JSAE Paper Number 20025335

7/30/2019 JSAE Paper Number 20025335

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20025335

324 A Class of Optimal Broadband Beamformer

Geometries Designed for Easy Construction

Jacob J. CHRISTENSEN and Jrgen HALD

Brel & Kjr Sound & Vibration Measurement A/S, Skodsborgvej 307, DK-2850 Nrum, Denmark

Very regular beamformer array geometries will have Grating Lobes in their directivity

pattern when applied above a certain frequency. The grating lobes can be suppressed by

removing periodicities, typically by the use of random arrays or spiral arrays. Large

irregular arrays are, however, difficult to build due to their complicated support structure

and cabling. The present paper describes a novel array design, which maintains the low

grating lobe level of irregular arrays, but which has a regularity that allows a much simpler

support structure and cabling. The performance is compared with that of comparable

irregular arrays, and verified through practical measurements

Keywords: Noise Source Location, Phased Array, Beamforming.

1. INTRODUCTION

As illustrated in Figure 1, we consider a planar array of

Mmicrophones at locations in the

xy-plane of our coordinate system. When such an arrayis applied for Delay-and-Sum Beamforming, the

measured pressure signals are individually delayed

and then summed, [1]:

),...,2,1( Mmm =r

mp

. (1)=

=M

m

mm tptb1

))((),(

The individual time delays are chosen with the aim

of achieving selective directional sensitivity in a specific

direction, characterized here by a unit vector

m

. Thisobjective is met by adjusting the time delays in such a

way that signals associated with a plane wave, incidentfrom the direction , will be aligned in time before theyare summed. Geometrical considerations (see Figure 1)show that this can be obtained by choosing:

c

mm

r = , (2)

where c is the propagation speed of sound. Signals

arriving from other far-field directions will not bealigned before the summation, and therefore they willnot coherently add up.

The frequency domain version of expression (1) for theDelay-and-Sum beamformer output is:

(3)

=

=

=

=

M

m

jm

M

m

jm

m

m

eP

ePB

1

1

)(

)(

)(),(

rk

Here, is the temporal angular frequency, k k isthe wave number vector of a plane wave incident from

the direction

in which the array is focused seeFigure 1 and ck= is the wave number. In

equation (3) an implicit time factor equal to is

assumed. Through our choice of time delays

tje

)(m , or

equivalently of the preferred wave number vector

k k , we have tuned the beamformer on the far-field direction . Ideally we would like to measure

only signals arriving from that direction, in order to get aperfect localization of the sound source. To investigate,

how much leakage we will get from plane wavesincident from other directions, we assume now a plane

wave incident with a wave number vector k differentfrom the preferred k

0

k . The pressure measured bythe microphones will then be:

jeP k

= 00

1

)( 0 rkk ej

=

M

m

j me1

rK

P

),( =B

W

r

m

)

0M

m

)K

)0

,x K

k

K

=

, (4)mmr(

which according to equation (3) will give the followingoutput from the beamformer:

. (5)(0 k WPP

Here, the function W

(6)(

is the so called Array Pattern, defined entirely by thearray geometry. It has the form of a generalized spatialDFT of a weighting function, which equals one over thearray area and zero outside. Because the microphone

positions havez-coordinate equal to zero, the Array

Pattern is independent of . We shall therefore

consider the Array Pattern W only in the (

plane, i.e. we consider the projections of the wave

m

zK

)y

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number vectors onto that plane. There, W has an areawith high values around the origin with a peak value

equal to M at . According to

equation (5), this peak represents the high sensitivity toplane waves coming from the direction

)0,0(),( =yx KK

, in which the

array is focused. Figure 1 contains an illustration of thatpeak, which is called the mainlobe. Other directionalpeaks, which are calledsidelobes, will cause waves from

such directions to leak into the measurement of themainlobe direction . This will produce falsepeaks/images in a measured directional source map. A

good phased array design can therefore be characterizedby having low Maximum Sidelobe Level (MSL),measured relative to the mainlobe level. With reference

to equation (5), both kand k have length equal to the

wave numberk, and they can have opposite direction pointing for example in the positive and negative x-axis

directions meaning that

0

k20 kk . Therefore, at

a given frequency only the section ck 22 =K

of the Array Pattern will be visible. And if the Array

Pattern has low MSL for cmax2=K maxK ,

then a beamformer application will provide accuratedirectional source maps with a low degree of false

images up to the frequency cmaxK21

max = . If the

beamformer will be focused on directions not more

than 30 off-axis, then the projection of the wave

number vector k on the array plane will be

shorter than

0)( k

k23 . Therefore only the section

ck 23

23 =K will be visible, so the upper

limiting frequency becomes cKmax32

max )30( = .

