Independent Encoding for the Broadcast Channel

1. UCLA Electrical Engineering Department – Communication Systems Laboratory Independent Encoding for the Broadcast Channel Bike Xie Miguel Griot Andres I. Vila Casado Richard D. Wesel Communication Systems Laboratory, UCLA

2. Introduction • Broadcast Channels • One transmitter sends independent messages to several receivers which decode without collaboration. • Stochastically Degraded Broadcast Channels • The worse channel is a stochastically degraded version of the better channel , i.e., such that . Y1 X Y2 X Y1 Y2 Communication Systems Laboratory, UCLA

3. X2 Y2 X Y1 Stochastically Degraded Broadcast Channels • Capacity Region [Cover72][Bergmans73][Gallager74] • The capacity region is the convex hull of the closure of all rate pairs (R1, R2) satisfying for some joint distribution . • Joint encoding and successive decoding are used to achieve the capacity region. Communication Systems Laboratory, UCLA

4. Broadcast Z Channels • Broadcast Z Channels • Broadcast Z channels are stochastically degraded broadcast channels. 1 Y1 1 0 X 0 1 Y2 0 Y1 Y2 X Communication Systems Laboratory, UCLA

5. Capacity Region • Implicit expression of the capacity region • The capacity region is the convex hull of the closure of all rate pairs (R1,R2) satisfyingfor some probabilities , and . • In general, joint encoding is potentially too complex. Y1 Y2 X X2 Communication Systems Laboratory, UCLA

6. Capacity Region • Explicit expression of the capacity region • The boundary of the capacity region iswhere parameters satisfy Communication Systems Laboratory, UCLA

7. X2 Y2 X Y1 R2 (R1,R2) R1 Optimal Transmission Strategy • An optimal transmission strategy is a joint distribution that achieves a rate pair (R1,R2) which is on the boundary of the capacity region. Communication Systems Laboratory, UCLA

8. Y1 Y2 X X2 Optimal Transmission Strategy • The optimal transmission strategies for broadcast Z channels are • All rate pairs on the boundary of the capacity region can be achieved with these strategies. Communication Systems Laboratory, UCLA

9. Optimal Transmission Strategy • These optimal transmission strategies are independent encoding schemes since . Y1 Y2 X X2 OR OR OR N1 Y1 X1 X N2 X2 Y2 Communication Systems Laboratory, UCLA

10. Sketch of the Proof Y1 Y2 X X2 • W.O.L.G assume • To prove • Lemma 1: any transmission strategy with is not optimal. • Lemma 2: any rate pair (R1,R2) achieved with or can also be achieved with Communication Systems Laboratory, UCLA

11. Sketch of the Proof • To prove Lemma 1 • Point A is achieved with • Slightly change the strategy to achieve • The shaded region is achievable. • To prove Lemma 2 • When or , the rate for user 2 is . • Point B can be achieved with the strategy , and Communication Systems Laboratory, UCLA

12. Sketch of the Proof • To prove the constraints on and • Solve the maximization problem for any fixed • Time sharing gets no benefit. Communication Systems Laboratory, UCLA

13. Communication Systems Successive Decoder Encoder 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. • User 2 with the worse channel decodes the message W2 directly. • User 1 with the better channel needs a successive decoder. Communication Systems Laboratory, UCLA

14. Successive Decoder • Decoder structure of the successive decoder for user 1 Communication Systems Laboratory, UCLA

15. Nonlinear Turbo Codes • Nonlinear turbo codes can provide a controlled distribution of ones and zeros. • Nonlinear turbo codes designed for Z channels are used. [Griot06] • Encoding structure of nonlinear turbo codes Communication Systems Laboratory, UCLA

16. 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. Communication Systems Laboratory, UCLA