Digital Signal Processing Reference
In-Depth Information
TABLE 1.1
Expressions for
h
G
and
h
U
for Some Error
Nonlinearities
Algorithm
Error Nonlinearity
{
h
G
,
h
U
}
h
G
=
1
LMS
f
[
e
]
=
e
E
e
a
(
2
v
h
U
=
i
)
+
σ
2
π
1
E
e
a
(
i
)
+
σ
h
G
=
sign-
LMS
f
[
e
]
=
sign[
e
]
2
v
h
U
=
1
3
E
e
a
(
i
)
+
σ
v
2
h
G
=
15
E
e
a
(
3
e
3
LMF
f
[
e
]
=
2
v
h
U
=
i
)
+
σ
Note:
In the least-mean-fourth (LMF) case, we assume Gaussian
noise for simplicity.
where
k
(
·
)
is some function of
y
+
z
. Using this result, together with the
equality
e
(
i
)
=
e
a
(
i
)
+
v
(
i
)
, we obtain
=
E
e
a
(
)
·
E
e
a
(
i
)
f
[
e
(
i
)
]
E
e
a
(
E
e
a
(
)
(
)
=
)
(
)
)
(
i
f
[
e
i
]
i
e
a
i
i
e
a
i
h
G
,
E
e
a
(
i
)
where the function
h
G
is defined by
E
e
a
(
i
)
f
[
e
(
i
)
]
h
G
=
.
(1.61)
E
e
a
(
i
)
Clearly, because
e
a
(
i
)
is Gaussian, the expectation E
e
a
(
i
)
f
[
e
(
i
)
] depends on
only through its second moment, E
e
a
(
e
a
(
i
)
i
)
. This means that
h
G
itself is only
a function of E
e
a
(
i
)
. The function
h
G
[
·
] can be evaluated for different choices
of the error nonlinearity
f
[
·
], as shown in Table 1.1.
2
Σ
f
2
[
e
]
)
To evaluate the second expectation, E
1.12.1.2
Evaluation of
E
(
u
i
2
f
2
[
e
, we resort to a separation
assumption; i.e., we assume that the filter is long enough so that
(
u
i
(
i
)
]
)
2
f
2
[
e
u
i
and
(
i
)
] are uncorrelated.
(1.62)
This assumption allows us to write
E
]
=
E
·
E
f
2
[
e
]
=
E
·
2
f
2
[
e
2
2
u
i
(
i
)
u
i
(
i
)
u
i
h
U
,
where the function
h
U
is defined by
h
U
=
E
f
2
[
e
(
i
)
]
.
(1.63)
Again, since
e
a
is Gaussian and independent of the noise, the function
h
U
is
a function of
E
e
a
(
(
i
)
only. The function
h
U
can also be evaluated for different
error nonlinearities, as shown in Table 1.1.
i
)
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