Iterative Decoders Robust to Threshold Voltage Uncertainty D. - - PowerPoint PPT Presentation

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Iterative Decoders Robust to Threshold Voltage Uncertainty D. Declercq, B. Vasic, B. Reynwar, V. Yella, S. Planjery March 2018

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SLIDE 1

Iterative Decoders Robust to Threshold Voltage Uncertainty

                   

  • D. Declercq, B. Vasic, B. Reynwar, V. Yella, S. Planjery

March 2018

This work was supported by the National Science Foundation under SBIR Phase II Grant 1534760.

                   

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SLIDE 2

Model FAID Optimization FAID diversity Conclusion

Outline

1 Model for Threshold Estimation Mismatch 2 FAID optimization through Density Evolution 3 FAID diversity for Threshold Mismatch 4 Conclusion

Robust FAIDs |

  • D. Declercq

| NVM’2018 2 / 19

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SLIDE 3

Model FAID Optimization FAID diversity Conclusion

Outline

1 Model for Threshold Estimation Mismatch 2 FAID optimization through Density Evolution 3 FAID diversity for Threshold Mismatch 4 Conclusion

Robust FAIDs |

  • D. Declercq

| NVM’2018 3 / 19

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SLIDE 4

Model FAID Optimization FAID diversity Conclusion

Problem addressed

Discrete Flash channel with low precision We assume that the Flash channel is quantized with only 2 bits of precision

1 bit for the hard decision, coming from the first read of the Flash, 1 extra bit for the soft decision, coming from 2 extra reads of the Flash,

For each stored bit b, the Flash channel output is denoted : u = (2 b − 1) + n where the additive white noise n is not necessarily Gaussian. To get 2 bits from u, 3 quantization thresholds are needed : {−T, 0, T}.

the hard decision threshold is assumed to be always correctly estimated, equal to {0}, the soft decision thresholds {−T, T} could be wrongly estimated.

Proposed study The simplest baseline model : Gaussian noise n ∼ N (0, σ2) with perfect threshold estimation T ∗, The proposed model : Gaussian noise n ∼ N (0, σ2) with threshold estimation mismatch T = T ∗.

Robust FAIDs |

  • D. Declercq

| NVM’2018 4 / 19

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Model FAID Optimization FAID diversity Conclusion

Equivalent Discrete Model

The Baseline model is a 4-levels quantized Additive White Gaussian Noise (AWGN) Model.

The 4 levels are denoted {−C2, −C1, +C1, +C2}. We do not assume particular numerical values for Ci , i.e. Ci are not based on LLR computation. The 4 transition probabilities are denoted (α2, α1, β1, β2), and fully describe the discrete channel. The Raw Bit Error Rate (RBER) is equal to (α1 + α2). α 1 α 2 β2 β1 +C2 +C1 − C1 − C2 1

Robust FAIDs |

  • D. Declercq

| NVM’2018 5 / 19

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Model FAID Optimization FAID diversity Conclusion

Non-Gaussian Voltage Distribution vs. Threshold Variation

Noise coming from Flash reads is usually non-Gaussian

We can consider generalized distributions, with asymetric shapes and/or with heavier tails than the Gaussian Distribution, Non-Gaussianity can be captured by changing the threshold locations : −T + δ1 and T + δ2, For analysis of LDPC codes, the equivalent discrete channel needs to be weakly symmetric ⇒ same conditional distribution for input-bit values 0 and 1, Under the weakly symmetric assumption, we will consider only symmetric threshold variations −T − δ and T + δ.

Robust FAIDs |

  • D. Declercq

| NVM’2018 6 / 19

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Model FAID Optimization FAID diversity Conclusion

Effect of Threshold Mismatch

Capacity Loss

Under the AWGN model, the optimum thresholds can be computed by Mutual Information maximization of the discrete channel X(ω) → Y(ω) T ∗ = arg max

T

{H(Y) − H(Y|X)} where both H(Y) and H(Y|X) depend on the discrete distribution (α, β) = (α2, α1, β1, β2). When the threshold T = T ∗, there is an unrecoverable capacity loss.

3.5 4 4.5 5 5.5 6 6.5 Gaussian Channel SNR : (E

b/N 0)dB

0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 Capacity

  • Optim. = T*

T* + 0.2 T* - 0.2 Thresholds 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

RBER 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 Capacity

  • Optim. = T*

T* + 0.2 T* - 0.2 Thresholds

Robust FAIDs |

  • D. Declercq

| NVM’2018 7 / 19

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Model FAID Optimization FAID diversity Conclusion

Optimization of Decoders for Threshold Mismatch

Performance prediction for LDPC decoding

LDPC iterative decoding limits can be predicted by the so-called Density Evolution (DE) technique, for a given LDPC ensemble and a given iterative decoder, Density Evolution predicts the asymptotic gap between iterative decoding and capacity : C − λ. The LDPC codes/decoders with the smallest gap C − λ show the best waterfall performance.

