Analysis and Design of Mappings for Iterative Decoding of Bit-Interleaved Coded Modulation*

1. 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. System model: BICM and BICM-ID e.g. QPSK, 16QAM AWGN, OFDM, ISI, MIMO Code: Convolutional, Turbo, LDPC Le(C) La(C) Interleaver c data Mapper Encoder Interleaver Channel data estimate De-interleaver DemapperDetector/ Equalizer Decoder

3. 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. Euclidean distance spectrum 00 10 00 10 Gray 01 11 01 11 00 11 00 11 Anti-Gray 01 10 01 10 1st bit 2nd bit QPSK, no a priori information at the demapper.

5. Euclidean distance spectrum 00 10 00 10 Gray 01 11 01 11 00 11 00 11 Anti-Gray 01 10 01 10 1st bit 2nd bit QPSK, no a priori information at the demapper.

6. Euclidean distance spectrum 00 10 00 10 Gray 01 11 01 11 00 11 00 11 Anti-Gray 01 10 01 10 1st bit 2nd bit QPSK, no a priori information at the demapper.

7. Euclidean distance spectrum 00 10 00 10 Gray 01 11 01 11 00 11 00 11 Anti-Gray 01 10 01 10 1st bit 2nd bit QPSK, no a priori information at the demapper.

8. Euclidean distance spectrum 00 10 00 10 Gray 01 11 01 11 00 11 00 11 Anti-Gray 01 10 01 10 1st bit 2nd bit QPSK, no a priori information at the demapper. Note that without a priori information, the distances d2might not be relevant. An expurgated distance spectrum would be more precise.

9. Euclidean distance spectrum 00 10 00 10 Gray 01 11 01 11 00 11 00 11 Anti-Gray 01 10 01 10 2nd bit 1st bit QPSK, no a priori information at the demapper. QPSK, ideal a priori information at the demapper : signal constellation is reduced to a symbol pair.

10. Euclidean distance spectrum 00 10 00 10 Gray 01 11 01 11 00 11 00 11 Anti-Gray 01 10 01 10 2nd bit 1st bit QPSK, no a priori information at the demapper. QPSK, ideal a priori information at the demapper : signal constellation is reduced to a symbol pair.

11. Euclidean distance spectrum 00 10 00 10 Gray 01 11 01 11 00 11 00 11 Anti-Gray 01 10 01 10 2nd bit 1st bit QPSK, no a priori information at the demapper. QPSK, ideal a priori information at the demapper : signal constellation is reduced to a symbol pair.

12. EXIT chart QPSK Average mutual information between coded bits C at the transmitter and LLRs L at the receiver: QPSK, AWGN channel

13. EXIT chart QPSK Average mutual information between coded bits C at the transmitter and LLRs L at the receiver: QPSK, AWGN channel

14. Bit-wise EXIT chart QPSK Compare to multilevel codes! QPSK, AWGN channel

15. Analytic EXIT chart QPSK Analytic and numeric computation with BEC a priori information.

16. 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. 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. 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: 2·1013 possible mappings

19. 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: 2·1013 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. QAP Algorithms 0111 0110 0010 0011 0100 0001 0101 0000 1100 1001 1101 1000 1111 1110 1010 1011 • 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. • Other possibilities: • Tabu search • Simulated annealing approaches • Integer Programming • …

21. Cost function Possible distinct Euclidean distances Frequency of distance dkin Euclidean distance spectrum mapping • Cost function based on Euclidean distance spectrum • AWGN channel: • Fading channel: • Optimized mapping:

22. Euclidean distance spectrum 16QAM no a priori ideal 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 Gray M16a

23. Euclidean distance spectrum 16QAM no a priori ideal 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 Gray M16a

24. Euclidean distance spectrum 16QAM no a priori ideal 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 Gray M16a

25. EXIT chart, 16QAM • AWGN channel

26. Error rate, 16QAM • BER for AWGN channel, 4-state, rate ½ conv. code, interleaver length 10000 bits

27. 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. 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