Cryptography Reference
In-Depth Information
coded according to different schemes was simulated (Figures 5.16 and 5.17):
classical (non-recursive non-systematic), non-recursive systematic and recursive
systematic.
The blocks were constructed following the classical trellis termination tech-
nique for non-recursive codes whereas the recursive code is circular tail-biting
(see Section 5.5). The decoding algorithm used is the MAP algorithm.
The BER curves are in perfect agreement with the conclusions drawn during
the analysis of the free distance of codes and of their transfer function: the
systematic code is not as good as the others at high signal to noise ratio and
the classical code is then slightly better than the recursive code. At low signal
to noise ratios, the hierarchy is different: the recursive code and the systematic
code are equivalent and better than the classical code.
Comparing performance as a function of the size of the frame (53 and 200
bytes) shows that the performance hierarchy of the codes is not modified. More-
over, the bit error rates are almost identical. This was predictable as the sizes
of the frames are large enough for the transfer functions of the codes not to be
affected by edge effects. However, the packet error rate is affected by the length
of the blocks since although the bit error probability is constant, the packet
error probability increases with size.
The comparisons above only concern codes with 8 states. It is, however, easy
to see that the performance of a convolutional code is linked with its capacity
to provide information on the succession of data transmitted: the more the
code can integrate successive data into its output symbols, the more it improves
the quality of protection these data. In other words, the greater the number
of states (therefore the size of the register of the encoder), the more ecient
a convolutional code is (within its category). Let us compare three recursive
systematic codes:
4 states
[1
,
(1 +
D
2
)
/
(1 +
D
+
D
2
)
],
•
8 states
[1
,
(1 +
D
2
+
D
3
)
/
(1 +
D
+
D
3
)]
•
and 16 states
[1
,
(1 +
D
+
D
2
+
D
4
)
/
(1 +
D
3
+
D
4
)]
.
•
Their performance in terms of BER and PER were simulated on a Gaussian
channel and are presented in Figure 5.18.
Thehigherthenumberofstatesofthecode,thelowertheresidualerrorrates
are. For a BER of
10
−
4
, 0.6 dB are thus gained when passing from 4 states to
8 states and 0.5 dB when passing from 8 states to 16 states. This remark is
coherent with the qualitative justification of the interest of a large number of
states. It would therefore seem logical to choose a convolutional code with a large
number of states to ensure the desired protection, especially since such codes
offer the possibility of producing redundancy on far more than two components,
and therefore of providing even higher protection. Thus, the
Big Viterbi Decoder
project at
NASA
's
Jet Propulsion Laboratory
used for transmissions with space
