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NEAR-FIELD LOCALIZATION OF ACOUSTICSOURCES WITH MULTIPLE SENSOR ARRAYS
AND IMPERFECT SPATIAL COHERENCE
Richard J. KozickBucknell University
Acknowledgements: Brian Sadler (ARL), Tien Pham (ARL),Keith Wilson (ARL), Nino Srour (ARL), Doug Deadrick (Sanders)
June 14, 2001
1
FEATURES OF MEASURED DATA
7100 7200 7300 7400 7500 7600 7700 7800 7900 8000 8100
9200
9300
9400
9500
9600
9700
9800
9900
EAST (m)
NO
RT
H (
m)
GROUND VEHICLE PATH AND ARRAY LOCATIONS
1
3
4
5
340 − 350 SEC
VEHICLE PATH 10 SEC SEGMENTARRAY 1 ARRAY 3 ARRAY 4 ARRAY 5
3
4
ARRAYS 1 AND 3
0 50 100 150 200 250 30060
65
70
75
80
FREQUENCY (Hz)
dB
MEAN PSD, ARRAY 1
0 50 100 150 200 250 30060
65
70
75
80
FREQUENCY (Hz)
dB
MEAN PSD, ARRAY 3
0 50 100 150 200 250 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FREQUENCY (Hz)
CO
HE
RE
NC
E |γ
|
MEAN SHORT−TIME SPECTRAL COHERENCE, ARRAYS 1 & 3
Significant coherence near 40 Hz and harmonicsSource distance > 100 m
5
DOPPLER EFFECTS (≈ ±1 Hz)
330 335 340 345 350 355 360−8
−6
−4
−2
0
2
4
6
8RADIAL VELOCITY OF SOURCE
TIME (SEC)
SP
EE
D (
m/s
)
ARRAY 1ARRAY 3ARRAY 4ARRAY 5
330 335 340 345 350 355 3600.98
0.99
1
1.01
1.02DOPPLER SCALING FACTOR α
TIME (SEC)
α
ARRAY 1ARRAY 3ARRAY 4ARRAY 5
7
ARRAYS 1 AND 3: DOPPLER COMPENSATION
0 50 100 150 200 250 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FREQUENCY (Hz)
CO
HE
RE
NC
E |γ
|
MEAN SHORT−TIME SPECTRAL COHERENCE, ARRAYS 1 & 3
0 50 100 150 200 250 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FREQUENCY (Hz)
CO
HE
RE
NC
E |γ
|
MEAN SHORT−TIME SPECTRAL COHERENCE, ARRAYS 1 & 3
WITHOUT DOPPLER COMPENSATION
With Doppler compensation Without Doppler compensation
8
COHERENCE WITHIN ARRAY 1
0 50 100 150 200 250 300 350 400 450 5000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FREQUENCY (Hz)
CO
HE
RE
NC
E |γ
|
MEAN SHORT−TIME SPECTRAL COHERENCE, ARRAY 1, SENSORS 1 & 5
10
CROSS-CORRELATION FUNCTIONS AT 342 SEC
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−1
−0.5
0
0.5
1CROSS−CORRELATION: ARRAYS 1 AND 3
LAG (SEC)
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−1
−0.5
0
0.5
1GENERALIZED CROSS−CORRELATION: ARRAYS 1 AND 3
LAG (SEC)−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1CROSS−CORRELATION: ARRAY1, SENSORS 1,5
LAG (SEC)
Between arrays 1 & 3 (> 200 m) Within array 1 (< 3 m)
What are conditions for accurate time-delay estimation?
11
KEY ISSUES
• Objective: Estimate source location (xs, ys) using “array of arrays”
• Signals at distinct arrays:
– Partially coherent due to random propagation effects
– Power spectrum variations due to aspect angle and propagation
• Individual arrays have small aperture
• Limited communication bandwidth to fusion center=⇒ perform some processing at arrays
• Source motion: Doppler, tracking
• How to exploit large baseline between arrays?When is joint processing beneficial?
