Cryptography Reference
In-Depth Information
which is sucient to enable the capacity to be evaluated by using (3.14).
In the case where the attenuation is known, the probability density for a partic-
ular realization of
α
can be written:
√
2
πσ
2
exp
2
1
−
|
y
−
α
x
i
|
p
(
y
|
x
i
,α
)=
2
σ
2
The instantaneous capacity
C
α
for this particular realization of
α
is first calcu-
lated and then we have to average
C
α
over the set of realizations of
α
in order
to obtain the capacity of the channel:
x
i
,α
)log
2
p
(
y
d
y
+
∞
M
1
M
x
i
,α
)
p
(
y
|
C
α
=
p
(
y
|
|
α
)
i
=1
−∞
+
∞
C
=
C
α
p
(
α
)
d
α
=
E
[
C
α
]
0
3.3
Practical limits to performance
In the sections above, we obtained the theoretical limits for performance which
are subject to certain hypotheses that are not realistic in practice, in particular
the transmission of infinite length data blocks. In the great majority of commu-
nication systems today, it is a sequence of data blocks that is transmitted, these
blocks being of very variable size depending on the system implemented. Logi-
cally, limited size block transmission leads to a loss of performance compared to
infinite size block transmission, because the quantity of redundant information
contained in the codewords is lower.
Another parameter used to specify the performance of real transmission sys-
tems is the packet error rate (PER), which corresponds to the proportion of
blocks of wrong data (containing at least one binary error after decoding).
What follows contains some results on the Gaussian channel, for two cases:
the binary input and continuous output Gaussian channel, and the continuous
input and output Gaussian channel. The case of the continuous input can be
assimilated to that of a modulation with an infinite number of states
M
.The
fewer states we have to describe the input, the less ecient the communication.
Consequently, a binary input channel gives a lower bound on the practical per-
formance of the set of modulations, whereas a continuous input channel gives
its higher limit.
3.3.1 Gaussian binary input channel
Initial work on this channel was done by Gallager [3.2]. We denote again
p
(
y
x
)
the probability of transition on the channel, and we consider information mes-
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