turbo codes and principles and applications - imt … turbo codes and principles and applications...

1

Turbo Codes Turbo Codes and and

Principles and applicationsPrinciples and applications

[email protected]

Universitatea Tehnica Cluj-Napoca8th April 2005

Catherine DouillardENST Bretagne

CNRS TAMCIC UMR 2872

Turbo Codes: Contents

• Concatenated codes

• The turbo encoder

• Turbo decoding

• Performance

• Recent improvements in turbo coding

• Conclusion

2

Concatenated Codes

Decoder 1(inner)

Encoder 2(outer)

Concatenated Codes : Divide and Conquer

π Encoder 1(inner)

π−1Decoder 2(outer)

Channel

• First proposed by Forney in 1966 [1]• Divide complexity between inner and outer decoding• π and π-1 : de-interleaving breaks up error bursts at inner decoder

output• A standard in many communication systems : CCSDS, DVB-S …

interleaver

de-interleaver

Algebraic (RS)or

Convolutional Convolutional

3

Concatenated Codes : Performance Improvements

Performance of concatenated codes suffers from the fact that inner decoder only provides “hard” (0 or 1) decisions to outer decoder

Step 1 Soft decision decoding (~1.5 dB gain)• Soft Output Viterbi Algorithm (SOVA): Battail (1987) [2] and

Hagenauer (1989) [3]• Maximum a Posteriori (MAP) or BCJR algorithm (1974) [4]

Step2 “Turbo” decoding: Soft Iterative Decoding (<1 dB from channel capacity)

• Turbo Codes (Parallel Concatenation of Convolutional Codes): Berrou and Glavieux (1993)

• Turbo Product Codes: Pyndiah (1995)• Iterative decoding of Serially Concatenated Convolutional

Codes: Benedetto et al. (1996) …

^

Iterative (“Turbo”) Decoding of Concatenated Codes: principle (I)

Decoder 1(SISO)

Decoder 2(SISO)

xi

y1i

y2i

xi = (2Xi-1) + ni (Xi=di)y1i = (2Y1i-1) + n1iy2i = (2Y2i-1) + n2i

di^π-1

- Based on a Soft Input Soft Output (SISO) elementary decoder- Iterative decoding process (turbo effect)

πdi ^

4

Iterative (“Turbo”) Decoding of Concatenated Codes: principle (II)

- Based on a Soft Input Soft Output (SISO) elementary decoder- Iterative decoding process (turbo effect)

Decoder 1 Decoder 2(SISO) (SISO)

First iteration

xi

y1i

y2i

π-1 π+-

Z: extrinsic information

Zi1

+-

Iterative (“Turbo”) Decoding of Concatenated Codes: Principle (II)

Decoder 1 Decoder 2(SISO) (SISO)

Iteration p

xiy1i

y2i π

+-

π-1

di^

ZipZi

p-1

π++

5

Parallel Concatenated Codes

Global code rate: k/(k+r1+r2)

Data

π

Code 1

Code 2

k bits

k bits

r1 bits

r2 bits

• Parallel concatenation of two (or more) systematic encoders separated by interleavers.

• Component codes can be algebraic or convolutional codes• Historical Turbo Codes: code 1 and code 2 are two

identical Recursive Systematic Convolutional (RSC) codes

rate : k/r1

rate : k/r2

Serial Concatenated Codes

• Component codes can be algebraic or convolutional codes• SCCC: Serial Concatenation of Convolutional codes• Some hybrid schemes (serial -//) can also be investigated

rate : k/p

πCode 1 Code 2

rate : p/n

Global rate : k/n

k bits

p bitsk p-k p bits

n bitsp n-p

6

Parallel Concatenation vs Serial Concatenation: Performance Comparison under Iterative Decoding

5

5

5

5

5

5

5

5

10-1

10-3

10-4

10-6

10-7

10-8

20 1 3 4 5 6 7 8 9 10 11 12

BER

uncoded

Typical serial concatenation behaviour

Typical parallel concatenation behaviour

10-2

10-5

∫∞

=−=

xt

b

dttxerfc

withNEerfc

)exp(2)(

)(21

2

0

π

Convergence abscissa

"Waterfall" region

"Error floor" region

)log(10 minRdGa =

Convolutional Turbo Codes

7

Turbo Codes Encoder

The turbo encoder involves a parallel concatenation of at least two elementary Recursive Systematic Convolutional (RSC) codes separated by an interleaver (π)

