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 Richard Wesel Bike Xie

2. Progress during this period • Journal Paper Publications and Submissions. • Conference Paper Submissions. • Expanding into related problems: • Broadcast Channels:. Bike Xie.

3. Journal Paper Publications/Submissions • A Tighter Bhattacharyya Bound for Decoding Error Probability, M. Griot, W.Y. Weng, R.D. Wesel. IEEE Communications Letters, Apr. 2007. • Nonlinear Trellis Codes for Binary-Input Binary-Output Multiple Access Channels with Single-User Decoding, M.Griot, A.I. Vila Casado, R.D. Wesel. Submitted to IEEE Transactions in Communications, March 15. • Nonlinear Turbo codes for the OR Multiple Access Channel and the AWGN Channel with High-Order Modulations, M. Griot, A.I. Vila Casado, R.D. Wesel. Soon to be submitted to TCOM. • Bike Xie: working on journal paper on Broadcast Z Channels.

4. Conference Paper Submission/ Preparation • On the Design of Arbitrarily Low-Rate Turbo Codes, M. Griot, A.I. Vila Casado, R.D. Wesel, submitted to GlobeCom 2007. • Optimal Transmission Strategy and Capacity region for the Broadcast Z Channel, B. Xie, M. Griot, A.I. Vila Casado, R. Wesel. Accepted in Information Theory Workshop, Sep. 2007. • Nonlinear Turbo Codes for High-Order Modulations over the AWGN channel, M. Griot, R.D. Wesel. Soon to be submitted to Allerton Conference 2007.

5. 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

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

7. 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 (8PSK, APSK) the mapper must be nonlinear. • Using a linear code + a mapper could be a limitation. Trellis coded modulation Mapper CC Interleaver CC Mapper

8. 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.

9. BER bounding for AWGN • We have developed an extension of Benedetto’s uniform interleaver analysis for nonlinear code over any channel. • Design Criteria: Maximize the effective free distance of each constituent code. • Effective free distance: output distance (for AWGN squared euclidean distance) of any two possible codewords produced by data-words with Hamming distance equal to 2. • We show that nonlinear code can increase the effective free distance of a constituent code.

10. 8PSK, 16-state turbo code, rate 2 bits/symbol • Linear [1]: • Nonlinear: • Constrained capacity 2.8dB [1] Turbo-Encoder Design for Symbol-Interleaved Parallel Concatenated Trellis-Coded Modulation. C. Fragouli, R.D. Wesel, IEEE Trans. In Comm, March 2001.

11. Design of Arbitrarily Low-Rate Turbo Codes. Miguel Griot Andres Vila Casado Richard Wesel

12. Low-rate turbo code, design criteria • We can see the general structure of a rate 1/n constituent code as: • Assuming that branches leaving a same state or merging to a same state are antipodal. • Goals: • Given certain values of n and m, maximize the minimum distance between output labels. This is equivalent to a (n,m-1) code design. • Given a certain m, choose the rate 1/n such that the performance is optimized in terms of BER vs. Eb/No.

13. Low-rate turbo code design over AWGN • The performance of a code in terms of Eb/No is driven by the term: • In our case, k = m-1 fixed. Hence, the objective is to maximize the term . • Theorem 1: • Theorem 2: BCH codes satisfy the upper bound with equality. A concatenation with a repetition code maintains the equality.

14. Optimal code is linear • Optimal structure:

15. Results:

16. Results:

17. Optimal Transmission Strategy for the Broadcast Z Channel Bike Xie Miguel Griot Andres Vila Casado Richard Wesel

18. Broadcast Z Channel Y1 Y2 X 1 X2 Y1 1 0 X 0 1 Y2 0 The capacity region is the convex hull of the closure of all rate pairs (R1,R2) satisfying for some probabilities and

19. Optimal Transmission Strategy Y1 Y2 X X2 N1 Y1 X1 The optimal transmission strategy is proved to be X N2 X2 Y2 OR OR OR The curve of the capacity region follows from with the optimal transmission strategy.

20. Communication System Encoder 1 Decoder 1 OR OR OR OR Decoder 2 Encoder 2 • It is an independent encoding scheme. • The one’s densities of X1 and X2 are p1 and p2 respectively. • The broadcast signal X is the OR of X1 and X2. • Nonlinear turbo codes that provide a controlled distribution of ones and zeros are used. • User 2 with the worse channel decodes message W2 directly. • User 1 with the better channel has a successive decoding scheme.

21. Simulation Results • The cross probabilities of the broadcast Z channel are • The simulated rates are very close to the capacity region. • Only 0.04 bits or less away from optimal rates in R1 • Only 0.02 bits or less away from optimal rates in R2

22. Future Work Gaussian channels with MPSK modulation: We have proved that the optimal surface of the capacity region can be achieved with independent encoding and group addition. Nonlinear turbo codes will also be used.