ACTIVE CONTROL OF ENVIRONMENTAL NOISE

Journal of Sound and Vibration (1996) 190(3), 565–585

ACTIVE CONTROL OF ENVIRONMENTAL NOISE,† S. E. W  B. V University of Huddersfield, School of Engineering, Huddersfield HD1 3DH, England (Received 1 November 1995) Most of the current research on active noise control is confined to restricted spaces such as earphones, active silencers, air-conditioning ducts, truck cabins and aircraft fuselages. In this paper the basic concepts of environmental noise reduction by using active noise control in unconfined spaces are explored. The approach is to develop a controlled acoustic shadow, generated by a wall of secondary sources, to reduce unwanted sound in the direction of a complaint area. The basic acoustic theory is considered, followed by computer modelling, and some results to show the effectiveness of the approach. EA Technology and Yorkshire electric in the United Kingdom are supporting this work. 7 1996 Academic Press Limited

1. INTRODUCTION

It is difficult to reduce low frequency sound from large, complex primary source structures by using conventional methods. One approach would be to reduce actively the vibrations on the structure, by using secondary vibration sources or absorbers. Unfortunately, this approach is not very practical in many applications, mainly because of the large number of drivers needed to produce a considerable reduction in sound, and the large forces/energy involved. Also in many cases, all around (360°) sound reduction from the primary source is not needed; only reduction over a given angle, covering a complaint area, is usually all that is required. The approach described here, therefore, is to investigate the acoustic properties of free standing walls of secondary acoustic sources. These have the ability to absorb/redistribute the acoustic energy, and therefore produce a controlled acoustic shadow in the direction of the complaint area. There is need to develop this technology, particularly, to reduce noise from low frequency sources, for example, heavy machinery in manufacturing and vibrating structures, such as power transformers, in substation yards. Classical pioneer work on one or two channel outdoor noise suppression is available [1–3]. This early work was not very effective, because of the low number of secondary sources, and the unavailability of inexpensive digitally controlled systems. More recently, preliminary work has been carried out successfully on generating acoustic shadows, from extended sources outdoors, by using multiple channels [4]. Considerable effort has gone into considering single channel feedforward and feedback systems, for

†This paper is the first of several companion papers. A second paper, ‘‘Active control of non-compact acoustic sources’’ has been submitted to JSV for a later publication.

565 0022–460X/96/080565+21 $12.00/0

7 1996 Academic Press Limited

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sound reduction in one dimension, such as in a duct [5–7] and in local spaces such as earphones [8, 9]. Some theoretical work on reflecting and absorbing point multi-pole source arrays has been considered [10]. Presently, multi-channel research [11] is ongoing to reduce or modify sound in enclosed spaces such as aeroplane, car and truck cabins. With inexpensive digital computer controlled electronic systems rapidly becoming available, it now appears practical and cost effective to investigate active wall technology to reduce sounds in unconfined spaces. The problem with conventional walls and enclosures is that they do not work well at low frequencies. Effective structures are very expensive to install and are difficult to remove once built. On the other hand, electronic walls could work well, particularly at low frequencies, can be tailored to a particular application, and are light and easily removed to a new site of application.

2. COMPUTER MODELLING

To investigate the properties of electronically controlled shadows, the primary source distribution, the active wall of secondary sources and the microphone array can be simulated through computer modelling, as illustrated in Figure 1. Here, both the primary and secondary sources are represented by point sources that radiate independently, and thus do not affect each other. Also, linear acoustic theory is used, with resistive losses neglected. A primary source is simulated by an array of point monopole sources the frequency, phase and amplitude of which can be controlled to represent various modal source distributions. The acoustic radiation from this primary source distribution is computed in the far field over given control angles, both in azimuth and in elevation. A wall of secondary sources (cancellers), the phase and amplitude of which can be varied, is positioned at some distance from the primary source plane and enclosed in the same control angles. An array of microphones is defined within the control angles, to monitor and control the sound field over the observer plane. The phase and amplitude of the secondary sources are computed through matrix algebra by using various forms of least squares algorithms to give minimum sound at the microphone array.

Figure 1. The spatial arrangement of the primary, secondary, microphone and observer planes.

