Digital Signal Processing Reference
In-Depth Information
where
g
(
i
)
l
,l
=
0
,
1
,
...
,N
are of size
n
×
1. Define
g
(
i
)
0
g
(
i
)
1
g
(
i
N
···
0
···
0
.
.
.
.
0
(
i
)
0
g
(
i
)
1
g
(
i
N
···
G
i
=
.
(8.22)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
0
(
i
)
0
g
(
i
)
1
g
(
i
N
···
···
0
that for each column of the matrix
H
,
h
i
,
i
It can be shown
35
,
39
=
1
,
...
,m
,
h
i
r
h
i
H
i
1
G
G
=
0
.
i
=
i
Moreover, the dimension of the null space of the matrix
r
H
i
C =
1
G
G
i
i
=
is
m
. This implies that
BR
−
1
,
H
=
where
B
is a
n
corresponding to
the eigenvalues that are equal to zero, and
R
is an invertible
m
(
L
+
1
)
×
m
matrix of the eigenvectors of
C
m
matrix. In
other words, by using the noise subspace eigenvectors, we may determine the
channel matrix up to a right multiplication of an invertible
m
×
×
m
ambiguity
matrix, i.e., determine
B
=
HR
.
(8.23)
8.4.2.3 Equalization Using Subspace Method and Blind Source Separation
Assume that we know the channel matrix up to a right multiplication of an
invertible
m
×
m
matrix
R
; i.e., we know
B
=
HR
. Then
R
,
H
(
)
= H
(
)
B
H
N
N
where
R
is an
m
(
L
+
N
+
1
)
×
m
(
L
+
N
+
1
)
block diagonal matrix
.
R0
···
0
0R
···
0
R
.
=
.
.
.
0
00
···
R
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