The active minimization of harmonic enclosed sound fields, part III: Experimental verification

Journal

of Sound and Vi’ibrarion (1987) 117(l),

THE

ACTIVE

HARMONIC

35-58

MINIMIZATION

ENCLOSED

OF

SOUND

PART III: EXPERIMENTAL

FIELDS,

VERIFICATION

S. J. ELLIOTT, A. R. D. CURTIS, A. J. BULLMORE AND

P. A. NELSON

Institute of Sound and Vibration Research, The University of Southampton, Southampton SO9 5 NH, England (Received 15 May 1986) The principle objective of this paper is to compare the measured results of active minimization experiments in an enclosed sound field with those predicted from theory. The enclosure used was essentially two dimensional over the frequency range of interest and was only lightly damped. A practical control system was built which minimized the sum of the squares of a number of microphone outputs by adjusting the outputs of a number of secondary loudspeakers at a single frequency. Various approaches to designing the algorithm which controls such a system are discussed, including matrix inversion, gradient descent methods, and pattern search methods. Although some problems with coupling between the acoustic and structural modes were initially encountered, the response of the experimental enclosure was very close to that predicted by the computer model when these problems were overcome. The pressure field inside the enclosure was measured at 200 points when excited both on resonance and off resonance, and the form of the pressure field was also found to be very similar to that predicted by the computer model. The conditions under which significant reductions in the total acoustic potential energy in the enclosure could be achieved by the action of a number of secondary sources were experimentally investigated. It was found that, in general, large reductions can be achieved only when the enclosure is excited on resonance. The secondary source does not have to be within half a wavelength of the primary to give good reductions, provided it is able to couple in to the most strongly excited modes.

1. INTRODUCTION

reduction of total acoustic potential energy has been suggested as a criterion by which the effectiveness of active noise control systems in enclosures can be evaluated [l]. This quantity, Ep, is proportional to the volume integrated mean square acoustic pressure in an enclosure, being defined as

The

Ep = (1/4~&)

c 1~1’d V

J”

Here p0 and co are the ambient density and speed of sound respectively, and p is the acoustic pressure at some point in the enclosure which has a volume V. This may be approximated [2] by a quantity proportional to the sum of the mean square pressures at a finite number of discrete points, Jp, defined as

Jp = ( V/~PO&)

,$

IPII*.

(2)

35 0022-460X/87/160035+24$03.00/0

@ 1987 Academic Press Limited

36

S. J. ELLIOTT

ET AL.

Here PI is the pressure at the Ith position in the enclosure, and L measurements are made. It is clear that the normalized sum of the squares of the pressures at a large number of uniformly spaced positions will be a good approximation to the volume integral. It has also been shown, by using computer simulations [2], that at low modal densities in a lightly damped enclosure the sum of the squares of the pressures at a relatively small number of carefully selected positions is also a good approximation to the volume integral. Minimizing this sum of mean square pressures, by adjusting the strengths of a number of secondary sources placed in the enclosure, would thus closely approximate the aim of minimizing the total acoustic potential energy. If p’ is the vector of the pressures at a number of positions (L) with no secondary qs is the vector of the M secondary source strengths (where M < L) sources operating, and Z is the L x M matrix of transfer impedances relating each pressure to each source, then the net pressure with the secondary sources in operation can be calculated by using the principle of superposition to be p, where p=p’+Zq,.

(3)

It is assumed that a linear, pure tone, acoustic field is being considered quantities are complex. Furthermore, Jp may now be written as Jp = ( V/4Wl%)PH where p” is the conjugate of the transpose source strengths [2] given by q = -[Z”

and that all

P,

of p. This is minimized z]-’ ZH p’.

(4) by a unique

set of

(5)

Although this equation is of considerable importance in determining the optimum source strengths theoretically, it is of more practical importance to recognize, from equations (3) and (4), that .Jp is a quadratic function of the real and imaginary parts of each source strength, and so simple gradient descent algorithms could be used to minimize Jp with respect to these variables. In practice the pressures at each point are measured with a number of microphones, and the secondary sources may be loudspeakers, for example, with finite internal impedance. One can, however, reformulate the minimization problem in entirely electrical terms. Suppose one has L microphones acting as sensors producing voltage outputs described by the complex elements of the vector v,,,. In the presence of only the original, primary, field this vector is equal to vb,,,. To minimize the sum of the squares of the outputs from these microphones, M loudspeakers are introduced, where M < L, which are driven by sinusoidal input voltages, the in-phase and quadrature amplitudes of which are given by the complex elements of the vector Vi,. If C is the Lx M matrix of complex electrical transfer functions relating each microphone output to each loudspeaker input at the excitation frequency in the absence of the primary field, then one can again use the principle of superposition to obtain V ,,u, =

’ V,“l

+cv;,.

(6)

only additional assumption needed is that of linearity of the electroacoustic, as well as the acoustic, components. Equation (6) has exactly the same form as equation (3), so that the sum of the squares of the output voltages will be a quadratic function of the in-phase and quadrature input voltage applied to each loudspeaker. Consequently a unique set of input voltages will lead to a “global” minimum in the sum of the squares of the output voltages. Because of the finite internal impedance of the loudspeakers, each will experience loading due to the presence of the enclosure and of other sources. These The

ENCLOSED

SOUND

FIELD

MINIMIZATION,

111

37

effects mean that the source strength of each loudspeaker is not directly proportional to its input voltage, but will, in general, depend on the input voltages to all the other loudspeakers. However equation (6) takes all of these mutual loading effects into account, and they do not interfere with the quadratic nature of the error surface [3]. If pressure microphones with equal sensitivities are used, then v~~,v~,,, will be proportional to J,,, and the argument above suggests that it is perfectly sufficient to adjust the electrical inputs to each of the loudspeakers to minimize this quantity, to obtain the minimum value of Jp which is equal to that obtained by using the optimal source strengths in equation (5). Consequently a minimum in Ep can also be obtained, under the conditions discussed in reference [2]. It is the object of this paper to discuss practical methods of minimization, and to present results of experiments in which Jp was minimized in a simple rectangular enclosure at low modal densities. These results, together with the form of the residual sound field, are compared with computer simulations obtained by using the results described in Part II [2]. 2. PRACTICAL 2.1.

