Punctured vs Rateless Codes for Hybrid ARQ

Transcription

1 Punctured vs Rateless Codes for Hybrid ARQ Emina Soljanin Mathematical and Algorithmic Sciences Research, Bell Labs Collaborations with R. Liu, P. Spasojevic, N. Varnica and P. Whiting Tsinghua University 10/30/2006 Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

2 AUTOMATIC REPEAT REQUEST The receiving end detects frame errors and requests retransmissions. If P e is the frame error rate, the average number of transmissions is 1 (1 P e ) + + n P n 1 e (1 P e ) + = 1 1 P e Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

3 AUTOMATIC REPEAT REQUEST The receiving end detects frame errors and requests retransmissions. If P e is the frame error rate, the average number of transmissions is 1 (1 P e ) + + n P n 1 e (1 P e ) + = 1 1 P e Hybrid ARQ uses a code that can correct some frame errors. In (type I and II) HARQ schemes the average number of transmissions is reduced, but each transmission carries redundant information. Incremental redundancy (IR) HARQ is appropriate for time-varying channels (mobile wireless), used in 3G wireless standards based on Turbo codes. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

4 IR-HARQ BASED ON PUNCTURING Information bits are encoded by a (low rate) mother code. Information and a selected number of parity bits are transmitted. If a retransmission is not successful: transmitter sends additional selected parity bits receiver puts together the new bits and those previously received. Each retransmission produces a codeword of a stronger code. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

5 IR-HARQ BASED ON PUNCTURING Information bits are encoded by a (low rate) mother code. Information and a selected number of parity bits are transmitted. If a retransmission is not successful: transmitter sends additional selected parity bits receiver puts together the new bits and those previously received. Each retransmission produces a codeword of a stronger code. TSINGHUA Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

6 IR-HARQ BASED ON PUNCTURING Information bits are encoded by a (low rate) mother code. Information and a selected number of parity bits are transmitted. If a retransmission is not successful: transmitter sends additional selected parity bits receiver puts together the new bits and those previously received. Each retransmission produces a codeword of a stronger code. TSINGHUA UNIVERSITY Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

7 IR-HARQ BASED ON PUNCTURING Information bits are encoded by a (low rate) mother code. Information and a selected number of parity bits are transmitted. If a retransmission is not successful: transmitter sends additional selected parity bits receiver puts together the new bits and those previously received. Each retransmission produces a codeword of a stronger code. TSINGHUA UNIVERSITY NORTH WEST Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

8 IR-HARQ BASED ON PUNCTURING Information bits are encoded by a (low rate) mother code. Information and a selected number of parity bits are transmitted. If a retransmission is not successful: transmitter sends additional selected parity bits receiver puts together the new bits and those previously received. Each retransmission produces a codeword of a stronger code. TSINGHUA UNIVERSITY NORTH WEST BEIJING Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

9 IR-HARQ EXAMPLE Punctured Rate 1/5 Mother Code TSINGHUA UNIVERSITY NORTH WEST BEIJING at the transmitter Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

10 IR-HARQ EXAMPLE Punctured Rate 1/5 Mother Code TSINGHUA UNIVERSITY NORTH WEST BEIJING at the transmitter transmission # 1 at the receiver TSINGHUA Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

11 IR-HARQ EXAMPLE Punctured Rate 1/5 Mother Code TSINGHUA UNIVERSITY NORTH WEST BEIJING at the transmitter transmission # 1 transmission # 2 at the receiver TSINGHUA UNIVERSTY Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

12 IR-HARQ EXAMPLE Punctured Rate 1/5 Mother Code TSINGHUA UNIVERSITY NORTH WEST BEIJING at the transmitter transmission # 1 transmission # 2 transmission # 3 at the receiver TSINGHUA UNIVERSTY NORTH WEST Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

13 IR-HARQ EXAMPLE Punctured Rate 1/5 Mother Code TSINGHUA UNIVERSITY NORTH WEST BEIJING at the transmitter transmission # 1 transmission # 2 transmission # 3 transmission # 4 at the receiver TSINGHUA UNIVERSTY NORTH WEST BEIJING Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

14 ABOUT THIS TALK If transmission m 1 fails, one has to decide how many bits to send in transmission m, which bits to send in transmission m, at which power to send the bits in transmission m, and we will look into these questions for two HARQ schemes. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

15 ABOUT THIS TALK If transmission m 1 fails, one has to decide how many bits to send in transmission m, which bits to send in transmission m, at which power to send the bits in transmission m, and we will look into these questions for two HARQ schemes. Talk outline: introduce linear codes define memoryless communications channels explain how Fountain codes work derive transmission rules for two schemes compare the schemes Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

16 BINARY LINEAR CODES Suppose we want to communicate k bits ω 1,...,ω k. If we encode ω 1,...,ω k into n = k + 1 bits x 1,...,x k, x k+1 s.t. x i = ω i for 1 i n 1 and x n = ω ω n 1, we can decode all ω 1,...,ω k even if one of x 1,...,x n is erased. The generator matrix of this (n, k) code is G = k n Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

17 BINARY LINEAR CODES G defines n equations featuring k information symbols. Which code features are important? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

18 BINARY LINEAR CODES G defines n equations featuring k information symbols. Which code features are important? performance Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

19 BINARY LINEAR CODES G defines n equations featuring k information symbols. Which code features are important? performance encoding/decoding complexity Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

20 BINARY LINEAR CODES G defines n equations featuring k information symbols. Which code features are important? performance encoding/decoding complexity Which code choices are good? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

21 BINARY LINEAR CODES G defines n equations featuring k information symbols. Which code features are important? performance encoding/decoding complexity Which code choices are good? random-like codes Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

22 BINARY LINEAR CODES G defines n equations featuring k information symbols. Which code features are important? performance encoding/decoding complexity Which code choices are good? random-like codes 3G wireless implements a turbo code based HARQ we will consider LDPC and Fountain codes Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

23 THE MEMORYLESS CHANNEL MODEL X W Y X and Y are random variables. We assume binary input alphabet {0, 1} and output alphabet Y. Constant in time with transition probabilities W(b 0) and W(b 1), b Y. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

24 THE MEMORYLESS CHANNEL MODEL X W Y X and Y are random variables. We assume binary input alphabet {0, 1} and output alphabet Y. Constant in time with transition probabilities W(b 0) and W(b 1), b Y. Bhattacharyya noise parameter γ = W(b 0)W(b 1). b Y Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

25 THE MEMORYLESS CHANNEL MODEL X W Y X and Y are random variables. We assume binary input alphabet {0, 1} and output alphabet Y. Constant in time with transition probabilities W(b 0) and W(b 1), b Y. Bhattacharyya noise parameter γ = W(b 0)W(b 1). b Y Time varying with transition probabilities at time i W i (b 0) and W i (b 1), b Y. W i ( 0) and W i ( 1) are known at the receiver. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

26 PERFORMANCE MEASURE Time-Invariant Channel An (n, k) binary linear code C with A d codewords of weight d. The union-bhattacharyya bound on word error probability: n PW C A d γ d, d=1 γ is the Bhattacharyya noise parameter γ = b Y W(b 0)W(b 1). Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

27 PERFORMANCE MEASURE Time-Invariant Channel An (n, k) binary linear code C with A d codewords of weight d. The union-bhattacharyya bound on word error probability: P C W n A d γ d, d=1 γ is the Bhattacharyya noise parameter γ = b Y W(b 0)W(b 1). Weight distribution A d for a given LDPC code? Consider the set of all codes [C] with a given degree distribution. Use the average A [C](n) d instead of A d for large n. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

28 THE THRESHOLD BEHAVIOR P [C](n) W n d=1 A [C](n) d γ d Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

29 THE THRESHOLD BEHAVIOR P [C](n) W n d=1 nθ 0 d=1 A [C](n) d γ d A [C](n) d + n d= nθ 0 +1 A [C](n) d γ d Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

30 THE THRESHOLD BEHAVIOR P [C](n) W n d=1 nθ 0 d=1 A [C](n) d γ d A [C](n) d + O(n 1 2 ) + n d= nθ 0 +1 n d= nθ 0 +1 A [C](n) d γ d θ 0, 0 < θ 0 < 1 exp[d(c [C] + log γ)], where c [C] 0 is the noise threshold of the code ensemble: c [C] 0 lim sup n max 0 nθ 0 <d n log A [C](n) d. d Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

31 AN HARQ STRATEGY We assume that the transmitter wants to maximize the throughput, Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

32 AN HARQ STRATEGY We assume that the transmitter wants to maximize the throughput, knows the mother code and data rates of the past transmissions, Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

33 AN HARQ STRATEGY We assume that the transmitter wants to maximize the throughput, knows the mother code and data rates of the past transmissions, is informed of channel information of the past transmissions. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

