Independent Encoding for the Broadcast Channel

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

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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

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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

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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

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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

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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)

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Optimal Transmission StrategyX Y1

1

Y2

X2

1p1q2q

2p

0,

1 1 1(1 ) (1 )1

11,

(1 )( 1)Hq

e

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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 .

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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

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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

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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.

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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.

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Successive Decoder Decoder structure of the successive decoder

for user 1

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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

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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.

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1 20.15, 0.6.