Parallel concatenation of two RSC codes or «convolutional» turbo code

Redundancy

natural coding rate : 1/3

Data

diY1i

Y2i

Xi

π

Code 1

Code 2

Recursive Systematic Convolutional (RSC) Codes

8

Xi

D Ddi

Yi

Xi=di

NSC(Non Systematic Convolutional)

D Ddi

YiSC

(Systematic Convolutional)

(Elias code)

Systematic Convolutional (SC) codes

SC codes Trellis diagram

NSC SC00

01

10

11

0011 11

011010

0001

00

01

10

11

0011 01

011000

1001

D D

Xi

di

Yi

D D

Xi=di

di

Yi

di =0

di =1

Xi Yi

Xi Yi

df = 5 df = 4

9

1,E-08

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

0 2 4 6 8 10 12

UncodedSCNSC

Eb/N0

BERSC codes Bit Error Rate

Is there any convolutional code able to combine

• the advantageous df of non systematic codes• the good behaviour at very low SNRs

????

The answer is : yes

10

Recursive Systematic Convolutional (RSC) codes

D Ddi

Yi

Xi

[1,(1+D+D2)/(1+D2)]

D Ddi

Yi

Xi

[1,(1+D2)/(1+D+D2)]

di

Xi

D D

YiNSC

[(1+D2),(1+D+D2)]

RSC Codes Trellis diagram

Xi

D Ddi

YiNSC RSC

di =0 Xi Yi

di =1 Xi Yi

00

01

10

11

0011 11

011010

0001

D Ddi

Yi

Xi

00

01

10

11

0011

10

01

11

00

01

10

df = 5 df = 5

11

RSC Codes Bit Error Rate

1,E-08

1,E-07

1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

0 2 4 6 8 10 12

UncodedSCNSCRSC

Eb/N0

BER

The permutation

12

xi = (2Xi-1) + ni (Xi=di)y1i = (2Y1i-1) + n1iy2i = (2Y2i-1) + n2i

Turbo Codes Permutation

Data

diY1i

Y2i

Xi

π

Code 1

Code 2

xi

y1i

y2i

Decoder 2

Decoder 1

π

The 1st function of Π : to compensate for the vulnerability of a decoder to errors occuring in bursts


Data

diY1i

Y2i

Xi

π

Code 1

Code 2

The 2nd function of Π: The way of Π is defined governs the behaviour at low BER (< 10-5)

The fundamental aim of the permutation : if the direct sequence is RTZ,minimize the probability that the permuted sequence is also RTZ and vice-versa.

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The fundamental aim of the permutation : if the direct sequence is RTZ,minimize the probability that the permuted sequence is also RTZ and vice-versa.

Data

diY1

i

Y2i

Xi

π

Code 1

Code 2

Codeword distance )()()( 221

1111 kkk YYwYYwXXwd LLL ++=

)erfc(21

0min

min

NERd

kwBER b

a ≈


Permutation Regular

Condition : k = M.N

0010110010...0110001...................................0100100100...10010100011110100...10110110101000110...0101001

M columns

N rows

natural order :address k-1

natural order :address 0

Row-wisewriting

Column-wisereading

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Permutation Regular

Another representation

i 0 1 2 … k-2 k-1Data in natural order

j 0 1 2 … k-2 k-1Data in interleaved order

i = Π(j)

kPjji mod)( =Π=

10for −= kj L

P and k are relatively prime integers

0k-1Π(0) = 0

Π(1) = P

Π(2) = 2P

Π(2) = 3P

Permutation Regular

0100000010...0000000...................................0000000000...00000000000000000...00000000000000000...0000000

M columns

N rowsperiod : 7

period : 7

Regular permutation is appropriate for input weight 2

k → ∞ ⇒ d(w=2) → ∞

15

Permutation Regular

0110100000...0000000...................................0000000000...00000000000000000...00000000000000000...0000000

M columns

N rowsperiod : 7

period : 7

Regular permutation is appropriate for input weight 3

k → ∞ ⇒ d(w=3) → ∞

Permutation Regular

0100000010...00000000000000000...00000000000000000...00000000000000000...00000000000000000...00000000000000000...00000000000000000...00000000100000010...0000000.....................................

