Digital Signal Processing Reference
In-Depth Information
can be reduced to
N
u
2
λ(k)V(k),P
D
(k),m
k
π(m
k
)
P
FA
(k)
=
arg max
P
FA
(6.35)
m
k
=
0
f
ROC
P
FA
(k),ζ(k)
and
subject to P
D
(k)
=
0
≤
P
FA
(k)
≤
1
,
i
.
e
.,
maximization of a weighted sum of information reduction factors for each pos-
sible values of m
k
with weights
1
P
D
(k)P
G
m
k
1
μ
F
m
k
;
λ(k)V(k)
π(m
k
)
+
λ(k)V(k)
−
(6.36)
where u
2
(
·
,
·
,
·
) and μ
F
(
·;·
) are the IRF given in
(
6.20
)
and the Poisson pmf defined
in
(
6.4
),
respectively
.
Proof
The proof is skipped.
The optimization problem given in (
6.35
) is solved using line search algorithms,
e.g., the Fibonacci Search method in [
2
], which results in the scheme
DYNAMIC-
HYCA-LS
in Fig.
6.2
.
6.4 Simulations
We consider the problem of tracking a single target in clutter using a 2D radar. The
target state vector is composed of the position and velocity components in East (
ξ
)
and North (
η
) directions:
ξ(k)
η(k)
T
.
ξ(k) η(k)
x(k)
˙
(6.37)
The target performs a
coordinated turn
[
9
, pp. 467] with a constant and
known
turn
rate:
⎡
⎤
⎡
⎤
sin
(ΩT)
Ω
1
−
cos
(ΩT)
Ω
T
2
/
20
T
0
0
T
2
/
2
0
1
0
−
⎣
⎦
⎣
⎦
0
cos
(ΩT)
0
−
sin
(ΩT)
F
=
,
G
=
,
(6.38)
1
−
cos
(ΩT)
Ω
sin
(ΩT)
Ω
0
1
T
0 i
(ΩT)
0
cos
(ΩT)
where the turn rate is selected as
Ω
1s.
Since we do not estimate the turn rate in the state vector, the state dynamics is
linear. This is adopted to decouple the maneuver problem from the clutter problem
on which our focus is. The process noise
v(k)
=
1 deg/s and the sampling period is
T
=
T
is a zero-mean white
Gaussian random vector sequence with covariance matrix
Q
[
]
v
ξ
(k) v
η
(k)
I
2
×
2
q
2
=
where
q
=
0
.
1m/s
2
for all
k
and
I
2
×
2
denotes the 2
×
2 identity matrix.
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