Baltimore, Maryland

NOISE-CON 2010 2010 April 19-21

A particle velocity gradient beam forming system Hans-Elias de Bree and Jelmer Wind Microflown Technologies HAN University, dpt. Vehicle Acoustics, Arnhem, the Netherlands

ABSTRACT The topic of this paper is the determination of the acoustic source distribution in the far field

with a small, three dimensional system consisting closely spaced sound pressure sensitive

microphones and particle velocity sensitive Microflowns1.

Sound pressure sensors do have a zero order directionality (that is no directionality). Particle

velocity vector sensors have a first order directionality (this is a cosine shape directionality).

With two closely spaced zero order sensors, a first order system can be created. Disadvantages

are the low sensitivity for low frequencies and a limited high frequency response.

With two closely spaced first order sensors a second order system is created. The directionality is

a squared cosine shape. With this higher directivity it is possible to create a very small 3D beam

forming system with a reasonable resolution which is the topic of this paper.

A second order system can be made with accelerometers or pressure sensors but the low

frequency response is very poor. The Microflown has a very high sensitivity at low frequencies

so the velocity gradient signal is good. In this paper the velocity gradient method is presented

and a 1D and 3D velocity gradient system is demonstrated.

1. INTRODUCTION Microphones are normally used to measure sound. These acoustic sensors do not have any

directionality (no sensitivity that is dependent on the direction of arrival (DOA) of a sound

wave).

For many applications directionality is important and sometimes even indispensible. An

increased directionality is important, for example, to increase speech intelligibility. In such cases

directional microphones are used with for instance a figure of eight, a cardioid, or a

hypercardioid directionality. Such microphones are made by crating a pressure gradient

microphone (having a figure of eight) and combine this signal with an omnidirectional

microphone.

For source localization applications, the directionality of an array is expressed as the directivity

index. This index is a direct measure of the quality of the source localization results in the

presence of white sensor noise. A highly directional array is not necessarily suitable for source

localization because a high directionality in all directions is required for source localization.

Systems become frequency dependent if directionality is created by spatial distribution. At low

frequencies such systems have a low signal to noise ratio and at high frequencies, aliasing

problems occur.

In this paper a system is proposed that makes use of acoustic sensors that are directional it self.

The so called Microflows1 have a figure of eight sensitivity measured at one point in space. A

higher directionality is reached if these systems are spatially distributed. The signal to noise ratio

of Microflowns is high at low frequencies so a proper performance can be expected even at low

frequencies

Analogue to a cardioid microphone that is constructed from summing a sound pressure and a

sound pressure gradient microphone, a unidirectional microphone can be constructed by

summing a Microflown and a velocity gradient response.

A single (one dimensional) velocity gradient microphone can be used for systems that require an

increased speech intelligibility and a three dimensional velocity gradient system can be used for

a 3D beamforming sound mapping system.

2. DIRECTIONALITY AND SENSOR TYPE Sound pressure sensors (i.e. microphones) do have a zero order directionality (that is no

directionality). Vector sensors (i.e. Microflowns) have a first order directionality (this is a cosine

shape directionality)2,4

.

With two closely spaced zero order sensors, a first order system can be created. Disadvantages

are the low sensitivity for low frequencies and the limited high frequency limitation. All other

systems (except for Microflowns) have a low sensitivity for low frequencies and the limited high

frequency limitation. (These systems are: pressure gradient (air and water) and accelerometers

for underwater6).

With two closely spaced first order sensors a second order system is created. The directionality is

a cosine square shape. With this higher directivity it is possible to create a very small 3D beam

forming system with a reasonable resolution which is the topic of this paper.

A second order system can be made with accelerometers or pressure sensors however their low

frequency response is very poor. The Microflown has a very high sensitivity at low frequencies

so the velocity gradient signal is good.

3. THE VELOCITY GRADIENT Some work with velocity gradient sensors is done in air

2 and also in water

5. In these cases the

velocity gradient is used to estimate the acoustic sound pressure. In both papers the velocity

gradient is measured and is behaving as expected. The velocity gradient signal is proportional to

the sound pressure.

