error signals is calculated as [
12]
Once the primary and secondary weights have been estimated separately using Eq. (
18), the pressure at the virtual location is estimated, as shown in
Fig. 8, using
where
and
are vectors containing the
optimal physical primary and secondary weights and
and
are vectors containing estimates of the primary and secondary disturbances at the
physical sensor locations.
2.6. The Kalman filtering virtual sensing method
The Kalman filtering virtual sensing method [
8] uses Kalman filtering theory to obtain estimates of the error signals at the virtual locations. In this virtual sensing method, the active noise control system is first modelled as a state space system whose outputs are the physical and virtual error signals. A Kalman filter is formulated to compute estimates of the plant states and subsequently estimate the virtual error signals using the physical error signals.
Figure 8.
Block diagram of the adaptive LMS virtual microphone technique [
38].
Figure 8.
Block diagram of the adaptive LMS virtual microphone technique [
38].
The active noise control system plant is described by the following state space model [
8,
12]
where
are the
N plant states,
are the physical measurement noise signals,
are the virtual measurement noise signals and
are the
K primary disturbance signals. In the state space model,
is the state matrix of size
in discrete form,
is the discrete secondary input matrix of size
,
is the discrete primary input matrix of size
,
is the discrete physical output matrix of size
,
is the discrete virtual output matrix of size
,
and
are the discrete physical feedforward matrices of size
and
respectively and
and
are the discrete virtual feedforward matrices of size
and
respectively. Inclusion of the measurement noise signals,
and
, in the state space model account for measurement noise on the microphones at the physical and virtual locations during the preliminary identification stage. Once the preliminary identification stage is complete, the microphones temporarily positioned at the virtual locations are removed.
Implementation of the Kalman filtering virtual sensing method is shown in the block diagram in
Fig. 9 (a). In this figure, G is the generalised plant of the acoustic system,
is an estimate of the generalised plant given by the state space model in Eq. (
23) and K are the Kalman filter gains. This is a form of the generalised control configuration with two sets of inputs and two sets of outputs [
53], as shown in
Fig. 9 (b). The generalised control configuration with two sets of inputs and two sets of outputs [
53] can therefore be interpreted as a virtual sensor arrangement.
The covariance properties of the stochastic signals
,
and
are required when using Kalman filtering theory to estimate the error signals at the virtual locations. These covariance properties and the state space model of the active noise control system plant are estimated during a preliminary identification stage with microphones temporarily positioned at the virtual locations. The primary disturbance signals,
, the physical measurement noise signals,
, and the virtual measurement noise signals,
, are all assumed to be zero mean white stationary random processes with the following covariance properties [
8,
12]
where
denotes the expectation operator,
is the identity matrix and
is the Kronecker delta function.
Figure 9.
Block diagram of (a) implementation of the Kalman filtering virtual sensing method and (b) the generalised control configuration with two sets of inputs and two sets of outputs [
53].
Figure 9.
Block diagram of (a) implementation of the Kalman filtering virtual sensing method and (b) the generalised control configuration with two sets of inputs and two sets of outputs [
53].
The term
in Eq. (
23) can be interpreted as process noise,
, and the combined influence of the measurement noise signals and disturbance signals can be interpreted as an auxiliary measurement noise signal,
, where
Using these definitions, the following covariance matrix can be defined
The covariance matrix
of the process noise
is given by
The covariance matrix
of the auxiliary measurement noise
is given by
The covariance matrix
between the process noise
and the auxiliary measurement noise
is given by
The virtual sensing algorithm in state space form, that estimates the virtual error signals
, given measurements of the physical error signals
up to
, is as follows [
8,
12]
where
is the Kalman gain matrix and
is the virtual innovation gain matrix. The Kalman gain matrix and the virtual innovation gain matrix are found by
with
, the unique solution to the discrete algebraic Riccati equation given by
where
is the covariance matrix of the innovation signals
given by
To implement the Kalman filtering virtual sensing method, the state space matrices
,
,
,
,
and
of the state space model in Eq. (
23) and the covariance matrices
,
,
and
need to be known [
12]. Together, the state space model in Eq. (
23) and covariance matrices describe the behaviour of the active noise control system and the covariance properties of the input signals. In practice, the behaviour of the active noise control system can be estimated in a preliminary system identification stage using subspace identification techniques [
54]. Subspace identification techniques estimate a model of the active noise control system in an innovations form [
54]. Therefore, the Kalman filtering virtual sensing method needs to be reformulated for practical implementation with an innovations model of the active noise control system. The steps to practical implementation of the Kalman filtering virtual sensing method using an innovations model of the active noise control system are as follows [
12]
Temporarily locate physical sensors at the spatially fixed virtual locations and measure an input-output data-set
Use subspace identification techniques [
54] to estimate an innovations model of the physical and virtual error signals
and estimate the covariance matrix of the white innovation signals
Implement the Kalman filtering virtual sensing method as
where the Kalman gain matrix
and the virtual innovation gain matrix
are calculated as follows
with
, the unique solution to the discrete algebraic Riccati equation given by
The Kalman filtering virtual sensing method is optimal in its estimation of the virtual error signals given a known or measured noise covariance. Also, instead of using a number of FIR or IIR filter matrices to compute an estimate of the virtual error signals, one compact state space model is used. This virtual sensing algorithm is also derived including measurement noise on the sensors [
8]. The Kalman filtering virtual sensing method is however, limited to use in systems of relatively low order.
