Convention Paper 6764

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Audio Engineering Society

Convention Paper 6764

Presented at the 120th Convention

2006 May 20–23 Paris, France

This convention paper has been reproduced from the author's advance manuscript, without editing, corrections, or consideration by the Review Board. The AES takes no responsibility for the contents. Additional papers may be obtained by sending request and remittance to Audio Engineering Society, 60 East 42nd_{Street, New York, New York 10165-2520, USA; also see www.aes.org.}

Journal of the Audio Engineering Society.

Optimisation of Co-centred Rigid and Open

Spherical Microphone Arrays

Abhaya Parthy1_{, Craig Jin}2_{, and André van Schaik}2

1_{School of Information Technology, The University of Sydney, NSW, 2006, Australia}

[email protected]

2 _{School of Electrical and Information Engineering, The University of Sydney, NSW, 2006, Australia}

{craig, andre}@ee.usyd.edu.au

ABSTRACT

We present a novel microphone array that consists of an open spherical array with a smaller rigid spherical array at its centre. The distribution of microphones, which results in the array having the largest frequency range, for a given beamforming order, was obtained by analysing microphone errors. For a fixed number of microphones, the results for several examples indicate that the maximum frequency range is obtained when the microphones are relatively evenly distributed between the open and rigid spheres.

1. INTRODUCTION

Many spherical microphone array configurations, such as that presented by Meyer and Elko [1] and Abhayapala and Ward [2], have a limited useable frequency range, typically 3-4 octaves depending on the number of microphones used. This frequency range limitation is due to spatial aliasing at the high frequencies and microphone positioning errors at the low frequencies. For a fixed number of microphones on a sphere, spatial aliasing can be reduced by reducing the

positioning error increases due to the smaller size of the sphere. The spatial aliasing error and the microphone positioning error for a given arrangement of microphones on a spherical microphone array is dependant on the dimensionless parameter kr, where k is the wave number and r is the radius of the sphere. Utilising multiple arrays of differing radii is a technique which allows a larger frequency range to be covered. Gover [3] uses two open spherical microphone arrays to capture a larger frequency range, a smaller spherical microphone array for capturing high frequencies and a larger spherical microphone array for capturing low

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field to be recorded at one location, however, open spherical microphone arrays are disadvantaged in that their error rises dramatically for certain values of kr. Rigid spherical microphone arrays do not have this problem [4]. It is preferable to use rigid spherical microphone arrays when recording a sound field for this reason. However, multiple rigid spherical microphone arrays can not be centred at the same point, and using multiple rigid spherical microphone arrays for recording a sound field at locations close to each other is not practical, as the rigid spheres will scatter sound and affect the sound field being recorded by the other rigid spherical microphone arrays.

We present a spherical microphone array configuration, which we have not seen reported previously in the literature, with microphones mounted on both an open and rigid sphere with a common centre. The smaller rigid spherical microphone array is used for recording high frequencies, while the larger open spherical microphone array is used for recording low frequencies. The sound scattered by the central rigid sphere can be calculated analytically at any point surrounding the sphere, and thus the sound field at the surface of the open sphere is known [1]. This configuration allows sound field recording at one location while retaining the advantages of the rigid spherical microphone array. Building a spherical microphone array with this configuration, using a fixed number of microphones, requires that a number of microphones be placed on the rigid spherical microphone array and the remaining microphones be placed on the open spherical microphone array. In addition, the frequency ranges that the two spherical microphone arrays cover must overlap. We present an optimisation algorithm that calculates the number of microphones that should be placed on the rigid and open spherical microphone arrays to maximise the frequency range of the combined arrays for a given spherical harmonic order of the beamformer.

2. METHODS

The optimisation program was written using the MATLAB software environment. The useable kr range is defined as the kr range for the spherical microphone array for which the microphone positioning error and the spatial aliasing error remain below a fixed value. The inputs for the optimisation algorithm are

• the total number of microphones,

• the maximum tolerable signal error, due to microphone positioning error and spatial aliasing error, in the spherical microphone array, expressed as a noise-to-signal ratio,

• the range for the uniform distribution of the random microphone placement error for the rigid and open spherical microphone arrays, and

• the spherical harmonic order of the beamformer. The optimisation algorithm calculates

• the number of microphones that should be placed on the rigid and open spheres,

• the ratio of the open sphere radius to the rigid sphere radius, and

• the useable kr range for the open array and the useable kr range for the rigid array.

