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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