When the channel is degraded, the DE threshold loss is not necessarily equal to the capacity loss. Robustness to Threshold Mismatch

C(T ∗) : capacity of the discrete channel under perfect thresholding, λ(T ∗) : corresponding DE threshold, C(T) : capacity of the discrete channel under wrong thresholding, λ(T) : corresponding DE threshold,

  • ur goal is to optimize the iterative decoders such that
  • λ(T ∗) − λ(T)

< C(T ∗) − C(T)

  • Robust FAIDs

|

  • D. Declercq

| NVM’2018 8 / 19

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SLIDE 9

Model FAID Optimization FAID diversity Conclusion

Outline

1 Model for Threshold Estimation Mismatch 2 FAID optimization through Density Evolution 3 FAID diversity for Threshold Mismatch 4 Conclusion

Robust FAIDs |

  • D. Declercq

| NVM’2018 9 / 19

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Model FAID Optimization FAID diversity Conclusion

Density Evolution for LDPC Iterative decoding

Definition

D(φv , φc) defines a Finite Alphabet Iterative Decoder (FAID), with variable node update φv and check-node update φc.

v

φ

c

φ

Interconnexion Network

Channel Values

m m

vc cv

Messages in the decoder are interpreted as Random Variables

p(l)

vc : density of messages mvc from variable nodes to check nodes at iteration (l),

p(l)

cv : density of messages mcv from check nodes to variable nodes at iteration (l),

pC(α, β) : transition probabilities of the channel, p(l)

APP : density of a posteriori probabilities at iteration (l). Robust FAIDs |

  • D. Declercq

| NVM’2018 10 / 19

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Model FAID Optimization FAID diversity Conclusion

Density Evolution for FAID decoders

The Density Evolution analysis requires the all-zero codeword assumption, i.e. bn = +C1

DE Initialization

The density evolution is initialized with the transition probabilities pC(α, β) of the channel. pC(u = −C2) = α2 pC(u = −C1) = α1 pC(u = +C1) = β1 pC(u = +C2) = β2

DE Recursion

[VNU] p(l)

vc = Function

p(l−1)

cv

, pC(α, β), φv [CNU] p(l)

cv = Function

p(l)

vc, φc

DE Convergence

Convergence is declared at iteration (l) if the following property holds for the APP density :

  • i≥0

p(l)

APP(i) = 0 Robust FAIDs |

  • D. Declercq

| NVM’2018 11 / 19

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SLIDE 12

Model FAID Optimization FAID diversity Conclusion

FAID decoders D(φv, φc) can be optimized with Density Evolution

Evolution of the APP density for a Good FAID : Convergence

  • 4
  • 2

2 4 0.2 0.4 0.6 0.8 1 Initialization

  • 4
  • 2

2 4 0.2 0.4 0.6 0.8 1 3 iterations

  • 4
  • 2

2 4 0.2 0.4 0.6 0.8 1 10 iterations

  • 4
  • 2

2 4 0.2 0.4 0.6 0.8 1 50 iterations

Evolution of the APP density for a Min-Sum decoder : non-Convergence

  • 4
  • 2

2 4 0.2 0.4 0.6 0.8 1 Initialization

  • 4
  • 2

2 4 0.2 0.4 0.6 0.8 1 3 iterations

  • 4
  • 2

2 4 0.2 0.4 0.6 0.8 1 10 iterations

  • 4
  • 2

2 4 0.2 0.4 0.6 0.8 1 50 iterations

Robust FAIDs |

  • D. Declercq

| NVM’2018 12 / 19

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Model FAID Optimization FAID diversity Conclusion

Density Evolution Threshold and FAID optimization

Definition of DE threshold : all or nothing behavior

∃ RBER∗ such that

  • i≥0

p(l)

APP(i) = 0

if RBER < RBER∗

  • i≥0

p(l)

APP(i) = 0

if RBER > RBER∗

RBER∗ is called DE threshold.

FAID optimization using DE

The DE threshold RBER∗ depends on three components : (i) the channel model pC(α, β), (ii) the FAID update rules φv and φc, and (iii) the LDPC code ensemble, defined by its connexion degree distribution. We choose here to optimize the FAID update rules φv and φc, for fixed :

  • channel model pC(α, β) : Gaussian channel with Wrong Threshold Estimation,
  • LDPC ensemble : regular (dv , dc) = (4, 40) ensemble, with rate R = 0.9.

