Active noise control in a pure tone diffuse sound field using virtual sensing D. J. Moreau, a) J. Ghan, B. S. Cazzolato, and A. C. Zander School of Mechanical Engineering, The University of Adelaide, SA 5005, Australia (Dated: June 20, 2008) Virtual sensing in diffuse sound fields 1
Abstract Local active noise control systems generate a zone of quiet at the physical error sensor using one or more secondary sources to cancel acoustic pressure and its spatial derivatives at the sensor location. The resulting zone of quiet is generally limited in size and as such, placement of the error sensor at the location of desired attenuation is required, which is often inconvenient. Virtual acoustic sensors overcome this by projecting the zone of quiet away from the physical sensor to a remote location. The work described here investigates the effectiveness of using virtual sensors in a pure tone diffuse sound field. Stochastically optimal virtual microphones and virtual energy density sensors are developed for use in diffuse sound fields. Analytical expressions for the controlled sound field generated with a number of control strategies are presented. These expressions allow the optimal control performance to be predicted. Results of numerical simulations and experimental measurements made in a reverberation chamber are also presented and compared. PACS numbers: 43.50.Ki 2
I. INTRODUCTION Local active noise control systems reduce the sound field at a number of points within the acoustic domain to create localised zones of quiet at the error sensors. While significant attenuation is achieved at the error sensor locations, the zone of quiet is generally small and impractically sized. Elliott et al. 6 investigated the spatial extent of the zone of quiet generated at the error sensor when controlling pressure with a single secondary source in a pure tone diffuse sound field. The zone of quiet generated at the microphone was found to be defined by a sinc function with the primary sound pressure level reduced by 10 db over a distance of λ/10, where λ is the excitation wavelength. In an attempt to broaden the zone of quiet, Elliott and Garcia-Bonito 5 extended previous theory to the control of both pressure and pressure gradient in a diffuse sound field with two secondary sources. Minimising both the pressure and pressure gradient along a single axis produced a 10 db zone of quiet over a distance of λ/2 in the direction of pressure gradient measurement. This is a significant size increase in comparison to the zone of quiet obtained by cancelling pressure alone. Virtual acoustic sensors are used in active noise control systems to shift the zone of quiet to a desired location that is remote from the physical sensor. Garcia-Bonito and Elliott 9 and Garcia-Bonito et al. 11 investigated the performance of virtual sensors in a diffuse sound field using the virtual microphone arrangement 4. The virtual microphone arrangement projects the zone of quiet away from the physical microphone using the assumption of equal primary pressure at the physical and virtual locations. A preliminary identification step is required in this virtual sensing method in which models of the transfer functions between the secondary source and microphones located at the physical and virtual locations are estimated. Garcia-Bonito et al. 11 investigated the performance of the virtual microphone arrangement in a local active headrest in a diffuse sound field. At low frequencies, below 500 Hz, the attenuation achieved at the virtual location with the virtual microphone arrangement is comparable to directly minimising the signal of a physical microphone located there. At a) danielle.moreau@mecheng.adelaide.edu.au 3
higher frequencies, however, limited attenuation is achieved at the virtual location due to the assumption relating to the similarity of the primary field at the physical and virtual locations being invalid. The remote microphone technique 23 is an extension to the virtual microphone arrangement that uses an additional filter to compute an estimate of the primary pressure at the virtual location from the primary pressure at the physical microphone location. The forward-difference extrapolation technique 2,12 15,18 20 is an alternative virtual sensing method that has several advantages. Firstly the assumption of equal primary 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 error sensors and the sources. Furthermore, this is a fixed gain prediction technique that can adapt to physical system changes, such as observer head movement, that may alter the complex transfer functions between the errors sensors and the control sources. In this paper, the performance of virtual sensors is explored in a diffuse sound field. Stochastically optimal virtual microphones and virtual energy density sensors that use both pressure and pressure gradient sensors are developed specifically for use in pure tone diffuse sound fields. Of considerable significance is that this stochastically optimal virtual sensing technique, like the forward difference extrapolation technique, does not require a preliminary identification stage nor models of the complex transfer functions between the error sensors and the sources. The nine local control strategies analysed in this paper are listed in Table I. The analytical expressions for the controlled sound field generated with these control strategies can be found in Moreau et al. 21 or derived using the procedure outlined in Section II. For brevity, only the analytical expressions for control strategies 1, 6 and 8 are included in this paper. These expressions predict the optimal control performance obtained in a pure tone diffuse sound field and thus set performance limits for the maximum achievable attenuation for each of the sensing and control strategies. The optimal expressions for the controlled sound field generated with all nine control strategies are validated in numerically simulated and post-processed experimental control. 4
