1. Independent Encoding for the Broadcast Channel
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. Communication Systems Laboratory, UCLA
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. 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. X Y1 Y2 X2
Communication Systems Laboratory, UCLA

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

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

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

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

8. Optimal Transmission StrategyX Y1 Y2 X2
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 X Y1 Y2 X2 N1 Y1 X1 OR X X2 OR N2 Y2 OR
Communication Systems Laboratory, UCLA

10. Communication Systems Laboratory, UCLA
Sketch of the Proof X Y1 Y2 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. Communication Systems Laboratory, UCLA
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. Communication Systems Laboratory, UCLA
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
Encoder 1
Successive Decoder OR
Encoder 2 OR OR OR
Decoder 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. Communication Systems Laboratory, UCLA
Successive Decoder
Decoder structure of the successive decoder for user 1
Communication Systems Laboratory, UCLA

15. Communication Systems Laboratory, UCLA
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. Communication Systems Laboratory, UCLA
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