Digital Signal Processing Reference
In-Depth Information
plot in the figure shows the channel input power per symbol as a function of the
symbol error probability, which is specified to be identical for all subchannels
(i.e.,
P
e
(
k
) is identical for all
k
). The plot shows the minimum power
P
min
of
the optimized system as well as the power
P
brute
for the system with no bit
allocation (
b
k
=
b
for all
k
), both divided by
M
to get the per-symbol value. In
this example the coding gain is
P
brute
P
min
=19
.
17
,
which corresponds to 12
.
8 dB for any symbol error probability. Thus the sepa-
ration between the two curves in the bottom plot (which has a log-log axis) is
identical for all symbol error probabilities.
Example 17.4: Channel with deeper nulls
In this example all quantities are as in the preceding example, but the channel
is different. We take
C
(
z
)=
C
0
(
ze
−jω
0
)
,
where
ω
0
=6
π/M
as before, and
C
0
(
z
)=0
.
1734 + 0
.
1634
z
−
1
+0
.
1664
z
−
2
+0
.
1651
z
−
3
+0
.
1621
z
−
4
+0
.
1696
z
−
5
.
Figure 17.24 shows the channel response and the required channel input power
as a function of error probability. In this example the coding gain is
P
brute
P
min
= 127
,
which corresponds to 21 dB. The higher coding gain is a consequence of the
fact that the channel nulls in this example are deeper than in Example 17.3.
As a result some of the DFT coe
cients
C
[
k
] have very small values, and the
set of numbers 1
/|C
[
k
]
|
2
have large variation. This makes their arithmetic to
geometric mean ratio (
AM/GM
) quite large. Recall here that the coding gain
due to bit allocation is precisely the
AM/GM
ratio (Sec. 14.7):
M−
1
1
M
1
|
C
[
k
]
|
2
=
P
brute
P
min
k
=0
G
=
1
/M
M−
1
1
|
C
[
k
]
|
2
k
=0
The aforementioned variation of
C
[
k
] can be seen qualitatively from the following
table. Indeed, for
k
= 11 the second channel has a much smaller value of
C
[
k
]
,
though for other values of
k
the two channels have comparable magnitudes.
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