    

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3. PARAMETERS UNDER INVESTIGATION

By using this basic structure, and assuming steady state conditions, initially the following parameters are being investigated for a given frequency and control (shadow) angles. Nominally, angles up to 360° could be investigated, but of most interest are angles between 15° and 180° in azimuth, and 15° to 30° in elevation. The parameters are as follows: (1) size and complexity (modal order) of the primary source in three dimensions; (2) position, number and multipole order of secondary sources within the control angles. Examples of multi-pole configurations to be investigated are omnidirectional (monopole) and directional (monopole/dipole combination); (3) the position and number of microphones within the control angle; (4) the distances of the secondary source array and microphones from the primary source distribution; (5) the shadow profile (shape and depth) as a function of distance into the far field from the microphone array; (6) the effect of acoustic interference within the secondary source array and between the secondary sources and primary source distributions; (7) the degree of absorption and reflection of acoustic power between the primary and secondary sources; (8) the adaptability of the secondary sources to optimize to particular primary source situations; (9) the effect of reflection, temperature inversion and wind speed. From this extensive modelling, the basic acoustic properties of shadows generated by active walls is being established and theory formulated to represent them. The resulting extensive parameter study can be used to predict and optimize the performance of subsequent active wall development.

4. THEORETICAL DEVELOPMENT

The one-dimensional wave equation for airborne sound can be written as 1 2p/1t 2=c 21 2p/1x 2

(1)

and the form of its general solution is p(x, t)=F1 (t−x/c)+F2 (t+x/c),

(2)

where p is the sound pressure at the observed point in space (Pa), c is the wave propagation velocity at 20 °C (m/s), F1 and F2 are arbitrary functions representing the outward- and inward-going waves respectively, x is the location of the observer point from the source (m) and t is the time (s). In terms of the radial co-ordinate r (m) to describe the location of the observer, the corresponding solution for waves in three-dimensional space can be written as p(r, t)=

F1 (t−r/c) F2 (t+r/c) + . r r

(3)

For a harmonic outward-going wave only equaton (3) becomes p(r, t)=A ej(vt−kr)/r,

(4)

where v=2pf is the angular frequency of the harmonic fluctuations (rad/s) with frequency f, k is the wavenumber k=v/c=2p/l, l is the wavelength, and kr is the phase angle with distance r. A is a complex number which specifies both the amplitude and phase of the pressure fluctuations at the source point.

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Figure 2. The definition of (a) the elevation and (b) the azimuth shadow angles a and b.

For a small (point) spherical source having radius al and with the time varying term suppressed, the complex sound pressure is given by p(r)=vrq e−jkr/4pr,

(5)

where r is the density of the air at 20°C (kg/m ), q is the complex source strength, volume rate (m3/s) defined as 3

q=Ua 4pa 2

(6)

and Ua (m/s) is the surface velocity of the point source, the phase and amplitude of which can be varied. By using the relation (5) for the sound field of a point monopole source and the principle of pressure superposition, it is possible to simulate the sound field from a collection of time-harmonic point monopole sources arranged anywhere in space. In our application we arrange the point sources in two planes, the secondary source plane a distance rs from the primary source plane, as shown in more detail in Figure 2. The complex sound pressures at a particular microphone m in the microphone plane, at a distance rm from the primary source array, from a single primary or secondary source, respectively, can be expressed as ppm=

vr e−jkrpm qp , 4prpm

psm=

vr e−jkrsm qs , 4prsm

(7, 8)

where rpm and rsm are the distances between the corresponding point sources (primary or secondary) and microphones, and qp , qS are the strengths of the corresponding primary and secondary sources respectively. The total complex sound pressure at microphone m from all primary and secondary sources can be expressed as pTm=s ppm+s psm . p

s

(9)

    

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By using complex matrix notation equation (9) can be written as PTm=Ppm+Psm ,

(10)

where the contributions from the primary and secondary sources respectively are Ppm=Cpm Qp , cpm=

vr e−jkrpm , 4prpm

csm=

vr e−jkrsm , 4prsm

Psm=Csm Qs , qp=Up 4pap2 ,

(11, 12) qs=Us 4pas2 .

