Active noise control in a free field with virtual sensors - Acoustical ...

Active noise control in a free field with virtual sensors Colin D. Kestell, Ben S. Cazzolato, and Colin H. Hansen Department of Mechanical Engineering, University of Adelaide, South Australia 5005, Australia

共Received 6 October 1999; revised 21 September 2000; accepted 25 September 2000兲 The zone of local control around a ‘‘virtual energy density sensor’’ is compared with that offered by an actual energy density sensor, a single microphone, and a virtual microphone. Intended as an introduction to the concept of forward difference prediction and a precursor to evaluating the virtual sensor control algorithms in damped enclosures, this paper investigates an idealized scenario of a single primary sound source in a free-field environment. An analytical model is used to predict the performance of the virtual error sensors and compare their control performance with their physical counterparts. The model is then experimentally validated. The model shows that in general the virtual energy density sensor outperforms the actual energy density sensor, the actual microphone, and the virtual microphone in terms of centering a practically sized zone of local control around an observer who is remotely located from any physical sensors. However, in practice, the virtual sensor algorithms are shown to be sensitive 共by varying degrees兲 to short wavelength spatial pressure variations of the primary and secondary sound fields. © 2001 Acoustical Society of America. 关DOI: 10.1121/1.1326950兴 PACS numbers: 43.50.Ki 关CBB兴 I. INTRODUCTION

The simplest strategy for active noise control within an enclosure is to minimize the squared pressure at a single microphone location by means of a secondary control source. However, the zone where the noise attenuation is evident may be so small and limited to the immediate vicinity of the error sensor, that a nearby observer may not perceive any improvement in noise reduction 关Fig. 1共a兲兴. However, energy density is more spatially uniform than squared pressure and can also be measured at single locations. In numerical simulations, Sommerfeldt and Nashif 共1991兲 found that minimizing the energy density at a discrete location significantly outperformed the minimization of squared pressures in terms of the size of the attenuation zone around the error sensor. Total energy density, which is the summation of kinetic energy density and potential energy density, can be easily calculated via the measurement of particle velocity 共proportional to the acoustic pressure gradient兲 and pressure. Elliott and Garcia-Bonito 共1995兲 and GarciaBonito and Elliott 共1995兲 demonstrated that in a diffuse sound field, minimizing both the pressure and pressure gradient along one axis rather than simply pressure, resulted in a significant increase in the size of the 10-dB ‘‘zone of quiet.’’ Nashif 共1992兲 applied the principle of energy density minimization to a one-dimensional enclosure, also demonstrating that the use of an energy density sensor can overcome the observability difficulties 共such as pressure nodes兲 that are inherent in the use of microphones as error sensors in active noise control systems. Since a phase difference of 90° exists between the velocity nodes and pressure nodes of any enclosure resonance, the weighted summation of pressure and velocity are much more spatially uniform. Parkins 共1998兲 extended the work to a three-dimensional enclosure, verifying the previous findings. Remotely placed energy density sensors used for active noise control can extend the zone of local control sufficiently to encompass a nearby head, but the size of the zone is 232

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frequency dependent on the maximum attenuation still occurring at the sensor location 关Fig. 1共b兲兴. An alternative approach has been to move the location of maximum attenuation away from the error sensor by using a ‘‘virtual microphone.’’ This involves estimating the cost function at some desired location 共the occupant’s head兲 via a remotely placed physical microphone 关Fig. 2共a兲兴. GarciaBonito et al. 共1996兲 and Garcia-Bonito et al. 共1997兲 modified the pressure measured from an error sensor to estimate the pressure at a nearby desired 共virtual兲 location. They state that at low frequencies the spatial rate of pressure change due to the primary field is small enough to assume that the primary source pressure component is the same at both the virtual and actual location. Close to the secondary sound source the actual sensor and the virtual error sensor to secondary source transfer impedance functions are significantly different. The prior measurement of this difference can then be used as an operator on the actual error signal to estimate the pressure at the virtual location. Rafaely et al. 共1999兲 showed that this principle could also be applied to the active control of broadband noise. However, in enclosures, higher-order modes may significantly affect the spatial rate of pressure change, even over relatively small distances with respect to wavelength. Therefore, the assumption that the primary source pressure component is the same at both the actual and virtual error sensor locations may not be accurate. It is also possible that any movement within the vicinity of the sensors will significantly alter the impedance transfer functions. Carme and De Man 共1998兲 also used a virtual microphone to improve the performance of a pair of ANC ear defenders using the same transfer function prediction techniques developed by Garcia-Bonito et al. 共1997兲. However, while improving the performance of a pair of ear defenders they still restrict the occupant to the use of ear defenders and all of the problems associated with them. Using the development of a new virtual microphone extrapolation technique 共different to that described earlier兲 and