Optimization of FAID : D1(φv , φc) > D2(φv , φc) if RBER∗

1 >RBER∗ 2 . Robust FAIDs |

  • D. Declercq

| NVM’2018 13 / 19

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SLIDE 14

Model FAID Optimization FAID diversity Conclusion

Outline

1 Model for Threshold Estimation Mismatch 2 FAID optimization through Density Evolution 3 FAID diversity for Threshold Mismatch 4 Conclusion

Robust FAIDs |

  • D. Declercq

| NVM’2018 14 / 19

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Model FAID Optimization FAID diversity Conclusion

Mismatch Model

Model for Threshold Wrong Estimation

We assume that for each and every codeword, the thresholds {−T, T} are noisy T = T ∗ + δ δ ∼ N (0, σ2

δ)

This model account for a different threshold mismatch at each and every codeword read, We do not assume that the thresholds are always over- or under-estimated.

Robust FAIDs |

  • D. Declercq

| NVM’2018 15 / 19

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Model FAID Optimization FAID diversity Conclusion

Concept and Use of FAID diversity

Criterion to select the different FAIDs

In order to optimize FAIDs decoders for a given threshold mismatch, the equivalent channel pC(α, β) must be fixed, However, in our model, the noise threshold changes from one codeword to another. We propose the following approach to robutness : 1

  • ptimize one FAID for left-shift thresholds, i.e. T = T ∗ − 0.2 : D−δ(φv , φc)

⇒ This will create more strong errors −C2 than weak errors −C1 2

  • ptimize another FAID for right-shift thresholds, i.e. T = T ∗ + 0.2 : D+δ(φv , φc)

Definition FAID diversity [2]

The two optimized FAIDs are used sequentially : 1 for a given noisy codeword u = c + e, run decoder D−δ(φv , φc) for a given number of iterations Nit , 2 if the first decoder fails to correct the error event e, run the second decoder D+δ(φv , φc) using u, for another Nit iterations. The two decoders are optimized for different channel models, which means that they typically correct different error events, Decoder diversity does not impact the decoder hardware complexity or the average throughput, Only the worst case latency is impacted : 2 Nit instead of Nit .

[2] D. DECLERCQ, B. VASIC, SK. PLANJERY, E. LI, “FINITE ALPHABET ITERATIVE DECODERS-PART II : TOWARDS GUARANTEED ERROR CORRECTION OF LDPC CODES VIA ITERATIVE DECODER DIVERSITY”, IEEE Transactions on Communications 61 (10), 4046-4057, 2013 Robust FAIDs |

  • D. Declercq

| NVM’2018 16 / 19

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Model FAID Optimization FAID diversity Conclusion

Results

Simulation settings

We compared three different decoders for the mismatch model : (i) Min-Sum quantized on 3 bits, (ii) FAID optimized for the AWGN model and (iii) FAID robust to Threshold mismatch, We used the noisy threshold model T = T ∗ + δ, with σ2

δ = 0.03 and σ2 δ = 0.045,

Layered decoding with a maximum of Nit = 20 iterations. Rate R=0.89 regular LDPC code, with N = 2kB length. Min-Sum and classical FAID D−δ(φv , φc) + D+δ(φv , φc)

0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 RBER 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Frame Error Rate

FAID - 2 = 0 FAID - 2 = 0.03 FAID - 2 = 0.045 MinSum - 2 = 0 MinSum - 2 = 0.03 MinSum - 2 = 0.045 Threshold Mismatch Variance

0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 RBER 10-6 10-5 10-4 10-3 10-2 10-1 100 Frame Error Rate

Robust FAID - 2 = 0 Robust FAID - 2 = 0.03 Robust FAID - 2 = 0.045 Threshold Mismatch Variance

Robust FAIDs |

  • D. Declercq

| NVM’2018 17 / 19

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Model FAID Optimization FAID diversity Conclusion

Outline

1 Model for Threshold Estimation Mismatch 2 FAID optimization through Density Evolution 3 FAID diversity for Threshold Mismatch 4 Conclusion

Robust FAIDs |

  • D. Declercq

| NVM’2018 18 / 19

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Model FAID Optimization FAID diversity Conclusion

Conclusion

What we have shown When the characteristics of the channel are known, we can optimize a FAID specifically for this model with Density Evolution, We optimized two FAIDs for right-shift and left-shift threshold errors, We used FAID Decoder Diversity to combine the advantages of the two decoders. Way forward Combine robustness in Threshold mismatch and Trapping Set mitigation for low error floors, Consider other channel impairments, such as fully asymmetric models.

Robust FAIDs |

  • D. Declercq

| NVM’2018 19 / 19