GSoC ’13LDPC Codes and More FEC in GNU Radio

Manu T SMentor : Dr.-Ing. Jens Elsner

IIT Bombay

October 1, 2013

Manu T S (IITB) GSoC ’ 13 October 1, 2013 1 / 21

An Evaluation

The Original Proposal

Generic Encoder/Decoder for LDPC codes

Algorithms for generating LDPC codes

Improve RS codes, add BCH etc

The Outcomes

Generic Encoder/Decoder for LDPC codes

2 Algorithms for generating LDPC codes

An attempt to incorporate LDPC into FECAPI

Manu T S (IITB) GSoC ’ 13 October 1, 2013 2 / 21

What is an LDPC Code

A linear error correcting code

All codewords should satisfyHx = 0 (1)

Any vector that satisfy above equation is a codeword

A block codeIf H is M x N and has rank R then the code has dimension

K = N − R

Block length of the code is N.

Sparse H

Here we are dealing with binary LDPC codes, so H has few 1s rest all zeros

Manu T S (IITB) GSoC ’ 13 October 1, 2013 3 / 21

Tanner Graph

H =

1 1 0 1 1 0 01 0 1 1 0 1 00 1 1 1 0 0 1

(2)

x7

x6

x5

x4

x3

x2

x1

f3

f2

f1

(a)

Manu T S (IITB) GSoC ’ 13 October 1, 2013 4 / 21

How we split up the work

Tracie / ManuConstruction of LDPC codes.

Regular LDPC matrix

LDPC codes based on Reed-Solomon codes

Progressive Edge Growth Algorithm

Encoding Techniques.

Transform parity check matrix into approximate upper triangular matrix

Obtain G = [I P] followed by matrix multiplication

Decoding Techniques.

Bit-Flip Decoding

Message Passing decoder for BSC

Message Passing decoder for AWGN Channel

Manu T S (IITB) GSoC ’ 13 October 1, 2013 5 / 21

The Algorithms - Encoding

If the parity check matrix is given as [I P], where I is N − K x N − K identitymatrix, P is some N − K x K matrix, a codeword x =

[xT

p xTd

]T can beformed by assigning

xp = P · xd

We obtain G = [I P] from H by

Column permutation

Row additions

Row permutations

Suppose f represents the permutation in obtaining G from H, and G · x = 0,then H · f (x) = 0

Manu T S (IITB) GSoC ’ 13 October 1, 2013 6 / 21

The Algorithms - Encoding

x0 x1 x2 x3 x4 x5 x6

xp xd

x

2 0 3 1 5 4 6f

x1 x3 x0 x2 x5 x4 x6f (x)

Manu T S (IITB) GSoC ’ 13 October 1, 2013 7 / 21

The Algorithms - Encoding

std::vector <char > cldpc:: encode(std::vector <char > dataword) {

if (dataword.size() == K) {

GF2Vec x(N);

GF2Vec data(K);

data.set_vec(dataword );

for ( int i = rank_H; i < N; i++ ) {

x[i] = dataword[i - rank_H ];

}

for ( int i = 0; i < rank_H; i++ ) {

x[i] = G[i]. sub_vector(N-K, N)*data;

}

GF2Vec y(N);

for ( int i = 0; i < N; i++ ) {

y[permute[i]] = x[i];

}

return y.get_vec ();

}

}

Manu T S (IITB) GSoC ’ 13 October 1, 2013 8 / 21

The Algorithms - Message Passing

7654321

3

2

11

�

check-to-variable

7654321

3

2

13

(0 0)(? ?)(0 0)(1 1)(? ?)(? ?)(0 0)

2

�

variable-to-check

(0 0)(? ?)(0 0)(1 1)(? ?)(1 ?)(0 0)

4

Manu T S (IITB) GSoC ’ 13 October 1, 2013 9 / 21

The Algorithms - Message Passing

7654321

3

2

15

7654321

3

2

17

(0 0)(? ?)(0 0)(1 1)(0 ?)(1 ?)(0 0)

6(0 0)(1 ?)(0 0)(1 1)(0 ?)(1 ?)(0 0)

8

Manu T S (IITB) GSoC ’ 13 October 1, 2013 10 / 21

The Algorithms - Message Passing

std::vector <char > awgn_bp :: decode(std::vector <float > rx_word ,

int *niteration) {

*niteration = 0;

compute_init_estimate(rx_word );

if (is_codeword ()) {

return estimate;

}

else {

rx_lr_calc(rx_word );

spa_initialize ();

while (* niteration < max_iterations) {

*niteration += 1;

update_chks ();

update_vars ();

decision ();

if (is_codeword ()) {

break;

}

}

return estimate;

}

}

Manu T S (IITB) GSoC ’ 13 October 1, 2013 11 / 21

The Algorithms - Generation of LDPC codes

Regular and Irregular Progressive Edge Growth Tanner Graphs

Suboptimal solution for a hard Combinatorial problem

Initialized from N, M, and symbol node degree sequence

The insertion of new edge has minimum effect on the girth of the graph

Both regular and irregular LDPC codes can be constructed

LDPC Codes based on RS Codes with 2 information symbols

Constructs a regular LDPC codes.

