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Iterative Soft Decoding of Reed-Solomon Convolutional Concatenated Codes

Iterative Soft Decoding of Reed-Solomon Convolutional Concatenated Codes. Li Chen Associate Professor School of Information Science and Technology, Sun Yat-sen University, China Institute of Network Coding, the Chinese University of Hong Kong 22nd, Jan, 2014. Outline. Introduction

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Iterative Soft Decoding of Reed-Solomon Convolutional Concatenated Codes

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  1. Iterative Soft Decoding of Reed-Solomon Convolutional Concatenated Codes • Li Chen • Associate Professor • School of Information Science and Technology, Sun Yat-sen University, China • Institute of Network Coding, the Chinese University of Hong Kong • 22nd, Jan, 2014

  2. Outline • Introduction • Encoding of Reed-Solomon Convolutional Concatenated (RSCC) Codes • Iterative Soft Decoding • The EXtrinsic Information Transfer (EXIT) Analysis • Implementation Complexity • Performance Evaluations and Discussions • Conclusions

  3. I. Introduction • The RSCC codes • The current decoding scheme: Viterbi-BM algorithm • Application of the RSCC codes Good at correcting spreaded bit errors Good at correcting burst errors The proposed work can be used to update the decoding system on earth!

  4. II. Encoding of RSCC Codes • Let γ denote the index of the RS codeword • The generator matrix of an (n, k) RS code is • With being the γth message vector, the γth RS codeword is generated by α is the primitive element of Fq! I

  5. II. Encoding of RSCC Codes • Given the depth of the block interleaver (I) is D, D interleaved RS codewords are then converted into Dnω interleaved RS coded bits as • They form the input to a conv. encoder with constraint length + 1, yielding the conv. codeword as q = 2ω! … to be modulated and transmitted through the channel. The number of states of the inner code is .

  6. III. Iterative Soft Decoding • Iterative soft decoding block diagram • SISO decoding of the inner code: the MAP algorithm • Input: channel observations and the a priori prob. of intl. RS coded bits ( ) ; • Output: extrinsic prob. of intl. RS coded bits ; • SISO decoding of the outer code: the ABP-KV algorithm • Input:a priori prob. of RS coded bits ( ) : ; • Output: extrinsic prob. of RS coded bits (estimated by the ABP algorithm) or the deterministic prob. of RS coded bits (estimated by the KV algorithm) θ [0, 1] I-1 I

  7. III. Iterative Soft Decoding • SISO decoding of the inner code • In light of the rate 1/2 conv. code with trellis • After the forward and backward traces, the a posteriori prob. of can be determined, and the extrinsic prob. of is: cj’ / b2j-1b2j χj χj+1 The state transition prob. is determined by …… …… A priori prob. of : At iteration 1, , at iteration v > 1, is updated by the outer decoding feedback . Channel observations:

  8. III. Iterative Soft Decoding • SISO decoding of the outer code • In light of decoding an (n, k) RS code • Functional blocks of the ABP-KV decoding • Parity-check matrix of an (n, k) RS code Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding KV decoding (×) KV decoding (√) A is the companion matrix of the primitive polynomial of Fq!

  9. III. Iterative Soft Decoding Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding • Bit reliability sorting:bit LLR values Pa,j1(0) = 0.49 |La,j1| = 0.04 Bit cj1 Pa,j1(1) = 0.51 Bit cj2 is more reliable! Pa,j2(0) = 0.93 |La,j2| = 2.59 Bit cj2 Pa,j2(1) = 0.07 A priori LLR vector: Sorted a priori LLR vector: UR = {δ1, δ2, δ3. ……, δ(n-k)w} The (n – k)ω least reliable bits

  10. III. Iterative Soft Decoding Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding Sorted a priori LLR vector: The (n – k)ω least reliable bits In Hb, reduce col. δ1 to [1 0 0 …… 0]T, col. δ2 to [0 1 0 …… 0]T, col. δ(n-k)ω to [0 0 0 …… 1]T. yielding a reduced density (adapted) parity-check matrix Hb’ …… Gaussian eliminations:

  11. III. Iterative Soft Decoding Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding • Belief propagation (BP): Based on Hb’, extrinsic LLR of bit is calculated by The a posteriori LLR of bit is calculated by The a posteriori LLR vector can be formed η (0, 1] is the damping factor. If there are multiple Gau. eliminations, utilized by KV decoding.

  12. III. Iterative Soft Decoding Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding unreliable bits 5/0 4/1 5/2 5/2 3/2 3/2 reliable bits Le,5 Le,7 Why the BP process has to be performed on an adapted H’b ?

