2003 MSS BA C-81

Acoustic Source Estimation with Acoustic Source Estimation with Doppler ProcessingDoppler Processing

Richard J. KozickRichard J. Kozick

Bucknell UniversityBucknell University

Brian M. SadlerBrian M. Sadler

Army Research LaboratoryArmy Research Laboratory

2003 MSS BA C-82

Why Doppler?

x

y

SourcePath

Sensor 1fd,1

Sensor 5fd,5

Sensor 2fd,2

Sensor 3fd,3

Sensor 4fd,4

2003 MSS BA C-83

Outline

• Model for sensor data– Sum-of-harmonics source– Propagation with atmospheric scattering

• Frequency estimation w/ scattered signals– Cramer-Rao bounds, differential Doppler– Varies with range, frequency, weather cond.– Examples, measured data processing

• Extension: Localization accuracy with Doppler

2003 MSS BA C-84

Source Signal Models

• Sum of harmonics– Internal combustion engines (cylinder firing)– Tread slap, tire rotation– Helicopter blade rotation

• Broadband spectra from turbine engines– Time-delay estimation may be feasible

• Focus on harmonic spectra in this talk– Differential Doppler estimation localization

2003 MSS BA C-85

Signal Observed at One Sensor

• Sinusoidal signal emitted by moving source:

• Phenomena that determine the signal at the sensor:

1. Transmission loss

2. Propagation delay (and Doppler)

3. Additive noise (thermal, wind, interference)

4. Scattering by turbulence (random)

tfSts o2cos)( refref

2003 MSS BA C-86

Transmission Loss

• Energy is diminished from Sref (at 1 m from source) to value S at sensor:– Spherical spreading– Refraction (wind & temperature gradients)– Ground interactions– Molecular absorption

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

2003 MSS BA C-87

Propagation Delay & Doppler ososs

ososs

ttyyty

ttxxtx

,

,

)(

)(

oooro tttv

ct

c

tdt 18if

1)(

)()(

:n timePropagatio

2/121,

21, :Distance yyxxtd ososo

ososso

oss

o

osor tytxy

td

yyx

td

xxtv sincos

: velocityRadial

1,1,

Source Path: (xs(t), ys(t))

Sensor at (x1, y1)

toto + T

ot

otd

2003 MSS BA C-88

No Scattering

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

• Complex envelope at frequency fo

(i.e., spectrum at fo shifted to 0 Hz):

oor

o

o

oo

ttc

tvtt

tfSts

Tttttwttstz

)()(

2cos)(

),()()(

2003 MSS BA C-89

No Scattering

• Complex envelope at frequency fo:

• Pure sinusoid in additive noise

• Doppler frequency shift is proportional to the source frequency, fo

shiftfrequency Doppler

)(~2expexp

)(~2expexp)(~

oor

d

od

ooor

oo

fc

tvf

twttfjjS

twttfc

tvjtjStz

2003 MSS BA C-810

Signal Observed at One Sensor

• Sinusoidal signal emitted by moving source:

• Phenomena that determine the signal at the sensor:

1. Transmission loss

2. Propagation delay (and Doppler)

3. Additive noise (thermal, wind, interference)

4. Scattering by turbulence (random)

tfSts o2cos)( refref

2003 MSS BA C-811

With Scattering

• A fraction of the signal energy is scattered from a pure sinusoid into a zero-mean, narrowband random process [Wilson et. al.]

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

conditions (sunny, cloudy)

• Easier to see with a picture:

)(~2expexp)(~

2expexp1)(~

twttfjjtvS

ttfjjStz

od

od

2003 MSS BA C-812

Power Spectrum (PSD)

Freq.