20p

i cu

ω

ρ= ∇⋅ (1)

where ω is the radian frequency of the acoustic wave c is the speed of sound in the fluid, ρ0 is the

density of the fluid and u is the velocity vector. A finite difference approximation to the velocity

gradient yields:

20 yx z

uu up

x y z

i c

ω

ρ ∆ ∆ ∆≈ + +

∆ ∆ ∆ (2)

where ∆ux is the difference of the x-component of acoustic particle velocity between two

locations separated by a distance ∆x (similar definitions apply in the y and z directions). The

pressure depends upon the components of velocity in all three directions. Thus in a three-

dimensional acoustic field, all three components of velocity at each of two locations are required

to measure the acoustic pressure.

In this case the use is not to calculate the sound pressure out of six velocity signals but to apply a

gradient signal in one (i.e. x) direction.

1 2

20 x x

x

u up

x

i c

ω

ρ − ≈ ∆

(3)

This (velocity gradient) sensor, consisting of two closely spaced and in the x-direction oriented

velocity probes, has a second order directivity pattern (cosines squared) and the phase of the

sensor is positive compared to a sound pressure sensor. A first order velocity probe has a phase

that alters sign each 180 degrees.

0

30

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0

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0

30

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++

+

+_

Figure 1: left: zeroth

order sensor (no directionality), middle: first order sensor (cosine shape directionality) and

right: second order sensor (cosine squared shape directionality).

4. CREATING A UNIPOLAR SENSOR A system based on a single AVS with an exact unipolar directivity is presented in

3. A nonlinear

method in the frequency domain is used to create this directionality. It is possible to use this

method to find sources in 3D however the resolution of such system is low.

If the velocity gradient sensor is summed with the velocity signal in the same direction, the

directivity resembles a unipolar sensor Ux.

1 2 1 20

20

2

x x x x

x

u u u uU c

x

i cρ

ω

ρ − + = + ∆

(4)

For each direction a unipolar signal can be constructed. This semi-unipolar directionality is

obtained in the time domain.

0

30

60

90

120

150

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270

300

330

Figure 2: A semi-unipolar directivity is created with the velocity gradient sensor.

It is possible to optimize the directivity by the use of adding some signal from velocity probes

from other directions.

5. MATHEMATICAL ROTATION OF VELOCITY PROBES

If signals are obtained from two orthogonally oriented particle velocity sensors, it is possible

to mathematically rearrange the vector orientation. For Microflown signals this is e.g. proven

in3,7

. It is therefore possible to mathematically ‘rotate’ the probe in any orientation. The

mathematical rotation is demonstrated for two orthogonal particle velocity sensors. It is possible

to create a particle velocity sensor that has a sensitivity in any direction that is desired. Assume

two sensors that are oriented in the x- and y-axis, see Figure 3 (left). A rotated figure of eight

directionality is obtained if the signals of the two probes are processed in the following way:

( ) cos( ) sin( )x y

u t u uθ θ θ= + (5)

The response for a rotation of θ=45o is shown in Figure 3 (right).

0.2

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0.8

1

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180 0

0.2

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0.8

1

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180 0

Figure 3: The particle velocity response in any direction can be obtained by combining two perpendicular particle

velocity sensors.

With this technique the probe orientation can be rotated in any desirable direction. This

mathematical rotation can also be applied in three dimensions. The velocity signals in three

directions are available as average signals in each direction. The velocity signal in any direction

is still a signal in time domain.

Since the velocity is a vector, the velocity gradient is a second order tensor. Commonly

represented by a 3×3 matrix, it contains 6 independent quantities. Hence at least 9 velocity

sensors are necessary to measure the three components of the velocity and the 6 elements of the

velocity gradient. It is therefore not trivial to rotate a velocity gradient microphone

mathematically.