(13–16)

cpm and csm are referred to as propagation coefficients. To optimize the shadow depth over all microphones a least squares algorithm can be applied by introducing a cost function as the sum of the squared moduli of the outputs from all m microphones see, for example, reference [12] for further information regarding optimization processes): i.e., H J=s =pTm =2=PTm PTm ,

(17)

m

where H denotes the Hermitian transpose of the matrix (complex conjugate of the matrix transpose). The optimization is performed by using the least squares method; i.e., the complex source strengths of the s secondary sources are chosen to minimize the sum of the squared outputs from each of the m microphones. Substituting equations (11) and (12) in equation (17) results in the Hermitian quadratic form (see the Appendix for further details) H H H H J=Ppm Ppm+Ppm Csm Qs+QsH Csm Ppm+QsH Csm Csm Qs .

(18)

By differentiating equation (18) with respect to the real and imaginary parts of the vector QS and setting these derivatives to zero the minimum value of J can be found. After mathematical manipulation (see, e.g., reference [13]), the resulting set of optimum secondary source strengths (QS )opt which minimizes J and gives the maximum shadow depth is given by H H (Qs )opt=−(Csm Csm )−1Csm Ppm ,

(19)

and, for the special case when m=s, (Qs )opt=−C−1 sm Ppm .

(20)

Thus the optimum secondary source strengths depends on the sound pressure at the microphones from the primary sources and the propagation distances between the sources and microphones. 5. COMPUTATION

The wave propagation distance r is calculated by using r(x, r)=z(x−xs )2( y−ys )2+(z−zs )2 .

(21)

Here x, y and z are the Cartesian co-ordinates of the measurement point and xS , yS and zS are the Cartesian co-ordinates of the source point (primary and secondary). Since we limit our calculation to parallel source and measurement planes, the y co-ordinate, as shown in Figure 2, is the same for all sources in the same plane, while the x and z co-ordinates have to be calculated according to the sources’ numbers and positions in the

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plane. By calculating the propagation distances r between each source and microphone, according to equations (13) and (14), the matrices Cpm and Csm are formed. Given qp from the relation (15), then by using relations (11) and (13) the vector Ppm can be calculated. With values of Csm and Ppm the set of optimal secondary source strengths (qS )opt is then calculated from equations (19) or (20). By substituting this value of (qS )opt back in equation (12) the vector Psm is calculated. By adding the corresponding values from vectors Ppm and Psm the total sound pressure at each microphone position PTm can be found. Similarly, Cpo , Cso , Ppo , Pso and PTo for any observer position can also be found. Finally, to find the overall sound pressure level the 20 log (P/Pref ) operation is performed, where P is the modulus of the complex sound pressure and Pref=20 mPa is the reference sound pressure. MATHCAD 5.0 is used to manipulate matrices and to generate surface and contour matrix plots. With this approach it is possible to determine the sound level for any plane in the system parallel to the primary and secondary sources’ planes. Calculations for two distinct planes at distances rm (microphone plane) and ro (observer plane) from the primary source plane can then be performed. For large angles, spherical secondary source, microphone and observer surfaces are used. For small angles, of course, there is virtually no difference between the two surfaces.

6. RESULTS

With the foregoing optimization procedures the following shadow characteristics have been computed. 6.1.   A 15°×15° unit shadow is selected as the standard case. By establishing the property of this unit shadow, the characteristics of any size of shadow can be estimated using multiples of this basic buliding bock. Unless other stated, the following standard parameters are used: p=36, qp=0·028 m3/s, total Qp=pqp=1 m3/s (primary source size

Figure 3. Primary sources (p) and secondary sources (s) and microphones (m) surrounded by a 360° observer strip.

    

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Figure 4. A high resolution 15°×15° unit shadow.

D=2 m square, 6×6 equally spaced source configuration, equal amplitude and phase for all p), s=m=9 (3×3 equally spaced secondary source and microphone configuration), d=D/(N−1)=1 (approximate secondary source spacing based on the primary source size, where N is number of secondary sources in a row or column), f=100 Hz,