MATRIX

METHODS

OF MINIMIZATION

INVERSION

If the matrix of transfer functions between each loudspeaker input and each microphone output can be accurately measured at the frequency of excitation, then the optimal set of input voltages to minize v,,* Hvou, is, by analogy with equation (5), Vin = -[C” In the case where L = M, C becomes

cl-’ CH Vb”,.

a square

matrix

Vin = -c-’

Vb”,.

and equation

(7) (7) reduces

to (8)

If this set of input voltages is applied to the loudspeakers the output voltages from each microphone would be driven to zero. In practice the matrix of transfer functions will inevitably be measured with some error. Let the measured matrix of transfer functions be C. It is assumed that the initial set of output voltages from the microphones can be perfectly measured, however, so the set of input voltages applied to the loudspeakers will be Vi~(l)=-?-‘V~ur.

This will cause a residual

set of output

voltages

(9) at the microphones,

V,,,(l)=V~u,-CC-lv;u,=[I-CC-‘]V~,,,

(IO)

where I is the Lx L identity matrix. It has been suggested [4] that this residual output may be reduced by using iteratively equation (9) above: i.e., by applying an additional input voltage given by V,(2) = -C-’ Thus

V,,,(l).

(11)

the total vi,, is equal to the sum Of Vin( 1) and v;,(2). The new residual v,,,(2)=v,,,(l)--CC%

and for the nth iteration

[I-C

C-‘I

will be

,,,(l)=[l-CC-‘]vO,,(l)=[l-CC-‘]*v~,,,

of this process v,,,(n)

If the matrix

output

is expressed

= [I - CC-l]nv:“, in “normal”

[I-CC-‘]=QAQ-‘,

.

(12)

form [5], (13)

38

S. J.

where Q is the matrix eigenvalues, then

of normalized

[I -C

ET AL.

ELLIOTT

eigenvectors

and

A is the diagonal

matrix

&-‘ln = QA” Q-‘.

of

(14)

The m$rix A” will also be a diagonal matrix whose elements are the eigenvalues of [I-C C-‘1 raised to the nth power. If all these eigenvalues have a magnitude which is less than unity then all the elements of A” will tend to zero after a large number of iterations. Consequently the elements of the vector of residFa1 outputs, v,,,(n), will also all tend to zero. If however there are eigenvalues of [I-C C-‘1 which have a magnitude of greater than unity, A” will not tend to zero with increasing n, so that the elements of v,,,(n) increase without bound and the system becomes unstable. If C=C+AC=[I+F]C, where AC is the absolute

error matrix

(15)

and F is the fractional

error matrix,

I-C?=F[I+F]-‘. The fractional

error matrix

can also be expressed

then (16)

in normal

F=P@P-’

form, (17)

so that F[I+F]-‘=P[cP[I+@]-‘]P-‘. If the eigenvalues of F, i.e., the diagonal elements that the modulus of the eigenvalues of equation than zero, becomes

(18)

of @, are Ai, then the stability criterion, (16) and hence equation (18) be less

Ihi/(l-hi)l
(19)

This implies that the modulus of the largest real part of any of the eigenvalues, hi must be less than l/2. The modulus of the largest hi must however be greater than the modulus of its largest real part so the stability condition becomes [Ai1< l/2. If If,,,j

is the modulus

of the largest

of F then

element

IAil
Then the stability

criterion

must be satisfied

(20)

i.

(21)

if

Ufmaxl <1; i.e.,IfmaxI < l/2.&

(22)

which represents a sufficient condition for convergence. This, however will be a rather pessimistic, worst case, condition and a numerical simulation was performed to find a more practical criterion. The elements of a number of relative error matrices of various orders were simulated by using a numerical random number generator. The generator summed the outputs of six independent random number generators, which individually produced a uniform probability distribution function, to give a reasonable approximation to a zero mean Gaussian distribution [6]. By examining the distribution of eigenvalues of the matrix F [I+ F]-’ for a large number of such matrices, the probability that the conditions for convergence would be satisfied for a given order of matrix and given standard deviation of the elements was estimated. The results of such a simulation are presented in Figure 1 for matrix orders of 2,4, 8 and 16. Each of these curves are the result of inverting 1000

ENCLOSED

SOUND

FIELD

S D of modulus

MINIMIZATION.

of error

motnx

III

39

elements

Figure 1. A graph of the probability of the largest eigenvalue of the matrix E [I+ E]-’ being greater than unity, and thus of an iterative matrix inversion algorithm being unstable, against the rms value of the elements of E for various matrix orders.

complex matrices of the relevant order, the real and imaginary parts of whose elements were taken from the random number generator. It can be seen, for example, that if the order of the matrix is 16 and the standard deviation of the complex elements of F is 0.15, then there is a 60% chance of an iterative matrix inversion algorithm being unstable. It should be noted that the relative error matrix can be written as F = C-’ AC. The magnitude of the but also the inverse of F will result. This will or if the acoustic field 2.2.

GRADIENT

(23:)

elements of F thus depend not only on the absolute error matrix C. Consequently if the matrix C is ill-conditioned, large values of occur, for example, if two microphones are placed close together, is dominated by a single, lightly damped mode.

