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In [19, 20] a minimal partial realization algorithm for vector sequences was
proposed based on a lattice reduction algorithm in function fields. However, the
generalization to the MIMO systems is not clear. Therefore in this paper we
extend the algorithm to the matrix sequences. Furthermore, we characterize all
the minimal partial realizations and give the sucient and necessary condition
for the unique issue. Our point of view is rather algebraic, however, the different
ways of transforming a given basis into a reduced one lead to different partial
realization algorithms and so our technique provides a unified approach to the
minimal partial realization problem.
2
The Realization Algorithm
We shall restrict our attention to an arbitrary field IF, the polynomial ring IF[
z
],
the rational function field IF(
z
), the field of formal Laurent series
K
=IF((
z
−
1
)).
There is a valuation
v
on
K
whereby for
α
=
j
=
j
0
a
j
z
−j
∈
K
we put
v
(
α
)=
max
if
α
= 0. It can be seen as
the generalization for the degree of a polynomial. For any two positive integers
k
and
n
,the
valuation v
(
A
)ofa
k
{−
j
∈
ZZ
:
a
j
=0
}
if
α
=0and
v
(
α
)=
−∞
×
n
matrix
A
=(
α
ij
)
k×n
over
K
is defined
and define
Az
−h
=(
a
ij
z
−h
)
1
≤i≤k,
1
≤j≤n
where
h
is an arbitrary integer so that
T
(
z
) can be seen as a
p
as max
{
v
(
α
ij
):1
≤
i
≤
k,
1
≤
j
≤
n
}
m
matrix
over
K
. In this paper we mainly use the valuation of a column vector. In the
sequel we often use the
projection θ
:
K
n
×
IF
n
→
such that
γ
=(
α
i
)
1
≤i≤n
→
(
a
1
,−v
(
γ
)
,...,a
n,−v
(
γ
)
)
t
,where
α
i
=
j
=
j
0
a
i,j
z
−j
,
1
≤
i
≤
n
,and
t
denotes
the transpose of a vector.
For any positive integers
k
and
s
, we denote the identity matrix of order
k
by
I
k×k
,the
k
×
s
zero matrix by
0
k×s
and the zero vector with
k
components
by
0
k
.
A nonzero polynomial column vector
c
(
z
)inIF[
z
]
m
canbewrittenas
c
(
z
)=
i
=0
c
i
z
i
where
c
0
,...,
c
d
∈
IF
m
and
d
=
v
(
c
(
z
)). Define
m
classes [
β
1
]
,...,
[
β
m
]
IF
m
,
1
,b
1
,...,b
m−i
)
t
by [
β
i
]=
{
(0
,...,
0
i−
1
∈
:
b
j
∈
IF f o r 1
≤
j
≤
m
−
i
}
,
i
=1
,...,m
.
Definition 1.
A nonzero polynomial column vector
c
(
z
)=
i
=0
c
i
z
i
is called
an Nth right i-annihilating polynomial column vector of T if the kth discrepancy
δ
k
(
c
(
z
)
,T
)=
0
p
for all
1
≤
k
≤
N,where
d
δ
k
(
c
(
z
)
,T
)=
T
k
c
d
+
T
k−
1
c
d−
1
+
...
+
T
k−d
c
0
=
T
k−d
+
i
c
i
,
i
=0
and θ
(
c
(
z
))
[
β
i
]
.TheNth right i-minimal polynomial column vector of T is
the Nth right i-annihilating polynomial column vector with the least valuation.
∈
Similarly, we can define the
left i-annihilating polynomial row vector
and
left
i-minimal polynomial row vector
. In this paper we only discuss the right case
and it is easy to get the corresponding results for the left case.