PSD

AWGN, 2No

-B/2 B/20

(1- )S

-fd

Bv = Bandwidth of scattered component

Area= S

B = Processing bandwidth

-fd = Doppler freq. shift

SNR = S / (2 No B)

2003 MSS BA C-813

2No

-B/2 B/20

(1- )S

-fd

Bv

S

-B/2 B/20

(1- )S

-fd

Bv

S

Strong Scattering: ~ 1

• Study estimation of Doppler, fd, w/ respect to– Saturation, (analogous to Rayleigh/Rician fading)– Processing bandwidth, B, and observation time, T– SNR = S / (2 No B)– Scattering bandwidth, Bv (correlation time ~ 1/Bv)

• Scattering ( > 0) causes signal energy fluctuations;may have low signal energy if (Bv T) is small

Weak Scattering: ~ 0

2003 MSS BA C-814

PDF of Signal Energy at Sensor

-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

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

= 0.02

0.04

0.08

0.20

0.50

1.00

2003 MSS BA C-815

Saturation vs. Frequency & Range

50 100 150 200 250

40

60

80

100

120

140

160

180

200F

RE

QU

EN

CY

(H

z)

RANGE (m)

SATURATION () CONTOURS, MOSTLY SUNNY

= 0.02

0.05

0.1

0.2

0.3

0.9

0.4

0.5

0.6

0.7

0.8

2003 MSS BA C-816

Model for Sensor Samples

• Gaussian randomprocess with non-zero mean

• Sample at rate Fs = B, spacing Ts =1/B

• Observe for T sec, so N = BT samples with– Independent AWGN

– Correlated scattered signal (Ts < 1/ Bv)

2No

-B/2 B/20

(1- )S

-fd

Bv

S

2003 MSS BA C-817

Model for Sensor Samples

• Vector of samplesis complex Gaussian:

2No

-B/2 B/20

(1- )S

-fd

Bv

S

BfNj

Bfj

d

d

/)1(2exp

/2exp

1

a

IaaRaz BNSSe

TNz

Tz

z

oH

vj

s

s 2,-1CN~

1~

)(~)0(~

~~

Mean Covariance ofscattered samples

AWGN

2003 MSS BA C-818

Cramer-Rao Bound (CRB)

• CRB is a lower bound on the variance of unbiased estimates of fd

• Schultheiss & Weinstein [JASA, 1979] provided CRBs for special cases:– = 1 (fully saturated, random signal)– = 0 (no scattering, deterministic signal)

• We evaluate CRB for 0 < < 1 with discrete-time (sampled) model

2003 MSS BA C-819

2No

-B/2 B/20

S

-fd -B/2 B/20 -fd

Bv

S

Fully Saturated: = 1No Scattering: = 0 dv ffGS ~

S

N

Tf od 322

3ˆCRB

12

0

1 )(logˆCRB

dxxGdx

d

T

Bf vd

vvv B

fG

BfG 1~

1

High SNR = S/(2 No B), Large (Bv T)

Schultheiss & Weinstein [JASA, 1979]

2003 MSS BA C-820

Example 1: Vary Bv &

• SNR = 28.5 dB

• B = 7 Hz

• T = 1 sec

• Bv from 0.1 Hz to 2.0 Hz

• True fd = -0.2 Hz

2No

-B/2 B/20

(1- )S

-fd

Bv

S

2003 MSS BA C-821

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.05

0.1

0.15

0.2

0.25

0.3

SATURATION

sqrt

(CR

B)

(Hz)

CRB on fd

Bv = 0.1

0.5

1.0

1.5

2.0

SAMPLEDSCHULTHEISS-WEINSTEIN

(Bv T)is notlarge

2003 MSS BA C-822

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10-2

10-1

SATURATION

sqrt

(CR

B)

(Hz)

CRB on fd

SAMPLEDSCHULTHEISS-WEINSTEIN

2003 MSS BA C-823

Example 2: Vary T &

• SNR = 28.5 dB

• B = 7 Hz

• Bv = 1 Hz

• T from 0.5 sec to 10 sec

• True fd = -0.2 Hz

2No

-B/2 B/20

(1- )S

-fd

Bv

S

2003 MSS BA C-824

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

SATURATION

sqrt

(CR

B)

(Hz)

CRB on fd

T = 0.5

1.0

1.5

2.0

5.010

SAMPLEDSCHULTHEISS-WEINSTEIN

(Bv T)is large

2003 MSS BA C-825

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-4

10-3

10-2

10-1

SATURATION

sqrt

(CR

B)

(Hz)

