Digital Signal Processing Reference
In-Depth Information
In this case, the value of
that guarantees the stability in the mean depends on
the statistical properties of the input through the matrix
L
x
. In general, it is not
possible to obtain closed form solutions for this matrix. An important case which
leads to a closed form result is when
x
μ
is a zero mean Gaussian vector with
correlation matrix
R
x
. In that case, we obtain:
(
n
)
2
πσ
L
x
=
R
x
.
(4.78)
x
This result can be obtained with the well known
Price theorem
[
36
] which permits
to calculate expectations of certain nonlinear functions of correlated Gaussian
random variables. In this manner, the stability condition (
4.77
) can be expressed
as:
2
x
πσ
0
<μ<
eig
max
[
R
x
]
.
(4.79)
4.5.2.2 MSD Stability
In order to obtain a better picture of the way in which an adaptive filter works, besides
the mean behavior, we need to look into the mean square performance. To obtain
results regarding the MSD stability of the algorithm we need to take squared norms
on both sides of equation (
4.57
) to obtain:
n
2
c
T
A
T
c
T
A
T
˜
c
(
n
)
= ˜
(
−
1
)
(
n
,
0
)
A
(
n
,
0
)
˜
c
(
−
1
)
−
2
0
˜
(
−
1
)
(
n
,
0
)
A
(
n
,
j
+
1
)
j
=
c
T
c
T
T
n
n
f
f
×
(
x
(
j
))
v
(
j
)
−
(
1
−
α)
+
(
x
(
i
))
v
(
i
)
−
(
1
−
α)
i
=
0
j
=
0
f
c
T
A
T
×
(
n
,
i
+
1
)
A
(
n
,
j
+
1
)
(
x
(
j
))
v
(
j
)
−
(
1
−
α)
.
(4.80)
If we take expectations on both sides, using the independence between the input and
the noise and the fact that the noise is i.i.d., we obtain:
