HYBRID ARQ IN WIRELESS NETWORKS Emina Soljanin Mathematical - - PowerPoint PPT Presentation

HYBRID ARQ IN WIRELESS NETWORKS

Emina Soljanin Mathematical Sciences Research Center, Bell Labs March 19, 2003

HYBRID ARQ IN WIRELESS NETWORKS

Emina Soljanin Mathematical Sciences Research Center, Bell Labs March 19, 2003

• R. Liu and P. Spasojevic

WINLAB

HYBRID ARQ IN WIRELESS NETWORKS

Emina Soljanin Mathematical Sciences Research Center, Bell Labs March 19, 2003

• R. Liu and P. Spasojevic

WINLAB

1

ACKNOWLEDGEMENTS

Alexei Ashikhmin Jaehyiong Kim Sudhir Ramakrishna Adriaan van Wijngaarden

2

AUTOMATIC REPEAT REQUEST

• The receiving end detects frame errors and requests retransmissions.
• Pe is the frame error rate, the average number of transmissions is

1 · (1 − Pe) + · · · + n · P n−1

e

(1 − Pe) + · · · = 1 1 − Pe

2

AUTOMATIC REPEAT REQUEST

• The receiving end detects frame errors and requests retransmissions.
• Pe is the frame error rate, the average number of transmissions is

1 · (1 − Pe) + · · · + n · P n−1

e

(1 − Pe) + · · · = 1 1 − Pe

• Hybrid ARQ uses a code that can correct some frame errors.

2

AUTOMATIC REPEAT REQUEST

• The receiving end detects frame errors and requests retransmissions.
• Pe is the frame error rate, the average number of transmissions is

1 · (1 − Pe) + · · · + n · P n−1

e

(1 − Pe) + · · · = 1 1 − Pe

• Hybrid ARQ uses a code that can correct some frame errors.
• In HARQ schemes

– the average number of transmissions is reduced, but – each transmission carries redundant information.

3

THROUGHPUT IN HYBRID ARQ BPSK, AWGN, BCH Coded

0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 uncoded

Es/N0 [dB]

3

THROUGHPUT IN HYBRID ARQ BPSK, AWGN, BCH Coded

0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 uncoded

Es/N0 [dB]

0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 [255,247,3]

3

THROUGHPUT IN HYBRID ARQ BPSK, AWGN, BCH Coded

0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 uncoded

Es/N0 [dB]

0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 [255,247,3] 0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 [255,215,11]

3

THROUGHPUT IN HYBRID ARQ BPSK, AWGN, BCH Coded

0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 uncoded

Es/N0 [dB]

0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 [255,247,3] 0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 [255,215,11] 0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 [255,177,23]

3

THROUGHPUT IN HYBRID ARQ BPSK, AWGN, BCH Coded

0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 uncoded

Es/N0 [dB]

0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 [255,247,3] 0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 [255,215,11] 0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 [255,177,23] 0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 cutoff rate

3

THROUGHPUT IN HYBRID ARQ BPSK, AWGN, BCH Coded

0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 uncoded

Es/N0 [dB]

0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 [255,247,3] 0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 [255,215,11] 0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 [255,177,23] 0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 cutoff rate 0.2 0.4 0.6 0.8 1

• 6
• 4
• 2

2 4 6 8 10 capacity

4

TYPE II HYBRID ARQ Incremental Redundancy

• Information bits are encoded by a (low rate) mother code.
• Information and a selected number of parity bits are transmitted.

4

TYPE II HYBRID ARQ Incremental Redundancy

• 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.
• Family of codes obtained by puncturing of the mother code.

5

INCREMENTAL REDUNDANCY A Rate 1/5 Mother Code

at the transmitter

5

INCREMENTAL REDUNDANCY A Rate 1/5 Mother Code

at the transmitter transmission # 1 at the receiver

5

INCREMENTAL REDUNDANCY A Rate 1/5 Mother Code

at the transmitter transmission # 1 at the receiver transmission # 2

5

INCREMENTAL REDUNDANCY A Rate 1/5 Mother Code

at the transmitter transmission # 1 at the receiver transmission # 2 transmission # 3

5

INCREMENTAL REDUNDANCY A Rate 1/5 Mother Code

at the transmitter transmission # 1 at the receiver transmission # 2 transmission # 3 transmission # 4

6

THROUGHPUT IN HYBRID ARQ

−8 −6 −4 −2 2 4 6 8 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Es/N0 (dB) Throughput HARQ Scheme based on Turbo codes in AWGN Channel Throughout of new puncturing scheme Throughout of standard BPSK Capacity Cutoff Rate

7

RANDOMLY PUNCTURED CODES

• The mother code is an (n, k) rate R turbo code.
• Each bit is punctured independently with probability λ.

7

RANDOMLY PUNCTURED CODES

• The mother code is an (n, k) rate R turbo code.
• Each bit is punctured independently with probability λ.
• The expected rate of the punctured code is R/(1 − λ).
• For large n we have

turbo code k bits puncturing device n bits ( 1−λ)n bits

8

A FAMILY OF RANDOMLY PUNCTURED CODES Rate Compatible Puncturing

• The mother code is an (n, k) rate R turbo code.
• λj for j = 1, 2, . . . , m are puncturing rates, λj > λk for j < k.
• If the i-th bit is punctured in the k-th code and j < k,

then it was punctured in the j-th code.