Table I also provides a summary of prior research conducted in a pure tone diffuse sound field, including analytical, numerical or experimental analysis of any of the nine control strategies. This paper compares all nine control strategies using all three analysis methods. As shown in Table I, expressions for the controlled sound field generated with control strategies 1, 2 and 3 have been previously derived, and control strategies 1 and 3 have been experimentally validated. Also, control strategy 6 has been previously derived using the virtual microphone arrangement; however it is investigated in this paper using the derived stochastically optimal virtual sensing method for pure tone diffuse sound fields. As shown in Table I, this paper validates and extends previous research to analytically, numerically and experimentally analyse a broader range of local control strategies in a pure tone diffuse sound field. On a final note, a diffuse sound field is the most complex sound field to control and hence provides a lower limit on the performance that can be expected in practice 5 and the results presented here provide a guide to the minimum control performance that is likely to be achieved in any sound field. 5
II. THEORETICAL BACKGROUND This section presents the background theory required for deriving the analytical expressions for the controlled sound field generated with any of the nine control strategies listed in Table I. Analytical expressions for the controlled sound field generated with control strategies 1, 6 and 8 are presented in Sections II.A - II.C. The analytical expressions for the controlled sound field generated with the remaining control strategies can be found in Moreau et al. 21 or derived using the procedure outlined here. Derivation of the stochastically optimal diffuse field virtual sensing technique for control strategies 6 and 8 are also presented in Sections II.B and II.C. For the present study, the primary acoustic field is considered diffuse and the sound field contributions due to each of the secondary sources are modelled as uncorrelated single diffuse acoustic fields. The secondary acoustic sound fields may be modelled as diffuse if each of the secondary sources is several wavelengths away from the cancellation region and remote from all other sources 5. In the following, a single diffuse acoustic field is denoted by the subscript i and the total acoustic field, given by superposition of each of the single diffuse acoustic fields, is indicated by a lack of subscript. The pressure at a point x in a single diffuse acoustic field is given by p i (x) and the x-axis component of pressure gradient at a point x in this field is given by g i (x). For a displacement vector, r = r x i + r y j + r z k, the following functions are defined: A(r) = sinc(k r ), (1) B(r) = A(r) ( ) ( ) sinc(k r ) cos(k r ) rx = k, (2) r x k r r C(r) = 2 A(r) rx 2 ( ) 2 ( ) = k [sinc(k r ) ( ( ) )] 2 2 rx sinc(k r ) cos(k r ) rx + 1 3. (3) r (k r ) 2 r The correlations between the pressures and pressure gradients at two different points x j and 6
x k separated by r are given by 5 p i (x j )p i(x k ) = A(r) p i 2, (4) p i (x j )gi (x k ) = B(r) p i 2, (5) g i (x j )p i(x k ) = B(r) p i 2, (6) g i (x j )gi (x k ) = C(r) p i 2, (7) where denotes spatial averaging and indicates complex conjugation. In the case that x j and x k are the same point, the limits of A(r), B(r) and C(r) as r 0 must be taken, yielding p i (x j )p i(x j ) = p i 2, p i (x j )g i (x j ) = 0, g i (x j )p i(x j ) = 0, (8) (9) (10) g i (x j )g i (x j ) = ( k 2 /3 ) p i 2. (11) If there are m sensors in the field, then define p as an m 1 matrix whose elements are the relevant pressures or pressure gradients measured by the sensors. The pressure and the pressure gradient at any point in the diffuse sound field can be expressed as the weighted sum of the m components, each of which are perfectly correlated with a corresponding element of p i, and a component which is perfectly uncorrelated with each of the elements. In a single diffuse acoustic field, the pressure and pressure gradient at each position x can be written as 6 p i (x) = H p (x)p i + p i,u (x), (12) g i (x) = H g (x)p i + g i,u (x), (13) where H p (x) and H g (x) are matrices of real scalar weights which are functions of the position x only and p i,u (x) and g i,u (x) are perfectly uncorrelated with the elements of p i. It can be shown, by postmultiplying the expressions for p i (x) and g i (x) by p H i and spatially averaging, 7
that H p (x) = L p (x)m 1, H g (x) = L g (x)m 1, (14) (15) where L p (x) = p i(x)p H i, p i 2 L g (x) = g i(x)p H i, p i 2 M = p ip H i p i 2. (16) (17) (18) L p (x), L g (x) and M can be determined using Eqs. (4)-(11) and therefore weight matrices H p (x) and H g (x) can also be found. By postmultiplying the expression for p i (x) in Eq. (12) by its adjoint and spatially averaging, it can be shown that the uncorrelated component of pressure at the position x is p i,u (x) 2 = ( 1 L p (x)m 1 L H p (x) ) p i 2, (19) where M is self-adjoint. The total acoustic field, produced by superposition of the primary acoustic field and the acoustic field contributions due to each of the secondary sources, is p(x) = i p i (x) = H p (x)p + i p i,u (x). (20) If the secondary sources are used to drive each of the elements of p to zero, then only the uncorrelated components remain and the resulting acoustic field is given by p(x) = p u (x) = i p i,u (x). (21) The uncorrelated components of each of the diffuse acoustic fields are uncorrelated to each other if the assumption is made that the secondary sources are all several wavelengths apart. 8
Therefore the mean squared pressure at the point x can be be expressed as p(x) 2 = i p i,u (x) 2 = ( 1 L p (x)m 1 L H p (x) ) p i 2 i = ( 1 L p (x)m 1 L H p (x) ) p 2, (22) where p 2 is the mean squared pressure after control. A. Control strategy 1: Cancelling the pressure at a point with one control source Control strategy 1 involves cancelling the pressure at a point with a single control source. The pressure at a single location x 1 is to be sensed and therefore the matrix p is given by [ p = p(x 1 ) ]. When a single control source is used to drive p(x 1 ) to zero, applying Eq. (22) and using Eqs. (4) and (8) to find L p (x) and M, gives the mean squared pressure at the point x as (23) p(x) 2 = ( 1 A 2 (x x 1 ) ) p 2 = ( 1 sinc 2 (k x x 1 ) ) p 2, (24) which is the same as that previously derived by Elliott et al. 6. The mean squared pressure of the total acoustic field, p 2, is given by the sum of the mean squared pressure of the primary diffuse acoustic field, p p 2, and the mean squared pressure of the secondary diffuse acoustic field, p s 2. The distribution of the random variable β = p s 2 / p p 2 is the f 2,2 distribution 25, if the source and cancellation points are randomly selected, as previously shown by Elliott et al. 6. The increase in mean squared pressure of the acoustic field after control, α, is α = p 2 p p 2 = p p 2 + p s 2 p p 2 = 1 + β. (25) 9