12
OUTLINE
• Mathematical model for sensor data at distributed arrays
– Wideband sources, aspect angle variation, partial coherence, . . .
• Source localization performance accuracy vs. communicationsbandwidth (Cramer-Rao bound (CRB))
– Nearly optimum performance with bearing estimation (each array)and time-delay estimation (between arrays)
• Tighter bounds for time-delay estimation with partially coherentsignals (modified Ziv-Zakai bound)
– Quantify required SNR, coherence, fractional bandwidth,time-bandwidth product =⇒ “threshold coherence”
• Examples from measured data
• Continuing work
13
SOURCE AND ARRAY GEOMETRY
SOURCE(x_s, y_s)
x
y
ARRAY 1
ARRAY H
(x_1, y_1)
ARRAY 2(x_2, y_2)
(x_H, y_H)
FUSIONCENTER
• Source location (xs, ys) is unknown
• H arrays, where each array h ∈ {1, . . . , H} has:
– Nh sensors with known locations
– Reference sensor located at (xh, yh)
– Sensors located at (xh + ∆xhn, yh + ∆yhn), n = 1, . . . , Nh
14
MODELING ASSUMPTIONS
• Individual arrays have aperture ∼ 1 meter
– Source is far-field =⇒ bearings φ1, . . . , φH
– Perfect wavefront coherence across each array
• Distributed arrays are spaced ∼ 10’s to 100’s of meters
– Source location is near-field w.r.t. spacing between arrays
– Signal power spectrum varies due to aspect angle and propagation
– Partial wavefront coherence from array to array
– Coherence varies with frequency
• Assume target position is fixed (small variation in ∼ 1 sec)
• Model source signal and noise at sensors as (wideband) Gaussianrandom processes (zero mean, wide-sense stationary, continuous-time)
– Coherence loss is modeled in Gaussian framework
15
MATHEMATICAL MODEL FOR ARRAY DATA
•Waveform received at sensor n on array h:
zhn(t) = sh(t− τh − τhn) + whn(t),h = 1, . . . , Hn = 1, . . . , Nh
• Propagation delay from source to reference sensor on array h:(near-field)
τh =1
c
[(xs − xh)
2 + (ys − yh)2]1/2
• Propagation delay from reference sensor on array h to sensor n:(far-field)
τhn ≈ −1
c[(cosφh)∆xhn + (sinφh)∆yhn]
16
SOURCE SIGNAL MODEL
• Vector of source signals received at the H arrays:
s(t) =[s1(t) s2(t) · · · sH(t)
]T
• Cross-spectral density matrix characterizes propagation and coherence:
Gs(ω) =
Gs,11(ω) Gs,12(ω) · · · Gs,1H(ω)Gs,21(ω) Gs,22(ω) · · · Gs,2H(ω)
... · · · ... ...Gs,H1(ω) Gs,H2(ω) · · · Gs,HH(ω)
• Gs,hh(ω) = power spectral density function of source signal at array h(may differ from array to array due to propagation, aspect angle, etc.)
• Cross-spectral density functions have the form
Gs,gh(ω) = γs,gh(ω) [Gs,gg(ω)Gs,hh(ω)]1/2
• γs,gh(ω) = coherence of source signal at arrays g and h, with0 ≤ |γs,gh(ω)| ≤ 1 (due to random phenomena in propagation medium)
17
COMPLETE MODEL FOR ARRAY DATA
• Form vector of data zhn(t) = sh(t− τh− τhn) +whn(t) from all sensors:
zh(t) =
zh1(t)...
zh,Nh(t)
, Z(t) =
z1(t)
...zH(t)
• Array manifold for array h at frequency ω:
ah(ω) =
exp[jωc ((cosφh)∆xh1 + (sinφh)∆yh1)
]...
exp[jωc
((cosφh)∆xh,Nh
+ (sinφh)∆yh,Nh
)]
(Recall assumption of perfectly coherent plane waves across each array)
• Dgh = τg − τh = relative propagation delay between arrays g and h
18
• Cross-spectral density matrix of Z(t):
GZ(ω) =
G11 G12 · · · G1H
G21 G22 · · · G2H... · · · . . . ...
