Digital Signal Processing Reference
In-Depth Information
Let us rewrite Equation 7.13 in the vector form
h
i
(
y
i
(
n
)
=
n
)
x
i
(
n
)
,
(7.23)
(
)
−
+
where
h
i
n
is the vector formed with the
N
i
1 coefficients of the
i
th
channel
]
T
h
i
(
n
)
=
[
h
(
0
,i
−
1
)
h
(
1
,i
)
···
h
(
N
−
i, N
−
1
)
.
(7.24)
The input vector
x
i
(
n
)
formed with
N
−
i
+
1 entries, with 1
≤
i
≤
M,
is
defined as
x
(
n
)
x
(
n
−
i
+
1
)
x
(
n
−
1
)
x
(
n
−
i
)
x
i
(
n
)
=
.
(7.25)
.
x
(
n
−
N
+
i
)
x
(
n
−
N
+
1
)
=
k
=
1
(
Let us define two vectors of
K
N
−
k
+
1
)
elements
)
=
h
1
(
)
T
h
T
M
(
h
(
n
n
)
···
n
(7.26)
)
=
x
1
(
)
T
x
T
M
(
x
(
n
n
)
···
n
(7.27)
formed with the partial vectors
h
i
(
n
)
and
x
i
(
n
)
. Then, the output of the pure
quadratic filter can be written as
h
T
y
(
n
)
=
(
n
)
x
(
n
).
(7.28)
The aim of the AP algorithm of order
L
is to find the minimum norm of
the coefficient increments that set to zero the last
Laposteriori
errors at time
n
−
j
+
1
h
T
n
+
1
(
n
−
j
+
1
)
=
d
(
n
−
j
+
1
)
−
(
n
+
1
)
x
(
n
−
j
+
1
)
,
(7.29)
where
j
,L
.Inamore explicit form, the following
L
constraints should
be verified for a
L
th order AP algorithm
=
1
,
...
h
T
(
n
+
1
)
x
(
n
)
=
d
(
n
)
,
.
h
T
(
n
+
1
)
x
(
n
−
L
+
1
)
=
d
(
n
−
L
+
1
).
(7.30)
The function
J
(
n
)
to be minimized is
L
1
λ
j
d
)
,
h
T
h
T
J
(
n
)
=
δ
(
n
+
1
)δ
h
(
n
+
1
)
+
(
n
−
j
+
1
)
−
(
n
+
1
)
x
(
n
−
j
+
1
j
=
(7.31)
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