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Error Control Coding
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Parity nodes (Row no.) 3
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Illustrating a Tanner graph.
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lines (technically referred to as edges) join the bit nodes to their respective parity check nodes. Edges occur wherever a 1 appears in the H matrix. Thus for row 0, 1s appear in the c0, c1, c2 and c9 positions [(and as shown by the first parity line in Eq. (11.26)]. In Fig. 11.13, only the parity equations for rows 0, 1, and 3 are shown for clarity, but the complete Tanner graph would show all the edges. Messages pass along the edges. Initially, the output from the channel demodulator provides the first soft estimate of a bit. If p1 is the probability that, the bit is a 1, the probability that it is a zero is 1 p1. These estimates are sent to their respective parity check equations where the equation is used to derive probability estimates for a bit. Considering the first equation for example, an estimate for the probability that c0 is a 1 can be obtained from the probabilities for c1, c2, and c9 being 1. For the group c1c2c9 the combinations that would result in c0 being 1 are 100, 010, 100, and 111. The sum of the corresponding probabilities gives an estimate, p for the 0 probability of c0 being 1: pr0 p1 Q1 1 p9 Q1
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The estimates from the parity nodes are returned to the respective bit nodes. The bit node now has estimates from the parity check nodes and from the channel, which enables a new estimate for probability to be calculated. For example if bit node 0 receives estimates from parity check nodes A, B, and C, denoted by p1 , p1 , p1 respectively, and p1 from the A B C CH channel, the new estimates sent to these parity check nodes are the 1 1 products K( p1 p1 pC) to parity node A, K( p1 p1 pC) to parity node B, and CH B CH A 1 K(p CH pA pB) to parity node C, where K is a normalizing constant. It will
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Code rate vs Performance Eb/N0 at BER = 10 8 1.0 0.9 0.8 Code rate 0.7 0.6 0.5 0.4 0.3 0.2 0.0 1.0 2.0 3.0 Eb/N0 (dB)
Figure 11.14 Shannon limit (BPSK) TPC 4K blocks TPC 16K blocks Reed-Solorn on/Vlterbi LDPC 30K LDPC 16K
FEC performance of Comtech AHA 4701 LDPC coder. (Courtesy of A. Summers Comtech AHA.)
be noticed that the estimate from any given parity node is not included in the new estimate sent to that parity node. This process is repeated a number of times (iterated) before a final hard decision is made. Turbo codes and LDPC codes are being employed in a number of satellite systems with a resulting improvement in channel performance, for example the Digital Video Broadcast S2 standard (DVB-S2) employs LDPC as the inner code in its FEC arrangement (Breynaert, 2005, Yoshida, 2003), and the DVB-RCS plans to use turbo codes (talk Satellite, 2004). The performance of a number of codes is shown in Fig. 11.14. 11.12 Automatic Repeat Request (ARQ) Error detection without correction is more efficient than FEC in terms of code utilization. It is less complicated to implement, and more errors can be detected than corrected. Of course, it then becomes necessary for the receiver to request a retransmission when an error has been detected. This is an automatic procedure, termed automatic repeat request (ARQ). The request for retransmission must be made over a low-bit-rate channel where the probability of bit error can be kept negligibly small. Because of the long round-trip delay time, on the order of half a second or more, encountered with geostationary satellites, ARQ is only suitable for transmission that is not sensitive to long delays. ARQ is normally used with block encoding. An estimate of the probability of an error remaining undetected can be made. With an (n, k) block code, the number of datawords is 2k, and