Digital Signal Processing Reference
In-Depth Information
q
(
n
)
J
s
(
n
)
s
(
n
)
M
P
M
F
G
H
precoder
channel
equalizer
Figure 16.1
. A transceiver with precoder
F
and equalizer
G
.
and that the components of
q
(
n
) are Gaussian.
1
In this section we consider the
zero-forcing (ZF) transceiver, which satisfies
GHF
=
I
.
(16
.
2)
In view of zero forcing, the reconstruction error
e
s
(
n
)=
s
(
n
)
−
s
(
n
)
(16
.
3)
is nothing but
q
(
n
) filtered through
G
.
So,
e
s
(
n
) has zero mean, and the com-
ponents of
e
s
(
n
) are Gaussian. The variances of these components are
|
2
]
.
E
k
=
E
[
|
s
k
(
n
)
−
s
k
(
n
)
(16
.
4)
So the probability of error in the
k
th symbol is of the form (Sec. 11.5)
A
√
E
k
,
P
e
(
k
)=
c
Q
(16
.
5)
where the constants
c
and
A
depend on the type of symbol constellation used
for
s
k
(
n
) (e.g., PAM, QAM, and so forth). For example, in a
b
-bit QAM system,
2
−b/
2
)and
A
=
3
σ
s
2
b
c
=4(1
−
1
.
(16
.
6)
−
The average symbol error probability therefore takes the form
A
√
E
k
.
M−
1
c
M
P
e
=
Q
(16
.
7)
k
=0
16.2.1 Introducing the unitary matrix U
Now consider Fig. 16.2, where we have inserted a unitary matrix
U
(i.e., a matrix
satisfying
U
†
U
=
I
) at the receiver, and its inverse
U
†
at the transmitter. Given
any zero-forcing pair
,
we will show how
U
should be chosen such that
the average symbol error probability is minimized.
{
F
,
G
}
1
For the PAM case the noise
q
(
n
) is assumed to be real and Gaussian, whereas for QAM
we assume the noise is circularly symmetric Gaussian (Sec. 2.3.2).
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