Digital Signal Processing Reference
In-Depth Information
17.9 Power minimization using bit allocation
We now demonstrate the optimal DMT design with optimal bit allocation (Sec.
17.5). Consider the channel with a complex coe
cient
0
.
9
jz
−
1
C
(
z
)=1
−
with a zero at 0
.
9
j,
rather close to the unit circle. We choose the block size
M
=4(any
M>L
= 1 will work). We use a square QAM constellation. For
this example we assume
σ
q
=0
.
01 (complex, circular Gaussian noise), and
b
=8
.
The required error probability is assumed to be
P
e
(
k
)=10
−
5
for all
k.
With
optimal bit allocation the DMT system has symbol power
P
min
/M
=30
.
32. If
we do not perform bit allocation and assume
b
k
=
b
for all
k
then the symbol
power is
P
brute
/M
= 450
.
7. So the
coding gain
, defined in Sec. 14.7 to be the
ratio
P
brute
/P
min
,
is given by
P
brute
P
min
=14
.
86
,
which corresponds to a gain of 11
.
7 dB. The optimal bit allocation obtained from
the theoretical formula is as follows:
b
0
=9
.
6260
,b
1
=2
.
1261
,b
2
=9
.
6260
,b
3
=10
.
6220
.
Since the numbers are not integers, we have to truncate them to integers. Trun-
cation ensures that the error probability is at least as small as the specified
values (in this case 10
−
5
for all
k
), for the same power
P
min
=30
.
32
.
In fact,
since a square QAM constellation is assumed, we have to truncate
b
k
to the
nearest
even
integer value. So in our example the final allocation is
b
0
=8
,
1
=2
,
2
=8
,
3
=10
.
With this the average bit rate is
8+2+8+10
4
=7
,
whichissmallerthanthedesiredaveragerate
b
=8
.
Thus the process of trun-
cation to even integers has resulted in a loss of bit rate.
Example 17.3: Channel with deep nulls
In this and the following examples we have
b
=8
,M
=16
,
and
σ
q
=0
.
01 (com-
plex, circular Gaussian noise). First consider the channel
C
(
z
)=
C
0
(
ze
−jω
0
)
,
where
C
0
(
z
)=0
.
1653 + 0
.
1549
z
−
1
+0
.
1700
z
−
2
+0
.
1714
z
−
3
+0
.
1593
z
−
4
+0
.
1790
z
−
5
and
ω
0
=6
π/M.
The channel
C
(
z
) has a complex impulse response, and its
frequency response magnitude is shown in Fig. 17.23 (top and middle) for 0
≤
ω<
2
π.
The channel has rather deep nulls at several frequencies. The bottom
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