Digital Signal Processing Reference
In-Depth Information
2
s
1
I
s
2
I
E
s
G
2
Imag
G
−
{
y
2
}=
+
Imag
{
n
2
}
(2.64)
where
Real
denote the real and imaginary parts of vector
z
, respectively.
So we can use the Maximum-Likelihood method to detect
{
z
}
,
Imag
{
z
}
(
c
1
R
,
c
2
R
)
,
(
c
1
I
,
c
2
I
)
,
(
s
1
R
,
s
2
R
)
,
(
s
1
I
,
s
2
I
)
separately. For example, by (
2.61
), we can detect
(
c
1
R
,
c
2
R
)
by
H
−
2
c
1
R
c
2
R
E
s
H
2
1
2
Real
1
c
1
R
,
c
2
R
=
arg min
c
1
R
,
c
2
R
{
y
1
}−
(2.65)
F
Similarly, using (
2.62
-
2.64
), we can detect all other codewords.
2.4 Proof of Full Diversity
Diversity is usually defined as the exponent of Signal-to-Noise-Ratio (SNR) in the
error rate expression at high-SNR. Mathematically, the diversity order can be defined
as
log
P
e
log
d
=−
lim
ρ
→∞
(2.66)
ρ
where
ρ
denotes the SNR and
P
e
represents the probability of error. We first consider
(
2.59
) to analyze the diversity for User 1. Here we add a unitary rotation
R
to
c
1
c
2
.
R
c
1
c
2
andwe define the errormatrix
c
1
c
2
.
c
1
Thus, the data vector
d
=
ε
=
−
c
2
By (
2.59
), the pairwise error probability (PEP) can be given by the Gaussian tail
function as [
2
]
⎛
⎝
⎞
⎠
1
2
R
ρ
||
H
2
F
ε
||
|
H
P
(
d
→
d
)
=
Q
(2.67)
4
Now we assume
H
1
and
H
2
have the following singular value decompositions
H
1
=
U
1
Λ
1
V
1
=
U
1
diag
{
λ
11
,λ
12
}
V
1
(2.68)
H
2
=
U
2
Λ
2
V
2
=
U
2
diag
{
λ
21
,λ
22
}
V
2
(2.69)
Since
H
†
1
Λ
1
V
1
and
H
†
V
1
Λ
†
V
2
Λ
†
1
H
1
=
2
H
2
=
2
Λ
2
V
2
are both block-circulant matri-
11
1
[
3
]. We let
V
1
=
V
2
=
11
1
and
Λ
1
=
1
√
2
1
ces,
V
1
=
V
2
=
√
2
Λ
1
=
−
1
−
1
{
λ
11
,λ
12
}
,
Λ
2
=
1
{
λ
21
,λ
22
}
diag
√
2
Λ
2
=
diag
. Therefore, (
2.67
) can be written as
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