Recent Advancement of LDPC Codes for High-Speed Optical ...

MITSUBISHI ELECTRIC RESEARCH LABORATORIES! Cambridge, Massachusetts!

Recent Advancement of LDPC Codes for High-Speed Optical Communications Toshiaki Koike-Akino (MERL)

© MERL 1/14/16

MITSUBISHI ELECTRIC RESEARCH LABORATORIES!

Outline •  Basics of coded modulation with low-density parity-check (LDPC) codes –  Coded modulation –  Decoding algorithm: Belief propagation, bit flipping, linear programing –  Code design: Density evolution, extrinsic information transfer (EXIT) chart

•  Iteration-dependent LDPC code design –  –  –  –  – 

More iteration, better performance while more complexity and more power Trajectory optimization for BICM Threshold analysis Performance evaluations Extension to Pareto-optimal code design

•  Modulation-dependent LDPC code design for high-dimensional modulation –  –  –  –  – 

High-dimensional modulation (HDM) Trajectory optimization for BICM-ID Threshold analysis Performance evaluations Extension to nonbinary LDPC codes

•  Summary

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Coded Modulation Schemes •  Trellis-coded modulation (TCM) [Ungerboeck, TIT1982] –  Best convolutional codes are no longer best for different modulations –  Proposed set partitioning design methods

•  Multi-level/layer coding (MLC) –  Optimal for infinite lengths, while codeword shortening issue

•  Bit-interleaved coded modulation (BICM) [Caire et al. TIT1998] –  Best codes matching to mutual information of LLR output

•  BICM with Iterative demodulation (BICM-ID) –  Soft-decision feedback to demodulator

•  Nonbinary-input coded modulation (NBICM) –  No feedback, while more complex nonbinary decoding

Set partitioning

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Modern Coded Modulations •  Communications system ENC

INT

Channel

C = sup I(Y ; X)

Design MOD-DEM (Higher mutual information) Design ENC-DEC (Matching to mutual information) © MERL 1/14/16

Mutual Information (b/s/Hz)

1 , . . .)

CBICM = I(L; B)

2. 

DEC

L BICM

16QAM

Pr(X)

1. 

DeINT

DEM Y

X MLC

CMLC = I(Y ; B) X = I(Y ; Bi |Bi

INT

Shannon

MOD B

BICM-ID

Labeling dependent

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LDPC Codes •  In 1963, R. Gallager (MIT) proposed in his thesis –  Sparse parity-check matrix –  Bit flipping decoder –  Minimum distance increases linearly with code length

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1 1 H = 40 0 1 0

3

1 1 0 0 1 1 0 15 0 1 1 0

•  In 1996, D. MacKay and R. Neal rediscovered the excellence of LDPC codes. –  Capacity approaching codes alternative to turbo codes.

•  In 2001, Chung-Forney-Richardson-Urbanke demonstrated that an irregular LDPC code achieves 0.0045dB from Shannon limit

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Tanner Graph •  Bipartite graph representation of linear codes [Tanner, TIT1981] Lower degree, more reliable

Check nodes

2

1 1 H = 40 0 1 0

3

1 1 0 0 1 1 0 15 0 1 1 0

Deg-3 Deg-2

Variable nodes Higher degree, more reliable

Parity-check matrix Code rate: R = 1

dv dc

Tanner graph Irregular

Regular

Code design à Degree distribution optimization

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Belief Propagation (BP) Decoding •  Message passing between variable-node decoders (VND) and check-node decoder (CND) •  Sum-product algorithm (SPA): linear complexity –  Probability domain –  Log-likelihood ratio (LLR) domain

•  Scheduling –  Flooding, random (shuffle), layered, adaptive, …

CND

VND

L0k = Lch +

X i6=k

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Li

L0k = 2tanh

1

✓Y

i6=k

tanh

⇣ L ⌘◆ i

2

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Cycle Avoidance: Girth Maximization •  BP decoding achieves maximum a posteriori (MAP) performance for loop-less tree graphs •  To avoid cycles: –  Quasi-cyclic (QC) girth checking [Wang-Draper-Yedidia, TIT2013] –  Progressive edge growth (PEG) [Hu-Eleftheriou-Arnold, TIT2005]