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FIG. 1. A schematic representation of the local region of control that may occur around either a single microphone error sensor or an energy density error sensor. 共a兲 A single microphone error sensor, 共b兲 an energy density cost function.

combining the benefits of an energy density sensor, the study described here investigates the use of a ‘‘virtual energy density sensor’’ to improve the zone of localized noise attenuation offered by existing active control systems 关Fig. 2共b兲兴. While a virtual microphone may be capable of placing a very high level of attenuation at the observer location, the zone may be impractically small and would not allow for any head movement. It will therefore be shown that it is not only possible to extend the zone of quiet around an observer beyond that offered by either the virtual microphone technique or remotely placed energy density sensors, but to also track any movement of an observer and thus continue to place the center of maximum noise attenuation at his or her location 共Fig. 3兲. The final objective of the research is to develop a nonintrusive virtual error sensor for use in a highly damped enclosure such as a light aircraft cabin. To fully investigate the concept and its limitations, the virtual sensors are initially evaluated analytically under the ideal condition of a single tonal noise source in a free-field environment. Future research shall extend the system complexity to reactive enclosures so that it is applicable to a light aircraft cabin. II. THEORY

In practical applications of active noise control it is not always possible to locate an error sensor at the desired control location, usually the observer’s head. Transducers located immediately adjacent to the ears may be even more restrictive than the use of personal hearing protection. However, moving the error sensor away from the desired control location often leads to very poor local control at the observer location. The virtual microphone sensor developed by Garcia-Bonito et al. 共1996兲 predicts the pressure at the observer’s head but requires measuring the secondary source to virtual microphone pressure transfer function prior to imple-

menting a control algorithm. The modified squared pressure cost function also tends to produce a small zone of local control. An alternative approach to the Garcia-Bonito et al. 共1996兲 transfer function based virtual microphone is to use forward difference extrapolation 共Fig. 4兲. The sound pressure at the observer 共virtual microphone兲 location is estimated in real time by extrapolating the signal from remotely placed microphones. This eliminates the need for the prior measurement of the complex acoustic transfer impedance function and allows the prediction technique to adapt to any physical system changes such as head movement or any other mechanism that may alter the error sensor to control source complex acoustic transfer impedance function. The theory for a first-order 共two-microphone兲 and a second-order 共three-microphone兲 forward difference extrapolation virtual microphone will now be derived. The same approach will be extended to a first-order 共two-microphone兲 and a second-order 共three-microphone兲 virtual energy density sensor with the intention of achieving a broader zone of local control to allow a practical amount of comfortable head movement. A. Virtual microphone

The spatial rate of change of the sound pressure between relatively closely spaced locations 共in terms of wavelength兲 in free space, in a duct, or in an enclosure will be small and hence predictable. By fitting either a straight line or a curve between pressures measured at fixed locations, the pressures at other locations may be estimated by either interpolation or extrapolation. Since in this research it is intended that an observer is to be remote from the physical sensors, rather than between them, the following equations are extrapolation based.

FIG. 2. A schematic representation of the region of control that may occur by estimating a cost function at the observer location. The ghosted sensors represent the ‘‘virtual sensor’’ location.

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p x⫽

p 2⫺ p 1 1 ⫹ p 2⫽ 共 3 p 2⫺ p 1 兲; 2 2

共2兲

if the separation distance is increased to x⫽2h then, p x ⫽2 p 2 ⫺ p 1 .

共3兲

2. Three-microphone second-order pressure prediction

FIG. 3. The concept of movement tracking.

1. Two-microphone first-order pressure prediction

Figure 4共a兲 illustrates that the pressure at location x can be approximated by the first-order finite difference estimate ( p⫽(dp/dx)x⫹c, where dp/dx is constant兲 from two remote microphones, separated by a distance of 2h by measuring pressures p 1 and p 2 , respectively, at the two microphone locations as follows: p x⫽

共 p 2 ⫺p 1 兲 x⫹p 2 . 2h

共1兲

To maintain consistency when comparing this method to the three-microphone second-order method 共discussed later兲, 2h is chosen as a separation distance. If x⫽0, or if x⫽⫺2h, then Eq. 共1兲 reduces to p x ⫽ p 2 or p 1 , respectively, as expected. But more practically, if the separation distance x between the observer and the nearest transducer is equal to ⫹h then Eq. 共1兲 reduces to

The use of a third intermediately placed microphone allows an estimation of the rate of change of the pressure gradient, which enables a greater prediction accuracy 关Fig. 4共b兲兴. This second-order approximation, where d 2 p/dx 2 is assumed constant 共a constant rate of pressure gradient change兲, can be integrated to determine the relationship between the pressure (p x ) at the virtual location 共x兲 and the pressures measured at the three actual microphone locations p 1 , p 2 , and p 3 as follows: p x⫽ ⫽ ⫽

