Green Codes : Energy-efficient short-range communication

Transcription

1 Green Codes : Energy-efficient short-range communication Pulkit Grover Department of Electrical Engineering and Computer Sciences University of California at Berkeley Joint work with Prof. Anant Sahai

2 Motivation : Understand processing power consumed in communicating Fixed Rate Fixed message size processor with heat sink small sensors 2

3 Motivation : Understand processing power consumed in communicating Fixed Rate Fixed message size processor with heat sink small sensors Moore s law : decreasing implementation complexity 2

4 Motivation : Understand processing power consumed in communicating Fixed Rate Fixed message size processor with heat sink small sensors Moore s law : decreasing implementation complexity - significant power consumed in computations 2

5 Motivation : Understand processing power consumed in communicating Fixed Rate Fixed message size processor with heat sink small sensors Moore s law : decreasing implementation complexity - significant power consumed in computations total power for communicating 2

6 Motivation : Understand processing power consumed in communicating Fixed Rate Fixed message size processor with heat sink small sensors Moore s law : decreasing implementation complexity Small battery operated wireless sensors - significant power consumed in computations total power for communicating 2

7 Motivation : Understand processing power consumed in communicating Fixed Rate Fixed message size processor with heat sink small sensors Moore s law : decreasing implementation complexity - significant power consumed in computations Small battery operated wireless sensors - energy at a premium. total power for communicating 2

8 Motivation : Understand processing power consumed in communicating Fixed Rate Fixed message size processor with heat sink small sensors Moore s law : decreasing implementation complexity - significant power consumed in computations total power for communicating Small battery operated wireless sensors - energy at a premium. - flexibility in rate. 2

9 Motivation : Understand processing power consumed in communicating Fixed Rate Fixed message size processor with heat sink small sensors Moore s law : decreasing implementation complexity - significant power consumed in computations total power for communicating Small battery operated wireless sensors - energy at a premium. - flexibility in rate. total energy per bit 2

10 Promise of Shannon Theory Fixed Rate: Shannon waterfall!$!$"#.*/0 $! 1!0), 0"02!%!%"#!&!&"#!(!("# :6;*<,<08-56=>?==?* *60758,-95..!#!#"# R = 1/3!'!!"# $ $"# % %"# & &"# )*+,- 3

11 Promise of Shannon Theory Fixed Rate: Shannon waterfall Fixed message size : Verdu On channel capacity per unit cost.*/0 $! 1!0), 0"02!$!$"#!%!%"#!&!&"#!(!("# :6;*<,<08-56=>?==?* *60758,-95..!#!#"# R = 1/3!'!!"# $ $"# % %"# & &"# )*+,- 3

12 Promise of Shannon Theory Fixed Rate: Shannon waterfall Fixed message size : Verdu On channel capacity per unit cost.*/0 $! 1!0), 0"02!$!$"#!%!%"#!&!&"#!(!("# :6;*<,<08-56=>?==?* *60758,-95..!#!#"# R = 1/3!'!!"# $ $"# % %"# & &"# )*+,- Long distance communication - processing power transmit power -- Shannon theory works! Short distance communication - Processing power can be substantial [Agarwal 98, Kravertz et al 98, Goldsmith et al 02, Cui et al 05] 3

13 Information theory + processing power =? Fixed rate!$!$"#.*/0 $! 1!0), 0"02!%!%"#!&!&"#!( :6;*<,<08-56=>?==?* *60758, =?!("#!#!#"#!'!!"# $ $"# % %"# & &"# )*+,- processor with heat sink 4

14 Information theory + processing power =? Fixed rate!$!$"#.*/0 $! 1!0), 0"02!%!%"#!&!&"#!( :6;*<,<08-56=>?==?* *60758, =?!("#!#!#"#!'!!"# $ $"# % %"# & &"# )*+,- Fixed message size processor with heat sink + =? 4 small sensors

15 Talk Outline Motivation: Power consumption - Fixed rate and fixed message size problems. Decoding power using decoding complexity. Complexity-performance tradeoffs. - our bounds for iterative decoding. Fixed rate -- lower bounds on total power. Fixed message size (Green codes) -- lower bounds on min energy. How tight are our bounds : Related coding-theoretic literature 5

16 Modeling processing power through decoding complexity Encoder Decoder 6

17 Modeling processing power through decoding complexity Encoder Decoder power consumed in decoding: model using the decoding complexity - decoding complexity : number of operations performed at the decoder - constant amount of energy per operation. 6

18 Modeling processing power through decoding complexity Encoder Decoder power consumed in decoding: model using the decoding complexity - decoding complexity : number of operations performed at the decoder - constant amount of energy per operation. the common currency: power 6

