Design of Energy-Efficient LDPC Codes and Decoders Elsa Dupraz - - PowerPoint PPT Presentation

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Design of Energy-Efficient LDPC Codes and Decoders Elsa Dupraz - - PowerPoint PPT Presentation

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Design of Energy-Efficient LDPC Codes and Decoders Elsa Dupraz 16/04/2019 Section 1: Introduction 2 OUTLINE Introduction LDPC codes and decoders Perf. analysis of faulty decoders Effect of faults in the decoders Conclusion 1.

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SLIDE 1

Design of Energy-Efficient LDPC Codes and Decoders Elsa Dupraz 16/04/2019

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SLIDE 2

Section 1: Introduction 2

OUTLINE

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

  • 1. Introduction
  • 2. LDPC codes and decoders
  • 3. Perf. analysis of faulty decoders
  • 4. Effect of faults in the decoders
  • 5. Conclusion

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 3

Section 1: Introduction 3

Acknowledgments

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ ANR JCJC project EF-FECtive (January 2018 - December 2020) ◮ Fangping Ye, Mohamed Yaoumi, Zeina Mheich ◮ Franc ¸ois Leduc-Primeau, David Declercq, Valentin Savin, Bane Vasic, Lav Varshney, Emanuel Popovici, Frederic Guilloud ...

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 4

Section 1: Introduction 4

Moore’s Law

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ In 1965, Moore predicted that the number of transistors on processors was going to double every 2 years

1970 1980 1990 2000 2010 2020 1e+02 1e+04 1e+06 1e+08 1e+10 1e+12 Year Number of transistors Moore Actual Intel

4004 8088 Pentium 80386DX Pentium 4 Core i7 Core 2 Duo

◮ What about energy consumption?

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 5

Section 1: Introduction 5

5G

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

In the 5G standardization process ◮ Huge increase of number of users, terminals, etc. ◮ Need to improve environmental footprint, battery lifetime

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 6

Section 1: Introduction 6

Faulty computation operations

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Hardware energy consumption has become a major issue ◮ Energy consumption can be reduced by

  • Aggressive voltage scaling
  • Increased sampling frequency

◮ Problem : this may introduce faults in the computation operations

t V

Vdd

Value 1 td Value 0

faulty

pxor = P(˜ c = a ⊕ b)

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 7

Section 1: Introduction 7

Energy-efficient LDPC codes

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ In this talk, focus on channel coding

Channel

faulty faulty

D E

◮ Noisy vs Faulty ◮ Family of error-correction codes : LDPC codes

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 8

Section 1: Introduction 7

Energy-efficient LDPC codes

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ In this talk, focus on channel coding

Channel

faulty faulty

D E

◮ Noisy vs Faulty ◮ Family of error-correction codes : LDPC codes ◮ Objectives

  • Study the effect of faults in LDPC decoders
  • Design fault-tolerant LDPC decoders

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 9

Section 1: Introduction 8

Outline

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

  • 1. Introduction
  • 2. LDPC codes and decoders
  • 3. Perf. analysis of faulty decoders
  • 4. Effect of faults in the decoders
  • 5. Conclusion

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 10

Section 2: LDPC codes and decoders 9

OUTLINE

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

  • 1. Introduction
  • 2. LDPC codes and decoders
  • 3. Perf. analysis of faulty decoders
  • 4. Effect of faults in the decoders
  • 5. Conclusion

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 11

Section 2: LDPC codes and decoders 10

Block channel codes

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

E D

Channel

◮ Channel : P(Y|X)

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 12

Section 2: LDPC codes and decoders 10

Block channel codes

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

E D

Channel

◮ Channel : P(Y|X)

Encoding uk : information sequence (k) G : generator matrix (n × k) xn = Guk Decoding xn : codeword (n) H (n × m) : parity check matrix

HTxn = 0 LDPC codes : H sparse, optimized for good perf.

