Digital Signal Processing Reference
In-Depth Information
E
2
E
A
T
c
T
˜
c
(
n
)
= ˜
(
−
1
)
(
n
,
0
)
A
(
n
,
0
)
c
˜
(
−
1
)
E
A
T
c
T
n
c
T
+
2
(
1
−
α)
0
˜
(
−
1
)
(
n
,
0
)
A
(
n
,
j
+
1
)
j
=
E
f
T
n
2
v
A
T
)
f
+
σ
(
x
(
j
))
(
n
,
j
+
1
)
A
(
n
,
j
+
1
(
x
(
j
))
j
=
0
c
T
E
A
T
c
T
.
n
n
2
+
(
1
−
α)
(
n
,
k
+
1
)
A
(
n
,
j
+
1
)
(4.81)
j
=
0
k
=
0
)
, which is a positive definite matrix.
From the definition (
4.58
) and the independence assumption we obtain:
E
A
T
We can define
D
(
n
,
j
)
=
(
n
,
j
)
A
(
n
,
j
D
E
A
T
B
k
−
j
(
n
,
k
)
,
k
≥
j
x
(
n
,
k
)
A
(
n
,
j
)
=
B
x
j
−
k
D
k
,
(4.82)
(
n
,
j
),
j
>
where
B
x
is given by (
4.62
). With the above equation we can write the first, second,
third and fourth terms in the right hand side of (
4.81
)as:
c
T
˜
(
−
1
)
D
(
n
,
0
)
˜
c
(
−
1
),
(4.83)
B
x
j
+
1
n
c
T
2
(
1
−
α)
0
˜
(
−
1
)
D
(
n
,
j
+
1
)
c
T
,
(4.84)
j
=
tr
F
x
D
n
2
v
σ
(
n
,
j
+
1
)
,
(4.85)
j
=
0
⎡
⎤
n
n
j
−
1
c
T
B
x
j
−
k
B
k
−
j
2
⎣
c
T
D
⎦
,
(
1
−
α)
(
n
,
k
+
1
)
c
T
+
D
(
n
,
j
+
1
)
c
T
x
j
=
0
k
=
j
k
=
0
(4.86)
E
, is assumed to be positive definite.
11
respectively, where
F
x
=
f
))
f
T
(
x
(
j
(
x
(
j
))
Given that
μ
i
>
0
,
i
=
1
,
2
,...,
L
, the MSD stability of
w
˜
(
n
)
is equivalent to the
MSD stability of
˜
c
(
n
)
. The latter can be characterized by the following theorem:
Theorem 4.1
[Mean square stability] A necessary and sufficient condition for
lim
n
→∞
E
˜
2
<
∞
c
(
n
)
for every
c
˜
(
−
1
)
and
c
T
, is given by the satisfaction
of (
4.65
) and
11
In (
4.85
) we used the fact that
A
(
n
,
j
+
1
)
and
f
(
x
(
j
))
are independent and that for two matrices
A
and
B
of appropriate dimensions, tr [
AB
]
=
tr [
BA
].
