September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Irregular Structured LDPC Codes and Structured Puncturing

Victor Stolpman, Nico van Waes, Tejas Bhatt,

Charlie Zhang, and Amitabh Dixit

This presentation accompanies submission

IEEE 802.11-04/948

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Overview• LDPC Introduction

– Regular versus Irregular Irregular codes have better performance

– Structured versus Unstructured Structured codes have better latency

• Irregular Structured LDPC Codes– Seed and Spreading Matrices – Building blocks for structured codes

– Expanded and Exponential Matrices – LDPC code construction

• Simulations– BLER in AWGN Performance improves with codeword length

– Conventional BP versus Layered BP Layered BP offers good performance with fast convergence and efficient silicon solutions

– Significant performance improvement over the legacy FEC solution for both small and large packet sizes in 802.11n channels

• Structured Puncturing

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Low-Density Parity-Check (LDPC) Codes

• What is a LDPC code?– A LDPC code is simply a block code defined by a parity-check matrix

that has a low density of ones (i.e. mostly zeros)

– Decoding is done iteratively using Belief Propagation (BP) – passing of extrinsic information between codeword elements and parity check equations

• Why do you want to use LDPC codes?– Best performing forward error correction code available

– Designs have approached capacity within 0.0045dB

– Structured designs offer the great performance with faster convergence and attractive silicon solutions

– For 802.11n, structured LDPC is a viable and attractive solution with significant gains over the legacy FEC system

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Regular vs. Irregular LDPC Codes

• Regular LDPC Codes– First developed in early 1960’s by Robert Gallager

– Each column of the parity-check matrix has the same number of ones

– Each row of the parity-check matrix has the same number of ones

• Irregular LDPC Codes– Superior performance over regular LDPC constructions

– Outperform Turbo-codes – especially at high code rates!

– Column-weight may vary across columns of the parity-check matrix

– Row-weight may vary across rows of the parity-check matrix

– Can be designed for particular channel statistics (e.g. AWGN, BEC, Rayleigh, etc.)

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Unstructured vs. Structured LDPC Codes

• Unstructured LDPC Codes – “Random” Constructions– Randomly constructed parity-check matrix

– No structure to exploit in decoding limited decoding choices

– Each codeword length requires another construction limited block sizes or high storage requirements for multiple code lengths along with complex interconnect

• Structured LDPC Codes – “Architecture Aware” Constructions– Reduction of 75% or more in memory requirements!

– Offers additional decoding choices that have fast convergence (e.g. Layered Belief Propagation) high performance with low latency!

– Supports many block sizes reduction in zero-padding inefficiencies

– Efficient decoder designs resulting in cheaper silicon solutions with lower power consumption and shorter interconnects

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Overview• LDPC Introduction

– Regular versus Irregular Irregular codes have better performance

– Structured versus Unstructured Structured codes have better latency

• Irregular Structured LDPC Codes– Seed and Spreading Matrices – Building blocks for structured codes

– Expanded LDPC and Exponential Matrices – Constructing a code

• Simulations– BLER in AWGN Performance improves with codeword length

– Conventional BP versus Layered BP Layered BP offers good performance with fast convergence and efficient silicon solutions

– Significant performance improvement over the legacy FEC solution for both small and large packet sizes in 802.11n channels

• Structured Puncturing

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Parity-Check “Seed” Matrix• Small binary matrix low storage costs• Acts as a blueprint to the structure of the expanded LDPC code• Constructed from an edge-distribution with good asymptotic

properties for the desired channel (e.g. AWGN, BEC, Fading, MIMO, etc.)

• Expanded using permutation matrices (e.g. circular-shift matrices) to construct the LDPC code used for FEC

• After expansion, the final LDPC matrix will be of the same code ensemble as the seed matrix with the same asymptotic performance

110100

100110

011011

001001

SEEDH6SEED N

2SEED K 3

1

SEED

SEED N

KR

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Permutation “Spread” Matrices• Finite set of matrices consisting of circular-shift matrices, the identity matrix, and

the all zeros matrix• Act as building blocks for the expanded LDPC matrix• Each is indexed using their exponent values (i.e shift-coefficients)

00100

00010

00001

10000

01000

2SPREADP

01000

00100

00010

00001

10000

1SPREADP

10000

01000

00100

00010

00001

0SPREADP

00000

00000

00000

00000

00000

SPREADP

00010

00001

10000

01000

00100

3SPREADP

00001

10000

01000

00100

00010

4SPREADP

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Expanded LDPC Matrix

• In matrix notation, we write

• “Expanded” LDPC matrix whose sub-matrices belong to

• Thus, the final exponents (i.e. shift-coefficients) are of the finite set:

SEED),SEEDSEED(2),SEEDSEED(1),SEEDSEED(

SEED,22,21,2

SEED,12,11,1

SPREADSPREADSPREAD

SPREADSPREADSPREAD

SPREADSPREADSPREAD

NKNKNKN

N

N

FFF

FFF

FFF

PPP

PPP

PPP

H

1SPREAD

2SPREAD

1SPREAD

0SPREADSPREAD ,,,,, pPPPPP

},,1,...,1,0{, pF ji ,,,2,1 SEEDSEED KNi SEED,,2,1 Nj

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Universal Exponential Matrix

• Exponential matrix definition used for all structured LDPC codes

• Because it is “rule-based” and not tied to a particular “seed” matrix, it offers forward-compatibility and hardware reuse for different device classes

• Supports all codeword lengths and code rates without additional storage for exponent values (i.e. shift-coefficients)

)2(),1(1),1(

)1(,4)2(,42,41,4

,3)1(,33,32,31,3

)1(,2)(,24,23,22,2

EXPONENT

SEEDSEEDSEEDSEEDSEED

SEEDSEEDSEED

SEEDSEEDSEED

SEEDSEEDSEED

KKNKN

NKN

NKN

NKN

EE

EEEE

EEEEE

EEEEE

E

pjiE ji mod)1)(1(,

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Final Exponential Matrix• Constructed via masking the “seed” matrix with the “universal”

exponent matrix (Note: operations can be reduced to just the ones locations in the seed parity-check matrix)

• We mask the seed matrix with the universal exponential:

SEEDSEEDSEEDSEEDSEEDSEEDSEED

SEED

SEED

,2,1,

,22,21,2

,12,11,1

NKNKNKN

N

N

FFF

FFF

FFF

F

0,SEED jiH

1,SEED jiH

ji ,F

jiji ,EXPONENT, EF

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Small Construction Example

110100

100110

011011

001001

SEEDH 6SEED N

010

001

1001SPREADP 3SPREAD N

1840

19630

1086420

654321

EXPONENTE

180

130

8620

41

F

010001000100000000

001100000010000000

100010000001000000

010000000100100000

001000000010010000

100000000001001000

000001100000001100

000100010000100010

000010001000010001

000000010000000010

000000001000000001

000000100000000100

H

11 ppN 2SEED

pN SPREAD

Parity Systematic

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Overview• LDPC Introduction

– Regular versus Irregular Irregular codes have better performance

– Structured versus Unstructured Structured codes have better latency

• Irregular Structured LDPC Codes– Seed and Spreading Matrices – Building blocks for structured codes

– Expanded and Exponential Matrices – LDPC code construction

• Simulations– BLER in AWGN Performance improves with codeword length

– Conventional BP versus Layered BP Layered BP offers good performance with fast convergence and efficient silicon solutions

– Significant performance improvement over the legacy FEC solution for both small and large packet sizes in 802.11n channels

• Structured Puncturing

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

BPSK-AWGN Simulations• Simulated codeword lengths:

– {576,720, 768, 864, 960, 1008, 1152, 1296, 1344, 1440, 1536, 1584, 1728, 1872, 1920, 2016, 2112, 2160, 2304}

– Larger codeword lengths are already supported by the specified seed matrices

• Permutation spreading sub-matrix dimensions:– {12,15,16,18,20,21,24,27,28,30,32,33,36,39,40,42,44,45,48}

• Rate 1/2 seed matrices of dimension (24x48)– 3 seed matrices (all 3 from the same ensemble)

• Rate 2/3 seed matrices of dimension (16x48)– 4 seed matrices (all 4 from the same ensemble)

• Rate 3/4 seed matrices of dimension (12x48)– 4 Seed matrices (all 4 from the same ensemble)

• 50 iterations of conventional belief propagation (i.e. Sum-Product-Algorithm (SPA))

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

0 0.5 1 1.5 2 2.5 3 3.5 4

10-4

10-3

10-2

10-1

100

BLER R1/2

Eb/No [dB]

BLE

RN= 576, K= 288,( 36 bytes,Ns=12)N= 720, K= 360,( 45 bytes,Ns=15)N= 768, K= 384,( 48 bytes,Ns=16)N= 864, K= 432,( 54 bytes,Ns=18)N= 960, K= 480,( 60 bytes,Ns=20)N=1008, K= 504,( 63 bytes,Ns=21)N=1152, K= 576,( 72 bytes,Ns=24)N=1296, K= 648,( 81 bytes,Ns=27)N=1344, K= 672,( 84 bytes,Ns=28)N=1440, K= 720,( 90 bytes,Ns=30)N=1536, K= 768,( 96 bytes,Ns=32)N=1584, K= 792,( 99 bytes,Ns=33)N=1728, K= 864,( 108 bytes,Ns=36)N=1872, K= 936,( 117 bytes,Ns=39)N=1920, K= 960,( 120 bytes,Ns=40)N=2016, K=1008,( 126 bytes,Ns=42)N=2112, K=1056,( 132 bytes,Ns=44)N=2160, K=1080,( 135 bytes,Ns=45)N=2304, K=1152,( 144 bytes,Ns=48)

Rate 1/2 BLER – AWGN BPSK

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

1 1.5 2 2.5 3 3.5 4

10-4

10-3

10-2

10-1

100

BLER R2/3

Eb/No [dB]

BLE

R

N= 576, K= 384,( 48 bytes,Ns=12)N= 720, K= 480,( 60 bytes,Ns=15)N= 768, K= 512,( 64 bytes,Ns=16)N= 864, K= 576,( 72 bytes,Ns=18)N= 960, K= 640,( 80 bytes,Ns=20)N=1008, K= 672,( 84 bytes,Ns=21)N=1152, K= 768,( 96 bytes,Ns=24)N=1296, K= 864,( 108 bytes,Ns=27)N=1344, K= 896,( 112 bytes,Ns=28)N=1440, K= 960,( 120 bytes,Ns=30)N=1536, K=1024,( 128 bytes,Ns=32)N=1584, K=1056,( 132 bytes,Ns=33)N=1728, K=1152,( 144 bytes,Ns=36)N=1872, K=1248,( 156 bytes,Ns=39)N=1920, K=1280,( 160 bytes,Ns=40)N=2016, K=1344,( 168 bytes,Ns=42)N=2112, K=1408,( 176 bytes,Ns=44)N=2160, K=1440,( 180 bytes,Ns=45)N=2304, K=1536,( 192 bytes,Ns=48)

Rate 2/3 BLER – AWGN BPSK

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

1.5 2 2.5 3 3.5 4 4.5 5

10-4

10-3

10-2

10-1

100

BLER R3/4

Eb/No [dB]

BLE

R

N= 576, K= 432,( 54 bytes,Ns=12)N= 720, K= 540,( 67.5 bytes,Ns=15)N= 768, K= 576,( 72 bytes,Ns=16)N= 864, K= 648,( 81 bytes,Ns=18)N= 960, K= 720,( 90 bytes,Ns=20)N=1008, K= 756,( 94.5 bytes,Ns=21)N=1152, K= 864,( 108 bytes,Ns=24)N=1296, K= 972,(121.5 bytes,Ns=27)N=1344, K=1008,( 126 bytes,Ns=28)N=1440, K=1080,( 135 bytes,Ns=30)N=1536, K=1152,( 144 bytes,Ns=32)N=1584, K=1188,(148.5 bytes,Ns=33)N=1728, K=1296,( 162 bytes,Ns=36)N=1872, K=1404,(175.5 bytes,Ns=39)N=1920, K=1440,( 180 bytes,Ns=40)N=2016, K=1512,( 189 bytes,Ns=42)N=2112, K=1584,( 198 bytes,Ns=44)N=2160, K=1620,(202.5 bytes,Ns=45)N=2304, K=1728,( 216 bytes,Ns=48)

