Coding for Noncoherent M-ary Modulation Matthew Valenti Shi Cheng West Virginia University Morgantown, WV {mvalenti,shic}@csee.wvu.edu 11/10/2004

Motivation Objective: M-ary Noncoherent FSK Questions: The objective is to design methods for communicating over a noncoherent (random phase) channel at low Eb/No. M-ary Noncoherent FSK Coherent reception not always possible: Rapid relative motion between transmitter and receiver. Phase noise in local oscillators. A natural choice is noncoherent FSK. M-ary FSK allows bandwidth efficiency to be traded for energy efficiency. Questions: What is the information theoretic limit of M-ary NFSK? How can we approach that limit in practice? 11/10/2004

Capacity of M-ary NFSK in AWGN 15 Reference: W. E. Stark, “Capacity and cutoff rate of noncoherent FSK with nonselective Rician fading,” IEEE Trans. Commun., Nov. 1985. 10 Noncoherent combining penalty Minimum Eb/No (in dB) M=2 5 M=4 M=16 M=64 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rate R (symbol per channel use)

Bit Interleaved Coded Modulation Binary to M-ary mapping Binary Encoder Bitwise Interleaver M-ary- modulator Random Phase AWGN Soft-In Binary Decoder LLR Bit Metric Calculation Receiver front end Bitwise Deinterleaver Caire G. Caire, G. Taricco, E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inform. Theory, May 1998 1998

M-FSK: Noncoherent Channel LLR To determine the LLR of bit k, 1  k  log2M Let Sk(1) be the set of symbol indices for which the kth bit is a one, and Sk(0) the set of symbols indices for which the kth bit is a zero. Assume that the bits other than k are equally likely to be 0 or 1. Then: For BFSK this becomes: 11/10/2004

Turbo Coded 16-ary NFSK 10 10 10 BER 10 10 2 2.5 3 3.5 4 4.5 5 10 Capacity limit is 2.07 dB -1 # iterations = {1, 2, 3, 4, 5, 10, 16} 10 Performance using Rate 1/2 cdma2000 Turbo Code 6138 data bits 16 iterations log-MAP -2 10 BER -3 10 -4 10 1.75 dB from capacity at BER 10-5 2 2.5 3 3.5 4 4.5 5 Eb/No(in dB)

BICM-ID: Bit Interleaved Coded Modulation with Iterative Decoding Binary to M-ary mapping Binary Encoder Bitwise Interleaver M-ary- modulator Random Phase AWGN Soft-In Binary Decoder LLR Bit Metric Calculation Receiver front end Bitwise Deinterleaver Li and Ritcey indicate a 1 dB gain from hard decision feedback in Rayleigh fading for 8-PSK and r=2/3 convolutional coding Bitwise Interleaver Soft-Output Estimates of Coded Bits

Noncoherent M-FSK Using A Priori Probabilities Earlier we assumed that all modulated symbols were equally likely and obtained the bit LLR: However, we can use the bit probabilities derived from the decoder to improve the bit LLRs: 11/10/2004

Computing the A Priori Probabilities We want to find p(si|ck’) by using the extrinsic bit information from the decoder. Let pj be the decoder’s estimate that the probability of the jth bit is a one: Then if si  [b1i b2i … bmi] 11/10/2004

Simplified Expression The LLR can also be expressed as: Where: 11/10/2004

16-NFSK: BICM vs. BICM-ID 10 BICM BICM ID 10 10 BER 10 10 2 2.5 3 3.5 10 BICM BICM ID -1 # iterations = {1, 2, 3, 4, 5, 10, 16} 10 Performance using Rate 1/2 cdma2000 Turbo Code 6138 data bits 16 iterations log-MAP -2 10 BER -3 10 -4 10 1.1 dB from capacity at BER 10-5 2 2.5 3 3.5 4 4.5 5 Eb/No(in dB)

Convergence Analysis: BICM 2.5 Rate 1/2 cdma2000 Turbo Code Gaussian Approximation for Decoder Output Shown: Eb/No = 3.8 dB Threshold: Eb/No = 3.69 dB Capacity: Eb/No = 2.07 dB 2 1.5 SNR out 1 0.5 11/10/2004 0.5 1 1.5 2 2.5 SNR in

Convergence Analysis: BICM-ID 1.5 Shown: Eb/No = 3.2 dB Threshold: Eb/No = 3.03 dB Capacity: Eb/No = 2.07 dB 1 SNR out 0.5 11/10/2004 0.5 1 1.5 SNR in

16-NFSK: BICM vs. BICM-ID 10 BICM BICM ID 10 10 BER 10 10 2 2.5 3 3.5 10 BICM BICM ID -1 10 -2 10 BER -3 10 -4 10 2 2.5 3 3.5 4 4.5 5 Eb/No(in dB)

Conclusions Feeding back from decoder to demod can improve the performance of noncoherent M-FSK. For M=16 and r=1/2 coding, the improvement is 0.65 dB in AWGN. Other possible benefits Reduce number of iterations from 16 to 4 Reduce signal constellation size from 64 to 16 The additional complexity is negligible No extra iterations needed. Only need to update demod metrics during each iteration Need to perform channel interleaving/deinterleaving during each iteration. 11/10/2004

Ongoing and Future Work Try to close gap further Optimize interleaver design. Consider symbol-interleaving and nonbinary codes. More iterations. Fading With and without amplitude estimates (CSI). Ergodic vs. block fading. Other applications Cooperative diversity systems for sensor networks. Performance in FH systems with partial band jamming. 11/10/2004

Capacity of M-ary NFSK in Rayleigh Fading 15 Ergodic Capacity (Fully interleaved) Assumes perfect fading amplitude estimates available to receiver 10 M=2 Minimum Eb/No (in dB) M=4 5 M=16 M=64 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rate R (symbol per channel use)

BER of Noncoherent 16-FSK in AWGN with UMTS Turbo Code 10 BICM # iterations = {1, 2, 3, 4, 5, 10, 16} BICM-ID -1 10 -2 10 BER -3 10 -4 10 capacity = 2.3 dB 5114 bit data word 3 3.5 4 4.5 5 5.5 Eb/No (dB)