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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!

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