M columns

N rowsperiod : 7

period : 7

Regular permutation is not appropriate for input weight 4

k → ∞ ⇒ d(w=4) is limited

16

Permutation Regular

0110100000...00000000110100000...00000000000000000...00000000110100000...00000000000000000...00000000000000000...00000000000000000...00000000000000000...0000000.....................................

M columns

N rowsperiod : 7

period : 7

Regular permutation is not appropriate for other values of w

So, let us introduce disorder, but not in any manner !!

A good permutation must ensure :(1) maximum scattering of data(2) maximum disorder in the permuted sequence

(these two conditions are in conflict)

17

Permutation pseudo-random (= controlled disorder)

Everyone has his own tricksFor instance, the turbo code permutation

DVB-RCS/RCT (ETSI EN 301 790 and 301 958)

NQPjji mod)( +=Π=10for −= Nj L

P and N are relatively prime integers

Built from a regular permutation

Introduction of controlled disorder (here with max degree 4)

4

3

2

1

34mod24mod14mod04mod

PQjPQjPQjPQj

=⇒==⇒==⇒==⇒=

P1… P4 are small integers depending on the block size

N-1 0 Π(0) = Q

Π(2) = 2P+QΠ(3) = 3P+Q

Π(1) = P+Q

Permutation

In conclusion :

No ideal permutation for the time being (does it exist ?)

18

Turbo Decoding

Turbo Decoding

Soft-In Soft-Out (SISO) algorithms

19

Turbo decoding SISO algorithms

Soft-Input Soft-Output (SISO) decoding algorithms are necessary to turbo decoding.

Two families of decoding algorithms:

•The Viterbi based SISO algorithms [2][3]

•The MAP algorithm and its approximations [4]

• Also known as BCJR (Bahl, Cocke, Jelinek, Raviv), APP (A Posteriori Probability), or Backward-Forward algorithm

• Aim of the algorithm: computation of the Logarithm of Likelihood Ratio (LLR) relative to data

Turbo decoding SISO algorithms: MAP

The MAP (Maximum A Posteriori) algorithm

id

}0{Pr

}1{Prln)(

1

1k

i

ki

id

dd

R

R

=

==Λ

k1R is the noisy received sequence

),( with } ,,{ 11 iiikk yxRRR == LR

Sign ⇒ hard decisionMagnitude ⇒ reliability

20

• The principle of MAP algorithm: processing separately data available between steps 1 to i and between steps i +1 to k.

=> Introduction of forward state probabilitiesand backward state probabilities

kii L1),( =α mkii L1),( =β m


}{Pr)( 1 mSRm ==β + ikii

},{Pr)( 1i

ii RmSm ==α

• One can show that:

where represents the branch likelihoodbetween states m' and m when the received symbol is at step i.

),,( mm′γ ij RiR

=>∑ ∑ β′α′γ

∑ ∑ β′α′γ=Λ

′−

′−

m m

m mmmmm

mmmm

)()(),,(

)()(),,(ln)(

10

11

iii

iii

i R

Rd


21

Example:

L)()(),,()()(),,()()(),,()()(),,(ln)(

1010

11111313000012120101

iiiiii

iiiiiii RR

RRdβαγ+βαγβαγ+βαγ

=Λ−−

−−

0

1

2

3

0

1

2

3

0=id1=id

i-1 i

)()(),,()()(),,()()(),,()()(),,(

1010

11113232212133332020

iiiiii

iiiiiiRRRR

βαγ+βαγ+βαγ+βαγ+

−−

−−L

)()(),,()( 100 21212 iiii R βαγ=λ −*


* )()(),,()( 111 20202 iiii R βαγ=λ −

• Forward recursion: computation of )(miα

∑ ∑ ′γ′α=α′

−m

mmmmj

ijii R ),,()()( 1

can be recursively computed from)(miα

)(miα are all computed during the forward recursion in the trellis from initial values )(0 mα


