Cryptography Reference
In-Depth Information
Figure 3.2 - Normalized Shannon limit.
discrete values,
M
being the modulation order, and have dimension
N
,thatis
x
i
=[
x
i
1
,x
i
2
,
···
,x
iN
]
. The transition probability of the Gaussian channel is:
√
πN
0
exp
N
N
x
in
)
2
1
−
(
y
n
−
p
(
y
|
x
i
)=
p
(
y
n
|
x
in
)=
N
0
n
=1
n
=1
and we assume inputs taking the
M
different possible values equiprobably. De-
noting
d
ij
=(
x
i
−
x
j
)
/
√
N
0
the vector of dimension
N
relative to the distance
between the symbols
x
i
and
x
j
,and
t
an integration vector of dimension
N
,we
obtain a simplified expression of (3.14), representing the capacity of a discrete
input Gaussian channel, for any type of modulation:
C
=log
2
(
M
)
M
2
d
t
+
∞
+
∞
exp
2
log
2
j
=1
exp
i
=1
(
√
π
)
−N
M
M
−
···
−|
t
|
−
2
td
ij
−|
d
ij
|
(3.16)
−∞
−∞
N
times
C
being expressed in bit/symbol. We note that
d
ij
increases when the signal
to noise ratio increases (
N
0
decreases) and the capacity tends towards
log
2
(
M
)
.
The different possible modulations only appear in the expression of
d
ij
.The
discrete sums from 1 to
M
represent the possible discrete inputs. For the final
calculation, we express
d
ij
as a function of
E
s
/N
0
according to the modulation,
E
s
being the energy per symbol, and the capacity of the channel can be deter-