Digital Signal Processing Reference
In-Depth Information
pendency between symbols is introduced, the mapper acts as the source of
arandomprocess
a
[
n
]
which is also i.i.d. The power of the output sequence
can be expressed as
l
g
r
,
y
i
d
.
,
©
,
L
s
=
E
a
[
n
]
2
2
a
σ
G
0
α
∈
α
2
=
p
a
(
α
)
(12.10)
G
0
σ
2
α
=
(12.11)
where
p
a
(
α
)
is the probability assigned by the mapper to symbol
α
∈
;
the distribution over the alphabet
is one of the design parameters of the
mapper, and is not necessarily uniform. The variance
2
α
σ
is the intrinsic
power of the alphabet and it depends on the alphabet size (it increases ex-
ponentially with
M
), on the alphabet structure, and on the probability dis-
tribution of the symbols in the alphabet. Note that, in order to avoid wast-
ing transmission energy, communication systems are designed so that the
sequence generated by the mapper is
balanced
,i.e.itsDCvalueiszero:
E
[
α
[
n
]]=
α
p
a
(
α
)=
0
α
∈
Using (8.25), the power of the transmitted signal, after upsampling andmod-
ulation, is
ω
max
2
G
e
j
ω
)
1
π
1
2
2
s
G
0
σ
2
α
σ
=
(
(12.12)
ω
min
The shaper is designed so that its overall energy over the passband is
G
2
=
2
π
andwecanexpressthisasfollows:
2
s
G
0
σ
2
α
σ
=
(12.13)
In order to respect the power constraint, we have to choose a value for
G
0
and design an alphabet
so that:
2
s
σ
≤
P
max
(12.14)
where
P
max
is the maximum transmission power allowed on the channel.
The goal of a data transmission system is to maximize the
reliable
through-
put but, unfortunately, in this respect the parameters
2
α
and
G
0
act upon
conflicting priorities. If we use (12.9) and boost the transmitter's bitrate by
increasing
M
,then
σ
2
α
grows and we must necessarily reduce the gain
G
0
to fulfill the power constraint; but, in so doing, we impair the reliability of
the transmission. To understand why that is, we must leap ahead and con-
sider both a practical alphabet and themechanics of symbol decoding at the
transmitter.
σ
