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
Good at correcting
burst errors
Good at correcting
spreaded bit errors

The current decoding scheme: Viterbi-BM algorithm

Application of the RSCC codes
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
α is the primitive
element of Fq!


With
codeword is generated by
being the γth message vector, the γth RS
I
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
q = 2ω !

They form the input to a conv. encoder with constraint length  + 1,
yielding the conv. codeword as
… to be modulated
and transmitted
through the channel.
The number of states of the inner code is
 2

.
III. Iterative Soft Decoding

Iterative soft decoding block diagram

SISO decoding of the inner code: the MAP algorithm



I
Input: channel observations
and the a priori prob. of intl. RS coded bits
( )
; θ  [0, 1]
Output: extrinsic prob. of intl. RS coded bits
;
SISO decoding of the outer code: the ABP-KV algorithm


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


SISO decoding of the inner code
In light of the rate 1/2 conv. code with trellis
cj’ / b2j-1 b2j
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:
After the forward and backward traces, the a posteriori prob. of
determined, and the extrinsic prob. of is:
can be
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
Bit reliability
sorting
Gaussian
elimination
Belief
Propagation
KV decoding (×)

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

Bit reliability sorting:
Pa,j1(0) = 0.49
Bit cj1
Bit cj2
Pa,j1(1) = 0.51
Pa,j2(0) = 0.93
Gaussian
elimination
Belief
Propagation
KV list
decoding
bit LLR values
|La,j1| = 0.04
Bit cj2 is more reliable!
|La,j2| = 2.59
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
III. Iterative Soft Decoding
Bit reliability
sorting

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. 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’
III. Iterative Soft Decoding
Bit reliability
sorting

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
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

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
5/2
Le,5
Le,7
3/2
3/2
5/0
reliable bits
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
Symbol wise APP values
Reliability transform + Interpolation + Factorization
transmitted message
.
III. Iterative Soft Decoding
Bit reliability
sorting
Gaussian
elimination
Belief
Propagation
KV decoding (×)

KV decoding (√)
ABP-KV decoding feedback


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
1
2
3
4
5
6
7
8
9
Undecoded RS codeword
Decoded RS codeword
γ=4
γ=5
γ=6
γ=7
γ=8
γ=9
γ = 10
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 (×)

KV list
decoding
KV decoding (√)
Performance improving approaches

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, 16,
8, 4,1,21,
3, 17, 7, 9, 10, 6, 11, 15, 13, 12, 14, 19, 18
Hb’
BP + KV
UR

Strengthen the KV process by increasing its factorization output list size (OLS)
Fac. OLS|L | = 2, L =
Uˆ 1
Uˆ 2
|L | = 5, L =
Uˆ 1
Uˆ 2
Uˆ 3
Uˆ
4
Uˆ 5
IV. The EXIT Analysis

Investigate the interplay between the two SISO decoders


Predict the error-correction performance
Design of the concatenated code
Mr. RS

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

SNR off
SNR (dB)
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
floating oper.
MAP
decoding
I-1
Bit reliability
sorting
binary oper.
×D
Gaussian
elimination
Belief
Propagation
floating oper.
×D
KV list
decoding
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

The number of RS decoding events reduces as the iteration progresses
1
2
3
4
5
6
7
8
9
Nr. RS decodings: 10
8
6
6
5
4
2
2
1
Iterations:
Undecoded RS codeword
Decoded RS codeword
V. Implementation Complexity

Complexity and Latency Reductions

Replace KV decoding by BM decoding
Bit reliability
sorting

Gaussian
elimination
Parallel outer decoding
Belief
Propagation
ABP-BM decoding
ABP-BM decoding
MAP
decoding
I-1
ABP-BM decoding
…
ABP-BM decoding
BMKV
decoding
list
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
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.
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
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.

Comparing RS (15, 11)-conv.(5, 7) code with other popular coding schemes
Code rate 0.367, codeword length 1200 bits
1.E-01
Viterbi-BM
MAP-KV
1.E-02
MAP-ABP-KV
LDPC (1200, 404)
Iterative (2)
Iterative (5)
Polar (1024)
1.E-03
Iterative (10)
Iterative (20)
BER

Iterative (30)
Iterative (50)
1.E-04
Damping factor = 0.10
Turbo (6 iter.)
1.E-05
Turbo (18 iter.)
1.E-06
1
1.5
2
2.5
3
3.5
SNR (dB)
4
4.5
5
5.5
6
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
Wireless Sensor Networks
High Mobility Communications
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.

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

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!