Unformatted text preview: shown that the probability of the
1
third event, i.e., an incorrect decoding event, is less than E!. Therefore, for the practical range
of interest in error probability performance, it almost surely can be assumed that only the
first and second events happen. This conclusion is much less sure for the recommended
(255,239) RS code with E = 8.
If it can be assumed that symbol errors occur independently with probability Vs at the RS
decoder input, then the probability Pw of undecodable word error at the output of the RS
decoder is given by
⎛n⎞
Pw (n, E) = n ⎝ j ⎠ Vsj (1 – Vs)n–j,
(3)
j = E+1
∑ n–k
where E= 2 is the number of correctable errors. This expression for Pw counts codeword
errors for every occurrence of either the second or third event above.
The RS decoder output symbol error probability can be approximated by
⎛n – 1⎞
⎜
Ps ≈ Vs Pw (n – 1, E – 1) = Vs n–1 i ⎟ Vsi(1 – Vs)n – i – 1.
⎝
⎠
∑ i= E (4) This approximate expression for Ps assumes that nearly all of the symbol errors come from
the second event above, and in this case it counts all of the erroneous symbols in the raw
(undecoded) information portion of the RS codeblock. CCSDS 130.1G1 Page 57 June 2006 TM SYNCHRONIZATION AND CHANNEL CODING —SUMMARY OF CONCEPT AND RATIONALE Finally, the bit error probability at the RS decoder output is given approximately by
Vb
Pb ≈ V Ps
s where Vb is the bit error probability on the channel. On the AWGN channel, Vs = 1–(1– Vb)J,
1 and Vb = Q( 2Es/N0), where Q(x) = 2erfc(x/ 2) is the unit Gaussian complementary
cumulative distribution function and Eb/N0 is the channel symbol signaltonoise ratio. This
expression for Pb relies on the same assumptions as for Ps , and also on the assumption that
the density of bit errors inside an erroneous undecodable Jbit RS symbol is the same as the
density of bit errors inside any Jbit RS symbol regardless of whether the RS codeword is
decodable or not and whether the particular RS symbol is erroneous or not.
The performance of the recommended RS codes with E = 16 and E = 8 is shown in figures 55
and 56, respectively, as a function of the channel symbol error probability Vs at the input of
the decoder. This figure shows the bit, symbol, and word error probabilities, Pb , Ps , and Pw ,
respectively, at the output of the decoder, as computed from the formulas above. Figure 55: CCSDS 130.1G1 Pw, Ps and Pb for the (255,223) RS Code with E=16 Page 58 June 2006 TM SYNCHRONIZATION AND CHANNEL CODING —SUMMARY OF CONCEPT AND RATIONALE Figure 56: Pw, Ps and Pb for the (255,239) RS Code with E=8 Figures 57 and 58 show BER and WER performance curves for the recommended RS
codes as a function of the normalized bit signaltonoise ratio Eb/N0 on the AWGN channel.
Note that the WER curve for RS codes on the AWGN channel does not depend on the
interleaving depth I, but for concatenated systems WER does depend on I. The WER curves
in Figures 57 and 58 are the same as FER curves for interleaving depth I = 1. Figure 57: BER and WER Performance of the CCSDS E=16 ReedSolomon Code
(255,223): Simulated and Analytical Results for the AWGN Channel CCSDS 130.1G1 Page 59 June 2006 TM SYNCHRONIZATION AND CHANNEL CODING —SUMMARY OF CONCEPT AND RATIONALE Figure 58: BER and WER Performance of the CCSDS E=8 ReedSolomon Code
(255,239): Simulated and Analytical Results for the AWGN Channel
Finally figure 59 illustrates the effects of shortening the recommended E=16 and E=8 ReedSolomon codes. On the AWGN channel shortening may actually improve the performance
(This is not the case for the recommended concatenated system). The best performance on
the AWGN channel is achieved by a nonstandard (255,173) RS code with E=41.
1
10  2
10  BER 3
10  E=8
4
10  E=16
2 55,
239 E=8
5
10  255,
223 E=16
204,
188 E=8
255,
173 E=41
97, E=8
81 E=41 157,
125 E=16
6
10 4 5 6 7 8 Eb /No (dB) Figure 59: BER Performance Comparison of Shortened and NonShortened ReedSolomon Codes on the AWGN Channel CCSDS 130.1G1 Page 510 June 2006 TM SYNCHRONIZATION AND CHANNEL CODING —SUMMARY OF CONCEPT AND RATIONALE 6
6.1 CONCATENATED CODES: REEDSOLOMON AND
CONVOLUTIONAL
INTRODUCTION One method to build a strong code while maintaining manageable decoding complexity is to
concatenate two codes, an ‘outer code’ and an ‘inner code’. This section discusses a
particular concatenated coding scheme of importance to space communications (low SNR).
The recommended concatenated coding system consists of a ReedSolomon outer code and a
convolutional inner code (which is Viterbi decoded). Typically, the inner convolutional code
corrects enough errors so that a highcoderate outer code can reduce the error probability to
the desired level. The reader may wish to consult reference [26] for the theory of
concatenated coding and references [9] and [27] for more information on the ReedSolomon/Viterbi concatenated code.
The concatenated code in the Recommended Standard (reference [3]...
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