Makes sure that there are no length 4 cycles

Manu T S (IITB) GSoC ’ 13 October 1, 2013 12 / 21

Introducing gr-ldpc OOT Module

The classes and blocks

Class for GF(2) Vector arithmetic.

Class for GF(2) Matrix arithmetic.

Class for accessing LDPC codes in “alist-file” format.

Class for holding an LDPC code.

Class for message passing mechanisms for BSC and AWGN channels.

LDPC Encoder block.

LDPC Decoder blocks, for both float inputs and byte inputs.

Two algorithms for construction of LDPC codes

Codes available in github

Manu T S (IITB) GSoC ’ 13 October 1, 2013 13 / 21

Introducing gr-ldpc OOT Module

Manu T S (IITB) GSoC ’ 13 October 1, 2013 14 / 21

Bit Error Rate Performance

(3, 6) Regular Code of block length N = 96, dimension K = 50,Number of parity checks M = 48, 107 Bytes Transferred

No Code With LDPCsigma Errors BER Errors BER0.3 4338 0.0004338 0 00.4 61814 0.0061814 0 00.5 227228 0.0227228 31 3.1e-060.6 477744 0.0477744 65855 0.00658550.7 765326 0.0765326 572393 0.05723930.8 1055848 0.1055848 968856 0.09688560.9 1332039 0.1332039 1291208 0.12912081.0 1585653 0.1585653 1565803 0.1565803

Manu T S (IITB) GSoC ’ 13 October 1, 2013 15 / 21

Block Error Rate Performance

LDPC generated using PEG Algorithm,Block length N = 1008, Dimension K = 504Number of Parity Checks M = 504

No Code With LDPCsigma Errors BER Errors BER

0.3 3898 0.196461 0 00.4 18938 0.954488 0 00.5 19840 0.999994 0 00.6 19841 1.0 17 0.00085680.7 19841 1.0 19841 1.0

Manu T S (IITB) GSoC ’ 13 October 1, 2013 16 / 21

BER Plot

0 2 4 6 8 10 12Eb/N0

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16BE

RBER Comparison

No CodeWith LDPC

Manu T S (IITB) GSoC ’ 13 October 1, 2013 17 / 21

Current Issues & Future Work

Current Issues

Have not finished porting them to fecapi

Initialization process is very slow.

LDPC codes saved as alist-file is only accepted.

Future Work

Finish porting the codes to fecapi

Optimization of message passing algorithms, initialization steps

Benchmarking the algorithms

VOLK for encoding, GF(2) matrix multiplication,

Bring more of FEC into fecapi framework

Manu T S (IITB) GSoC ’ 13 October 1, 2013 18 / 21

Lessons Learned and Experiences

How to use gr modtool.

fecapi framework.

Importance of testing the code.

Keep the mentor posted about what you do.

Improved my coding skills.

Manu T S (IITB) GSoC ’ 13 October 1, 2013 19 / 21

Acknowledgements

I take this opportunity to thank

Jens

Nick, Tracie

Prof. Sibi Raj B Pillai, IIT Bombay

Prof. Radford Neal, University of Toronto

Prof. Eric G. Larsson, Linkoping University

Sreeraj

GNU Radio Developers and Users

Ettus Research

GNU Radio Mailing List, USRP Users Mailing List

Google SoC Team

Manu T S (IITB) GSoC ’ 13 October 1, 2013 20 / 21

References

T. Richardson and R. Urbanke, “Modern Coding Theory”, Cambridge 2003.

I. Djurdjevic, Jun Xu, K. Abdel-Ghaffar and Shu Lin, “A class of low-density parity-check codes constructed based on Reed-Solomon

codes with two information symbols”, IEEE Communication Letters, July 2003.

T. Richardson, M Amin Shokrollahi and R. Urbanke,“Design of Capacity Approaching Irregular Low-Density Parity-Check Codes”,

Transactions On Information Theory, February 2001.

Xiao-Yu Hu, Evangelos Eleftheriou and Dieter M. Arnold, ”Regular and Irregular Progressive Edge-Growth Tanner Graphs”,Transactions on Information Theory, January 2005.

Jun Xu, Lei Chen, I. Djurdjevic, Shu Lin and K. Abdel-Ghaffar, “Construction of Regular and Irregular LDPC codes: GeometryDecomposition and Masking”, Transactions on Information Theory, January 2007.

David J. C. MacKay, “Good Error-Correcting Codes Based on Very Sparse Matrices”, Transactions on Information Theory, March1999.

Richard E. Blahut, “Algebraic Codes for Data Transmission”, Cambridge University Press 2003.

F.J. MacWilliams and N.J.A. Sloane. “Theory of Error-Correcting Codes”, North-Holland Publishing Company 1977.

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