PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Sound field reproduction in a cabin using loudspeakers
Dong WANG1; Guosong FENG1; Gongbo MA1; Xinming LI1
1

Beijing Institute of Spacecraft Environment Engineering, Beijing, China

ABSTRACTTo evaluate the interior sound quality of an aircraft during flight, multiple loudspeakers are used to reproduce
the sound field inside a mock-up cabin which is used for the passengers to experience on ground. The
reproduction is based on the principle of the frequency-domain pressure matching approach. To reduce theinfluence of measurement errors in acoustic transfer functions, the case of more loudspeakers than thematching points is studied, which is different from most studies before. Some important aspects, including
the number of loudspeakers and the loudspeakers placement are discussed. Both numerical and experimental
examples are both described in this paper.
Keywords: Sound field reproduction, Pressure matching, Inverse problem, Aircraft noise

1. INTRODUCTION
The sound field reproduction discussed in this paper, is to use the secondary sound sources to makethe same sound field as the preliminary sound sources in a given space. Sound field reproduction could
be used to represent the real sound field to make the ‘passengers’ feel as if they were personally on thereal plane to evaluate the sound effects and sound quality. Besides, the produced sound field could be
useful for the design and test of the noise control system.
There are three major categories of sound field reproduction methods commonly researched inliterature. The first category is based on the Huygens’ principle, which reproduces the sound field
inside the target area through reproducing the sound pressure and particle velocity on the its boundary,such as the wave field synthesis (WFS) method (1, 2). The second category is through the mode
matching approach, such as ambisonics method (3, 4). In this approach, the sound field is describedusing the basis functions in spherical coordinate and their coefficients. The drive signals of thesecondary sound sources are calculated through minimizing the difference of the mode coefficients
between the preliminary and secondary sound field. The third category is the sound pressure matching
method (5), in which the sound pressure difference of the control points inside the target area between
the preliminary and secondary sound field is minimized to calculate the drive signals. Compared withthe first two categories, there is no limit for the location and array form of the secondary sound sourcesin the sound pressure matching method, so it is more convenient for engineering practice. In literature,
this method is adopted by most practical cases (6, 7).

2. THEORY BASIS FOR SOUND FIELD REPRODUCTION
2.1 Principle of sound pressure matching
As in Figure 1, several target points are arranged in the reproduction area. The sound pressure ofthese target points has been measured in the preliminary sound filed for acting as the reproduction
objectives. The sound sources, which could be loudspeakers or shakers (excite the structure to producesound) are arranged around the target area. In our opinion, loudspeakers are preferable because thesound characteristic of shakers is coupled with the structure.

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dongwang0312@163.com

Sound sorces...
Reproduction area...
Target points

Sound sorces

...
...

Figure 1 – Diagram for sound field reproduction
For a linear and time-invariant system, the relationship between the drive of the sound sources andthe sound pressure of the target points could be described in frequency domain as:P f   G f S  f (1)where f is the frequency points, and is omitted for clarity later. P is a M×1 column vector of the soundpressure of the target points, which has been given as the reproduction target. M is the number of thetarget points. S is a N×1 column vector of the drives. N is the number of the sound sources. G is a M×Ntransfer function matrix.
In most frequencies, the transfer function matrix G is full rank.
if M=N, Eq. (1) has the exclusive solution.
if M>N, Eq. (1) has the least square solution.
if M<N, Eq. (1) has infinite solutions, and the general solution is:
S  G  P   I  G G  Y(2)where G  is the Moore-Pseudo inverse of G. Y is an arbitrary N×1 vector.
The case of M>N is commonly researched in literature. It is treated as an optimization problem byintroducing in constraint conditions, such as the number of loudspeakers, the maximum power, thetotal power. And the reproduction error is minimized by methods of least-square (LS), Lasso,Lasso-LS (8, 9).
In the application background of this paper, there is no quantity restriction for loudspeakers, andhigh-power loudspeakers could be used to avoid the power restrictions, so the situation of M<N isstudied here.
2.2 Sound Pressure Matching Method when M<N
When M<N, Eq. (1) has infinite solutions, among which the drive of sound sources could bearbitrarily chosen to meet the reproduction requirement accurately. However, in the measurement of
the transfer function matrix G, measurement error is inevitable. Besides, the real transfer function may
change with environment. To cope with the influence of errors in G on the reproduction effect, proper
test design should be implemented.
Suppose the real transfer function is Ĝ , and its relationship with G is:(3)Ĝ  G  G
where G is the measurement error.
In Eq. (2), the drive S is calculated with the measured transfer function. When it is applied to theactual system, the sound pressure of the target points is:
ˆ(4)P̂  GS
The difference between the real sound field and the desired sound field is:P  GS(5)
The elements in P should be reduced. The 2-norm is a measure of length of a vector. So it could
be used to evaluate the reproduction accuracy. The following aspects would be discussed to reduceP 2 .