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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Section 2: LDPC codes and decoders 11

Code construction

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ H is a sparse parity check matrix, HTxn = 0. HT =   1 1 1 1 1 1 1 1 1   ◮ v1, v2, · · · , vn : Variable Nodes (VN), degrees dvi c1, c2, · · · , cm : Check Nodes (CN), degrees dcj ◮ Regular codes : constant degrees dv, dc R = k n = 1 − dv dc

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 14

Section 2: LDPC codes and decoders 12

LDPC code construction

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ H is a sparse parity check matrix, HTx6 = 03 HT =   1 1 1 1 1 1 1 1 1   ◮ v1, v2, · · · , vn : Variable Nodes (VN), degrees dvi c1, c2, · · · , cm : Check Nodes (CN), degrees dcj ◮ Regular codes : constant degrees dv, dc R = k n = 1 − dv dc

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 15

Section 2: LDPC codes and decoders 13

LDPC decoders

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Ex : Gallager decoder, hard-decision decoder CN c1

1

VN v1 1

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 16

Section 2: LDPC codes and decoders 14

LDPC decoders

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Ex : Gallager decoder, hard-decision decoder CN c1

1 1

VN v1 1

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 17

Section 2: LDPC codes and decoders 15

LDPC decoders

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Ex : Gallager decoder, hard-decision decoder CN c1

1 1

VN v1 1

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 18

Section 2: LDPC codes and decoders 16

LDPC decoders

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Ex : Gallager decoder, hard-decision decoder CN c1

1 1

VN v1 1

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 19

Section 2: LDPC codes and decoders 17

LDPC decoders

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Ex : Gallager decoder, hard-decision decoder CN c1

1 1

VN v1 1 1

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 20

Section 2: LDPC codes and decoders 18

LDPC decoders

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Ex : Gallager decoder, hard-decision decoder CN c1

1 1

VN v1 1 1

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 21

Section 2: LDPC codes and decoders 18

LDPC decoders

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Ex : Gallager decoder, hard-decision decoder CN c1

1 1

VN v1 1 1 ◮ Hard-decision decoders : binary messages ◮ Soft-decision decoders : LLR messages, e.g., log P(X=0|y)

P(X=1|y)

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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Section 2: LDPC codes and decoders 19

LDPC decoders

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ VN update function : γ(ℓ) = Φv(µ(ℓ)

0 , µ(ℓ) 1 , · · · , µ(ℓ) dv−1)

... ◮ CN update function : µ(ℓ+1) = Φc(γ(ℓ)

1 , · · · , γ(ℓ) dc−1)

...

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 23

Section 2: LDPC codes and decoders 20

LDPC decoders

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ VN update function : γ(ℓ) = Φv(µ(ℓ)

0 , µ(ℓ) 1 , · · · , µ(ℓ) dv−1)

◮ CN update function : µ(ℓ+1) = Φc(γ(ℓ)

1 , · · · , γ(ℓ) dc−1)

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 24

Section 2: LDPC codes and decoders 20

LDPC decoders

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ VN update function : γ(ℓ) = Φv(µ(ℓ)

0 , µ(ℓ) 1 , · · · , µ(ℓ) dv−1)

◮ CN update function : µ(ℓ+1) = Φc(γ(ℓ)

1 , · · · , γ(ℓ) dc−1)

◮ APP computation : α(ℓ) = Φa(µ(ℓ)

0 , µ(ℓ) 1 , · · · , µ(ℓ) dv )

Decide ˆ X = 0 if α(ℓ) > 0

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 25

Section 2: LDPC codes and decoders 20

LDPC decoders

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ VN update function : γ(ℓ) = Φv(µ(ℓ)

0 , µ(ℓ) 1 , · · · , µ(ℓ) dv−1)

◮ CN update function : µ(ℓ+1) = Φc(γ(ℓ)

1 , · · · , γ(ℓ) dc−1)

◮ APP computation : α(ℓ) = Φa(µ(ℓ)

0 , µ(ℓ) 1 , · · · , µ(ℓ) dv )

Decide ˆ X = 0 if α(ℓ) > 0 ◮ Hard-decision decoders : binary messages ◮ Soft-decision decoders : LLR messages, e.g., µ0 = log P(X=0|y)

P(X=1|y)

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 26

Section 2: LDPC codes and decoders 21

Faulty LDPC decoders

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Faulty VN update function : γ(ℓ) = Φv(µ(ℓ)

0 , ˜

µ(ℓ)

1 , · · · , ˜

µ(ℓ)

dv−1),

P(˜ γ(ℓ)|γ(ℓ))

