Information Technology Reference
In-Depth Information
Therefore pol(
T
(
z
)
M
N
(
z
))
M
−
N
(
z
)isan
N
th partial realization of
T
.Conversely,
let Pol(
T
(
z
)
M
N
(
z
))
M
−
N
(
z
)bean
N
th partial realization of
T
.Since
M
N
(
z
)is
nonsingular, it is easy to multiply
M
N
(
z
) by elementary matrices such that the
projection of each column of the obtained matrix is in [
β
i
] respectively. Therefore
the
i
th column is an
N
th right
i
-annihilating polynomial column vector of
T
for
1
≤
i
≤
m
. With the above discussions we obtain the following theorem.
Theorem 1.
If M
N,i
(
z
)
,
1
m,isanNth right i-minimal polynomial col-
umn vector of T and M
N
(
z
)=(
M
N, i
(
z
))
1
≤i≤m
,then
pol(
T
(
z
)
M
N
(
z
))
M
−
1
≤
i
≤
(
z
)
N
is an Nth minimal partial realization of T .
Therefore the minimal partial realization problem is reduced to finding
m
min-
imal polynomial column vectors with their projections in distinct classes.
Here we need to use the lattice theory.
In the sequel we always let
n
=
m
+
p
. A subset
Λ
of
K
n
is called an IF[
z
]-
lattice
if there exists a basis
ω
1
,...,ω
n
of
K
n
such that
n
n
Λ
=
IF [
z
]
ω
i
=
f
i
ω
i
:
f
i
∈
IF [
z
]
,i
=1
,...,n
.
i
=1
i
=1
In this situation we say that
ω
1
,...,ω
n
form a
basis
for
Λ
.Abasis
ω
1
,...,ω
n
is
reduced
if
θ
(
ω
1
)
,...
,
θ
(
ω
n
) are linearly independent over IF. The
determinant
of
the lattice is defined by det(
Λ
)=
v
(det(
ω
1
,...,ω
n
)) and is independent of the
choice of the basis. In [14, 15] it was proved that
n
v
(
ω
i
)=det(
Λ
)
(3)
i
=1
if
ω
1
,...,ω
n
are reduced for a lattice
Λ
. For any vector
γ
,let
γ
be the vector
containing only the last
m
components of
γ
. The reduced basis is normal if
v
(
ω
1
)
≤
...
≤
v
(
ω
p
),
θ
(
ω
i
)=
0
m
for 1
≤
i
≤
p
and
θ
(
ω
p
+
i
)
∈
[
β
i
]for1
≤
i
≤
m
.
Consider the matrix
−
0
m×p
I
m×m
z
−N−
1
,
I
p×p
T
(
z
)
and denote its
n
columns by
−
ε
1
,
···
,
−
ε
p
,α
N,
1
,
···
,α
N,m
, which span an IF[
z
]-
lattice, simply denoted by
Λ
later.
By means of a lattice basis reduction algorithm [14, 17], we can transform
the initial basis into a reduced one. Then it is easy to obtain a normal basis
by performing some elementary transformations on the reduced basis. In the
following we will show that the information we want about
T
must appear in a
normal basis of
Λ
.
The mapping
η
:
Λ
IF [
z
]
m
is given by
η
(
−f
1
(
z
)
ε
1
−···−f
p
(
z
)
ε
p
+
f
p
+1
(
z
)
α
N,
1
+
···
+
f
n
(
z
)
α
N,m
)=(
f
p
+1
(
z
)
, ···,f
n
(
z
))
t
,
→
