Digital Signal Processing Reference
In-Depth Information
of Neyman-Pearson detector under
HOG
SQL
I
assumption
11
as
[
(
1
+
ζ)/(
0
.
57
−
ζ)
0
.
37
N
C
(k)(ζ
−
1
.
57
)
]
if ζ
≥
1
.
57
+
1
/
[
0
.
37
N
C
(k)
]
,
P
FA
(k)
=
1
otherwise
,
(6.31)
where N
C
(k)
V(k)/V
C
is the number of resolution cells enclosed by the valida-
tion gate at time k of the PDAF
.
Proof
A functional approximation for the IRF in (
6.11
) was proposed in [
40
]as
q
2
λ V(k),P
D
≈ˆ
q
2
λ V(k),P
D
=
0
.
997
P
D
λ V(k)
.
(6.32)
0
.
37
P
−
1
.
57
D
1
+
The ROC curve relation for the Neyman-Pearson (NP) detector under HOG
SQL
I
is
given by [
65
]
P
1
/(
1
+
ζ)
FA
P
D
=
.
(6.33)
Using (
6.32
) and (
6.33
) in solving the constraint optimization problem defined in
(
6.30
), after some elaboration, results in the closed-form solution given in (
6.31
).
A more detailed explanation can be found in [
5
].
We refer to this closed-form solution given in (
6.31
)as
DYNAMIC-MRE-CF
in
Fig.
6.2
. This expression gives some useful insights into dynamic detection thresh-
old optimization. Consider, e.g., the plot of the optimal
P
FA
surface as a function of
ζ
and
N
C
which is illustrated in Fig.
6.5
(a) where the third data dimension (opti-
mal
P
FA
values) are represented by colors. Note that the optimization consistently
suggests increasing
P
FA
when the SNR decreases or the filter goes from its tran-
sient operation to its steady-state operation.
12
Note also that, considering a practical
operating region, where SNR values are below 20 dB, the optimization suggests
considerably higher
P
FA
values than the ones used commonly in practice (i.e., be-
tween 10
−
8
and 10
−
4
[
57
]). Similar values like 10
−
8
are only suggested when the
SNR is very high (
>
60 dB) and the gate volume is large, i.e., in the transient phase
of the filter. This clearly shows that the practically chosen
P
FA
values are far from an
optimal setting in terms of the overall radar system tracking performance. The main
11
This covers
homogeneous
and
Gaussian
background detector noise, a Swerling-I target fluctu-
ation and
square-law
detection scheme. In the radar detection theory, such assumptions are made
frequently when obtaining the ROC curves for a specific detector [
65
]. We refer to this joint as-
sumption shortly as HOG
SQ
I
.
12
It can be argued that a decreasing value of
N
C
, namely decreasing the number of resolution
cells falling inside a validation gate, suggests that the gate volume (hence the Gaussian hyper-
ellipse suggested by the filter covariance) is diminishing. This in turn suggests the convergence of
the filter to its steady-state although this may not be guaranteed to be the correct state estimate.
Conversely, by the same argument, a large value of
N
C
suggests a large gate volume, which in turn
suggests that the filter is comparatively in its transient phase.
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