24 August 2004 1 2004 MSS AC C-05

Acoustic Sensor Arrays with Small Baseline

Richard J. Kozick, Bucknell UniversityBrian M. Sadler, Army Research Lab

Sandra L. Collier , Army Research Lab

Acknowledgments: D.K. Wilson and T. Pham

24 August 2004

24 August 2004 2 2004 MSS AC C-05

Motivation

• Freq. in [30, 250] Hz in [1.3, 11] m

• Angle of arrival (AOA) accuracy w.r.t.

– Array aperture size– Turbulence (, )

• Small aperture:– Cheaper, disposable– Easier to deploy– More covert– Better coherence– How small can we go?

SenTech HE01 acoustic sensor [Prado2002]

24 August 2004 3 2004 MSS AC C-05

Outline• Brief review of source characteristics (ground

vehicles, aircraft)• Physics-based statistical model for turbulence

(saturation=coherence=• AOA estimation accuracy:

– Cramer-Rao bounds (CRBs)– Performance of practical algorithms (achieve CRB?)

• Questions:– What is the achievable accuracy with small-baseline

acoustic arrays?– When is the ideal plane wave model valid (i.e.,

turbulence is negligible)?• Implications for system design

24 August 2004 4 2004 MSS AC C-05

Source Characteristics• Ground vehicles (tanks, trucks), aircraft (rotary,

jet), commercial vehicles LOUD• Main contributors to source sound:

– Rotating machinery: Engines, aircraft blades– Tires and “tread slap” (spectral lines)– Vibrating surfaces

• Internal combustion engines: Sum-of-harmonics due to cylinder firing

• Turbine engines: Broadband “whine”• Key features: Spectral lines and high SNR

24 August 2004 5 2004 MSS AC C-05

Signal Model at One SensorSinusoidal signal emitted by moving source: tfSts o2cos)( refref

)()()( twtstz Signal at the sensor:

1. Propagation delay, 2. Additive noise 3. Transmission loss4. Turbulent scattering

24 August 2004 6 2004 MSS AC C-05

Transmission Loss

• Energy is diminished from Sref (at 1 m from source) to S at sensor:

– Spherical spreading– Refraction (wind, temp. gradients)– Ground interactions– Molecular absorption

• We model S as a deterministic parameter: Average signal energy

LowPassFilter

NumericalSolution[Wilson2002]

+/- 125 m from CPA

[Embleton1996]

24 August 2004 7 2004 MSS AC C-05

Sensor Signal: No Scattering

• Sensor signal with transmission loss,propagation delay, and AWG noise:

• Complex envelope at frequency fo

– Spectrum at fo shifted to 0 Hz

– FFT amplitude at fo

n timepropagatio ,2cos)(

),()(

tfSts

Tttttwtstz

o

oo

)(~exp

)(~exp)(~

twjS

twjStz o

24 August 2004 8 2004 MSS AC C-05

Sensor Signal: With Scattering• A fraction, , of the signal energy is scattered

from a pure sinusoid into a zero-mean, narrowband, Gaussian random process, :

• Saturation parameter, in [0, 1] – Varies w/ source range, frequency, and

meteorological conditions (sunny, windy)– Based on physical modeling of sound propagation

through random, inhomogeneous medium

• Illustration in the frequency domain:

)(~exp)(~exp1)(~ twjtvSjStz

)(~ tv

[Norris2001, Wilson2002a]

24 August 2004 9 2004 MSS AC C-05

AWGN, 2No

-B/2 B/20

(1- )S

Bv

S

-B/2 B/20

(1- )S

Bv

S

• Important quantities:– Saturation, (analogous to Rayleigh/Rician fading in comms.)– Processing bandwidth, B, and observation time, T– SNR = S / (2 No B)

– Scattering bandwidth, Bv < 1 Hz (correlation time ~ 1/Bv > 1 sec)

– Number of independent samples ~ T/Bv often small

• Scattering ( > 0) causes signal energy fluctuations

Weak Scattering: ~ 0 Strong Scattering: ~ 1

Freq.

Power SpectralDensity (PSD)

24 August 2004 10 2004 MSS AC C-05

Probability Distributions

• Complex amplitude has complex Gaussian PDF with non-zero mean:

• Energy has non-central -squared PDF with 2 d.o.f.