Faults

◮ Faulty CN update function : µ(ℓ+1) = Φc(˜ γ(ℓ)

1 , · · · , ˜

γ(ℓ)

dc−1),

P(˜ µ(ℓ)|µ(ℓ))

Faults

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 27

Section 3: Perf. analysis of faulty decoders 22

OUTLINE

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

  • 1. Introduction
  • 2. LDPC codes and decoders
  • 3. Perf. analysis of faulty decoders
  • 4. Effect of faults in the decoders
  • 5. Conclusion

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 28

Section 3: Perf. analysis of faulty decoders 23

LDPC codes performance

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Two areas of performance :

BER SNR

Error floor Waterfall

◮ Error Floor : Avoid short cycles in the code (PEG algorithm) ◮ Waterfall : Optimize the code threshold (density evolution)

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 29

Section 3: Perf. analysis of faulty decoders 24

All-zero codeword assumption

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Symmetry conditions [Richardson01], [Varshney11] ◮ Channel : P(Y|X = 0) = P(−Y|X = 1) ◮ VN function : Φv(−µ0, −µ1, · · · , −µdv−1) = −Φv(µ0, µ1, · · · , µdv−1) ◮ CN function : Φc(b1γ1, · · · , bdc−1γdc−1) = (

i bi) Φc(γ1, · · · , γdc−1)

◮ Fault model : P(−˜ µ|µ) = P(˜ µ| − µ)

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 30

Section 3: Perf. analysis of faulty decoders 24

All-zero codeword assumption

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Symmetry conditions [Richardson01], [Varshney11] ◮ Channel : P(Y|X = 0) = P(−Y|X = 1) ◮ VN function : Φv(−µ0, −µ1, · · · , −µdv−1) = −Φv(µ0, µ1, · · · , µdv−1) ◮ CN function : Φc(b1γ1, · · · , bdc−1γdc−1) = (

i bi) Φc(γ1, · · · , γdc−1)

◮ Fault model : P(−˜ µ|µ) = P(˜ µ| − µ) Examples ◮ BSC : α = P(Y = 0|X = 1) = P(Y = 1|X = 0) ◮ Φv(µ0, µ1, · · · , µdv−1) = µi

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 31

Section 3: Perf. analysis of faulty decoders 25

All-zero codeword assumption

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Symmetry conditions [Richardson01], [Varshney11] ◮ Channel : P(Y|X = 0) = P(−Y|X = 1) ◮ VN function : Φv(−µ0, −µ1, · · · , −µdv−1) = −Φv(µ0, µ1, · · · , µdv−1) ◮ CN function : Φc(b1γ1, · · · , bdc−1γdc−1) = (

i bi) Φc(γ1, · · · , γdc−1)

◮ Fault model : P(−˜ µ|µ) = P(˜ µ| − µ)

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 32

Section 3: Perf. analysis of faulty decoders 25

All-zero codeword assumption

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Symmetry conditions [Richardson01], [Varshney11] ◮ Channel : P(Y|X = 0) = P(−Y|X = 1) ◮ VN function : Φv(−µ0, −µ1, · · · , −µdv−1) = −Φv(µ0, µ1, · · · , µdv−1) ◮ CN function : Φc(b1γ1, · · · , bdc−1γdc−1) = (

i bi) Φc(γ1, · · · , γdc−1)

◮ Fault model : P(−˜ µ|µ) = P(˜ µ| − µ) All-zero codeword assumption [Richardson01], [Varshney11] ◮ The decoder performance does not depend on the codeword xn ◮ All-zero codeword assumption : xn = 0

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 33

Section 3: Perf. analysis of faulty decoders 26

Density Evolution

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ VN messages probability distributions P(γ(ℓ)|X = 0), P(˜ γ(ℓ)|X = 0)

Faults

◮ CN messages probability distributions P(µ(ℓ)|X = 0), P(µ(ℓ)|X = 0)

Faults

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 34

Section 3: Perf. analysis of faulty decoders 27

Error probability

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Message error probabilities

  • recall LLR : log P(X=0|y)

P(X=1|y)

  • Pn,(ℓ)

e

(α) = P(γ(ℓ) < 0|X = 0) ˜ Pn,(ℓ)

e

(α, ǫ) = P(˜ γ(ℓ) < 0|X = 0) [Richardson01], [Varshney11], [Huang14], [Ngassa15], [Leduc18], etc.