DESCENT

METHODS

It has been shown that the sum of the mean square outputs from each of the microphones is a quadratic function of the amplitudes of the in-phase and quadrature components of the inputs to the loudspeakers. If the sum of the mean squared outputs is plotted against any one of these variables a minimum is seen in the error curve. However the position of this minimum will, in general, depend on the values of all the other input voltages. If a multidimensional graph is plotted of the sum of the squared outputs against all of the components of the input voltages, then this “error surface” will have a unique global minimum. Any algorithm which adjusts each of the input parameters in such a way as to decrease the error will, eventually, converge to give the smallest possible error: in other words, it will find its way to the bottom of the error surface. A large number of algorithms have been suggested to perform such a minimization. A simple “trial and error’ adjustment of the input voltages as suggested, for example, by Chaplin [7], would achieve this objective. If there are as many loudspeakers as microphones, and a number of independent control loops are implemented to minimize the mean square voltage at a single microphone by adjusting a single loudspeaker, then this too will converge to the minimum of the total error surface described above, provided each individual control loop is allowed to converge sequentially a sufficiently large number of times. So even if cross coupling exists between

40

S. 3.

ELLIOTT

ET AL.

the control loops, provided each is allowed to converge in isolation, as described above, the system will be stable and, eventually, lead to exactly the same minimum total error as more sophisticated systems. It should be noted however that equation (6) is defined only for steady state sinusoids. Consequently, sufficient time must be allowed, after any changes to the inputs to the loudspeakers for any acoustic transients to settle down before another measurement is taken of the total error. Such a new measurement of the error is generally necessary to judge the efficacy of the last change before a new change can be contemplated. This settling time will be of the order of the reverberation time of the enclosure. To keep the total time of convergence to a minimum it is thus necessary to select a method of descending this error surface which requires the smallest number of steps. Consequently an important criterion by which to judge any iterative procedure which minimizes the total error is the likely number of steps which have to be made. The method of steepest descent, in which the step size is proportional to the gradient of the error would, for example, be more efficient than a trial and error method, in which the step size is fixed. In practice however it has been found that simple algorithms based on the method of steepest descent are rather prone to observation noise, and an algorithm based on the “pattern search” technique has been developed and is described in the Appendix. This algorithm is used to perform the practical minimization necessary in the experiments described below. Another algorithm, with an instantaneous gradient estimate [lo], has also been used in this application more recently and has been found to give very rapid and robust convergence. 3. EXPERIMENTAL ARRANGEMENT 3.1.

EXPERIMENTAL

ENCLOSURE

rectangular experimental enclosure was designed to be much smaller in one dimension than it was in the other two. This was to ensure that a substantially two dimensional sound field was excited in the enclosure over a reasonable frequency range, which could be easily visualized. The enclosure was constructed of 18 mm medium density fibreboard with internal dimensions 0.668 m x O-265 m x 0.050 m and is illustrated in Figure 2. The dimensions of the enclosure were chosen so that over the frequency range of interest the natural frequencies of the modes were reasonably well spaced. It was found that the relatively thick walls of the enclosure were necessary to keep the mode shapes in the experimental enclosure of the same form of those of the computer model [2], for which hard-walled mode shapes were assumed. Even with 18 mm thickness walls it was necessary to use five “G” cramps to clamp the outside of the two larger sides of the enclosure to suppress coupling between the structural and acoustic responses over the frequency range used. The size of the enclosure used in the experiments was considerably smaller than that used in the initial computer simulation [2]. This is because the coupling between the acoustic and structural modes was found to be very significant in the original, larger, enclosure. Holes were positioned on one of the larger sides of the box for five Briiel and Kjaer (B & K) 12 mm microphones (type 4133), and four dome loudspeakers (KEF Electronics type T27a, which have a dome diameter of 20 mm) were mounted on the opposite wall. The position of each of the microphones and loudspeakers are indicated as Ml to MS and P, Sl, 52 and S3 in Figure 2, and the exact co-ordinates of their centres are given in Table 1. The loudspeakers were chosen such that their mechanical resonance frequency (quoted as being at 1200 Hz by the manufacturers) was above the frequency range of interest, since it was found that the loudspeakers were the dominant source of The

ENCLOSED

3

i2

SOUND

FIELD

MINIMIZATION.

41

III

0

s3

SI

0

6

MI

M2 x2

(a)

*2

(b)

Figure2. The positions of (a) the four dome loudspeakers used as sources, P, SI, S2 and S3, and (b) the five microphones used as sensors, Ml to MS, on opposite sides of the experimental enclosure.

TABLE

1

The co-ordinates of the centres of microphones and loudspeakers in the experimental enclosure Source/sensor

xl (m)

P Sl s2 s3

0.025 0.135 0.643 0.643

0.025 o-025 0.240 0.133

o-0 0.0 0.0 0.0

Ml M2 M3 M4 M5

0.005 0.663 0.663 0.005 0.350

0.005 0.005 0.260 0.260 0.133

0.05 0.05 0.05 o-05 0.025

x2

(4

x3

(4

damping, due to their low mechanical impedance, at frequencies near their mechanical resonance. On one of the longer sides of the enclosure, twenty evenly spaced holes were drilled for a 2 mm probe microphone (made from a B & K probe attachment kit UA0040 attached to a B & K 4133 microphone), so that by traversing the microphone to ten evenly spaced positions in the x2 direction in each of the holes, the pressure could be measured at 200 points in the enclosure. This procedure was partly automated by monitoring the d.c. output of the microphone amplifier (B & K type 2609) with an analogue to digital converter (part of a 3D “Inlab” interface system) which was read by a small computer (HP85), which stored the results, and which was later used to plot the pressure field. A small Perspex window was set into the side of the enclosure opposite the loudspeaker marked P, so that the velocity of its dome could be measured by using a laser Doppler velocimeter. By exciting the loudspeaker P electrically and measuring the transfer function between the output of the microphone ‘Ml’ and the output of the velocimeter, a measurement of

42

S. J. ELLIOTI-

ET AL.

