1. III. Turbo Codes

2. Applications of Turbo Codes
Worldwide Applications & Standards
Data Storage Systems
DSL Modem / Optical Communications
3G (Third Generation) Mobile Communications
Digital Video Broadcast (DVB)
Satellite Communications
IEEE WiMAX …

3. Basic Concept of Turbo Codes
Invented in 1993 by Alian Glavieux, Claude Berrou and Punya Thitimajshima
Basic Structure:
a) Recursive Systematic Convolutional (RSC) Codes
b) Parallel concatenation and Interleaving
c) Iterative decoding

4. Recursive Systematic Convolutional Codes
Non Systematic Convolutional Code
Recursive Systematic Convolutional Code c1 c1 + + + + + c2 c2
g1=[1 1 1]
g2=[1 0 1]
g1=[1 1 1]
g2=[1 0 1]
G=[g1 g2]
G=[1 g2/g1]

5. Non-Recursive Encoder Trellis Diagram00 00 00 00 11 10 01
s0 (0 0) 11 11 11
s1 (1 0) 11 00 10 10
s2 (0 1) 01 01 01
s3 (1 1) 10

6. Systematic Recursive Encoder Trellis Diagram00 00 00 00 11 10 01
s0 (0 0) 11 11 11
s1 (1 0) 11 00 10 10
s2 (0 1) 01 01 01
s3 (1 1) 10

7. Non-Recursive & Recursive Encoders
Non Recursive and recursive encoders both have the same trellis diagram structure
Generally Recursive encoders provide better weight distribution for the code
The difference between them is in the mapping of information bits to codewords

8. Parallel Concatenation with Interleavingxk xk
RSC Encoder 1
yk(1) π
RSC Encoder 2
yk(2)
1/3 Turbo Encoder
Two component RSC Encoders in parallel separated by an Interleaver
The job of the interleaver is to de-correlate the encoding process of the two encoders
Usually the two component encoder are identical
xk : Systematic information bit
yk(1) : Parity bit from the first RSC encoder
yk(2) : Parity bit from the second RSC encoder

9. Interleaving
Permutation of data to de-correlate encoding process of both encoders
If encoder 1 generates a low weight codeword then hopefully the interleaved input information would generate a higher weight codeword
By avoiding low weight codewords the BER performance of the turbo codes could improve significantly

10. Turbo Decoder rk
RSC Decoder 1
xk*
π-1 π
RSC Decoder 2
Iteratively exchanging information between decoder 1 and decoder 2 before making a final deciding on the transmitted bits

11. Turbo Codes Performance
Principles of Turbo Codes: Keattisak Sripiman 

12. Turbo Decoding
Turbo decoders rely on probabilistic decoding of component RSC decoders
Iteratively exchanging soft-output information between decoder 1 and decoder 2 before making a final deciding on the transmitted bits
As the number of iteration grows, the decoding performance improves

13. Turbo Coded System Model
u=[u1, …, uk, …,uN] xs ys
Turbo
Decoding
RSC Encoder 1
x1p
y1p x y u*
MUX
Channel
DEMUX
N-bit
Interleaver π
RSC Encoder 2
x2p
y2p
u = [u1, …, uk, …,uN] : vector of N information bits xs = [x1s, …, xks, …,xNs] : vector of N systematic bits after turbo-coding of u x1p = [x11p, …, xk1p, …,xN1p] : vector of N parity bits from first encoder after turbo-coding of u x2p = [x12p, …, xk2p, …,xN2p] : vector of N parity bits from second encoder after turbo-coding of u x = [x1s, x11p, x12p, …, xNs, xN1p, xN2p] : vector of length 3N of turbo-coded bits for u y = [y1s, y11p, y12p, …, yNs, yN1p, yN2p] : vector of 3N received symbols corresponding to turbo-coded bits of u ys = [y1s, …, yks, …,yNs] : vector of N received symbols corresponding to systematic bits in xs y1p = [y11p, …, yk1p, …,yN1p] : vector of N received symbols corresponding to first encoder parity bits in x1p y2p = [y12p, …, yk2p, …,yN2p] : vector of N received symbols corresponding to second encoder parity bits in x2p u* = [u1*, …, uk*, …,uN*] : vector of N turbo decoder decision bits corresponding to u
Pr[uk|y] is known as the a posteriori probability
of kth information bit

14. Log-Likelihood Ratio (LLR)
The LLR of an information bit uk is given by:
Given that Pr[uk=+1]+Pr[uk=-1]=1

15. Maximum A Posteriori Algorithm
u=[u1, …, uk, …,uN] xs ys
Turbo
Decoding
RSC Encoder 1
x1p
y1p x y u*
MUX
Channel
DEMUX
N-bit
Interleaver π
RSC Encoder 2
x2p
y2p

16. Some Definitions & Conventions
u=[u1, …, uk, …,uN] xs ys
Turbo
Decoding
y1p
RSC Encoder 1
x1p x y u*
MUX
Channel
DEMUX
N-bit
Interleaver π
RSC Encoder 2
x2p
y2p
uk ykp
Sk-1 Sk s0 s0 s1 s1 s2 s2
uk=-1
uk=+1 s3 s3

17. Derivation of LLR uk ykp
Define S(i) as the set of pairs of states (s’,s) such that the transition from Sk-1=s’ to Sk=s is caused by the input uk=i, i=0,1, i.e.,
S(0) ={(s0 ,s0), (s1 ,s3), (s2 ,s1), (s3 ,s2)}
S(1) ={(s0 ,s1), (s1 ,s2), (s2 ,s0), (s3 ,s3)}
Sk-1 Sk s0 s0 s1 s1 s2 s2
uk=-1
uk=+1 s3 s3
Remember Bayes’ Rule

18. Derivation of LLR
Define

19. Derivation of LLR
depends only on Sk and is independent on Sk-1, yk and
Sk , yk depends only on Sk-1 and is independent on

20. Derivation of LLR

21. Derivation of LLR S0
Sk-1 Sk
Sk+1 SN s0 s1 s2 s3 y= y1
yk-1 yk
yk+1 yN

22. Computation of αk(s)
Example

23. Forward Recursive Equation
Computation of αk(s)
Forward Recursive Equation
Given the values of γk(s’,s) for all index k, the probability αk(s) can be forward recursively computed. The initial condition α0(s) depends on the initial state of the convolutional encoder
The encoder usually starts at state 0

24. Backward Recursive Equation
Computation of βk(s)
Backward Recursive Equation
Given the values of γk(s’,s) for all index k, the probability βk(s) can be backward recursively computed. The initial condition βN(s) depends on the final state of the trellis
First encoder usually finishes at state s0
Second encoder usually has open trellis

25. Computation of γk(s’,s)
Note that

26. Computation of γk(s’,s)