Digital Signal Processing Reference
In-Depth Information
Thus, insertion of the unitary matrices does not affect the zero-forcing property,
the transmitted power, or the average MSE. For example, if
{
F
,
G
}
is an MMSE
{
FU
†
,
UG
}
transceiver, then so is the modified transceiver
.
Although
U
does
not change the MSE, we will see that we
can
reduce the average error probability
(16.7) by optimizing
U
.
16.2.2 Minimizing error probability by optimizing U
The average symbol error probability in Fig. 16.2 is, from Eq. (16.7),
A
E
s,k
,
M−
1
c
M
P
e
(
y
)=
Q
(16
.
14)
k
=0
where the argument vector
y
is defined as
y
=[
E
s,
0
E
s,
1
...
E
s,M−
1
]
.
(16
.
15)
(
A/
√
y
)isa
convex
function of
y
as long as the signal-to-error ratio exceeds a certain threshold. This idea was
also discussed in Chap. 11 (Sec. 11.5). More precisely
It is shown in Sec. 21.2.3 of Chap. 21 that
Q
is
convex in
A
E
s,k
E
s,k
<A
2
/
3
E
s,k
for
Q
(16
.
16)
E
s,k
>A
2
/
3
.
concave in
E
s,k
for
Assuming that the errors
E
s,k
are su
ciently small, the convexity condition
holds. In Sec. 11.5 of Chap. 11 we presented examples of typical signal-to-
noise ratios for which convexity holds, and found that the assumption is often
reasonable.
Assuming the convexity condition holds, we conclude from Chap. 21 (Sec.
21.4) that
P
y
(
y
) in Eq. (16.14) is a
Schur-convex
function of
y
. By definition,
what this means is that, if a vector
y
1
is majorized by
y
2
(for definitions see
Sec. 21.3, Chap. 21), then
≤P
e
(
y
2
)
.
(16
.
17)
Now refer to Eq. (16.10). The freedom to change the unitary matrix
U
gives
us the freedom to adjust the elements [
E
s
]
kk
, but only subject to the constraint
that their sum be constant (from Eq. (16.12)). We now use a result from Chap.
21 (Lemma 21.1) which says that the vector
P
e
(
y
1
)
α
[1
1
...
1]
,
α>
0
,
is majorized by any vector of the form
0
,
with identical sum, that is,
k
y
k
=
αM.
Thus the unitary
U
that minimizes
P
e
(
y
) is the one that equalizes the diagonal elements of
E
s
,thatis,
[
y
0
y
1
...
y
M−
1
]
,
k
≥
⎡
⎤
1
×
...
×
⎣
×
1
...
×
⎦
UE
x
U
†
=
α
=
E
s
.
(16
.
18)
.
.
.
.
.
.
××
...
1
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