Cryptography Reference
In-Depth Information
introduced by the encoder and the decoder, the degree of flexibility of the code
(in particular its ability to conform to different lengths of message and/or to
different coding rates) are also to be considered more or less closely, depending
on the specific constraints of the communication system.
The residual errors that the decoder has not managed to correct are measured
by using two parameters. The binary error rate (BER) is the ratio between the
number of residual binary errors and the total number of bits of information
transmitted. The word, block or packet error rate (PER) is the number of
codewords badly decoded (at least one of the bits of information is wrong) out
of the total number of codewords transmitted. The ratio between BER and PER
is the average density of errors
δ
e
in the systematic part of a badly decoded word:
w
k
BER
PER
δ
e
=
=
(1.13)
where
w
=
kδ
e
is the average number of erroneous information bits in the
systematic part of a badly decoded block.
Figure 1.5 gives a typical example of the graphic representation of the per-
formance of error correction coding and decoding. The ordinate gives the BER
on a logarithmic scale and the abscissa carries the signal to noise ratio
E
N
0
,ex-
pressed in decibels (dB).
N
0
is defined by (1.9) and
E
b
is the energy received
per bit of information. If
E
s
is the energy received for each of the symbols of
the codeword,
E
s
and
E
b
are linked by:
E
s
=
RE
b
(1.14)
The comparison of different coding and decoding processes or the variation in
performance of a particular process with the coding rate are always defined
with the same global reception energy. When there is no coding, the energy
per received codeword is
kE
b
. With coding, which increases the number of
bits transmitted, the energy
kE
b
is to be distributed between the
n
bits of
the codeword, which justifies relation (1.14). The reference of energy to be
considered, independent of the code and of the rate, is therefore
E
b
.
In Figure 1.5 are plotted the curves for the error correction of the (8, 4, 4) and
(7, 4, 3) Hamming codes that have been dealt with throughout this introduction
on a Gaussian channel. Hard-input decoding according to (1.4) and soft-input
decoding according to (1.11) are considered. Also shown in the diagram is the
curve for the binary error probability
P
e
that is obtained on this channel without
using coding
2
. This curve is linked to the complementary error function erfc(
x
)
given by the relation (2.74) of Chapter 2:
P
e
=
2
erfc
E
N
0
. With a low error
1
2
The distinction between
P
e
and BER is only traditional: the value of
P
e
is given by an
equation whereas the BER is obtained by measuring or simulation.
