1

Low Complexity LDPC Codes for Partial Response Channels

Hongwei Song

Vijayakumar Bhagavatula

Data Storage Systems Center

Carnegie Mellon University

Pittsburgh, PA 15213

November 20, 2002

Hongwei@ece.cmu.edu

2

As hard disk drive densities are pushed towards Tb/in2, channels must cope with

Low SNRs (7 to 12 dB)

High ISI (PW50/T >3)

Data rates needed are Gbps and faster

Corrected BER requirements are 10-12 to 10-15

Current PRML approaches will not be good enough

Codes must be of high rate (8/9 and higher)

Investigating the use of low-complexity, high-rate LDPC codes to meet above goals

Motivation

3

Systematic construction of column weight j=2 LDPC codes

Girth and distance property of constructed j=2 codes.

Density evolution analysis of LDPC codes concatenated with partial response channels

Simulation results

Outline

4

Quasi-cyclic j=2 LDPC Codes

Typically, LDPC codes with column weights > 2 are considered; here we look at j=2 codesLow complexity encoding and decoding implementation (bandwidth and memory).Lower computational complexity due to smaller column weight j=2.Short cycle free, i.e., girth g=8 or 12.Minimum distance dmin and its multiplicity A(dmin) can be easily computed.

1111

1111

1111

1111

1111

1111

1111

1111

Parity check matrix

5

Quasi-cyclic j=2 LDPC Code Construction

Disjoint difference sets (DDS) LDPC codesPermutation matrix based LDPC codesGraph based LDPC codesRandom interleaver used with structured codes

1111

1111

1111

1111

1111

1111

1111

1111

1111

1111

1111

1111

1111

1111

1111

1111

1 2 t=H [H H H ]1 2 1pσ σ σ −

=

I I I IH

I

6

Graph Based Large Girth j=2 LDPC Codes

Case 2…

Case 1 Case 2…

Case 1

Lemma 1. A j=2 LDPC code having girth g=12 and block length exists, if k-1 is a prime number, where k is the row weight.

Lemma 2. A j=2 LDPC code constructed in Lemma 1 has number of codewords with minimum distance , where k is row weight.

2( 1)n k k k= − +

3 2min( ) ( 1) ( 1) / 6A d k k k k= − − +

min 6d =

Must avoid Case 1 and Case 2 to achieve girth 12

7

Encoding Algorithm

. . .

1ix

2ix

1kix

−

ip

'iq

t

1 2 1

1 2 1' ' ' '

1 2 1

k

k

i i i i

i i i i

k

p x x x

q p p p

t q q q

−

−

−

= ⊕ ⊕ ⋅⋅ ⋅ ⊕

= ⊕ ⊕ ⋅⋅ ⋅ ⊕

= ⊕ ⊕ ⋅⋅ ⋅⊕2# (1 2( 1) 2( 1) )( 2) 2op k k k n= + − + − − ≈

The parity bits can be computed from the information bits as follows:

linear encoding complexity!

⊕

8

Word Error Rates for j=2 codes over AWGN

3.5 4 4.5 5 5.5 6 6.510

-5

10-4

10-3

10-2

10-1

100

Eb/N0 (dB)

WER

ML Bound, code(456, k=8)Simulation, code(456, k=8)ML Bound, code(5526, k=18)Simulation, code(5526, k=18)

( ) ( )min

0 min min 0( ) 2 / ( ) 2 /N

E b bd d

P A d Q d R E N A d Q d R E N=

≤ ⋅ ⋅ ≈ ⋅ ⋅∑

ML union bound

Iterative soft decoding is fairly close to ML decoding for these codes

Rate R = 1-(2/k)

9

Density Evolution Analysis

Channel Detector

LDPC Decoder

0/bE N

C

inSNR

C

outSNR L

inSNR L

outSNR

0/( , )C C

out in bCSNR SNR E Nf=( )L L

out inLSNR SNRf=

Track the evolution of the probability density function (pdf)

Assume infinite block length and Gaussian pdf, define 2 2/SNR m σ=

LLR-20 -10 0 10 20

0

0.5

1

No

rmal

ized

pd

f

DE of EPR4, SNR=5.0dB, rate8/9

Iter 1...5

Empirically measured histogramsof an EPR4 channel detectorat SNR = 5.0 dB

10

0 1 2 3 4 5 60

1

2

3

4

5

6

7

8

SNRCin (dB)

SNRCout (dB)

PR4EPR4EEPR4PR1PR2PR3EPR4, 1/(1+D2)

Extrinsic Information Transfer Functions

0/( , 4.75 )C C

out in bCSNR SNR E Nf dB= =

3 3.5 4 4.5 5 5.5 60

1

2

3

4

5

6

7

8

SNRLi n

SN

RL o

ut

j=2j=3j=4j=6

( )L L

out inLSNR SNRf=

LDPC Decoder

L

inSNR L

outSNR

ChannelDetector

0/b

E N

C

inSNR

C

outSNR

Transfer functions of PR channel (computed via Monte-Carlo) without precoder flatten out and saturate to the same output.