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2.2.1 Choice of solutions
For ΔG is a random matrix, the way to decrease P 2 is to decrease the element in vector S.
Among all solutions in (2), the minimal 2-norm solution is:(6)
S =G + P
This solution is recommended in the reproduction process. It should be noted that, for a given ΔG,
this solution may not be the best choice to minimize P 2 , however, when many probabilities of ΔG
are given, the usage of (6) leads to a minimized average value of P 2 .
2.2 The number of secondary sound sources
The singular value decomposition of G is:(7)
G =U V H
where U is a M×M unitary matrix. V is a N×N unitary matrix, and the superscript H indicates conjugatetranspose.  is a M×N matrix, and only the diagonal elements of the first M columns, which are called
the singular value of G, are none-zeros.
The Moore-Penrose generalized inverse matrix of G could be expressed as:(8)
G + =V *U H
where  * is a N×M unitary matrix, and only the diagonal elements of the first M lines are none-zeros.
The non-zero elements of  * and  are reciprocal.
Substituting (8) into (6):(9)
S + =V *U H P
HV and U are unity matrix. When a vector is multiplied with a unity matrix, its 2-norm would notchange. So, in Eq. (9), the change of 2-norm happens only when multiplying  * .  * better has small
elements to reduce the 2-norm of S.
When an extra sound source is added, G has a new column. As a result, the value of all singularvalues of G increase, while the number of singular values of G remains the same. So it is better to havemore sound sources.
2.3 Location of secondary sound sourcesˆ ˆ   I , we have:
Substitute Eq. (3) and Eq. (6) into (5), and notice GG+=I and GG

P̂  Gˆ  G G P  Gˆ G  Gˆ P

(10)

If matrix G is badly conditioned, G is greatly varied from Ĝ . The condition number of G is related
to the location of the sound sources. The location with smaller condition number is preferable.

3. SOUND FIELD REPRODUCTION PROCEDURE
Flowchart of an open-loop sound field reproduction method based on sound pressure matching isshown in Figure 2.

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Target PSD
Phase randomization

Transfer function
test

Target spectrum

Sound pressure
matching method

Drive Spcetrum

Time sequence ofdrive

OverlappingEndFigure 2 – Flowchart of sound field reproduction(1) Transfer function test
The transfer function from the driving voltage of each sound source to the sound pressure of eachtarget point shall be measured. As mentioned above, it is better to have more sound sources than thetarget points. There were mature methods for transfer function measurement. For the measurement ofdriving voltage could be accurate, the H 2 estimator could be used:G  m, n   S P P / S U P(11)m mn mwhere G(m,n) is the (m,n) element of G, S P P is the averaged auto-power of sound pressure of them mm-th target point, and S U P is the averaged cross-power of sound pressure of the m-th target point andn mdriving voltage of the n-th sound source.
(2) Target spectrum calculation
For the noise inside a cabin is close to random signal, the target sound field is usually described by
the power spectral density (PSD) of the target points. However, the phase information of the target
sound field is used in (2). So firstly, the PSD spectrum should be transformed to Fourier spectrum byintroducing random phase.
The amplitude spectrum is calculated as:
P  m   sm f(12)where | | indicates the amplitude, sm is the target PSD spectrum of the m-th target point, f is thefrequency spacing of sm . ( sm is defined on discrete frequency points.)
Then random phase is introduced into (12) to have:P  m   P  m  e j(13)where, e is the natural logarithm. j is the imaginary unit.  is the random phase, which is a uniform
distribution between  to  .
(3) Drive spectrum calculation
Based on the sound pressure matching method, the drive spectrum S is calculated by (6). In theresults, there may be one or more drive has an amplitude too big for the output channels of theequipment used. Each of these drives could be multiplied by a factor smaller than 1 to make its
amplitude equal to the threshold. Then these drives are treated as known drives. Substitute them into
(6), the other drives are calculated.
(4) Time sequence of drive
The drive spectrum could be transformed into time signal by inverse Fourier transform. The lengthof this time signal is T=1/ f .

