Digital Signal Processing Reference
In-Depth Information
and
E
E
2
1
x
(
n
)
,
(4.140)
+
δ
2
2
x
(
n
)
+
δ
2
x
(
n
)
can be evaluated with the input vector pdf. However, if
δ
=
0, we have:
2
v
μσ
ξ
∞
=
−
μ
,
(4.141)
2
which, as in the stability condition seen before, does not depend on any character-
istic of the input vectors pdf. The misadjustment for the NLMS algorithm in the
case of
δ
=
0is:
μ
M
=
−
μ
,
(4.142)
2
μ
which depends only on
. If we use the independence assumption and consider
x
I
L
, it is easy to show that:
that
R
x
=
σ
E
2
2
v
μ
σ
lim
n
˜
w
(
n
)
=
x
.
(4.143)
2
−
μ
σ
→∞
x
I
L
, we can only bound the behavior of the
asymptotic MSD as in (
4.119
). It is interesting to observe how the final EMSE and
misadjustment reflect the stability bound
If we cannot assume that
R
x
=
σ
μ<
2 in their expressions, in a similar
way as in the LMS algorithm.
As in the LMS case, even though these expressions have been derived under some
particular assumptions, they have proved to be quite accurate when compared with
results obtained from using the NLMS algorithm in real problems.
η
μ
f
w
T
•
Sign Data algorithm
:Wealsohave
(
n
)
=
μ
˜
(
n
−
1
)
sign [
x
(
n
)
]. Equation
(
4.129
) can be written as:
E
w
T
]
x
T
0
=−
2
μ
lim
n
˜
(
n
−
1
)
sign [
x
(
n
)
(
n
)
˜
w
(
n
−
1
)
→∞
v
E
2
2
E
2
2
2
+
μ
σ
sign
[
x
(
n
)
]
+
μ
sign
[
x
(
n
)
]
ξ
∞
.
(4.144)
2
Using that
sign [
x
(
n
)
]
=
L
we can write:
2 lim
n
→∞
E
˜
)
−
μ
w
T
]
x
T
2
v
)
˜
(
n
−
1
)
sign [
x
(
n
)
(
n
w
(
n
−
1
L
σ
ξ
∞
=
.
(4.145)
μ
L
This is the general expression for the steady state behavior of the SDA. In order
to obtain a more explicit expression we can assume that the input regressors are
Gaussian, and the fact that when
n
is large
˜
(
−
)
(
)
w
n
1
is MSU with respect to
x
n
(using the same reasoning presented before). With these assumptions,
