ECE 580 / Math 587
SPRING 2011
Correspondence # 10
March 15, 2011
Solution Set 4
29.
i)
Let
M
=
z
∈
L
2
(Ω
,
P
,
IR
n
) :
z
=
i
∑
j
=0
K
j
y
j
;
K
j
∈ M
nm
.
This is a closed linear subspace of
L
2
(Ω
,
P
; IR
n
) which is a Hilbert space. Hence, from the
Projection Theorem, there exists a
unique
ˆ
x
∈
M
, expressed as ˆ
x
=
±
i
j
=0
ˆ
K
j
y
j
, with
inf
K
j
∈M
nm
j
=0
,...,i
∥
x
−
i
∑
j
=0
K
j
y
j
∥
=
∥
x
−
ˆ
x
∥
and a necessary and suﬃcient condition for ˆ
x
to be the minimizing solution is
(
x
−
ˆ
x,z
) = 0
∀
z
∈
M
⇔
E
[
x
T
QK
ℓ
y
ℓ
] =
i
∑
j
=0
E
[
y
T
j
ˆ
K
T
j
QK
ℓ
y
ℓ
]
∀
K
ℓ
∈ M
nm
ℓ
= 0
,
1
,...,i
⇔
Tr
{
E
[
y
ℓ
x
T
QK
ℓ
]}
=
i
∑
j
=0
Tr
{
E
[
y
ℓ
y
T
j
ˆ
K
T
j
QK
ℓ
]}
∀
K
ℓ
∈ M
nm
.
Since
Q
is positive deﬁnite,
QK
ℓ
∈ M
nm
whenever
K
ℓ
∈ M
nm
, and vice versa, and hence
the earlier condition is
Tr
Λ
ℓx
−
i
∑
j
=0
Λ
ℓj
ˆ
K
T
j
K
= 0
∀
K
∈ M
nm
ℓ
= 0
,
1
,...,i
where
Λ
ℓx
∆
=
E
[
y
ℓ
x
T
]
;
Λ
ℓj
∆
=
E
[
y
ℓ
y
T
j
]
.
Now, two random vectors are uncorrelated if their components (considered as random vari-
ables) are uncorrelated. Furthermore, since
E
[
y
ℓ
] = 0, we have
Λ
ℓj
=
²
Λ
ℓℓ
j
=
ℓ
0
otherwise
.
Hence, the condition now becomes
Tr
{(
Λ
ℓx
−
Λ
ℓℓ
ˆ
K
T
ℓ
)
K
}
= 0
∀
K
∈ M
nm
, ℓ
= 0
,...,i
;