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c=343 m/s, l=3·43 m, r=1·2 kg/m3, a=15°, b=15°, rs=1·715=l/2 (plane surface), rm=ro=50 m (spherical surface), scanning step sizes for observer surface Do=4°. 6.2.   The measurement geometry used in computing the following figures is illustrated in Figure 3. It shows the positions of the primary, secondary and microphone positions surrounded by an observer strip 360° long×60° wide with the primary source at the centre. The shadow which is generated in detail in front of the source, can also be monitored around the whole 360° in aximuth. 6.3. 15°   In Figure 4 is shown a remarkable, 100 Hz acoustic shadow in relief and contour. The shadow, which is the standard case, is over a 15°×15° unit control angle and observed over a 50°×50° window at an observer distance of 50 m from the primary source surface. The primary source, 2 m×2 m square (D=2 m, l/D=1·71) was simulated by an array of 6 rows×6 columns of equispaced in-phase discrete sources. The secondary source surface comprises 3×3 equispaced discrete sources (N=3), making a separation distance between sources of approximately 1 m (l/d=3·43). The secondary sources are positioned at a distance of l/2 (1·71 m) from the primary source surface. The microphone measurement surface comprises 3×3 equispaced microphones situated at 50 m from the primary source. Both the secondary source and micrphone arrays lie within the 15°×15° control angle. 6.4.     In Figure 5 is shown a cross-section of the shadow bottom and the effect of increasing the number of microphones beyond the number of cancellers. The high resolution observer plots (scanning step=0·5°) show that the sound for equal number of microphones and secondary sources (s=m=9) produces zero sound pressure at three microphones positions. Increasing the number of microphone rows to six, but still three deep, produces

Figure 5. M=S and MqS effects on the shadow shape.

    

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Figure 6. 15°×15° unit shadow observed over 360°.

finite sound at the microphones, but not necessarily minima. This is demonstrated by the four minima not occurring at the six equispaced microphone positions. Using more microphones smooths the shadow bottom, as is to be expected, but does not increase the average shadow depth.

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6.5. 360°   In Figure 6 is shown the acoustic field computed all around the source (360° azimuthal angle) and over a 60° observer elevation angle, for the standard 15°×15° unit shadow. The relief and contour plots (a) and (b) show the shadow positioned in front of the source at about 0°. The radiation to the side (290°) and behind the source (2180°) are of

Figure 7. A unit shadow at large distance from the source (ro=5000 m).

    

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Figure 8. A unit shadow with reduced canceller distance (l/8).

particular interest. The shadow averaged over 15° in elevation, plot (c), shows the sound from the primary source before and after cancellation. The uncancelled sound is some 10 dB lower to the side than in front and behind. The cancelled sound produces a shadow depth of approximately 75 dB with a 15° wide bottom. A weaker shadow (25 dB) is also generated behind the source. This is because rs=l/2, or multiples of l/2, for optimum

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phasing, produces 180° phase behind the source, also. The cancelled radiation to the side is the same amplitude as the uncancelled sound in front of the source. 6.6.     In Figure 7 is shown the effect on the 15°×15° unit shadow of moving the observer distance d0 from 50 m to 5000 m. Very importantly, it can be seen that the shadow does

Figure 9. A unit shadow with only four cancellers.

    

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Figure 10. A 60°×15° shadow.

not ‘‘fill in’’. That is, once generated the shadow propagates to large distances from the source little attenuated. Its depth is still about 60 dB, but of course it has become smoother. 6.7.    In Figure 8 is shown the unit shadow with the cancelling sources moved closer to the primary source surface, from l/2 to l/8 (0·428 m). The main effect here is that the shadow

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depth now increases by 20 dB, from 75 dB to 95 dB. Also there is no longer a shadow behind the source, or increased radiation to the side. 6.8.    In Figure 9 is shown the unit shadow with only four cancelling sources. Even here a respectible 25 dB shadow depth is still generated. Note the minima at 80° due to the interference between the individual sources. 6.9. 60°   In Figure 10 is shown a nominal 60° azimuthal by 15° elevation shadow constructed from three rows by nine columns of equispaced discrete sources. It can be seen that the resulting 75 dB shadow is equivalent to four 15°×15° unit shadows. This supports the construction of arbitary shadow sizes by using a single control surface and the unit shadow as a building block. The secondary sources were computed on a sphere as well as on a

Figure 11. A summary of unit shadow depths for various canceller distances and numbers of cancellers. (a) S=M=2×2: (b) S=M=3×3; (c) S=M=4×4; (d) S=M=5×5; (1) rs=l/8; (2) rs=l/2; (3) rs=2l.