GH1 GH2 · · · GHH
+ Gw(ω)I.
Gw(ω) = noise power spectral density, independent of source signals
• Diagonal blocks of GZ(ω) have the form
Ghh = ah(ω)ah(ω)HGs,hh(ω),
which is the standard model in coherent, far-field array processing
• Off-diagonal blocks of GZ(ω) have the form
Ggh = ag(ω)ah(ω)H︸ ︷︷ ︸coherent,far− field,
φg, φh
· [Gs,gg(ω)Gs,hh(ω)]1/2︸ ︷︷ ︸deterministicpropagation
(aspect angle)
· exp(−jωDgh)︸ ︷︷ ︸time− delay,near− field
· γs,gh(ω)︸ ︷︷ ︸randomprop.
19
CRB ON LOCALIZATION ACCURACY
• Objective is to estimate the source location parameter vector
Θ = [xs, ys]T
• Let Θ be an unbiased estimate based on T samples recorded at eachsensor, with samples spaced by Ts = 2π/ωs sec
• The Cramer-Rao bound (CRB) matrix C has the property that thecovariance matrix of Θ satisfies Cov(Θ)−C ≥ 0
• The CRB matrix C = J−1 where J is the Fisher information matrix(FIM) with elements
Jij =T
2ωs
∫ ωs
0 tr
∂GZ(ω)
∂ θiGZ(ω)−1∂GZ(ω)
∂ θjGZ(ω)−1
dω, i, j = 1, 2
20
CRB FOR SUB-OPTIMUM SCHEMES
• CRB result assumes data from all sensors is communicated to thefusion center (centralized processing)
– Allows optimum processing (best performance)
– Requires largest communication bandwidth
•We derived CRB for two other schemes with distributed processing:
1. Individual arrays transmit bearings to fusion center
* Ignores signal coherence at distributed arrays
* Low communication bandwidth; simple triangulation of bearings
2. Individual arrays transmit bearings and the raw data from onesensor to the fusion center
* Fusion center: time-delay estimation between array pairs
* Moderate communication bandwidth; triangulation of bearingsand time-delays
21
EXAMPLE: CRB EVALUATION
• Evaluate CRB on source localization accuracy for H = 3 arrays forvarious levels of coherence
• Individual arrays are circular with 7 elements and 4 foot radius
• Processing is narrowband in the range from 49.5 to 50.5 Hz
• SNR is 16 dB at each sensor: Gs,hh(ω)/Gw(ω) = 40 for h = 1, . . . , Hand 2π(49.5) < ω < 2π(50.5) rad/sec.
• T = 1000 samples at each sensor, with 2000 samples/sec (0.5 sec)
• Source location (xs, ys) = (200, 300) meters
• Array locations:(x1, y1) = (0, 0), (x2, y2) = (400, 400), (x3, y3) = (100, 0) for H = 3
22
H = 3 ARRAYS
−50 0 50 100 150 200 250 300 350 400 450
0
50
100
150
200
250
300
350
400
X AXIS (m)
Y A
XIS
(m
)
CRB ELLIPSES FOR COHERENCE 0, 0.5, 1.0 (JOINT PROCESSING)
TARGETARRAY
140 160 180 200 220 240 260240
260
280
300
320
340
360
X AXIS (m)Y
AX
IS (
m)
CRB ELLIPSES FOR COHERENCE 0, 0.5, 1.0 (BEARING + TD EST.)
RMS source localization error ellipses from FIM:[x y
]J
xy
= 1
23
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
30
35
40
45
COHERENCE
RM
S E
RR
OR
(m
)
CRB ON ELLIPSE "RADIUS": H = 3 ARRAYS
JOINT PROCESSING BEARING + TD EST.