Several assumptions were made in the design of the optimisation algorithm and are detailed in the following paragraphs. Firstly, we assume that the spherical harmonic order for the beamformer remains constant to ensure reasonably constant gain across the spherical microphone array’s useable kr range. The gain of the microphone array is related to the directivity index which is defined as the peak-to-average ratio of the beam pattern expressed in decibels [3]. For spherical microphone arrays processed using spherical harmonics, the directivity index is related to the spherical harmonic order of the beamformer and increases as the spherical harmonic order of the beamformer is increased. The directivity index remains relatively constant for a large range of kr values. By only beamforming at one order on both arrays, the directivity index will remain approximately constant across the entire useable frequency range.

A second assumption is that the microphones will be arranged on the open and rigid spheres with a nearly uniform spatial sampling scheme [5]. Nearly uniform spatial sampling schemes have been shown to be the most efficient in terms of the number of microphones required [4]. In addition, only spatial sampling schemes that satisfy the discrete spherical harmonic

orthonormality criterion (see [4]) are used:

( ) ( ) m m j n j n j nn mm nn mm j Y Y α ′ ∗ δ δ ε ′ Ω Ω = ′ ′+ ′ ′

∑

, (1)

(3)

where Ω =j ( , )θ ϕj j are the sample positions on a unit

sphere in spherical coordinates, αj are the weights for

those sample positions, m n

Y is the spherical harmonic function of order n and mode m, δ is the Kronecker delta function, ∗ denotes complex conjugation, and

nn mm

ε ′ ′ denotes the error in the sum for n n m m, , ,′ ′. Only

spatial sampling position lists which satisfy the spherical harmonic orthonormality criterion with

6

3 10

nn mm

ε −

′ ′ ≤ × for all n n m m, , ,′ ′ such that n n N, ′ ≤ ,

where N is the spherical harmonic order of the beamformer, are used for arranging the microphones on the spherical microphone arrays. Several spatial sampling position lists exist that do not satisfy the orthonormality criterion (1) at a specified order, although there are other sampling position lists, with fewer positions, that do satisfy the criterion at the same order. Spatial sampling position lists that do not satisfy the criterion are replaced with a spatial sampling position list, with a lower number of positions, which does satisfy the criterion. For example, at 4th order, we have found spatial sampling position lists with 37, 38, 39 and 41 positions that do not satisfy the orthonormality criterion, but a spatial sampling position list with 36 positions that does satisfy the criterion. Finally, we also assume that measurement noise is independent of the configuration of the spherical microphone array and do not include it in our signal error calculations.

2.1. Optimisation Algorithm

The optimisation algorithm begins with all microphones considered to be on the rigid spherical microphone array. For each iteration of the algorithm, the number of microphones on the rigid spherical microphone array decreases by one and the number of microphones on the open spherical microphone array increases by one. This iteration continues until all microphones are on the open spherical microphone array. For each iteration, the sum of the microphone positioning error and the spatial aliasing error, herein referred to as PA error, for both the open and rigid spherical microphone arrays is computed. The PA error is calculated assuming a single far-field, plane-wave source and that the beamformer is steered in the direction of the incoming plane-wave. The spatial aliasing error, Ea, is due to spatial sampling

of the sound field on the surface of the spherical microphone array. Spatial sampling limits the order to

which a sound field can be decomposed on the surface of the sphere and the spherical harmonic decomposition of the sound field is truncated. The spatial aliasing error [4] is defined as 2 0 1 1 2 2 1 2 1 4 4 (cos ) (cos ) N n n n N n M j n j n j j a s b n n b P P E y π π α ∞ ′ ′ = = + ′ = ′ + + × Θ Θ =

∑ ∑

∑

, (2)

where N is the beamforming order, M is the number of microphones, Θjis the angle between the incoming

plane wave and the sampling position Ωj, 2 s

y is the signal power, ₍_N₊_{1) (4 )}4 _π 2_{, and}

n b is defined as ( ) ' ( ) 4 ( ( ) n ( )) n j ka n n h ka n i j kr h kr π ₋ ′ _{, (3)}

where jn, hn are the spherical Bessel and Hankel

functions respectively, jn′, hn′ are their derivatives,

1

i= − , and a r≤ is the radius of the central rigid sphere. The spatial aliasing error is dependant on the beamforming direction, thus it is calculated for 625 incoming plane-wave directions, distributed around the sphere as in [6], and then averaged. It was found, empirically that after averaging across 625 incoming wave directions the spatial aliasing error varied insignificantly as more incoming wave directions were added and averaged.

