pose α
k
= α(τ
k
), β
k
= β(τ
k
) and ϑ
k
= ϑ(τ
k
) are
real-valued random variables having ﬁnite second
moments. We write x = [α
1
,α
2
,. .. ,α
m
]
T
y =
[β
1
,β
2
,. .. ,β
n
]
T
and z = [ϑ
1
,. .. ,ϑ
r
]
T
.
Let
˜
z be a compressed form of data y deﬁned
by
˜
z = B(y) with
˜
z = [
˜
ϑ
1
,. .. ,
˜
ϑ
r
]
T
, and
˜
x be a de-
compression of
˜
z deﬁned by
˜
x = A(
˜
z) with
˜
x =
[
˜
α
1
,. .. ,
˜
α
m
]
T
.
In many applications
2
, to obtain
˜
ϑ
k
for k =
1,...,r, it is necessary for B to use only a limited
number of input components, η
B
k
= 1,...,r. A num-
ber of such input components η
B
k
is here called a kth
local memory for B.
To deﬁne a notation of memory for the compres-
sor B, we use parameters p and g which are positive
integers such that 1 ≤ p ≤ n and n−r+2≤ g ≤ n.
Deﬁnition 1. The vector η
B
= [η
B
1
,. .. ,η
B
r
]
T
∈ R
r
is called a variable memory of the compressor B. In
particular, η
B
is called a complete variable memory if
η
B
1
= g and η
B
k
= n when k = n−g+1,...,n. Here, p
relates to the last possible nonzero entry in the bottom
row of B and g relates to the last possible nonzero
entry in the ﬁrst row.
The notation η
A
= [η
A
1
,. .. ,η
A
m
]
T
∈ R
m
has a sim-
ilar meaning for the de-compressor A, i.e., η
A
is a
variable memory of the de-compressor A. Here, η
A
j
is the jth local memory of A.
The parameters q and s, which are positive inte-
gers such that 1 ≤ q ≤ r and 2 ≤ s ≤ m, are used
below to deﬁne two types of memory for A.
Deﬁnition 2. Vector η
A
is called a complete variable
memory of the de-compressor A if η
A
1
= q and η
A
j
= r
when j = s+ r − 1, .. ., m. Here, q relates to the ﬁrst
possible nonzero entry in the last column of A and s
relates to the ﬁrst possible nonzero entry in the ﬁrst
column.
The memory constraints described above imply
that certain elements of the matrices B = {b
ij
}
r,n
i, j=1
and A = {a
ij
}
m,r
i, j=1
must be set equal to zero. In this
regard, for matrix B with r ≤ p ≤ n, we require that
b
i, j
= 0
if j = p− r+ i+ 1,...,n,
for
p = r,... ,n − 1,
i = 1,. .. ,r
and
p = n,
i = 1,... ,r− 1,
2
Examples include computer medical diagnostics (Gi-
meno, 1987) and problems of bio-informatics (H. Kim,
2005).
and, for 1 ≤ p ≤ r− 1, it is required that
b
i, j
= 0
if
i = 1,...,r− p,
j = 1,...,n,
and
i = r − p+ 1,...,r,
j = i− r+ p+ 1, .. .,n.
For matrix A with r ≤ p ≤ n, we require
a
i, j
= 0 (11)
ifj = q + i,...,rforq = 1,...,r− 1,i = 1,...,r− q,
and, for 2 ≤ s ≤ m, it is required that
a
i, j
= 0
if j = s+ i, .. ., r for s = 1,...,m, i = 1,...,s+ r − 1,
The above conditions imply the following deﬁnitions.
Deﬁnition 3. A matrix B satisfying the constraint
(11)–(11) is said to be a causal operator with the
complete variable memory η
B
= [g,g + 1,...,n]
T
.
Here, η
B
k
= n when k = n − g + 1,..., n. The set of
such matrices is denoted by M
C
(r,n,η
B
).
Deﬁnition 4. A matrix A satisfying the constraint
(11)–(11) is said to be a causal operator with the
complete variable memory η
A
= [r − q + 1,...,r]
T
.
Here, η
A
j
= r when j = q, .. ., m. The set of such ma-
trices is denoted by M
C
(m,r,η
A
).
3.1 Solution of Problems (8) and (10)
To proceed any further we shall require some more
notation. Let
hα
i
,β
j
i =
Z
Ω
α
i
(ω)β
j
(ω)dµ(ω) < ∞, (12)
E
xy
= {hα
i
,β
j
i}
m,n
i, j=1
∈ R
m×n
,
y
1
= [β
1
,. .. ,β
g−1
]
T
, y
2
= [β
g
,. .. ,β
n
]
T
, (13)
z
1
= [ϑ
1
,. .. ,ϑ
g−1
]
T
and z
2
= [ϑ
g
,. .. ,ϑ
n
]
T
.
(14)
The pseudo-inverse matrix (Golub and Loan,
1996) for any matrix M is denoted by M
†
. The symbol
O designates the zero matrix.
Lemma 1. (Torokhti and Howlett, 2007) If we de-
ﬁne w
1
= y
1
and w
2
= y
2
− P
y
y
1
where
P
y
= E
y
1
y
2
E
†
y
1
y
1
+ D
y
(I − E
y
1
y
1
E
†
y
1
y
1
) (15)
with D
y
an arbitrary matrix, then w
1
and w
2
are mu-
tually orthogonal random vectors.
ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics
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