冕冕 冕冕 冕

d2p • 共 dx 兲共 dx 兲 dx 2 k 1 • 共 dx 兲共 dx 兲

共 k 1 x⫹k 2 兲 • 共 dx 兲 ⫽

k 1x 2 ⫹k 2 x⫹k 3 . 2

共4兲

The constants of integration k 1 , k 2 , and k 3 can be found by applying Eq. 共4兲 to the pressure and location of each of the three microphones at x 3 ⫽0, x 2 ⫽⫺h, and x 1 ⫽⫺2h; thus, p 3 ⫽k 3 , p 2⫽

k1 2 h ⫺k 2 h⫹k 3 , 2

共5兲

p 1 ⫽2k 1 h 2 ⫺k 2 h⫹k 3 . Solving the equations allows the value for each of these constants to be calculated. When substituted into Eq. 共4兲 this yields p x⫽

冉

冊 冉

冊

p 1 ⫺2 p 2 ⫹ p 3 x 2 p 1 ⫺4 p 2 ⫹3 p 3 x⫹p 3 . ⫹ h2 2 2h

共6兲

Collecting like terms to calculate the weighting factors for each actual microphone 共which is more practical for hardware design兲, results in the pressure at location x being expressed as p x⫽

x 共 x⫹h 兲 x 共 x⫹2h 兲 共 x⫹2h 兲共 x⫹h 兲 p 1⫹ p 2⫹ p3 . 2 2 2h ⫺h 2h 2

共7兲

Once again, in order to confirm the equation, if x⫽0, or, x ⫽⫺h or x⫽⫺2h, then Eq. 共7兲 reduces to p x ⫽p 3 , p 2 , or p 1 , respectively. If x⫽⫹h, then Eq. 共7兲 reduces to p x ⫽ p 1 ⫺3 p 2 ⫹3 p 3 ⫽ p 1 ⫹3 共 p 3 ⫺ p 2 兲 .

共8兲

If x⫽2h, then Eq. 共7兲 reduces to FIG. 4. Forward difference prediction. 共a兲 First-order forward prediction, 共b兲 second-order forward prediction. 234

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p x ⫽3 p 1 ⫺8 p 2 ⫹6 p 3 ⫽ p 1 ⫺2 共 p 2 ⫺ p 1 兲 ⫹6 共 p 3 ⫺p 2 兲 .

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共9兲 234

In the next section this forward prediction approach is adapted to derive a virtual energy density sensor, with the aim of achieving broad zones of reduced noise centered around the head of the observer.

1 2␳c2

共 p 2⫺ p 1 兲 x⫹ p 2 2h

⫽

1 2␳c2

1⫹

⫹

B. Virtual energy density sensor

Using the same forward difference prediction method as introduced in the previous section, the pressure gradient at some point located away from the sensing microphones may also be estimated and this can be used to derive the equations to define a first and second-order ‘‘virtual energy density sensor.’’

For the two-microphone sensor shown in Fig. 4共a兲, the best estimate of pressure at a distance x from the second microphone is given by Eq. 共1兲, i.e., p x ⫽(( p 2 ⫺p 1 )/2h)x ⫹p 2 , and the particle velocity is obtained from the best estimate of the pressure gradient given by dp p 2 ⫺p 1 ⫽ . dx 2h

共10兲

Euler’s equation relates particle velocity to spatial pressure gradient with the general relationship dv dp ⫽⫺ ␳ , dx dt

共11兲

which for monotone sound fields reduces to v ⫽⫺

1 dp . j ␻␳ dx

共12兲

Therefore, at any given frequency ␻ 共radians/s兲, the particle velocity estimate for a two-microphone sensor is obtained by multiplying the pressure gradient in Eq. 共10兲 by ⫺1/j ␳␻ , i.e., v x ⫽⫺

冉 冊

1 dp p 2 ⫺p 1 ⫽ . j ␳␻ dx j2h ␳␻

共13兲

The instantaneous energy density 关Nashif and Sommerfeldt 共1992兲兴, which is the sum of instantaneous potential and kinetic energy density at a point x, is given as

␳ v 2x 1 E Dx⫽ ⫹ ⫽ 关 p 2 ⫹ ␳ 2 c 2 v 2x 兴 . 2␳c 2 2␳c2 x p 2x

共14兲

It should be noted that the terms outside of the square brackets equally weight both the pressure and the velocity. The terms inside the square brackets define the relative magnitude of the pressure and velocity components and must therefore be accurately incorporated in the design of any energy density sensor. Substituting the pressure 关Eq. 共1兲兴 and velocity 关Eq. 共13兲兴 estimates into Eq. 共14兲 and simplifying with k⫽ ␻ /c, an estimate for the energy density at some virtual location 共x兲 in terms of the sound pressures at the two microphones is obtained, 235

