1. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. C H A P T E R 15 ERROR CORRECTING CODES

2. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.1 Encoder for linear block codes.

3. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.2 Encoder for systematic cyclic code.

4. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.3 Performance comparison of coded (doshed) and uncoded (solid) systems.

5. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.4 Burst error detection.

6. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.5 Convolutional encoder.

7. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.6 Code tree for the encoder in Fig. 15.5.

8. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.7 (a) State and (b) state transition diagram of the encoder in Fig. 15.5.

9. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.8 Trellis diagram for the encoder in Fig. 15.5.

10. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.9 A recursive systematic convolutional (RSC) encoder.

11. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.10 Viterbi decoding example in Fig. 15.5: (a) stage 1 and 2; (b) stage 3; (c) stage 4; (d) stage 5; (e) stage 6.

12. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.10 Continued

13. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.10 Continued

14. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.11 Convolutional encoder.

15. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.12 Code tree for the encoder in Fig. 15.11.

16. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.13 Setting the threshold in sequential decoding.

17. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.14 Trellis diagram of a Hamming (7, 4, 3) code with parity check matrix of Eq. (15.34).

18. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.15 A block (nonrandom) interleaver for correcting random and burst errors.

19. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.16 Product code formed by two encoders separated by a block interleaver.

20. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.17 Concatenated code with a nonbinary outer code and a binary inner code.

21. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.18 Block diagram of Chase soft-decoding algorithms.

22. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.19 Block diagram of Chase soft decoding algorithms.

23. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.20 Parallel concatenated turbo code: (a) rate 1/3 turbo encoder; (b) Implementation of recursive systematic convolutional (RSC) encoder g 1 (D)=1+D+D 4, g 2 (D)=1+ D 2 +D 3 +D 4.

24. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.21 Exchange of extrinsic information between two component CJR decoders for iterative turbo decoding.

25. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.22 The decoding performance of a rate 1/2 Turbo code is shown to be very close to the theoretical limit. (Reproduced with copyright permission from IEEE from Ref. 14.)

26. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.23 Tanner graph of the (7, 4, 3) Hamming code.

27. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.24 Message passing in the sum-product algorithm.

28. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure 15.25 Comparison of bit error rates of uncoded polar signaling transmission and polar signaling transmission of Hamming (7, 4) encoded [(dashed) and uncoded (solid) message bits.

29. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure P.15.5-2

30. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure P.15.5-3

31. Modern Digital and Analog Communication Systems Lathi Copyright © 2009 by Oxford University Press, Inc. Figure P.15.5-5