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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