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Belief-Propagation with Information Correction: Near Maximum-Likelihood Decoding of LDPC Codes

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Belief-Propagation with Information Correction: Near Maximum-Likelihood Decoding of LDPC Codes

Ned Varnica+, Marc Fossorier#, Alek Kavčić+

+Division of Engineering and Applied Sciences

Harvard University

#Department of Electrical Engineering

University of Hawaii

- Motivation – BP vs ML decoding
- Improved iterative decoder of LDPC codes
- Types of BP decoding errors
- Simulation results

. . .

. . .

- Parity check matrix
- H

- Bipartite Tanner code graph G = (V,E,C)
- Variable (symbol) nodes vi V, i = 0, 1, …, N-1
- Parity check nodes cj C, j = 0 , 1, … , Nc-1

Nc x N

A non-zero entry in H

an edge in G

- Code rate
- R = k/N, k N-Nc

- Belief Propagation
- Iterative propagation of conditional probabilities

- Locally operating
- optimal for cycle-free graphs
- Optimized LDPC codes (Luby et al 98, Richardson, Shokrollahi & Urbanke 99, Hou, Siegel & Milstein 01, Varnica & Kavcic 02)

- sub-optimal for graphs with cycles

- optimal for cycle-free graphs
- Good finite LDPC have an exponential number of cycles in their Tanner graphs (Etzion, Trachtenberg and Vardy 99)
- Encoder constructions

- BP to ML performance gap due to convergence to pseudo-codewords (Wiberg 95, Forney et al 01, Koetter & Vontobel 03)

- Short Codes
- e.g. Tanner code with N = 155, k = 64, diam = 6, girth = 8,dmin = 20

- Long Codes
- - e.g. Margulis Code with N = 2640k = 1320

- Construct decoder
- Improved BP decoding performance
- More flexibility in performance versus complexity
- Can nearly achieve ML performance with much lower computational burden
- Reduce or eliminate LDPC error floors

- Applications
- Can use with any “off-the-shelf” LDPC encoder
- Can apply to any communication/data storage channel

^

x(L) {0,1}N

Definition 1:SUC graphGS(L) = (VS(L), ES(L), CS(L))is graph induced by SUCCS(L)

rRN

x {0,1}N

BCJR detector

BP decoder

channel

transmitted binary

vector

received

vector

decoded vector

after L iterations

^

- Syndrome s = H x(L)
- CS(L) - Set of unsatisfied check nodes(SUC) CS(L) = {ci :(Hx(L))i 0}
- VS(L) - Set of variable nodes incident to c CS(L)
- ES(L) - Set of edges connecting VS(L)and CS(L)

^

- dGs(v) - Degree in SUC graphGS(L)for v V
- dGs(v) dG(v)

Observation 1: The higher the degree dGs(v) of a node v Vs(L) the more likely is v to be in error

e.g. Statistics for Tanner (155,64) code blocksfor which BP failed on AWGN channel at SNR = 2.5 dB

- Select v node
- Perform information correction

Strategy 1: Determine SUC graph and select the node with maximal degree dGs in SUC graphGS(L)

Select node v0 or v2 or v12

Definition 2: Nodes v1 and v2 are neighbors with respect to SUC if there exist cCS(L) incident to both v1 and v2

CS(L)

- nv(m) - number of neighbors of v with degree dGs = m

. . .

nv(2) = 1 and nv(1) = 4

. . .

Observation 2: The smaller the number of neighbors (wrt to SUC graph)

with high degree, the more likely v is to be in error

Strategy 2: Among nodes with maximal degree dGs select a node with minimal number of highest degree neighbors

nv0(2)= nv12(2)= 1; nv2(2)= 2

nv0(1)= 4; nv12(1)= 6

Select node v0

max

- dGs = max dGs(v)
- Set of suspicious nodes Sv= {v : dGs(v) = dGs }
- Edge penalty function r(v,c) =
(Nc- set of v nodes incident to c)

- Penalty function R(v) = r(v,c) – r(v,c)
- Select vp Svas vp = argmin R(v)
- Numerous related approaches possible

v V

max

max

max dGs(vn); if Nc \ {v}

vn Nc\{v}

0 ; if Nc \ {v} =

c Cs

c Cs

max

max

v Sv

- Decoder input on node vi

- Memoryless AWGN channel:

Observation 3: A variable node v is more likely to be incorrect if its decoder

input is less reliable, i.e., if |O(v)| is lower

Strategy 3: Among nodes with maximal degree dGs select node with minimal input reliability|O(v)|

- Set of log-likelihood ratios messages on v nodes:
M = (C,O)

- Decoder input:
O = [O (v0 ), …, O (vN-1)]

- Channel detector (BCJR) input
B = [B (v0 ), …, B (vN-1)]

. . .