24
OBSERVATIONS
• CRBs indicate potential for significantly improved localizationaccuracy:
– if distributed arrays are processed jointly, as long as signals arepartially coherent
– combination of bearing estimation & time-delay estimation performsnearly as well as processing all data
• But, look what happens with measured data for a similar scenario . . .
25
EXAMPLE
7100 7200 7300 7400 7500 7600 7700 7800 7900 8000 8100
9200
9300
9400
9500
9600
9700
9800
9900
EAST (m)
NO
RT
H (
m)
GROUND VEHICLE PATH AND ARRAY LOCATIONS
1
3
4
5
340 − 350 SEC
VEHICLE PATH 10 SEC SEGMENTARRAY 1 ARRAY 3 ARRAY 4 ARRAY 5
27
COHERENCE
0 50 100 150 200 250 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FREQUENCY (Hz)
CO
HE
RE
NC
E |γ
|
MEAN SHORT−TIME SPECTRAL COHERENCE, ARRAYS 1 & 3
0 50 100 150 200 250 300 350 400 450 5000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FREQUENCY (Hz)
CO
HE
RE
NC
E |γ
|
MEAN SHORT−TIME SPECTRAL COHERENCE, ARRAY 1, SENSORS 1 & 5
Between arrays 1 & 3 (> 200 m) Within array 1 (< 3 m)
28
CROSS-CORRELATION FUNCTIONS AT 342 SEC
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−1
−0.5
0
0.5
1CROSS−CORRELATION: ARRAYS 1 AND 3
LAG (SEC)
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−1
−0.5
0
0.5
1GENERALIZED CROSS−CORRELATION: ARRAYS 1 AND 3
LAG (SEC)−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1CROSS−CORRELATION: ARRAY1, SENSORS 1,5
LAG (SEC)
Between arrays 1 & 3 (> 200 m) Within array 1 (< 3 m)
Need to analyze time-delay estimation with partially coherent signals.
29
FUNDAMENTAL LIMITS IN TIME DELAY ESTIMATION
Threshold SNR is required to attain the CRB (Weiss & Weinstein, 1983):
Narrowband correlation Threshold bound on delay estimation
Narrow band signal: ambiguity prone delay estimationWe have extended this analysis to partially coherent signals
30
TIME DELAY ESTIMATION WITH TWO SENSORS
Source Sensor 1
z1(t) = s1(t) + w1(t)
z2(t) = s2(t - D) + w2(t)
Sensor 2
Additive noise at sensor
Cross-spectral density of
s1(t)s2(t−D)
, with partial coherence γs,12(ω):
Gs,11(ω) e+jωDγs,12(ω)Gs,11(ω)1/2Gs,22(ω)1/2
e−jωDγs,12(ω)∗Gs,11(ω)1/2Gs,22(ω)1/2 Gs,22(ω)
Gs,kk(ω) = power spectral density of sk(t)
31
EQUIVALENT MODEL: 1
s(t)
h1(t)
h2(t)"Coherent"part ofs1(t), s2(t)
Filters that model sourceaspect angle &deterministic propagationeffects
+
+
n1(t)
n2(t)
"Excess" additivenoise causingpartial coherence(due to random propagationeffects)
+
+
w1(t)
w2(t)
z1(t)
z2(t)
Additivesensornoise