By substituting β = α - 1 into the f 2,2 distribution, α has probability density and cumulative distribution functions given by f α1 (α) = 1 α2, α 1, (26) F α1 (α) = α 1, α α 1. (27) The relative change in mean squared pressure at the point x, as a function of α, is therefore given by p(x) 2 p p (x) 2 = ( 1 sinc 2 (k x x 1 ) ) p 2 p p 2 = ( 1 sinc 2 (k x x 1 ) ) α. (28) B. Control strategy 6: Cancelling the pressure at a virtual location with one control source using the pressure and pressure gradient at a point Instead of cancelling the measured quantities (pressure and pressure gradient), control strategy 6 involves minimising the pressure at a virtual location x 0 with a single control source. To create a virtual microphone, the pressure at the virtual location, p(x 0 ), must be estimated from measured quantities, in this case p(x 1 ) and g(x 1 ). The matrix p is given by [ ] T p = p(x 1 ) g(x 1 ). (29) As stated in Eq. (12), the pressure at any point x is given by p(x) = H p (x)p + p u (x). (30) Therefore the pressure at any point x can be expressed as the sum of two components, one of which is perfectly correlated with the elements of p, and a perfectly uncorrelated component. If only the measured quantities p(x 1 ) and g(x 1 ) are known, the best possible estimate of p u (x) is zero since it is perfectly uncorrelated with the measured signals. Therefore, the best estimate of the pressure, p(x), is given by the component perfectly correlated with the elements of p. Subsequently the best estimate of the pressure at the virtual location, x 0, is ˆp(x 0 ) = H p (x 0 )p. (31) 10
The matrix of weights H p (x 0 ) is found to be [ H p (x 0 ) = L p (x 0 )M 1 = A(x 0 x 1 ) 3 k 2 B(x 0 x 1 ) ], (32) where Eqs. (4)-(11) have been used to find matrices L p (x) and M. The best estimate of the pressure at the virtual location, x 0, is therefore given by ˆp(x 0 ) = A(x 0 x 1 )p(x 1 ) 3 k 2B(x 0 x 1 )g(x 1 ). (33) This diffuse field extrapolation method fits a sinc type function to the known data unlike the traditional forward-difference virtual microphone prediction technique that fits a polynomial to the measured quantities. As the distance between x 0 and x 1 increases, the pressure estimate, ˆp(x 0 ), approaches zero. This is because the pressure at the virtual location and the known quantities become uncorrelated as the distance between x 0 and x 1 increases. This is the case for any virtual sensor in a diffuse sound field. 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. When the estimate of the pressure at the virtual location, ˆp(x 0 ), given in Eq. (33), is cancelled in a pure tone diffuse sound field, the total complex pressure at a position x, where x = x 0 + x, is given by p(x). The total complex pressure, p(x), can be decomposed into two components. The first is a component perfectly spatially correlated with the estimate of the pressure at the virtual location, ˆp(x 0 ), and the second is a component perfectly spatially uncorrelated with ˆp(x 0 ). The total complex pressure at any point x is the sum of these two components and may be written as p(x) = h pp (x)ˆp(x 0 ) + p uu (x), (34) where h pp (x) is a function of the distance x and p uu (x) is perfectly uncorrelated with ˆp(x 0 ). By postmultiplying the expression for p(x) by ˆp (x 0 ) and spatially averaging, it can be 11
shown that p(x)ˆp (x 0 ) = h pp (x) ˆp(x 0 ) 2. (35) Using Eqs. (4) and (5), the correlation between the total complex pressure at x and the estimate of the pressure at the virtual location can be written as p(x)ˆp (x 0 ) = ( A(x 0 x 1 )A(x x 1 )+ 3 ) p k 2B(x 2 0 x 1 )B(x x 1 ). (36) Using Eqs. (8) - (11) and Eq. (33), it can be shown that the mean squared pressure estimate at the virtual location is related to the total mean squared pressure by ˆp(x0 ) 2 = ( A 2 (x 0 x 1 ) + 3 k 2B2 (x 0 x 1 )) p 2. (37) By substituting Eqs. (36) and (37) into Eq. (35), the weight function h pp (x) is found to be h pp (x) = A(x 0 x 1 )A(x x 1 )+ 3 B(x k 2 0 x 1 )B(x x 1 ) A 2 (x 0 x 1 ) + 3. (38) B k 2 (x 2 0 x 1 ) The space-average mean squared pressure at x, which is uncorrelated with ˆp(x 0 ), is obtained by multiplying both sides of Eq. (34) by their conjugates and spatially averaging. Using Eqs. (37) and (38), the space-average uncorrelated pressure may be written as ( ( puu (x) 2 A(x0 x 1 )A(x x 1 )+ 3 B(x k = 1 2 0 x 1 )B(x x 1 ) ) ) 2 p 2 A 2 (x 0 x 1 ) + 3. (39) B k 2 (x 2 0 x 1 ) If ˆp(x 0 ) is cancelled with a single secondary source, the residual pressure at x is, according to Eq. (34), p uu (x) only. Therefore, the mean squared pressure at a position x in the controlled sound field is given by Eq. (39). When the pressure at the virtual location, x 0, estimated using the pressure and pressure gradient at the point x 1, is cancelled with a single secondary source, the relative change in mean squared pressure at a point x in the controlled sound field is given by p(x) 2 pp 2 = (1 ( A(x0 x 1 )A(x x 1 )+ 3 B(x k 2 0 x 1 )B(x x 1 ) ) ) 2 A 2 (x 0 x 1 ) + 3 α. (40) B k 2 (x 2 0 x 1 ) If x 0 = x 1, then ˆp(x 0 ) = p(x 1 ) and this control strategy is equivalent to control strategy 1, minimising the pressure at a point with a single control source. 12
