Digital Signal Processing Reference
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with the weighted term E
w
i
−
1
R
u
. This term can be deduced from Relation
1.78 by writing it for
=
R
u
, which leads to
2
E
2
R
u
g
2
[
u
i
]
E
u
i
2
2
2
2
2
v
E
w
i
R
u
=
E
w
i
−
1
R
u
−
2
µ
h
G
E
w
i
−
1
R
u
+
µ
w
i
−
1
R
u
+
σ
2
with the weighted term E
R
u
and so forth. The procedure terminates and
leads to the following state-space model:
w
i
=
F +
µ
e
2
W
2
2
2
v
Y
W
Y
+
µ
σ
,
(1.82)
i
i
−
1
where the
M
×
1 vectors
{W
i
,
Y}
are defined by
E
g
2
[
u
i
]
2
2
u
i
/
E
w
i
E
g
2
[
u
i
]
2
R
u
2
E
w
i
u
i
R
u
/
.
.
i
=
W
=
Y
,
,
(1.83)
E
g
2
[
u
i
]
2
R
M
−
2
u
2
R
M
−
2
u
E
w
i
u
i
/
E
g
2
[
u
i
]
2
R
M
−
1
u
E
w
i
2
R
M
−
1
u
u
i
/
the
M
×
M
matrix
F
is given by
1
−
2
µ
h
G
0
1
−
2
µ
h
G
0
0
1
−
2
µ
h
G
=
F
0
0
1
−
2
µ
h
G
2
µ
p
0
h
G
2
µ
p
1
h
G
...
2
µ
p
M
−
2
h
G
1
+
2
µ
p
M
−
1
h
G
and
=
{
...
}
.
e
2
col
0
,
1
,
0
,
,
0
Also,
E
1
g
[
u
i
]
h
G
=
.
The evolution of the top entry of
W
i
describes the mean-square deviation of
2
, while the evolution of the second entry of
the filter, E
i
relates to
the learning behavior of the filter. The model (Equation 1.82) is an alternative
to Equation 1.22 for adaptive filters with data nonlinearities; it is based on
assumptions in Listing 1.76.
w
i
W
1.12.5.3 Steady-State Performance
The variance Relation 1.78 can also be used to approximate the steady-state
performance of data-normalized adaptive filters. Writing it for
=
I
,
2
E
E
2
u
i
2
2
2
2
2
v
E
w
i
=
E
w
i
−
1
−
2
µ
h
G
E
w
i
−
1
R
u
+
µ
w
i
−
1
R
u
+
σ
g
2
[
u
i
]
(1.84)
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