Part 1: Overview of Low Density Parity Check(LDPC) codes.

Part 1:Overview of Low Density

Parity Check(LDPC) codes

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












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


• Low density parity check

- H matrix has a large number of columns (n> 1000 or 10,000)

- Number of 1’s in H is small ( <<1%)

- H is constructed pseudorandomly subject to some constraints


• Low density parity check

- H is constructed pseudorandomly subject to some constraints:

- fixed number of rows and columns - this fixes the rate

- randomly fill H with 1’s

- e.g. fixed number of 1’s per row/column

Encoding (Cont’d)

• Encoding in graph

m1,1 m1,2 m1,k-1 p1

Degree k

m1,1 m1,2 m1,k-1 p1

Degree k

m2,1 m2,2 m2,c-1 p2

Degree c

Decoding

• Message passing decoder– Iterative algorithm

• Amenable to highly parallelized hardware implementation

Code design

‘Optimal’ LPDC codes can be describedby distributions (x) and (x)

Optimization

• Basic idea– Optimize the triple ( (x), (x), (x) )

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f (0)(v) = π jΔ(v)+ g(0)(v)

x)

x)

Optimization recursion

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f (0)(v) = π jΔ(v)+ g(0)(v)

x)

x)

Simulation Results

Simulation Results

One encoder/one decoder

2-IID capacity

Simulation Results

One encoder/one decoder

A single LDPC code designed for one rate and then punctured for a range of rates canbe optimal for “all rates”

2-IID capacity

Part 2:Details of Low Density Parity Check(LDPC) codes: binary

symmetric channels

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

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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

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

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rn


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

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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

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

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

Receive

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rn

Bit flipping decoder

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

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

Receive

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rn

0 0 0 ••• 0 0 1 0 ••• 1

Motivating example

• Hard decoder: bit flipping decoder

Bit (variable)nodes

Checknodes

Motivating 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 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0Rec’d

Assume all 0’s codeword and a single error


0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0Rec’d

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 00 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0Rec’d

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 0Rec’d

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 0Rec’d

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 0Rec’d




0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0Rec’d



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 0Rec’d



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 0Rec’d



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 0Rec’d

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 0Rec’d


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 0Rec’d


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 0Rec’d

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 0Rec’d



0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Rec’d


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

Date post:	18-Jan-2016
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Part 1: Overview of Low Density Parity Check(LDPC) codes.

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