Cryptography Reference
In-Depth Information
Figure 3.5 - Penalties in Eb/N 0
for the transmission of finite length packets for the
continuous input channel and the binary input channel as a function of size k (infor-
mation bits), for a coding rate
5 / 6
and different PER.
3.4
Minimum distances required
So far, we have highlighted the theoretical limits and they have been calculated
for the Gaussian channel. These limits determine boundaries, expressed in signal
to noise ratio, between transmission channels at the output of which it is possible
to correct the errors and channels for which this correction cannot be envisaged.
Assuming that codes exist whose decoding can be performed close to these limits,
the question now arises about how we can know which minimum Hamming
distances (MHD) these codes should have in order to satisfy a given objective
of error rates.
Here we present some results for the Gaussian channel and modulations
currently used: 4-PSK, 8-PSK and 16-QAM.
3.4.1 MHD required with 4-PSK modulation
With maximum likelihood decoding after transmission on a Gaussian channel,
the PER has a known upper bound, called the union bound:
N ( d ) erfc dR E b
N 0
1
2
PER
(3.21)
d≥d min
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