the acoustic impedance of the enclosure could be made, with the microphone and velocimeter assumed to have flat responses. Since Ml is nearly coincident with source P, this approximately corresponds to the “input” impedance at this point. The measured impedance is plotted in Figure 3 for frequencies up to 1 kHz. Below about 30 Hz the loudspeaker response is considerably reduced and the signal to noise ratio of the measurement becomes poor causing the coherence between the two measured signals to fall. Also plotted in Figure 3 is the ratio of the pressure at the position corresponding to the centre of Ml, to the volume velocity of a square piston source (20 mm x 20 mm) operating at the same position as the source P, calculated by using the computer simulation described in reference [2]. The damping ratio of all modes used in the computer simulations was set to O-01. This was the average of those calculated by using the centre frequencies and half power bandwidths of the peaks in the measured response. The measured and predicted input impedance curves are seen to be in good agreement. The frequencies of the response maxima from the measured and predicted results are given for each mode in Table 2. 3.2.

THE

MINIMIZATION

PROCEDURE

The loudspeaker denoted P is always used as the source which excites the primary field in the enclosure. The object of the minimization is to reduce, as far as possible, the sum of the squared outputs from some combination of the microphones Ml to M5 by

500

Frequency

(Hz)

Figure 3. The input impedance of the experimental enclosure at source P; the solid curve is the measured data and the dotted curve is that calculated by using the modal summation model. TABLE

Frequencies Mode number

1,090 29% 0 O,l,O l,l,O 3,0,0 2,&O

Computed

frequency 255 510 643 692 765 821

2

of peaks in response (Hz)

Measured

frequency 248 514 634 680 756 812

(Hz)

ENCLOSED

SOUND

FIELD

MINIMIZATION,

43

III

adjusting the input voltages to some combination of the other loudspeakers, used as secondary sources: Sl, S2 and S3. Each of the microphones to be used (all type B & K 4133), are connected to measuring amplifiers (B & K type 2609), adjusted so that each channel has an equal sensitivity. The d.c. outputs of these measuring amplifiers are proportional to the rms pressures at each microphone position, and these are monitored by using the analogue to digital converters in a 3D “Inlab” interface system, as indicated in Figure 4. The digital value of each of these voltages is read by the HP85 and the sum of their squares is used as a variable proportional to the total error. The single frequency excitation is originally derived from a two phase oscillator (Feedback VBF 602), which is adjusted so as to produce quadrature outputs. One output is used to drive the primary loudspeaker via a power amplifier. Both outputs are connected to a bank of programmable attenuators driven by a digital output from the interface device. These attenuators are constructed by using 12 bit multiplying digital to analogue converters (Analogue Devices type AD 7541JN) with the “reference” signal fed by the oscillator. The output of these devices is proportional to the “reference” input signal multiplied by the digital input, and the converters thus act as linear attenuators. The outputs from a pair of these devices, driven from the two phases of the oscillator, are added together electronically to produce an output which is fed to one of the secondary sources via another power amplifier. A program running on the HP85 computer which implemented the pattern search algorithm was used to minimize the sum of the squared output voltages from the microphones by adjusting the programmable attenuators drivmg the secondary loudspeakers.

D

2

3

D

:

3

D

-

D

D

2

Computer

$ Meosqing Amplifiers

Microphones MI to M4

\ ”

Y

ti

A/D Converters _____________

V’

,-\

IEEE 488 Bus

INTERFACE SYSTEM

: Loudspeakers SI to 53

B Loudspeaker P

=

< Power Amplifiers

Figure 4. Block diagram of the equipment

used for the minimization

procedure.

44

S. J. ELLIOTT

ET AL.

4. EXPERIMENTAL 4.1.

MlNIMlZATION

AT AN

ACOUSTIC

RESULTS

RESONANCE

It has been suggested in reference [2] that if an enclosure is driven near a resonance, then minimizing the pressure at a small number of microphone positions by using few secondary sources should give significant reductions in the total acoustic potential energy, provided the positions of the microphones and sources are not close to nodal planes of the resonance being excited. These findings were investigated experimentally, and the form of the residual pressure field, after cancellation, was measured and compared with that predicted from a computer simulation. The frequency chosen for this set of experiments was 680 Hz, which was the frequency of the maximum of the measured input impedance (see Figure 3) for the third (1, 1,0) mode. A sketch of the nodal planes of this mode, and of the fourth (3,0,0) mode which will be used later, are shown in Figure 5. The pressure at 200 points in the enclosure were measured with the primary loudspeaker driven by a voltage amplifier fed by a pure tone of frequency 680 Hz, and the terminals of all other loudspeakers shorted together. This procedure was adopted to ensure that each of the loudspeakers presented a constant acoustic impedance to the enclosure, whether they were driven or not. The results of these measurements, together with the pressure field predicted from the computer simulation at this resonance, are presented in Figure 6, and the shapes of the two fields are seen to be in good agreement. The average levels of the two pressure fields are adjusted to be equal to aid comparison of the measured and predicted fields after cancellation. 14

M:

M?

z-E-lM4 -

I

-------I -----I ________------ 8

II

M2r

(4

M:

+

AMI

(b)

Figure 5. The nodal planes of (a) the (1, 1,O) mode and (b) the (3,0,0) mode, together with the positions of the sources and sensors. The relative phase of the pressure field when excited by source P is indicated by + and - signs.

