Digital Signal Processing Reference
In-Depth Information
4. Compute the weights by
w
(
m
)
(
n
)
/ w
(
m
)
(
n
1)
f
( y(
n
)
j
x
(
m
)
n
(0 :
n
), y(1 :
n
1))
where
ð
f
( y(
n
)
j
x
(
m
)
(
m
)
(
m
)
l
f
( y(
n
)
j
x
n
(0 :
n
), y(1 :
n
1))
¼
n
(
n
), x
(
n
))
(
m
)
l
(
m
)
l
(
m
)
n
f
(x
(
n
)
j
x
(0 :
n
), y(1 :
n
1))
d
x
(
n
)
:
(
m
)
l
(
n
) inside the integral are Gaussian densities,
the integral can be solved analytically. We obtain
Again, if the two densities of x
(
m
)
y
(
m
)
n
(
m
)
y
f
( y(
n
)
j
x
(0 :
n
), y(1 :
n
1))
¼N
(
m
(
n
),
S
(
n
))
where
(
m
)
y
(
m
)
n
(
n
))
þ
A
(
m
)
(
m
)
l
2
(
n
) x
m
(
n
)
¼ g
2
(x
(
n
)
(
m
)
x
l
(
m
)
y
(
m
)
2
(
n
) C
(
m
)
`
2
S
(
n
)
¼
A
(
n
)A
(
n
)
þ
C
v
2
:
5. Finally we carry out the measurement update of the linear states
(
m
)
l
(
m
)
l
(
m
)
2
(
m
)
l
(
m
)
(
n
)( y(
n
)
g
2
(x
(
m
)
n
x
(
n
)
¼
x
(
n
)
þ
K
(
n
))
A
(
n
) x
(
n
))
(
m
)
x
l
(
m
)
x
l
(
m
)
(
n
)
¼
C
(
n
) C
(
m
)
`
2
(
m
)
2
(
m
)
2
(
n
)
þ
C
v
2
)
1
K
(
n
)A
(
n
)(A
(
n
)A
(
m
)
x
l
(
m
)
x
l
C
2
(
n
)) C
(
m
)
(
n
)A
(
m
)
(
n
)
¼
(I
K
(
n
)
:
In general, the Rao-Blackwellization amounts to the use of a bank of KFs, one for
each particle stream. So, with
M
particle streams, we will have
M
KFs.
B
EXAMPLE 5.11
The problem of target tracking using biased measurements presented in
Example 5.9 can be addressed using a Rao-Blackwellized scheme. Here we
outline the steps of the scheme. First, we assume that at time instant
n
2
1we
have the random measure
x
(
n
1)
¼
{x
(
m
)
(
n
1),
w
(
m
)
(
n
1)}
m¼
1
and the esti-
mates of the bias in each particle stream
u
(
m
)
(
n
1) and the covariances of the
(
m
)
u
C
estimates
(
n
1).
At time instant
n
, we first generate the particles of the dynamic variables
(
m
)
(
n
)
N
(Ax
(
m
)
(
n
1), BC
v
1
B
`
)
:
x
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