Phased Array of Microphones

rm

Plane wave

k k=

Origin

Main

lobe

Figure 1: Illustration of a phased microphonearray, a directional sensitivity represented by a

mainlobe, and a Plane wave incident from the

direction of the mainlobe.

The width of the mainlobe of the Array Pattern can beestimated from the similarity of the expression (5) with a

2D DFT of a rectangular type of spatial window

function covering the area of the array: The mainlobewidth will be inversely proportional to the diameterD ofthe array, and the first null will be approximately at

DK 2min ==K , [1].

2. OPTIMAL ARRAY GEOMETRIES

Very regular phased array geometries, with a highdegree of periodicity, will have so-called grating lobesin their directivity pattern, when they are applied above acertain frequency. A grating lobe is a sidelobe withamplitude of the same size as the mainlobe, created byspatial aliasing. Consider as an example a rectangular

array with constant microphone spacing din both the x-and they-direction. For such an array, the Array PatternW of equation (6) will have a first grating lobe at

)0,2(),( dKK yx = . According to the above

description, this grating lobe will enter the visible

region at the frequency for which dc /22 = .

At that frequency the microphone spacing d is equal tohalf a wavelength. At higher frequency there may befalse images of the same magnitude as the true image inmeasured directional source maps.

Grating lobes can be avoided by removing theperiodicities, for example by the use of randomgeometries, which will provide a variety of differentspatial sampling intervals and thereby suppress thespatial aliasing problems, [1]. Large random arrays are,however, much more difficult to build than more regulararrays. Both the support structure and the cabling aremore complicated and as a consequence the operation inthe practical measurement situation is difficult or

tedious. Various non-random designs have beenintroduced, which provide also a variety of different

spatial sampling intervals and therefore rather lowMSLover a wide frequency range for example arraygeometries based on one or several concentriclogarithmic spirals, [2], or on an Archimedes spiral, [3].But the spiral arrays are still quite complicated to builddue to the complicated curved geometries. See Figure 2.Also, the need for high resolution in the far field canonly be met with relatively large dimensions of thearrays. Thus, an array with a diameter of several metersis often required. In connection with outdoorapplications it is therefore of practical importance thatthe array construction allows for easy assembly anddisassembly at the site of use, and for easy transport.



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Figure 2: Schematic representation of a 90 ch.

Archimedes spiral with the same diameter (2.43m)

as the B&K wheel array [Fig.3]. The microphone

spacing, measured as arc length along the spiral

curve, is 0.3 m.

Figure 3 shows an example of a new-patented

Wheel Array design, which can be optimized to havevery lowMSL over a wide frequency band, but which isbuilt of a set of identical line arrays, simplifying themechanical construction and the cabling. The patenteddesign consists of typically an odd numberN of smallidentical line arrays arranged around a center as spokes

in a wheel, with identical angular spacing between thespokes. All spokes are tilted the same angle away fromradial direction. The geometry is invariant under a

rotation around the center, n being any

integer. The mechanical design of the array shown inFigure 3 is fully modular: all spokes can easily bedetached from the supporting inner and outer ringswhich themselves can be disassembled. Thus the whole

array structure can be disassembled and transported in astandard size flight case. Regarding the cabling each

spoke mounts 6 microphones, which through integratedcabling, are connected to a common plug (LEMO-connector). Each spoke is then connected to a 6-channelinput module through a single cable. Despite itsirregularity the Wheel Array has its sensors grouped ineasily identifiable logical units. In a practical

measurement situation, which requires channel

detection, calibration and occasionally detection ofhardware faults, this is a great advantage.

Nn /360

Minimize Max |W(K)| over the area

cD

max22 K , (7)

Figure 3: Example of a 90-channel Brel & Kjr

wheel array for beamforming. The wheel array

depicted has 15 identical spokes, with integrated

cabling, each carrying 6 microphones. The wheel

diameter is 2.43 m (patent pending).

