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Optimization of LDPC Codes for Joint Detection and Decoding

This study explores the optimization of LDPC codes for generalized joint detection and decoding, focusing on turbo demapping and the EXIT function of the demapper. The code design, stability condition, code search, and simulation results are discussed.

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Optimization of LDPC Codes for Joint Detection and Decoding

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  1. Optimization of LDPC codes for generalized joint detection and decoding Göttfried Lächner, Ingmør Lønd, Jössy Säyir Newcom SPW1 Meeting December 15, 2005

  2. Outline • Introduction – Turbo Demapping • EXIT function of demapper • LDPC code design • Stability condition • Code search • Simulation results

  3. Example: “Turbo-Demapping” u x t y Binary Source Channel Encoder Bit- Interl. QAM Mapper Channel y xM xD û Soft Demapper Bit- Deinterl. Channel Decoder Hard Decision Sink Bit- Interl. For LDPC codes, bit-interleaving can be omitted.

  4. EXIT Function of Demapper • 16 QAM signal constellation • set partitioning mapping • AWGN channel =0.53 • a-priori messages modelled according to Gaussian distribution

  5. LDPC Code Design check nodes variable nodes I*VC I*CV

  6. LDPC EXIT Functions • Approximations:- Gaussian densities- Duality property • EXIT function of variable nodes: • EXIT function of check nodes:

  7. i is the node perspective of the variable node distribution LDPC EXIT Functions • The intersection point of the curves can be found by solvingfor the smallest I*CV in the intervall [0,1]. • The transfer function of the code is then given by

  8. Code Design • LDPC code design aims to maximize the rateunder the constraint that • Joint optimization of  and  is a hard problem. • For a fixed , optimization of  is still hard. • For a fixed , optimization of  is a linear optimization problem.

  9. Stability Condition • In order to converge to zero error probability, the stability condition has to be satisfied • For Gaussian message distributions, this condition can be written as constraint on 2 given  constraint on  given 2

  10. Code Search – Fixed  • In order to get practical distributions, we limit the search space to  = [0 23 0 0 0 0 0 0 10] R=0.4624

  11. Code Search – Optimized  • We do the same search but now  is optimized (using linear programming) for every  R=0.5065

  12. Optimized Distributions

  13. EXIT Chart • LDPC code optimized for demapper • threshold = 2.5dB • LDPC code optimized for AWGN channel • threshold = 5.5dB

  14. Simulation Results

  15. Conclusion • LDPC codes can be matched to the EXIT function of an inner component by jointly optimizing variable and check node distributions. • Optimization of the variable node distribution is hard to perform. • Optimization of the check node distribution is a linear optimization problem. • The overall code search can be performed efficiently by reducing the search space for the variable node distribution.

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