z1(t) = s1(t) + w1(t) = (h1 ∗ s)(t) + n1(t) + w1(t)z2(t) = s2(t−D) + w2(t) = (h2 ∗ s)(t−D) + n2(t) + w2(t)
32
EQUIVALENT MODEL: 2
Components of the model are chosen as follows:
• s(t), n1(t), n2(t) are independent, zero mean, Gaussian randomprocesses, with power spectral densities
PSD [s(t)] = Gs(ω) = |γs,12(ω)|PSD [n1(t)] = G1(ω) = Gs,11(ω) [1− |γs,12(ω)|]PSD [n2(t)] = G2(ω) = Gs,22(ω) [1− |γs,12(ω)|]
• The filters have frequency response
H1(ω) = Gs,11(ω)1/2, H2(ω) =γs,12(ω)∗
|γs,12(ω)|Gs,22(ω)1/2
• SNR of “coherent” component at sensor k is limited:
|γs,12(ω)|1− |γs,12(ω)| +
Gs,kk(ω)
Gw(ω)
−1 ≤
|γs,12(ω)|1− |γs,12(ω)|
33
ALTERNATIVE MODEL FOR PARTIAL COHERENCE
s(t)
h1(t)
h2(t)Sound emittedby source(Gaussian)
+
+
w1(t)
w2(t)
z1(t)
z2(t)
Additivesensornoise
Random filters modeldeterministic and randompropagation effects as"multiplicative noise"
• A narrowband version of the above model has been used by manyresearchers (Paulraj & Kailath, 1988; Song & Ritcey, 1996; Wilson,1998; Gershman, Bohme, et. al., 1997; Swami & Ghogo, 2000; etc.)
• Sensor signals z1(t), z2(t) are partially coherent and non-Gaussian
• The sensor signals in our model are partially coherent and Gaussian
•Which model is “better”????
34
NEW LIMITS FOR PARTIALLY COHERENT SIGNALS
Signal bandwidth ∆ω, centered at ω0, with observation time T seconds.
Gs,11(ω0) = Gs,22(ω0) = Gs, Gw(ω0) = Gw, γs,12(ω0) = γs
SNR(γs) =
1
|γs|21 +
1
(Gs/Gw)
2
− 1
−1
SNRthresh =6
π2(∆ωT2π
) ω0
∆ω
2
φ−1
(∆ω)2
24ω20
2
SNR(γs) ≥ SNRthresh for CRB attainability
|γs|2 ≥(1 + 1
(Gs/Gw)
)2
1 + 1SNRthresh
≥ 1
1 + 1SNRthresh
asGs
Gw→∞
35
THRESHOLD COHERENCE w/ BANDWIDTH & SNR
0.94 0.95 0.96 0.97 0.98 0.99 110
12
14
16
18
20
22
24
26
28
30
SIGNAL COHERENCE MAGNITUDE, | γs |
Gs /
Gw
WIT
H P
AR
TIA
L C
OH
ER
EN
CE
REQUIRED Gs / G
w FOR REGIME OF CRB ATTAINABILITY
ω0 = 2 π 50 rad/sec
∆ ω = 2 π 10 rad/sec
T = 2 sec
| γs,min
| = 0.9314
0 10 20 30 40 50 60 70 800.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BANDWIDTH (Hz), ∆ ω / (2 π)
TH
RE
SH
OLD
CO
HE
RE
NC
E, |
γs |
ω0 = 2 π 50 rad/sec, T = 2 sec
Gs / G
w = 0 dB
Gs / G
w = 10 dB
Gs / G
w → ∞
Need |γs| ≈ 1 for narrowband, |γs| < 1 ok with wider bandwidth
Coherence-limited performance, despite SNR = Gs/Gw →∞36
REVISIT CRB EXAMPLE
• Recall CRBs on source localization accuracy for H = 3 arrays:
– Narrowband: ω0 = 2π50 rad/sec and ∆ω = 2π rad/sec
– SNR is 16 dB at each sensor: Gs/Gw = 40
– T = 0.5 sec observation time
– Threshold coherence is very close to 1 !! =⇒ CRB is optimistic.