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Decoding Algorithms L0k = 2tanh

•  Bit flipping (BF) •  Simplified CND:

1

✓Y

tanh

i6=k

–  Min-sum (MS), offset min-sum (OMS), modified min-sum (MMS), delta-min (DM)

⇣ L ⌘◆ i

2

•  Divide-and-concur (DC) [Yedidia-Wang-Draper, TIT2011] •  Analog decoding, stochastic decoding •  Linear programming (LP) [Feldman et al. TIT2005] –  Lower error floor

MS SPA

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DM

OMS

LP

SPA

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Code Design Methods •  Degree distribution determines bit-error rate (BER) performance •  Density evolution (DE) •  EXIT chart [ten Brink, Electron.Lett.1999]

CND VND

BP iterations BP iterations

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d n r

l a i , a c

e e i c t

In the -value notation of [37], this can also be written “box-plus” operation


EXIT Chart Analysis •  EXIT curves of VND and CND

VND

We again model as the output -value of an AW channel whose input is the th interleaver bit transmitted u BPSK. The check node EXIT curves can be computed in clo form [38], [46] or by simulation. Alternatively, for the bin erasure channel, a duality property exists [5], [30] that gives of the length single parity-check c EXIT curve of the length (or in terms of the EXIT curve CND ) repetition code, i.e.,

This property is not exact for BPSK/AWGN a priori inputs, it is very accurate [38], [46]. For convenience, we use (8) write Weighted average Weighted average Fig. 2. VND EXIT curves foraccording dB . to and according to degree distribution (x) degree distribution ⇢(x) To compute an EXIT function, we model as the output -value of an AWGN channel whose input is the th interleaver where the second step follows from (4) with . bit transmitted using BPSK. The EXIT function of a degreefurther useful to express (9) in terms of its inverse function, variable node is then

, -

)

(4) Fig. and are given in the Ap-3 plots several check node curves. Observe that the cu where the functions are similar to the VND curves of Fig. 2, except that11they all pendix (see 1/14/16 also [6]). Fig. 2 plots several variable node curves © MERL


Conventional Code Design with EXIT Curve Fitting •  Area property: rate loss [Haganeur 2004] •  Curve fitting: Linear programming, line search, differential evolution, … Optimize degree distribution to minimize area between VND and CND

Rate loss

VND

CND

Check-regular Triple-weight

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•  Summary

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Best code should be changed if the number of iterations is reduced

~ 1.8dB gain 13


Practical Code? •  Problems of conventional design methods: –  Assumed infinite codeword length: >109 length, >100 max deg. –  Assumed infinite precision decoding: >9-bit precision –  Assumed infinite number of BP iterations: 2000 iterations

8000 max deg. 0.0045dB from limit

We consider max degree of 16 as in [Sugihara et al. OFC2013] © MERL 1/14/16

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EXIT Trajectory Across Decoding Iteration •  EXIT curve fitting: infinite-iteration decoding is required •  Code design based on EXIT trajectory rather than curve fitting 7-ite 6-ite

1-ite

2-ite

3-ite

4-ite

5-ite

Very large number of iterations

Best code based on curve fitting is no longer best for finite-iteration decoding © MERL 1/14/16

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Iteration-Aware Optimized Degree •  Optimized degree for triple-weight check-concentrated LDPC codes with rate 0.8 Conventional (infinite iteration): 0.07dB from limit 2 3 12 1 (x) = 0.15 x + 0.31 x + 0.54 x

Deg2

Iteration-aware EXIT trajectory optimization 32-ite:

32 (x)

= 0.03 x2 + 0.46 x3 + 0.51 x16

16-ite:

16 (x)

= 0.43 x3 + 0.12 x4 + 0.45 x16

8 (x)

= 1.00 x4

4-ite:

4 (x)

5

= 0.51 x + 0.49 x

2-ite:

2 (x)

= 0.85 x10 + 0.15 x11

1-ite:

2 3 1 (x) = 0.99 x + 0.01 x

8-ite:

Deg12

Conv.

