2003 Internoise CrossSpecBF

7/28/2019 2003 Internoise CrossSpecBF

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The 32nd International Congress and Exposition on Noise Control En

[N356] Improvements of cross spectral beamforming

Jacob Juhl Christensen

Brel & Kj ement A/S

Jrgen Hald

Brel & Kjr Sou easurement A/S

ABST ACT

conventional delay-and-sum beamforming, individual delays are applied to the transducer

ing, Sensor Array Signal Processing.

gineeringJeju International Convention Center, Seogwipo, Korea,

August 25-28, 2003

r Sound & Vibration Measur

Skodsborgvej 307, DK-2850 Nrum, Denmark

Email address: [email protected]

nd & Vibration M

Skodsborgvej 307, DK-2850 Nrum, Denmark

R

In

signals, prior to summation, in order to focus the beamformer in a given direction. For the

case of measurements at a finite distance, amplitude corrections are often also applied, i.e.

corrections for the fact that different positions on the assumed source plane have different

distances to the array transducers and therefore are attenuated by different amounts. It is not,

however, easy to find an optimal way of doing this amplitude correction. The present paper

describes a cross-spectral beamforming algorithm intended for a planar transducer array and a

stationary sound field. Assuming a model where the recorded field is generated by a

distribution of monopole point sources, an error function between measured and modeled

cross spectra is formed. Minimization of this error function leads to a cross-spectral imaging

function, which automatically includes amplitude correction. Furthermore, by excluding the

auto spectra, disturbing self-noise is avoided, and also the dynamic capability of the

beamformer is improved.

KEYWORDS: Beamform


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INTRODUCTION

We consider a planar array of microphones at locations ( 1, 2,..., )m m M=

r in the xy-plane of our coordinate system [Figure 1]. When applied for Delay-and-Sum Beamforming

[1], the measured pressure signalsm

p are individually delayed and then summed

(1)1

( , ) ( ( )).M

m m

m

b t p t =

= 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

. This objective is met

by adjusting the time delays in such a way that signals associated with a plane wave, incident

from the direction , will be aligned in time before they are summed. Geometrical

considerations show that this can be obtained by choosing:

/ ,m m c= r (2)

where is the propagation speed of sound. Signals arriving from other far-field directions

will not be aligned before the summation, and therefore they will not coherently add up .

c

The frequency domain expression for the Delay-and-Sum beamformer output is:

(3)( )

1 1

( , ) ( ) ( ) .mM M

j

m m

m m

B P e P

= =

= = k r mje

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

wave incident from the direction in which the array is focused [Figure 1] and k c= is

Figure 1. Illustration of a phased microphone

array, a directional sensitivity represented by a

mainlobe and sidelobes, and a plane wave

incident from the direction of the mainlobe.

Figure 2. In near field focusing, spherical waves

emitted by a monopole source at the focus point r is

assumed. Signal delays are computed according to

equation (14).


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the wave number. In equation (3) an implicit time factor equal to j te is assumed.

For a given array geometry {rm} the structure of the directional sensitivity is contained in the

Array Pattern function [1] defined in wave number space as

(4)1

( ) .mM

j

m

W e

=

K rK

It has the form of a generalized spatial DFT of a weighting function, which equals one over

the array area and zero outside. Because the microphone positions have z-coordinate

equal to zero, the Array Pattern is independent of . We shall therefore consider the Array

Pattern Wonly in the (K

mr

zK

x,Ky) plane. There, Whas an area with high values around the origin

with a peak value equal to M at (Kx,Ky) = (0,0). This peak represents according to the

following section the high sensitivity to plane waves coming from the direction , in which

the array is focused. Figure 1 contains an illustration of that peak, which is called the

mainlobe. Other directional peaks are called sidelobes and a good phased array design is

characterized by having low Maximum Sidelobe Level (MSL), measured relative to the main

lobe level [2]. The highest sidelobe is identified as the highest secondary peak in thePower

Array Pattern

(5)( )2

, 1

( ) | ( ) | ,m nM

j

m n

U W e

=

= K r rK K

for |K| < 2max/c, max being the upper frequency of the arrays intended use.