    

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Figure 12. A construction of 15°, 30° and 60° shadows using one, two and four, 15°×15° unit shadows for various numbers of cancellers. (a) a=15°, S=2×2 (1), S=3×3 (2); (b) a=30°, S=3×2 (2), S=5×3 (3); (c) a=60°, S=5×2 (2), S=9×3 (3).

plane with similar results. Of course, for angles approaching 180°, there will be progressively more distortion for measurements in a plane than for those on a sphere. 6.10.      The unit shadow levels averaged over 15° in elevation are summarized in Figure 11. The shadow depths in dB for various canceller distances rs=l/8, l/2 and 2l from the primary source and various canceller arrays s=m=2×2, 3×3, 4×4 and 5×5 are shown. Generally, the shadow depth increases with increasing the number of cancellers or non-dimensional canceller spacing l/d, where d is the separation distance between the cancellers. Also, the shadow depth reduces with increasing primary source–canceller distance rs . Apart from the rs=l/8 case, the sound is reduced behind the primary source as well as in front, and is increased to the sides. For canceller arrays larger than 5×5, producing cancellations q−130 dB, radiation to the side (290°) increases and the cancellation begins to fail. This is presumably through a combination of matrix ill-conditioning (too many small and similar microphone values) and computer numerical resolution limitations. 6.11. 15°, 30°  60°   In Figure 12 are shown 15°, 30° and 60° azimuth shadows averaged over 15° shadow elevation. Generally, the shadow width increases linearly with increase in aximuthal cancellers, while maintaining constant shadow depth. The shadow depth generally increases with increase in the number of cancellers or non-dimensional cancellers spacing

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(l/d). This figure again supports the notion of unit shadow addition for f=100 Hz and D=2 m. 6.12.     In Figure 13 is shown the unit shadow for various observer (ro ) and microphone (rm ) distances. For the observer closer to the primary source than the microphones, interesting curves are produced, but with poor shadow depths. Generally, the shadow depth increases with microphone distance and the sound pressure generally reduces with observer distance. However, the shadow depth is maintained. 6.13.     The shadow depth with microphone distance from the primary source is given in Figure 14 for various observer distances. Note that minimums occur when r0=rm . For the normal case of observer distances greater than the microphone distance, the shadow depth increases linearly in dB, as the microphone distance increases from l to 100l (dotted

Figure 13. A summary of unit shadow depths for various observer and microphone distances. (a) ro=l; (b) ro=10l; (c) ro=100l; (d) ro=1000l. (1) rm=l; (2) rm=10l; (3) rm=100l.

    

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Figure 14. 100 Hz distance curves showing average shadow depth (dB) as a function of microphone distance rm for various observer distances ro . ro values in wavelengths (l): ×, 1; q, 5; e, 10; +, 20, ×, 100; q, 1000. D=2m, l=3·43 m, rs=l/2, d=1 m, l/D=1·71, l/d=3·43, s=3×3.

line). Here the following equation can be used to predict the minimum shadow depth with microphone distance for observer distances greater than 100l: dB=−30 log rm−15.

(22)

For rmq100l, the shadow depth levels out to about 75 dB. The standard case rm=ro=50 m (14·5l) is shown as a circle. If the observer moves to 100l, for instance, this will reduce the shadow depth by approximately 20 dB (75–55), and the microphone would then need to move out to about 60l to restore the depth. 6.14.    Finally, the basic attenuation curves of actively controlled shadows are summarized in Figure 15. Here the average shadow depth in dB is plotted as a function of the log of the non-dimensional canceller spacing l/d. The shadow depths are shown for various canceller – primary source separation distances l/8, l/2 and 2l. The various canceller matrix sizes are shown at the top of the figure for the 15°×15° unit shadow. It can be seen that the relationships are linear in logarithmic units, converging at approximate zero dB, and l/d of unity. Thus cancellation starts to be effective when the canceller spacing becomes less than the wavelength and increases with increasing canceller density. The data suggest the following shadow depth prediction formula for ro=rm=50 m and l/D=1·715: dB=−10n log (l/d)+c, with approximate values of the constants c and n as given in Table 1.

(23)

. .   . 