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(5) Windowing and overlapping
The length of one frame is usually not long enough for reproduction practice. Repeat step (2) – (4)for several times, then the time sequences are overlapped to form a long-time drive signal in time
domain.
To avoid saltation at the splice points, windowing and overlapping are necessary. Figure 3 shows
this process.

tOverlapping

Window

Figure 3 – Windowing and overlapping
Suppose x1 , x2 … are time sequences form step (4), w is the window function. The time signal x
after overlapping is:x  x1w1  x2 w2 (14)Different window functions could be used. Half sine window is recommended:tsin 0t T0t  T,t  0wt   T(15)The time interval of w1 , w 2… is T/2.

4. EXAMPLES
4.1 Numerical example
The conclusions in section 2.2 are verified through Monte Carlo simulation.
A random 6×8 complex matrix (both the real and image part of each element are between -100 to100) is generated as the transfer function matrix G1 , and a random 6×1 complex vector is generated as
the reproduction target p.
(1) Choice of infinite solutions
Step 1: Using Eq. (6), the minimal 2-norm solution S 1 is calculated. Besides, 100 different
solutions S 2, S 3 , …S 101 using Eq. (2) are also calculated.
Step 2: Ĝ1 is generate by adding uniformly distributed noise to each element of G1.
Step 3: Substituting G 1, Ĝ1 , S 1, S 2, …, S 101 to Eq. (3) and Eq. (5),

P1 2 ,calculated.
Repeat step 2 and 3 above for 1000 times. The result reveals thatP2 2 ,…,

P101 2 areP1 2 is not always thesmallest, but the average value of P1 2 is the smallest, as in Figure 4. So solution (6) is the bestchoice.

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Figure 4 – Averaged reproduction errors with different solution of S
(2) Number of sound sources
Step 1: G 2 is generated by add four random columns to G 1.
Step 2: Substituting G 1 and G 2 into Eq. (6) respectively, and solving for S yields S 1 and S2 .
Step 3: G1 and G2 are generated, and the first 8 columns of G2 are the same as G1 . Theelements are also uniformly distributed noise.
Step 4: Substituting G1 and S 1 into Eq. (5) yields P1 2 , and substituting G2 and S2 into Eq.
(5) yields P2 2 .
Repeat step 3 and 4 above for 1000 times. The result reveals thatP2 2 is smaller than P1 2for most times as shown in Figure 5, and the average value of P2 2 is the smaller than that of P1 2 .
So it’s better to use more sound sources.

Figure 5 – Comparison of reproduction errors
4.2 Experimental example
The method of this paper was applied to sound field reproduction inside a simulated cabin as Figure
4. The out wall is made of alloy aluminum skin, the inside wall is micro-perforated panel, and theinterlayer is melamine foam.
As seen in Figure 6, a loudspeaker was placed outside the cabin to produce the preliminary soundfield. The sound pressure of three target points was measured by three microphones hanged inside thecabin.

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Figure 6 – Photograph of the test setup
Five loudspeakers both inside and outside the cabin were used to reproduce this sound field, asshown in Figure 7. The method raised in this paper is used, and the reproduction result is shown in
Figure8 (take point 1 for example). In all 1/3 octave between 40 to 5000Hz, the reproduction error isgenerally smaller than 2dB.
Loud
speaker 5

Targetpoint 1

Targetpoint 2
Targetpoint 3Loud
speaker 4

A2
Loud
speaker 3

Loud
speaker 1

Loud
speaker 2

Figure 7 – The location of loud speakers and target points
80
TargetPadB

Reproduction

30
44.67

Octave 1/3
HzFigure 8 – Reproduction result4376

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5. CONCLUSIONThe sound pressure matching method is adopted to reproduce the sound field. The case of moresound sources than target points is studied. The advantage of more sound sources is its insensitivity tothe measurement noises in transfer functions. The best choice in infinite solutions is given. The moresound sources used, the better. After the theory basis, the detailed reproduction procedure is given. The
method is validated by both numerical and experimental examples.

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