An initial series of experiments was performed with loudspeaker Sl as a secondary source, this being the one closest to the primary source. Firstly the pressure at microphone M3, in the furthest corner, was minimized by adjusting the input voltage to this loudspeaker. The sound pressure was again measured at 200 positions in the enclosure and is plotted in Figure 7, together with the residual pressure field calculated by using the computer simulation for this case. By calculating the reduction of the sum of the squared sound pressures at each of the 200 measurement points, an estimate of the reduction in

ENCLOSED

SOUND

FIELD

MINIMIZATION,

111

45

Figure 6. The pressure distribution in the enclosure at 680 Hz when driven only by source P, whose position is indicated by the arrow. Experimental measurements at 200 points in the field are used to construct the measured field (a) and the computer model was used to obtain the theoretically predicted field (b).

total acoustic potential energy (E,) can be obtained. This quantity is termed JP2,,0 and the reduction in this case was measured to be 16.8 dB, compared to a predicted reduction, using the computer simulation, of 16.0 dB. The predicted values of E,, and JPzM)calculated by using the computer simulation were within O-5 dB for these measurements, indicating that Jp200 is a useful measure of Ep. The reductions in JPzoopredicted in the computer simulation are not strictly comparable with those directly measured, because of the finite internal acoustic impedance of the loudspeaker used as the primary source. For the computer simulations a constant volume velocity primary source is assumed: i.e., one with an injnite internal acoustic impedance. The change in velocity of the primary source due to the action of the secondary sources was, however, measured and found to be less than O-1 dB for all the experiments, and this effect can thus be ignored in this case. The shapes of the measured and predicted residual fields in Figure 7 are in good agreement. The pressure at microphone position M3 has obviously been reduced and in achieving this the shape of the pressure field has been significantly altered by the action of the secondary. The residual pressure field may be thought of as being made up of contributions from a number of different modes each excited by the primary and secondary sources. It is clear that the residual field contains little contribution from the 1, 1,O mode and although other modes are excited to some extent by the action of the secondary

46

S. .I. ELLIOTT

E7- AL.

Figure 7. The pressure distribution in the enclosure at 680 Hz after minimization of the pressure M3 by using secondary source Sl; the positions, together with that of source P, are indicated Measured; (b) predicted.

at microphone by arrows. (a)

source, these are not as strongly excited as was the original 1, 1,O mode, and the overall pressure field is decreased. This is not the case however if secondary source Sl is driven so as to minimize the pressure at microphone position MS, in the centre of the enclosure. The pressure in this position was measured by using the probe microphone described above. In this case a slight redistribution of the pressure field is produced which actually increases the measured value of JPZoOby 5.6 dB. The resulting pressure field in this case, together with that predicted by the computer simulation, is shown in Figure 8. The shape of the pressure field is very similar to that due to the primary only, in Figure 6. The shapes of the predicted and measured fields are also very similar although the level of the predicted residual field is somewhat higher than that measured, giving an increase of 11.3 dB in J,,zoo over the original, primary, field. This experiment demonstrates the dangers of trying to minimize the pressure too near a nodal plane. The discrepancy between the overall levels of the predicted and measured sound fields, is due to the secondary source strength being critically dependent on the exact position of the microphone M5, and, in practice, the position of the probe microphone is subject to uncertainties of several milbrnetres in each direction. In each of the experiments described above the secondary loudspeaker was positioned within half a wavelength of the primary. In order to investigate minimization when using a “remote” secondary source a series of experiments was conducted with source S2, the position of which is shown in Figure 2. The input voltage to this loudspeaker was initially adjusted to minimize the pressure at the microphone position adjacent to this source: i.e., M3. The residual pressure field, together with that predicted by the computer

ENCLOSED

SOUND

FIELD

MINIMIZATION,

Figure 8. The pressure distribution in the enclosure at 680 Hz after minimization M5 by using secondary source Sl. (a) Measured; (b) predicted.

111

of the pressure

47

at microphone

simulation,

is shown in Figure 9. Both the shapes of the sound fields and the reductions 10.9 dB measured and 11.6 dB predicted, are in good agreement in this case. in Jpzoo, The best possible predicted reduction in EP when using this secondary source, that which could be obtained by using an infinite number of measurement microphones, is 12.0 dB. Although this is closely approached in the case above it is of interest to investigate the effect on the reduction and residual pressure field of minimizing at a number of microphone positions. The microphone positions, discussed in reference [2], which appear to give a close approximation to the best possible reductions are in the corners of an enclosure of this form. Figure 10 shows the measured and predicted residual pressure fields after the input to source S2 has been adjusted to minimize the outputs of microphones Ml, M2, M3 and M4. The measured and predicted reductions in Jpzoo are 1 l-4 dB and 1 l-7 B respectively. The use of four measurement microphones is thus seen to give closer approximations to the maximum possible reductions in overall level, although since the field is excited exactly on resonance in this case, minimization at a single microphone placed in a comer (as in Figure 9) gives similar overall reductions. 4.2.

MINIMIZATION

AT AN

ACOUSTIC

ANTIRESONANCE

It is expected from the computer simulations that little reduction in the overall sound field is possible if the enclosure is excited off resonance [2]. Such a case is worthy of experimental investigation since in some ways it is a more severe test of the applicability

48

S. J. ELLIOT-I-

ET AL.

Figure 9. The pressure field in the enclosure at 680 Hz after minimizing the pressure using secondary source S2, remote from the primary. (a) Measured; (b) predicted.

at microphone

M3 by

of the computer model than when the enclosure is excited on resonance. This is because contributions from the residual modes will play a more important role in the primary as well as the residual pressure field. A series of experiments was thus performed at the minimum between the third and fourth resonances, corresponding to the 1, 1,O and 3,0,0 modes respectively, of the input impedance. This minimum is at a frequency of 723 Hz according to the measured input impedance curve of Figure 3 and at 730 Hz according to the predicted one. The pressure field at this frequency with only the primary source in operation is shown in Figure 11, which also shows the field predicted from the computer simulation. The shapes of the curves are seen to be very similar, indicating that the correct mixture of residual modes is being predicted by the model. The average amplitudes of the two fields are again set equal to aid comparison. In the first experiment in this series the same secondary source and microphone positions (Sl and M3) were used as for Figure 7, for which a 16.8 dB reduction in JPZoowas measured when the enclosure was excited on resonance. In this anti-resonant case the measured value of JP2,,,,increased by 7.5 dB after minimization, compared with a predicted increase of 3.6 dB. The overall level of the sound field has increased in this case since the two dominant modes are out of phase, relative to the primary excitation, at this microphone position, as indicated in Figure 5. Consequently the pressure at this microphone position can have a low value even if both modes are strongly excited. The measured and predicted resulting pressure fields are shown in Figure 12, which are again seen to be broadly similar. The difference in overall level is probably due to slight differences in

