Digital Signal Processing Reference
In-Depth Information
n
+
1
P
−
1
If
λ
i
<
1
,
i
=
1
,...,
L
it is clear that
P
c
˜
(
−
1
)
goes to zero as
n
→∞
.
At the same time, each of the diagonal entries in
j
=
0
1
j
converges to
−
λ
i
.This
means that every component in lim
n
→∞
E
˜
)
is finite. On the other hand, if for
1
(
c
n
λ
i
≥
some
i
1, we have that both terms in the right hand side of (
4.64
) are unbounded,
which means that lim
n
→∞
E
˜
)
=∞
c
(
n
.
Using (
4.62
) we can set the necessary and sufficient condition for the mean sta-
bility of
w
˜
(
n
)
as:
α
−
eig
i
2
<
1
/
2
C
x
μ
1
/
μ
1
,
i
=
1
,
2
,...,
L
.
(4.65)
As we will see, in several cases of interest the matrix
C
x
will be positive definite,
which implies that eig
i
2
1
/
2
C
x
μ
1
/
μ
>
0
,
i
=
1
,...,
L
. In these cases, the mean
stability condition can be written as:
eig
i
2
1
/
2
C
x
μ
1
/
0
<
μ
<
1
+
α,
i
=
1
,
2
,...,
L
.
(4.66)
It is clear from (
4.66
) that a careful choice of
μ
can guarantee the mean stability.
From the above proof we can see that the limiting value of
E
˜
)
can be put as
c
(
n
E
˜
)
=
(
I
L
−
)
−
1
P
−
1
c
T
,
(
−
α)
(
lim
c
n
1
P
(4.67)
n
→∞
or equivalently
E
˜
)
=
(
1
/
2
P
I
L
−
)
−
1
P
−
1
μ
−
1
/
2
w
T
.
lim
n
w
(
n
1
−
α)
μ
(
(4.68)
→∞
Evidently, if
α
=
1 the adaptive filter will give a biased estimation when
n
→∞
.
This is the case of the Leaky LMS, as previously discussed.
In the followingwe particularize these results for some of the algorithms presented
in the previous sections:
•
LMS algorithm
: For the LMS we have
α
=
1,
μ
=
μ
I
L
and
f
(
x
(
n
))
=
x
(
n
)
.It
is easy to see that condition (
4.66
) can be expressed as:
0
<μ
eig
i
[
R
x
]
<
2
,
i
=
1
,
2
,...,
L
,
(4.69)
which leads to
2
eig
max
[
R
x
]
.
0
<μ<
(4.70)
as the one for the stability of the SD algorithm. In fact the convergence dynamics
of the mean weight error vector for the LMS algorithm (under the independence
assumption) is the same as the one of the SD algorithm.
