Task, or Objective:
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Transcript Task, or Objective:
Acoustic Source Estimation with
Doppler Processing
Richard J. Kozick
Bucknell University
Brian M. Sadler
Army Research Laboratory
1
2003 MSS BA C-8
Why Doppler?
y
Sensor 2
fd,2
Sensor 1
fd,1
Source
Path
Sensor 3
fd,3
Sensor 5
fd,5
Sensor 4
fd,4
x
2
2003 MSS BA C-8
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
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2003 MSS BA C-8
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
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2003 MSS BA C-8
Signal Observed at One Sensor
•
Sinusoidal signal emitted by moving
source: s (t ) S cos2f t
Phenomena that determine the signal at
the sensor:
ref
•
1.
2.
3.
4.
ref
o
Transmission loss
Propagation delay (and Doppler)
Additive noise (thermal, wind, interference)
Scattering by turbulence (random)
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2003 MSS BA C-8
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
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2003 MSS BA C-8
Propagation Delay & Doppler
Source Path: (xs(t), ys(t))
to
to + T
xs (t ) xs ,o xs t to
ys (t ) ys ,o y s t to
P ropagation time:
d (t )
1
(t )
(to ) vr to t to if
c
c
d to
18o
Distance : d to xs ,o x1 ys ,o y1
2
2 1/ 2
to
Radial velocity:
xs ,o x1
ys ,o y1
vr to
xs
y s xs cos to y s sin to
Sensor at (x1, y1)
d to
d to
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2003 MSS BA C-8
No Scattering
• Sensor signal with transmission
loss,propagation delay, and additive noise:
z (t ) st (t ) w(t ), to t to T
s(t ) S cos2f ot
vr to
t t
(t ) (t )
o
o
c
• Complex envelope at frequency fo
(i.e., spectrum at fo shifted to 0 Hz):
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2003 MSS BA C-8
No Scattering
• Complex envelope at frequency fo:
v t
~(t )
~
z (t ) S exp j o to exp j 2 r o f o t to w
c
~(t )
S exp j exp j 2 f t t w
d
fd
o
vr to
f o Dopplerfrequencyshift
c
• Pure sinusoid in additive noise
• Doppler frequency shift is proportional to
the source frequency, fo
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2003 MSS BA C-8
Signal Observed at One Sensor
•
Sinusoidal signal emitted by moving
source: s (t ) S cos2f t
Phenomena that determine the signal at
the sensor:
ref
•
1.
2.
3.
4.
ref
o
Transmission loss
Propagation delay (and Doppler)
Additive noise (thermal, wind, interference)
Scattering by turbulence (random)
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2003 MSS BA C-8
With Scattering
•
A fraction of the signal energy is scattered from a
pure sinusoid into a zero-mean, narrowband
random process [Wilson et. al.]
~
z (t )
•
exp j exp j 2 f d t to
WS v~ (t ) exp j exp j 2 f t t
1 WS
d
o
~ (t )
w
Saturation parameter, W in [0, 1]
– Varies w/ source range, frequency, and meteorological
conditions (sunny, cloudy)
•
Easier to see with a picture:
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2003 MSS BA C-8
Power Spectrum (PSD)
PSD
(1- W)S
Area
= WS
-B/2
0
B = Processing bandwidth
-fd = Doppler freq. shift
AWGN, 2No
-fd
B/2 Freq.