The microphone positioning error, EΩ, is due to errors

in the placement of the microphones on the spherical microphone array. The microphone placement error, Δ, is the deviation from the ideal microphone position,

j

Ω , to the position, Ωj′, given by

and sin

j j j j j

θ′=θ + Δ ϕ′=ϕ + Δ θ . (4) The microphone placement error, Δ, is assumed to be uniformly distributed such that Δ ≤0.005radians. This range of microphone placement error seemed reasonable given the size of the spherical microphone array. Microphone positioning error [4] is defined as

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2 0 0 1 2 2 1 2 1 4 4

(cos )[ (cos ) (cos )]

N n n n n M j n j n j n j j s b n n b P P P E y π π α ∞ ′ ′ = = ′ ′ = Ω ′ + + ′ × Θ Θ − Θ =

∑∑

∑

,(5)

where Θj is the angle between the specified

microphone position and the incoming wave, and Θj′ is

the angle between the actual microphone position and the incoming wave. The microphone positioning error is dependant on the direction of beamforming, so an average error is calculated across the sphere for a number of beamforming directions. The positioning error for the spherical microphone array is calculated for 100 realisations of the random microphone placement error for each of 121 incoming plane-wave directions, distributed around the sphere as in [6], and then averaged. It was found empirically that after averaging 100 realisations of the microphone placement error for each of 121 incoming wave directions the error varied insignificantly as more realisations of the placement error and incoming wave directions were added and averaged.

For each iteration, the PA error with the specified maximum tolerable noise-to-signal ratio is used to calculate:

• firstly, the useable kr range for the rigid spherical microphone array,

• secondly, the largest ratio of the radius of the open sphere to the radius of the rigid sphere, for which the PA error is less than the maximum noise-to-signal ratio and such that the highest kr value for the open spherical microphone array is identical to the lowest kr value for the rigid spherical microphone array, • finally, the useable kr range for the open spherical

microphone array.

It should be noted that the useable kr range for the open spherical microphone array is computed so that the largest value of kr lies before the first local maximum, in the alias error curve, which is greater than the specified noise-to-signal ratio. As shown in Fig. 1, the spatial aliasing error curve for the open sphere contains numerous ranges of kr for which the error rises dramatically. 100 101 -60 -40 -20 0 20 40 60 kr N ois e-to -S ig na l R atio (d B )

Figure 1: Spatial aliasing error curve is shown averaged over 625 incoming wave directions, for open spherical

array with 26 microphones, having a radius 7 times greater than the rigid sphere located at its centre.

3. RESULTS

The optimisation algorithm was executed to find the optimal distribution for rigid-open spherical microphone array configurations with 96, 64, 32 and 24 microphones. All spherical microphone array configurations were designed to have a maximum error noise-to-signal ratio of -30 dB.

The first spherical microphone array configuration consists of 96 microphones operating at 4th order. A minimum of 36 microphones are required to satisfy the spherical harmonic orthonormality criterion for this configuration. Thus, when the rigid or open spherical microphone array contains less than 36 microphones, that spherical microphone array cannot be used for beamforming. Referring to Fig. 2, the frequency (i.e., kr) range, for this configuration, is at a maximum of 5.06 octaves when 46 microphones are placed on the open spherical microphone array and 50 microphones are placed on the rigid spherical microphone array. The ratio of the open sphere radius to the rigid sphere radius, with this distribution of microphones, is 5.66.

The frequency range curve shown in Fig. 2 is not smooth when plotted against the number of

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0 20 40 60 80 100 2 2.5 3 3.5 4 4.5 5 5.5

Number of Microphones on Rigid Sphere

Fr equ ency R an ge ( O ct ave s)

Figure 2: The frequency range is shown for a microphone array configuration with 96 microphones and working to 4th order with maximum noise-to-signal

ratio of -30 dB.

microphones. This is caused by the spatial aliasing error which is highly non-linear across kr and changes in a non-linear fashion as the number of microphones are increased or reduced. The frequency range curves shown in Figs. 3, 4 and 5 are not smooth for the same reason.

The second spherical microphone array configuration consists of 64 microphones operating at 3rd order. A minimum of 26 microphones is required to satisfy the spherical harmonic orthonormality criterion for this configuration. The frequency range for this configuration is at a maximum of 6.32 octaves when 32 microphones are placed on the open spherical microphone array and 32 microphones are placed on the rigid spherical microphone array. The ratio of the open sphere radius to the rigid sphere radius, with this distribution of microphones, is 7.53.

