Digital Signal Processing Reference
In-Depth Information
where
H
is the
M
t
×
M
r
complex channel matrix,
g
is the
T
×
M
t
complex matrix of the
transmitted signals,
X
is the
T
×
M
r
complex matrix of the received signals, and
N
is the
T
×
M
r
matrix of noise.
Let us denote complex information symbols prior to space-time encoding as
s
1
, …,
s
J
and define the
J
× 1 symbol vector
s
[
s
1
, …,
s
J
]
T
. he
T
×
M
t
matrix
g
=
g
(
s
) is called an
orthogonal space-time block code (OSTBC) if [16-18]:
•
All elements of
g
(
s
) are linear functions of the
J
complex variables
s
1
, …,
s
J
and
their complex conjugates.
For any
•
s
, it satisfies
g
H
(
s
)
g
(
s
) =
s
2
I
.
Assuming the OSTBC signaling, the matrix
g
(
s
) can be written as [16, 61, 62]
J
l
(
)
gs
()
=
C
Re
{}
s
+
D
Im
{}
s
,
(8.55)
l
l
l
l
=
1
where
C
l
g
(
e
l
),
D
l
g
(
j
e
l
),
e
l
is the
J
× 1 vector having one in the
l
th
position and
zeros elsewhere, and
j
−1. Using (8.55), the MIMO model (8.54) can be rewritten as
[59, 62]
Xs
=+
A
N
,
(8.56)
where the “underline” operator for any matrix
P
is defined as
,
P
vecRe
vecIm
{()}
{()}
P
P
(8.57)
vec{·} is the vectorization operator stacking all columns of a matrix on top of each other,
and the 2
M
r
T
× 2
J
real matrix
A
=
A
(
H
) is given by
.
A
=
CH
, , , ,,
…
CHDH
…
DH
(8.58)
1
J
1
J
This matrix can be shown to satisfy the following orthogonality property (that, in
turn, is due to the orthogonality property of the space-time code):
2
A
T
= .
HI
(8.59)
From (8.58), it can be seen that the matrix
A
in (8.56) captures both the effects of the
OSTBC and the channel, while the vector
s
depends on the information symbols only.
The columns of
A
can be viewed as spatio-temporal signatures that describe the receiver
response to each entry of
s
(i.e., to real or imaginary parts of each information symbol).
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