Optimal Data Shaping Code Design Yi Liu, Pengfei Huang, Alexander W. Bergman and Paul H. Siegel Center for Memory and Recording Research, UC San Diego Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 1 / 17
Outline Introduction 1 Type-I and Type-II Minimization 2 Encoder Design 3 Experiment Results on MLC Shaping Codes 4 Conclusion 5 Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 2 / 17
Introduction Introduction ◮ Flash memory: the most widely used non-volatile memory ◮ fast read/write speed ◮ low power consumption ◮ Flash memory cells gradually wear out during program-erase (P/E) cycling. ◮ Damage from programming the cell depends on the cell level. Programming a cell to higher level induces more damage. ◮ Enhancing lifetime by using shaping codes ◮ Endurance code 1 : shapes random (unstructured) data with a given rate ◮ Direct shaping code 2 , 3 : shapes structured data with rate 1 1 A. Jagmohan, M. Franceschini, L. A. Lastras-Montano and J. Karidis, "Adaptive endurance coding for NAND Flash," 2010 IEEE Globecom Workshops , Miami, FL, 2010, pp. 1841-1845. 2 E. Sharon, et al., Data Shaping for Improving Endurance and Reliability in Sub-20nm NAND, presented at Flash Memory Summit, Santa Clara, CA, August 4-7, 2014. 3 Y. Liu and P. H. Siegel, “Shaping codes for structured data,” in Proc. IEEE Globecom , Washington, D.C., Dec. 4-8, 2016, pp. 1–5. Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 3 / 17
Type-I and Type-II Minimization Definition of General Shaping Codes Definition ◮ Let X = X 1 X 2 . . . be an i.i.d source with alphabet X = { α 1 , . . . , α u } . The distribution of X will be denoted by P ( P 1 ≥ P 2 ≥ . . . P u ). Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 4 / 17
Type-I and Type-II Minimization Definition of General Shaping Codes Definition ◮ Let X = X 1 X 2 . . . be an i.i.d source with alphabet X = { α 1 , . . . , α u } . The distribution of X will be denoted by P ( P 1 ≥ P 2 ≥ . . . P u ). ◮ Let Y = { β 1 , . . . , β v } be an alphabet and Y ∗ the set of all finite sequences over Y , including the null string λ of length 0. Every β i corresponds to a cost U i ( U 1 ≤ U 2 ≤ . . . ≤ U v ). Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 4 / 17
Type-I and Type-II Minimization Definition of General Shaping Codes Definition ◮ Let X = X 1 X 2 . . . be an i.i.d source with alphabet X = { α 1 , . . . , α u } . The distribution of X will be denoted by P ( P 1 ≥ P 2 ≥ . . . P u ). ◮ Let Y = { β 1 , . . . , β v } be an alphabet and Y ∗ the set of all finite sequences over Y , including the null string λ of length 0. Every β i corresponds to a cost U i ( U 1 ≤ U 2 ≤ . . . ≤ U v ). A shaping code is defined as a prefix-free mapping φ : X q → Y ∗ which maps x q 1 to a variable length sequence y ∗ . Example ◮ Input: X = { 0 , 1 } , X ∼ Ber ( 1 2 ) ◮ Outut: Y = { 0 , 1 } , U 0 = 0 . 585 and U 1 = 1 . 585 ◮ Shaping code defined by mapping { 11 → 111 , 10 → 110 , 01 → 10 , 00 → 0 } . Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 4 / 17
Type-I and Type-II Minimization Expansion Factor Definition ◮ The expected length of a codeword is � P ( x q 1 ) L ( φ ( x q E ( L ) = 1 )) . (1) x q 1 ∈X q Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 5 / 17
Type-I and Type-II Minimization Expansion Factor Definition ◮ The expected length of a codeword is � P ( x q 1 ) L ( φ ( x q E ( L ) = 1 )) . (1) x q 1 ∈X q ◮ We define the expansion factor of a shaping code to be f = E ( L ) (2) q Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 5 / 17
Type-I and Type-II Minimization Expansion Factor Definition ◮ The expected length of a codeword is � P ( x q 1 ) L ( φ ( x q E ( L ) = 1 )) . (1) x q 1 ∈X q ◮ We define the expansion factor of a shaping code to be f = E ( L ) (2) q Example ◮ Shaping code defined by mapping { 11 → 111 , 10 → 10 , 01 → 10 , 00 → 0 } . Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 5 / 17
Type-I and Type-II Minimization Expansion Factor Definition ◮ The expected length of a codeword is � P ( x q 1 ) L ( φ ( x q E ( L ) = 1 )) . (1) x q 1 ∈X q ◮ We define the expansion factor of a shaping code to be f = E ( L ) (2) q Example ◮ Shaping code defined by mapping { 11 → 111 , 10 → 10 , 01 → 10 , 00 → 0 } . ◮ E ( L ) = 1 4 ( 3 + 3 + 2 + 1 ) = 2 . 25 ◮ f = E ( L ) = 1 . 125 q Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 5 / 17