CROSS SPECTRAL BEAMFORMING WITH AUTO-SPECTRA EXCLUSION

For a stationary sound field it is natural to consider the time average power output,

( ) ( )2 *

, 1 , 1

( , ) | ( , ) | ( ) ( ) ( ) ,m n m nM M

j

m n nm

m n m n

V B P P e C e

= =

= = k r r k r r j (6)

of the beamformer, where we have introduced the cross-spectrum matrix

*( ) ( ) ( )nm m n

C P P . We may split (6) in an auto-spectrum part and a cross-spectrum part,

(7)( )

1

( , ) .m nM M

j

mm nm

m m n

V C C e

=

= + k r r

Here, the auto-spectra C will contain self-noise from the individual channels such as

wind-noise and electronic noise from the data acquisition hardware. For that reason it would

be desirable to omit the first sum in equation (7). Ideally, the cross-spectra , are

not affected by the self-noise, because the self-noise in one channel is generally incoherent

with the self-noise in any other channel. Under that condition, averaging will suppress

contributions from self-noise in the cross-spectra. We can assess the effect of excluding the

mm

,nmC m n


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auto-spectra by relating the plane wave response of the cross-spectral beamformer to the

power array pattern. For a unit amplitude plane wave with wave number vector k0 the

spectrum recorded by the mth microphone is 0exp( )mP j m=

k r

0 ( ) ( )

, 1

m

M Mj

m n

e e

= =

= = k k r r

'( , )M M

j

m n

V

0'( ).V U

. Insertion of that in equation(6) leads to the following expression for the beamformer power output:

k k

( ) .MK K

2 /10 M

0

1010 logM M

=

2 /1010MSL M

(8)0( ) ( )0

, 1

( , ) ( ).m n m n nj j

m n

V e U =k r r k r r k k

)j

where we have used formula (5) for the power array pattern U. In a similar way we see that

the self-term free versions of the power array pattern (5) and the cross-spectral beamformer

response (7),

(9)( ) ('( ) andm n m nnmm n

U e C e K r r k r rK

are for plane waves related by

'( , ) = (10)

Thus, removal of the auto-spectral terms from the cross-spectral beamformer (7) corresponds

to omitting the self-terms from the definition of the power array pattern (5). Provided the

reduced array pattern has lower sidelobe level than U, we can therefore reduce the level

of ghost images in cross-spectral beamformer output by omitting the auto-spectra.

'U

Comparing the definitions of the array pattern Uand the reduced version Uwe find that

'( )U U= (11)

The mainlobe is therefore reduced fromM2 (forU) toM2M(forU), and the highest sidelobe

is reduced from to2 10MSLM /10 10MSL . Assuming first that U does not

become negative, this leads to the followingMaximum Sidelobe LevelforU:

2 /1 /10

10 2

10 10 1' 10 log ,

1

MSL MSLMMSL

M M M

=

which is easily shown to be always smaller (better) than MSL. As an example, assume a 90-

channel array with anMSL equal to 15 dB. In that caseMSLequals -16.83 dB, meaning that

the highest sidelobe has been reduced by 1.83 dB. If the power array pattern U contains

values less than M then the reduced array pattern U will have areas with negative values.

Worst case is when Uhas a null. In that case the minimum value ofUequals M, which

will have the same effect as a sidelobe with amplitude equal to M. Such a sidelobe will not

affectMSLas long asMis smaller than . This condition has been fulfilled

for all the arrays that we have been designing [2]. And additionally, this worst-case condition

will not occur, when only array geometries without redundant spacing vectors are used.


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Near Field Beamforming

Up to now we have considered only the case of sources in the far field. In that case eachsource will create a plane wave in the region occupied by the array, meaning that different

sources can be located by identifying associated plane waves. For sources in the near field this

will not be the case, and we assume instead a distribution of monopole point sources on the

focus plane. In this case the pressure measured by the microphones will be

[ ( ) ]( ) / ( ),m i ij krm i mi

P Pe r+= i r r (12)

where ri is the source positions,Pi and i are the individual source strengths and phases and

rm(ri) = |rm ri| is the microphone to source distance. The expression for delay-and-sum

beamforming (3) must be restated for point focusing:

(13)( )

1

( , ) ( ) ,mM

j

m

m

B P e

=

= rr

where we have replaced the delays (2) with the form

( ) (| | ( )) / ,m mr c= r r r (14)

appropriate for a spherical wave [Figure 2]. The near field version of equation (7) for the

beamformer power appears as

(15)

[ ( ) ( )]

1( , ) .m n

M Mj

mm nmm m nV C C e

=

= +

r r

r

CROSS SPECTRAL BEAMFORMING WITH AMPLITUDE CORRECTION

Equation (15) for finite distance beamforming contains no compensation for the fact that

different positions on the assumed source plane have different distances to the array

transducers and therefore are attenuated by different amounts. For a single source at ri, a

possible correction could be to replace the cross-spectrum matrix by the scaled version

. The introduction of a scaled cross-matrix is, however, an ad-hoc correction

with uncontrolled effects. A sound approach can be achieved by assuming a model where the

recorded sound field is generated by a monopole distribution. Based on this assumption we

determine the distribution of source positions and amplitudes which minimizes an error

function between the measured cross-spectra and the model cross-spectra. The approach is

inspired by reference [3].

( ) ( )nm m i n iC r rr r

Let be the transducer coordinates and let r be the position of a monopole.