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Figure 15. 100 Hz performance curves showing average shadow depth (dB) as a function of non-dimensionalized canceller spacing (l/d) for various canceller distances: rs=l/8 (1); rs=l/2 (2); rs=2l (3). ro=rm=50 m (14·6l), D=2 mg l=3·43 m, l/D=1·71.

As an example, a 40 dB 100 Hz shadow is needed over a 15°×30° control angle. The primary source size is D=3 m square, and canceller – primary source spacing is rs=l/2. From equation (23) one has log (l/d)=(40+5)/130=0·346,

l/d=2·22,

d=3·43/2·22=1·5 m,

N=(D/d)+1=3. Thus for a 15°×15° shadow, s=3×3=9 cancellers are needed, and for a 15°230° shadow, s=18 cancellers are predicted. 7. CONCLUSIONS

The feasibility of actively controlled environmental noise has been investigated. A method has been described that generates electronically controlled shadows from primary acoustic sources. Computer modelling shows that this approach is physically sound and has considerable potential to reduce environmental noise. T 1 Approximate values of the prediction constants c and n for shadow depth prediction equation (23) rs

c (dB)

n

l/8 l/2 2l

5 5 5

16 13 10

    

583

As a consequence, the concept of generating an acoustic shadow in the direction of the complaint area, has resulted in the acoustic properties of a 15°×15° unit shadow being established in detail. It appears that any arbitrary shadow can then be constructed by an addition of these unit shadows, the shadow depth depending on the density of the cancellers per unit angle. Deep shadows in access of 100 dB are predicted, making practical shadows from real sources a possibility. ACKNOWLEDGMENTS

The authors would like to thank Yorkshire Electric and EA Technology for supporting this work, in particular Mr P. M. Brown. REFERENCES 1. W. B. C 1965 Noise Control 2, 78–82. Fighting noise with noise. 2. A. R. D J 1975 D.Sc. Thesis, George Washington University. Active reduction of low frequency sound from large vibrating surfaces. Ann Arbor, Michigan: University Microfilms International. 3. M. H 1978 Applied Acoustics 1, 27–34. Investigation of noise reduction on 100 kVA transformer tank by means of active methods. 4. O. L. A and S. E. W 1990. Inter-Noise 90, The 1990 International Conference on Noise Control Engineering, Gothenburg, Sweden. Active cancellation of the hum of a simulated electric transformer. 5. P. L 1936 U.S. Patient, No 2,043,416. Process of silencing sound oscillations. 6. M. A. S 1973 Journal of Sound and Vibration 27, 411–436. The active control of sound propagation in long ducts. 7. G. B. B. C and R. A. S 1979 U.K. Patient 1, 555–760. Implementings in the relating to active methods for attenuating compression waves. 8. H. F. O and E. G. M 1953 Journal of the Acoustical Society of America 25, 1130–1136. Electronic sound absorber. 9. C. C 1987 The`se de Docteur de l’Universite D’Aix-Marseille II, Faculte´ des Sciences de Luminy. Absorption acoustique active dans les cavitie´s. 10. P. A. N, A. R. D. C and S. T. E 1986 Proceedings of Euromech Colloquium 213, Marseille. Optimal multiple source distributions for the active suppression and absorption of acoustic radiation. 11, S. J. E, I. A. S and P. A. N 1987 IEEE Transactions on Acoustic, Speech and Signal Processing, ASSP-35(10), 1423–1434. A multiple error LMS algorithm and its applications to the active control of second and vibration. 12. L. C. W. D (1972) Nonlinear Optimization. English University Press. 13. L. A. P 1963 Matrix Methods for Engineers. Englewood, Cliffs, New Jersey: Prentice-Hall. see pp. 38–65. APPENDIX: MATRIX MANIPULATION

Details of the derivation of equations (18), (19) and (20) in the main text are given below. To help with the manipulations, it may be useful to remind onself that if a vector x1 then its transpose is x T=[x1 x2 · · ·], (A1) x= x2 , *

&'

the scalar product is hence n

x Tx=x12+x22+· · ·+xn2= s xi2 i=1

(A2)

. .   . 

584 and

(xA+xB )T=xAT+xBT ,

(xA xB )T=xBT xAT .