• has Rice PDF

2,-1CN~~ SSez j

-20 -15 -10 -5 0 5 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

ENERGY, 10 log10

(P) (dB)

PR

OB

AB

ILIT

Y D

EN

SIT

Y

PDF OF RECEIVED ENERGY (S=1, SNR = 30 dB)

= 0.02

0.04

0.08

0.20

0.50

1.00

2~zP

2,-1CN~~ SSez j

P

(Experimental validation in [Daigle1983, Bass1991, Norris2001])

24 August 2004 11 2004 MSS AC C-05

Saturation vs. Frequency & Range• Saturation depends on [Ostashev & Wilson]:

– Weather conditions (sunny/cloudy), but varies little with wind speed

– Source frequency and range do

21

27

27

weather

Hz]500,30[2

,

cloudymostly ,2

1042.1

sunnymostly ,2

1003.4

2exp1

(rad/sec)frequency (m), source of range

o

o

d

d

Theoretical form

Constants from numerical evaluation of particular conditions

24 August 2004 12 2004 MSS AC C-05

50 100 150 200 250 300 350 400 450 5000

0.5

1

SA

TU

RA

TIO

N,

MOSTLY SUNNY

do = 10 m50 m

100 m

200 m500 m

50 100 150 200 250 300 350 400 450 5000.2

0.4

0.6

0.8

1

FREQUENCY (Hz)

SA

TU

RA

TIO

N,

MOSTLY SUNNY

1 km

2 km

5 km

10 km

50 100 150 200 250 300 350 400 450 5000

0.5

1

SA

TU

RA

TIO

N,

MOSTLY CLOUDY

do = 10 m

50 m100 m200 m500 m

50 100 150 200 250 300 350 400 450 5000.2

0.4

0.6

0.8

1

FREQUENCY (Hz)

SA

TU

RA

TIO

N,

MOSTLY CLOUDY

1 km

2 km

5 km

10 km

Turbulence effects are small only for very short range and low frequency

Fully scattered

Saturation variesover entire range[0, 1] for typicalvalues

24 August 2004 13 2004 MSS AC C-05

Signal Model forTwo Sensors

[0,1] Coherence ,1

1

[0,1] Saturation

sin)/(phase

arcsinAOA,

exp

1

,-1CN~~

~~ 2

2

1

Γ

a

IaaΓaz

o

o

Hj

c

c

j

SSez

z

sensor spacing

AOA

Turbulence effects

Perfect plane wave: = 0 or 1 = 1

24 August 2004 14 2004 MSS AC C-05

Model for Coherence,

• Assume AOA = 0, freq. in [30, 500] Hz

• Recall saturation model:

• Coherence model [Ostashev & Wilson 2000]:

sensor spacing

do = range

od21 weather2exp1

2

2

2

2

22

3/522

3

22137.0weather

10

,1weatherexp

o

v

o

T

o

effo

c

C

T

C

c

Ld

Velocityfluctuations

Temperaturefluctuations

0 with freq., sensorspacing, and range

24 August 2004 15 2004 MSS AC C-05

2

2

2

2

22

3/522

3

22137.0weather

10

,1weatherexp

o

v

o

T

o

effo

c

C

T

C

c

Ld

Velocityfluctuations

Temperaturefluctuations

Depends on wind leveland sunny/cloudy

24 August 2004 16 2004 MSS AC C-05

50 100 150 200 250 300 350 400 450 5000.97

0.98

0.99

1

CO

HE

RE

NC

E,

MOSTLY SUNNY, MODERATE WIND

50 m100 m

200 m

500 m

50 100 150 200 250 300 350 400 450 5000.5

0.6

0.7

0.8

0.9

FREQUENCY (Hz)

CO

HE

RE

NC

E,

MOSTLY SUNNY, MODERATE WIND

do = 10 m

1 km2 km

5 km

10 km

50 100 150 200 250 300 350 400 450 5000.97

0.98

0.99

1

CO

HE

RE

NC

E,

MOSTLY CLOUDY, MODERATE WIND50 m

100 m200 m

500 m

50 100 150 200 250 300 350 400 450 500

0.7

0.8

0.9

1

FREQUENCY (Hz)

CO

HE

RE

NC

E,

MOSTLY CLOUDY, MODERATE WIND

do = 10 m

1 km2 km

5 km

10 km

> 0.99 for range < 100 mIs this “good”?

Curves moveup w/ less wind,down w/ more wind

Coherence, , versus frequencyand range forsensor spacing = 12 inches

24 August 2004 17 2004 MSS AC C-05

Impact on AOA Estimation

• How does the turbulence (, ) affect AOA estimation accuracy?

– Cramer-Rao lower bound (CRB), simulated RMSE– Achievable accuracy with small arrays?

