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Independent Encoding for the Broadcast Channel

This document provides an overview of independent encoding techniques for broadcast channels, particularly focusing on stochastically degraded channels. It discusses the capacity region, which encompasses all achievable rate pairs (R1, R2) using joint encoding and successive decoding strategies. The optimal transmission strategies for various configurations, including Broadcast Z channels, are analyzed, demonstrating their efficiency. Simulation results reveal that the rates achieved are close to optimal, illustrating the effectiveness of nonlinear turbo codes in enhancing communication performance.

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Independent Encoding for the Broadcast Channel

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  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

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