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x 2h

x 2h

2

p 21 ⫺

冊

2

p 22 ⫺

冊 冉

冉 冊

2

⫹ ␳ 2c 2

p 2 ⫺p 1 j2h ␳␻

x x 1⫹ p p h 2h 1 2

冊册 2

册

1 共 p 2 ⫺2 p 1 p 2 ⫹ p 21 兲 . 共 2hk 兲 2 2

共15兲

For a separation distance between the nearest transducer and the observer of x⫽0 this reduces to E Dx⫽

冋

册

1 1 2 共 p 2 ⫺2 p 1 p 2 ⫹ p 21 兲 , 2 p 2⫺ 2␳c 共 2hk 兲 2 2

or for x⫽h, E Dx⫽

1. Two-microphone first-order prediction

冋冉 冋冉 冉 冊

E Dx⫽

1 2␳c2 ⫺

冋 冉冉 冊 冉 冊 3 2

2

p 2⫺

冉冊 冊

3 1 2 p p ⫹ p 2 1 2 4 1

册

1 共 p 2 ⫺2 p 1 p 2 ⫹ p 21 兲 , 共 2hk 兲 2 2

共17兲

or if x⫽2h, E Dx⫽

共16兲

冋

册

1 1 4 p 2 ⫺4 p 1 p 2 ⫹ p 21 ⫺ (p 2 ⫺2 p 1 p 2 ⫹p 21 ) . 2␳c2 共 2hk 兲 2 2 共18兲

2. Three-microphone second-order prediction

For the three-microphone sensor shown in Fig. 4共b兲, the pressure gradient estimate is obtained by differentiating the pressure estimate shown in Eq. 共6兲, i.e., d dp ⫽ dx dx

冋冉

冊 冉 冊 冉

冊 册

p 1 ⫺2 p 2 ⫹ p 3 x 2 p 1 ⫺4 p 2 ⫹3 p 3 ⫹ x⫹p 3 2 h 2 2h

⫽

冋冉

⫽

1 2x⫹h 2x⫹3h p 1 ⫺ 共 2x⫹2h 兲 p 2 ⫹ p3 . 2 h 2 2

p 1 ⫺2 p 2 ⫹ p 3 p 1 ⫺4 p 2 ⫹3 p 3 x⫹ h2 2h

冋

冊册

册

共19兲

In the same manner as the two-microphone method, the particle velocity estimate for the three-microphone sensor is obtained by multiplying the pressure gradient in Eq. 共19兲 by 1/j ␳␻ . This can then be substituted into Eq. 共14兲 along with the virtual pressure estimate 关Eq. 共7兲兴, to estimate the virtual energy density 关Eq. 共20兲兴. E Dx⫽

1 2␳c2 ⫹

冋冉

共 x⫹2h 兲共 x⫹h 兲 p3 2h 2

⫹ ␳ 2c 2 ⫹ ⫽

x 共 x⫹h 兲 x 共 x⫹2h 兲 p 1⫹ p2 2 2h ⫺h 2

冉

冋冉

2

1 2x⫹h p 1 ⫺ 共 2x⫹2h 兲 p 2 2 j ␳␻ h 2

2x⫹3h p3 2

1 2␳c2

冉 冊冊 册

冊

2

x 共 x⫹h 兲 x 共 x⫹2h 兲 p 1⫹ p2 2 2h ⫺h 2

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冊

⫹

共 x⫹2h 兲共 x⫹h 兲 p3 2h 2

⫺

1 2x⫹h 2x⫹2h 2x⫹3h p ⫺ p 2⫹ p3 k 2 2h 2 1 h2 2h 2

冉

2

冊册 2

.

共20兲 This may also be customized for various set distances of x in terms of h. C. Higher-order prediction methods

From Eq. 共1兲 and Fig. 4共a兲 it is evident that the twosensor prediction method is fundamentally based on the assumption of a constant pressure gradient between the sensor locations and the observer, or that the estimate has a constant first-order derivative. The two-microphone forward difference prediction method is therefore described as a ‘‘firstorder’’ estimate. In the three-sensor prediction method 关Eq. 共7兲 and Fig. 4共a兲兴, it is assumed that the pressure gradient changes at a constant rate, or that the second-order derivative is constant. Three sensor forward difference predictions are therefore termed ‘‘second-order’’ estimates. From Figs. 4共a兲 and 共b兲, it can be seen that the second-order method is theoretically more accurate than the first-order method at defining a waveform that is remote from the sensors. This indicates that the theoretical prediction accuracy increases with the order of the prediction method. In fact, any waveform ( f (x)) can be explicitly defined by a Taylor series expansion, f 共 x⫹h 兲 ⫽ f 共 x 兲 ⫹h• f ⬘ 共 x 兲 ⫹

