Digital Signal Processing Reference
In-Depth Information
It is interesting to quantify how the value of
J
differs from the optimum value
of the Wiener filtering
J
min
, which in this case is equal to
∞
v
. This leads us to the
σ
definition of the
misadjustment
:
J
∞
−
J
min
J
min
ξ
∞
J
min
.
M
=
(4.134)
For the LMS algorithm we obtain:
μ
tr [
R
x
]
M
=
tr [
R
x
]
,
(4.135)
2
−
μ
which does not depend on the noise variance. It is not surprising to see that the
misadjustment is increasing with
.
Finally we can see that when the input is white, i.e.,
R
x
=
σ
μ
x
I
L
,from(
4.120
)it
is easy to see that the final MSD is:
E
2
2
v
μ
L
σ
lim
n
˜
w
(
n
)
=
x
,
(4.136)
2
−
μ
L
σ
→∞
which is an increasing function of
μ
and
L
.
μ
η
μ
f
•
NLMS algorithm
:Wehave
(
n
)
=
+
δ
η(
n
)
. Equation (
4.129
) can be
2
x
(
n
)
written as:
E
v
E
2
2
η
(
n
)
x
(
n
)
2
2
0
=−
2
μ
lim
n
+
μ
σ
x
(
n
)
2
+
δ
(
x
(
n
)
2
+
δ)
2
→∞
2
E
2
x
(
n
)
+
μ
ξ
∞
.
(4.137)
(
x
(
n
)
2
+
δ)
2
As reasoned before, we can make the following approximation:
E
E
2
η
(
n
)
1
≈
ξ(
n
).
(4.138)
2
2
x
(
n
)
+
δ
x
(
n
)
+
δ
With (
4.138
) we can obtain
ξ
∞
as:
v
E
2
x
(
n
)
μσ
+
δ
2
2
x
(
n
)
E
+
δ
2
.
ξ
∞
=
(4.139)
2
E
2
x
(
n
)
1
−
μ
2
x
(
n
)
+
δ
2
x
(
n
)
The quantities:
