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.
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