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111

49

Figure 10. The pressure field in the enclosure at 680 Hz after minimizing the sum of the squares of the pressures at microphones Ml, M2, M3 and M4 by using secondary source S2. (a) Measured; (b) predicted.

the relative frequencies used to obtain the measured and predicted results. If the enclosure is not excited exactly at the frequency corresponding to the minimum of the input impedance, then the relative contributions of the two nearest modes will be significantly altered. Although the excitation frequency was held constant to within 0.1 Hz during the course of the experiments described in this section, a temperature drift of about 1°C was observed. This temperature drift corresponds to an equivalent change in mode frequency of about 1.2 Hz. To reduce the overall level of the pressure field in this case it is expected that a number of microphones will be needed [2]. If the sum of the squares of the pressures at four microphone positions in the corners of the enclosure (M 1, M2, M3 and M4) are minimized by adjusting the input voltage to secondary source Sl, then a measured reduction in JPZoo of 0.75 dB is obtained. The predicted reduction under these conditions is l-94 dB. The resulting pressure field in the two cases are shown in Figure 13. It is interesting that in this case little reduction in JPZoo has been achieved despite the secondary source being within half a wavelength of the primary. This is because a nodal line of the 3,0,0 mode lies between the two sources (as in Figure 5), so that the secondary source cannot excite both modes with the same phase as the primary. The excitation of the secondary source is thus a compromise between cancelling the 1, 1,0 mode and increasing the excitation of the 3,0,0 mode or vice versa, with the result that this source can effect little net cancellation. The maximum co_mputed reduction in Ep with this source is only 2.0 dB because of this effect, so that mnnmizing with respect to four microphones in the corners is still giving near-optimal results. When the sum of the squared pressures in all four corners is minimized by using source S2, in the opposite corner, similar results are obtained. The residual field, together with that predicted is plotted in Figure 14, and the measured and predicted changes in JPZoo are an increase of O-49 dB and a decrease of 0.01 dB respectively. Again the secondary

50

S. J. ELLIOTT

Figure 11. The pressure distribution in the enclosure primary source. (a) Measured; (b) predicted.

ET AL.

off resonance,

near 730 Hz, when driven

only by the

source cannot cancel modes without exciting the other because they are out of phase, relative to the excitation due to the primary, at the position of source S2, as indicated in Figure 5. Source S3 operating in isolation can achieve some overall cancellation, since it is able to cancel the 3,0,0 mode without unduly exciting the 1, 1,0 mode. This is because it is positioned near a nodal plane of the latter mode. The measured and predicted residual fields are shown in Figure 15, and the measured and predicted reductions in JPZo,, are 1.75 dB and 0.68 dB respectively. Further reductions in overall level are expected if the secondary sources S2 and S3 are both driven to minimize the pressures at all four comers of the enclosure. This is because S3 is expected to cancel the component of the field due to the 3,0,0 mode while S2 can couple into both. Thus if only these two modes significantly contribute to the sound field, both could be cancelled by a combination of secondary sources S2 and S3. However the measured and predicted reductions in _I,,,, after minimizing at all four corners are 1.96 dB and 1.57 dB respectively. The form of the residual fields is shown in Figure 16. The maximum reduction in Ep which can be obtained with this combination of secondaries, and upon assuming an effectively infinite number of microphones, is calculated to be 2.8 dB. This rather modest reduction is due to the contributions from the “tails” of the other modes which significantly contribute to the net field at frequencies between resonances, as illustrated in Figure 17. 4.3.

MINIMIZATION

OF THE

PRESSURE

AT EACH

CORNER

OVER

A RANGE

OF FREQUENCIES

Minimization of the sum of the squares of the pressures in each of the four comers of the enclosure has been shown above to give reductions in E, which are close to those

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51

111

Figure 12. The pressure distribution in the enclosure off resonance after minimization microphone M3 by using secondary source Sl. (a) Measured; (b) predicted.

of the pressure

at

which would be obtained with an effectively infinite number of microphones, at both resonance and anti-resonance frequencies. This section presents the results of an investigation of the effect of minimizing the sum of the squares of these four pressures, a quantity termed JP4, on the rest of the sound field over a range of frequencies. The frequency range chosen was from 600 to 750 Hz, which includes the cases discussed in sections 4.1 and 4.2 above. Measurements of Jp4 were made at 5 Hz intervals over this range firstly with only the primary exciting the enclosure, and secondly after Jp4 had been minimized by adjusting the input voltage to secondary source S2. The velocity of the dome of the loudspeaker used as the primary source was measured by using a laser Doppler velocimeter during the initial part of these measurements. The values of Jp4 due to the primary alone were then normalized by using these velocity measurements to give values, at each frequency, corresponding to a constant volume velocity excitation. These measured values of Jp4, before and after cancellation, are shown in Figure 18. The normalization of Jp4 with the volume velocity of the primary allows direct comparison to be made with the values of Jp4 computed by using the computer simulation. These are shown in Figure 19. Also shown in this figure is the predicted value of Jp4 after it has been minimized by using source S2 in the simulation. Apart from the difference in relative amplitude of the two peaks in the two curves of Jp4 due to the primary alone, which was also present in the input impedance curve of Figure 3, the results in Figures 18 and 19 are very similar. This similarity gives sufficient confidence in the computer model to use it in computing the total acoustic potential energy, I$, before and after the minimization described above. These are shown, together with

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S. J. ELLIOTT

ET AL.