8

A FAMILY OF RANDOMLY PUNCTURED CODES Rate Compatible Puncturing

• The mother code is an (n, k) rate R turbo code.
• λj for j = 1, 2, . . . , m are puncturing rates, λj > λk for j < k.
• If the i-th bit is punctured in the k-th code and j < k,

then it was punctured in the j-th code.

• θi for i = 1, 2, . . . , n are uniformly distributed over [0, 1].
• If θi < λl, then the i-th bit is punctured in the l-th code.

9

MEMORYLESS CHANNEL MODEL

• Binary input alphabet {0, 1} and output alphabet Y.
• Constant in time with

transition probabilities W(b|0) and W(b|1), b ∈ Y.

9

MEMORYLESS CHANNEL MODEL

• Binary input alphabet {0, 1} and output alphabet Y.
• Constant in time with

transition probabilities W(b|0) and W(b|1), b ∈ Y.

• Time varying with

transition probabilities at time i Wi(b|0) and Wi(b|1), b ∈ Y.

• Wi(·|0) and Wi(·|1) are known at the receiver.

10

PERFORMANCE MEASURE Time Invariant Channel

• Sequence x ∈ C ⊆ {0, 1}n is transmitted, and x′ decoded.
• Sequences x and x′ are at Hamming distance d.

10

PERFORMANCE MEASURE Time Invariant Channel

• Sequence x ∈ C ⊆ {0, 1}n is transmitted, and x′ decoded.
• Sequences x and x′ are at Hamming distance d.
• The probability of error Pe(x, x′) can be bounded as

Pe(x, x′) ≤ γd = exp{−dα}, where γ is the Bhattacharyya noise parameter: γ =

• b∈Y
• W(b|x = 0)W(b|x = 1)

10

PERFORMANCE MEASURE Time Invariant Channel

• Sequence x ∈ C ⊆ {0, 1}n is transmitted, and x′ decoded.
• Sequences x and x′ are at Hamming distance d.
• The probability of error Pe(x, x′) can be bounded as

Pe(x, x′) ≤ γd = exp{−dα}, where γ is the Bhattacharyya noise parameter: γ =

• b∈Y
• W(b|x = 0)W(b|x = 1)

and α = − log γ is the Bhattacharyya distance.

11

PERFORMANCE MEASURE

• An (n, k) binary linear code C with Ad codewords of weight d.

11

PERFORMANCE MEASURE

• An (n, k) binary linear code C with Ad codewords of weight d.
• The union-Bhattacharyya bound on word error probability:

P C

W ≤ n

• d=1

Ade−αd.

11

PERFORMANCE MEASURE

• An (n, k) binary linear code C with Ad codewords of weight d.
• The union-Bhattacharyya bound on word error probability:

P C

W ≤ n

• d=1

Ade−αd.

• Weight distribution Ad for a turbo code?

11

PERFORMANCE MEASURE

• An (n, k) binary linear code C with Ad codewords of weight d.
• The union-Bhattacharyya bound on word error probability:

P C

W ≤ n

• d=1

Ade−αd.

• Weight distribution Ad for a turbo code?
• Consider a set of codes [C] corresponding to all interleavers.
• Use the average A

[C](n) d

instead of Ad for large n.

12

TURBO CODE ENSEMBLES A Coding Theorem by Jin and McEliece

• There is an ensemble distance parameter c[C]

s.t. for large n A

[C](n) d

≤ exp

• dc[C]
• for large enough d.

12

TURBO CODE ENSEMBLES A Coding Theorem by Jin and McEliece

• There is an ensemble distance parameter c[C]

s.t. for large n A

[C](n) d

≤ exp

• dc[C]
• for large enough d.
• For a channel whose Bhattacharyya distance α > c[C]

0 , we have

P

[C](n) W

= O(n−β).

• c[C]

is the ensemble noise threshold.

13

PUNCTUREDTURBO CODE ENSEMBLES ITW, April 2003

• c[CP ]

is the punctured ensemble noise threshold: A

[CP ](n) d

≤ exp

• dc[CP ]
• for large enough n and d.

13

PUNCTUREDTURBO CODE ENSEMBLES ITW, April 2003

• c[CP ]

is the punctured ensemble noise threshold: A

[CP ](n) d

≤ exp

• dc[CP ]
• for large enough n and d.
• If log λ < −c[C]

0 ,

13

PUNCTUREDTURBO CODE ENSEMBLES ITW, April 2003

• c[CP ]

is the punctured ensemble noise threshold: A

[CP ](n) d

≤ exp

• dc[CP ]
• for large enough n and d.
• If log λ < −c[C]

0 ,

c[CP ] ≤ log

• 1 − λ

exp (−c[C]

0 ) − λ

• .

