Theory and Technology of Error Control Coding

1. Chapter 5 Turbo Code Theory and Technology of Error Control Coding

2. Outline • Introduction • Structure of Turbo Code • Turbo decoding • Performance Bound Discussion • Application • Open Problem

3. Introduction • First proposed by C.Berrou in 1993 . • A new version of modified BCJR algorithm was introduced by Hagenauer in 1994 . • In 1996, S.Benedetto shed some light on the structure of Turbo Code . • 2001, Frey proposed factor graphs and sum-product algorithm for Turbo code. • Defined in the 3GPP LTE specifications.

4. Outline • Introduction • Structure of Turbo Code • Turbo decoding • Performance Bound Discussion • Application • Open Problem

5. Structure of Turbo Code • Turbo code exploits the parallel concatenation of at least two usually identical recursive systematic convolutional (RSC) encoders with interleaving. We can see from Fig. 1. Puncturing And Multiplexing u RSC code 1 Interleaver RSC code 2 Fig.1 Turbo code encoder

6. T T T Structure of Turbo Code • Recursive System Convolutional Code (RSC) The structure of RSC is illuminated in Fig.2. We can get some properties of RSC. T Fig.2 Recursive systematic code

7. 1. The role of design interleaver . To improve the free-distance of codeword with the length of interleaver increasing To diffuse the “1” in low-weight bits sequence and increase the weight of codeword. . 2. The commonly used interleaver . Uniform interleaver Pseudo-random interleaver. . Structure of Turbo Code • Interleaver The function of the interleaver is to rearrange the incoming block of N data bits in a pseudo-random fashion prior to encoding by the second encoder.

8. S-random interleaver Code-Matched interleaver Structure of Turbo Code

9. Structure of Turbo Code • Puncturer From Fig.1 we can see each constituent encoder produces a parity code, so the every bit inputed there are three bits outputed, while the overall rate of the encoder is 1/3. We can use puncturer to get high rate. for example: The output sequence of two RSC is : puncture matrix is : the output will be :

10. Outline • Introduction • Structure of Turbo Code • Turbo decoding • Performance Bound Discussion • Application • Open Problem

11. Turbo decoding • Why Turbo The Turbo engine uses a feedback to increase the performance of the overall system. The Turbo decoder works similar to the Turbo engine .

12. De-Interleaver DEC 2 DEC 1 Interleaver Interleaver De-Interleaver Turbo decoding • Structure of the Turbo decoder Fig.3 Structure of the Turbo decoder

13. if if Turbo decoding In Fig.3: andis called the extrinsic information which are part of the log-likelihood ratio (LLR) The final decision is made from ,

14. Turbo decoding • MAP(BCJR) • MAX-Log-MAP • Log-MAP • SOVA

15. Turbo decoding • MAP Algorithm This algorithm was proposed in 1974,modified and applied to iteration decoding

16. Turbo decoding • MAX-Log-MAP Getting rid of the drawback of MAP algorithm --- lots of exponential operation lead to the instability.

17. Turbo decoding

18. Turbo decoding • Log-MAP algorithm

19. Turbo decoding • Combining the MAX-Log-MAP and the correction function we can obtain the Log-MAP algorithm • needn’t be calculated for every x, but instead of a look-up table.

20. Turbo decoding • SOVA algorithm • Fundamental Viterbi algorithm principle: searching for the state sequence S(m) or the information U(m) that maximize the APP • SOVA algorithm principle: First-the path metric be the function of the a-prior information between the two decoders;Second providing soft output LLR

21. Turbo decoding • is the path metric; is time t,state k path metric; respectively for surviving path and competing path .

22. Turbo decoding • Features of the Turbo decoder Serial concatenation Probability decoding algorithm Extrinsic information Iterative decoding

23. Outline • Introduction • Structure of Turbo Code • Turbo decoding • Performance Bound Discussion • Application • Open Problem

24. Performance bound discussion • A performance bound The performance of a Turbo code with maximum-likelihood (ML) decoding can also be bounded using the union bound of a convolutional code with ML decoding on an additive white Gaussian noise

25. Performance bound discussion where the Hamming weight of the information sequence the total Hamming weight of the codeword function is defined as

26. Performance bound discussion Denote the number of codewords with Hamming weight d as , and the total information weight of codewords with Hamming weight d as . Then we define the average information weight per codeword as : Then the upper bound can be re-written as :

27. Performance bound discussion here is the free distance of the code. is called the effective multiplicity of codewords of weight d. We can have following conclusions : 1. Since Q function decrease drastically when its parameter increase, the larger the information weight d, the less significant the term with weight d affects the performance. Therefore, to improve the performance, one way is to maximize the free distance.

28. Performance bound discussion 2. the effective multiplicity also affect the performance. We can enlarge the length N while maintaining small. • Some discussions For Turbo codes, the free distance depends both on the structure of the constituent encoder and the pattern of interleaving. Some researches show that the free distance of Turbo codes is usually not very large. However, at low SNR, when the sequence length N is large, minimizing the effective multiplicity is even more important than maximizing the free distance.

29. Outline • Introduction • Structure of Turbo Code • Turbo decoding • Performance Bound Discussion • Application • Open Problem

30. Applications • Space communication • Combine with other coding technology • WCDMA • The iterative decoding idea • 3GPP LTE

31. Outline • Introduction • Structure of Turbo Code • Turbo decoding • Performance Bound Discussion • Application • Open Problem

32. Open Problem • RSC • Length of interleaver • Algorithm of Interleaver • Theoretical analysis • Decoding algorithm