Figure 13. The pressure distribution in the enclosure off resonance after minimizing the sum of the squares of the pressures at the four corners of the enclosure Ml, M2, M3 and M4 by using secondary source Sl. (a) Measured; (b) predicted.

optimum reductions which can be achieved in Ep, in Figure 20. Although the reduction in Ep achieved by minimizing Jp4 is near its optimal value over the most of this frequency range, there is a range of about 20 Hz, centred on 650 Hz, over which E, actually increases due to the minimization of Jp4. If the values of Jp4 over this frequency range are examined in Figure 19, it is seen that very little change in J Pa, typically O-01 dB, is achieved as a result of the minimization. The secondary source strength is not, however, correspondingly small over this region: it was measured to be approximately equal to the primary source strength. Consequently considerable effort is expended in achieving very small reductions in pressure at the corners, which, over this frequency region, increases the value of Ep. One method of avoiding this problem is to minimize a more complicated cost function which penalizes “effort” as well as “error”. For example, a total cost function, JT, can be defined as JT = ( V/~P&)[P”

QP+ qHRql,

(24)

by analogy with that used in optimal control [8]. The diagonal matrix Q weights each of the pressures at the L error microphone (the elements of the vector p) and the diagonal matrix R (of appropriate dimensions) weights the output volume velocities of each of the secondary sources (the elements of q). A simplified form of this equation has been considered in the case with the four error microphones and one secondary above, where all microphone outputs have equal, unity, weighting and the source strength is weighted by a factor r. Thus, JT

=

(

V/4&&)[RH

P+ ‘9*4].

(25)

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111

Figure 14. The pressure distribution in the enclosure off resonance after minimization comers (Ml-M41 by the action of secondary source S2. (a) Measured; (b) predicted.

of the pressure

in the

An example of the effect on Ep on minimizing a cost function of this form in the simulation are shown in Figure 21, in which the values of Ep due to the primary alone and after minimization by using S2 are shown. Although the objective of reducing EP in the region about 659 Hz has been achieved, the value of Ep after minimization has increased at other frequencies when compared with the result of minimizing Jp4 in Figure 20. In fact if the integral of Ep, after minimization, is taken over this frequency range (600 to 750 Hz) as a measure of the overall level of suppresssion, then its value monotonically increases as the value of r is increased from zero in equation (25). 5. CONCLUSIONS A control system has been built which operates at single frequencies, and will minimize the sum of the squares of a number of microphone outputs by adjusting the inputs to a number of loudspeakers acting as secondary sources. It is shown that this minimization problem is quadratic with a unique global minimum even with practical transducers which may experience acoustic loading. The computer algorithm which controls the system makes use of the quadratic form of the error surface by using a pattern search technique. Because the principle object of this paper is to compare the measured acoustic field in a simple enclosure after such a minimization with that predicted from a computer model, there is no necessity for the control system to converge quickly to the global minimum, provided it is reliable in eventually achieving this goal.

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S. J. ELLIOTT

ET AL.

Figure 15. The pressure distribution in the enclosure off resonance after minimization comers (Ml-M4) by the action of secondary source S3. (a) Measured; (b) predicted.

of the pressure

in the

The experimental enclosure used was designed such that the pressure field was essentially two dimensional over the frequency range of interest, so that it was relatively easy to visualize. After some initial problems with coupling between the acoustic and structural modes were overcome, the behaviour of the experimental enclosure was very close to that expected from the computer model previously developed [2], for which hard-walled mode shapes were assumed. The damping ratio in the experimental enclosure was found to be about 0.01 over the frequency range considered. A series of experiments were then performed which compared the form of the pressure field, measured at 200 points in the experimental enclosure, with that predicted from the computer model. These measurements also allowed a comparison to be made between an estimate of the total acoustic potential energy in the experimental enclosure, obtained from the 200 pressure measurements, and the acoustic potential energy in the enclosure as predicted by the computer model. Such measurements were made. at single frequencies which corresponded to (a) the resonance due to the 1, 1, 0 mode in the enclosure and (b) the minimum in the response between the 1, 1, 0 and 3, 0, 0 modes. The form of the pressure fields in both cases, with either the primary loudspeaker only in operation or after minimization with a number of secondary sources, were very similar to those predicted by the computer model. This can be considered as a convincing validation of the computer model for this case of a lightly damped enclosure at low modal densities. The magnitudes of the measured reductions in the total acoustical potential energy in the enclosure were generally similar to those predicted by the computer model. Differences

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55

III

Figure 16. The pressure distribution in the enclosure off resonance after minimization of the pressure comers (Ml-M4) by the action of both secondary sources S2 and S3. (a) Measured; (b) predicted.

in the

30

Frequency

Figure 17. The contributions (E,,), when driven by source arrow.

(Hz)

of the individual modes to the total acoustic potential energy in the enclosure P. The excitation frequency for the off resonance experiments is indicated by an

56

ET

S. J. ELLIOTT

AL.

+ +

+ +

T-

+

+

+

dB +

+ +

+

ooo

+

+

Q

+

+oo

+ 0

1-$0 1)

+ o 0

o ooo

600

I 650

I

+

+ o

,

1

0

I,

0 /

700

Frequency

0

;&o

o

1

I 750

(Hz)

Figure 18. The measured sum of the squares of the pressures in the corners of the enclosures (J,,,) plotted in dB over a range of frequencies with the primary source only exciting the enclosure (+) and with JP4 minimized by the action of secondary source S2 (0).