34 AN HARQ STRATEGY We assume that the transmitter wants to maximize the throughput, knows the mother code and data rates of the past transmissions, is informed of channel information of the past transmissions. Strategy: The transmitter will send only as many codeword symbols as necessary to insure a high probability of successful ML decoding assuming a high SNR channel during the current transmission. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

35 HARQ MODEL There are at most M transmissions. I = {1,...,n} is the set indexing the bit positions in a codeword. I is partitioned in M subsets I(j), for 1 j M. Bits at positions in I(j) are transmitted during j-th transmission. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

36 HARQ MODEL There are at most M transmissions. I = {1,...,n} is the set indexing the bit positions in a codeword. I is partitioned in M subsets I(j), for 1 j M. Bits at positions in I(j) are transmitted during j-th transmission. The channel remains constant during a single transmission. γ j is the Bhattacharyya noise parameter for transmission j. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

37 PERFORMANCE MEASURE Time-Varying Channel Assume that all M transmissions have taken place. The union bound on the ML decoder word error probability: P C W I(1) d 1 =1 I(M) M A d1...d M d M =1 j=1 γ d j j, A d1...d M is the number of codewords with weight d j over I(j). Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

38 PERFORMANCE MEASURE Time-Varying Channel Assume that all M transmissions have taken place. The union bound on the ML decoder word error probability: P C W I(1) d 1 =1 I(M) M A d1...d M d M =1 j=1 γ d j j, A d1...d M is the number of codewords with weight d j over I(j). Problems: How to deal with the split-weight enumerator A d1...d M? How to evaluate the error-rate at the m-th transmission for m < M? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

39 HARQ PERFORMANCE A bit is assigned to transmission j with probability α j. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

40 HARQ PERFORMANCE A bit is assigned to transmission j with probability α j. The bits which are not yet transmitted are transmitted over a really bad channel, that is, a channel with the noise parameter equal to 1. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

41 HARQ PERFORMANCE A bit is assigned to transmission j with probability α j. The bits which are not yet transmitted are transmitted over a really bad channel, that is, a channel with the noise parameter equal to 1. The performance after m transmissions, 1 m M, is P [C](n) W n d=1 ( A [C](n) d γ d m, γ m = 1 m j=1 α j ) 1 + m γ j α j. j=1 Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

42 HARQ PERFORMANCE A bit is assigned to transmission j with probability α j. The bits which are not yet transmitted are transmitted over a really bad channel, that is, a channel with the noise parameter equal to 1. The performance after m transmissions, 1 m M, is P [C](n) W n d=1 Require γ m < exp( c [C] 0 ). ( A [C](n) d γ d m, γ m = 1 m j=1 α j ) 1 + m γ j α j. j=1 Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

43 THE m-th TRANSMISSION The average Bhattacharyya parameter after the m-th transmission: γ m = α 1 γ α m γ m + (1 α 1 α m ) 1. Choose α m or γ m or both so that γ m < exp( c [C] 0 ). Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

44 THE m-th TRANSMISSION The average Bhattacharyya parameter after the m-th transmission: γ m = α 1 γ α m γ m + (1 α 1 α m ) 1. Choose α m or γ m or both so that γ m < exp( c [C] 0 ). The tradeoff between the rate and the power: α m (1 γ m ) > 1 exp( c [C] m 1 0 ) α j (1 γ j ). j=1 Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

45 LT CODES Encoding x 1 x 2 x 3 x 4 x 5 x 6 Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

46 LT CODES Encoding x 1 x 2 x 3 x 4 x 5 x 6 y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

47 LT CODES Encoding x 1 x 2 x 3 x 4 x 5 x y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

48 LT CODES Encoding x 1 x 2 x 3 x 4 x 5 x y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 face Ω Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

49 LT CODES Encoding x 1 x 2 x 3 x 4 x 5 x y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 face Ω Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

50 LT CODES Encoding x 1 x 2 x 3 x 4 x 5 x y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 face Ω Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

51 LT CODES Encoding x 1 x 2 x 3 x 4 x 5 x y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 face Ω Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

52 LT CODES Encoding x 1 x 2 x 3 x 4 x 5 x y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 face Ω Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

53 LT CODES Encoding x 1 x 2 x 3 x 4 x 5 x y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 face Ω Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

54 LT CODES Encoding x 1 x 2 x 3 x 4 x 5 x y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 face Ω Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