C. Control strategy 8: Cancelling the pressure and pressure gradient at a virtual location with two control sources using the pressures and pressure gradients at two points Control strategy 8 involves minimsing the pressure and pressure gradient at a virtual location x 0, generating a virtual energy density sensor. In order to do this, the pressure, p(x 0 ), and pressure gradient, g(x 0 ), at the virtual location must be estimated from measured quantities, in this case p(x 1 ), p(x 2 ), g(x 1 ) and g(x 2 ). The matrix p is given by [ ] T p = p(x 1 ) p(x 2 ) g(x 1 ) g(x 2 ). (41) With reference to Eqs. (12) and (13), the pressure and pressure gradient at the virtual location x 0 can be expressed as the sum of two components, one of which is perfectly correlated with the elements of p, and an uncorrelated component. If only the measured quantities p(x 1 ), p(x 2 ), g(x 1 ) and g(x 2 ) are known, the best possible estimates of p u (x) and g u (x) are zero since they are perfectly uncorrelated with the measured signals. Therefore, the best estimates of the pressure and pressure gradient at the virtual location are given by ˆp(x 0 ) = H p (x 0 )p, ĝ(x 0 ) = H g (x 0 )p. (42) (43) Weight matrices H p (x 0 ) and H g (x 0 ) can be found as H p (x 0 ) = L p (x 0 )M 1 [ = H pp1 (x 0 ) H pp2 (x 0 ) H pg1 (x 0 ) H pg2 (x 0 ) H g (x 0 ) = L g (x 0 )M 1 [ = H gp1 (x 0 ) H gp2 (x 0 ) H gg1 (x 0 ) H gg2 (x 0 ) ], (44) ], (45) where matrices L p (x 0 ), L g (x 0 ) and M are found using Eqs. (4) - (11). Weight matrices H p (x 0 ) and H g (x 0 ) can be used in Eqs. (42) and (43) to give estimates of the pressure and pressure gradient at the virtual location. 13
When the estimates of the pressure and pressure gradient at x 0 are cancelled in a pure tone diffuse sound field, the total complex pressure at a position x, where x = x 0 + x, is given by p(x). The total complex pressure, p(x), can be decomposed into three components. The first is a component perfectly spatially correlated with the estimate of the pressure, ˆp(x 0 ), the second is a component perfectly spatially correlated with the estimate of the pressure gradient, ĝ(x 0 ), and the third is a component perfectly spatially uncorrelated with both ˆp(x 0 ) and ĝ(x 0 ), p uu (x). The total complex pressure at any point x is the sum of these three components and may be written as p(x) = h pp (x)ˆp(x 0 ) + h pg (x)ĝ(x 0 ) + p uu (x), (46) where the weight functions h pp (x) and h pg (x) are functions of the distance x, and may be found using the procedure outlined in Section II.B, and are given by h pp (x) = P px P 0, h pg (x) = P gx G 0, (47) (48) where P px =H pp1 (x 0 )A(x x 1 ) + H pp2 (x 0 )A(x x 2 ) H pg1 (x 0 )B(x x 1 ) H pg2 (x 0 )B(x x 2 ), (49) P gx =H gp1 (x 0 )A(x x 1 ) + H gp2 (x 0 )A(x x 2 ) H gg1 (x 0 )B(x x 1 ) H gg2 (x 0 )B(x x 2 ), (50) and 14
P 0 = (H pp1 (x 0 )H pp2 (x 0 ) + H pp2 (x 0 )H pp1 (x 0 ))A(x 2 x 1 ) + (H pp2 (x 0 )H pg1 (x 0 ) H pg2 (x 0 )H pp1 (x 0 ))B(x 1 x 2 ) + (H pp1 (x 0 )H pg2 (x 0 ) H pg1 (x 0 )H pp2 (x 0 ))B(x 2 x 1 ) H pg1 (x 0 )H pg2 (x 0 )C(x 2 x 1 ) H pg2 (x 0 )H pg1 (x 0 )C(x 1 x 2 ) +H 2 pp1(x 0 ) + H 2 pp2(x 0 ) + k2 3 G 0 =(H gp1 (x 0 )H gp2 (x 0 ) + H gp2 (x 0 )H gp1 (x 0 ))A(x 2 x 1 ) + (H gp2 (x 0 )H gg1 (x 0 ) H gg2 (x 0 )H gp1 (x 0 ))B(x 1 x 2 ) + (H gp1 (x 0 )H gg2 (x 0 ) H gg1 (x 0 )H gp2 (x 0 ))B(x 2 x 1 ) ( H 2 pg1 (x 0 ) + H 2 pg2(x 0 ) ), (51) H gg1 (x 0 )H gg2 (x 0 )C(x 2 x 1 ) H gg2 (x 0 )H gg1 (x 0 )C(x 1 x 2 ) + H 2 gp1(x 0 ) + H 2 gp2(x 0 ) + k2 3 ( H 2 gg1 (x 0 ) + H 2 gg2(x 0 ) ). (52) The space-average mean squared pressure at x, which is uncorrelated with both ˆp(x 0 ) and ĝ(x 0 ), is obtained by multiplying both sides of Eq. (46) by their conjugates and spatially averaging. Using Eqs. (47) and (48), the space-average uncorrelated pressure is found to be puu (x) 2 = ( ( P 2 px 1 + P 2 )) gx p 2. (53) P 0 G 0 If ˆp(x 0 ) and ĝ(x 0 ) are both cancelled with two secondary sources, the residual pressure at a point x is, according to Eq. (46), p uu (x) only. Therefore, the mean squared pressure at a position x in the controlled sound field is given by p(x) 2 = ( ( P 2 px 1 + P 2 )) gx p 2. (54) P 0 G 0 When the pressure and pressure gradient at the virtual location, x 0, estimated using the pressures and pressure gradients at the points x 1 and x 2, are cancelled with two secondary sources, the relative change in mean squared pressure is given by p(x) 2 ( P 2 (1 pp 2 = px + P 2 )) gx α. (55) P 0 G 0 15
III. DIFFUSE FIELD SIMULATIONS The performance of the nine local active noise control strategies were evaluated using a MATLAB simulation. Quadratic optimisation 3 was used to simulate control, giving the limit on the maximum achievable feedforward control performance. The objective of quadratic optimisation is to calculate the optimal control source strengths required to minimise a desired cost function, in this case the total physical or virtual quantities measured at the sensors. The acoustic field was simulated using the analytical model described by Bullmore et al. 1. This analytical model assumes that the pressure at any point in the acoustic enclosure can be calculated using a finite modal summation. The complex pressure amplitude at any point x is given by N p(x,ω) = ψ n (x)a n (ω) = Ψ T a, (56) n=0 where the summation consists of N normal modes with normalised mode shape functions ψ n (x) and complex modal amplitudes a n (ω). Nth order vectors of these quantities are given by Ψ and a, whose nth order components are ψ n (x) and a n (ω) respectively. The vector a can be considered as a linear superposition of contributions from a primary source and a series of M secondary sources. Therefore a may be written as a = a p +Bq s, (57) where a p is the vector of complex modal amplitudes a pn (ω) due to the primary source, q s is the Mth order vector of complex secondary source strengths q sm (ω) and B is the N M matrix of modal excitation coefficients B nm (ω), connecting the excitation of the nth mode to the mth secondary source. Morse s solution 17 was used to describe the form of the sound field in the rectangular enclosure. The normalised mode shape functions are given by ψ n (x) = ε n1 ε n2 ε n3 cos(n 1 πx 1 /L 1 ) cos(n 2 πx 2 /L 2 )cos(n 3 πx 3 /L 3 ), (58) 16