The field,p

, 1, ,m m =r M

m, recorded by the mth transducer is then 0 0( ) ( ) ( )m m mp p v p v = r r r r . wherep0

is the source strength and is the steering vector given by( )v r


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(16)| |( ) / | | .jkv e= rr r

According to our model the cross-spectrum, , between channel m and n ismodnmC

(17)mod * *( ) ( ),nm n m n mC p p a v v= r r

where a is a reel amplitude coefficient. Then we define an error function, , between

the model cross-spectra and the measured cross-spectra, C

( , )E a r

nm,

2mod *

, 1 , 1

( , ) ( ) ( ) .M M

nm nm nm n m

m n m n

E a C C C av v= =

= = r2

r r

.v v

(18)

We can simplify this expression by introducing the column matrices

(19)*

[ ] and [ ]nm n mC= =g h

Then (18) appears as

2 2( , ) ( ) ,E a a a a= = + +r g h g g h g g h h h (20)

where we have used that a is real. Minimizing first with respect to a, we find ag h which

upon multiplication from left with leads toh

/a = h .g h h (21)

We have to make sure that the right-hand side is real. Appealing to the fact that the cross-

spectral matrix is Hermitian and to the definition (19) ofh and g we see that

* * * *

, 1 , 1 , 1

,M M M

nm n m mn m n nm n m

m n m n m n

C v v C v v C v v= = =

= = = h =g g h (22)

implying that h g is real. With this observation and by insertion of (21) into (20) the error

function (20) can be rewritten as

( )2

2 ( , ) 1 / / .E a a = = r

g g h h g g g g h g h h (23)

Minimizing the error function over all r thus corresponds to maximizing the Imaging

Function, ( , )I r ,

( )2

2 24 * *

, 1 , 1

( , ) / ( ) ( ) ( ) / ( ) ( ) ,M M

nm m n n m

m n m n

I C v v= =

= r h v vg h h r r r r (24)

over all r (We choose the definition I4 since (24) has unit of power squared). In practice

( , )I r is computed over a discrete mesh covering the focus area. In the resulting map, peaks

are interpreted as areas with a high probability of finding a source. This interpretation can be

justified if we compare the imaging function in the far field with the corresponding expression

(3) for the Delay-And-Sum beamformer. For large | |R

r the approximation


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| |m m

R R r r is valid. In the far field limit (24) can therefore be approximated by

4

(r

2 2

2 *

2, 1 , 1 ( )

222

2 * , 1

, 1 , 1

1.

m n

n m

n m

M MjkR jkR

nm m n nm Mm n m n jk R R

nmM MjkR jkR m n

n m

m n m n

R C v v C e eI C

MR v v e e

= =

=

= =

= =

(25)e

mNow, using the fact that the difference in travel paths nR R equals the projection

difference [Figure 3] we find that the beamformer power (7) is)m n r

( ) ( )

, 1 , 1

.m n n mM M

jk jk R R

nm nm

m n m n

V C e C e

= =

= = r r

Obviously we have 2I V= showing us that apart from a constant factor the imaging

function in the far field equals the output of the Delay-And-Sum beamformer, which justifies

the chosen interpretation. Due to this connection with the plane wave case we can expect

improved side lobe levels from the self-term free version of the imaging function (24):

Figure 3.For a source in the extreme far field the

difference,Rn-Rm, in the propagation path length to the

transducers at rn and rm can be calculated from the

vector diagram.

Figure 4. An example of a planar 66-channel

beamformer. The microphone positions () are

randomly distributed inside the disc.


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Figure 5. Comparison of the output of three different beamforming algorithms for a configuration

with two incoherent 3 kHz monopole sources of equal strength. The data were generated using the

array shown in Figure 4. In the legend I refers to the full cross-spectral imaging function (24), J is

the cross-spectral imaging function (26) which excludes the auto-spectra, and V is the near-field

delay-and-sum algorithm (15). All curves are normalized to 0 dB maximum level.

22

4 * *( , ) ( ) ( ) ( ) / ( ) ( ) .M M

nm m n n m

m n m n

J C v v v v

r r r r r

(26)

A comparison of the self-term free algorithm (26) and the full cross-spectrum methods (15)

and (24) confirms that auto-spectra exclusion provides lower sidelobe levels [Figure 5].

SUMMARY

In this paper we have discussed the possible benefits of excluding the auto spectra in cross-

spectral beamforming algorithms for stationary sound fields. Furthermore we have presented

a self-contained derivation of a near field cross-spectral beamforming algorithm, which

includes amplitude corrections.

REFERENCES

1. D. H. Johnson and D. E. Dudgeon,Array Signal Processing(Prentice Hall, New Jersey, 1993).

2. J. Hald and J.J. Christensen, A class of optimal broadband phased array geometries designed for easy

construction, Proceedings of Internoise 2002.

3.G. Elias, Proceedings of Internoise 1995, p.1175-1178.

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2003 Internoise CrossSpecBF