CRB on fd

SAMPLEDSCHULTHEISS-WEINSTEIN

2003 MSS BA C-826

Example 3: Vary SNR &

• T = 1 sec

• B = 7 Hz

• Bv = 1 Hz

• SNR from -1.5 dB to 38.5 dB

• True fd = -0.2 Hz

2No

-B/2 B/20

(1- )S

-fd

Bv

S

2003 MSS BA C-827

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

SATURATION

sqrt

(CR

B)

(Hz)

CRB on fd

SNR = -1.5 dB

8.5 dB 18.5 dB

38.5 dB

SAMPLEDSCHULTHEISS-WEINSTEIN

SNRfloor

2003 MSS BA C-828

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-3

10-2

10-1

SATURATION

sqrt

(CR

B)

(Hz)

CRB on fd

SAMPLEDSCHULTHEISS-WEINSTEIN (Bv T)

is notlarge

No SNRfloor

2003 MSS BA C-829

CRBs with Saturation Model

• Value of harmonics for Doppler est.?

• Fundamental frequency = 15 Hz

• Process harmonics 3, 6, 9, 12 45, 90, 135, and 180 Hz

• Range: 5 to 320 m

• SNR ~ (Range)-2

50 100 150 200 250

40

60

80

100

120

140

160

180

200

FR

EQ

UE

NC

Y (

Hz)

RANGE (m)

SATURATION () CONTOURS, MOSTLY SUNNY

= 0.02

0.05

0.1

0.2

0.3

0.9

0.4

0.5

0.6

0.7

0.8

T=1 s, B=10 Hz, Bv=0.1 Hz

2003 MSS BA C-830

5 m 10 m 20 m 40 m 80 m 160 m

320 m

45 Hz

.004 .008 .02 .03 .06 .12 .23

90 Hz

.02 .03 .06 .12 .23 .41 .65

135 Hz

.04 .07 .13 .25 .44 .69 .90

180 Hz

.06 .12 .23 .41 .65 .88 .98

2003 MSS BA C-831

CRB 5 m 10 m 20 m 40 m 80 m 160 m

320 m

45 Hz

.006 .009 .01 .02 .04 .07 .13

90 Hz

.01 .01 .02 .03 .05 .09 .19

135 Hz

.01 .02 .03 .04 .05 .09 .20

180 Hz

.02 .02 .03 .04 .05 .09 .21

2003 MSS BA C-832

Differential Doppler Estimation

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

VEHICLE PATH10 SEC SEGMENTARRAY 1ARRAY 3ARRAY 4ARRAY 5

1

3

4

5

2003 MSS BA C-833

Differential Doppler Estimation

340 341 342 343 344 345 346 347 348 349-1.25

-1.2

-1.15

-1.1

-1.05

-1

-0.95

-0.9

TIME (sec)

FR

EQ

UE

NC

Y S

HIF

T (

Hz)

DIFFERENTIAL DOPPLER FREQUENCY SHIFT FOR ARRAYS 1 AND 3

SQRT(CRB) = 0.1 Hz

GPS GROUND TRUTHESTIMATESMEAN ESTIMATE

2003 MSS BA C-834

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

VEHICLE PATH10 SEC SEGMENTARRAY 1ARRAY 3ARRAY 4ARRAY 5

1

3

4

5

340 341 342 343 344 345 346 347 348 349-1.25

-1.2

-1.15

-1.1

-1.05

-1

-0.95

-0.9

TIME (sec)

FR

EQ

UE

NC

Y S

HIF

T (

Hz)

DIFFERENTIAL DOPPLER FREQUENCY SHIFT FOR ARRAYS 1 AND 3

SQRT(CRB) = 0.1 Hz

GPS GROUND TRUTHESTIMATESMEAN ESTIMATE

2003 MSS BA C-835

Continuing Work

• ACIDS database, exploiting >1 harmonic

• Extend CRBs from differential Doppler to source localization with >= 5 sensors

• Use CRBs to test the value of using differential Doppler with bearings for localization– Include coherence losses due to scattering in the

bearing results– Frequency estimates may already be available at the

nodes

• Use Doppler to help data association?

2003 MSS BA C-836

Bearings & Doppler

x

y

SourcePath

Sensor 1fd,1

Sensor 5fd,5

Sensor 2fd,2

Sensor 3fd,3

Sensor 4fd,4