where n 1, n 2 and n 3 are integer modal indices and L 1, L 2 and L 3 are the enclosure dimensions. The normalisation factors are ε ν = 1 if ν = 0 and ε ν = 2 if ν > 0, so that V ψ2 ndv = V, where V is the enclosure volume. In the computer simulation, the primary and secondary sources are assumed to be point monopoles at positions y p and y s respectively. Therefore the complex amplitude of the nth mode due to the primary source, with source strength q p, is a pn (ω) = ρv c ω 2ξ n ω n ω j(ω 2 n ω 2 ) ψ n(y p )q p, (59) and the modal excitation coefficients of the mth secondary source are given by B nm (ω) = ρv c ω 2ξ n ω n ω j(ω 2 n ω 2 ) ψ n(y sm ), (60) where ρ and c are the density and speed of sound in the medium and ξ n and ω n are the damping ratio and natural frequency of the nth mode respectively. The enclosure model used in this simulation was that described by Elliott et al. 6 and was rectangular in shape with dimensions π m e m 1 m. The constant damping ratio was set to be 0.0014 and the Schroeder frequency 24 was calculated as 400 Hz. An excitation frequency of 1.5 khz was used and all modes with natural frequencies below 2 khz were included so that the total number of modes was approximately 8000. A number of sources (dependent on the control strategy) were randomly located within the enclosure such that they were not within a wavelength of one another, or the enclosure walls. The average mean squared pressure 6 of the total controlled pressure field was computed over 200 source configurations at various distances in the x-direction from the point of cancellation. The results from these simulations are presented in Section V.A. IV. DIFFUSE FIELD EXPERIMENTS Experiments were also conducted to validate the analytical expressions and the numerical simulations. Post-processed control was implemented computationally using transfer functions experimentally measured in the reverberation chamber in the School of Mechanical Engineering at the University of Adelaide. The chamber has dimensions 6.840 m 5.660 17
m 4.720 m, a volume of 183 m 3 and a Schroeder frequency 24 of 391 Hz. In the chamber, a condenser microphone and a Microflown, to measure pressure and pressure gradient respectively, were mounted to a stepper-motor traverse. This enabled measurement over a 465 mm 360 mm 320 mm volume, with the target zone of quiet being approximately located at the centre of the volume. Six loudspeakers located near the corners of the room and sufficiently far apart as to produce uncorrelated sound fields, were used to either generate the primary sound field or to act as control sources. Such an arrangement allowed for a large number of combinations of primary and secondary sources to be used. The sources were driven with a multi-tonal signal containing tones from 800 Hz to 3000 Hz in 50 Hz increments using a dspace DS1104 card. For each speaker acting as the source in turn, the sound field in the room was allowed to stabilise, then the complex transfer functions between each of the sources and the sensors were measured. The average diffuse field zone of quiet was estimated by calculating the mean squared average of the controlled sound fields over a number of data sets (dependent on control strategy) and dividing it by the mean squared average of the primary fields. To obtain a large number of data sets to provide the spatial average, a number of different points in the field were selected as the sensor location, while ensuring that the relative arrangement of sensors remained constant. At each of the sensor locations, one of the loudspeakers was selected as the primary source and then a suitable number of secondary sources (dependent on control strategy) were selected from the remaining loudspeakers. This process was repeated for a large number of primary and secondary source combinations at each sensor location. V. RESULTS Figs. 1 and 2 show the control profiles obtained with each of the nine control strategies in a pure-tone diffuse sound field. The zones of quiet generated in numerically simulated and post-processed experimental control are shown, together with the analytical expressions for the relative change in mean squared pressure after control. In Figs. 1 and 2 the solid 18
vertical lines indicate the positions of the physical sensors while the dashed vertical line indicates the virtual location. Using control strategy 6 as an illustrative case, the control profiles obtained in analytical, numerically simulated and post-processed experimental control are shown in Fig. 1 (b), where a single secondary source has been used to minimise the pressure at the virtual location, estimated using the pressure and pressure gradient at a point. The analytical zone of quiet in this figure has been generated using Eq. (40) for the relative change in mean squared pressure after control. The numerically simulated and post-processed experimental control profiles have been generated using Eq. (33) to estimate the pressure at the virtual location. A. Numerically simulated results Fig. 1 (a) shows the relative change in the mean squared pressure of the sound field when the pressure at a point is cancelled with a single control source. The mean squared pressure change is plotted against the distance from the point of pressure cancellation, for a value α = 2, the 50th percentile value of α obtained in the numerical simulation. Fig. 1 (a) reveals that the simulated results close to the point of cancellation are a good fit to the analytical function. When minimising the pressure at a point with a single control source, the random variable α has the cumulative distribution given in Eq. (27). Such a distribution means that the increase in space-average mean squared pressure after control does not have a finite mean value. However, in practice, the strength of the secondary source is limited which will prevent α from having a theoretically infinite mean value. Fig. 3 shows the cumulative distribution function of the random variable α observed in the numerical simulation compared to the analytical distribution function. The distribution computed from 600 simulations is seen to be a good fit to the analytical function and indicates that sufficient modes have been included in the simulations. The 50th percentile value of α is approximately 2 for cancellation of pressure at a point with a single control source as seen in Fig. 3. For a value of α = 2, the 10 db zone of quiet observed in the simulation is a sphere 19