LDPC code with column weight j=2 provides higher output SNR than codes with j>2 for low input SNR; situation reverses at high input SNR

SNR

Cou

t

11

Estimated Vs. Simulated BER for AWGN

Larger girth estimates and simulation results are closer than for small girth

3 4 5 6 7 8 910

-8

10-7

10-6

10-5

10-4

10-3

10-2

Eb/N

0 (dB)

BE

R

Estimate, 2 iterEstimate, 4 iterEstimate, 6 iterEstimate, 7 iterS imulated, 2 iterS imulated, 4 iterS imulated, 6 iterS imulated, 7 iter

4 26

j=2, LDPC

N=5526

Girth g=12

Rate=8/9

3 4 5 6 7 8 9 1010

-10

10-8

10-6

10-4

10-2

Eb/N

0 (dB)

BE

R

Estimate , 1 iterEstimate , 2 iterEstimate , 3 iterS imulated, 1 iterS imulated, 2 iterS imulated, 3 iter

j=4, LDPC

N= 4360

Girth g=6

Rate = 0.9394

12

BER for PR Channels

2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

18

20

SNRLin

, SNRCout

SN

RL o

ut, S

NR

C in

j=2 LDPC decoderchannel detector

fixed points

PR4 predicted BER and simulated BER match well

At high SNRs, all three PR targets yield similar BER performance

( ) ( )L L

out inSNR SNR fixed SNR fixed= + ( )BER Q SNR=

0

4.0 : 0.5 : 7.0bEdB

N=

4 4.5 5 5.5 6 6.5 7

10-6

10-5

10-4

10-3

10-2

10-1

Eb/N

0(dB)

BE

R

PR4EPR4PR2Predicted (PR4)

3 Channeldecoding iterations

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Single Convolutional Code as Outer Code

BER curves based on single rate 8/9 RSC (31,23)8 concatenated with PR

At low SNR, PR1 and PR4 are better

Different PR channels have similar BER performance at high SNR

j=2 LDPC code worse than the convolutional code for low SNR, better at high SNR; both suffer error floor

3 4 5 6 7 8

10-6

10-5

10-4

10-3

10-2

10-1

SNR (dB)

BER

PR4EPR4EEPR4E3PR4PR1PR2PR3j=2 code

3 4 5 6 7 8 9 100

5

10

15

20

25

30

SNRCin

SN

RCo

ut

Conv.(31,23)8j=2 LDPC codej=6 LDPC code

BE

R

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Density Evolution Vs. Simulations

2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

10

SNRLin, SNRC

out

SN

RL o

ut,

SNR

C in

EPR4, w/o precodingEPR4, w precodingj=2 LDPC code, 3 iterj=3 LDPC code, 3 iter

4 4.5 5 5.5 6 6 .5 7 7.5 810

- 6

10- 5

10- 4

10- 3

10- 2

10- 1

Eb/N0 (dB)

BE

R

uncoded EPR4j=2 DD S LDPC, no precodingj=2 DD S LDPC, 1/(1+D2)j=3 random LDPCj=4 random LDPC

j=2 LDPC code with proper precoding outperforms LDPC codes with j≥3, while precoding adds little complexity to encoding and decoding.

LDPC codes with j≥3 have narrow “tunnel” compared to j=2 code, which results in significantly different block error statistics.

Inclusion of precoding narrows the “tunnel”.

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Block Error Statistics

j=2 LDPC code exhibits block error statistics more compatible with an outer Reed-Solomon (RS) code, due to wider “tunnel”.

0 5 10 15 20 250

5000

10000

15000

SNR=5.5 BER=9.6e-005 Blocks=167072

0 2 4 6 8 10 12 14 160

5000

10000

15000 SNR=5.625 BER=6.1e-005 Blocks=228894

# o

f blo

cks

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140

2000

4000

6000 SNR=5.75 BER=3.7e-005 Blocks=155269

# of errors

0 20 40 60 80 100 120 140 1600

50

100

SNR=5.375 BER=8.4e-5 Blocks=38599

0 50 100 1500

10

20

30

SNR=5.5 BER=7.6e-6 Blocks=19728

# of

blo

cks

0 20 40 60 80 1000

10

20

30

SNR=5.625 BER=8.9e-7 Blocks=162492

# of errors

j=2 LDPC code j=3 LDPC code

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Systematic construction of column weight j=2 LDPC codes

Investigated girth and distance properties

Sum-product algorithm is fairly close to ML decoding for these codes.

j=2 LDPC code exhibits block error statistics more compatible with an

outer RS code, attractive for magnetic recording channels.

Conclusions