55 LT CODES Decoding?????? ?? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

56 LT CODES Decoding????? ?? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

57 LT CODES Decoding????? ?? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

58 LT CODES Decoding???? ?? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

59 LT CODES Decoding???? ?? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

60 LT CODES Decoding??? ?? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

61 LT CODES Decoding??? ?? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

62 LT CODES Decoding?? ?? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

63 LT CODES Decoding?? ?? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

64 LT CODES Decoding? ?? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

65 LT CODES Decoding? ?? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

66 LT CODES Decoding ?? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

67 LT CODES Decoding ?? Code design: Ω has to be chosen to enable simple decoding under erasures. Raptor codes, fountain codes. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

68 HARQ WITH RAPTOR CODES k information bits are pre-coded by an (n, k) code. An LDPC code ensemble will be used to model the precode. For transmission i, N i Raptor bits are generated by LT of the n LDPC bits. γ j is the Bhattacharyya noise parameter for transmission j. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

69 HARQ WITH RAPTOR CODES k information bits are pre-coded by an (n, k) code. An LDPC code ensemble will be used to model the precode. For transmission i, N i Raptor bits are generated by LT of the n LDPC bits. γ j is the Bhattacharyya noise parameter for transmission j. Suppose that m transmissions have taken place. The union bound on the average ML decoder word error probability: where B w1...w m = P R W n d=1 N 1 w 1 =1 A [C](n) d m i=1 N m m B w1...w m w m=1 j=1 ( Ni w i γ wm j, ) p(h/n) w i ( 1 p(h/n) ) N i w i. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

70 HARQ PERFORMANCE The union bound on the average ML decoder word error probability: P R W n d=1 where p(d/n) = A [C](n) d m [ 1 p(d/n)(1 γi ) }{{} i=1 h Ω h(1 2d/n) h. ] Ni Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

71 HARQ PERFORMANCE The union bound on the average ML decoder word error probability: P R W n d=1 where p(d/n) = A [C](n) d m [ 1 p(d/n)(1 γi ) }{{} γ i i=1 h Ω h(1 2d/n) h. ] Ni Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

72 HARQ PERFORMANCE The union bound on the average ML decoder word error probability: P R W n d=1 where p(d/n) = A [C](n) d m [ 1 p(d/n)(1 γi ) }{{} γ i i=1 h Ω h(1 2d/n) h. Compare with the punctured scheme counterpart: P [C](n) W n d=1 A [C](n) d γ d m. ] Ni Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

73 HARQ PERFORMANCE There exist θ 0, 0 < θ 0 < 1 and s.t. p(θ) π Ω min θ 0 <θ 1 θ P R W n d=1 A [C](n) d O(n 1 2 ) + c [C] 0 lim sup n max nθ 0 <d n m [ 1 p(d/n)(1 γi ) ] N i i=1 n d= nθ 0 +1 { [ exp d c [C] 0 π Ω m i=1 log A [C](n) d. d 1 γ i R li ]}, where R li = n/n i. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

74 THE m-th TRANSMISSION The tradeoff between the rate and the power: 1 γ m R lm > c[c] m γ j. π Ω R j=1 lj Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

75 THE m-th TRANSMISSION The tradeoff between the rate and the power: 1 γ m R lm > c[c] m γ j. π Ω R j=1 lj Compare with the punctured scheme counterpart: α m (1 γ m ) > 1 exp( c [C] m 1 0 ) α j (1 γ j ). j=1 Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

76 Punctured vs Rateless Codes for Hybrid ARQ Throughput BPSK Capacity Reg LDPC (R = 0.4) Irreg LDPC (R = 0.5) Reg LDPC (R = 0.8) Reg LDPC (R = 0.9) Raptor with Reg LDPC (R = 0.9) Raptor with Reg LDPC (R = 0.8) Raptor with Reg LDPC (R = 0.4) SNR[dB] Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

77 Punctured vs Rateless Codes for Hybrid ARQ Rateless schemes win when the information about the range of SNR is not available, the SNR range is wide. Punctured schemes win when the SNR range is known and narrow, changing the mother code (precode) is not possible. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

78 Punctured vs Rateless Codes for Hybrid ARQ Rateless schemes win when the information about the range of SNR is not available, the SNR range is wide. Punctured schemes win when the SNR range is known and narrow, changing the mother code (precode) is not possible. Encoding/Decoding complexity. Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23

79 Punctured vs Rateless Codes for Hybrid ARQ Rateless schemes win when the information about the range of SNR is not available, the SNR range is wide. Punctured schemes win when the SNR range is known and narrow, changing the mother code (precode) is not possible. Encoding/Decoding complexity. Hybrid, hybrid ARQ schemes? Emina Soljanin (Bell Labs) Punctured vs Rateless Codes for Hybrid ARQ Tsinghua, October / 23