analytic and experimental results of spatial correlations of vector intensity sensors

Pacific Rim Underwater Acoustics Conf.

Vancouver, BC, Canada

03-05 Oct 2007

Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670

Analytic and Experimental Results of Spatial Correlations of

Vector Intensity Sensors

Nathan K. NaluaiGraduate Program in AcousticsPennsylvania State University

University Park, PA [email protected]



03-05 Oct 2007


“Diffuse Field” Model of Isotropic Noise

• Generally assumes sound coming from all directions

• One model definition (Jacobsen)– Sound field in unbounded medium– Generated by distant, uncorrelated sources– Sources uniformly distributed over all directions– Field would be homogeneous and isotropic– Time-averaged intensity is zero at all positions



03-05 Oct 2007


kij

θi

φj

ux(rb ,t)

uz(ra ,t)

x

y

z

p(0,t)

Coordinate System Orientation

tutprR azpu ,,0E),(||

r

Notation conventions:

tutprR bzpu ,,0E),( r



03-05 Oct 2007


Analytic Solutions for Spatial Correlationsof Separated Sensors in Isotropic Noise

kd

kddpp

sin

2

cossin3

|| kd

kdkdkddpu

3

2 sin2cos2sin3

|| kd

kdkdkdkdkdduu

3

cossin3

kd

kdkdkdduu



03-05 Oct 2007


Instantaneous Intensity:

The correlation between spatially separated intensity sensors is:

Intensity Correlation Derivations

ttpt ,,),( rurrI

2211

2211

21

E

,,,,E

),(),(E),(

upup

ttpttp

ttrRII

rurrur

rIrI



03-05 Oct 2007


For four Gaussian random variables [Bendat & Piersol]:

Can re-write the intensity correlation expression as

Intensity Correlation Derivations

xywzxzwyyzwxwxyz EEEEEEE

122121212211 EEEEEE),( upupuuppupuprRII

0EE

)(

)cos()()sin(

2limE

2211

2

2

011

upup

kr

krkrkr

c

Aup

r



03-05 Oct 2007


Analytic Expressions for Spatial Correlationsof Intensity Sensors in Isotropic Noise

yxII IIdyx

where,0

4

2 cossinsin3

kd

kdkdkdkddII

4

22 sincos2sin23

|| kd

kdkdkdkdkddII



03-05 Oct 2007


Correlations for Separated Sensors in Isotropic Noise

0 0.5 1 1.5-1

-0.5

0

0.5

1

Cor

rela

tion

Coe

ffic

ient

Spacing (in wavelengths, λ)

uu

||pu

pp

||uu

0 0.5 1 1.5-1

-0.5

0

0.5

1

Spacing (in wavelengths, λ)

Cor

rela

tion

Coe

ffic

ient

II

||IIpp



03-05 Oct 2007


N4

N7

N10

Computational Experiment Design/Layout

• Computational Simulation in MATLAB environment

• Source distribution determined by variable M, (no. of sources about “equator”)

• Each source generating noise (0-6.4kHz band)

• Signals oversampled to allow for 1mm separation resolution

• Assumptions– Plane wave superposition– Sensor separation: 7cm– Air-like medium (c, ρ)

r1

r2



03-05 Oct 2007


0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

Pressure-Pressure Spatial Correlations

ρ pp2

kd

SimulationTheory

Input Parameters for Computational Experiment

• Where S can be considered the number of sample “locations” in field.

• Resulting curve is the average over those locations

51M

Hz12800sf

256FFTN

64avgN

10S

m07.0d dd xyxy ,, 22



03-05 Oct 2007


Equal Amplitude Distribution (Ideal Case)

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

xy2

kd

Simulation

Theory

2||pu

2uu

2||uu

2pp



03-05 Oct 2007


Equal Amplitude Distribution (Ideal Case)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

kd

Simulation

Theory

xy2 2

II

2||II



03-05 Oct 2007


Random Amplitude Source Weighting

0 1 2 3 4 5 6 7 80

0.5

1

kd

Random Amplit.TheoryCOS Weighting2

||pu

0 1 2 3 4 5 6 7 80

0.5

1

kd

Random Amplit.TheoryCOS Weighting

2

||uu



03-05 Oct 2007


Effect of Inter-channel Phase Offsets on Correlation

0 0.5 1 1.5 2 2.5 3 3.5 400.10.20.30.40.50.6

kd

TheoryMismatched

2

||pu

0 0.5 1 1.5 2 2.5 3 3.5 400.10.20.30.40.50.6

kd

TheoryMismatched

2

||pu

Random Phase held fixed over averaging period

Phase shift applied on every 3rd average



03-05 Oct 2007


Physical Correlation Measurements

• Reverberant Acoustic Test Tank (ASB-PSU)

• Two Lubell LL-9162 sources (uncorrelated noise)– Low freq. rolloff at 1-kHz

• pa-probe (McConnell)– Sensitivity axes aligned

• Outputs recorded at four separate locations in tank.– 64 avgs at each location

5.5m

8.5m

6.1 m

0.197 m

to Amp (R-Ch)

to Amp (L-Ch)



03-05 Oct 2007


0 1 2 3 4 5 6 7 80

0.5

1

kd

TheorySimulationExperimental

0 1 2 3 4 5 6 7 80

0.5

1

kd



2pp

2

||pu



03-05 Oct 2007


0 1 2 3 4 5 6 7 80

0.5

1

kd


0 1 2 3 4 5 6 7 80

0.5

1

kd



2

uu

2

||uu



03-05 Oct 2007


0 1 2 3 4 5 6 7 80

0.5

1

kd


0 1 2 3 4 5 6 7 80

0.5

1

kd



2

||II

2

II



03-05 Oct 2007


Summary

• Analytical solutions for spatially separated Intensity measurements have been derived and verified experimentally

• Constant phase offsets have no effect on the agreement between the coherence and the theoretical predictions

• Intensity measurements demonstrate shorter correlation lengths than the component measures

• Suggest that intensity processing of vector sensor arrays may be less susceptible to ambient noise contamination than traditional pressure hydrophone array.

• Examine performance of intensity vector sensor arrays– Possible gains in directivity

analytic and experimental results of spatial correlations of vector intensity sensors

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