14

PUNCTUREDTURBO CODE ENSEMBLES

2 4 6 8 10 12 10

−3

10

−2

10

−1

10

Es/N0 (dB) FER R=0.7 k=384 R=0.7 k=3840 R=0.8 k=384 R=0.8 k=3840 R=0.9 k=384 R=0.9 k=3840

15

PUNCTUREDTURBO CODE ENSEMBLES

−6 −4 −2 2 4 6 8 10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

← R=0.82 Es/N0 (dB) Throughput Punctured TC k=384 Punctured TC k=3840 BPSK Capacity Cutoff Rate

16

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.

16

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:

γi = γ(j) for all i ∈ I(j).

17

PERFORMANCE MEASURE Time Varying Channel

• Let W n(y|x) = n

i=1 Wi(yi|xi).

17

PERFORMANCE MEASURE Time Varying Channel

• Let W n(y|x) = n

i=1 Wi(yi|xi).

• Sequence x ∈ C ⊆ {0, 1}n is transmitted, and x′ decoded.
• The probability of error Pe(x, x′) can be bounded as

Pe(x, x′) ≤

• y∈Yn
• W n(y|x)W n(y|x′)

=

n

• i=1
• b∈Y
• Wi(b|xi)Wi(b|x′

i)

• ≤
• i:xi=x′

i

γi

18

HARQ PERFORMANCE

• dj is the Hamming distance between x and x′ over I(j).
• The probability of error Pe(x, x′) can be bounded as

Pe(x, x′) ≤

m

• j=1

γ(j)dj

18

HARQ PERFORMANCE

• dj is the Hamming distance between x and x′ over I(j).
• The probability of error Pe(x, x′) can be bounded as

Pe(x, x′) ≤

m

• j=1

γ(j)dj

• Ad1...dm is the number of codewords with weight dj over I(j).
• The union bound on the ML decoder word error probability:

P ≤

|I(1)|

• d1=1

· · ·

|I(m)|

• dm=1

Ad1...dm

m

• j=1

γ(j)dj

19

HARQ PERFORMANCE Random Transmission Assignment

• A bit is assigned to transmission j with probability αj.
• d is the weight of the original codeword.
• dj is the weight of the d-th transmission sub-word.
• The probability that the sub-word weights are d1, d2 . . . , dm is

d d1 d − d1 d2

• . . .

d − d1 · · · − dm−1 dm

• αd1

1 αd2 2 . . . αdm m

20

HARQ PERFORMANCE Random Transmission Assignment

• The union bound on the ML decoder word error probability:

P ≤

|I(1)|

• d1=1

· · ·

|I(m)|

• dm=1

Ad1...dm

m

• j=1

γ(j)dj

20

HARQ PERFORMANCE Random Transmission Assignment

• The union bound on the ML decoder word error probability:

P ≤

|I(1)|

• d1=1

· · ·

|I(m)|

• dm=1

Ad1...dm

m

• j=1

γ(j)dj

• The expected value of the union bound is
• d

Ad m

• j=1

γ(j)αj h .

• The average Bhattacharyya noise parameter:

γ =

m

• j=1

γ(j)αj

21

A RANDOMLY PUNCTURED TURBO CODE An Example of Random Transmission Assignment

• The puncturing probability is λ.
• Transmission over the channel with noise parameter γ.

21

A RANDOMLY PUNCTURED TURBO CODE An Example of Random Transmission Assignment

• The puncturing probability is λ.
• Transmission over the channel with noise parameter γ.
• Equivalent to having two transmissions:

– first with assignment probability (1 − λ) and noise parameter γ; – second with assignment probability λ and noise parameter 1.

21

A RANDOMLY PUNCTURED TURBO CODE An Example of Random Transmission Assignment

• The puncturing probability is λ.
• Transmission over the channel with noise parameter γ.
• Equivalent to having two transmissions:

– first with assignment probability (1 − λ) and noise parameter γ; – second with assignment probability λ and noise parameter 1.

• The average noise parameter is γ = (1 − λ)γ + λ.
• Requirement − log γ > c[C]

21

A RANDOMLY PUNCTURED TURBO CODE An Example of Random Transmission Assignment

• The puncturing probability is λ.
• Transmission over the channel with noise parameter γ.
• Equivalent to having two transmissions:

– first with assignment probability (1 − λ) and noise parameter γ; – second with assignment probability λ and noise parameter 1.

• The average noise parameter is γ = (1 − λ)γ + λ.
• Requirement − log γ > c[C]

0 translates into

− log γ > log

• 1 − λ

exp (−c[C]

0 ) − λ

• .

22

INCREMENTAL REDUNDANCY Concluding Remarks

at the transmitter

22

INCREMENTAL REDUNDANCY Concluding Remarks

at the transmitter transmission # 1 at the receiver

22

INCREMENTAL REDUNDANCY Concluding Remarks

at the transmitter transmission # 1 at the receiver transmission # 2

22

INCREMENTAL REDUNDANCY Concluding Remarks

at the transmitter transmission # 1 at the receiver transmission # 2 transmission # 3

22

INCREMENTAL REDUNDANCY Concluding Remarks

at the transmitter transmission # 1 at the receiver transmission # 2 transmission # 3 transmission # 4