Presentasi LDPC

LDPC(LOW DENSITY PARITY CHECK CODES)

Oleh :Agung AgeyadiRidwan NMDianthy Marya

EVOLUSI ERROR CORRECTING CODES

ERROR CODING FAMILIES

BLOCK CODES

BHC BOSE-CHAUDHURI

& …

RSREED SOLOMON

BLOCK TURBO CODE

LDPCLOW DENSITY PARITY

CODE

CONVOLUTIONAL CODES

ODSOPTIMUM DISTANCE SPECTRUM

RCCRATE COMPATIBLE CONVOLUTIONAL

NESTED CONVOLUTIONAL CODES

RCPCRATE COMPATIBLE PUNCTURED

WHAT IS AN LDPC CODE ?

• Kode LDCP merupakan kode blok linier yang diperoleh dari sparse bipartite graph (Tanner graph).

• H (n-k, n) is the Parity Check Matrix and it is SPARSE.

• Variable Nodes ( in the left) correspond to the columns of H, there are n of them.

• Check Nodes( in the right) correspond to the rows, there are n-k in number.

• Valid code words are those vectors (c1,c2,…,cn) such that for all check nodes the sum of the neighboring positions among the message nodes is zero.

• Hence LDPC Code ~ H.cT = 0

Low Density Parity Check Codes

0521 ccc0641 ccc

06321 cccc

Consider the 6 bit long codeword in the form

which satisfies 3 parity check equations as shown below.

654321 ,,,,, ccccccc

We can now define 3x6 parity check matrix as,

100111

101001

010011

H

521 ccc

The density of ‘1’s in LDPC code parity check matrix is very low

Column weight - number of ‘1’s in a column

Number of times a symbol taking part in parity checks cw

Row weight - number of ‘1’s in a row Number of

Number of symbols taking part in a parity check rw

If the parity check matrix has uniform row weight and uniform column weight (same number of ‘1’ in a column and same number of ‘1’ in a row) we call that a regular parity check matrix.

and changes, therefore this is an irregular parity check matrix

rw cw


100111

101001

010011

HThe parity check matrix defines a rate , code where

NKR KN ,MNK

NMH

Codeword is said to be valid if it satisfies the syndrome calculation

0. THcz

We can generate the codeword in by multiplying message with generator matrixm GGmc .

We can obtain the generator matrix from parity check matrix by, G H

1.) Arranging the parity check matrix in systematic form using row and column operations

2.) Rearranging the systematic parity check matrix

3.) We can verify our results as

110100

111010

101001

sysH KMMsys PIH

KTMK IPG

100111

010110

001011

G

0. THG


100111

101001

010011

HTanner graph is a graphical representation of parity check matrix specifying parity check equations.

Tanner graph consists of N number of variable nodes and M number of check nodes

In Tanner graph mth check node is connected to nth variable node if and only if nth element in mth row in parity check matrix is a ‘1’.mnh H

1c 3c2c 4c 5c 6c

1z 3z2z

The marked path z2 → c1 → z3 → c6 → z2 is an example for short cycle of 4

The number of steps needed to return to the original position is known as the girth of the code

Low density parity check codes

R. G. Gallager, “Low-Density Parity Check Codes,” in 1962,

Simple parity-check code specified by a parity-check matrix or Tanner graph.

An ‘optimal’ LDPC can get within ~.005 db of channel capacity

Details: Encoding

• Encoder is derived from the the parity check matrix H• Row reduction of H into systematic form -- get G from this• If H is sparse then with high probability G will be dense• Not addressed here, but this continues to be a topic of great concern

rn

Details: Decoding

• the channel output can be either hardor soft information• Use the properties of the graph to decode• Decoding will be done in an iterative way: iterate between variable (bit) nodes and checks nodes

rn (r1,r2 ,...,rn )

rn


rn (r1,r2 ,...,rn )

If p(x j 0 | rn ) p(x j 1| rn ) choose x j 0

If p(x j 1| rn ) p(x j 0 | rn ) choose x j 1

Consider the MAP rule discussed in the context of convolutional codes


Transmit

cn (c1,c2 ,...,cn )

rn (r1,r2 ,...,rn )

Receive

rn

Bit flipping decoder

cn (c1,c2 ,...,cn )

rn (r1,r2 ,...,rn )

Receive

rn

0 0 0 ••• 0 0 1 0 ••• 1

example

• All 0’s codeword is sent0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Bit (variable)nodes

Checknodes

Example 1: single error

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Assume all 0’s codeword and a single error


0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Step 1: Check node: Identify which parity checks are in error


0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Step 1: Parity node check:Identify which parity checks are in error


0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Step 1: Parity node check:Identify bits that are connected to those checks


0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Step 2: Bit node check:For each bit node that is potentially in error -

identify number of unsatisfied checks for that bit node


0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0




0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0



0 2 0 1 1 1 0 0 0 0 1 0 1 1 0 0

Number of unsatisfied checks for this bit


0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0



0 2 0 1 1 1 0 0 0 0 1 0 1 1 0 0

Number of unsatisfied checks for this bit


0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0



0 2 0 1 1 1 0 0 0 0 1 0 1 1 0 0

Number of unsatisfied checks for all bits


0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Step 2: Bit node check:Flip the bits with the most unsatisfied checks

0 2 0 1 1 1 0 0 0 0 1 0 1 1 0 0


0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0


0 2 0 1 1 1 0 0 0 0 1 0 1 1 0 0


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


0 2 0 1 1 1 0 0 0 0 1 0 1 1 0 0


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Step 3: parity node check:Check if all parities are satisfied


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0



0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0


Done!

Example 2: Double error

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Iteration #1


1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Iteration #1


1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Iteration #1

2 0 1 1 0 1 1 1 0 2 1 1 0 1 1 2


Iteration #1

2 0 1 1 0 1 1 1 0 2 1 1 0 1 1 2

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0


Iteration #2

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0


Iteration #2

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0


Iteration #2

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0


Iteration #2

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0


Iteration #2

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

1 0 0 0 0 1 1 0 0 2 0 1 0 1 0 1


Iteration #2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 1 1 0 0 2 0 1 0 1 0 1


Iteration #2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Done

Thank You

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Presentasi LDPC