Digital Signal Processing Reference
In-Depth Information
Appendix to Chapter 17
17.A Error probability versus channel-output SNR
The bottom plot in Fig. 17.17 shows the error probability as a function of channel
output SNR. Even though such a plot is sometimes included in discussions it
should be warned that comparisons based on this plot have only limited value.
Some conclusions can be misleading. For example, the error probability for the
pure-MMSE system is “worse” than that of the ZF-MMSE system in some parts
of the figure!
The reason is that, for fixed
p
0
(channel input power) the channel output
power is not the same for all four methods. This is because, the covariance of
the signal at the channel input is not identical in all these methods, even though
the trace of this covariance is fixed at
p
0
. Even though the covariance of the input
s
(
n
)is
σ
s
I
, the above differences in covariance arise because the precoders are
different in these systems. To analyze this, consider again the optimal (MMSE)
cyclic-prefix system in Fig. 17.7(a). The diagonal matrices
Λ
a
and
Λ
b
depend
on whether the MMSE system has the zero-forcing property or not. Since
s
(
n
)
is assumed to have covariance
σ
s
I
, the covariance of the channel input is
σ
s
M
W
†
Λ
a
Λ
a
W
.
The channel input power, which is the trace of this, is given by
R
in
=
M
Tr
=
M
Tr
WW
†
Λ
a
Λ
a
=
σ
s
Tr
Λ
a
Λ
a
,
σ
s
σ
s
W
†
Λ
a
Λ
a
W
p
0
=
where we have used Tr (
AB
)=Tr(
BA
)
,
and the fact that
WW
†
=
M
I
.
The
channel output has covariance
R
out
=
HR
in
H
†
, so the channel-output power is
M
Tr
HW
†
Λ
a
Λ
a
WH
†
.
σ
s
p
out
=
To simplify this, recall that in cyclic-prefix systems
H
is circulant so that
H
=
W
†
Λ
c
W
M
where
Λ
c
is diagonal. Using
W
†
W
=
WW
†
=
M
I
,
we therefore obtain
HW
†
=
W
†
Λ
c
,
so that
M
Tr
σ
s
W
†
Λ
c
Λ
a
Λ
a
Λ
c
W
p
out
=
M
Tr
WW
†
Λ
c
Λ
a
Λ
a
Λ
c
=
σ
s
Tr
Λ
c
Λ
a
Λ
a
Λ
c
.
σ
s
=
Summarizing, we have the channel-input power
p
0
=
σ
s
Tr
Λ
a
Λ
a
=
σ
s
M−
1
λ
a,k
|
2
,
|
(17
.
62)
k
=0
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