The third spherical microphone array configuration consists of 32 microphones operating at 2nd order. A minimum of 12 microphones is required to satisfy the spherical harmonic orthonormality criterion for this configuration. The frequency range, for this configuration, is at a maximum of 11.19 octaves when 16 microphones are placed on the open spherical microphone array and 16 microphones are placed on the rigid spherical microphone array. The ratio of the open sphere radius to the rigid sphere radius, with this distribution of microphones, is 50.0. 0 10 20 30 40 50 60 70 3 3.5 4 4.5 5 5.5 6 6.5

Figure 3: The frequency range is shown for a microphone array configuration with 64 microphones, and working to 3rd order with maximum noise-to-signal

ratio of -30 dB. 0 5 10 15 20 25 30 35 5 6 7 8 9 10 11 12

Figure 4: The frequency range is shown for a microphone array configuration with 32 microphones

and working to 2nd order with maximum noise-to-signal ratio of -30 dB.

The fourth spherical microphone array configuration consists of 24 microphones operating at 2nd order. A minimum of 12 microphones is required to satisfy the spherical harmonic orthonormality criterion for this configuration. The frequency range, for this configuration, is at a maximum of 9.90 octaves when 12 microphones are placed on the open spherical microphone array and 12 microphones are placed on the rigid spherical microphone array. The ratio of the open sphere radius to the rigid sphere radius, with this distribution of microphones, is 32.7.

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0 5 10 15 20 25 4 5 6 7 8 9 10

Figure 5: The frequency range is shown for a microphone array configuration with 24 microphones

and working to 2nd order with maximum noise-to-signal ratio of -30 dB.

3.1. Example Microphone Array

An example spherical microphone array is discussed to illustrate the practical considerations that are required when constructing a spherical microphone array. The example spherical microphone array configuration has 32 microphones on the open sphere and 32 microphones on the rigid sphere, and has the highest frequency range of all the configurations with 64 microphones. The kr range for the rigid microphone array is 0.4186 – 4.777, and the kr range for the open microphone array is 0.4507 – 3.155.

First of all, the radius is related to f the frequency by

2 krc r f π = , (6)

where c is the speed of sound. Thus, if we would like the array to work to a maximum frequency of 14.0 kHz, the radius of the rigid sphere would have to be 1.87 cm and the radius of the open sphere would then be 14.1cm. With this radius, the open spherical microphone array can work to a low frequency limit of 175 Hz. However, when building the rigid microphone array using DPA type 4060-BM microphones, which have a diameter of 0.54 mm and a height 12.7 mm, it is not be possible to place 32 microphones on a sphere of radius 1.87 cm and the radius of the rigid sphere has to be increased to accommodate 32 microphones. We find that a sphere with a radius of at least 2.04 cm has to be used to

accommodate 32 microphones. With this radius for the rigid sphere, the high frequency limit of the rigid spherical microphone array becomes 12.8 kHz. The radius of the open sphere then becomes 15.4 cm, and the low frequency limit of the open spherical microphone array is 160 Hz.

4. CONCLUSION

From the results presented above, it is evident that the largest useable frequency range for a concentric rigid/open spherical microphone array beamformer that operates to a constant order is achieved when microphones are placed both on the rigid sphere and the open sphere. The results indicate that a relatively even distribution of microphones, between the open and rigid spheres, produces the highest frequency range.

5. REFERENCES

[1] J. Meyer and G. W. Elko, “A highly scalable spherical microphone array based on an orthonormal decomposition of the soundfield,” in Proc. ICASSP, vol. II, 2002, pp. 1781–1784. [2] T. D. Abhayapala and D. B. Ward, “Theory and

design of high order sound field microphones using spherical microphone array,” in Proc. ICASSP, vol. II, 2002, pp. 1949–1952.

[3] B. N. Gover, J. G. Ryan, and M. R. Stinson, “Microphone array measurement system for analysis of directional and spatial variations of sound fields,” J. Acoust. Soc. Amer., vol. 112, no. 5, pp. 1980–1991, 2002.

[4] B. Rafaely, “Analysis and design of spherical microphone arrays,” IEEE Trans. on Speech and Audio Processing, vol. 13, no. 1, pp. 135-143, 2005.

[5] R. H. Hardin and N. J. A. Sloane, “McLaren’s improved snub cube and other new spherical designs in three dimensions,” Discrete Computational Geometry, vol. 15, pp. 429–441, 1995.

[6] J. Fliege and U. Maier, “A two-stage approach for computing cubature formulae for the sphere,” Ergebnisberichte Angewandte Mathematik, No. 139T. Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany. September 1996.