andimperfectspatialcoherence ...kozick/acoustic/dls01_vug.pdftime spectral coherence, arrays 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

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

|


0 50 100 150 200 250 3000

0.1

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0.3

0.4

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0.9

1

FREQUENCY (Hz)

CO

HE

RE

NC

E |γ

|


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

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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

|


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


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)


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

48

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

49

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

50

VERY WIDE SENSOR SPACING

See PPT slides from Barry Black

51

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

52

• 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

53

andimperfectspatialcoherence ...kozick/acoustic/dls01_vug.pdftime spectral coherence, arrays 1 &...

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