Digital Signal Processing Reference
In-Depth Information
3.4.1 Maximizing the Mahalanobis Distance
To decide whether a target is present or not in the range cell under test, the standard
procedure is to construct a decision problem that chooses between two possible hy-
potheses: the null hypothesis
H
0
(target-free hypothesis) or the alternate hypothesis
H
1
(target-present hypothesis). The problem can be expressed as
H
0
:
y
=
e
,
(3.18)
H
1
:
y
=
x
+
e
,
and the measurement
y
is distributed as
)
.
To distinguish between these two distributions, one standard measure is the squared
Mahalanobis distance, defined as
C
N
LN
(
0
,
I
N
⊗
)
or
C
N
LN
(
x
,
I
N
⊗
d
2
x
H
H
(
I
N
⊗
)
−
1
x
=
N
−
1
x
H
(n)
H
A
H
−
1
A
(n)
x
.
=
(3.19)
n
=
0
Then, to maximize the detection performance, we can formulate an optimization
problem as
N
−
1
x
H
(n)
H
A
H
−
1
A
(n)
x
subject to
a
(
1
)
a
H
a
=
arg max
a
=
1
.
(3.20)
∈C
L
n
=
0
After some algebraic manipulations (see [
48
, App. C]) we can rewrite this problem
as
a
H
N
−
1
−
1
a
(n)
xx
H
(n)
H
T
a
(
1
)
a
H
a
=
arg max
a
subject to
=
1
.
L
∈C
n
=
0
(3.21)
Hence, the optimization problem reduces to a simple eigenvalue-eigenvector prob-
lem, and the solution of (
3.21
) is the eigenvector corresponding to the largest eigen-
value of
N
−
1
−
1
(n)
xx
H
(n)
H
T
.
n
=
0
However in practical scenarios, to obtain
a
(
1
)
by solving (
3.21
), we need to estimate
the values of
v
,
x
, and
.
3.4.2 Minimizing the Weighted Trace of CRB Matrix
To characterize the accuracy of the estimation process, we compute the CRBs on
the target velocity,
v
, and scattering-parameters,
x
. For mathematical simplicity, we
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