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 35

Section 3: Perf. analysis of faulty decoders 28

Performance criterion

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Noiseless threshold : worst channel parameter α for which lim

n,ℓ→∞ Pn,(ℓ) e

(α) = 0 [Richardson01] ◮ Faulty threshold : worst channel parameter α for which lim

n,ℓ→∞

˜ Pn,(ℓ)

e

(α, ǫ) < η [Varshney11], [Dupraz15]

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 36

Section 3: Perf. analysis of faulty decoders 29

Threshold comparison

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Ex : Binary Symmetric Channel α = P(Y = 0|X = 1) = P(Y = 1|X = 0)

10-4 10-3 10-2 10-1 100 0.02 0.04 0.06 0.08 0.1

Pe

1e-3 1e-2 1e-1 0.02 0.04 0.06 0.08 0.1 0.12 eps01 Threshold

(3,4)-code (3,4)-code All Zero (3,5)-code (3,5)-code All Zero (3,6)-code (3,6)-code All Zero (3,12)-code (3,12)-code All Zero

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 37

Section 4: Effect of faults in the decoders 30

OUTLINE

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

  • 1. Introduction
  • 2. LDPC codes and decoders
  • 3. Perf. analysis of faulty decoders
  • 4. Effect of faults in the decoders
  • 5. Conclusion

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 38

Section 4: Effect of faults in the decoders 31

Decoder optimization for fault-tolerance

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Min-sum decoder with LLR messages quantized on q bits ◮ i.i.d. fault model, P(˜ µ|µ) , P(˜ γ|γ)

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 39

Section 4: Effect of faults in the decoders 31

Decoder optimization for fault-tolerance

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Min-sum decoder with LLR messages quantized on q bits ◮ i.i.d. fault model, P(˜ µ|µ) , P(˜ γ|γ) Method [Dupraz15], [Nguyen-Li16]

  • A wide range of quantization functions
  • Performance evaluation with faulty Density Evolution
  • Optimization of the quantization function for fault-tolerance

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 40

Section 4: Effect of faults in the decoders 32

Decoder optimization for fault-tolerance

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Optimization results

Histogram of thresholds for 5192 decoders (different sets of quantization parameters)

0.02 0.04 0.06 0.08 0.1 200 400 600 800 1000 Functional threshold Number of decoders

Robust Decoders Non-robust decoders

0.01 0.02 0.03 0.04 0.05 0.06 10

−8

10

−6

10

−4

10

−2

10 Bit Error Rate (BER) Channel error probability

Φ(v,SP)

non−robust noisy

Φ(v,SP)

  • pt

, noisy Φ(v,SP)

robust, noisy

Φ(v,SP)

non−robust noiseless

Φ(v,SP)

robust, noiseless

Φ(v,SP)

  • pt

, noiseless

Without faults With faults

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 41

Section 4: Effect of faults in the decoders 32

Decoder optimization for fault-tolerance

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Optimization results

Histogram of thresholds for 5192 decoders (different sets of quantization parameters)

0.02 0.04 0.06 0.08 0.1 200 400 600 800 1000 Functional threshold Number of decoders

Robust Decoders Non-robust decoders

0.01 0.02 0.03 0.04 0.05 0.06 10

−8

10

−6

10

−4

10

−2

10 Bit Error Rate (BER) Channel error probability

Φ(v,SP)

non−robust noisy

Φ(v,SP)

  • pt

, noisy Φ(v,SP)

robust, noisy

Φ(v,SP)

non−robust noiseless

Φ(v,SP)

robust, noiseless

Φ(v,SP)

  • pt

, noiseless

Without faults With faults

◮ Conclusion : Careful quantizer design is sufficient to ensure fault-tolerance

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 42

Section 4: Effect of faults in the decoders 33

LDPC decoders under timing errors

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Timing errors in the decoder [Brkic15]

. . . . . .

