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UCLA Electrical Engineering Department – Communication Systems Laboratory. Independent Encoding for the Broadcast Channel. Bike Xie Miguel Griot Andres I. Vila Casado Richard D. Wesel. Introduction. Broadcast Channels - PowerPoint PPT Presentation
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Independent Encoding for the Broadcast Channel
UCLA Electrical Engineering Department – Communication Systems LaboratoryUCLA Electrical Engineering Department – Communication Systems Laboratory
Bike Xie Miguel Griot
Andres I. Vila Casado Richard D. Wesel
1Communication 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 .
XY1
Y2
1 |p y x
2 |p y x
2Communication Systems Laboratory, UCLA
2 |p y x
1 |p y x
1 1
2 1 2 1| ( | ) ( | )y
p y x p y x q y yy
2 1( | )q y y
1 |p y x 2 1|q y yX Y1 Y2
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.
1 1 2( ; | ),R I X Y X
2 2 2( ; ),R I X Y
3Communication Systems Laboratory, UCLA
2 2 1 2( ) ( | ) ( , | )p x p x x p y y x
1 |p y x 2 1|q y yX Y1 Y2 2|p x xX2
Broadcast Z Channels Broadcast Z Channels
Broadcast Z channels are stochastically degraded broadcast channels.
X
Y1
Y2
1
0
1
0
1
0
1
2 1 20 1
X Y1
1
Y2
2 1
11
4Communication 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.
1 21 1 2 , ,( ; | ) |q qR I X Y X
1 22 2 2 , ,( ; ) |q qR I X Y
X Y1
1
Y2
X2
1p1q2q
2p 2 1
11
1q 2q
1 1 2 2 1p q p q
5Communication Systems Laboratory, UCLA
Capacity Region Explicit expression of the capacity region
The boundary of the capacity region is
where parameters satisfy
1 2 1 1 2 1 1
2 2 1 2 2 1 2
( (1 )) ((1 ))
( (1 )) ( (1 ))
R q H q q q H
R H q q q H q
1 1 1(1 ) /(1 )1
11,
(1 )( 1)Hq
e
6Communication Systems Laboratory, UCLA
1 2,q q
2 1 2 1 21 2 1 2 1 1 1 1
2 1 2 1 1
1 (1 ) log(1 (1 ))( (1 )) (1 ) log ( ( (1 )) (1 )).
(1 ) log(1 (1 ))
q q qH q q H q q H
q q q
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.
1 |p y x 2 1|q y yX Y1 Y2 2|p x xX2
2 2( ) ( | )p x p x x
2 21 1 2 ( ) ( | )( ; | ) |p x p x xR I X Y X
2 22 2 2 ( ) ( | )( ; ) |p x p x xR I X Y
R1
R2
(R1,R2)
7Communication Systems Laboratory, UCLA
Optimal Transmission StrategyX Y1
1
Y2
X2
1p1q2q
2p
0,
1 1 1(1 ) (1 )1
11,
(1 )( 1)Hq
e
8Communication Systems Laboratory, UCLA
The optimal transmission strategies for broadcast Z channels are
2 1 2 1 21 2 1 2 1 1 1 1
2 1 2 1 1
1 (1 ) log(1 (1 ))( (1 )) (1 ) log ( ( (1 )) (1 )).
(1 ) log(1 (1 ))
q q qH q q H q q H
q q q
All rate pairs on the boundary of the capacity region can be achieved with these strategies.
Optimal Transmission Strategy These optimal transmission strategies are
independent encoding schemes since .
Communication Systems Laboratory, UCLA 9
X Y1
1
Y2
X2
1p1q2q
2p
X2 OR
X1
XOR
ORY2
N1
Y1
N2
0
1 1Pr( 1)X p
2 2Pr( 1)X p
1 1Pr( 1)N
2 2Pr( 1)N
0
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
Sketch of the Proof
10Communication Systems Laboratory, UCLA
0
2 10 1,0q q
1.q
X Y1
1
Y2
X2
1p1q2q
2p
2 20, 1q q 1q 0.
1 2( , , )q q
Δ2
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 11
Δ1
R1
R2
A
B
1 21 2 , ,: , |q qR R A
1 2 2( ; )R I X Y
2 1 2( ; | )R I X Y X
2 20, 1q q 1q
2 0R
0 2 1q
1 1 1arg max( ( (1 )) (1 )).q H x xH
2 10 1,0 .q q
2 10 1,0q q
1 2( , , )q q 1 2, .
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 12
1q 2q
1 2
1 2 1 1 2 1 1
2 2 1 2 2 1 2
1 2
maximize
subject to ( (1 )) ((1 ))
( (1 )) ( (1 ))
0 , 1
R R
R q H q q q H
R H q q q H q
q q
0.
Communication Systems
Encoder 2
Encoder 1
OR
OR
OR OR
Successive Decoder
Decoder 2
1W
2W
1W
2W2X
1X
X 2Y
1Y
N1N
1N
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.
13Communication Systems Laboratory, UCLA
Successive Decoder Decoder structure of the successive decoder
for user 1
Communication Systems Laboratory, UCLA 14
1 21 1 2
2
ˆif 0ˆ( , )
ˆif 1
y xy e y x
erasure x
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
15Communication Systems Laboratory, UCLA
Simulation ResultsThe 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.
16Communication Systems Laboratory, UCLA
1 20.15, 0.6.