1. Forschungszentrum Telekommunikation Wien [Telecommunications Research Center Vienna] Göttfried Lächner, Ingmør Lønd, Jössy Säyir Optimization of LDPC codes for generalized joint detection and decoding Newcom SPW1 Meeting December 15, 2005

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

3. -3- Example: “Turbo-Demapping” Binary Source Channel Encoder Bit- Interl. QAM Mapper uxt Channel y Soft Demapper Bit- Deinterl. Channel Decoder Hard Decision Sink yxDxD ûxMxM Bit- Interl. For LDPC codes, bit-interleaving can be omitted.

4. -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. -5- LDPC Code Design I * CV I * VC check nodes variable nodes

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

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

8. -8- Code Design  LDPC code design aims to maximize the rate under 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. -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. -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. -11- Code Search – Optimized   We do the same search but now  is optimized (using linear programming) for every R=0.5065

12. -12- Optimized Distributions

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

14. -14- Simulation Results

15. -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.