Mean squared pressure difference, db 10 5 0 5 10 15 20 25 30 Analytical Numerical Experimental 1 0.5 0 0.5 1 Normalised distance, x/λ (a) Mean squared pressure difference, db 10 5 0 5 10 15 20 25 Analytical Numerical Experimental 1 0.5 0 0.5 1 Normalised distance, x/λ (b) Mean squared pressure difference, db 10 5 0 5 10 15 20 25 Analytical Numerical Experimental 1 0.5 0 0.5 1 Normalised distance, x/λ (c) FIG. 1. Control profiles achieved in analytical, numerically simulated and post-processed experimental control with control strategies employing a single source and error sensor: (a) Control strategy 1. (b) Control strategy 6. (c) Control strategy 7. Solid vertical lines indicate the positions of the physical sensors while the dashed vertical line indicates the virtual location. of approximately λ/10, as shown in Fig. 1 (a). The same result was found by Elliott et al. 6. Fig. 2 (a) illustrates the control profile achieved with an energy density sensor (pressure and pressure gradient) and two control sources in a diffuse sound field, for a value of α = 3.4. The analytical function clearly gives a good prediction of the zone of quiet obtained in the numerical simulation. The distribution of the random variable α observed in the numerical simulation, when two control sources are used, is shown in Fig. 4 (a). Again, α does not have a finite mean value, however, in practice the mean squared pressure after control will 20
Mean squared pressure difference, db 10 0 10 20 30 40 50 Analytical Numerical Experimental 1 0.5 0 0.5 1 Normalised distance, x/λ (a) Mean squared pressure difference, db 10 0 10 20 30 40 50 Analytical Numerical Experimental 1 0.5 0 0.5 1 Normalised distance, x/λ (b) Mean squared pressure difference, db 10 0 10 20 30 40 50 Analytical Numerical Experimental 1 0.5 0 0.5 1 Normalised distance, x/λ (c) Mean squared pressure difference, db 10 0 10 20 30 40 50 Analytical Numerical Experimental 1 0.5 0 0.5 1 Normalised distance, x/λ (d) Mean squared pressure difference, db 10 0 10 20 30 40 50 Analytical Numerical Experimental 1 0.5 0 0.5 1 Normalised distance, x/λ (e) Mean squared pressure difference, db 10 0 10 20 30 40 50 Analytical Numerical Experimental 1 0.5 0 0.5 1 Normalised distance, x/λ (f) FIG. 2. Control profiles achieved in analytical, numerically simulated and post-processed experimental control with control strategies employing multiple sensors and secondary sources: (a) Control strategy 2. (b) Control strategy 3. (c) Control strategy 4. (d) Control strategy 5. (e) Control strategy 8. (f) Control strategy 9. Control profiles obtained with postprocessed experimental data are generated by removing any ill-conditioning. Solid vertical lines indicate the positions of the physical sensors while the dashed vertical line indicates the virtual location. 21
1 Cumulative distribution 0.8 0.6 0.4 0.2 0 Analytical Numerical Experimental 2 4 6 8 10 α FIG. 3. Cumulative distributions for α obtained in numerically simulated and post-processed experimental control when using control strategy 1 compared to the analytical F 2,2 distribution. The 50th percentile value of α is indicated by a solid vertical line. be limited by the secondary source strengths. The 50th percentile value of α observed in the numerical simulation is approximately 3.4 for cancellation of pressure and pressure gradient at a point with two control sources. For a value of α = 3.4, the 10 db zone of quiet observed in the simulation has a diameter of λ/2 in the direction of pressure gradient cancellation. The same result was found by Elliott and Garcia-Bonito 5 and Garcia-Bonito and Elliott 9. This is a five fold increase in the size of the zone of quiet compared to that obtained by cancelling pressure alone. Similar control performance is achieved by minimising the pressures at two points with two control sources as shown in Fig. 2 (b). Again, this is the same result found by Elliott and Garcia-Bonito 5 and Garcia-Bonito and Elliott 9. Superior control performance is achieved using energy density sensors at two points with four control sources, as shown in Fig. 2 (c), for a value of α = 5.6. Again, numerically simulated results close to the point of cancellation are a good fit to the analytical function. When four control sources are used, the random variable α has the distribution shown in Fig. 4 (a). Similarly to the previous two distributions, it does not have a finite mean value, but will in practice be limited by the strengths of the secondary sources. The 50th percentile value of α observed in the numerical simulation is approximately 5.6 for cancellation of 22
1 1 (a) (b) 0.8 0.8 Cumulative distribution 0.6 0.4 0.2 0 One Numerical One Experimental Two Numerical Two Experimental Four Numerical Four Experimental 2 4 6 8 10 α Cumulative distribution 0.6 0.4 0.2 0 One Numerical One Experimental Two Numerical Two Experimental 2 4 6 8 10 α FIG. 4. Cumulative distributions for α obtained in numerically simulated and post-processed experimental control: (a) Conventional control strategies 1 (one control source), 2 or 3 (two control sources) and 4 or 5 (four control sources). (b) Virtual control strategies 6 or 7 (one control source) and 8 or 9 (two control sources). 50th percentile values of α are indicated by solid vertical lines. pressure and pressure gradient at two points with four control sources. For a value of α = 5.6, the 10 db zone of quiet observed in the simulation has a diameter of λ in the direction of pressure gradient measurement, as shown in Fig. 2 (c). This is a 10 fold increase compared to using a single microphone with one control source or a doubling compared to using a single energy density sensor with two control sources. The same result can be achieved by minimising the pressures at four points with four control sources, as shown in Fig. 2 (d). Control with a virtual microphone, using the measured pressure and pressure gradient at a point, is shown in Fig. 1 (b), for a value of α = 2. Fig. 1 (b) reveals that the numerically simulated results close to the point of cancellation are a good fit to the analytical function. The distribution of the random variable α, observed in the simulation of virtual microphone control, is equal to the F α,1 distribution given in Eq. (27) as demonstrated in Fig. 4 (b). For the 50th percentile value of α = 2, a maximum attenuation of 24 db and a 10 db zone of quiet with diameter of approximately λ/10 is generated at the virtual location in 23