Rate 3/4 BLER – AWGN BPSK

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Layered Belief Propagation

010001000100000000

001100000010000000

100010000001000000

010000000100100000

001000000010010000

100000000001001000

000001100000001100

000100010000100010

000010001000010001

000000010000000010

000000001000000001

000000100000000100

H

• Parity-check matrix is partitioned into layers and messages are passed between

• Speeds convergence time significantly High performance with low latency

• Significant reduction in memory requirements (75% reduction)

• Most structured LDPC codes can implement layered-BP in cost effective solutions

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.810

-4

10-3

10-2

10-1

100

Eb/N0 (dB)

BLE

RComparing Conventional and Layered Belief Propagation, AWGN, BPSK, N=1152

Nokia, Conventional BPNokia, Layered BPTI, Conventional BPTI, Layered BP

Conventional BP : 50 IterationsLayered BP : 15 Iterations

Layered vs. Conventional BP (Rate 1/2)

Layered BP(15 iterations)

Conventional BP(50 iterations)

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Structured LDPC, N=1920, Different Code-Rates

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.210

-3

10-2

10-1

100

Eb/N0 (dB)

BLE

RLayered Belief Propagation, 12-iterations, AWGN, BPSK, N=1920

R-1/2R-2/3R-3/4

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.810

-4

10-3

10-2

10-1

100

Eb/N0 (dB)

BLE

RComparing Conventional and Layered Belief Propagation, AWGN, BPSK, N=1920, R-1/2

Conventional BP, 12-iterLayered BP, 12-iterLayered BP, 8-iterParallel Layered BP, 12-iter

Structured LDPC, N=1920, Rate 1/2

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

802.11n Channel Simulations

• Channel B– 2x2 MIMO with 2 spatial streams in 20MHz

– 30 iterations of conventional belief propagation (i.e. SPA)

– Large packet sizes using concatenated codewords of length 2304

• Channel D– 1x1 SISO in 20MHz

– 20 iterations of conventional belief propagation (i.e. SPA)

– Small packet sizes using a single codeword of length 2304

• Channel E– 2x2 MIMO with 2 spatial streams in 20MHz

– 30 iterations of conventional belief propagation (i.e. SPA)

– Large packet sizes using concatenated codewords of length 2304

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Channel B 2x2 Simulation Results

Over 3dB gain in

2x2

2 4 6 8 10 12 14 16 18 20 22

10-2

10-1

100

SNR [dB]

PE

R

Packet Error Rate for LDPC Codes in ChB 2x2 using N=2304

CC .11a ChB R1/2 2x2 4-QAM 24Mbps (1008 bytes)LDPC(30-SPA) ChB R1/2 2x2 4-QAM 24Mbps (1008 bytes)CC .11a ChB R3/4 2x2 16-QAM 72Mbps (1080 bytes)LDPC(30-SPA) ChB R3/4 2x2 16-QAM 72Mbps (1080 bytes)CC .11a ChB R2/3 2x2 64-QAM 96Mbps (960 bytes)LDPC(30-SPA) ChB R2/3 2x2 64-QAM 96Mbps (960 bytes)

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Channel D 1x1 Simulation Results

5 10 15 20 25

10-3

10-2

10-1

100

SNR [dB]

PE

R

Packet Error Rate for LDPC Codes in ChD 1x1 using N=2304

CC .11a ChD R1/2 1x1 4-QAM 12Mbps (144 bytes)LDPC(20-SPA) ChD R1/2 1x1 4-QAM 12Mbps (144 bytes)CC .11a ChD R3/4 1x1 16-QAM 36Mbps (216 bytes)LDPC(20-SPA) ChD R3/4 1x1 16-QAM 36Mbps (216 bytes)CC .11a ChD R2/3 1x1 64-QAM 48Mbps (192 bytes)LDPC(20-SPA) ChD R2/3 1x1 64-QAM 48Mbps (192 bytes)

~2dB Gainin 1x1

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Channel E 2x2 Simulation Results

3 4 5 6 7 8 9 10

10-2

10-1

100

SNR [dB]

PE

R

Packet Error Rate for LDPC Codes in ChE 2x2 using N=2304

CC .11a ChE R1/2 2x2 4-QAM 24Mbps (1008 bytes)LDPC(30-SPA) ChE R1/2 2x2 4-QAM 24Mbps (1008 bytes)