• Backward recursion: computation of )(m′βi

∑ ∑ ′γβ=′β′

++m

mmmmj

ijii R ),,()()( 11

can be recursively computed from)(m′βi

)(m′βi are all computed during the backward recursion into the trellis from initial values )(mkβ

22

• Branch likelihoods: computation of ),,( mm′γ ij R

σ

−−

πσ×==′γ 2

2

2 2exp

21j}{Pr),,( ii

iijCR

dR mm

0),,( =′γ⇒ mmij R

If there is no transition between m' and m in the trellis or if the transition is not labeled with .jdi =

else, for a transmission over a Gaussian channel

Where Ci is the expected symbol along the branch from state m' to state m and Ri is the received symbol .


Solution 1: LOG-MAP

)1ln(),max(),(max*)ln( baba ebabaee −−∆++=≈+

Term is precomputed and stored in a lookup table)1ln( bae −−+

Multiplications => additions Exponentials in branch probabilities disappearAdditions => ?

The MAP algorithm in the logarithmic domain (LOG-MAP) [12]

Turbo decoding SISO algorithms: LOG-MAPand Max-Log-MAP

☺ Performance MAPKnowledge of σ is necessary

23

The MAP algorithm in the logarithmic domain (MAX-LOG-MAP) [12][13]

Turbo decoding SISO algorithms: LOG-MAPand Max-Log-MAP

Performance < MAP (a few tenths of dB)☺ Knowledge of σ is not necessary

Solution 2: MAX-LOG-MAP

),max()ln( baee ba ≈+

• In the Log domain, one can show that Λ(di) can be written as:

iii Zxd +=Λ )(

is the extrinsic information provided by the decoderiZ

Turbo decoding SISO algorithms: Log-MAPand Max-Log-MAP

xi

yiDecoder

24

Turbo decoding The turbo principleThe turbo decoder structure

xi

y1i

y2iZ1

Z2 di^Decoder 1

(SISO)

+-

+-π

Decoder 2(SISO)π

π-1

x1

x2

Through extrinsic information, each decoder uses both redundancies.Fundamental principle: a decoder must not reuse a piece of information provided by itself .

Turbo codes

Performance

25

1.0e-05

1.0e-04

1.0e-03

1.0e-02

1.0e-01

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6Eb/N0 (dB)

Gaussian channel, code rate=1/2

Turbo Codes Performance

Theoretical limit (binary input channel): 0,2 dB

uncoded

Iteration #1

#3

#2#6

#18

Gaussian channelQPSK modulationMAP decoding

BER

Interleaving256X256

S1 S3S2 S4

S1 S3S2 S4

diXi

Y1i

Y2i

Yi

RSC37,21

RSC37,21

ICC’93

Recent improvements in turbo codes

26

Circular Recursive Systematic Convolutional (CRSC) Codes

In many applications, block coding is required: How to transform a convolutional code into a

block code?

Improvements Trellis termination

Inserting tail bits: encoder state returns to 0☺ easy to implement

the transmission of ν additional bits is neededinitial and final states are singular states

Adopting circular encoding (= tail-biting)[9][10]☺ coding rate remains unchanged☺ the trellis can be regarded as a circle without any

singularitya precoding step is necessary

27

000001010011100101110111 0 1 1 1 1 10 0

Improvements Circular encoding

D DD

Yi

di

D DD

000001010011100101110111

Yi

di


28


It can be shown that, provided the data block length k is not a multiple of the LFSR period Penc,

• For a given information block, there is one and only one state Sc (circulation state) such as Sc= S0=Sk

• Sc can be easily computed from

Sc=f(k mod Penc, Sk0)

Sk0 is the final encoder state when encoding starts

from state 0

000001010011100101110111 0 1 1 1 1 10 0

- Encode the information block Sk = Sk0

- Compute Sc

• Pre-coding :

• Set the encoder to Sc

• Encode

- Set the encoder to 0 (= 000)