. . .

C

. . .

V

. . .

. . .

O

. . .

. . .

T

T

T

j = 1

j = 2

j = 3

7

3

8

1

9

4

10

start

11

5

12

2

13

6

14

- Replace decoder and detector input LLRs corresponding to selected vp
- O (vp) = +Sand B (vp) = +S
- O (vp) = –Sand B (vp) = –S

- Perform correction in stages
- Test 2j combinations at stage j
- For each test perform additional Kjiterations
- Max number of attempts (stages) jmax

- “codeword listing” approach
- Test all 2jmax possibilities
- W – collection of valid codeword candidates
- Pick the most likely candidate
- e.g. for AWGN channel set
x = argmin d(r,w)

- e.g. for AWGN channel set

j = 1

j = 2

j = 3

7

3

8

1

9

^

4

wW

start

10

- “first codeword” approach
- Stop at a first valid codeword
- Faster convergence, slightly worse performance for large jmax

11

5

12

2

13

6

14

j = 1

j = 2

j = 3

j = 1

j = 2

j = 3

7

3

3

2

8

4

1

1

9

6

4

5

10

7

start

11

10

5

9

12

11

2

8

13

13

12

6

14

14

start

j = 1

j = 2

j = 3

7

3

8

1

9

4

10

start

11

5

12

2

13

6

14

- Decoding continued
- M need to be stored
- storage (2jmax)
- lower Kj required
- “first codeword” procedure - fastest convergence

- Decoding restarted
- M need not be stored
- higher Kjrequired

ML decoder

0

10

“codeword listing” procedure

original BP (max 100 iter)

-1

10

-2

10

WER

-3

10

-4

10

-5

10

0

0.5

1

1.5

2

2.5

3

3.5

4

E

/ N

[dB]

b

0

Fact 1: As jmax N, “codeword listing” algorithm with Kj = 0, for j < jmax, and Kjmax = 1 becomes ML decoder

- For low values of jmax (jmax << N) performs very close to ML decoder

- Tanner (N = 155, k = 64) code
- jmax = 11, Kj = 10
- Decoding continued
- faster decoding
- M need to be stored

- ML almost achieved

- Pseudo-codewords compete with codewords in locally-operating BP decoding (Koetter & Vontobel 2003)
- c - a codeword in an m-cover of G
- i - fraction of time vi V assumes incorrect value in c
- = (0,1, …,N-1) - pseudo-codeword
- pseudo-distance (for AWGN)
- Eliminate a large number of pseudo-codewords by forcing symbol ‘0’ or symbol ‘1’ on nodes vp
- Pseudo-distance spectra improved
- Can increase min pseudo-distance if jmax is large enough

- Very high SNRs (error floor region)
Stable errors on saturated subgraphs:

- decoder reaches a steady state and fails
- messages passed in SUC graph saturated

Definition 3: Decoder D has reached a steady state in the interval [L1,L2] if Cs(L) = Cs(L1) for all L [L1,L2]

- 2. Medium SNRs (waterfall region)
- Unstable Errors:
- decoder does not reach a steady state

Theorem 1: In the error floor region

Corollary: For regular LDPC codes with

- Information correction for high SNRs (error floor region)
- Pros:
- Small size SUC
- Faster convergence

- Cons:
- dGsplays no role in node selection

- Pros:

ML decoder

0

“codeword listing” procedure

10

“first codeword” procedure

original BP (max 100 iter)

-1

10

-2

10

WER

-3

10

-4

10

-5

10

0

0.5

1

1.5

2

2.5

3

3.5

4

E

/ N

[dB]

b

0

- Tanner (155,64) code
- Regular (3,5) code
- Channel: AWGN
- Strategy 3
- jmax = 11, Kj = 10
- More than 1dB gain
- ML almost achieved

- Tanner (155,64) code
- Regular (3,5) code
- Channel: AWGN
- Strategy 3
- “First codeword” procedure
- jmax = 4,6,8 and 11
- Kj = 10

- Margulis (2640,1320) code
- Regular (3,6) code
- Channel: AWGN
- Strategy 3
- “First codeword” procedure
- jmax = 5, Kj = 20
- More than 2 orders of magnitudes WER improvement

- Tanner (155,64) code
- Channels:
- Dicode (1-D)
- EPR4 (1-D)(1+D)2

- Strategy 2
- jmax = 11, Kj = 20
- 1dB gain
- 20 % of detected errors are ML

- Information correction in BP decoding of LDPC codes
- More flexibility in performance vs complexity
- Can nearly achieve ML performance with much lower computational burden
- Eliminates a large number of pseudo-codewords
- Reduces or eliminates LDPC error floors

- Applications
- Can use for any “off-the-shelf” LDPC encoder
- Can apply to any communication/data storage channel