• CRB is meaningful with wider bandwidth:
– Wideband: ω0 = 2π50 rad/sec and ∆ω = 2π20 rad/sec
– SNR is 16 dB at each sensor: Gs/Gw = 40
– T = 1.0 sec observation time
– Threshold coherence is ≈ 0.6
– Joint processing still improves performance,and bearing + time-delay estimation nearly optimum
37
H = 3 ARRAYS, ∆ω = 2π20
−50 0 50 100 150 200 250 300 350 400 450
0
50
100
150
200
250
300
350
400
X AXIS (m)
Y A
XIS
(m
)
CRB ELLIPSES FOR COHERENCE 0, 0.5, 1.0 (JOINT PROCESSING)
TARGETARRAY
195 196 197 198 199 200 201 202 203 204 205
296
297
298
299
300
301
302
303
304
X AXIS (m)
Y A
XIS
(m
)
CRB ELLIPSES FOR COHERENCE 0, 0.5, 1.0 (BEARING + TD EST.)
38
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
COHERENCE
RM
S E
RR
OR
(m
)
CRB ON ELLIPSE "RADIUS": H = 3 ARRAYS
JOINT PROCESSING BEARING + TD EST.
39
OBSERVATIONS
•We now have a test to determine when joint processing is beneficial:
– Measure SNR Gs/Gw, fractional bandwidth ∆ω/ω0, andtime-bandwidth product ∆ω · T=⇒ Compute threshold coherence value
– If measured coherence exceeds threshold, then time-delay estimationbetween distributed arrays is feasible
– If not, then incoherent triangulation of bearings is essentiallyoptimum
• Next: Examples from measured data with accurate time-delayestimation using widely-spaced sensors
– Synthetically-generated sounds from Sanders Corporation
– M1 tank with array separations of 25, 50, and 75 ft
40
DATA FROM SANDERS CORP.
0 50 100 150 200 250 300 350−200
−150
−100
−50
0
50
100
150
200RELATIVE NODE LOCATIONS FOR SOURCE 1, 2, 3, 3A, AND 7
EAST
NO
RT
H
NODE 0NODE 1NODE 2NODE 3
0 20 40 60 80 100 120 140 160 180 2000
20
40
60
80
100
120
140
160
180
200
Frequency (Hz)
Am
plitu
de (
dB)
FFT OF 10s WAVEFORM
Coherent sampling at 4 sensors, Synthetic wideband signal (FFT)spaced from 233 ft to 329 ft
41
PSD, COHERENCE, AND CROSS-CORRELATION
0 50 100 150 200−30
−25
−20
−15
−10
−5
0PSD AT SENSORS 0 AND 1 (NORMALIZED)
FREQUENCY (Hz)
PS
D
NODE 0NODE 1
0 50 100 150 2000
0.2
0.4
0.6
0.8
1COHERENCE BETWEEN SENSORS 0 AND 1
FREQUENCY (Hz)
|γs|
−4 −3 −2 −1 0 1 2 3−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
LAG (sec)
GENERALIZED CROSS−CORRELATION
PSD and coherence: transmitfrom node 2 and receive at nodes0 & 1
Generalized cross-correlation(peak at 0 is correct)
42
COMPARISON WITH THRESHOLD COHERENCE
0 10 20 30 40 50 60 70 800.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
BANDWIDTH (Hz), ∆ ω / (2 π)
TH
RE
SH
OLD
CO
HE
RE
NC
E, |
γs |
ω0 = 2 π 100 rad/sec, T = 1 sec
Gs / G
w = 0 dB
Gs / G
w = 10 dB
Gs / G
w → ∞
44
TX FROM NODE 0, RX AT NODES 1 & 3
0 50 100 150 200 250 300 350 400 450 500−80
−70
−60
−50
−40
−30
−20PSD of x1
Frequency (Hz)
Ene
rgy
of x
1 (d
B)
0 50 100 150 200 250 300 350 400 450 500−80
−70
−60
−50
−40
−30
−20PSD of x3
Frequency (Hz)
Ene
rgy
of x
3 (d
B)
Note different shapes of PSDs
45
TX FROM NODE 0, RX AT NODES 1 & 3
0 100 200 300 400 500 6000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1COHERENCE BETWEEN SENSORS 1 AND 3, Time−Aligned
Frequency (Hz)
Coh
eren
ce
−5000 −4000 −3000 −2000 −1000 0 1000 2000 3000 4000−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
INDEX
GENERALIZED CROSS−CORRELATION
MA
GN
ITU
DE
Reasonable coherence ismaintained over signal bandwidth
Generalized cross-correlationmaintains a clear peak
46
HYPERBOLIC TRIANGULATION
−50 0 50 100 150 200 250 300 350−400
−300
−200
−100
0
100
200
300
400
500Hyperbolic Triangulation of Time−Delays from Processed Data
East (ft)
Nor
th (
ft)
r12r13r23
−10 −8 −6 −4 −2 0 2 4 6 8 10−6
−4
−2
0
2
4
6Hyperbolic Triangulation of Time−Delays from Processed Data
East (ft)
Nor
th (
ft)
r12r13r23
Transmit from node 0, receive at nodes 1, 2, 3.Triangulate the three time-delay estimates.True source location is within 1 foot of (−3, 0) !
