Digital Signal Processing Reference
In-Depth Information
The important thing here is that the stability limit decreases with the increasing
length of the adaptive filter. That is, longer filters will have a more restricted interval
of valid
values. Although this was derived from condition (
4.96
), which is valid
under the independence assumption, the result in (
4.106
) (i.e., that the maximum
μ
μ
for which the filter remains stable is inversely proportional to the filter length) is
observed to be true in general, even when that assumption is not satisfied. The reader
should be cautioned about the last discussion. Although our conclusion was derived
in an intuitive way, it can be justified in rigorous mathematical terms. However, those
mathematical developments, although important on their own as Theorem
4.1
,are
not directly linked with our interest in adaptive filters, and for this reason we chose
to present the above important result in the way we did.
In the case where the input vectors pdf is Gaussian with covariance matrix
R
x
,
we can obtain an explicit and exact bound for
. We can use the result known as
Gaussian factoring theorem
15
for the obtention of
S
x
, leading to:
μ
2
R
x
+
S
x
=
R
x
tr [
R
x
]
.
(4.107)
Then, using (
4.96
) we can show that:
2
2eig
max
[
R
x
]
0
<μ<
tr [
R
x
]
.
(4.108)
+
This a sufficient condition for the MSD stability of the LMS with Gaussian and
independent input regressors. However, it is close enough to the necessary and suf-
ficient condition that can be obtained from a more elaborated model of the transient
behavior of the LMS algorithm under Gaussian signaling [
19
,
40
]. In fact, for white
Gaussian signals condition (
4.108
) is also necessary.
A more restrictive but useful bound for the stability of the LMS can be obtained
from the fact that eig
max
[
R
x
]
≤
tr [
R
x
], and therefore
2
3tr [
R
x
]
.
0
<μ<
(4.109)
This bound was obtained with a different approach in [
31
]. Obviously the case
μ
=
0,
although it gives eig
max
[
G
x
]
1, is a stable point. However, it is useless because it
means the algorithmwill not move and stay forever at the initial condition. The reader
can also see that equation (
4.109
) has the same asymptotic behavior as (
4.106
). In
the general non-Gaussian case, the sufficient condition on
=
μ
for MSD stability has
to be derived from (
4.96
).
15
For Gaussian random variables we have the following result [
39
]:
E
[
x
1
x
2
x
3
x
4
]=
E
[
x
1
x
2
]
E
[
x
3
x
4
]+
E
[
x
1
x
3
]
E
[
x
2
x
4
]+
E
[
x
1
x
4
]
E
[
x
2
x
3
]
.
