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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 Fq!
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 !
The number of states of the inner code is .
2
… 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:
……
……
cj’ / b2j-1 b2j The state transition prob. is determined byχj+1
Channel observations:A priori prob. of :At iteration 1, , at iteration v > 1, is updated by the outer decoding feedback .
χj
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!
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
|La,j1| = 0.04
|La,j2| = 2.59Bit cj2 is more reliable!
Pa,j1(0) = 0.49
Pa,j1(1) = 0.51Pa,j2(0) = 0.93
Pa,j2(1) = 0.07
Bit cj1
Bit cj2
UR = {δ1, δ2, δ3. ……, δ(n-k)w}
III. Iterative Soft Decoding
Gaussian eliminations:
Sorted a priori LLR vector:
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’
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 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
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
Le,7Le,5
4/1 5/2 5/2 3/2 3/2 5/0
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.
1Iterations: 2 3 4 5 6 7 8 9
γ = 1
γ = 2γ = 3
γ = 4
γ = 5γ = 6
γ = 7γ = 8
γ = 9
γ = 10
III. Iterative Soft DecodingBit reliability
sortingGaussian
eliminationBelief
PropagationKV 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’ BP + KV16, 1, 3,
8, 4, 21,
Fac. OLS|L | = 2, L = 1U
2U|L | = 5, L =
1U
2U
3U
4U
5U
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 cj is decoded, -- deterministic prob.
If bit cj is not decoded, -- 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.
SNR (dB)B
ER
SNR off
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
V. Implementation Complexity
Bit reliability sorting
Gaussian elimination
Belief Propagation
KV list decoding
MAP decoding
I-1
floating oper.
floating oper.binary 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.
The number of RS decoding events reduces as the iteration progresses
V. Implementation Complexity
1Iterations: 2 3 4 5 6 7 8 9
Undecoded RS codeword
Decoded RS codeword
Nr. RS decodings:
10 8 6 6 5 2 24 1
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
ABP-BM decoding
ABP-BM decoding
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 NGR 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 Hb (Hb
’).
VI. Performance Eva. & Discuss.
Code Codeword length
Coding gain
RS (15,11)-conv. (5,7)8 1200 bits 1.8dB
RS (63, 50)-conv. (15, 17)8 7560 bits 1.3dB
RS (255, 239)-conv. (133, 171)8 40800 bits 0.5dB
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
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
SNR (dB)
BE
R
Viterbi-BM
MAP-KV
MAP-ABP-KV
Iterative (2)
Iterative (5)
Iterative (10)
Iterative (20)
Iterative (30)
Iterative (50)
Damping factor = 0.10
LDPC (1200, 404)
Turbo (6 iter.)
Turbo (18 iter.)
Polar (1024)
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 CommunicationsWireless 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 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
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!