Deg3

Deg2

6

Deg16

32-ite

Deg3

Deg16

16-ite

Deg3

Deg4 © MERL 1/14/16

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Threshold vs. Iterations •  Required SNR (threshold) for finite-iteration decoding

1 (x) 16 (x) 4 (x)

8 (x)

Limit

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Finite-Iteration Decoding BER Performance •  38400 bits, PEG

Risky in keeping the same code for different iteration decoding

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Novel Extension to Pareto-Optimal LDPC Codes •  The complexity of BP decoding is proportional to the number of iterations and the average degree •  We can reduce the power Conv consumption by using lower degrees •  Pareto design [Koike-Akino et al. JLT16] Ite-Optimal Pareto-Optimal –  Additional 2dB gain –  50% power reduction

•  More detail in JLT16 special issue

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•  Summary

1.  2. 

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Design MOD-DEM (Higher mutual information) Design ENC-DEC (Matching to mutual information)

0.5~2.0dB gain

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Modulation Formats •  •  •  • 

Amplitude-shift keying (ASK): 1D Phase-shift keying (PSK): 2D (Topological 1D) Quadrature-amplitude modulation (QAM): 2D High-dimensional modulation (HDM): >2D •  2D lattice

128-ary QAM

0.8 dB gain

Densest Lattice A2 © MERL 1/14/16

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High-Dimensional Modulation (HDM) •  Set-partitioned HDM •  Block-coded HDM [Millar et al. Opt.Expr.2013] –  Extended Hamming, extended Golay, …

•  Lattice-packed HDM [Koike-Akino&Tarokh, ICC2009] –  –  –  –  –  – 

Hexagonal A2 Checker-board D4 Diamond E8 Coxeter-Todd K12 Barnes-Wall L16 Leech L24

E8 lattice generator matrix

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Lattice Packing for HDM Design •  Greedy lattice packing for HDM design [Koike-Akino&Tarokh, ICC2009]

4-ary D4 lattice (simplex): 1.8dB gain

Hyper-sphere cutting 16-ary E8 lattice: 3dB gain

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LDPC Code Design for HDM •  We use a method combining VND and DEM [ten Brink et al. TCOM2004] •  We first analyze EXIT curve for DEM •  We combine EXIT curves for DEM+VND BICM-ID

A Channel

DEM Y

VND

CND

L

Mutual information analysis from LLR histogram

VND

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Modulation-Dependent Optimized Degree •  Optimized degree for triple-weight check-concentrated LDPC codes with rate 8/9 1D:

1D (x)

= 0.12 x2 + 0.32 x3 + 0.56 x13

16D:

2 3 13 16D (x) = 0.71 x + 0.16 x + 0.13 x

24D:

24D (x)

Deg2

= 0.74 x2 + 0.001 x3 + 0.26 x16

Deg12

Deg3

24D Deg2

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Deg3

Deg13

Deg16 Deg3

1D

16D Deg2 25


Threshold vs. Dimension •  Sphere-cutting lattice modulation: more dimension, higher gain

BICM

BICM-ID

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HDM BER Performance •  Codelength: 38400 bits, PEG design, BICM-ID, 32 iterations

24D BICM-ID 24D BICM-ID (optimized) 24D (x)

© MERL 1/14/16

24D BICM 1D BICM 1D (x)

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Nonbinary LDPC for HDM •  Joint use of HDM and Nonbinary LDPC codes offers a significant gain [KoikeAkino et al. SPPCom15] –  No iterative demodulation is required

•  Lattice-packed 8D modulation •  Spatially-coupled QC-NB-LDPC 38400 bits (3,15,20,192)

8D Binary BICM-ID

8D Binary

BPSK Binary

8D NonBinary 1.2dB

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Summary •  We introduced recent advancement of coded modulations and LDPC codes •  We proposed design method based on EXIT trajectory analysis –  We showed the benefit of iteration-dependent LDPC code design –  We showed the benefit of modulation-dependent LDPC code design

•  Key message: –  Degree distribution of LDPC codes should be re-designed if we change the modulation formats or the number of decoding iterations.

