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Multilevel Coding and Iterative Multistage Decoding ELEC 599 Project Presentation

Multilevel Coding and Iterative Multistage Decoding ELEC 599 Project Presentation. Mohammad Jaber Borran Rice University April 21, 2000. q 1 K 1. N x 1. M -way Partitioning of data. E 1 (rate R 1 ). Mapping (to 2 M -point constellation). data bits from the

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Multilevel Coding and Iterative Multistage Decoding ELEC 599 Project Presentation

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

  2. q1 K1 N x1 M-way Partitioning of data E1 (rate R1) Mapping (to 2M-point constellation) data bits from the information source q2 K2 N x2 Signal Point E2 (rate R2) qM KM N xM EM (rate RM) Multilevel Coding A number of parallel encoders The outputs at each instant select one symbol

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

  4. Probability of error for Fading Channels • Rayleigh fading with independent fading coefficients Chernoff bound L’: effective length of the error event (Hamming distance) dk(ci,cj): distance between the kth symbols of the two sequences

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

  6. Decoding Criterion • For a fast fading channel, or a slowly fading channel with interleaving/deinterleaving (akis the fading coefficient for kth symbol) • Maximizes the likelihood function

  7. Decoding • Optimum decoder: Maximum-Likelihood decoder • If the encoder memories are n1, n2, …,nM, the total number of states is 2n, where n = n1 + n2 + … + nM. • Complexity  Need to look for suboptimum decoders

  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. Decoder D1 Decoder D2 Y Decoder DM Multistage Decoding • At stage i, decoder Di processes not only the sequence of received signal points, but also decisions of decoders Dj, for j = 1, 2, …, i-1.

  10. ... Decoder Di Y • 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.

  11. Optimal Decoding • Ai(x1,…, xi) is the subset determined by x1,…, xi • fY|A(y|a) is the transition probability (determined by the channel)

  12. Decoder D1 Decoder D2 Y Decoder DM Rate Design Criterion then the rate of the code at level i, Ri, should satisfy

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

  14. R2 I(Y;X2|X1) I(Y;X2) R1 I(Y;X1|X2) I(Y;X1) Rate Design Criterion Using the multiaccess channel analogy, if optimal decoding is used,

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

  16. Two level Code • R1I(Y;X1|X2) • Decoder D1: Iterative Multistage Decoding Assuming then the a posteriori probabilities are This expression, then, can be used as a priori probability of point a for the second decoder.

  17. Probability Mass Functions Error free decoding Non-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

  26. 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.

  27. 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)

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

  29. 8-PSK, 2-level, 4-state, 2-state

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