Page 358 - Satellite Communications, Fourth Edition

338 Chapter Eleven

Fig. 11.10. Note that although the graph shows E /N in decibels ([E /N ]
0
b
0
b
in our previous notation) the power ratio must be used in evaluating
Eq. (11.22).
In any practical system there will be a finite probability of bit error,
and to see how this fits in with the Shannon limit, consider the BER
graph of Fig. 10.17, which applies for BPSK and QPSK. From Fig. 10.17
[or from calculation using Eq. (10.18)], for a probability of bit error (or
BER in this case) of 10 5 the [E /N ] is about 9.6 dB. (See also the
b
0
uncoded curve of Fig. 11.8). As shown in Sec. 10.6.3 the bit rate to band-
width ratio is 1/(1 ) for BPSK and 2/(1 ) for QPSK. For purposes
of comparison ideal filtering will be assumed, for which 0. Thus on
Fig. 11.10, the points (1, [9.6]) for BPSK and (2, [9.6]) can be shown. At
an [E /N 0 ] of 9.6 dB the Shannon limit indicates that a bit rate to band-
b
width ratio of about 5.75 : 1 should be achievable, and it is seen that
BPSK and QPSK are well below this. Alternatively, for a bit-rate/band-
width ratio of 1, the Shannon limit is 0 dB (or an E b /N 0 ratio of unity),
compared to 9.6 dB for BPSK.

11.11 Turbo Codes and LDPC Codes

Till 1993 all codes used in practice fell well below the Shannon limit.
In 1993, a paper (Berrou et al., 1993) presented at the IEEE
International Conference on Communications made the claim for a
digital coding method that closely approached the Shannon limit (a pdf
file for the paper will be found at www-elec.enst-bretagne.fr/equipe/
berrou/ Near%20Shannon%20Limit%20Error.pdf). Subsequent test-
ing confirmed the claim to be true. This revitalized research into
coding, resulting in a number of “turbo-like” codes, and a renewed
interest in codes known as low density parity check (LDPC) codes (see
Summers, 2004).
Turbo codes and LDPC codes use the principle of iterative decoding in
which “soft decisions” (i.e., a probabilistic measure of the binary 1 or 0
level) obtained from different encoding streams for the same data, are
compared and reassessed, the process being repeated a number of times
(iterative processing). This is sometimes referred to as soft input soft
output to describe the fact that during the iterative process no hard deci-
sions (binary 1 or 0) are made regarding a bit. Each reassessment gen-
erally provides a better estimate of the actual bit level, and after a certain
number of iterations (fixed either by convergence to a final value, or by
a time limit placed on the process) a hard decision output is generated.
Turbo codes are so named because the iterative or feedback process was
likened to the feedback process in a turbo-charged engine (see Berrou
et al., 1993). The turbo principle can be applied with concatenated block