Digital Signal Processing Reference
In-Depth Information
x
(
n
)
•
NLMS algorithm
: In this case we have
f
(
x
(
n
))
=
,
μ
=
μ
I
L
, and
2
x
(
n
)
+
δ
α
=
1. Replacing everything in (
4.66
), the step size
μ
needs to satisfy
2
eig
i
[
K
x
]
,
0
<μ<
i
=
1
,...,
L
,
(4.71)
where
E
x
x
T
(
n
)
(
n
)
K
x
=
.
(4.72)
x
(
n
)
2
+
δ
The condition (
4.71
) can be compactly written as:
2
eig
max
[
K
x
]
.
0
<μ<
(4.73)
As in the case of the LMS, if (
4.73
) is satisfied, the NLMSwill be an asymptotically
unbiased estimator of the optimal Wiener filter, as it can be seen from (
4.68
). The
exact calculation of eig
max
[
K
x
] requires knowledge of the input vector probability
density function (pdf). However, it is possible to bound it in a useful and appropriate
way. We can write the following, given the fact that
K
x
is a symmetric positive
definite matrix:
E
a
T
x
2
(
n
)
a
T
K
x
a
eig
max
[
K
x
]
=
max
≤
max
<
1
,
(4.74)
2
x
(
n
)
+
δ
a
∈R
L
:
a
=
1
a
∈R
L
:
a
=
1
which means that we can modify condition (
4.73
) by the following more restrictive
one:
0
<μ<
2
.
(4.75)
Although condition (
4.75
) is only sufficient for the mean stability, whereas con-
dition (
4.73
) is necessary and sufficient, it is by far more useful. Notice that it
does not depend on any statistical information about the input vector. In addition,
condition (
4.75
) is valid when
δ
=
0.
•
Sign Data algorithm
: For the sign data algorithm, we have
f
(
x
(
n
))
=
sign [
x
(
n
)
],
μ
=
μ
I
L
, and
α
=
1. If we define
E
sign [
x
]
x
T
L
x
=
(
n
)
(
n
)
,
(4.76)
and assume that this matrix is positive definite, (
4.66
) can be compactly put as:
2
eig
max
[
L
x
]
.
0
<μ<
(4.77)
