Acoustic Source Localization and Beamforming: Theory and Practice

Acoustic Source Localization
and Beamforming: Beamforming:
Theory and
Practice
150
210
120
240
Chen, Yao and Hudson
Presented by Ran Kaftory
90
270
1
0.5
2
1.5
180 0
60
300
30
330

“ML ML Source-Localization Source Localization and its
lower bound” bound
Definitions
Cramer-Rao Cramer Rao bound on Source-Localization
Source Localization
ML Source-Localization
Source Localization
Example
Demonstration

Definitions – planar sensor array
Array center:
Diffused noise:
Near-Field
Far-Field
Near Field
Far Field
r
c
w p
~
1
R
N
p
R
1
r
( 0,
p
2
)
r 1 =[x 1 ,y 1 ] T
r 4 =[x 4 ,y 4 ] T
r c =[x c ,y c ] T
r R =[x R ,y R ] T
s
r 2 =[x 2 ,y 2 ] T
r s =[x s ,y s ] T
r 3 =[x 3 ,y 3 ] T

t
cp
( x
Definisions – collected data
Far-field Far field case
Relative time delay:
c
x
p
) sin
s
v
( y
Collected data:
x p c cp p
c
y
p
) cos
( n)
s ( n t ) w ( n)
Frequency domain:
X c
( k)
D(
k)
S ( k)
( k)
D(
k)
e
2 kt
n
j2
kt
n
j c1
cR
,.., e
T
s
Near-field Near field case
Relative time delay:
t
cp
Collected data:
x p
p c cp p
r
s
v
( n)
a s ( n t ) w ( n)
Frequency domain:
X c
D(
k)
( k)
D(
k)
S ( k)
( k)
a
1
e
2 kt
n
r
p
R
j2
kt
n
j 1
R
,.., a
e
T

Cramer-rao Cramer rao bound
“ Provides a lower bound on the estimator variance of an
unbiased estimator” estimator
Frequency domain in matrix form:
X c
( k)
D(
k)
S ( k)
( k)
while
Cramer-rao Cramer rao bound:
X
G(
T
H 2
[ t cp,
v,
sc
,...] R E[
] L I
s
while H
while
G G G
F 2 Re[ H R H ] H , , ,...
s v
1
F
s
)
c

Examples of CRB for DOA
estimation (far-field (far field case)
Known speed, unknown DOA:
Geometry factor:
SNR factor:
s
p
R
L
1
1
[( x
c
2 N
2
v
2
k
x
/ 2
1
p
( 2
) cos
k
S
c
s
( k)
( y
c
/ N )
y
2
p
) sin
s
2
]

Examples of CRB for DOA
estimation (far-field (far field case)
Unknown speed, unknown DOA:
s
Penalty term due to the unknown speed:
z
v
p
R
1
1
t
2
cp
p
R
1
[( x
c
x
p
(
1
) cos
zv
s
)
( y
c
y
p
) sin
s
] t
cp
2

Examples of CRB for source
localization (near-field (near field case)
RMS error in source localization:
2
2
x
2
y
1
1
[ Fr
, s ] 11 [ F
s 0
r , s0
d s
Known speed, unknown location:
A
Frs as in the near field case
p
R
1
a
2
p
A
u
p
u
T
p
]
22

Examples of CRB for source
localization (near-field (near field case)
Unknown speed, unknown location:
when
T
Z v ( 1/
t Aat
)
p
s
p
F s ,
r
v
t
T
A
A
a
U
1
T
t
UA
T
A
1
[ r , v ] 11:
DD ( A Zv
F s
UA
s
a
tt
T
p
A
a
U
2 2
u ( r r ) / r r A
diag([
a ,..., a ])
T
a
a
)
t
1
1 R

Circular-array Circular array (far-field (far field case)
r [ sin , cos ]
p
Geometry factor:
p
p
T
2
R /
Penalty term due to the unknown speed:
SNR factor:
2
2
2
2 N
The CRB for DOA estimation is
independent of the speed and
source direction !
L
v
k
/ 2
1
( 2
k
S
c
p
zv
( k)
/ N )
s
1
2
0

