Digital Signal Processing Reference
In-Depth Information
•
NLMS algorithm
: For the MSD stability of the NLMS algorithm we need to
consider the matrix
G
x
, that in this case can be written as:
2
H
x
,
G
x
=
I
L
−
2
μ
K
x
+
μ
(4.110)
where
E
2
(
)
x
n
x
T
H
x
=
(
)
(
)
.
+
δ
2
x
n
n
(4.111)
2
x
(
n
)
By applying Lemma
4.3
, the resulting stable range of
μ
is
2
eig
max
K
−
1
H
x
.
0
<μ<
(4.112)
x
For the case
δ
=
0, it is easy to see that
H
x
=
K
x
, leading to
<μ<
.
0
2
(4.113)
The condition on the step size of the NLMS algorithm for mean square stability is
independent of any statistical characteristic of the input vector and is not restricted
to the Gaussian case only. This is the same sufficient condition for the MSD
stability of the NLMS algorithm obtained in [
40
-
42
], and [
43
], where specific
distributions for the input vector were used. Here we do not use any statistical
information about the input vectors; just the independence assumption which is
used in those works as well. In the case
0, the stability bound (
4.113
) is still
sufficient. Notice, that in contrast to the LMS algorithm, the stability result for the
NLMS algorithm does not depend on the filter length
L
.
δ
=
•
Sign Data algorithm
:Matrix
G
x
can be written for the SDA as:
2
L
R
x
,
G
x
=
I
L
−
2
μ
L
x
+
μ
(4.114)
where we used the fact that
E
)
=
2
x
x
T
L
R
x
. Application of
Lemma
4.3
leads us to the following bound for the MSD stability:
sign [
x
(
n
)
]
(
n
)
(
n
2
L
eig
max
L
−
1
<μ<
R
x
.
0
(4.115)
x
In order to obtain a more explicit result we need to find
L
x
. A closed form expres-
sion of this quantity is not possible in general, but we can always use computer
simulations to have and estimate. However, using the expression for
L
x
obtained
in (
4.78
), for the important Gaussian case (
4.115
) can be put as
