Analytic and Experimental Results of Spatial Correlations of Vector Intensity Sensors
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Transcript of 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]
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
“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
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
Analytic Expressions for Spatial Correlationsof Intensity Sensors in Isotropic Noise
yxII IIdyx
where,0
4
2 cossinsin3
kd
kdkdkdkddII
4
22 sincos2sin23
|| kd
kdkdkdkdkddII
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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)
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
TheorySimulationExperimental
Physical Correlation Measurements
2pp
2
||pu
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
TheorySimulationExperimental
Physical Correlation Measurements
2
uu
2
||uu
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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
TheorySimulationExperimental
Physical Correlation Measurements
2
||II
2
II
Pacific Rim Underwater Acoustics Conf.
Vancouver, BC, Canada
03-05 Oct 2007
Naluai – NAVAIRAcoustics SystemsPatuxent River, MD 20670
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