Nonlinear Turbo Codes for the broadcast Z Channel

Transcription

1 UCLA Electrical Engineering Department Communication Systems Lab. Nonlinear Turbo Codes for the broadcast Z Channel Richard Wesel Miguel Griot Bike ie Andres Vila Casado Communication Systems Laboratory, UCLA

2 Outline The stochastically degraded Broadcast channel. The broadcast Z channel (B-Z channel) Optimal transmission strategy. Capacity region. Channel coding design, nonlinear turbo codes: Controlled ones density. Designed for the Z channel and the Z channel with erasures. Simulation results. Conjecture: optimal transmission strategy for a particular set of broadcast channels. Conclusions Communication Systems Laboratory, UCLA

3 The stochastically degraded broadcast channel ( x ) p y Y Decoder ( W, W) Encoder ( x ) p y Y Decoder Stochastically degraded if: ( ) ( ) ( ) ( ) = p ' y y such that p y x p y x p' y y y ( ) Y ( ) p y x p ' y y Y Communication Systems Laboratory, UCLA 3

4 The stochastically degraded channel Capacity region for sending independent information over the degraded channel is the convex hull of the closure of the rate pairs R optimal surface R I( ; Y ) R I( ; Y ) Y Y R p( x ) ( x x ) p" ( ) Y ( ) p y x p ' y y Y Communication Systems Laboratory, UCLA 4

5 ( W, W) Encoder Encoder Encoder ( W, W) The broadcast Z channel Encoder Encoder 0 α Y Y Decoder Communication Systems Laboratory, β UCLA Decoder 5 0 α < β Y N Bernoulli( α N Bernoulli( ) α ) Y N Bernoulli( β ) Decoder Decoder Theorem: Optimal surface can be achieved by: 0 0 Y ~ Bernoulli ( p ) ( p ) β α N3 Bernoulli α Bernoulli Y

6 Optimal transmission strategy Sketch of proof: p p General case p q α Y Y γ R I( ; Y ) R I( ; Y ) We need to prove that q =0 (or p =) Without loss of generality Consider any (R,R ) point achieved with p 0, p, q 0, p, q p q p Communication Systems Laboratory, UCLA 6

7 Proof for the B-Z channel p p : q q ( p) ε, ε > 0 p p pε A : R, R p q p ( ),, R I( ; Y) R I( ; Y ) p 0, p, q 0, p, q p p p + ( p ˆ ) ε ˆ : q q, ε > 0 p p ˆ + ( p q) ε Communication Systems Laboratory, UCLA 7

8 Perceived channels Receiver : Z channel R I( ; Y ) p p ( )( )( β ) ( ) ( )( β ) = H p p p H p Receiver : Z channel + erasure channel. p β Y Y Y R I( ; Y ) ( p ) H[ ( p )( α )] ( p ) H[ α ] { } = p p α p e Communication Systems Laboratory, UCLA 8

9 Implementation Encoding: of two parallel concatenated nonlinear trellis codes [GlobeCom 06]. W PC-NLTC ~ p W PC-NLTC Decoding receiver (hard): ~ p e Decoder Y ˆ Decoder Communication Systems Laboratory, UCLA 9 x y ˆ if x = 0, = e if xˆ =.

10 Parallel Concatenated Nonlinear Trellis Codes Presented in GlobeCom 06 (for Z channel). The NLTC consists of: A ν -state trellis structure (block S). A look-up table (LUT) stores an output per branch. The outputs satisfy the required ones density p (nonsystematic) PC-NLTC: Two constituent ( n0, k0) non-linear trellis codes (NLTC) linked by an interleaver (Π) of length K. k0 rate = n 0 Π k 0 LUT S NLTC Communication Systems Laboratory, UCLA 0 n 0

11 Example Capacity region and simulated rates, α = 0.5, β = 0.6 (/,/5) (/6,/6) Capacity surface simulated rates optimal rates R 0. (/3,/9) R (/,/) Communication Systems Laboratory, UCLA

12 Results 8-state nonlinear turbo codes. k 0 = R R p p K K BER BER / / /6 / /3 / / / Communication Systems Laboratory, UCLA

13 The broadcast Z channel ( W, W) Encoder Y Y W Encoder W Encoder N Bernoulli( α ) Optimal surface can be achieved by: β α N3 Bernoulli α Also true for (AWGN, + operator), (BSC, ). Communication Systems Laboratory, UCLA 3 ~ Bernoulli ( p ) ( p ) Bernoulli

14 Conjecture f( x, n ) Y ( ) N p n f( x, n ) Y f( x, n3 ) Y f( x, n ) Y ( ) N p n = Y N p( n ) = Y = N = N Points on the optimal surface can be achieved by: ( ) N p n W Encoder W Encoder f( x, x ) f( x, n ) Y f( x, n3 ) Y Communication Systems Laboratory, UCLA ( ) N p n ( ) N p n

15 Conclusions We have presented an optimal transmission strategy for the Broadcast Z Channel. Simple encoding and decoding. A practical implementation that works close to capacity has been presented. Nonlinear turbo codes, specifically designed for the Z channel and the Z channel + erasures, have been designed. Conjecture: simple transmission strategy could be used to a set of stochastically degraded broadcast channels. Communication Systems Laboratory, UCLA 5