  13. III. Iterative Soft Decoding Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding • KV list decoding By converting the a posteriori LLR into the a posteriori prob. of bits as We can then obtain the reliability matrix ∏ whose entry is defined as Reliability transform + Interpolation + Factorization transmitted message . Symbol wise APP values

  14. III. Iterative Soft Decoding Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding KV decoding (×) KV decoding (√) 1 2 3 4 5 6 7 8 9 Iterations: Undecoded RS codeword Decoded RS codeword γ = 1 γ = 2 γ = 3 γ = 4 The decoded RS codeword will not be decoded in the following iterations. γ = 5 γ = 6 γ = 7 γ = 8 γ = 9 γ = 10 • ABP-KV decoding feedback • KV output validation can be realized by the ML criterion or the CRC code. • A successive cancellation decoding manner

  15. III. Iterative Soft Decoding Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding • Performance improving approaches • Strengthen the ABP process by regrouping the unreliable bits • Strengthen the KV process by increasing its factorization output list size (OLS) KV decoding (×) KV decoding (√) In decoding the RS (7, 5) code, the sorting outcome is: 2, 5, 20, BP + KV 16, 1, 3, 8, 4, 21, 8, 4, 21, 16, 1, 3, 17, 7, 9, 10, 6, 11, 15, 13, 12, 14, 19, 18 Hb’ UR |L | = 5, L = Fac. OLS|L | = 2, L =

  16. IV. The EXIT Analysis • Investigate the interplay between the two SISO decoders • Predict the error-correction performance • Design of the concatenated code • The EXIT analytical model Mr. RS Miss. Conv. Represent the iterated (a priori/ext.) probs. by their mutual information. Ext. mutual information of the ABP-KV decoding is determined by taking the decoding outcome of D codewords as an entity I-1 MAP (1) ABP-KV (2) I If bit cj is decoded, -- deterministic prob. If bit cj is not decoded, -- extrinsic prob.

  17. IV. The EXIT Analysis • EXIT chart for iterative decoding of the RS (63, 50)-conv.(15, 17)8 code SNRoff: the SNR threshold at which an exit tunnel starts to exist between the EXIT curves of the two decoders. SNR off BER SNR (dB)

  18. IV. The EXIT Analysis Given the RS (63, 50) code as an outer code, choose a suitable inner code Code design: (1) SNRoff; (2) Free distance of the inner code

  19. V. Implementation Complexity floating oper. MAP decoding I-1 Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding binary oper. floating oper. Finite field oper. × D × D × D Note: Θ is the average row weight of matrix Hb’; Λ(M): interpolation cost of multiplicity matrix M.

  20. V. Implementation Complexity • The number of RS decoding events reduces as the iteration progresses 1 2 3 4 5 6 7 8 9 Iterations: Nr. RS decodings: 10 8 6 6 5 4 2 2 1 Undecoded RS codeword Decoded RS codeword

  21. V. Implementation Complexity • Complexity and Latency Reductions • Replace KV decoding by BM decoding • Parallel outer decoding BM decoding Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding ABP-BM decoding ABP-BM decoding MAP decoding I-1 ABP-BM decoding … ABP-BM decoding

  22. VI. Performance Eva. & Discuss. • Simulation platform: (1) AWGN channel; (2) BPSK modulation; • The RS (15, 11) – conv. (5, 7)8 code;

  23. VI. Performance Eva. & Discuss. The RS (15, 11) – conv. (5, 7)8 code; Performance improving approaches (increase NGR or |L |);

  24. VI. Performance Eva. & Discuss. The RS (63, 50) – conv. (15, 17)8 code;

  25. VI. Performance Eva. & Discuss. The RS (63, 50) – conv. (15, 17)8 code with different rates;

  26. VI. Performance Eva. & Discuss. • The RS (255, 239) – conv.(133, 171) code; In ABP decoding, the extrinsic LLR is determined by

  27. VI. Performance Eva. & Discuss. • The iterative soft decoding algorithm is more competent in improving the error-correction performance for small codes; • Numerical analysis: Iter. soft (20)’s coding gain over Viterbi-BM alg. • As the size of RS code increases, the APB algorithm becomes less effective in delivering extrinsic information as there are too many short cycles in a long RS code’s parity-check matrix Hb (Hb’).

  28. VI. Performance Eva. & Discuss. • Comparing RS (15, 11)-conv.(5, 7) code with other popular coding schemes • Code rate 0.367, codeword length 1200 bits

  29. VI. Performance Eva. & Discuss. • Powered by the iterative soft decoding algorithm, the RSCC codes can be a very good candidate for a certain communication scenario in which Data packet: small Energy budget: low Latency requirement: high High Mobility Communications Wireless Sensor Networks

  30. VII. Conclusions • An iterative soft decoding algorithm has been proposed for RSCC codes; • The inner code and outer code are decoded by the MAP algorithm and the ABP-KV algorithm, respectively. The ABP-KV algorithm feeds back both the extrinsic prob. and the deterministic prob. for the next round MAP decoding; • EXIT analysis has been conducted for the iterative decoding mechanism  design of the concatenated code; • Significant error-correction performance improvement over the benchmark schemes (e.g. Viterbi-BM); • The proposed algorithm is more competent in decoding RSCC codes with limited length.

  31. Acknowledgement • The National Basic Research Program of China (973 Program) with project ID 2012CB316100; From 2012. 1 to 2016. 12. Project: Advanced coding technology for future storage devices; ID: 61001094; From 2011. 1 to 2013. 12. Project: Spectrum and energy efficient multi-user cooperative communications; ID: 61372079; From 2014.1 to 2017.12. National Natural Science Foundation of China

  32. Related Publications • L. Chen, Iterative soft decoding of Reed-Solomon convolutional concatenated codes, IEEE Trans. Communications, vol. 61 (10), pp. 4076-4085, Oct. 2013. • L. Chen and X. Ma, Iterative soft-decision decoding of Reed-Solomon convolutional concatenated codes, the IEEE International Symposium on Information Theory (ISIT), Jul. 2013, Istanbul, Turkey. Thank you!

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