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

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

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 I α is the primitive element of F q !

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 = 2 ω ! The number of states of the inner code is. … to be modulated and transmitted through the channel.

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

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: …… c j ’ / b 2j-1 b 2j The state transition prob. is determined by χ j+1 Channel observations:A priori prob. of : At iteration 1,, at iteration v > 1, is updated by the outer decoding feedback. χjχj

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 F q !

III. Iterative Soft Decoding Bit reliability sorting: bit LLR values A priori LLR vector: Sorted a priori LLR vector: The (n – k)ω least reliable bits Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding |L a,j1 | = 0.04 |L a,j2 | = 2.59 Bit c j2 is more reliable! P a,j1 (0) = 0.49 P a,j1 (1) = 0.51 P a,j2 (0) = 0.93 P a,j2 (1) = 0.07 Bit c j1 Bit c j2 UR = {δ 1, δ 2, δ 3. ……, δ (n-k)w }

III. Iterative Soft Decoding Gaussian eliminations: Sorted a priori LLR vector: In H b, 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 H b ’ The (n – k)ω least reliable bits Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding

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

III. Iterative Soft Decoding Why the BP process has to be performed on an adapted H’ b ? reliable bits unreliable bits L e,7 L e,5 4/14/15/25/25/25/23/23/2 3/23/2 5/05/0 Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding

III. Iterative Soft 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 Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding

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

III. Iterative Soft Decoding Bit reliability sorting Gaussian elimination Belief Propagation KV list decoding KV decoding (×)KV 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) 2, 5, 20, In decoding the RS (7, 5) code, the sorting outcome is: 16, 1, 3,8, 4, 21,17, 7, 9, 10, 6, 11, 15, 13, 12, 14, 19, 18 UR Hb’Hb’ BP + KV 16, 1, 3,8, 4, 21, Fac. OLS |L | = 2, L = |L | = 5, L =

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 MAP (1) ABP-KV (2) I -1 I 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 If bit c j is decoded, -- deterministic prob. If bit c j is not decoded, -- extrinsic prob.

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

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

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

The number of RS decoding events reduces as the iteration progresses V. Implementation Complexity 1 Iterations: Undecoded RS codeword Decoded RS codeword Nr. RS decodings:

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

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

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

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

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

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

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 H b (H b ’ ). VI. Performance Eva. & Discuss. CodeCodeword length Coding gain RS (15,11)-conv. (5,7) bits1.8dB RS (63, 50)-conv. (15, 17) bits1.3dB RS (255, 239)-conv. (133, 171) bits0.5dB

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

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

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.

Acknowledgement The National Basic Research Program of China (973 Program) with project ID 2012CB316100; From to Project: Advanced coding technology for future storage devices; ID: ; From to Project: Spectrum and energy efficient multi-user cooperative communications; ID: ; From to National Natural Science Foundation of China

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