Digital Signal Processing Reference
In-Depth Information
8.4.2
Subspace Method for Blind Identification of MIMO Systems
In blind MIMO system identification, the input-output relationship may be
rewritten in the form of a low-rank model if a sufficient number of received
data samples are available. The fact that the column space of the received data
matrix and column space of the channel matrix span the same subspace may
then be exploited in determining the parameters of the FIR-MIMO model. A
well-known noise subspace method is used to determine the channel matrix
up to an ambiguity matrix. In blind identification some ambiguities always
remain unsolved. In blind subspace methods where the low-rank model is
used, these ambiguities may be described in a form of constant full-rank
m
×
m
ambiguity matrix, where
m
is the number of transmitted sources.
The remaining ambiguity matrix can be considered to perform instanta-
neous linear mixing of the equalized signals. Consequently, we have an I-
MIMO model with non-Gaussian communication signals that may be demixed
via blind source separation (BSS).
5
To accomplish the separation, the sources
are assumed to be statistically independent, which is reasonable if signals
originate from different users.
8.4.2.1 Signal Model
We assume the standard FIR-MIMO baseband signal model
39
with
m
trans-
mitters and
n
receivers, in which the received signal
y
(
k
)
having
n
components
arranged as a column vector is of the form
L
(
)
=
(
−
)
+
(
).
y
k
H
l
s
k
l
w
k
(8.17)
l
=
0
]
T
(
)
=
(
)
(
)
...
(
)
Here
s
k
[
s
1
k
,s
2
k
,
,s
m
k
is an
m
-dimensional signal vector
>
{
}
×
(
n
m
matrix-
valued impulse response coefficients associated with the transfer function
H
m
),
L
is the channel order,
H
l
L
are the unknown
n
l
=
0
,
...
l
=
0
H
k
z
−
l
, and
w
(
)
=
(
)
z
k
is noise. Let us make the following notation:
H
,
H
L
]
T
.
We assume that
=
[
H
0
,
H
1
,
...
H
rank
(
(
z
))
=
m
for each
z
(8.18)
H
(
L
)
is of full column rank
.
(8.19)
These assumptions are needed to ensure the channel identifiability by using
only second-order statistics of the received signal vector
y
(
k
)
; see Reference
39. By stacking
N
+
1 observations of Equation 8.17 into an
(
N
+
1
)
n
×
1 vector
[
y
T
,
y
T
,
y
T
]
T
Y
(
k
)
=
(
k
)
(
k
−
1
)
,
...
(
k
−
N
)
we may write
H
Y
(
k
)
= H
(
)
S
(
k
)
+
W
(
k
).
(8.20)
N
T
,
s
T
,
T
]
T
,
V
T
,
Here,
S
(
k
)
=
[
s
(
k
)
(
k
−
1
)
...
,
s
(
k
−
N
−
L
)
(
k
)
=
[
v
(
k
)
H
T
,
T
]
T
, and
v
(
k
−
1
)
...
,
v
(
k
−
N
)
H
(
)
is the
(
N
+
1
)
n
×
m
(
L
+
N
+
1
)
channel
N
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