FIG. 5. A schematic diagram of the modeled system.

ultrasonic sensor for example兲, then the prediction algorithm could be continually updated to effectively make the sound field minimum follow the observer. Although only a fixed virtual location is considered here, tracking an observer’s head movement may provide the basis of future research. III. METHOD

The system investigated consisted of a single frequency primary noise source located in a free field. Twenty-one measurement locations were chosen 2 m away, along a 0.5-m length at intervals of 25 mm 共referred to as h兲 and the control

h2 h3 f ⬙ 共 0 兲 ⫹ f ⵮ 共 0 兲 ⫹¯ . 2! 3! 共21兲

While Eq. 共21兲 has an infinite number of terms, it can be estimated by using n terms in the following equation: f 共 x⫹h 兲 ⫽ f 共 x 兲 ⫹h• f ⬘ 共 x 兲 ⫹ ⫹¯

h2 h3 f ⬙共 0 兲 ⫹ f ⵮共 0 兲 2! 3!

hn 共n兲 f 共 0 兲. n!

共22兲

Introducing higher-order terms to theoretically improve the accuracy of a cost function estimate at the observer location will require that the number of sensors proportionally increase to n⫹1. However, this theory does not consider any practical issues associated with implementation that may adversely affect the prediction accuracy. Therefore, in the chapters that follow, only first- and second-order virtual error sensors shall be evaluated and compared, in order to observe the practical benefits of using higher-order terms in the estimation of the cost function at the observer location. D. Movement tracking

The first- and second-order methods of forward difference prediction presented in the previous sections all depend on the measurement of pressure from physical sensors placed at some distance x from the observer. It therefore follows that if a varying separation distance were measured 共via an 236

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FIG. 6. The experimental configuration in the anechoic chamber. 共a兲 The primary sound source and the measurement microphone, 共b兲 the two secondary sound sources and the measurement microphone 共foreground兲. Kestell et al.: Virtual sensing in a free field

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FIG. 7. Comparing the pressure minimization at a single location with energy density control using only one control source. The vertical line is the observer 共desired control兲 location and the physical sensor locations are shown with a circle. 共a兲 A single microphone error sensor, with increasing separation distances from the observer, 共b兲 energy density control, with increasing separation distances from the observer.

noise sources 共control speakers兲 were positioned at 4.5 and 5 m from the primary source 共Fig. 5兲. The system was initially modeled and then the results were experimentally validated. A. The model

The system was modeled with the primary and control sources represented as point monopoles with a spherical pressure amplitude radiation pattern 关Hansen and Snyder 共1997兲兴, which is defined by p r⫽

j ␻␳ 0 qe ⫺ jkr , 4␲r

共23兲

where p is the pressure amplitude measured at a distance r from the source, ␻ is the rotational frequency, ␳ 0 is the air density, q is the source signal strength, and k is the wave number. The primary sound field was initially modeled and transfer functions were obtained between the sound pressure calculated at each source location and the sound pressure calculated at each of the 21 measurement locations. These locations were at 25-mm intervals 共h兲 along a 0.5-m length, 2 m from the nearest source 共Fig. 5兲. The procedure was repeated for each individually modeled control source sound field. The relevant cost functions at the error sensor location were then minimized via quadratic optimization. B. The experiment

The physical system 共representative of the previously discussed model兲 consisted of a single frequency primary noise source located in an anechoic chamber. The location of the measurement sensors and the 150-mm diameter enclosed primary source and control source speakers were identical to those locations used in the model 共Fig. 6兲. In each experiment, while only the primary noise source was driven with broadband random noise, transfer functions were measured between the signal from a microphone in the rear chamber of the primary noise source and the signal from the measurement microphone at each of the 21 locations. 237

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The procedure was repeated for each individually driven control source. To allow a direct comparison between the active noise control results from each model and its respective experiment and to eliminate any further differences that may be associated with the performance of a practical ANC controller, the cost functions in the experiment 共either from the physical sensors or the virtual sensors兲 were also minimized via quadratic optimization. In theory, infinite attenuation may be achieved at an error sensor when using quadratic optimization. In practice, active noise control systems would have inherent errors that would limit the amount of noise attenuation that could be achieved. Therefore, to force the optimization results to a more realistic magnitude and emulate the uncertainty that occurs in practice, transfer function errors were incorporated into the control algorithm limiting the amount of control to less than 40 dB.

