Digital Signal Processing Reference
In-Depth Information
)
=
X
T
)
−
1
. Under these conditions, it can be seen that the last
where
S
(
n
(
n
)
X
(
n
K
−
1 components of the vector
e
(
n
)
will be zero (as they are the first
K
−
1
components of
e
p
(
n
−
1
)
). Therefore, only the first column of
S
(
n
)
will be relevant
for the update.
Now, let the
L
matrix
X
×
(
K
−
1
)
(
n
)
include the input data vectors from time
n
−
1to
n
−
K
+
1, i.e.,
x
)
X
X
(
n
)
=
(
n
(
n
)
.
(4.155)
Using (
4.155
), the matrix
S
(
n
)
can be written as:
s
x
T
−
1
)
X
s
T
x
T
(
n
)
(
n
)
(
)
(
)
(
(
)
n
x
n
n
n
S
(
n
)
=
=
,
(4.156)
)
S
X
T
X
T
)
X
s
(
n
(
n
)
(
)
(
)
(
(
)
n
x
n
n
n
and the APA update takes the form:
x
e
)
+
X
X
(
n
)
S
(
n
)
e
(
n
)
=
(
n
)
s
(
n
(
n
)
s
(
n
)
(
n
).
(4.157)
In (
4.156
), the inverse on the right can be solved in terms of the original blocks,
leading to:
x
T
)
X
1
−
(
n
(
n
)
s
(
n
)
s
(
n
)
=
,
x
T
(
n
)
x
(
n
)
X
T
−
1
)
X
X
T
−
(
n
(
n
)
(
n
)
x
(
n
)
I
L
−
X
x
s
(
n
)
=
(4.158)
X
T
−
1
)
X
X
T
x
T
(
n
)
(
n
)
(
n
(
n
)
(
n
)
(
n
)
Replacing in (
4.157
), the APA update can be put as:
I
L
−
X
x
−
1
X
T
)
X
X
T
(
n
)
(
n
(
n
)
(
n
)
(
n
)
e
(
n
)
x
e
)
+
X
I
L
−
X
x
(
n
)
s
(
n
(
n
)
s
(
n
)
(
n
)
=
−
1
X
T
)
X
X
T
x
T
(
n
)
(
n
)
(
n
(
n
)
(
n
)
(
n
)
(
)
p
n
=
(
).
e
n
(4.159)
p
T
(
)
(
)
n
p
n
This implies that the direction of the update of the APA comes from the orthogonal
projection of the most recent input vector
x
(
n
)
onto the orthogonal subspace spanned
X
by the columns of
1 input vectors).
In order to show how this fact is related to the decorrelating property of the APA,
let the input signal come from an autoregressive process of order
K
(
n
)
(the previous
K
−
−
1[
39
], i.e.,
)
=
X
x
(
n
(
n
)
a
+ ˜
v
(
n
),
(4.160)