the simulation. Fig. 1 (c) shows similar control performance can be obtained using the pressures at two points to estimate the pressure at a virtual location. In comparison to the performance of conventional control strategies, a virtual microphone achieves higher attenuation at the virtual location than a single microphone and one control source or an energy density sensor and two control sources. Conventional control strategies employing four control sources achieve significantly higher attenuation at the virtual location than a virtual microphone and a single control source. The increase in attenuation at the virtual location achieved with four secondary sources is, however, most likely accompanied by an increase in sound pressure level away from the point of cancellation. This is indicated by the median (50th percentile) value of α being only 2 for a single control source and 5.6 for four control sources. As found in previous research, the size of the 10dB zone of quiet achieved at the virtual location with the stochastically optimal diffuse field virtual sensing method and the virtual microphone arrangement is similiar at low frequencies. However, at frequencies above 500Hz the stochastically optimal diffuse field virtual sensing method outperforms the virtual microphone arrangement because the assumption of equal sound pressure at the physical and virtual locations is no longer valid and the zone of quiet achieved at the virtual location with the virtual microphone arrangement is severely reduced 9,11. Control with a virtual energy density sensor and two control sources produces a superior control profile to that achieved with a virtual microphone and a single control source. This is indicated by the zone of quiet in Fig. 2 (e), for a value of α = 3.4, where the pressure and pressure gradient at a virtual location are estimated using the measured pressures and pressure gradients at two points. Again, numerically simulated results close to the point of cancellation are a good fit to the analytical function. A maximum attenuation of 50 db and a 10 db zone of quiet with diameter of approximately λ/2 is achieved at the virtual location in the simulation. A similar result is obtained using the pressures at four points to estimate the pressure and pressure gradient at the virtual location as shown in Fig. 2 (f). The distribution of the random variable α, when using two control sources and a virtual energy density sensor is shown in Fig. 4 (b). This figure indicates that the distribution for 24
α is equal when using either a physical or virtual energy density sensor and two control sources. Comparison of Figs. 4 (a) and (b) illustrates that the distribution of α is only dependent on the number of control sources and not on the control strategy. In comparing the performance of virtual energy density sensors to that of conventional control strategies, a virtual energy density sensor achieves significantly higher attenuation at the virtual location than a single microphone and one control source or an energy density sensor and two control sources. Conventional control strategies employing four control sources achieve similar levels of attenuation at the virtual location to a virtual energy density sensor and two control sources. The increase in mean squared pressure after control will, however, most likely be higher when four control sources are used compared to only two control sources. This is indicated by the median (50th percentile) value of α being approximately 3.4 for two control sources and 5.6 for four control sources. B. Experimental results Fig. 1 shows the performance of control strategies employing a single error sensor and control source, for a value of α = 2. Using the post-processed experimental data, minimising either the measured or estimated pressure generates a zone of quiet with diameter of λ/10 at the physical and virtual locations respectively. This agrees well with the analytical and the numerically simulated results presented earlier. The distributions of the random variable α observed in post-processed experimental control are shown in Figs. 3 and 4. The distributions are seen to be a good fit with those observed in the simulations. In comparison to previous experimental results, Garcia-Bonito and Elliott 8,9 also found that the 10dB zone of quiet obtained by minimising the pressure at a physical microphone is a sphere with a diameter of λ/10 in a pure tone diffuse sound field at high frequencies. It was also shown that minimising the pressure at the surface of a rigid object tends to extend the zone of quiet beyond that achieved in the absence of the diffracting object 8,9. As stated previously, minimising the pressure estimated at the virtual location with the 25
Mean squared pressure difference, db 10 0 10 20 30 40 50 Analytical Numerical Experimental 1 0.5 0 0.5 1 Normalised distance, x/λ FIG. 5. Example of the effect of ill-conditioning in post-processed experimental results using control strategy 2. The solid vertical line indicates the position of the physical sensors. stochastically optimal diffuse field virtual sensing method generates a zone of quiet the same size as that achieved by minimising the measured pressure. The same experimental result was observed at low frequencies by Garcia-Bonito et al. 11 for the virtual microphone arrangement. The attenuation achieved at the virtual location with the virtual microphone arrangement is, however, severely reduced at higher frequencies due to the assumption of equal sound pressure at the physical and virtual locations being invalid. While the performance of control strategies employing a single error sensor and control source is as expected, control strategies employing multiple sensors and secondary sources achieve poorer experimental control than expected from the analytical and numerically simulated results. An example of the poor experimental performance of control strategies employing multiple sensors and secondary sources is shown in Fig. 5. With the raw experimental post-processed data, minimising the pressure and pressure gradient at a point with two control sources generates a zone of quiet with a diameter of λ/5 in the direction of pressure gradient measurement. This is significantly smaller than the zone of quiet with diameter of λ/2 obtained in analytical and numerically simulated results. A 15 db reduction in maximum attenuation can also be seen at the sensor location in post-processed experimental results. 26