The performance of this virtual sensing algorithm in generating a zone of quiet at a virtual microphone 10 cm from a physical microphone has been investigated in real-time broadband feedforward experiments conducted in an acoustic duct over a 50 - 500 Hz frequency range [
8,
12]. The state space model of the plant was first estimated using subspace model identification techniques [
54] with a microphone temporarily placed at the virtual location. Combining this virtual sensing algorithm with the filtered-x LMS algorithm [
14] achieved an overall attenuation of 19.7 dB at the virtual location. This is compared to an attenuation of 25.1 dB being achieved by directly minimising the error signal at the virtual location. The 5.4 dB difference was attributed to the fact that the primary disturbances at the physical and virtual locations were not completely causally related, which is a requirement in this virtual sensing algorithm.
2.7. 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 in a single diffuse acoustic field is denoted , and 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,
, the following functions are defined:
where
k is the wavenumber. The correlations between the pressures and pressure gradients at two different points,
and
, separated by
, are given by [
3]
where
denotes spatial averaging and ☆ indicates complex conjugation. In the case that
and
are the same point, the limits of
,
and
as
must be taken. If there are
sensors in the field, each measuring pressure or pressure gradient, then define
as an
matrix whose elements are the relevant pressures and pressure gradients measured by the sensors. The pressure and pressure gradient at any point can be expressed as the weighted sum of the
components, each of which are perfectly correlated with a corresponding element of p and a component which is perfectly uncorrelated with each of the elements. Therefore, for each position
,
and
can be written as
where
and
are matrices of weights which are functions of the position
only and
and
are perfectly uncorrelated with the elements of
. It can be shown, by postmultiplying the expressions for
and
by
and spatially averaging, that
where
The aim here is to estimate the pressure and pressure gradient at a virtual location. In order to do this,
and
must be estimated from the known quantities in
. The pressure and pressure gradient at any point
are given by Eqs. (51) and (52). If only the measured quantities in
are known, then the best possible estimates of
and
are zero, since they are perfectly uncorrelated with the measured signals. Therefore the best estimates of pressure and pressure gradient at any point
are given by
Therefore, in a diffuse sound field, the pressure and pressure gradient at a virtual location can be estimated using Eqs. (58) and (59). This requires matrix
whose elements are the relevant pressures and pressure gradients measured by the sensors and calculation of the weight matrices
and
using matrices
,
and
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
from the physical sensor array has been investigated theoretically and using experimentally measured data [
9,
55]. Control at a virtual microphone, using the measured pressure and pressure gradient at a point, achieved a maximum attenuation of 24 dB at the virtual location and generated a 10 dB zone of quiet with a diameter of
. This is the same sized zone of quiet as that achieved by Elliott et al. [
1], when minimising the measured pressure at the physical sensor location with a single secondary source. Similar control performance was obtained using two closely spaced physical microphones to estimate the pressure at the virtual location. Minimising the pressure and pressure gradient at virtual location with two control sources, using the measured pressures and pressure gradients at two points, achieved a maximum attenuation of 45 dB and extended the zone of quiet to a diameter of
. This is the same sized zone of quiet as that achieved by Elliot and Garcia-Bonito [
3], when minimising the measured pressure and pressure gradient with two control sources. Similar control performance was also obtained using physical microphones at four closely spaced points to estimate the pressure and pressure gradient at the virtual location.