1. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Analysis and Design of Mappings for Iterative Decoding of Bit-Interleaved Coded Modulation* Frank Schreckenbach Institute for Communications Engineering Munich University of Technology, Germany Norbert Görtz School of Engineering and Electronics, University of Edinbrugh, UK * This work was supported by NEWCOM and DoCoMo Communications Laboratories Europe GmbH

2. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 System model: BICM and BICM-ID Encoder Interleaver Decoder De- interleaver data data estimate c Mapper Demapper Detector/ Equalizer L e (C) Interleaver L a (C) Channel Code: Convolutional, Turbo, LDPC e.g. QPSK, 16QAMAWGN, OFDM, ISI, MIMO

3. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Outline Consider mapping as coding entity: characterization with  Euclidean distance spectrum  EXIT charts Bit-Interleaved Coded Irregular Modulation (BICIM) Optimization of mapping: Quadratic Assignment Problem (QAP)  Binary Switching Algorithm Future work - Open problems

4. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Euclidean distance spectrum Distance Frequencyλ1λ1 λ2λ2 Gray Anti Gray QPSK, no a priori information at the demapper. 1101 1000 1101 1000 1001 1100 1001 1100 Gray Anti- Gray 1 st bit 2 nd bit

5. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Euclidean distance spectrum Distance Frequencyλ1λ1 λ2λ2 Gray4 Anti Gray QPSK, no a priori information at the demapper. 1101 1000 1101 1000 1001 1100 1001 1100 Gray Anti- Gray 1 st bit 2 nd bit

6. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Euclidean distance spectrum Distance Frequencyλ1λ1 λ2λ2 Gray44 Anti Gray QPSK, no a priori information at the demapper. 1101 1000 1101 1000 1001 1100 1001 1100 Gray Anti- Gray 1 st bit 2 nd bit

7. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Euclidean distance spectrum Distance Frequencyλ1λ1 λ2λ2 Gray44 Anti Gray6 QPSK, no a priori information at the demapper. 1101 1000 1101 1000 1001 1100 1001 1100 Gray Anti- Gray 1 st bit 2 nd bit

8. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Euclidean distance spectrum Distance Frequencyλ1λ1 λ2λ2 Gray44 Anti Gray62 QPSK, no a priori information at the demapper. 1101 1000 1101 1000 1001 1100 1001 1100 Gray Anti- Gray 1 st bit 2 nd bit Note that without a priori information, the distances d 2 might not be relevant. An expurgated distance spectrum would be more precise.

9. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Euclidean distance spectrum Distance Frequencyλ1λ1 λ2λ2 Gray44 Anti Gray62 QPSK, no a priori information at the demapper. Distance Frequencyλ1λ1 λ2λ2 Gray Anti Gray QPSK, ideal a priori information at the demapper : signal constellation is reduced to a symbol pair. 1101 1000 1101 1000 1001 1100 1001 1100 Gray Anti- Gray 1 st bit 2 nd bit

10. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Euclidean distance spectrum Distance Frequencyλ1λ1 λ2λ2 Gray44 Anti Gray62 QPSK, no a priori information at the demapper. Distance Frequencyλ1λ1 λ2λ2 Gray40 Anti Gray QPSK, ideal a priori information at the demapper : signal constellation is reduced to a symbol pair. 1101 1000 1101 1000 1001 1100 1001 1100 Gray Anti- Gray 1 st bit 2 nd bit

11. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Euclidean distance spectrum Distance Frequencyλ1λ1 λ2λ2 Gray44 Anti Gray62 QPSK, no a priori information at the demapper. Distance Frequencyλ1λ1 λ2λ2 Gray40 Anti Gray22 QPSK, ideal a priori information at the demapper : signal constellation is reduced to a symbol pair. 1101 1000 1101 1000 1001 1100 1001 1100 Gray Anti- Gray 1 st bit 2 nd bit

12. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 EXIT chart QPSK Average mutual information between coded bits C at the transmitter and LLRs L at the receiver: QPSK, AWGN channel

13. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 EXIT chart QPSK QPSK, AWGN channel Average mutual information between coded bits C at the transmitter and LLRs L at the receiver:

14. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Bit-wise EXIT chart QPSK Compare to multilevel codes! QPSK, AWGN channel

15. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Analytic EXIT chart QPSK Analytic and numeric computation with BEC a priori information.

16. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Bit Interleaved Coded Irregular Modulation (BICIM) Within one code block, use different  signal constellations: fine adaptation of data rate to channel characteristics with the modulation  mappings: optimization of iterative decoding procedure Basic idea similar to irregular channel codes Low complexity, good performance with low and medium code rates EXIT chart: linear combination of EXIT functions.

17. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Optimization of mapping Goal: find optimal assignment of binary indexes to signal points. Optimization for: No a priori information at the demapper (Gray mapping) Ideal a priori information at the demapper Trade off no/ideal a priori Optimization for bit positions

18. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Optimization of mapping Goal: find optimal assignment of binary indexes to signal points. Optimization for: No a priori information at the demapper (Gray mapping) Ideal a priori information at the demapper Trade off no/ideal a priori Optimization for bit positions Exhaustive search intractable for high order signal constellations: 2m! possible mappings. 16QAM: 2·10 13 possible mappings

19. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Optimization of mapping Goal: find optimal assignment of binary indexes to signal points. Optimization for: No a priori information at the demapper (Gray mapping) Ideal a priori information at the demapper Trade off no/ideal a priori Optimization for bit positions Exhaustive search intractable for high order signal constellations: 2m! possible mappings. 16QAM: 2·10 13 possible mappings Problem can be cast to a Quadratic Assignment Problem (QAP, Koopmans and Beckmann, 1957) QAP is NP-hard, i.e. not solvable in polynomial time. Famous applications are e.g. wirering in electronics or assignment of facilities to locations.

20. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 QAP Algorithms Binary Switching Algorithm (Zeger, 1990): try to switch the symbol with highest costs, i.e. the strongest contribution to a bad performance, with an other symbol such that the total cost is minimized. 0110011100110010 0100 0101 0001 0000 1100 1101 1001 1000 1110111110111010 Other possibilities: Tabu search Simulated annealing approaches Integer Programming …

21. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Cost function based on Euclidean distance spectrum AWGN channel: Fading channel: Optimized mapping: Cost function Possible distinct Euclidean distances Frequency of distance d k in Euclidean distance spectrum mapping

22. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Euclidean distance spectrum 16QAM Distance… Frequencyλ1λ1 λ2λ2 λ3λ3 …λ1λ1 λ2λ2 λ3λ3 λ4λ4 λ5λ5 … Gray243632…240000… SP563224…48808… MSP523824…02848… M16a564240…000164… I16524240…000168… GrayM16a no a prioriideal a priori SP: Set Partitioning MSP: Modified Set Partitioning M16a: optimized for ideal a priori information in AWGN channels I16: optimized for maximum sum of mutual info. without and with a priori

23. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Euclidean distance spectrum 16QAM Distance… Frequencyλ1λ1 λ2λ2 λ3λ3 …λ1λ1 λ2λ2 λ3λ3 λ4λ4 λ5λ5 … Gray243632…240000… SP563224…48808… MSP523824…02848… M16a564240…000164… I16524240…000168… GrayM16a no a prioriideal a priori SP: Set Partitioning MSP: Modified Set Partitioning M16a: optimized for ideal a priori information in AWGN channels I16: optimized for maximum sum of mutual info. without and with a priori

24. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Euclidean distance spectrum 16QAM Distance… Frequencyλ1λ1 λ2λ2 λ3λ3 …λ1λ1 λ2λ2 λ3λ3 λ4λ4 λ5λ5 … Gray243632…240000… SP563224…48808… MSP523824…02848… M16a564240…000164… I16524240…000168… GrayM16a no a prioriideal a priori SP: Set Partitioning MSP: Modified Set Partitioning M16a: optimized for ideal a priori information in AWGN channels I16: optimized for maximum sum of mutual info. without and with a priori

25. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 EXIT chart, 16QAM AWGN channel

26. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Error rate, 16QAM BER for AWGN channel, 4-state, rate ½ conv. code, interleaver length 10000 bits

27. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Conclusion Mapping has a big influence on the performance of iterative detection schemes. Consider mapping as coding entity: characterization with  Euclidean distance spectrum  EXIT chart Optimization of mapping: Quadratic Assignment Problem (QAP)  Binary Switching Algorithm Bit-Interleaved Coded Irregular Modulation (BICIM)

28. Frank Schreckenbach, Munich University of Technology NEWCOM 2005 Future work – Open problems Complexity: trade-off “cheep” outer code vs. number of required iterations Suboptimum demapping algorithms Combination of different (optimized) mappings with iterative MIMO detection, equalization, MU detection, … Further extensions: Investigations on signal constellations Multidimensional mappings: map a sequence of bits to a sequence of symbols