1. Improving conditioning The poor experimental performance of strategies employing multiple sensors and secondary sources is attributed to ill-conditioning. Quadratic optimisation was used to calculate the optimal secondary source strengths required to minimise the quantities measured at the physical or virtual sensors. As all control strategies employ the same number of secondary sources as sensors, the system is said to be fully-determined. In this case, the optimal set of secondary source strengths for which the value of the cost function is zero are 3 q c = Z 1 p p, (61) where Z is the secondary transfer matrix and p p is the primary sound field measured at the sensors. Calculation of the optimal source strengths therefore requires inversion of the secondary transfer matrix Z. When minimising the pressure at the physical or virtual location with a single secondary source, Z is scalar and hence the control source strength can be evaluated perfectly provided the impedance matrix is non-zero. All remaining control strategies, however, employ multiple sensors and secondary sources and the accuracy of the calculated secondary source strengths can be attributed to the conditioning of matrix Z. Distributions of the condition number, κ, of matrix Z have been examined for all control strategies employing multiple sensors and secondary sources. The spread of all condition number distributions is very wide and it is evident that certain configurations of secondary sources and sensors result in a high condition number and hence inaccurately calculated secondary source strengths. For the example of pressure and pressure gradient control given in Fig. 5, the median condition number is κ mean = 53 and the spread of all condition numbers is between κ min = 2 and κ max = 5012, as shown in Fig. 6. The ill-conditioning may be attributed to coherences less than unity in the measured transfer functions, thus introducing magnitude and more importantly phase errors. To improve the poor experimental performance of control strategies employing multiple sensors and secondary sources, ill-conditioning was addressed in two ways. Firstly, poorly conditioned data sets were removed from calculation of the mean squared pressure. Only 27
25 20 Configurations with κ 15 10 5 0 0 1 2 3 4 log(κ) FIG. 6. Distribution of the condition number, κ, for control strategy 2. samples of controlled sound field with a condition number below a certain threshold value were used in data averaging. The threshold value for each control strategy was selected to achieve the desired accuracy with the largest number of samples. For pressure and pressure gradient minimisation, shown in Fig. 5, the threshold value used was κ = 25 and therefore 42% of the total number of data sets were included in calculation of the mean squared pressure (where the total number of data sets was 3600). The alternative technique used to improve the conditioning was to add one more control source than necessary to the active control system. When a greater number of control sources than error sensors are present, the system is said to be underdetermined. In this case, the optimal secondary source strengths are found by applying an additional constraint on the control effort 22. Both of these methods for improving conditioning have been investigated and were found to produce essentially equivalent control profiles. Fig. 2 shows the experimental performance of control strategies employing multiple sensors and secondary sources with ill-conditioned data removed. By removing ill-conditioning, post-processed control with an energy density sensor and two control sources generates a zone of quiet of diameter 0.45λ as shown in Fig. 2 (a). This is an agreeable fit with the analytical and numerically simulated results in which a quiet zone of size λ/2 is achieved. Minimising the pressures at two points generates a similar sized zone of quiet to that with 28
an energy density sensor and two control sources as shown in Fig. 2 (b). The experimental results achieved here by minimising the pressures at two points are in agreement with those obtained by Miyoshi and Kaneda 16, however they plotted zones of quiet using contours of 6dB and 14.5dB attenuation. The superior control performance achieved with four control sources is indicated by the larger zones of quiet in Figs. 2 (c) and (d). By removing ill-conditioning, a zone of quiet of diameter 0.9λ is obtained. This is a good fit with the analytical and numerically simulated results in which the quiet zone has a diameter of λ. Control with a virtual energy density sensor is shown in Fig. 2 (e), where the pressure and pressure gradient at a virtual location are estimated using the measured pressure and pressure gradient at two points. By removing ill-conditioning, this virtual control strategy results in a 10 db zone of quiet with longest diameter approximately λ/2, which is a good fit with the analytical and numerically simulated results. A reduction in maximum attenuation is seen in post-processed control with 39 db of attenuation achieved at the virtual location compared to 50 db in the simulation. A similar level of control can be obtained using the pressures at four points to estimate the pressure and pressure gradient at the virtual location, as shown in Fig. 2 (f). Fig. 7 shows the distributions of the random variable α observed in post-processed control when any ill-conditioning is removed or when one more control source than necessary is added to the control system. With the conditioning improved, the increase in mean squared pressure away from the point of cancellation is likely to be much smaller. This is indicated by the 50th percentile α values in Fig. 7 being significantly smaller than those discussed in Section V.A. Fig. 7 shows that a smaller increase in mean squared pressure away from the point of cancellation can be expected when conditioning is improved by adding an extra control source instead of by removing any ill-conditioning. The distributions corresponding to the removal of ill-conditioning shown in Figs. 7 (a) and (b) can be improved and made to closely match those for addition of an extra control source, as shown in Figs. 7 (c) and (d), by reducing the threshold value of κ. However, reducing the threshold value means 29