Progress report

1. UCLA Graduate School of Engineering - Electrical Engineering Program Communication Systems Laboratory Progress report Miguel Griot Andres I. Vila Casado Wen-Yen Weng Richard Wesel

2. Results up to last meeting • Non-linear trellis codes for OR-MAC (Completed) • Design criteria. • BER analytical bounding technique. • Results for any number of users. • Parallel concatenated NLTC for OR-MAC • Design criteria. • BER analytical bounding technique. • Results for 6 and 24 users. • Theoretically achievable Sum-rates for more general channels, in particular coherent interference model. • Preliminary results for 6-user optical MAC with coherent interference, using NLTC.

3. Non-linear trellis codes for OR-MAC • Design Criteria: • Extension to Ungerboeck’s rules. • We maximize the minimum free distance of the code, using the proper directional distance definition for the Z-Channel. • BER bounding technique for Z-Channel • Transfer function bound technique.

4. Bit Error Rate Bound for the Z-Channel • We use the transfer function bound technique on [Viterbi ‘71] for linear codes, and extended by [Biglieri ‘90] for non-linear codes, modifying the pairwise error probability measure. • Given two codewords • Replace and the transfer function bound technique can be readily applied to the NLTC to yield an upper bound to its BER over the Z-Channel.

5. Bit Error Rate Bound for the Z-Channel • Product states: where and denote the state at the encoder and receiver respectively. G denotes a ‘good product-state’ and B denotes a ‘bad product-state’. • Transition matrix: • For each transition in the product-state diagram the branch is labeled as:

6. Bit Error Bound for the Z-Channel • Transfer function: where: • Then:

7. Results : 6-user OR-MAC

8. Large number of users • Main results: • For any number of users, we achieve the same sum-rate with similar performance. • Tight BER analytical bound for Z-Channel provided.

9. Concatenation with Outer Block Code • A concatenation of an NLTC with a high rate block code provides a very low BER, at low cost in terms of rate. • Results: • A concatenation of the rate-1/20 NL-TCM code with (255 bytes,247 bytes) Reed-Solomon code has been tested for the 6-user OR-MAC scenario. • This RS-code corrects up to 8 erred bits. • Although we don’t have simulations for the 100-user case, it may be inferred that a similar BER would be achieved. Block-Code Encoder NL-TC Encoder Z-Channel Block-Code Decoder NL-TC Decoder

10. Parallel Concatenated NL-TCs • 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. NL-TC NL-TC Interleaver

11. 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.

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

13. 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 General Model for Optical MAC

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

15. 1 1 0 0 Achievable sum-rates • n users with equal ones density p. • Joint Decoding • Treating other users as noise – Binary Asymmetric Channel:

16. Sum-rate for coherent interence We provide an analytical lower bound to the achievable sum-rate for ANY number of users, for both joint decoding and treating other users as noise.

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

18. Progress since last meeting • New FPGA demo for 6-user optical multiple access. • Design of NL-TC for optical MAC with coherent interference, for large number of users. • BER bounding technique for BAC. • (Ongoing work) Design of parallel concatenated NLTC for optical MAC with coherent interference.

19. Progress: publications & presentations • Trellis Codes with Low Ones Density for the OR Multiple Access Channel, M.Griot, A.Vila Casado, W-Y Weng, H. Chan, J.Basak, E.Yablanovitch, I.Verbauwhede, B. Jalali, R.Wesel. IEEE ISIT, Seattle, 9-14 July 2006. • Presented in IEEE ISIT 2006 by Miguel Griot. • Non-linear Turbo Codes for Interleaver-Division Multiple Access on the OR Channel, M.Griot, A.Vila Casado, R.Wesel. To be presented at IEEE GLOBECOM Technical Conference 2006, Nov. 27 – Dec. 1, San Francisco. • Presentation: Demonstration of Uncoordinated Multiple Access in Optical Communications, H.Chan, A.Vila Casado, J.Basak, M.Griot, W-Y Weng, E.Yablanovitch, I.Verbauwhede, R.Wesel. 2006 43rd Design Automation Conference, July 24-28, San Francisco. • Winner of the 2006 DAC/ISSCC Student Design Contest 1st Prize award, on the Operational System Design category. • Presented by Herwin Chan. • Journal Papers under preparation: • Non-linear Trellis Codes for Interleaver-Division Multiple Access on optical channels. (IEEE Trans. Telecommunications) • Includes material presented on ISIT 2006, and NL-TC codes for BAC. • Non-linear Turbo Codes for Interleaver-Division Multiple Access on optical channels. • Includes material to be presented on GlobeCom 2006, and PC-NLTC codes for BAC. (IEEE Trans. Telecommunications) • Demonstration of Uncoordinated Multiple Access in Optical Communications. • Includes material presented in DAC Conference 2006.

20. BER analytical bound

21. threshold Results for 6-user MAC • 6-user MAC • 64-state, rate 1/30 NLTC (Sum-rate = 0.2) • Coherent interference model (CI-MAC): • Z-Channel:

22. BER bound for 6-user CI-MAC • 64-state NL-TC

23. Results for Optical MAC • Model: Coherent interference • 128-state NL-TC • Sum-rate = 0.2

24. Simulator Features • Random data is generated and encoded • The signal passes through a parameterizable channel model • Probes are placed at different point of the receiver to see how the codes react to changes in the channel

25. Channel Model • a and b simulate the degradation of the transmitted signal due to interference from other transmitters • a – non-coherent combination Probability that a 0 bit turns into 1 • b – coherent combination Probability that a 1 bit turns into 0

26. FPGA Channel Simulator • Hardware transmitter, receiver and channel model simulated on a single FPGA • Effects of changing channel parameters can be evaluated in real time • New Channel codes can be easily tested FPGA BER < 10-5 transmitter Channel Model Reed Solomon Decoder (255,237) BER < 10-9 Trellis Decoder Rate:1/20 a b

27. Measurement Points FPGA BER < 10-5 transmitter Channel Model Reed Solomon Decoder (255,237) BER < 10-9 Trellis Decoder Rate:1/20 a b • Ones density • Channel Errors • One to zero transitions Total bit error rate Non-linear trellis code bit error rate

28. Simulation Interface • Real-time Matlab graphical user interface • Real-time control of channel parameters a and b Real time channel conditions Bit error rate measurement at receiver Channel parameter selection