1-2

2

2

22

SNR-1SNR2

-1-1SNR2

-1SNR2SNR1

-1SNR2

12

ˆCRBˆCRB,1ˆCRB

J

cJoLarger sensor

spacing, :DESIRABLE

BAD!

phase

arcsinAOA

exp

1

oc

ja

24 August 2004 18 2004 MSS AC C-05

Special Cases of CRB

• No scattering (ideal plane wave model):

• High SNR, with scattering:

SNR2J

SNR21 and 1 SNRfor -1

1-1

2

-1

-1-12

-1SNR2

-12

22

2

2

2

22

J

SNR-limitedperformance

Coherence-limitedperformance

If SNR = 30 dB, then < 0.9989995 limits performance!

24 August 2004 19 2004 MSS AC C-05

Phase CRB with Scattering (, )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

COHERENCE,

sqrt

[C

RB

()]

(ra

d)

SNR = 30 dB

= 00.05

0.25

0.5

0.75

0.9

0.95

1.0

Idealplanewave

Coherence loss < 1 is significantwhen saturation > 0.1

24 August 2004 20 2004 MSS AC C-05

50 100 150 200 250 300 350 400 450 5000

5

10

15

20

25

30

35

40

FREQUENCY (Hz)

sqrt

[C

RB

()]

(d

eg)

MOSTLY SUNNY, STRONG WIND

do = 10 m

100 m

500 m

1 km

2 km

5 km

10 km

CRB on AOA Estimation

SNR = 30 dB

for all ranges

Sensor spacing

= 12 in.Increasingrange (fixed SNR)

Aperture-limitedat low frequency

Ideal plane wave model is accurate for very short ranges ~ 10 m

Coherence-limited at larger ranges

24 August 2004 21 2004 MSS AC C-05

50 100 150 200 250 300 350 400 450 5000

2

4

6

8

10

12

14

16

18

20

FREQUENCY (Hz)

sqrt

[C

RB

()]

(d

eg)

MOSTLY CLOUDY, LIGHT WIND

do = 10 m

100 m

500 m1 km2 km

5 km

10 km

Cloudy and Less WindSNR = 30 dB

for all ranges

Sensor spacing

= 12 in.

Aperture-limitedat low frequency

Atmospheric conditionshave a large impact onAOA CRBs

Plane wave model isaccurate to 100 m range

24 August 2004 22 2004 MSS AC C-05

CRB Achievability

50 100 150 200 250 300 350 400 450 5000

0.5

1

FREQUENCY (Hz)

SA

TU

RA

TIO

N,

SNR = 40 dB, = 3 in, RANGE = 50 m

50 100 150 200 250 300 350 400 450 5000.9997

0.9998

0.9999

1

FREQUENCY (Hz)

CO

HE

RE

NC

E,

50 100 150 200 250 300 350 400 450 5000

5

10

15

FREQUENCY (Hz)

RM

SE

& s

qrt

[CR

B]

ON

AO

A,

(d

eg)

SNR = 40 dB, = 3 in, RANGE = 50 m

PD ESTIMATEML ESTIMATECRB

Coherence is high: > 0.999

Saturation issignificant for most offrequency range

AOA estimators break away from CRB approx.when > 0.1

Aperture-limited

Phase difference estimator:

PDo

PD

PD

c

zz

ˆarcsinˆ :AOA

ˆ :Phase 12

Turbulence prevents performance gain from larger aperture

Scenario:Small Sensor Spacing: = 3 in., SNR = 40 dB, Range = 50 m

24 August 2004 23 2004 MSS AC C-05

AOA Estimation for Harmonic Source

20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

30

35

AO

A (

deg

)

COMBINED AOA ESTIMATES AT FREQS. 50, 100, 150 Hz

RANGE (m)

RMSE, = 6 insqrt[CRB], = 6 inRMSE, = 3 insqrt[CRB], = 3 in

Equal-strength harmonicsat 50, 100, 150 Hz

SNR = 40 dB at 20 mrange, SNR ~ 1/(range)2

(simple TL)

Sensor spacing = 3 in. and 6 in.

Mostly sunny,moderate wind

One snapshot

= 6 in.

= 3 in.

RMSE

CRB

Achievable AOA accuracy ~ 10’s of degrees for this case

24 August 2004 24 2004 MSS AC C-05

Turbulence Conditions for Three-Harmonic Example

Coherence is close to 1, butstill limits performance.

Strongscattering

24 August 2004 25 2004 MSS AC C-05

Summary of AOA with Small Arrays• CRB analysis of AOA estimation

– Tradeoff: larger aperture vs. coherence loss– Ideal plane wave model is overly optimistic for

longer source ranges– Performance varies significantly with weather cond.– Important to consider turbulence effects

• AOA algorithms do not achieve the CRB in turbulence ( > 0.1) with one snapshot

• Similar results obtained for circular arrays with > 2 sensors (SNR gain) Continuing study