C. The number of control sources

To control pressure at a single location only requires one control source; additional sources are redundant 关Fig. 7共a兲兴. In a single control source system, however, there is little advantage in trying to minimize energy density if one is trying to activate local control 关Cazzolato 共1999兲兴. Attempts to minimize the pressure gradient can result in a local sound pressure increase. Using the free-field model data as an example, Fig. 7共b兲 shows that while the control source magnitude and phase 共with respect to the primary noise source兲 can be optimized to produce a zero pressure gradient between the two sensors, it is at the expense of constructive wave summation that results in an increase in the sound pressure level throughout the region of interest. Therefore, to effectively minimize energy density, the independent control of pressure and pressure gradient 共i.e., independent control of pressure at two locations兲 requires the use of a second control source. Minimizing energy density in a two-control source and two-sensor system, is the result of minimizing the acoustic Kestell et al.: Virtual sensing in a free field

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FIG. 8. A comparison of the primary and controlled sound pressure levels for first-order virtual energy density control at the observer location, first-order energy density control at the sensors, and minimizing the acoustic pressure at the two error sensors. There are two control sources and a primary noise tone at 100 Hz. The vertical line is the observer 共desired control兲 location and the physical sensor location is shown with a circle. The minima are shown between two of these in each case. 共a兲 First-order virtual energy density minimization at the observer location, 共b兲 energy density control at the sensors, 共c兲 pressure minimization at the two sensors.

pressure gradient between the two sensors and the mean acoustic pressure measured between them 关Eq. 共14兲兴. In the idealized systems considered here, where it is possible to equally minimize the signal from the two sensors, energy density control estimated via two microphones 关Fig. 8共a兲兴 is identical to simply minimizing the pressures at the twomicrophone locations 关Fig. 8共b兲兴. Practical systems would have more complex wave interaction and inherent errors and it is therefore unlikely that both pressures would be significantly equally minimized while the pressure gradient is also minimized. In a first-order virtual energy density sensor the pressure gradient is assumed to be spatially constant, i.e., the same at the sensor and observer locations 共Fig. 4兲. Because the pressure at the virtual location is also at a minimum when the the two error sensor pressures are at a minimum 关Eq. 共1兲兴, first-order virtual energy density control 关Fig. 8共a兲兴 will be identical to minimizing the energy density at the sensor location 关Fig. 8共b兲兴 or the pressure at the two error sensor locations 关Fig. 8共c兲兴. A second-order system that involves estimating the re238

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mote pressure and pressure gradient with three sensors results in a more accurate prediction of both pressure and pressure gradient 共in the absence of noise兲. The pressure at the observer location is estimated by extrapolating the pressure profile from three sensors and the pressure gradient is no longer considered constant, but assumed to have a constant rate of change. The use of only two control sources, however, is still pertinent since 共in these examples兲 the energy density has two independent contributors; namely pressure and pressure gradient. In the examples that follow, the use of a single control source will therefore be limited to observing the acoustic pressure minimization via a single microphone, a first-order virtual microphone, and a second-order virtual microphone. Energy density minimization 共identical to two-point pressure control and first-order virtual energy density control兲 and second-order virtual energy density control will be evaluated with two control sources. At system excitation frequencies of 100 and 400 Hz, the performance of the following error sensors, with increasing Kestell et al.: Virtual sensing in a free field

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FIG. 9. A 100-Hz primary sound source controlled via one control source. Measured along a 0.5-m length in an anechoic chamber, the actual sensors are marked with a circle and the observer location by a vertical line. 共a兲 Analytical model—pressure control at one microphone location; 共b兲 experimental results—pressure control at one microphone location; 共c兲 analytical model—first-order virtual microphone; 共d兲 experimental results—first-order virtual microphone; 共e兲 analytical model—second-order virtual microphone; 共f兲 experimental results—second-order virtual microphone.

separation distances between the physical sensors and the observer, are compared and discussed:

共5兲 second-order virtual energy density.

共1兲 共2兲 共3兲 共4兲

IV. RESULTS

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A. Control of a 100-Hz sinusoidal wave

Figure 9 shows the results that are obtained when controlling a 100-Hz monotone in both the free-field model and Kestell et al.: Virtual sensing in a free field

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FIG. 10. Examples of extrapolation 共prediction兲 error. 共a兲 An example of the greater theoretical accuracy of a second-order extrapolation technique in the absesnce of short wavelength noise; 共b兲 an example of when the first-order method may be more accurate in the presence of short wavelength noise.

the anechoic chamber experiment. Figures 9共a兲 and 共b兲 compare the modeled and experimental results for a conventional pressure squared cost function, where the sensor is incrementally moved farther from the observer location. These demonstrate that as the error sensor is moved away from the observer, the attenuation at the observer location decreases. Both figures are similar and show that the attenuation at the observer location reduces from 40 to 8 dB as the observer/ sensor separation distance increases from 0h to 4h 共100 mm兲.

