Reed-Solomon Error Probability
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Table of Contents
8.1.1 Reed-Solomon Error Probability
The Reed–Solomon (R–S) codes are particularly useful for burst-error correction;
that is, they are effective for channels that have memory. Also, they can be used efficiently
on channels where the set of input symbols is large. An interesting feature
of the R–S code is that as many as two information symbols can be added to an
R–S code of length n without reducing its minimum distance. This extended R–S
code has length n + 2 and the same number of parity check symbols as the original
code. From Equation (6.46), the R–S decoded symbol error probability, PE, in
terms of the channel symbol error probability, p, can be written as follows [2]:
where t is the symbol-error correcting capability of the code, and the symbols are
made up of m bits each.
The bit error probability can be upper bounded by the symbol error probability
for specific modulation types. For MFSK modulation with M = 2m, the relationship
between PB and PE as given in Equation (4.112) is repeated here:
Figure 8.1 shows PB versus the channel symbol error probability p, plotted
from Equations (8.7) and (8.8) for various t-error-correcting 32-ary orthogonal
Reed–Solomon codes with n = 31 (thirty-one 5-bit symbols per code block).
Figure 8.1 PB versus p for 32-ary orthogonal signaling and n = 31, t-error-correcting Reed–Solomon
coding. (Reprinted with permission from Data Communications, Networks and Systems, ed. Thomas C.
Bartee, Howard W. Sams Company, Indianapolis, Ind., 1985, p. 311. Originally published in J. P.
Odenwalder, Error Control Coding Handbook, M/A-COM LINKABIT,
Inc., San Diego, Calif., July 15, 1976, p. 91.)
Figure 8.2 shows PB versus Eb/N0 for such a coded system using 32-ary MFSK modulation
and noncoherent demodulation over an AWGN channel [2]. For R–S
codes, error probability is an exponentially decreasing function of block length, n,
and decoding complexity is proportional to a small power of the block length [1].
The R–S codes are sometimes used in a concatenated arrangement. In such a
Figure 8.2 Bit error probability versus Eb /N0 performance of several n = 31, t -error
correcting Reed–Solomon coding systems with 32-aryMFSK modulation over an AWGN channel.
(Reprinted with permission from Data Communications, Networks, and Systems, ed. Thomas C.Bartee,
Howard W. Sams Company, Indianapolis, Ind., 1985, p. 312. Originally published in J. P. Odenwalder,
Error Control Coding Handbook,M/A-COM LINKABIT, Inc. San Diego, Calif., July 15, 1976, p. 92.)
system, an inner convolutional decoder first provides some error control by operating
on soft-decision demodulator outputs; the convolutional decoder then presents
hard-decision data to the outer Reed–Solomon decoder, which further reduces the
probability of error. In Sections 8.2.3 and 8.3 we discuss further the use of concatenated
and R–S coding as applied to the compact disc (CD) digital audio system.
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