Type-I and Type-II Minimization Probability of Occurrence Definition ◮ Consider the first l symbols of φ ( X ) , denoted by y l 1 . Its probability is Q ( y l 1 ) Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 6 / 17
Type-I and Type-II Minimization Probability of Occurrence Definition ◮ Consider the first l symbols of φ ( X ) , denoted by y l 1 . Its probability is Q ( y l 1 ) ◮ We denote the number of β i in sequence y l 1 by N i ( y l 1 ) Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 6 / 17
Type-I and Type-II Minimization Probability of Occurrence Definition ◮ Consider the first l symbols of φ ( X ) , denoted by y l 1 . Its probability is Q ( y l 1 ) ◮ We denote the number of β i in sequence y l 1 by N i ( y l 1 ) The probability of occurrence ˆ Y in encoded sequences φ ( X ) is E ( N i ( Y l 1 )) P i = Pr ( ˆ ˆ � N i ( y l 1 ) Q ( y l Y = β i ) = lim 1 ) / l = lim . (3) l l →∞ l →∞ y l 1 Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 6 / 17
Type-I and Type-II Minimization Probability of Occurrence Definition ◮ Consider the first l symbols of φ ( X ) , denoted by y l 1 . Its probability is Q ( y l 1 ) ◮ We denote the number of β i in sequence y l 1 by N i ( y l 1 ) The probability of occurrence ˆ Y in encoded sequences φ ( X ) is E ( N i ( Y l 1 )) P i = Pr ( ˆ ˆ � N i ( y l 1 ) Q ( y l Y = β i ) = lim 1 ) / l = lim . (3) l l →∞ l →∞ y l 1 Lemma For a prefix-free shaping code φ : X q → Y ∗ , ˆ Y exists and 1 ˆ P i = E ( N i ( φ ( X q ))) (4) E ( L ) Once we know the probability of occurrence, we can calculate the cost per output i ˆ symbol � P i U i (we also call it average wear cost). Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 6 / 17
Type-I and Type-II Minimization Type-I and Type-II Minimization ◮ Data shaping codes try to reduce the wear cost, there are two different goals. Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 7 / 17
Type-I and Type-II Minimization Type-I and Type-II Minimization ◮ Data shaping codes try to reduce the wear cost, there are two different goals. ◮ The first goal is to minimize the average cost per output symbol (average cost), given a fixed expansion factor (Type-I minimization). Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 7 / 17
Type-I and Type-II Minimization Type-I and Type-II Minimization ◮ Data shaping codes try to reduce the wear cost, there are two different goals. ◮ The first goal is to minimize the average cost per output symbol (average cost), given a fixed expansion factor (Type-I minimization). ◮ We try to solve the following type-I minimization problem � ˆ minimize P i U i ˆ P i i Y ) ≥ H ( X ) H ( ˆ (5) subject to f ˆ � P i = 1 . i Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 7 / 17
Type-I and Type-II Minimization Type-I and Type-II Minimization ◮ Data shaping codes try to reduce the wear cost, there are two different goals. ◮ The first goal is to minimize the average cost per output symbol (average cost), given a fixed expansion factor (Type-I minimization). ◮ We try to solve the following type-I minimization problem � ˆ minimize P i U i ˆ P i i Y ) ≥ H ( X ) H ( ˆ (5) subject to f ˆ � P i = 1 . i ◮ High rate is required in flash memory device for low encoding/decoding time complexity and high storage capacity. Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 7 / 17
Type-I and Type-II Minimization Type-I and Type-II Minimization ◮ The second goal is to minimize the average cost per input symbol (total cost) and find the optimal expansion factor (Type-II minimization). Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 8 / 17
Type-I and Type-II Minimization Type-I and Type-II Minimization ◮ The second goal is to minimize the average cost per input symbol (total cost) and find the optimal expansion factor (Type-II minimization). ◮ We try to solve the following type-II minimization problem � ˆ minimize f P i U i f , ˆ P i i Y ) ≥ H ( X ) H ( ˆ (6) subject to f ˆ � P i = 1 . i Liu, Huang, Bergman, Siegel (CMRR) Optimal Shaping Codes March. 2018 8 / 17
Recommend
More recommend