47
MOVING TRACKED VEHICLE (M1 TANK)
• Measurements at Spesutie Island, Maryland, Sept. 1999
• Three 7-sensor arrays in a line
B ← 15 m→ A← 8 m→ C
• Vehicle is moving at broadside, range 140 m
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PSD AND COHERENCE
0 50 100 150 200 250 300 350 400 450 50010
20
30
40
50
60
70
80
Frequency (Hz)
Mag
nitu
de (
dB)
Plot of PSDs of Xa, Xb, & Xc
XaXbXc
0 100 200 300 400 500 6000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1COHERENCE BETWEEN SENSORS a, b, & c, Time−Aligned; With Doppler
Frequency (Hz)
Coh
eren
ce
CxabCxacCxbc
Strong harmonics due to treadslap?
Coherence is significant over alarge bandwidth =⇒ accuratetime-delay estimation possible
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CROSS-CORRELATION
−5000 −4000 −3000 −2000 −1000 0 1000 2000 3000 4000−5
0
5
10x 10
8
INDEX
CROSS−CORRELATION for Bigger Window − 10s
XC
orr
of X
a, X
b
−5000 −4000 −3000 −2000 −1000 0 1000 2000 3000 4000−5
0
5x 10
8
INDEX
XC
orr
of X
a, X
c
−5000 −4000 −3000 −2000 −1000 0 1000 2000 3000 4000−1
0
1
2x 10
9
INDEX
XC
orr
of X
b, X
c
Correlation peaks at correct delays
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VERY WIDE SENSOR SPACING
See PPT slides from Barry Black
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CONCLUDING REMARKS
• Source localization accuracy vs. communication bandwidth (CRB)
– Joint processing improves localization compared with bearings-onlytriangulation, if sufficient coherence bandwidth
– Bearing estimation with pairwise time-delay estimation:Saves communication bandwidth with little performance loss
• Characterized fundamental limitations on time-delay estimation withpartially coherent sources
– Yields required coherence as a function of fractional bandwidth,time-bandwidth product, and SNR, which can be measuredexperimentally
– Joint processing is not beneficial for some narrowband (harmonic)sources
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• Framework can be used to evaluate the effectiveness of sensorplacement geometries (e.g., smaller spacing between arrays, individualarrays with larger aperture, distributed sensor networks with less“clustering” of sensors)
• Showed examples with measured data to illustrate accurate sourcelocalization with widely-spaced sensors
• Continuing work:– MUSIC-like subspace algorithm for partially-coherent signals
– Harmonic source model: analysis & fundamental limitations
– Incorporate time-delay estimation into localization and tracking
– Multiple sources: incorporate beamforming/source separation
– Classification of targets:quantify the amount of additional information when aspect angle,range, heading/velocity, and signal coherence are exploited
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