Here, max is the upper limiting frequency for use of

the array. This optimization is performed by the use of a

Minimax optimization program.The Wheel Array of Figure 3 was optimized with

max = 3.0 kHz. For a given diameter and number of

channels the Archimedes spiral array can be optimizedfor a given frequency range by adjusting the microphonespacing, d (d = 0.3m was chosen for optimal

performance below 3 kHz).

The array diameter determines the width of themainlobe and thus the angular resolution. Havingselected the array diameterD, the number of spokes and

the number of microphones per spoke, the followingvariables are adjusted to provide the lowest possiblevalue of the MSL: 1) The rotation of the spokes awayfrom radial direction. 2) The position of the microphoneson a spoke. With 6 microphones on each spoke, wehave 7 adjustable parameters to minimize the MSL over

the following area in the Array Pattern (see section 1):

Figure 4 shows the MSL as a function of frequency forthe optimized Wheel Array of Figure 3 and thecomparable spiral array shown in Figure 2 These arrayshave approximately the same diameterD = 2.43m and

therefore the same mainlobe width (2/D) and spatialangular resolution. Both arrays are designed for the

frequency range 0.5 3.0 kHz but clearly the WheelArray is superior in performance.Due to the grating lobes, which always are present inregular arrays, the traditional X-shaped arrays and

rectangular grid-arrays perform even worse. Seereference [5] for a discussion.



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3. MEASUREMENTS

In order to demonstrate the capabilities of the 90 ch.Wheel Array of Figure 3 we have performed ameasurement on a car (Volvo S60). The vehicle wasinstalled in an in-door test bench and the engine wasoperated at 2500 RPM with no load. The Wheel Array

was placed in parallel with the side of the car at adistance of 2.9m and the stationary sound field wasrecorded for 5 sec. at 16 KHz sampling rate using theBrel & Kjr Acoustic Test Consultant (Type 7761)running on the PULSE measurement system. Data werepost-processed with the PULSE Beamformingapplication (Type 7768) and a representative result is

shown in Figure 5. Despite acoustically challengingmeasurement conditions (back ground noise andreflecting walls, ceiling and floor) a useful dynamicrange of 12 dB is achieved. The beamformingcalculation clearly reveals a leakage from the frontwheel box at high frequencies.

-18

-15

-12

-9

-6

-3

0

0 1000 2000 3000 4000 5000

f [Hz]

MSL

[dB]

90 ch. Archimedes

spiral

90 ch. B&K wheel

array

Figure 4: MSL as function of frequency for the B&K

Wheel array [Fig.3] and the spiral array [Fig.2].

Figure 5: Beamforming result using the 90 ch. B&K

Wheel Array shown in Figure 3. The upper display

shows the sound pressure levels resulting from the

beamforming calculation as a contour plot. The

lower display gives the spectrum in 1/3-octave bandsat a point on the front tire, and the highlighted

interval (2.03.15 kHz) is the selected frequency

range which the contour plot covers. (Screen-shot

from the PULSE Beamforming application).

REFERENCES

1. D. H. Johnson and D. E. Dudgeon, ArraySignal Processing: Concepts and Techniques,Prentice Hall, New Jersey, 1993.

4. CONCLUSION 2. J. R. Underbrink and R. P. Dougherty, Array

Design for Non-intrusive Measurement ofNoise Sources,Proceedings of Noise-Con 96,757-762, 1996.

The new-patented Wheel Array with tilted spokes can beoptimized to have low Maximum Sidelobe Level (MSL)over a very broad frequency range. Its performance issuperior to that of comparable Archimedes spiral arrays.The Wheel Array type has an irregular geometry but due

to its modular construction it is easy to manufacture,transport and operate. As demonstrated by practicalmeasurements the low sidelobe levels of the WheelArray makes its useful even in acoustically non-idealenvironments.

3. A. Nordborg, J. Wedemann and L. Willenbrink,Optimum Array Microphone Configuration,

Proceedings of Inter-Noise 2000.4. G. Elias, Source Localization with a Two-

dimensional Focused Array: Optimal SignalProcessing for a Cross-shaped Array,Proceedings of Inter-Noise 95, 1175-1178,1995.

5. J. Hald and J.J. Christensen, to appear inProceedings of Inter-Noise 2002.

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JSAE Paper Number 20025335