Bv = Bandwidth of
scattered component
SNR = S / (2 No B)
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2003 MSS BA C-8
Weak Scattering: W ~ 0
Strong Scattering: W ~ 1
(1- W)S
WS
WS
(1- W)S
2No
0
-B/2
-fd
B/2
-B/2
0
Bv
•
•
-fd
B/2
Bv
Study estimation of Doppler, fd, w/ respect to
–
–
–
–
Saturation, W (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 (W > 0) causes signal energy fluctuations;
may have low signal energy if (Bv T) is small
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2003 MSS BA C-8
PDF of Signal Energy at Sensor
PDF OF RECEIVED ENERGY (S=1, SNR = 30 dB)
0.45
W = 0.02
0.4
PROBABILITY DENSITY
0.35
0.04
0.3
0.25
0.08
0.2
0.20
0.15
0.50
0.1
1.00
0.05
0
-20
-15
0
-5
-10
ENERGY, 10 log10(P) (dB)
14
5
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2003 MSS BA C-8
Saturation vs. Frequency & Range
SATURATION (W ) CONTOURS, MOSTLY SUNNY
200
180
0.9
0.8
160
FREQUENCY (Hz)
0.7
0.6
140
0.5
0.4
120
0.3
100
0.2
80
0.1
60
0.05
40
W = 0.02
50
100
150
RANGE (m)
15
200
250
2003 MSS BA C-8
(1- W)S
Model for Sensor Samples
WS
2No
• Gaussian random
-B/2
0
-fd
B/2
process with nonB
zero mean
Sample at rate Fs = B, spacing Ts =1/B
Observe for T sec, so N = BT samples with
v
•
•
– Independent AWGN
– Correlated scattered signal (Ts < 1/ Bv)
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2003 MSS BA C-8
(1- W)S
Model for Sensor Samples
WS
2No
• Vector of samples
-B/2
is complex Gaussian:
-fd
0
B/2
Bv
1
Mean Covariance of
exp
j
2
f
/
B
d
scattered samples
a
AWGN
exp
j
2
(
N
1
)
f
/
B
d
z (0)
~
~
z
(
T
)
s
~z
~ CN e j 1 - W S a, WS R ~ aaH 2 N B I
v
o
~
z N 1Ts
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2003 MSS BA C-8
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:
– W = 1 (fully saturated, random signal)
– W = 0 (no scattering, deterministic signal)
• We evaluate CRB for 0 < W < 1 with
discrete-time (sampled) model
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2003 MSS BA C-8
No Scattering: W = 0
Fully Saturated: W = 1
S
S
S Gv~ f f d
2No
0
-B/2
-fd
B/2
1 f
Gv~ f G1
Bv Bv
3 No
ˆ
CRB f d
2 2T 3 S
Schultheiss &
Weinstein [JASA, 1979]
0
-B/2
Bv
ˆ
CRB f d
T
-fd
B/2
Bv
d
logG1 ( x) dx
dx
0
2
1
High SNR = S/(2 No B), Large (Bv T)
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2003 MSS BA C-8
Example 1: Vary Bv & W
(1- W)S
WS
2No
-B/2
• SNR = 28.5 dB
• B = 7 Hz
• T = 1 sec
• Bv from 0.1 Hz to 2.0 Hz
• True fd = -0.2 Hz
20
0
-fd
B/2
Bv
2003 MSS BA C-8
CRB on fd
0.3
SAMPLED
SCHULTHEISS-WEINSTEIN
0.25
2.0
sqrt(CRB) (Hz)
0.2
1.5
0.15
1.0
(Bv T)
is not
large
0.5
0.1
Bv = 0.1
0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
SATURATION W
21
0.7
0.8
0.9
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2003 MSS BA C-8
CRB on fd
SAMPLED
SCHULTHEISS-WEINSTEIN
-1
sqrt(CRB) (Hz)
10
-2
10
0
0.1
0.2
0.3
0.4
0.5
0.6
SATURATION W
22
0.7
0.8
0.9
1
2003 MSS BA C-8
Example 2: Vary T & W
(1- W)S
WS
2No
-B/2
• SNR = 28.5 dB
• B = 7 Hz
• Bv = 1 Hz
• T from 0.5 sec to 10 sec
• True fd = -0.2 Hz
23
0
-fd
B/2
Bv
