Digital Signal Processing Reference
In-Depth Information
f
k
x
j
1
.
x
j
1
,ω
j
k
=
(10.3)
k
−
k
−
The prior probability density of the current states is expressed as
p(
x
j
x
j
k
k
|
1
,
z
k
)
.
−
10.3.3 Weight Computation and Normalization
The current output,
z
k
, estimated by particle
j
at time
k
, is calculated as follows:
h
k
x
j
k
.
z
j
k
=
(10.4)
The observation likelihood for each particle is expressed as
p(
z
j
x
j
k
)
.Afterter-
mination of the measurement update, the weight of particle
j
at time
k
will be
assigned recursively as follows [
9
]:
k
|
p(
x
k
|
x
k
−
1
)p(
z
k
|
x
k
)
w
k
=
w
k
−
1
.
(10.5)
q(
x
j
x
j
k
k
|
1
,
z
k
)
−
x
k
−
1
,
z
k
)
is referred to as the importance, or proposal, density.
The choice of the importance density is one of the most critical issues in the design
of GPF. The optimal importance density, such as Gaussian distribution, minimizes
the variance of weights [
20
]. Also, the weight of particle
j
at time
k
is normalized
as follows:
The PDF
q(
x
k
|
w
k
j
=
1
w
j
w
k
=
¯
.
(10.6)
k
10.3.4 Resampling
After a few iterations, most particles will have negligible weights. Computational
effort for updating particles with small weight is bulky. This problem is called the
degeneracy phenomenon. To avoid the degeneration of particles, a resampling pro-
cedure is necessary. The degeneration can be measured in terms of the effective
sample size which can be estimated via [
21
]:
1
j
=
1
(
¯
N
eff
=
w
k
)
2
.
(10.7)
≤
N
eff
≤
N
with the following two extreme
cases: (i) if the weights are distributed uniformly (i.e.,
It is straightforward to verify that 1
w
j
¯
k
=
1
/N
for
j
=
1
,...,N
)
Search WWH ::
Custom Search