1. Progress Report for the UCLA OCDMA Project UCLA Graduate School of Engineering - Electrical Engineering Program Communication Systems Laboratory Miguel Griot Richard Wesel Andres Vila-Casado 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. CC Interleaver CC Mapper Trellis coded modulation

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

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 [1] Turbo-Encoder Design for Symbol-Interleaved Parallel Concatenated Trellis-Coded Modulation. C. Fragouli, R.D. Wesel, IEEE Trans. In Comm, March 2001. Linear [1]: Nonlinear: Constrained capacity 2.8dB

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:

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

18. Broadcast Z Channel X Y1Y1 Y2Y2 1 0 1 0 1 0 X Y1Y1 Y2Y2 X2X2 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 X Y1Y1 Y2Y2 X2X2 The optimal transmission strategy is proved to be The curve of the capacity region follows from with the optimal transmission strategy. X2X2 OR X1X1 X Y2Y2 N1N1 Y1Y1 N2N2

20. Communication System Encoder 2 Encoder 1 OR Decoder 1 Decoder 2 It is an independent encoding scheme. The one’s densities of X 1 and X 2 are p 1 and p 2 respectively. The broadcast signal X is the OR of X 1 and X 2. Nonlinear turbo codes that provide a controlled distribution of ones and zeros are used. User 2 with the worse channel decodes message W 2 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 R 2

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.