Cryptography Reference
In-Depth Information
Chapter 3
Theoretical limits
The recent invention of turbo codes and the rediscovery of LDPC codes have
brought back into favour the theoretical limits of transmission which were re-
puted to be inaccessible until now. This chapter provides the conceptual bases
necessary to understand and compute these limits, in particular those that cor-
respond to real transmission situations with messages of finite length and binary
modulations.
3.1
Information theory
3.1.1 Transmission channel
A channel is any environment where symbols can be propagated (telecommuni-
cations) or recorded (mass memories). For example, the symbols 0 and 1 of the
binary alphabet can be represented by the polarity of a voltage applied to one
end of a pair of conducting wires, stipulating for example that + V corresponds
to1and
V to 0. Then, the polarity measure at the other end will show which
binary symbol was emitted. At the emitter side, the polarity is changed at reg-
ularly spaced intervals to represent the bits of a message and will enable this
message to be reconstituted at the receiver side. This scheme is far too simple
to illustrate modern telecommunications systems but, generally, it is the sign
of a real physical value that represents a binary symbol at the output of the
channel. Usually, a binary symbol is represented by a certain waveform and the
operation that associates a sequence of waveforms with the sequence of symbols
of the message is the modulation. Modulation was the subject of the previous
chapter.
We consider a situation where the channel is not very reliable, that is, where
the observation at the receiving end does not enable the bit really emitted to
be identified with certitude because an interference value, noise, independent
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