000

010


29

Decoding CRSC codes

Double-binary Turbo Codes

30

C1

C2

Π

data (bits)X

Y1

Y2

C1

C2

Π

data (couples) A

Y1

Y2

B

R = 1/3 (1/2 //1/2) R = 1/2 (2/3 //2/3)

binary double-binary

Improvements Double binary codes

Qualitative Behaviour of Concatenated Codes under Iterative Decoding

5

5

5

5

5

5

5

5

10-1

10-3

10-4

10-6

10-7

10-8

20 1 3 4 5 6 7 8 9 10 11 12

BER

uncoded

Bad convergence, high asymptotic gain

Good convergence, low asymptotic gain

10-2

10-5

QPSK, AWGN channel, target BER : 10-8

Ga = 10 log (R.dmin)

∫∞

=−=

xt

b

dttxerfc

withNEerfc

)exp(2)(

)(21

2

0

π

31

Contribution to the convergence property of TCs

kk / 2

k / 2k

binary

double-binary :decreases the correlation when decoding

C1

C2

Π

C1

C2

Π

erroneouspath

lockedpatterns

k

k


Thanks to this technique, minimum distances of turbo codes are increased

Contribution to the minimum distance

Intra-symbol permutation

2 0 10 00 00 00 00 00 02 0 1

1 30 00 00 00 00 00 01 3

3 0 0 0 0 0 0 30 00 00 02 0 0 0 0 0 0 2

3 0 0 0 0 0 0 30 00 00 00 00 00 03 0 0 0 0 0 0 3

1 30 00 00 00 00 00 00 00 00 00 00 00 00 01 3

These patterns don’t exist anymore if couples are inverted

periodicallyRemaining error patterns

(A,B) becomes (B,A)before vertical encodingonce out of two


32

• Latency divided by 2 (encoding and decoding)• Data rate doubled (intrinsic parallelism)• Less degradation when MAP → Max-Log-MAP

Other avantages of double-binary codes


E /N

BER

10

10

10

10

10

10

0.1 0.2 0.3 0.4 0.5 0.6

-1

-2

-3

-4

-5

-6

iteration#1

#5

#10

#20

b 0

0.35 dB

Sha

nnon

lim

it(b

inar

y-in

put c

hann

el)

0.7

#15

The residual loss in turbo decoding

Example of performance

• QPSK modulation

• R=1/2

• AWGN channel

• Duo-binary 8-state turbo code

• Large block size

• MAP algorithm

• No quantization


33

Conclusion

Convolutional Turbo Codes Current standards

4/5, 6/78-state, duo-binaryEutelsat (Skyplex)

1/2

1/2, 3/4

1/3 up to 6/7

1/4, 1/3, 1/2

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

Rates

8-state, binaryUMTS (data), CDMA2000

16-state, binaryINMARSAT (M4)

8-state, duo-binaryDVB-RCT

8-state, duo-binaryDVB-RCS

16-state, binaryCCSDS

Turbo codeApplication

34

Convolutional Turbo Codes Example of performance

• QPSK modulation

• AWGN channel

• Duo-binary turbo codes

• 1504-bit data blocks

• Max-log-MAP algorithm

• 4-bit quantization

• 8 iterations

1.0e-08

1.0e-07

1.0e-06

1.0e-05

1.0e-04

1.0e-03

1.0e-02

1.0e-01

1.0e+00

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Eb/N0 (dB)

R=1/2: 8-state16-state



Theoretical limits: R=1/2 R=2/32

R=3/4

FER

Eb / N0 (dB)

FER

5

5

5

5

5

5

5

5

10-8

10-3

10-4

10-5

10-6

10-7

10-1

10-2

2 3 4 5 6 7

QPSK 8-PSK

8 9

64-QAM16-QAM

8-state

16-state

Gaussien channel, 188-byte blocks, R = 2/3, Max-Log-MAP decoding, 8 itérations

0,9 dB

0,8 dB

0,85 dB 0,9 dB

0,8 dB

0,6 dB

1,7 dB

0,8 dB

Turbo codes and modulations

35

Turbo codes : hot topics

Reduction of decoding complexity (decrease the number of iterations …)

Search for robust permutations

Method for quick computation of dmin

Design of analogue turbo decoders

Optimisation of turbo coded systems for non Gaussian channels

…

Turbo Communications• Turbo demodulation• Turbo equalisation• Turbo detection • Turbo synchronisation• Source turbo decoding

probabilisticprocessor

localinformationextrinsic

1

4

3

2

turbo processor

sharedintrinsicinformation

intrinsicinformation

36

Turbo codes ReferencesReferences

[1] G. D. Forney, Jr., Concatenated codes, MIT Press, 1966.[2] G. Battail, “Pondération des symboles décodés par l’algorithme de Viterbi,” Ann.

