Union Bound Analysis of Bit Interleaved Coded Orthogonal Modulation with Differential Precoding

1. Union Bound Analysis of Bit Interleaved Coded Orthogonal Modulation with Differential Precoding Shi Cheng and Matthew C. Valenti Lane Dept. of CSEE West Virginia University

2. Outline • Review of BICM • Bit Interleaved Coded Modulation (BICM) (Caire 1998) • BICM with Iterative Decoding (BICM-ID) (Li and Ritcey 1997) • Results of turbo coded BICOM • Union bound for convolutional coded BICOM • Convolutional Coded BICOM with Differential Precoding(DP) • Conclusions

3. BICM System • Use the off-the-shelf binary codes • For Gray labeled constellations, BICM capacity is close to the Coded Modulation (CM) capacity • Examples: 8PSK, 16QAM with Gray labeling Binary Encoder Bitwise Interleaver M-ary Modulator Channel Soft Decoder Soft Demodulator Bitwise Deinterleaver

4. Not Gray Labeling? • For the labeling approach other than gray mapping, BICM capacity is worse than the CM capacity • Examples: 16QAM with set partition labeling, Orthogonal modulation • Use BICM with iterative decoding (BICM-ID) Soft Decoder Soft Demodulator Bitwise Deinterleaver Channel Bitwise Interleaver Extrinsic Information feedback

5. BICM and CM Capacity of Orthogonal Modulation 12 Reference: M.C.Valenti and S. Cheng, “Iterative demodulation and decoding of turbo coded M-ary noncoherent orthogonal modulation” IEEE Journal on Selected Areas in Communications, Sept. 2005. CM BICM AWGN Channel, Noncoherent Detection M: Modulation Alphabet Size 10 Minimum Eb/No (in dB) 8 M = 2 6 M = 4 4 M = 16 2 M = 64 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Code Rate R

6. Turbo Code Simulation Results 10 Reference: M.C.Valenti and S. Cheng, “Iterative demodulation and decoding of turbo coded M-ary noncoherent orthogonal modulation” IEEE Journal on Selected Areas in Communications, Sept. 2005. BICM AWGN Channel, Noncoherent Detection Simulation is collected for BER = 10-4, with CDMA 2000 turbo code of length 6138 BICM-ID 9 8 M = 2 7 6 Eb/No in dB 5 M = 4 4 3 M = 16 M = 64 2 1 M = 64 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Code rate R

7. Summary • BICM-ID improves the performance up to 1dB against BICM without ID. • The gap between the CM capacity and the turbo code result is still large, for M=16 and 64. • Is there any better code?

8. Preliminary Results of Convolutional Codes (rate = ½) 0 0 10 10 C = 3 C = 3 C = 4 C = 4 C = 5 C = 5 C = 6 C = 6 Turbo code Turbo code -1 -1 10 10 -2 -2 10 10 BER BER -3 -3 10 10 Length 6138 Length 6138 -4 -4 10 10 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 1 1.5 2 2.5 3 3.5 4 Eb/No (dB) Eb/No (dB) M = 16 M = 64

9. Union Bound • Simple, asymptotically tight • Following Benedetto’s SCCC bound, • Channel coding -> outer code • Modulator and its precoding -> inner code • Uniform interleaving assumption • Outer code • Inner Code • Pairwise Error Probability (PEP) M-ary Modulator Binary Encoder Bitwise Interleaver Precoding

10. Inner Code Trellis • Trellis of the modulator and its precoding • Allow parallel Transitions 1/1 1/1 10/e 11/e 2 3 00/e 01/e 0/0 0/0 0 1 00/e 11/e 0/1 0/1 3 2 0 1 / 1 1 e / / 1 1 1 0 / 3 e D e 2 / 0 1 0 0 1 / / 1 e 1 / 1 0 0 0/0 0/0 00/e 11/e 0 1

11. Tail Terminated Error Events • For a recursive code, the tail termination input bits could have a positive weight. (i) W l,h,j ... j 1 2 3 Input weight l Output weight h (i) T l',h,j ... j 1 2 3 Input weight l’ Input weight l>l’ Output weight h

12. 10 -4 10 -5 10 -6 10 -7 10 -8 10 -9 10 -10 10 5 6 7 8 9 10 11 12 13 14 15 Example • Outer Code: dodd=7, deven=10 • Overall Effective Ouput Weight: hmin=4, with tail termination. hmin=5, without tail termination. 0 10 Bound no termination Bound termination -1 10 simulation -2 10 -3 Rayleigh R Noncoherent CSI E F BICOM-DP AWGN Outer Code Noncoherent g (o) = [ 1+D 4 , D+D 3 +D 4 ] Inner Code g (i) = 1/(1+D) M = 4, K = 200 Eb/No (dB)

13. Bound Results 0 10 K=300 Simulation Rate ½ Outer Code K=300 (o) 2 2 g = [ 1+D ,1+D+D ] K=600 K=3000 M = 8, AWGN K=6000 Coherent Detection EF bound -5 10 R BICOM E B (i) g =1 -10 10 BICOM-DP (i) g =1/(1+D) -15 10 0 1 2 3 4 5 6 7 8 9 10 Eb/No (dB)

14. Pairwise Error Probability • Equivalent to the PEP of binary FSK, not a function of M (modulation alphabet size) • Channel: AWGN and Rayleigh fading • Detection: • Coherent • Noncoherent with known fading amplitude information. • Noncoherent with Rayleigh fading statistics only. • Use Gauss Chebyshev quadratures if necessary

15. Bound Results 0 10 Rayleigh Noncoherent noCSI BICOM-DP Rayleigh Noncoherent CSI Rate ½ Outer Code Rayleigh Coherent g = [ 1+D +D , (o) 2 3 AWGN Noncoherent AWGN Coherent 1+D+D +D ] 2 3 M = 16, K =500 -5 10 R E F Square Law -10 Upper Bound 10 on AWGN Noncoherent Detection -15 10 0 5 10 15 Eb/No (dB)

16. -3 10 -4 10 -5 10 -6 10 -7 10 -8 10 -9 10 -10 10 1 1.5 2 2.5 3 3.5 4 4.5 5 Convolutional Code Results 0 10 C = 3 C = 4 -1 10 Turbo code -2 10 BICOM-DP Simulations BICOM Simulations BER BICOM Bounds Length 6138 Rate ½ M = 16 AWGN Channel Noncoherent Detection BICOM-DP Bounds Eb/No (dB)

17. Convolutional Code Results 0 10 C = 3 C = 4 -1 10 Turbo code -2 10 BICOM Simulations -3 10 -4 10 BICOM Bounds BER -5 10 BICOM-DP Simulations -6 10 -7 10 Length 6138 Rate ½ M = 64 AWGN Channel Noncoherent Detection -8 10 BICOM-DP Bounds -9 10 -10 10 1 1.5 2 2.5 3 3.5 4 4.5 5 Eb/No (dB)

18. Conclusions • Union bound is a simple method to get the asymptotic performance of BICOM system. • Tail termination bits for recursive code need to be considered in the bound. • Interleaving gain is offered by the recursive structure of inner code. The simplest structure is 1/(1+D). • Using convolutional code with differential precoding for M = 16 or 64 orthogonal modulation, we can get better performance than the turbo coded BICOM system.