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-
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