P(z(t) = w(t) |w(t), w(t−1)) = 1 − ε, P(z(t) = w(t−1)|w(t), w(t−1)) = ε.

t V

Vdd

Value 1 td Value 0

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 43

Section 4: Effect of faults in the decoders 33

LDPC decoders under timing errors

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Timing errors in the decoder [Brkic15]

. . . . . .

P(z(t) = w(t) |w(t), w(t−1)) = 1 − ε, P(z(t) = w(t−1)|w(t), w(t−1)) = ε.

t V

Vdd

Value 1 td Value 0

Main result [Dupraz17] If limℓ→∞ P(ℓ)

e (α) exists, then ∀ǫ,

lim

ℓ→∞

˜ P(ℓ)

e (α, ǫ) = lim ℓ→∞ P(ℓ) e (α)

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 44

Section 4: Effect of faults in the decoders 34

LDPC decoders under timing errors

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ L = 100 iterations

0.02 0.04 0.06 0.08 1e-7 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 1e+0 p BER (4,16), ε=0 (4,16), ε=0.2 (3,5), ε=0 (3,5), ε=0.2 (4,8), ε=0 (4,8),ε=0.2 (3,6), ε=0 (3,6),ε=0.2

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 45

Section 4: Effect of faults in the decoders 34

LDPC decoders under timing errors

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ L = 100 iterations

0.02 0.04 0.06 0.08 1e-7 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 1e+0 p BER (4,16), ε=0 (4,16), ε=0.2 (3,5), ε=0 (3,5), ε=0.2 (4,8), ε=0 (4,8),ε=0.2 (3,6), ε=0 (3,6),ε=0.2

◮ Conclusion : Timing errors do not affect the decoder performance

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 46

Section 4: Effect of faults in the decoders 35

Noisy Gallager B decoder

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Hard-decision decoders [Sundararajan14],[Vasic15] ◮ Fault model P(˜ µ = 1|µ = 0) = P(˜ µ = 0|µ = 1) = ǫ

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 47

Section 4: Effect of faults in the decoders 35

Noisy Gallager B decoder

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Hard-decision decoders [Sundararajan14],[Vasic15] ◮ Fault model P(˜ µ = 1|µ = 0) = P(˜ µ = 0|µ = 1) = ǫ ◮ LS(155,64)-code

10-3 10-2 10-1 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 BER p epsilon = 0 epsilon=1e-2

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 48

Section 4: Effect of faults in the decoders 35

Noisy Gallager B decoder

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

◮ Hard-decision decoders [Sundararajan14],[Vasic15] ◮ Fault model P(˜ µ = 1|µ = 0) = P(˜ µ = 0|µ = 1) = ǫ ◮ LS(155,64)-code

10-3 10-2 10-1 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 BER p epsilon = 0 epsilon=1e-2

◮ Conclusion : Faults in the decoder sometimes improve the decoder performance

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 49

Section 5: Conclusion 36

OUTLINE

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

  • 1. Introduction
  • 2. LDPC codes and decoders
  • 3. Perf. analysis of faulty decoders
  • 4. Effect of faults in the decoders
  • 5. Conclusion

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 50

Section 5: Conclusion 37

Conclusion

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Conclusions ◮ Density Evolution permits to analyze the performance of faulty LDPC decoders ◮ The robustness to faults depends on the decoder and on the fault model Other existing works ◮ LDPC encoders [Hachem13],[Yang14],[Dupraz16] ◮ LDPC decoders for faulty computation [Grandhi16],[Yang16] ◮ LDPC decoders in faulty memories [Chilappagari07],[Vasic07] ◮ Other families of error-correction codes [Balatsoukas18] ◮ Machine Learning Algorithms under faulty hardware [Yang16], [Leduc18] ,[Dupraz19]

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019

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SLIDE 51

Section 5: Conclusion 38

Perspectives

Introduction LDPC codes and decoders

  • Perf. analysis of faulty decoders

Effect of faults in the decoders Conclusion

Ongoing works and Perspectives ◮ Energy optimization of LDPC codes and decoders [Yaoumi19] ◮ Realistic energy-vs-faults models ◮ Practical implementations ◮ Energy-efficient Machine Learning algorithms

DESIGN OF ENERGY-EFFICIENT LDPC CODES AND DECODERS Elsa Dupraz 16/04/2019