Digital Signal Processing Reference
In-Depth Information
b
11
b
11
b
12
b
12
B
1
B
2
=
,
=
(4.6)
b
21
b
22
b
21
b
22
denote
the
precoders
of
User
2
in
time
slots
1
and
2,
respectively.
Here
A
i
2
F
B
i
2
F
||
||
=||
||
=
1,
i
=
1
,
2, in order to satisfy the normalization conditions
[
2
].
In time slots 1 and 2, the received signals are respectively denoted by
y
1
y
2
y
1
y
2
y
1
y
2
=
,
=
.
(4.7)
Then, in time slot 1, the signal model can be written as
E
s
HA
1
E
s
GB
1
c
1
s
1
y
1
W
1
=
+
+
.
(4.8)
c
2
s
2
In time slot 2, we have
E
s
HA
2
−
E
s
GB
2
−
c
2
s
2
y
2
W
2
=
+
+
(4.9)
c
1
s
1
n
1
n
2
,
where
E
s
denotes the total transmit energy of each user and
W
1
=
denote the noise at the receiver in time slots 1 and 2, respectively.
We assume that
n
1
,
n
2
,
n
1
n
2
W
2
=
n
1
,
n
2
are i.i.d complex Gaussian noises with mean 0 and
variance 1. In order to simplify the notation, we let
h
i
11
h
i
12
h
i
21
h
i
22
h
11
a
i
11
+
h
12
a
i
21
h
11
a
i
12
+
h
12
a
i
22
H
i
HA
i
=
,
i.e.
,
=
(4.10)
h
21
a
i
11
+
h
22
a
i
21
h
21
a
i
12
+
h
22
a
i
22
g
11
b
i
11
+
g
i
11
g
i
12
g
12
b
i
21
g
11
b
i
12
+
g
12
b
i
22
G
i
GB
i
=
,
i.e.
,
=
(4.11)
g
i
21
g
i
22
g
21
b
i
11
+
g
22
b
i
21
g
21
b
i
12
+
g
22
b
i
22
where
i
2. With these new notations, after applying some simple algebra to
Eqs. (
4.8
) and (
4.9
), we have
=
1
,
⎛
⎝
⎞
⎠
+
c
1
y
1
y
2
E
s
h
11
h
12
n
1
n
2
g
11
g
12
c
2
=
,
(4.12)
h
21
h
22
g
21
g
22
s
1
s
2
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