Iterative Soft Decoding of ReedSolomon Convolutional Concatenated Codes

Iterative Soft Decoding of Reed-Solomon Convolutional Concatenated Codes n n n 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 22 nd, Jan, 2014

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

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

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

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

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

III. Iterative Soft Decoding n n SISO decoding of the inner code In light of the rate 1/2 conv. code with trellis cj’ / b 2 j-1 b 2 j 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. n Channel observations: After the forward and backward traces, the a posteriori prob. of determined, and the extrinsic prob. of is: can be

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

III. Iterative Soft Decoding Bit reliability sorting n Bit reliability sorting: Bit cj 1 Bit cj 2 Pa, j 1(0) = 0. 49 Pa, j 1(1) = 0. 51 Pa, j 2(0) = 0. 93 Pa, j 2(1) = 0. 07 Gaussian elimination Belief Propagation KV list decoding bit LLR values |La, j 1| = 0. 04 |La, j 2| = 2. 59 Bit cj 2 is more reliable! A priori LLR vector: Sorted a priori LLR vector: UR = {δ 1, δ 2, δ 3. ……, δ(n-k)w} The (n – k)ω least reliable bits

III. Iterative Soft Decoding Bit reliability sorting n Gaussian elimination Belief Propagation KV list decoding Gaussian eliminations: Sorted a priori LLR vector: The (n – k)ω least reliable bits In Hb, reduce col. δ 1 to [1 0 0 …… 0]T, col. δ 2 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’

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

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

III. Iterative Soft Decoding Bit reliability sorting n Gaussian elimination Belief Propagation 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 Symbol wise APP values Reliability transform + Interpolation + Factorization transmitted message .

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

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

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

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. BER n SNR off SNR (d. B)

IV. The EXIT Analysis n n 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

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

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

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

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

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

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

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

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

VI. Performance Eva. & Discuss. n n n 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. Codeword length Coding gain RS (15, 11)-conv. (5, 7)8 1200 bits 1. 8 d. B RS (63, 50)-conv. (15, 17)8 7560 bits 1. 3 d. B RS (255, 239)-conv. (133, 171)8 40800 bits 0. 5 d. B 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’).

VI. Performance Eva. & Discuss. n n Comparing RS (15, 11)-conv. (5, 7) code with other popular coding schemes Code rate 0. 367, codeword length 1200 bits

VI. Performance Eva. & Discuss. n 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 Wireless Sensor Networks High Mobility Communications

VII. Conclusions n An iterative soft decoding algorithm has been proposed for RSCC codes; n 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; n EXIT analysis has been conducted for the iterative decoding mechanism design of the concatenated code; n Significant error-correction performance improvement over the benchmark schemes (e. g. Viterbi-BM); n The proposed algorithm is more competent in decoding RSCC codes with limited length.

Acknowledgement n The National Basic Research Program of China (973 Program) with project ID 2012 CB 316100; From 2012. 1 to 2016. 12. n National Natural Science Foundation of China 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.

Related Publications n L. Chen, Iterative soft decoding of Reed-Solomon convolutional concatenated codes, IEEE Trans. Communications, vol. 61 (10), pp. 4076 -4085, Oct. 2013. n 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!