In Figs. 9共c兲 and 共d兲 the modeled and experimental results for the first-order virtual microphone are compared. Since the algorithm adapts to an increasing separation distance there is only a negligible reduction in attenuation at the observer location. What small error there is, is due to the decrease in estimation accuracy as the separation distance increases. As the sensor is moved farther away, the sound pressure at the observer location is estimated by way of extrapolation. Both the model and the experiment show similar results, although the experiment shows that as the observer/

FIG. 11. A 100-Hz primary sound source controlled via two control sources. Measured along a 0.5-m length in an anechoic chamber, the actual sensors are marked with a circle and the observer location by a vertical line. 共a兲 Analytical model—energy density control 共and first-order virtual energy density control兲; 共b兲 experimental results—energy density control 共and first-order virtual energy density control兲; 共c兲 analytical model—second-order virtual energy density control; 共d兲 experimental results—second-order virtual energy density control. 240

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FIG. 12. A 400-Hz primary sound source controlled via one control source. Measured along a 0.5-m length in an anechoic chamber, the actual sensors are marked with a circle and the observer location by a vertical line. 共a兲 Analytical model—pressure control at one microphone location; 共b兲 experimental results—pressure control at one microphone location; 共c兲 analytical model—first-order virtual microphone; 共d兲 experimental results—first-order virtual microphone; 共e兲 analytical model—second-order virtual microphone; 共f兲 experimental results—second-order virtual microphone.

sensor separation distance increases to 100 mm, attenuation at the observer location reduces from 40 to 22 dB compared to only a negligible reduction in the performance of the model. However, this control strategy still demonstrates a practical advantage over the conventional remotely placed single microphone 关Fig. 9共b兲兴. Figure 9共e兲 illustrates that in theory second-order predic241

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tion is more accurate for the forward difference prediction of a pressure squared cost function, but in practice the experiment shows it is less accurate than the first-order method. The model 共theory兲 is based on sound fields that smoothly reduce at a rate of 6 dB per doubling in separation distance from the source whereas in practice there are likely to be relatively small spatial pressure variations due to reflections Kestell et al.: Virtual sensing in a free field

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FIG. 13. A 400-Hz primary sound source controlled via two control sources. Measured along a 0.5-m length in an anechoic chamber, the actual sensors are marked with a circle and the observer location by a vertical line. 共a兲 analytical model—energy density control 共and first-order virtual energy density control兲; 共b兲 experimental results—energy density control 共and first-order virtual energy density control兲; 共c兲 analytical model—second-order virtual energy density control; 共d兲 experimental results—second-order virtual energy density control.

inside the chamber as well as signal noise. These small spatial pressure variations can introduce errors into the extrapolation and the effect of these errors are amplified in the more sensitive second-order method 关Fig. 10共b兲兴, resulting in poorer performance in practice than achieved when using the first-order method. Results obtained by either controlling the pressure at two sensor locations or controlling direct or virtual firstorder energy density 共Sec. III C兲 are shown in Figs. 11共a兲 and 共b兲. This cost function produces a broader region of control 共when compared to that obtained using a single microphone and a single source兲 and hence maintains an attenuation envelope around the observer location as the sensors are moved farther away. This attenuation, however, reduces from 35 to 18 dB 共at a separation distance of 100 mm兲 at the observer location in the model. The cost function prediction in the experiment appears to be fairly stable up to an observer/ sensor separation distance of 75 mm, but prediction inaccuracies result in a gain of 8 dB when the observer/sensor separation distance increases to 100 mm. In Figs. 11共c兲 and 共d兲 the performance of the modeled 242

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and actual second-order virtual energy density sensors are compared. The experiment shows that the energy density cost function is more rugged in the presence of small spatial pressure variations and maintains the maximum attenuation at the observer location within a broad and practically sized zone of attenuation. B. Control of a 400-Hz sinusoidal wave

In Fig. 12 the results obtained when controlling a 400-Hz monotone in both the model and the experiment are shown. Figures 12共a兲 and 共b兲 again illustrate that the conventional pressure squared cost function produces similar results in both the model and the experiment. The size of the attenuation zone has been reduced with the increased frequency, so that the attenuation at the observer location now becomes a gain of 4 dB for an observer/sensor separation distance of 100 mm. The modeled and experimental results for the first-order virtual microphone are compared in Figs. 12共c兲 and 共d兲. The results of both the model and experiment show similar atKestell et al.: Virtual sensing in a free field

