1. Multilevel Coding and Iterative Multistage Decoding ELEC 599 Project Presentation Mohammad Jaber Borran Rice University April 21, 2000

2. Multilevel Coding A number of parallel encoders The outputs at each instant select one symbol M-way Partitioning of data data bits from the information source E 1 (rate R 1 ) E M (rate R M ) E 2 (rate R 2 ) q 1 K 1 N x 1 Mapping (to 2 M -point constellation) Signal Point q 2 K 2 q M K M N x 2 N x M

3. Minimum Hamming distance for encoder i: d Hi, Minimum Hamming distance for symbol sequences For TCM (because of the parallel transitions) d H = 1 MLC is a better candidate for coded modulation on fast fading channels Distance Properties

4. Probability of error for Fading Channels Rayleigh fading with independent fading coefficients Chernoff bound L’: effective length of the error event (Hamming distance) d k (c i,c j ): distance between the kth symbols of the two sequences

5. For a fast fading channel, or a slowly fading channel with interleaving/deinterleaving Design criterion (Divsalar) Design Criterion for Fading Channels For a slowly fading channel without interleaving/deinterleaving Design criterion

6. For a fast fading channel, or a slowly fading channel with interleaving/deinterleaving Decoding Criterion  k is the fading coefficient for kth symbol) –Maximizes the likelihood function

7. Optimum decoder: Maximum-Likelihood decoder If the encoder memories are 1, 2, …, M, the total number of states is 2, where  = 1 + 2 + … + M. Complexity  Need to look for suboptimum decoders Decoding

8. If A and Y denote the transmitted and received symbol sequences respectively, using the chain rule for mutual information: Suggests a rule for a low-complexity staged decoding procedure

9. Multistage Decoding At stage i, decoder D i processes not only the sequence of received signal points, but also decisions of decoders D j, for j = 1, 2, …, i-1. Decoder D 1 Decoder D 2 Decoder D M Y

10. The decoding (in stage i) is usually done in two steps –Point in subset decoding –Subset decoding This method is not optimal in maximum likelihood sense, but it is asymptotically optimal for high SNR. Decoder D i Y...

11. Optimal Decoding –A i (x 1,…, x i ) is the subset determined by x 1,…, x i –f Y|A (y|a) is the transition probability (determined by the channel)

12. Rate Design Criterion then the rate of the code at level i, R i, should satisfy Decoder D 1 Decoder D 2 Decoder D M Y

13. Two-level, 8-ASK, AWGN channel

14. Rate Design Criterion Using the multiaccess channel analogy, if optimal decoding is used, R1R1 R2R2 I(Y;X1)I(Y;X1) I(Y;X2)I(Y;X2) I(Y;X 2 |X 1 ) I(Y;X 1 |X 2 )

15. Two-level, 8-ASK, AWGN channel

16. Iterative Multistage Decoding Assuming This expression, then, can be used as a priori probability of point a for the second decoder. –Two level Code –R 1  I(Y;X 1 |X 2 ) –Decoder D 1 : then the a posteriori probabilities are

17. Probability Mass Functions Error free decodingNon-zero symbol error probability

18. Two-level, 8-ASK, AWGN channel

19. Two-level, 8-ASK, Fast Rayleigh fading channel

20. 8-PSK, 2-level, 4-state, uncoded, AWGN channel

21. 8-PSK, 2-level, 4-state, uncoded, fast Rayleigh fading channel

22. 8-PSK, 2-level, 4-state, zero-sum, fast Rayleigh fading channel

23. 8-PSK, 2-level, 4-state, 2-state, fast Rayleigh fading channel

24. 8-PSK, 2-level, fast Rayleigh fading

25. Higher Constellation Expansion Ratios For AWGN, CER is usually 2 –Further expanding  Smaller MSED  Reduced coding gain For fading channels, –Further expanding  Smaller product distance  Reduced coding gain –Further expanding  Larger Hamming distance  Increased diversity gain

28. Conclusion Using iterative MSD with updated a priori probabilities in the first iteration, a broader subregion of the capacity region of MLC scheme can be achieved.  Lower complexity multilevel codes can be designed to achieve the same performance. Coded modulation schemes with constellation expansion ratio greater than two can achieve better performance for fading channels.

29. Coding Across Time If channels are encoded separately, assuming –A slowly fading channel in each frequency bin, and –Independent fades for different channels (interleaving/deinterleaving across frequency bins is used)

30. Coding Across Frequency Bins If coding is performed across frequency bins, assuming independent fades for different channels (interleaving/deinterleaving across frequency bins is used)

31. 8-PSK, 2-level, 4-state, 2-state