Frequency

(Hz)

Figure 19. The predicted values of Jr, in dB, over the same range of frequencies as Figure 18. The upper curve is with the primary source only operating; the lower curve is with J,,4 minimized by the action of secondary source S2.

between these two quantities arose only when the exact form of the sound field was critically dependent on the precise positioning of a cancelling microphone or the precise frequency of operation. The magnitudes of these reductions were greatest at resonance provided the secondary source could efficiently couple into the mode excited by the primary source, and the error microphones were also not placed near nodal planes of this mode. Reductions of about 16 dB in the energy were achieved with a secondary source close to the primary (within half a wavelength) and of about 11 dB were achieved with a secondary source remote from the primary (greater than half a wavelength away). It was not possible to get significant reductions in the acoustic potential energy in the enclosure at the frequency between the two resonances. This is because there are more modes significantly contributing to the pressure field at this frequency and the secondary sources cannot couple into all of these modes with the necessary phase relative to the primary without being extremely close. It is not sufficient to be within half a wavelength in this case, as demonstrated by the poor reductions obtained in an experiment in which one source was within half a wavelength of the primary, but a nodal plane of a significantly

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Frequency

III

57

(Hz1

Figure 20. The predicted values of the total acoustic potential energy in the enclosure (E,) plotted in dB over the same range of frequencies as in Figures 18 and 19. The solid curve is the energy in the enclosure due to the primary source alone. The dashed curve is the energy in the enclosure after the pressure at the four comers (Jr,) has been minimized by using secondary source S2. The dotted curve is the energy in the enclosure after the energy has been minimized by the action of secondary source S2.

Frequency

(Hz)

Figure 21. The predicted values of the total acoustic potential energy in the enclosure. The solid curve and the dotted curve are respectively that due to the primary alone and that remaining after the minimization of the energy in the enclosure, by using source S2, as plotted in Figure 19. The dashed line is the energy in the enclosure after minimization of a cost function which penalizes both the sum of the squares of the pressures at the corners and the source strength of S2.

excited mode fell between them. The best reductions in the total acoustic potential energy which could be achieved in this case were of the order of 2 dB. In a final experiment, the reductions obtained with a fixed configuration of a remote secondary source and four error microphones positioned in the comers of the box, were investigated over a range of frequencies. These experiments confirmed that while appreciable reductions in the energy could be obtained near resonances, it was possible to increase the energy in the enclosure off resonance by minimizing the pressure at the comers. The minimization of a more complicated cost function which took the strength of the secondary source into account, as well as the sum of the squares of the pressures in the comers, was found to suppress this increase in the energy off resonance. However it also limited the reductions obtained at other frequencies.

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ET AL.

ACKNOWLEDGMENT S. J. Elliott and P. A. Nelson are supported by the Science and Engineering Research Council under the Special Replacement Scheme; the Department of Trade and Industry for a research grant which supports A. R. D. Curtis and A. J. Bullmore also is gratefully acknowledged. REFERENCES 1. P. A. NELSON, A. R. D. CURTIS, S. J. ELLIOTT and A. J. BULLMORE 1987Journal ofSound and Vibration 117, 1-13. The active minimization of harmonic enclosed sound fields, Part I: Theory. 2. A. J. BULLMORE, P. A. NELSON, A. R. D. CURTIS and S. J. ELLIOTT 1987Journal ofSound and Vibration 117, 15-U. The active minimization of harmonic enclosed sound fields, Part II: A computer simulation. 3. R. J. SILCOX and S. J. ELLIOTT 1985.Proceedings of Internoise ‘85. 587-590. Applicability of superposition and source impedance models of active noise control systems. 4. A. D. WHITE and D. G. COOPER 1984Applied Acoustics 17,99-109. An adaptive controller for multivariable active noise control. 5. B. NOBLE 1969Applied Linear Algebra. Englewood Cliffs, New Jersey: Prentice Hall. 6. L. R. RABINER and B. GOLD 1975 Thoery and Applications of Digital Signal Processing. Englewood Cliffs, New Jersey: Prentice Hall. 7. G. B. B. CHAPLIN 1983 Chartered Mechanical Engineer 30, 41-47. Anti-noise-the Essex breakthrough. 8. R. J. RICHARDS 1979An Introduction to Dynamics and Control. New York: Longman. 9. P. R. ADBY and M. A. H. DEMPSTER 1974 Introduction to Optimization Methods. London, Chapman and Hall. 10. S. J. ELLIOIT and P. A. NELSON 1985Electronics Letters 21,979-981. Algorithm for multichannel LMS adaptive filtering. APPENDIX:

THE PATTERN

SEARCH

METHOD

The pattern search algorithm, which is described briefly below and more fully in reference [9], is suitable for experimental optimization because it meets the following requirements. It is robust in the presence of noise. It tends to follow the line of steepest descent. Lower and upper bound variable constraints are easily accommodated. It does not stop when the noise floor is reached but maintains the search with a single bit step length. This enables the algorithm to track a drifting minimum. It is also possible to change the gains of the inputs and outputs as the algorithm is converging to make the best possible use of the dynamic range of the instrumentation. In the experiment each secondary loudspeaker channel has two programmable attenuators and each attenuator is arranged to have 256 possible settings. The vector of 2N attenuator settings is a position vector in a 2N dimensional space which together with Jp form a 2N + 1 dimensional space, with the experimental values of Jp for given attenuation vectors forming a concave surface. The least value of Jp is at the bottom of this surface. It would be possible to find this minimum by evaluating Jp at all the 2562N possible values of the attenuation vector but this is time consuming. A quicker strategy would be to search a small area of the surface for the lowest local point and then to search an area of the same size around that point for a new lowest local point and so on. This search would tend to follow the line of steepest descent and is essentially the basis of the pattern search which has a few additional features. If a new lowest local point is not found in the search area its size is reduced and the search continued. (If the search area is two bits across then it is reduced no further.) If a new lowest local point is found then it is assumed that the area beyond this point is a lower area and so the new search centre is specified as being the same distance again beyond the new lowest local point. If the extended search centre fails to produce a new lowest local point then the search continues around the last one. This feature allows the algorithm to accelerate down a steep slope.