L(
ML DOA Estimation
Frequency domain in matrix form:
)
X
G(
- Is zero-mean zero mean complex white Gaussian
log Likelihood(
)
ˆ L
log( f ( X
ML estimator: max ( )
max ( L
)
min
Which is equivalent to min f ( k)
s , S
c
N
k
/ 2
1
))
( ) k X
f c
( k)
X ( k)
D(
k)
S ( k)
X
G(
)
( ) c( ) k S k D
for all k bins
2
)
2
const

while
ML DOA Estimation
min ( k)
s
f ( k)
min X ( k)
D(
k)
Sc
, Sc ( k )
s , Sc
( k )
ˆ k
must satisfy 0
( k)
D ( k)
X ( )
H
1 H
D ( k)
[ D(
k)
D(
k)]
D(
k)
s
argmin
argmax
s
S
s
f H
c
( k)
( k)
N
k
N
k
/ 2
1
/ 2
1
[ I
[ D(
k)
D(
k)
D
2
D
( k)]
S c
is the pseudoinverse
( k)]
X(
k)
X ( k)
2
2

Circular array – ML algorithm
implementation (far-field (far field case)
Definition: Normalized root weighted mean
squared source frequency (nrwms ( nrwms): ):
k
nrwms
2
N
N / 2
k 1
N / 2
Definition: Effective beamwidth: beamwidth
BW
k
k
1
2
S
S
c
v
k
c
( k )
( k )
nrwms
2
2
p

Circular array – ML algorithm
implementation (far-field (far field case)
i
Define: i
Calculate:
iˆ
argmax
i
s
s
N
k
/ 2
1
360
BW
[ D(
k)
D
( k)]
X ( k)
2
BW
BW
BW
BW
North
Find ˆ
s by quadratic polynomial interpolation
iˆ
î 1
1
between and
s
s
î
s
BW
BW
BW
BW

3 arrays with 4 sensors
Locating the source through
cross bearing of 3 DOA’s DOA
Outdoor experiment:
White Gaussian signal from a
stationary load speaker
RMS error of 32 cm and 97 cm
Authors example

3 arrays with 4 sensors
Locating the source through
cross bearing of 3 DOA’s DOA
Outdoor experiment:
Vehicle signal from a moving
load speaker
Authors example

Demonstration
8 microphones circle array:
p
0.
25
m.
amplifiers
Sampling rate = 5 khz
8 channel
DAQ Board
Computer (Matlab)

Source signal:
Amplitude
Demonstration
3
2.5
2
1.5
1
0.5
Signal at Reference Microphone
0
0 500 1000 1500 2000 2500
Frequency (Hz)

Demonstration
Theoretical beam width:
BW
k
nrwms
345 /
0.
25
0.
1
5000
0.
09
0.
82

Simulated beam width:
Demonstration
150
210
ML Beamformer Beamshape
120
240
90
270
200
100
300
180 0
60
300
30
330

RMS error in CRB:
3
2.5
2
RMS (degrees) 1.5
1
0.5
0
Demonstration
1
SNR
array
R
SNR
CRB for DOA estimation vs. Simulation
SNR
2
3
BW
SNR
R
array
N
k
CRB
/ 2
1
S
c
Experimented
( k)
CRB
2
/
L
Experimented
2

My Experiment

RMS Error
(degrees)
Experiment Results
120
100
80
60
40
20
0
1
0.3
0
SNR
3.7
2
RMS Error in Expermint
0.9
0.7
112
3
2.7
2.3
106
CRB
Simulated
Experimented
CRB
Simulated
Experimented

Summary
Theoretical CRB for source localization and DOA estimation was
shown.
ML beamforming shown to be effective especially for wide-band wide band
signals.
Uniformly spaced circular array is preferred in most scenarios
because:
1. DOA variance does not degrade when speed of
propagation is unknown.
2. DOA variance does not depend on the source
DOA .
Simulation shows that ML beamforming is close to CRB on DOA
estimation.
Experiment shows that ML beamforming does not perform well in
negative SNR