Turbo codes for short and medium block length: the state of the art

Department 1

Paris, June 25, 2004

Claude Berrou, Catherine Douillard

GET-ENST Bretagne/PRACOM/CNRS 2658, Brest, France

Application turbo code

termination polynomials rates

CCSDS binary,16-state

tail bits 23, 33, 25, 37

1/6, 1/4, 1/3, 1/2

3G binary,8-state

tail bits 13, 15, 17 1/4, 1/3, 1/2

DVB-RCS duo-binary,8-state

circular 15, 13 1/3 up to 6/7

DVB-RCT duo-binary,8-state

circular 15, 13 1/2, 3/4

Inmarsat (M4) binary,16-state

no 23, 35 1/2

Eutelsat (Skyplex)

duo-binary,8-state

circular 15, 13 4/5, 6/7

Current known applications of convolutional TCs

Others: Echostar (Broadcom), 802.16, …

k binarydata

permutation

Y1

Y2

X

B

A

Y1

Y2

permutation

k/2 binarycouples

polynomials 15, 13 (or 13, 15)

k binarydata

permutation

Y1

Y2

X

B

A

Y1

Y2

permutation

k/2 binarycouples

polynomials 23, 35 (or 31, 27)

(a) (b)

(c) (d)

The TCs used in practice

Main progress in turbo coding/decoding since 1993

• Max-Log-MAP and Max*-Log-MAP algorithms

• Sliding window

• Duo-binary turbo codes

• Circular (tail-biting) encoding

• Permutations

• Computation or estimation of Minimum Hamming distances (MHDs)

• Stopping criterion

• Bit-interleaved turbo coded modulation

• …

Gaussian, Max-Log-MAP algorithm, 8 iterations, 4 or 5 bit-quantization

Typical performance

Eb/N0 (dB)

Frame Error Rate

5

5

5

5

5

5

5

5

10-8

10-3

10-4

10-5

10-6

10-7

10-1

10-2

1 2 3 4 5 6

8-PSK, 16-state duo-binary,R = 2/3, 188 bytes,pragmatic coded modulation

QPSK, 16-state duo-binary,R = 2/3, 188 bytes

QPSK, 8-state duo-binary,R = 2/3, 188 bytes

QPSK, 8-state binary,R = 1/3, 640 bits

theoretical limit

theoretical limit

Gaussian, 424 bits (53 bytes)

(reference: ESA MHOMS project)

TC is 16-state

Classical TC and LDPC

Eb/N0 (dB)0 1 2 3 4 5 6 7 8

100

10-1

10-2

10-3

10-4

10-5

10-6

FER

SCCCLDPC

PCCC

Eb/N0 (dB)0 1 2 3 4 5 6 7 8

100

10-1

10-2

10-3

10-4

10-5

10-6

FER

SCCCLDPC

PCCC

Eb/N0 (dB)0 1 2 3 4 5 6 7 80 1 2 3 4 5 6 7 8

100

10-1

10-2

10-3

10-4

10-5

10-6

100

10-1

10-2

10-3

10-4

10-5

10-6

FER

SCCCLDPC

PCCC

LDPCLDPC

TC

R=1/3 R=5/6 R=9/10

About the Rayleigh channel

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

10-6

10-5

10-4

10-3

10-2

10-1

10016QAM AWGN 51200 bits16QAM Rayleigh 51200 bitsCapacity RayleighCapacity AWGN

16 QAM rate 1/2

BER

Eb/N0 (dB)

Duo-binary 8-state TC

The ways we are investigating to reach FERs 10-9

• adopt 16-state duo-binary TCs and try to increase the MHD by 25%, by improving permutations

• keep 8-state duo-binary TCs and try to increase the MHD by 100%, by introducing a clever 3rd dimension

Example

Adding a rate-1 third dimension

X

Y2

permutation

data(bits)

C1

Y1

C2

patch

(k bits)

P/Spost-encoder

W'

(P bits)

puncturing

Pre-existing turbo encoder Patch:

- a parallel to serial (P/S) multiplexer

- permutation '

- rate-1 post-encoder

1-

1-

Combining parallel and (double) serial concatenation

BA

data(couples)

C1

C2

Y2

permutation

(N couples)

Y1

P/Spost-encoder

W

patch

'(P bits)

puncturing

Adding the rate-1 third dimension (to duo-binary TC)

Pre-existing turbo encoder Patch:

- a parallel to serial (P/S) multiplexer

- permutation '

- rate-1 post-encoder

1-

1-

Simulation results (8-state DVB-RCS TC + patch)

Max-Log-MAP component algorithm, 5 and 8 iterations

Eb/N0 (dB)

Frame Error Rate

5

5

5

5

5

5

5

5

10-8

10-3

10-4

10-5

10-6

10-7

10-1

10-2

1 2 3

188 bytesR=1/2

DVB-RCS

TL

3D TC

#5

#8

#8

Conclusions

• not much left to gain in convergence

• MHDs to be increased reasonably

• also, practical limits to be determined (Rayleigh)

• about complexity, latency, …