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

Outline Review of BICM Results of turbo coded BICOM 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 7/11/2006

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 Bitwise Deinterleaver Soft Demodulator

Extrinsic Information 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 Bitwise Deinterleaver Soft Demodulator Channel Bitwise Interleaver Extrinsic Information feedback

BICM and CM Capacity of Orthogonal Modulation 12 CM 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. AWGN Channel, Noncoherent Detection M: Modulation Alphabet Size BICM 10 Minimum Eb/No (in dB) 8 M = 2 6 M = 4 4 M = 16 2 M = 64 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Code Rate R

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. AWGN Channel, Noncoherent Detection Simulation is collected for BER = 10-4, with CDMA 2000 turbo code of length 6138 BICM 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 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Code rate R

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? 7/11/2006

Preliminary Results of Convolutional Codes (rate = ½) 10 10 C = 3 C = 3 C = 4 C = 4 C = 5 C = 5 C = 6 C = 6 10 -1 Turbo code Turbo code 10 -1 10 -2 10 -2 BER BER 10 -3 10 -3 Length 6138 Length 6138 10 -4 10 -4 1 1.5 2 2.5 3 3.5 4 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Eb/No (dB) Eb/No (dB) M = 16 M = 64 7/11/2006

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) Binary Encoder Bitwise Interleaver M-ary Modulator Precoding 7/11/2006

Symbol-wise Trellis (M=4) Inner Code Trellis Trellis of the modulator and its precoding Allow parallel Transitions Modulation Type Precoding Bit-wise Trellis Symbol-wise Trellis (M=4) BICOM BICOM-DP 1/1 1/1 10/e 11/e 2 3 0/0 0/0 00/e 01/e 1 0/1 0/1 00/e 11/e 3 2 1 / 1 1 / / e 1 1 1 D 3 / e / e 2 1 / 1 / 1 1 1 / e 0/0 0/0 00/e 11/e 1 7/11/2006

Tail Terminated Error Events For a recursive code, the tail termination input bits could have a positive weight. W (i) l,h,j ... 1 2 3 j Input weight l Output weight h T (i) l',h,j ... 1 2 3 j Input weight l’ Input weight l>l’ Output weight h 7/11/2006

Example Outer Code: dodd=7, deven=10 Overall Effective Ouput Weight: Outer Code: dodd=7, deven=10 Overall Effective Ouput Weight: hmin=4, with tail termination. hmin=5, without tail termination. 5 6 7 8 9 10 11 12 13 14 15 -10 -9 -8 -7 -6 -5 -4 Bound no termination Bound termination 10 -1 simulation 10 -2 -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) 7/11/2006

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

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 7/11/2006

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

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

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

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. 7/11/2006