2003 MSS BA C-8
CRB on fd
0.4
SAMPLED
SCHULTHEISS-WEINSTEIN
0.35
sqrt(CRB) (Hz)
0.3
0.25
T = 0.5
0.2
(Bv T)
is large
0.15
1.0
1.5
0.1
2.0
0.05
5.0
0
0.1
0.2
0.3
0.4
0.5
0.6
SATURATION W
24
0.7
0.8
10
0.9
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2003 MSS BA C-8
CRB on fd
SAMPLED
SCHULTHEISS-WEINSTEIN
-1
sqrt(CRB) (Hz)
10
-2
10
-3
10
-4
10
0
0.1
0.2
0.3
0.4
0.5
0.6
SATURATION W
25
0.7
0.8
0.9
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2003 MSS BA C-8
Example 3: Vary SNR & W
(1- W)S
WS
• T = 1 sec
-B/2
• B = 7 Hz
• Bv = 1 Hz
• SNR from -1.5 dB to 38.5 dB
• True fd = -0.2 Hz
26
2No
0
-fd
B/2
Bv
2003 MSS BA C-8
CRB on fd
0.5
SAMPLED
SCHULTHEISS-WEINSTEIN
0.45
0.4
sqrt(CRB) (Hz)
0.35
SNR
floor
0.3
SNR = -1.5 dB
0.25
0.2
18.5 dB
8.5 dB
0.15
38.5 dB
0.1
0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
SATURATION W
27
0.7
0.8
0.9
1
2003 MSS BA C-8
CRB on fd
SAMPLED
SCHULTHEISS-WEINSTEIN
(Bv T)
is not
large
-1
sqrt(CRB) (Hz)
10
No SNR
floor
-2
10
-3
10
0
0.1
0.2
0.3
0.4
0.5
0.6
SATURATION W
28
0.7
0.8
0.9
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2003 MSS BA C-8
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
T=1 s, B=10 Hz, Bv=0.1 Hz
SATURATION (W ) CONTOURS, MOSTLY SUNNY
200
180
•
0.8
160
0.7
FREQUENCY (Hz)
•
0.9
0.6
140
0.5
0.4
120
0.3
100
0.2
80
0.1
60
0.05
40
•
•
W = 0.02
50
29
100
150
RANGE (m)
200
250
2003 MSS BA C-8
W
5m
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
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2003 MSS BA C-8
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
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2003 MSS BA C-8
Differential Doppler Estimation
GROUND VEHICLE PATH AND ARRAY LOCATIONS
3
9900
9800
1
NORTH (m)
9700
9600
9500
9400
VEHICLE PATH
10 SEC SEGMENT
ARRAY 1
ARRAY 3
ARRAY 4
ARRAY 5
4
9300
9200
5
7100 7200 7300 7400 7500 7600 7700 7800 7900 8000 8100
EAST (m)
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2003 MSS BA C-8
Differential Doppler Estimation
DIFFERENTIAL DOPPLER FREQUENCY SHIFT FOR ARRAYS 1 AND 3
-0.9
GPS GROUND TRUTH
ESTIMATES
MEAN ESTIMATE
-0.95
FREQUENCY SHIFT (Hz)
SQRT(CRB) = 0.1 Hz
-1
-1.05
-1.1
-1.15
-1.2
-1.25
340
341
342
343
344
345
TIME (sec)
33
346
347
348
349
2003 MSS BA C-8
GROUND VEHICLE PATH AND ARRAY LOCATIONS
3
9900
9800
1
DIFFERENTIAL DOPPLER FREQUENCY SHIFT FOR ARRAYS 1 AND 3
-0.9
9600
9500
9400
VEHICLE PATH
10 SEC SEGMENT
ARRAY 1
ARRAY 3
ARRAY 4
ARRAY 5
SQRT(CRB) = 0.1 Hz
9300
9200
GPS GROUND TRUTH
ESTIMATES
MEAN ESTIMATE
-0.95
4
5
FREQUENCY SHIFT (Hz)
NORTH (m)
9700
-1
-1.05
7100 7200 7300 7400 7500 7600 7700 7800 7900 8000 8100
-1.1
EAST (m)
-1.15
-1.2
-1.25
340
34
341
342
343
344
345
TIME (sec)
346
347
348
349
2003 MSS BA C-8
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?
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2003 MSS BA C-8
Bearings & Doppler
y
Sensor 2
fd,2
Sensor 1
fd,1
Source
Path
Sensor 3
fd,3
Sensor 5
fd,5
Sensor 4
fd,4
x
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2003 MSS BA C-8