Digital Signal Processing Reference
In-Depth Information
TABLE 7.1
AP Adaptive Algorithm of Order
L
for a Volterra Filter
Implemented with
M
Diagonals
y
i
(
h
i
n
)
=
(
n
)
x
i
(
n
)
)
=
i
=
1
y
i
(
n
)
y
(
n
)
−
i
=
1
h
i
e
j
(
n
)
=
d
(
n
−
j
+
1
(
n
)
x
i
(
n
−
j
+
1
)
]
T
e
(
n
)
=
[
e
1
(
n
)
e
2
(
n
)
···
e
L
(
n
)
G
i
(
n
)
=
[
x
i
(
n
)
x
i
(
n
−
1
)
···
x
i
(
n
−
L
+
1
)
]
]
T
l
i
(
n
)
=
[
x
(
n
)
x
(
n
−
i
+
1
)
x
(
n
−
1
)
x
(
n
−
i
)
···
x
(
n
−
L
+
1
)
x
(
n
−
L
−
i
+
2
)
P
i
(
n
−
1
)
l
i
(
n
)
k
i
(
n
)
=
λ
l
i
−
λ
+
(
n
)
P
i
(
n
−
1
)
l
i
(
n
)
1
1
λ
(
l
i
P
i
(
)
=
P
i
(
−
)
−
k
i
(
)
(
)
P
i
(
−
))
n
n
1
n
n
n
1
h
i
(
+
)
=
h
i
(
)
+
µ
i
G
i
(
)
P
i
(
)
(
)
n
1
n
n
n
e
n
Initialization:
P
i
=
δ
I
,
h
i
=
0
∀
i
By defining the gain vector
k
i
(
n
)
as
(
−
)
(
)
P
i
n
1
l
i
n
(
)
=
k
i
n
,
(7.47)
λ
l
i
(
−
λ
+
n
)
P
i
(
n
−
1
)
l
i
(
n
)
1
the following recursive estimate for
P
i
(
n
)
is derived:
P
i
)
.
1
λ
l
i
(
P
i
(
n
)
=
(
n
−
1
)
−
k
i
(
n
)
n
)
P
i
(
n
−
1
(7.48)
G
T
))
−
1
given in Equation
7.48, the final updating expression for the
i
th channel of the diagonal realiza-
tion is
By replacing
(
(
n
)
G
(
n
in Equation 7.42 with
P
i
(
n
)
h
i
(
n
+
1
)
=
h
i
(
n
)
+
µ
i
G
i
(
n
)
P
i
(
n
)
e
(
n
)
(7.49)
for 1
M
. The equations employed for updating the coefficients and for
filtering the input signal using the diagonal realization are summarized in
Table 7.1. The algorithm can be applied to the whole quadratic filter simply
by setting
M
≤
i
≤
N
. Because this adaptive algorithm treats all channels of
the nonlinear filter separately, it is easy to extend this algorithm to a generic
Volterra filter with different order terms. It has also been shown
12
that the
complexity of this AP algorithm is about
L
times that of the LMS algorithm.
It is worth noting that for
L
=
1 the AP algorithm becomes in practice an LMS
algorithm, while for increasing value of the number
L
of the errors minimized
it tends to behave as an RLS algorithm. This is the reason for the improved
performance that is experienced.
=
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