Digital Signal Processing Reference
In-Depth Information
E
E
2
2
μ
2
e
2
2
μ
2
v
2
f
(
x
(
n
))
(
n
)
=
f
(
x
(
n
))
η
(
n
)
+
2
η(
n
)
v
(
n
)
+
(
n
)
E
v
E
2
2
μ
2
2
2
μ
=
f
(
x
(
n
))
η
(
n
)
+
σ
f
(
x
(
n
))
,
(4.128)
2
where we used
E
2
μ
(
)
)
η(
)
(
)
=
f
(
x
n
n
v
n
0. With the reasoning used above we
2
can write
E
E
2
E
η
)
,so(
4.124
) can be
2
μ
2
2
μ
2
f
(
x
(
n
))
η
(
n
)
≈
f
(
x
(
n
))
(
n
2
put as:
E
E
˜
2
≈
w
T
E
˜
)
η
μ
f
2
(
1
−
α
)
w
(
∞
)
lim
n
−
2
α
(
n
)
e
(
n
)
+
2
α(
1
−
α)
w
(
n
−
1
→∞
E
2
2
μ
w
T
μ
+
f
(
x
(
n
))
ξ(
n
)
−
2
(
1
−
α)
E
[
f
(
x
(
n
))
]
E
[
η(
n
)
]
v
E
2
2
2
2
μ
2
+
σ
f
(
x
(
n
))
+
(
1
−
α)
w
T
.
(4.129)
From this equation we can obtain steady state results for almost all the algorithms
presented throughout this chapter.
η
μ
f
,so(
4.129
) takes the form
17
•
LMS algorithm
: In this case
(
n
)
=
μη(
n
)
2
E
2
v
E
2
2
2
0
=−
2
μξ
∞
+
μ
x
(
n
)
ξ
∞
+
μ
σ
x
(
n
)
,
(4.130)
where we denoted
ξ
∞
=
lim
n
→∞
ξ(
n
)
. Rearranging terms in (
4.130
) we obtain:
2
v
μ
tr [
R
x
]
σ
ξ
∞
=
tr [
R
x
]
,
(4.131)
2
−
μ
where we have used that
E
2
=
x
(
n
)
tr [
R
x
]. Notice that the final EMSE is
increasing with
and the variance of the noise. This means that in order to have a
small EMSE the value of
μ
μ
should be small. In fact, if
μ
is small the final EMSE
can be approximated by:
v
ξ
∞
≈
μ
tr [
R
x
]
σ
.
(4.132)
2
From (
4.131
) we can also see that there is a maximum value
ξ
∞
becomes
infinite, and after which it would take negative values, which is meaningless. We
see that this value of
μ
before
coincides with the approximate upperbound we derived in
(
4.106
), which is satisfactory. Obviously, from (
4.117
) and (
4.131
)wealsohave:
μ
E
2
v
μ
tr [
R
x
]
σ
2
v
J
∞
lim
n
|
e
(
n
)
|
=
σ
+
tr [
R
x
]
.
(4.133)
2
−
μ
→∞
17
Although in (
4.130
) we should put
≈
, in an abuse of notation we state it as an equality.
