Digital Signal Processing Reference
In-Depth Information
As (
4.90
) is equivalent to asking for
13
G
x
<
I
L
(4.93)
we will need to look at
in order to guarantee this.
We will see that, although the exponential bound (
4.89
) might not be the tightest
one, we can find with this approach some well known results on the stability of
classical adaptive filters, which are actually very tight.
14
In the following we particularize these results for some of the algorithm presented
in the previous sections:
μ
•
LMS algorithm
: Remember that in this case
f
(
x
(
n
))
=
x
(
n
)
,
α
=
1, and
μ
=
μ
I
L
. We need to consider (
4.91
) and check when
G
x
<
I
L
. We can show that in
the LMS case, the matrix
G
x
can be written as:
2
S
x
,
G
x
=
I
L
−
μ
R
x
+
μ
2
(4.94)
where the matrix
S
x
is defined as:
E
2
x
x
T
S
x
=
x
(
n
)
(
n
)
(
n
)
.
(4.95)
The matrix
S
x
depends on the fourth order moments of the input signal which are
assumed to exist. The stability condition can be found in the following lemma:
Lemma 4.3
The matrix
G
x
in (
4.94
) will satisfy
eig
max
[
G
x
]
<
1
if and only if
2
eig
max
R
−
1
0
<μ<
S
x
.
(4.96)
x
Proof
Given that
G
x
is a symmetric positive definite matrix, we can set the problem
of bounding the maximum eigenvalue of
G
x
by 1 as [
1
]:
a
T
G
x
a
max
<
1
.
(4.97)
L
a
∈R
:
a
=
1
Using (
4.94
) we get:
2
a
T
L
a
−
μ
(
2
R
x
−
μ
S
x
)
a
<
1
,
∀
a
∈ R
,
a
=
1
,
(4.98)
or equivalently
a
T
L
−
μ
(
2
R
x
−
μ
S
x
)
a
<
0
,
∀
a
∈ R
,
a
=
1
.
(4.99)
13
We will use the usual partial ordering defined for symmetric positive definite matrices [
35
].
14
It is in this place where we only keep the sufficiency and lose the necessity.
