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

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.              

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

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

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 , σ 2 ) with perfect threshold estimation T ∗ , The proposed model : Gaussian noise n ∼ N (0 , σ 2 ) with threshold estimation mismatch T � = T ∗ . Robust FAIDs | D. Declercq | NVM’2018 4 / 19

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 {− C 2 , − C 1 , + C 1 , + C 2 } . We do not assume particular numerical values for C i , i.e. C i are not based on LLR computation. The 4 transition probabilities are denoted ( α 2 , α 1 , β 1 , β 2 ), and fully describe the discrete channel. The Raw Bit Error Rate (RBER) is equal to ( α 1 + α 2 ). +C2 β 2 β 1 0 +C1 α 1 − C1 1 α 2 − C2 Robust FAIDs | D. Declercq | NVM’2018 5 / 19

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 + δ 1 and T + δ 2 , 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

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 { H ( Y ) − H ( Y | X ) } T where both H ( Y ) and H ( Y | X ) depend on the discrete distribution ( α , β ) = ( α 2 , α 1 , β 1 , β 2 ). When the threshold T � = T ∗ , there is an unrecoverable capacity loss. 0.99 0.99 Thresholds 0.98 0.98 Optim. = T* T* + 0.2 0.97 T* - 0.2 0.97 0.96 0.96 Capacity Capacity 0.95 0.95 0.94 0.94 Thresholds 0.93 0.93 Optim. = T* T* + 0.2 0.92 0.92 T* - 0.2 0.91 0.91 3.5 4 4.5 5 5.5 6 6.5 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Gaussian Channel SNR : (E b /N 0 ) dB RBER Robust FAIDs | D. Declercq | NVM’2018 7 / 19

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, our 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

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

Model FAID Optimization FAID diversity Conclusion Density Evolution for LDPC Iterative decoding Definition D ( φ v , φ c ) defines a Finite Alphabet Iterative Decoder (FAID), with variable node update φ v and check-node update φ c . Channel Values φ v m vc Interconnexion Network m cv φ c Messages in the decoder are interpreted as Random Variables p ( l ) vc : density of messages m vc from variable nodes to check nodes at iteration ( l ), p ( l ) cv : density of messages m cv from check nodes to variable nodes at iteration ( l ), p C ( 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

Model FAID Optimization FAID diversity Conclusion Density Evolution for FAID decoders The Density Evolution analysis requires the all-zero codeword assumption, i.e. b n = + C 1 DE Initialization The density evolution is initialized with the transition probabilities p C ( α , β ) of the channel. p C ( u = − C 2 ) = α 2 p C ( u = − C 1 ) = α 1 p C ( u = + C 1 ) = β 1 p C ( u = + C 2 ) = β 2 DE Recursion p ( l ) vc = Function � p ( l − 1) , p C ( α , β ) , φ v � [ VNU ] cv p ( l ) cv = Function � p ( l ) vc , φ c � [ CNU ] DE Convergence Convergence is declared at iteration ( l ) if the following property holds for the APP density : � p ( l ) APP ( i ) = 0 i ≥ 0 Robust FAIDs | D. Declercq | NVM’2018 11 / 19

Model FAID Optimization FAID diversity Conclusion FAID decoders D ( φ v , φ c ) can be optimized with Density Evolution Evolution of the APP density for a Good FAID : Convergence Initialization 3 iterations 10 iterations 50 iterations 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 0 0 0 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 Evolution of the APP density for a Min-Sum decoder : non-Convergence Initialization 3 iterations 10 iterations 50 iterations 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0 0 0 0 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 Robust FAIDs | D. Declercq | NVM’2018 12 / 19

Model FAID Optimization FAID diversity Conclusion Density Evolution Threshold and FAID optimization Definition of DE threshold : all or nothing behavior ∃ RBER ∗ such that � p ( l ) if RBER < RBER ∗ APP ( i ) = 0 i ≥ 0 � p ( l ) if RBER > RBER ∗ APP ( i ) � = 0 i ≥ 0 RBER ∗ is called DE threshold. FAID optimization using DE The DE threshold RBER ∗ depends on three components : (i) the channel model p C ( α , β ), (ii) the FAID update rules φ v and φ c , and (iii) the LDPC code ensemble, defined by its connexion degree distribution. We choose here to optimize the FAID update rules φ v and φ c , for fixed : - channel model p C ( α , β ) : Gaussian channel with Wrong Threshold Estimation, - LDPC ensemble : regular ( d v , d c ) = (4 , 40) ensemble, with rate R = 0 . 9. Optimization of FAID : RBER ∗ 1 > RBER ∗ D 1 ( φ v , φ c ) > D 2 ( φ v , φ c ) if 2 . Robust FAIDs | D. Declercq | NVM’2018 13 / 19

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

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 , σ 2 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