19 Talk Outline Motivation: Power consumption - Fixed rate and fixed message size problems. Decoding power using decoding complexity. Complexity-performance tradeoffs. - our bounds for iterative decoding. Fixed rate -- lower bounds on total power. Fixed message size (Green codes) -- lower bounds on min energy. How tight are our bounds : Related coding-theoretic literature 7

20 Understanding decoding complexity : complexity - performance tradeoffs complexity-performance tradeoffs : - Required complexity to attain error probability P e and rate R. - Lower bounds : Abstract away from details of code structure. - Upper bounds : code constructions. e.g. block codes : P e exp( me r (R)) e.g. convolution codes : - error exponents with constraint length [Viterbi 67] - cut-off rate for sequential decoding [Jacobs and Berlekamp 67] 8

21 Understanding decoding complexity : complexity - performance tradeoffs complexity-performance tradeoffs : - Required complexity to attain error probability P e and rate R. - Lower bounds : Abstract away from details of code structure. - Upper bounds : code constructions. e.g. block codes : P e exp( me r (R)) e.g. convolution codes : - error exponents with constraint length [Viterbi 67] - cut-off rate for sequential decoding [Jacobs and Berlekamp 67] Want a similar analysis for iterative decoding. 8

22 9 Iterative decoding : Decoding by passing messages Output nodes Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 9 Decoder implementation graph

23 9 Iterative decoding : Decoding by passing messages Output nodes Y 1 Information nodes B 1 Y 2 B 2 Y 3 Y 4 Y 5 B 3 B 4 B 5 Y 6 Y 7 B 6 B 7 Y 8 Y 9 Decoder implementation graph

24 9 Iterative decoding : Decoding by passing messages Output nodes Y 1 Helper nodes Information nodes B 1 Y 2 B 2 Y 3 B 3 Y 4 B 4 Y 5 B 5 Y 6 B 6 Y 7 Y 8 B 7 Y 9 Decoder implementation graph

25 9 Iterative decoding : Decoding by passing messages Output nodes Y 1 Helper nodes Information nodes B 1 Y 2 B 2 Y 3 B 3 Y 4 B 4 Y 5 B 5 Y 6 B 6 Y 7 Y 8 B 7 Y 9 Decoder implementation graph

26 9 Iterative decoding : Decoding by passing messages Output nodes Y 1 Helper nodes Information nodes B 1 Y 2 B 2 Y 3 B 3 Y 4 B 4 Y 5 B 5 Y 6 B 6 Y 7 Y 8 B 7 Y 9 Decoder implementation graph

27 9 Iterative decoding : Decoding by passing messages Output nodes Helper nodes Information nodes Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 B 1 B 2 B 3 B 4 B 5 B 6 B 7 Each node consumes Joules of energy per iteration. After l iterations, the energy consumed is γ l # of nodes Each node is connected to at most other nodes -- an implementation constraint. γ α Y 9 Decoder implementation graph

28 9 Iterative decoding : Decoding by passing messages Output nodes Helper nodes Information nodes Y 1 Y 2 Y 3 Y 4 B 1 B 2 B 3 B 4 Each node consumes Joules of energy per iteration. After l iterations, the energy consumed is γ l # of nodes γ Y 5 Y 6 Y 7 B 5 B 6 B 7 Each node is connected to at most other nodes -- an implementation constraint. α Y 8 Y 9 Suffices now to find l Decoder implementation graph

29 Lower bound on l : Key Idea B i a 10

30 Lower bound on l : Key Idea B i a 10

31 Lower bound on l : Key Idea B i a 10

32 Lower bound on l : Key Idea B i a l < a l+1 10

33 Lower bound on l : Key Idea B i a l < a l+1 Channel needs to behave atypically only in the decoding neighborhood to cause an error 10

34 Lower bound on decoding complexity Result [Sahai, Grover, Submitted to IT Trans. 07] In the limit of small P e l 1 log(α) log ( log 1 P e (C R) 2 ) C = Channel capacity R = Rate P e = error probability α = maximum node degree 11

35 Lower bound on decoding complexity l 1 log(α) log ( log 1 P e (C R) 2 ) A general lower bound - applies to all (possible) codes with decoding based on passing messages. - applies regardless of the presence of cycles. - applies to all decoding algorithms based on passing messages. 12

36 Talk Outline Motivation: Power consumption - Fixed rate and fixed message size problems. Decoding power using decoding complexity. Complexity-performance tradeoffs. - our bounds for iterative decoding. Fixed rate -- lower bounds on total power. Fixed message size (Green codes) -- lower bounds on min energy. How tight are our bounds : Related coding-theoretic literature 13

37 Fixed Rate: Total power consumption k Encoder m m Decoder k # of nodes P total = P T + γ l m P T + γ l ( P T + γ log(α) log log 1 P e (C(P T ) R) 2 ) Minimize P total by optimizing over P T l = Number of iterations g = Energy consumed per node per iteration P T = Transmit power m = block-length 14