6. BEAMFORMING WITH THE 3D VELOCITY GRADIENT SYSTEM Any array can be used to perform source localization. Although the linear velocity gradient

approach yields a very directive sensor, more advanced nonlinear beamforming methods yield

an even higher directivity. Furthermore, multiple velocity gradient sensors can be combined to

make an array which makes it possible to determine the location of the source. A disadvantage of

these nonlinear methods is that, without further assumptions, it is not possible to listen to the

sound from each direction.

The source localization method used in this article is the Capon method, also known as the

Minimum Variance method. This method is compared to a number of other methods and other p-

u arrays in the ASA 2010 article by Wind et al.10

.

7. EXPERIMENTS Twenty four loudspeakers are positioned in an anechoic chamber around a 3D velocity gradient

system. The set up is described in detail in the ASA 2010 articles by Basten et al.8 and Wind et

al.10. Eight loudspeakers are configured in three planes.

Only the middle horizontal plane of (eight) loudspeakers is used to demonstrate the directivity of

the 1D velocity gradient setup. All loudspeakers are used to demonstrate the 3D velocity gradient

set up.

Figure 4 left: Twenty four loudspeakers in an anechoic room around the velocity gradient set up (shown in detail

right).

In Figure 5 the sensitivity as function of the direction of arrival is shown for a unipolar

microphone. The set up is made as explained in Eq. 4. As can be seen, the unipolar microphone

is only sensitive in the forward (0°) direction.

As can be seen, the method unipolar microphone works quite well in the 200Hz-2kHz range. For

lower frequencies the generated signal of the loudspeakers was too low to get reliable results. For

frequencies higher than 2kHz the method as explained in Eq. 4 fails because the sensors cannot

be considered spaced closely.

Figure 5: Sensitivity as function of angle. Left: color plot and right the relative sensitivity in dB.

Three dimensional results of the beamforming algorithm are shown in figure 6. In the figure on

the left, it can be seen that each of the two sources is localized accurately. In the figure on the

right, it can be seen that three of the four sources are localized accurately and one has an error of

about 30 degrees.

Figure 6:

Source localization results using six velocity sensors. White circles: actual locations of the sources. Red: likely

source locations.

8. CONCLUSION This article has shown that a highly directional sensor array can be made by computing the

velocity gradient. Experimental results in an anechoic chamber have shown that the theoretical

directionality can also be achieved in practice. Although the current measurement setup does not

make it possible to rotate the velocity gradient, the array is suitable to perform source

localization. Results of the Capon method have shown that it is possible to localize at least two

sources.

REFERENCES 1 H.E. de Bree et al., The Microflown: a novel device measuring acoustical flows, Sensors and Actuators A-

Physical, 54, pp 552-557, 1996

2 H.E. de Bree et al., A method to measure apparent acoustic pressure, Flow gradient and acoustic intensity

using two micro machined flow microphones, Eurosensors X, Leuven, 1996

3 H.E. de Bree H.E. de Bree, T. Basten, D. Yntema, A single broad banded 3D beamforming sound probe,

DAGA, 2008

4 Vladimir Shchurov, Vector acoustics of the ocean, Vladivostok Dalnauka, 2006

5 Kevin J. Bastyr, Gerald C. Lauchle, James A. McConnell, Development of a velocity gradient underwater

acoustic intensity sensor, J. Acoust. Soc. Am. 106 (6), December 1999

6 Bruce M. Abraham, Ambient Noise Measurements with Vector Acoustic Hydrophones, 2006 IEEE

7 Winkel, A. et al. A particle velocity based method for separating all multi incoherent sound sources, Fisita,

2006

8 Buye Xu et al, Amplitude, phase, location and orientation calibration of an acoustic vector sensor, part I:

theory, ASA 2010.

9 Tom Basten et al, Amplitude, phase, location and orientation calibration of an acoustic vector sensor, part

II: experiments, ASA 2010.

10 Jelmer Wind et al. 3D sound source localization and sound mapping using a p-u sensor array, ASA 2010.