Progress Report for the UCLA OCDMA Project

1. UCLA Graduate School of Engineering - Electrical Engineering Program Communication Systems Laboratory Progress Report for the UCLA OCDMA Project Miguel Griot Andres Vila-Casado Wen-Yen Weng Herwin Chan Richard Wesel

2. Progress during this period • Conference Presentations • Journal Paper preparation • Dissertation Preparation • Expanding into related problems

3. Conference Presentations • H. Chan, M. Griot, A. Vila Casado, R. Wesel, I. Verbauwhede "High Speed Channel Coding Architectures for the Uncoordinated OR Channel". IEEE 17th INTERNATIONAL CONFERENCE ON Application-specific Systems, Architectures and Processors (ASAP), Steamboat Springs, Colorado, September 2006. • M.Griot, A. I. Vila Casado and R. D. Wesel "Non-linear Turbo Codes for Interleaver-Division Multiple Access on the OR Channel". Globecom 2006, 27 Nov. - 1 Dec., San Francisco, USA.

4. Journal Paper Preparation • Journal paper on Trellis Codes nearly complete (attached). • Miguel is working towards new results for higher-order modulations before finalizing manuscript for journal paper on Turbo Codes.

5. Dissertation Preparation • During this reporting period Wen-Yen Weng completed his dissertation.

6. Expanding into related areas • An improvement in the Bhattacharya Bound • A technique for handling the broadcast Z channel • A new technique for turbo codes using higher order modulations

7. Factor of ½ improvement in Bhattacharya BER bound Miguel Griot Wen-Yen Weng Richard Wesel

8. Bound so far • (n,k) linear code over a symmetric channel (BSC, AWGN). • Denote u the input words, and x= x(u) the codeword. • The all-zero codeword can be assumed to be transmitted. Denote it . • Union bound:

9. Bhattacharyya Bound • Upper bound for :

10. But…

11. Hence…

12. UCLA Electrical Engineering Department-Communication Systems Laboratory The Z-Broadcast Channel Andres I. Vila Casado Miguel Griot Richard Wesel

13. 1 1 1 1 0 0 0 0 The 2-User Broadcast Z-Channel ?

14. 1 1 1 1 0 0 0 0 Degraded Broadcast Z-Channel • The broadcast Z-channel is a stochastically degraded broadcast channel ? • The capacity region is given by

15. 1 1 1 1 0 0 0 0 Capacity Region • Theoretically the capacity region is calculated by allowing any possible combination (joint distribution) of the messages (X1 and X2). • We conjecture that if chose to combine the messages with an OR gate we can still achieve every point of the capacity region. OR

16. Capacity Region • A numerical computation of the capacity regions show that our conjecture is correct • The equations then become: • Where p1 and p2 are the density of ones of X1 and X2 respectively

17. Capacity Region • For =0.1 and  =0.6 the capacity region was numerically computed:

18. Coding solution • The capacity region can be achieved using non-linear codes with the appropriate density of ones • Our solution is to use a Z-capacity achieving nonlinear code with density of ones p2 for X2 and a Z-capacity achieving nonlinear code with density of ones p1 for X1 transmitted only when X2 is zero

19. Parallel concatenated TCM for high-order modulations Miguel Griot Andres Vila Casado Richard Wesel

20. Applications for non-linear codes • The new family of non-linear codes we proposed for the Z-Channel: • their basic structure… • their design… • and the analytical BER bounds… • can be applied, with little modification, to other channels. • In general, over any asymmetric channel requiring non-linear codes.

21. High-order modulations • So far, for high-order modulations, a linear code with a bits-to-constellation point mapper has been used • However, in some constellations, the mapper must be nonlinear. • Using a linear code + a mapper could be a limitation. Trellis coded modulation Mapper CC Interleaver CC Mapper

22. TCM Interleaver TCM Parallel Concatenated TCM • Structure of PC-TCM: • Codeword : a set of constellation points • Rate : • Using directly a TCM there could be a gain in performance.

23. Uniform Interleaver Analysis for AWGN • The extension of Benedetto’s uniform interleaver analysis shown for PC-NLTC over the Z-Channel [GlobeCom’06], can be applied to any system in which non-linear codes are required (or used). • In particular, it can be applied to high-order modulations over the AWGN. • Changes: • Use squared Euclidean distance instead of directional distance for Z-Channel. • Evaluate final expression in instead of .

24. BER bounding analysis

25. Comments • We have a very general bounding technique under ML decoding for parallel concatenated non-linear (block or trellis) codes. • Still more work to do on the code design. • We believe we can improve the performance of previous works, using this more general approach.