Over 3dB gain in

2x2

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Features

• Forward compatibility and hardware reuse– Existing seed sets already support longer codeword lengths– Additional seed are easily added for different channel models, additional

code rates, and to accommodate tradeoffs in silicon

• “Architecture Aware” constructions that allow for Layered-BP– Fast convergence high performance and low latency– Efficient silicon solutions

• Wide range of block sizes reduces zero-padding inefficiencies• Upper triangular seed matrices linear time encoding• In the pipeline …

– Seed matrices for additional code rates 5/6 and 7/8– Additional seed sizes for different number of data sub-carriers (e.g

40MHz channel bonding)

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Overview• LDPC Introduction

– Regular versus Irregular Irregular codes have better performance

– Structured versus Unstructured Structured codes have better latency

• Irregular Structured LDPC Codes– Seed and Spreading Matrices – Building blocks for structured codes

– Expanded and Exponential Matrices – LDPC code construction

• Simulations– BLER in AWGN Performance improves with codeword length

– Conventional BP versus Layered BP Layered BP offers good performance with fast convergence and efficient silicon solutions

– Significant performance improvement over the legacy FEC solution for both small and large packet sizes in 802.11n channels

• Structured Puncturing

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Structured Puncturing of LDPC Codes

• Used to offer all possible code rates in between and above the basic code rate set {1/2,2/3,3/4,7/8}

• Puncturing does not require changing the parity-check connective net at either the encoder or decoder

• Supports easy link adaptation. In MIMO applications, puncturing allows for different spatial streams to have different code rates without using multiple coding blocks

• Approach can be reused in Hybrid-ARQ systems

• Structured approach reduces storage requirements and expands easily to multiple block lengths

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

0 1 2 3 4 5 6 710

-4

10-3

10-2

10-1

100

sim punc output N624 M312 53Nokia12 50iters.mat

BLE

R

Eb/No dB

Rate 0.500, N= 624, K= 312Rate 0.525, N= 594, K= 312Rate 0.553, N= 564, K= 312Rate 0.584, N= 534, K= 312Rate 0.619, N= 504, K= 312Rate 0.658, N= 474, K= 312Rate 0.703, N= 444, K= 312Rate 0.754, N= 414, K= 312Rate 0.813, N= 384, K= 312Rate 0.884, N= 353, K= 312

Rate 1/2 Puncture Example (Mother Code, N=624)

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

Summary

• Irregular Structured LDPC codes have great performance

• Offers forward-compatibility and hardware reuse

• Already supports codeword lengths greater than 2304

• “Architecture Aware” constructions Layered-BP decoding

• Efficient silicon solutions with high throughput and low latency

• Wide range of block sizes reduces zero-padding inefficiencies

• Upper triangular seed matrices linear time encoding

• Structured puncturing allows for additional code rates for use with spatial stream adaptation in MIMO systems

September, 2004

Victor Stolpman et. al

doc.: IEEE 802.11-04/992

Submission

References

1. T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of Capacity Approaching Irregular Low-Density Parity-Check Codes,” IEEE Transactions on Information Theory, vol. 47, pp. 619-637, Feb. 2001.

2. Sae-Young Chung, On the Construction of Some Capacity-Approaching Coding Schemes, PhD Dissertation, MIT, 2000.

3. J. Hou, P. Siegel, and L Milstein, “Performance Analysis and Code Optimization of Low Density Parity-Check Codes on Rayleigh Fading Channels,” IEEE J. Select. Areas Commun., Issue on The Turbo Principle: From Theory to Practice I, vol. 19, no. 5, pp. 924-934, May 2001.

4. M. M. Masour and N. R. Shanbhag, “Turbo decoder architectures for low-density parity check codes,” IEEE Global Comm. Conf. (GLOBECOM), Nov. 2002, pp. 1383-1388.

5. M. M. Mansour and N. R. Shanbhag, “Low power VLSI architectures for LDPC codes,” in 2002 International Low Power Electronics and Design, 2002, pp. 284-289.

6. D. E. Hocevar, “LDPC code construction with flexible hardware implementation,” Proc.: IEEE Int’l Conf. On Comm. (ICC), Anchorage, AK, May 2003.

7. M. M. Mansour and N. R. Shanbhag, “High-Throughput LDPC Decoders,” IEEE Trans. On VLSI Systems, vol. 11, No. 6, pp. 976-996, December 2003.