Image Processing Reference
In-Depth Information
with:
diag
cos
θ
0
,
cos
θ
1
,...,
cos
θ
k
Δ
θ
[6.33]
diag
r
−
1
y
(
k
)
(0)
,r
−
1
y
(1)
,...,r
−
1
y
Δ
r
This factorization is valid so long as each term
r
(
k
) is different from zero, which
is hardly restrictive. Let
O
be the factored observability matrix:
⎛
⎞
r
x
(0)
r
y
(0)
0
0
⎝
⎠
.
.
αr
x
(1)
αr
y
(1)
O
=
r
x
(
k
)
T
x
r
y
(
k
)
T
y
k
α
r
x
(
k
)
V
x
k
αr
y
(
k
)
V
y
O
.
We then have: rank
O
=rank
We easily infer from the previous calculations that the vectors
T
x
,
T
y
,
V
x
and
V
y
2
, and we have:
are linear combinations of the three vectors
1
,
Z
and
Z
⎧
⎨
⎩
T
x
=
r
x
(0)
1
+
αv
x
Z
+
α
2
v
x
Z
2
V
x
=
αr
x
(1)
Z
T
y
=
r
y
(0)
1
+
αv
y
Z
+
α
2
v
y
Z
2
V
y
=
αr
y
(1)
Z
with:
(1
,
1
,...,
1)
1
1
(0
,
1
,
2
,...,k
)
∗
Z
2
1))
∗
Z
(0
,
0
,
2
,...,k
(
k
−
O
) and therefore of the matrix
It is then obvious that the rank of the matrix (
O
is
equal to 3, except if we have
r
x
(0)
v
y
=
r
y
(0)
v
x
,
in which case the rank of
is only
2. This analysis might seem simplistic, however, it makes it possible to handle most
problems. Let us consider, for example, the case of a system comprised of two linear
sub-antennae located on a same line. Here, the previous calculations lead to:
O
=
O
1
O
2
O
,
[6.34]
=
O
1
Δ
+
β
OO O O
Δ
,
[6.35]
O
1
1O
α
Δ
ZO
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