(A3)

If the vector is complex, i.e., x=xR+jxl then its complex conjugate is x¯=xR−jxl .

(A4)

The transpose conjugate (Hermitian) of the vector x is now defined as x H=x¯ T=(xR−jxl )T=(xRT−jxlT )

(A5)

and the equivalent scalar product is n

n

i=1

i=1

x Hx=x¯ Tx=x¯1 x1+x¯2 x2+· · ·+x¯n xn= s (xR2 i+xIi2 )= s =xi =2.

(A6)

Upon returning to the main text and replacing x by PTm , the expression for the cost function J in equation (17) is obtained. From equations (10) and (12) the cost function becomes J=(Ppm+Csm Qs )H(Ppm+Csm Qs ), H H =(Ppm +QsH Csm )(Ppm+Csm Qs ),

H =(Ppm +(Csm Qs )H)(Ppm+Csm Qs )

(A7, A8)

H H H H =QsHCsm Ccm Qs+QsHCsm Ppm+Ppm CsmQs+Ppm Ppm ,

(A9, A10) which gives equation (18). Upon making the substitutions H a=aR+jaI=Csm Csm , H Csm , b=bR+jbI=Csm

H c=Ppm Ppm ,

(A11)

Qs=(QsR+jQsI ),

(A12)

where a is a matrix, c is scalar and b and QS are vectors, and also noting that for a complex matrix C and vector P (C HP)H=P HC,

(A13)

J=QsHaQs+QsHb+b HQs+c,

(A14)

equation (18) now becomes

or J=(QsR+jQsI )H(aR+jaI )(QsR+jQsI )+(QsR+jQsI )H(bR+jbI ) +(bR+jbI )H(QsR+jQsI )+c.

(A15)

Using equations (A5) yields T T J=(QsR −jQsIT )(aR+jaI )(QsR+jQsI )+(QsR −jQsIT )(bR+jbI )

+(bRT−jbIT )(QsR+jQsI )+c.

(A16)

The imaginary part of this expression is equal to zero since J is real according to its definition. Therefore one has T T T QsR aR QsR+QsIT aR QsI+QsIT aI QsR−QsR aI QsI+QsR bI+bRT QsR+bIT QsI+c.

(A17)

    

585

Since for two vectors QsT b=b TQS

(A18)

and for a Hermitian matrix a aRT=aR ,

a H=a, T sI

T

aIT=−aI , T sR

T I

T sR

Q aI QsR=(aI QsR ) QsI=(Q a )QsI=−Q aI QsI ,

(A19) (A20)

The cost function finally becomes T T J=QsR aR QsR+QsIT aR QsI−2QsR aI QsI+2bRT QsR+2bITQsI+c.

(A21)

Now differentiating J with respect to QS one obtains dJ/dQs=dJ/dQsR+j dJ/dQsI ,

(A22)

T T dJ/dQsR=1(QsR aR QsR )/1QsR−2 1(QsR aI QsI )/1QsR+2 1(bRT QsR )/1QsR ,

(A23)

dJ/dQsI=1(QsIT aR QsI )/1QsI−2 1(QsIT aI QsI )/1QsI+2 1(bIT QsI )/1QsI ,

(A24)

where

Noting that 1(QsTaQs )/1Qs=2aQs ,

T 1(QsR aI QsI )/1QsR=aI QsI ,

T 1(QsR aI QsI )/1QsI=−aI QsR ,

1(b TQs )/1Qs=b,

(A25)

one has dJ/dQsR=2aR QsR−2aI QsI+2bR ,

dJ/dQsI=2aR QsI+2aI QsR+2bI ,

(A26, A27)

giving dJ/dQs=2(aQs+b).

(A28)

The cost function J has its minimium value when dJ/dQS=0, and thus aQso+b=0,

which gives

Qso=−a−1b,

(A29)

or, in terms of the matrix Csm and vector Ppm , equation (19) is obtained: H H Csm )−1Csm Ppm . Qso=−(Csm

(A30)

For the same number of secondary sources s and microphones m, the matrix Csm is a square matrix and further simplification of the above expression is possible. Using the well known matrix identity H H −1 Csm )−1=C−1 (Csm sm (Csm )

(A31)

Qso=−C−1 sm Ppm .

(A32)

results finally in equation (20)