Télécomm., vol. 42, N°1-2, pp. 31-38, Jan. 1987.[3] J. Hagenauer and P. Hoeher, “A Viterbi algorithm with soft-decision outputs and its

applications,” in Proc. IEEE Globecom’89, Dallas, Texas, Nov. 1989, pp. 4711-4717[4] L.R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for

minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 284-287, 1974.

[5] C. Berrou, A. Glavieux and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo-codes,” Proc. ICC’93, Geneva, Switzerland, May 1993, pp.1064-1070.

[6] C. Berrou and A. Glavieux, “Near optimum error correcting coding and decoding: Turbo codes,” IEEE Trans. Commun., vol. 44, n° 10, pp. 1261-1271, Oct.1996.

[7] R. Pyndiah, “Near optimum decoding of product codes: Block turbo codes,” IEEE Trans. Commun., vol. 46, n° 8, pp. 1003-1010, Aug. 1998.

[8] S. Benedetto, D. Divsalar, G. Montorsi and F. Pollara, “Serial concatenation of interleaved codes: Performance analysis, design and iterative decoding,”, IEEE Trans. Inform. Theory., vol.44, n°3, pp. 909-926, May 1998.

Turbo codes ReferencesReferences

[9] C. Weiss, C. Bettstetter and S. Riedel, “Code construction and decoding of parallel concatenated tail-biting codes,” IEEE Trans. Inform. Theory., vol. 47, n° 1, pp. 366-386, Jan. 2001.

[10] C. Berrou, C. Douillard and M. Jézéquel, “Multiple parallel concatenation of circular recursive convolutional (CRSC) codes,” Annals of Telecommun., vol. 54, n° 3-4, pp. 166-172, 1989.

[11] C. Berrou, P. Adde, E. Angui, and S. Faudeil, “A low complexity soft-output Viterbidecoder architecture”, Proc. ICC’93, Geneva, Switzerland, May 1993, pp.737-740.

[12] P. Robertson, E. Villebrun, and P. Hoeher, “ A comparison of optimal and sub-optimal MAP decoding algorithms operating in the log domain”, Proc. ICC’95, Seattle, WA, 1995, pp.1009-1013.

[13] A. J. Viterbi, “An intuitive justification and a simplified implementation of the MAP decoder for convolutional codes”, IEEE Journal on Selected Areas in Comm., Vol. 16, N°2, Feb. 1998.

[14] D. Divsalar, S. Dolinar and F. Pollara, “Iterative turbo decoder analysisbased on density evolution”, IEEE JSAC, vol. 16, n°5, pp.891-907, May 2001.

[15] S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes”, IEEE Trans. Commun., vol. 49, pp.1727-37, Oct. 2001.

37

General information about Turbo Codes• C. Heegard, S. B. Wicker, Turbo Coding, Kluwer Academic Publishers, 1999

• Branka Vucetic, Jinhong Yuan, Turbo Codes, Principles and Applications, KluwerAcademic Publishers, 2000

• C. Schlegel, Trellis coding, IEEE Press, 1997

• B.J. Frey, Graphical Models for Machine Learning and Digital Communication, MIT Press, 1998

• R. Johannesson , K. Sh. Zignagirov, Fundamentals of Convolutional Coding, IEEE Press, 1999

• Hanzo, Lajos, Adaptative wireless transceivers, John Wiley & sons, 2002

• Hanzo, Lajos, Turbo coding, turbo equalisation and space-time coding for transmission over fading channels, John Wiley & sons, 2002

• IEEE Communications Magazine, vol. 41, n°8, August 2003Capacity approaching codes, iterative decoding, and their applications

• WEB sites : http://www-turbo.enst-bretagne.fr/2emesymposium/presenta/turbosit.htmis a good starting point

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