Joint Decoding on the OR Channel

1. UCLA Graduate School of Engineering - Electrical Engineering Program Communication Systems Laboratory Joint Decoding on the OR Channel Communication System Laboratory

2. Decoder DEC-N Decoder DEC-1 Joint Decoding Architecture • Decoding is done by performing belief propagation over the receiver graph • Performs well at very high Sum-Rates • High decoding complexity for a large number of users • Requires either bit synchronism or timing knowledge of all the transmitters Randomly picked (different with very high probability) Same Code Encoder 1 Interleaver 1 Encoder 2 Interleaver 2 Encoder N Interleaver N Elementary Multi-User Decoder (Threshold) De-Interl 1 Interleaver 1 De-Interl N Interleaver N

3. Joint Decoding Results 6 users

4. UCLA Graduate School of Engineering - Electrical Engineering Program Communication Systems Laboratory Turbo Codes for the OR Channel Communication System Laboratory

5. NL-TC NL-TC Interleaver Parallel Concatenated NL-TCs • Increases complexity and latency with respect to NLTC. • Capacity achieving. • Design criteria: • An extension of Benedetto’s uniform interleaver analysis for parallel concatenated non-linear codes has been derived. • This analysis provides a good tool to design the constituent trellis codes.

6. Parallel Concatenated NL-TCs • The uniform interleaver analysis proposed by Benedetto, evaluates the bit error probability of a parallel concatenated scheme averaged over all (equally likely) interleavers of a certain length. • Maximum-likelihood decoding is assumed. • However, this analysis doesn’t directly apply to our codes: • It is applied to linear codes, the all-zero codeword is assumed to be transmitted. The constituent NL-TCM codes are non-linear, hence all the possible codewords need to be considered. • In order to have a better control of the ones density, non-systematic trellis codes are used in our design. Benedetto’s analysis assumes systematic constituent codes. • An extension of the uniform interleaver analysis for non-linear constituent codes has been derived.

7. Results • Parallel concatenation • of 8-state, duo-binary • NLTCs. • Sum-rate = 0.6 • Block-length = 8192 • 12 iterations in message-passing algorithm 6 users

8. UCLA Graduate School of Engineering - Electrical Engineering Program Communication Systems Laboratory OR Channel when treating other users as noise: Can we provide the same sum-rate and performance for any number of users? Communication Systems Laboratory

9. Theoretical answer • Theoretically: YES.

10. Rate Sum-rate p BER 1/360 0.2778 0.006944 0.49837 1/400 0.25 0.006875 0.49489 Our Experience: NL-TCM • NL-TCM: looked like we don’t have a limit in the number of users. • Results for 100 user case: • And we were right in that case.

11. Comparison: Number of output bits n0 & number of ones M vs number of users

12. Number of ones is increasing increasing Comparison: n0(N) & M(N)

13. increasing Comparison: n0(N) & M(N) Same number of ones. Ungerboeck’s extension: moving deeper into the trellis.

14. increasing Comparison: n0(N) & M(N) All branches different Best code at this point….

15. increasing Comparison: n0(N) & M(N) is the best code at this point.

16. v=6 We can support any number of users in the OR-MAC with basically same decoding complexity for each user, and practically same performance.

17. Moreover: Unused bits (Bunch of zeros)

18. Moreover: • Denote N* the minimum number of users for which n0 > M. • For every N greater than N* we can use the same encoder and decoder • Design for N* . Encoder Add Zeros Interleaver Delete unused bits Decoder De-Interleaver

19. Limitation for Non-linear Turbo Codes • With 8-state constituent non-linear trellis codes: • 16-state constituent non-linear trellis codes should be used for more than 24 users.

20. Results • Parallel concatenation • of 8-state, duo-binary • NLTCs. • Sum-rate = 0.6 • Block-length = 8192 • 12 iterations in message-passing algorithm 6 users

21. With 16-state constituent NL-TCs For 50 users: For 100 users: Around 50 users should be supported.

22. UCLA Graduate School of Engineering - Electrical Engineering Program Communication Systems Laboratory Code design for the Binary Asymmetric Channel Communication System Laboratory

23. User 1 User 2 User N Receiver if all users transmit a 0 if one and only one user transmits a 1 if m users transmit a 1 and the rest a 0 Model for Optical MAC

24. threshold Model • The can be chosen any way, depending on the actual model to be used. • Examples: • Coherent interference: • constant

25. 1 1 0 0 Achievable sum-rates • n users with equal ones density p. • Joint Decoding • Treating other users as noise – Binary Asymmetric Channel:

26. Simulations

27. Simulations JD : Joint Decoding OUN: Other Users Noise

28. Simulations

29. Simulations

30. Simulations

31. 1 1 0 0 Achievable sum-rates • n users with equal ones density p. • Joint Decoding • Treating other users as noise – Binary Asymmetric Channel:

32. Lower bounds for Sum-rate (1) • Joint Decoding: • n users, with equal ones density p. • Using • Then: • For the worst case ( constant) the bound is actually very tight. • Note that for the case where • Also note that if (OR channel) , the lower bound becomes 1 for .

33. Lower bounds for Sum-rate (2) • Treating other users as noise: • n users, with equal ones density p. • Using • Then: • For the worst case ( constant) the bound is again very tight. • Note that if (OR channel) , the lower bound becomes log(2) for .

34. Lower bound for different • This figure shows the lower bounds and the actual sum-rates for 200 users for the worst case ( constant) . JD : Joint Decoding OUN: Other Users Noise

35. Lower bound for Sum-rate • For the Binary Asymmetric Channel, there is still a strictly positive achievable sum-rate for any number of users. • For the Coherent Interference Model, the lower bound for the achievable sum-rate is around 48% (vs. 70% for Z-Channel). • Our target sum-rate for Non-linear trellis codes is 20% (vs. 30% for Z-Channel). • For Parallel Concatenated NL-TCs, our goal will be to achieve a sum-rate of 40%. These codes are under design.

36. Metric for Z-Channel • We use a ‘greedy’ definition of distance (not the usual Hamming distance). • Directional distance between two codewords (denoted ) is the number of positions at which has a 0 and has a 1. • ‘Greedy’ definition of distance:

37. Design of NL-TCM for the BAC • The metric of the Viterbi decoder for the BAC is: • Where and are the number of 0-to-1 and 1-to-0 transitions from the codeword and the received word , respectively. • The decoded codeword is: • The directional distance between two codewords (denoted ) is the number of positions at which has a 0 and has a 1. • Both directional distances are relevant when computing the probability of error. • A good criteria is maximize the minimum of both directional distances: • This is exactly the same criteria used for NL-TCM codes for the Z-Channel

38. Design of NL-TCM for the BAC • Hence, although the metrics in the Viterbi decoder are different on the Z-Channel and the BAC, we use the same design technique for both cases. • However, since the achievable rate is lower for the BAC, our target rate will be lower. • We have designed codes for the Coherent Interference Model. Nevertheless, this design technique applies to any model for the 1-to-0 transition probabilies.

39. Design of NL-TCM for the BAC • Results (so far): • 6-user MAC • 128-state, rate 1/30 NLTC (Sum-rate = 0.2) • Coherent interference model. • In order to achieve the same BER than in the OR Channel case: • The number of states had to be increased from 64 to 128 (Increase in complexity). • The sum-rate was decreased from 0.3 to 0.2. • Simulations for larger number of users are running. • Parallel concatenated NL-TCs are being designed for this channel.