38 P total P T + Fixed Rate: Total Power Curves γ log(α) log ( log 1 P e (C(P T ) R) 2 )!# R = 1/3.!% /*0 "! 1.),.2!'!"(!$# 34566*6.758,-95//.! " # $ % & )*+,-. 15

39 P total P T + Fixed Rate: Total Power Curves γ log(α) log ( log 1 P e (C(P T ) R) 2 )!# R = 1/3.!% /*0 "! 1.),.2!'!"( 758,-:/;<, =>-?,!$# 34566*6.758,-95//.! " # $ % & )*+,-. 15

40 P total P T + Fixed Rate: Total Power Curves γ log(α) log ( log 1 P e (C(P T ) R) 2 )!# R = 1/3.!% /*0 "! 8-56:B;8.A*+,- 758,-:/;<, =>-?,!$# 34566*6.758,-95//.! " # $ % & )*+,-. 15

41 Fixed Rate: Summary Total power increases unboundedly as P e 0 Optimal transmit power strictly larger than the Shannon limit (transmit power - decoding power tradeoff) 16

42 Talk Outline Motivation: Power consumption - Fixed rate and fixed message size problems. Decoding power using decoding complexity. Complexity-performance tradeoffs. - our bounds for iterative decoding. Fixed rate -- lower bounds on total power. Fixed message size (Green codes) -- lower bounds on min energy. How tight are our bounds : Related coding-theoretic literature 17

43 Fixed message size : Green Codes Minimum energy per-bit k Encoder m m Decoder k E total = mp total = m P T + γ l # of nodes 18

44 Fixed message size : Green Codes Minimum energy per-bit k Encoder m m Decoder k E total = mp total = m P T + γ l # of nodes E per bit = E total k = P T # of nodes + γ l R k 18

45 Fixed message size : Green Codes Minimum energy per-bit k Encoder m m Decoder k E total = mp total = m P T + γ l # of nodes E per bit = E total k = P T # of nodes + γ l R k P T R + γ l max{k, m} k 18

46 Fixed message size: Minimum energy per bit curves!!(!&)!&( 45.!) 6"7, "8!$)!$(!')!'( 9:;++5+"42<23!()!((!"!#$%" &" $" '" ( *+,-./"0,-"123 19

47 Fixed message size: Minimum energy per bit curves!!(!&)!&( 45.!) 6"7, "8!$)!$(!')!'(!()!(( 9:;++5+"42<23 &" $" '" ( *+,-./"0,-"123 Black-box bounds : Based on [Massaad, Medard and Zheng] 19

48 Fixed message size: Minimum energy per bit curves!!( 45.!) 6"7, "8!&)!&(!$)!$(!')!'(!()!(( 9:;++5+"42<23 ""C")#& &" $" '" ( *+,-./"0,-"123 Black-box bounds : Based on [Massaad, Medard and Zheng] 19

49 Fixed message size: Optimal rate curves!)!!$!.+/ &! !(!!'!!151!"6!151!")!#!!%!!&!!!"#$!"%!"%$ & * +,- 20

50 Fixed message size: Summary Minimum energy per bit increases to infinity as P e 0 - compare with a constant, ln(4), in classical information theory. Optimizing rate converges to 1. - zero in classical information theory. 21

51 Talk Outline Motivation: Power consumption - Fixed rate and fixed message size problems. Decoding power using decoding complexity. Complexity-performance tradeoffs. - our bounds for iterative decoding. Fixed rate -- lower bounds on total power. Fixed message size (Green codes) -- lower bounds on min energy. How tight are our bounds : Related coding-theoretic literature 22

52 Lower bounds on complexity: how tight are they? l 1 log(α) log ( log 1 P e (C R) 2 ) y y = x Optimal behavior with respect to P e y = f! x " - regular LDPC s achieve this! [Lentmaier et al] gap = C R what about behavior with? x 23

53 Complexity behavior with gap = C R [Gallager, Burshtein et al, Sason-Urbanke] Lower bounds on density for LDPCs. [Pfister-Sason, Hsu-Anastastopoulos] Upper bounds. ( ) 1 Khandekar-McEliece conjecture: l Ω C R [Sason, Weichman] For LDPCs, IRAs, ARAs, if there are a nonzero fraction of degree 2 nodes, and the graph is a tree, the conjecture holds. ( ) 1 - but with degree-2 nodes, l log - and it seems that degree-2 nodes are needed to approach capacity. - from energy perspective, is it worth approaching capacity? P e 24

54 Thank you Full paper on arxiv - The price of certainty: Waterslide curves and the gap to capacity. Anant Sahai and Pulkit Grover. 25