1.8dB 0.5dB

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Acknowledgment •  David S. Millar (MERL) •  Keisuke Kojima (MERL) •  Kieran Parsons (MERL) •  •  •  •  • 

Kenya Sugihara (MELCO) Yoshikuni Miyata (MELCO) Wataru Matsumoto (MELCO) Takashi Sugihara (MELCO) Tsuyoshi Yoshida (MELCO)

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Greedy Sphere Packing •  Greedy method [Koike-Akino/Tarokh, ICC2009]

1.  Choose the tentative center 2.  Search for closest points inside a sphere by sphere decoding 3.  Move the center point to the centroid of those points 4.  Enlarge the sphere radius 5.  Continue 2,3,4 until we find all 2R signal points 6.  Take an offset of the centroid 7.  Normalize the energy

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Densest Known Lattices •  Generator matrices

Lattice Points

Generating Matrix

Integer Coordinates © MERL 1/14/16

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Protograph-Based LDPC Codes •  Protograph: Compact Tanner graph with a smaller set of nodes [Thorpe, JPL2003] –  Degree distribution can be determined –  Edge connection is represented

•  Example: quasi-cyclic (QC) base matrix •  Parity-check matrix is derived by graph lifting procedure with copy&permutate Check node

Protograph (3,6)-regular LDPC Variable node 1

Variable node 2

Non-parallel protograph (QC type) © MERL 6/28/15 SpS3D.5

2

3

1 1 1 1 1 1 B 0 = 4 1 1 1 1 1 15 1 1 1 1 1 1

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LDPC Convolutional Codes •  LDPC-CC: combined LDPC codes and convolutional codes [Felstrom-Zigagirov, TIT1999] –  Non-zero entry is structured to have memory

•  Variants: truncated, tail-biting, infinite, …

Memory size

Memory size (constraint length) © MERL 6/28/15 SpS3D.5

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Why LDPC-CC? •  Many benefits: –  Low-complexity encoding from convolutional structure –  Asymptotic regular –  Threshold saturation: Belief-propagation (BP) threshold converges to maximum a posteriori (MAP) threshold when codeword length increases [Kudekar-RichardsonUrbanke, TIT2011] –  Low-latency decoding with sliding window [Lentmaier-Prenda-Fettweis, ISIT2011]

2 4 6 6

Almost regular, but Inherently irregular

… 6 4 2 © MERL 6/28/15 SpS3D.5

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BER Convergence

BER

•  Bit-error rate (BER) over BP decoding iterations [Lentmaier et al. ISIT2011]

Variable node index © MERL 6/28/15 SpS3D.5

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Sliding Window Decoding •  Low-latency window decoding: sub-graph BP scheduling •  Window size: Check-constant, variable-constant, adaptive, …

Wv Wc

Low-latency regardless of total LDPC codeword length © MERL 6/28/15 SpS3D.5

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Protograph EXIT Chart •  Protograph-based extrinsic information transfer (P-EXIT) chart [Liva-Chiani, GLOBECOM2007] –  More accurate than EXIT chart –  Account for edge connection –  Idea: tracking mutual information updates at all edges

•  Extended to Nonbinary P-EXIT [Dolecek-Divsalar-Sun-Amiri, TIT2013]

Same degree distribution… but B1 is better than B2 P-EXIT but EXIT can analyze accurate threshold © MERL 6/28/15 SpS3D.5

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Threshold Analysis of Window Decoding

Complexity per Iteration

•  With P-EXIT, we analyzed NB-LDPC-CC [Wei-KoikeAkino-Mitchell-Fuja-Costello, ISIT2014]

W=3

BP

W=10

W=5

W=5

BP 2

4

8

32 16 64 Galois Field Size

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128

256

2

4

8

16

32

64

128

256 512 1024

Galois Field Size

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