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tenuation. For an observer/sensor separation distance of 100 mm, this control strategy shows an experimental improvement of approximately 10 dB when compared to the conventional single microphone error sensor 关Fig. 12共b兲兴. In Fig. 12共f兲 it is shown that the second-order virtual microphone control strategy is sensitive to higher-order spatial pressure variations, with no resulting improvement in control when compared to results obtained using the conventional pressure squared cost function. In Figs. 13共a兲 and 共b兲 the achievable active noise control results for a 400-Hz monotone with an energy density cost function and two control sources is shown. Once again 共as in the 100-Hz example兲, energy density control produces a broader region of control than achieved when using a single microphone and control source. The experiment shows that as the observer/sensor separation distance increases to 100 mm, the attenuation at the observer location is still approximately 8 dB, compared to an observer gain of 4 dB that occurs when using a single remotely placed microphone. The experimental performance appears somewhat better than theoretically possible 共when compared to the model兲. This is merely due to the more fortunate destructive wave interference of the experiment’s slightly more erratic pressure profiles for both the primary and secondary sound fields. The second-order virtual energy density sensor continues to contribute toward a superior control strategy 关Figs. 13共c兲 and 共d兲兴 with the maximum attenuation remaining at the observer location for relatively large observer/sensor separation distances. C. Conclusions

All of the virtual microphone systems investigated show the potential to outperform their physical counterpart, offering a higher level of attenuation at the observer location than by minimizing pressure at an equivalent observer/sensor separation distance. For the frequencies analyzed and for this particular environment, it has been demonstrated that the first-order virtual microphone 共based on forward difference prediction兲 outperforms a conventional microphone 共in terms of noise reduction at the observer location兲 with an equivalent observer/sensor location separation distance.

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While the highest attenuation in the model and at the observer location was in general achieved by using a secondorder virtual microphone, the size of the attenuation zone was narrow. The theoretically more precise prediction method of a second-order virtual microphone was found to be more sensitive to shorter wavelength spatial variations in an actual sound field, offering only a small practical advantage to using a conventional microphone. It has been shown that first-order prediction methods for energy density estimation at a remote location 共the observer兲 offer no advantage to controlling energy density directly at the remote sensor. In terms of both a high level of attenuation and a broad control zone around the location of the observer, the secondorder virtual energy density probe produced the most favorable results. Carme, C., and De Man, P. 共1998兲. ‘‘How to improve an ANC headset by using a virtual microphone,’’ in Proceedings of InterNoise 98, Christchurch. Cazzolato, B. S. 共1999兲. ‘‘Sensing systems for active control of sound transmission into cavities,’’ Ph.D. thesis, The University of Adelaide, Adelaide, South Australia, April. Elliott, S. J., and Garcia-Bonito, J. 共1995兲. ‘‘Active cancellation of pressure and pressure gradient in a diffuse sound field,’’ J. Sound Vib. 186共4兲, 696–704. Garcia-Bonito, J., and Elliott, S. J. 共1995兲. ‘‘Strategies for local active control in diffuse sound fields,’’ in Proceedings of Active 95, pp. 561–572. Garcia-Bonito, J., Elliott, S. J., and Boucher, C. C. 共1996兲. ‘‘A virtual microphone arrangement in a practical active headrest,’’ in Proceedings of InterNoise 96, pp. 1115–1120. Garcia-Bonito, J., Elliott, S. J., and Boucher, C. C. 共1997兲. ‘‘Generation of zones of quiet using a virtual microphone arrangement,’’ J. Acoust. Soc. Am. 101共6兲, 3498–3516. Hansen, C. H., and Snyder, S. D. 共1997兲. Active Control of Noise and Vibration 共E and FN Spon, London兲. Nashif, P. J. 共1992兲. ‘‘An energy density based control strategy for minimizing the sound field in enclosures,’’ Ph.D. thesis, Penn State University. Nashif, P. J., and Sommerfeldt, S. D. 共1992兲. ‘‘An active control strategy for minimizing the energy density in enclosures,’’ in Proceedings of InterNoise 92, pp. 357–361. Parkins, J. W. 共1998兲. ‘‘Active minimization of energy density in a three dimensional enclosure,’’ Ph.D. thesis, Penn State University. Rafaely, B., Garcia-Bonito, J., and Elliott, S. J. 共1999兲. ‘‘Broadband performance of an active headrest,’’ J. Acoust. Soc. Am. 106共2兲, 787–793. Sommerfeldt, S. D., and Nashif P. J. 共1991兲. ‘‘A comparision of control strategies for minimizing the sound field in enclosures,’’ in Proceedings of Noise-Con 91, pp. 299–306.

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