If it can be assumed that symbol errors occur

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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 130.1-G-1 Page 5-7 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 signal-to-noise 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 J-bit RS symbol is the same as the density of bit errors inside any J-bit 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 5-5 and 5-6, 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 5-5: CCSDS 130.1-G-1 Pw, Ps and Pb for the (255,223) RS Code with E=16 Page 5-8 June 2006 TM SYNCHRONIZATION AND CHANNEL CODING —SUMMARY OF CONCEPT AND RATIONALE Figure 5-6: Pw, Ps and Pb for the (255,239) RS Code with E=8 Figures 5-7 and 5-8 show BER and WER performance curves for the recommended RS codes as a function of the normalized bit signal-to-noise 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 5-7 and 5-8 are the same as FER curves for interleaving depth I = 1. Figure 5-7: BER and WER Performance of the CCSDS E=16 Reed-Solomon Code (255,223): Simulated and Analytical Results for the AWGN Channel CCSDS 130.1-G-1 Page 5-9 June 2006 TM SYNCHRONIZATION AND CHANNEL CODING —SUMMARY OF CONCEPT AND RATIONALE Figure 5-8: BER and WER Performance of the CCSDS E=8 Reed-Solomon Code (255,239): Simulated and Analytical Results for the AWGN Channel Finally figure 5-9 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 non-standard (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 5-9: BER Performance Comparison of Shortened and Non-Shortened ReedSolomon Codes on the AWGN Channel CCSDS 130.1-G-1 Page 5-10 June 2006 TM SYNCHRONIZATION AND CHANNEL CODING —SUMMARY OF CONCEPT AND RATIONALE 6 6.1 CONCATENATED CODES: REED-SOLOMON 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 Reed-Solomon outer code and a convolutional inner code (which is Viterbi decoded). Typically, the inner convolutional code corrects enough errors so that a high-code-rate 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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