Turbo codes for short and medium block length: the state of the art Department 1 Paris, June 25,...
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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, …