Group 02-Dark Knight

Anand Kotadiya_131003Devanshi Piprottar_Jaysheel Shah_Manindar Sambhi_Suhani Ladani_131057

Institute of Engineering and Technology

Linear Algebra_2014

LDPC Bit Flipping Decoding

Outline

History of LDPC Codes

Linear Block Codes

Properties of LDPC Codes

Decoding of LDPC Code

Tanner Graph

Hard Decision Decoding(Bit Flipping)

Applications of LDPC Code

References

History of LDPC Codes

Low Density Parity Check Code A class of Linear Block Codes

Invented by Robert Gallager in his 1960 MIT Ph. D. dissertation.

Being ignored for long time due to Requirement of high complexity computation

Introduction of Reed-Solomon codes

The concatenated RS and convolution codes were considered perfectly suitable for error control coding

The LDPC codes were rediscovered in mid 90s by R. Neal and D. Mackay at the Cambridge University.

LDPC codes are arguably the best error correction codes in existence at present.

Linear Block Codes

A Linear Code can be described by a generator matrix G or a parity check matrix H.

A (N,K) block encoder accepts K-bit input and produces N-bit codeword

c= xG, and cHT = 0 where c = codeword, x = information

G = Generator matrix, H = parity check matrix

G can be found by Gaussian elimination

H can be put in the form H = [PT : I].

The Generator Matrix G = [I : P].

Properties of LDPC Codes

LDPC codes are defined by a sparse parity-check matrix

Parity Check Matrix (H) for decoding is sparse Very few 1's in each row and column. Expected large minimum distance.

Regular LDPC codes H: m x n where (n-m) information bits are encoded into n

codewords H contains exactly Wc 1's per column and exactly Wr = Wc(n/m) 1's

per row, where Wc << m. The above definition implies that Wr << n. Wc ≥ 3 is necessary for good codes.

If the number of 1's per column or row is not constant, the code is an irregular LDPC code. Usually irregular LDPC codes outperforms regular LDPC codes.

Decoding of LDPC Codes

General decoding of linear block codes

Only if c is a valid codeword, we have

c HT = 0

For binary symmetric channel (BSC), the received codeword is cadded with an error vector e

The decoder needs to find out e and flip the corresponding bits

The decoding algorithm is based on linear algebra

Graph-based Algorithms

Sum-product algorithm for general graph-based codes

MAP algorithm for trellis graph-based codes

Bit Flipping(Message passing algorithm) for bipartite graph-based codes

Tanner Graph

Tanner showed parity check matrix can be represented effectively by a bipartite graph, now called a Tanner graph. a bipartite graph is an unidirectional graph whose nodes may be

separated into two classes, where edges only connect two nodes not residing in the same class

Tanner graph has two classes of nodes variable nodes (bit or symbol nodes)

check nodes (function nodes)

The Tanner graph is drawn according to the following rule check node j is connected to variable node i

whenever element hji in H is a 1.

Tanner Graph

Decoding of LDPC Codes Decoding complexity grows in O(n2)

Even sparse matrices don’t result in a good performance if the block length (n) gets very high

So iterative decoding algorithms are used

Those algorithms perform local calculations and pass those local results via messages

This step is typically repeated several times

It was observed that iterative decoding algorithms of sparse codes perform very close to the optimal decoder

Hard Decision Decoding(Bit Flipping) Let’s assume codeword c = [1 0 0 1 0 1 0 1] and

received codeword c’ = [1 1 0 1 0 1 0 1]

1. All v-nodes ci send a “message” to their c-nodes fj containing the bit they believe to be the correct one for them. At this stage the only information a v-node ci has is the corresponding received i-th bit of c.

Hard Decision Decoding(Bit Flipping)

2. Every check nodes fj calculate a response to every connected variable node. The response message contains the bit that fj believes to be the correct one for this v-node cj assuming that the other v-nodes connected to fj are correct.

Hard Decision Decoding(Bit Flipping)

3. The v-nodes receive the messages from the check nodes and use this additional information to decide if their originally received bit is OK. A simple way to do this is majority vote.

4. Go to step 2.

FPGA Implementation

Applications of LDPC Code Today Internet, communication and digital circuits are huge source of

data sharing. If I want to transmit my confidential data then I need to convert it into codeword(Encoding).

But while transmitting it through the transmission channel, it will give wrong information to the receiver due to noise in the channel.

So we need to use LBC which will encodes the data and transmit it over the channel and if the error occurs in the received data then it should be capable of finding the error and correcting it.

Some of the Real life Applications of LDPC Codes

1. COMUNICATION SYSTEMS/INTERNET Teletext systems, satellite communication, broadcasting (radio and digital TV),

telecommunications (digital phones)

Ethernet, Cellular wireless

2. INFORMATION SYSTEMS Logical circuits, semiconductor memories.

Data Storage-Magnetic disks (HD), optic reading disks (CD-ROM).

3. AUDIO AND VIDEO SYSTEMS

Digital sound (CD) and digital video(DVD)

References

“Lecture 10 on LDPC Codes”, Information Electronics Engineering,Ewha Womans University

Ryan, W., “An Introduction to Low Density Parity Check Codes”, UCLA Short Course Notes, April, 2001

R. Gallager, “Low-density parity-check codes”, IRE Trans. IT, Jan. 1962