Progress Report for the UCLA OCDMA Project

UCLA Graduate School of Engineering - Electrical Engineering ProgramUCLA Graduate School of Engineering - Electrical Engineering Program

Communication Systems Laboratory

Miguel Griot

Richard Wesel

Andres Vila-Casado

BikeXie

Progress during this period Journal Paper Publications and Submissions.

Conference Paper Submissions.

Expanding into related problems:

Broadcast Channels:. Bike Xie.

Journal Paper Publications/Submissions A Tighter Bhattacharyya Bound for Decoding Error

Probability, M. Griot, W.Y. Weng, R.D. Wesel. IEEE Communications Letters, Apr. 2007.

Nonlinear Trellis Codes for Binary-Input Binary-Output Multiple Access Channels with Single-User Decoding, M.Griot, A.I. Vila Casado, R.D. Wesel. Submitted to IEEE Transactions in Communications, March 15.

Nonlinear Turbo codes for the OR Multiple Access Channel and the AWGN Channel with High-Order Modulations, M. Griot, A.I. Vila Casado, R.D. Wesel. Soon to be submitted to TCOM.

Bike Xie: working on journal paper on Broadcast Z Channels.

Conference Paper Submission/ Preparation On the Design of Arbitrarily Low-Rate Turbo Codes,

M. Griot, A.I. Vila Casado, R.D. Wesel, submitted to GlobeCom 2007.

Optimal Transmission Strategy and Capacity region for the Broadcast Z Channel, B. Xie, M. Griot, A.I. Vila Casado, R. Wesel. Accepted in Information Theory Workshop, Sep. 2007.

Nonlinear Turbo Codes for High-Order Modulations over the AWGN channel, M. Griot, R.D. Wesel. Soon to be submitted to Allerton Conference 2007.

Expanding into related areas An improvement in the Bhattacharya Bound

A technique for handling the broadcast Z channel

A new technique for turbo codes using higher order modulations

Parallel concatenated TCM for high-order modulations

Miguel Griot

Andres Vila Casado

Richard Wesel

High-order modulations So far, for high-order modulations, a linear code with a

bits-to-constellation point mapper has been used

However, in some constellations (8PSK, APSK) the mapper must be nonlinear.

Using a linear code + a mapper could be a limitation.

CC

Interleaver

CC

0k

0k

kMapper

Mapper

Trellis codedmodulation

Structure of PC-TCM:

Codeword : a set of constellation points.

Rate :

Using directly a TCM there could be a gain in performance.

Parallel Concatenated TCM

TCM

Interleaver

TCM

0k

0k

k

0 / 2 bits/symbolk

BER bounding for AWGN We have developed an extension of Benedetto’s

uniform interleaver analysis for nonlinear code over any channel.

Design Criteria: Maximize the effective free distance of each constituent code.

Effective free distance: output distance (for AWGN squared euclidean distance) of any two possible codewords produced by data-words with Hamming distance equal to 2.

We show that nonlinear code can increase the effective free distance of a constituent code.

8PSK, 16-state turbo code, rate 2 bits/symbol

[1] Turbo-Encoder Design for Symbol-Interleaved Parallel Concatenated Trellis-Coded Modulation. C. Fragouli, R.D. Wesel, IEEE Trans. In Comm, March 2001.

2.8 3 3.2 3.4 3.6 3.8 410

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

8PSK 16-state turbo code, K = 10000, AWGN

BE

R

Eb/N

0 [dB]

Linear

Bound LinearNonlinear

Nonlinear bound

, 1.17eff freed

, 2eff freed

Linear [1]:

Nonlinear:

Constrained capacity 2.8dB

Design of Arbitrarily Low-Rate Turbo Codes.

Miguel Griot

Andres Vila Casado

Richard Wesel

Low-rate turbo code, design criteria We can see the general structure of a rate 1/n constituent code

as:

Assuming that branches leaving a same state or merging to a same state are antipodal.

Goals: Given certain values of n and m, maximize the minimum

distance between output labels. This is equivalent to a (n,m-1) code design.

Given a certain m, choose the rate 1/n such that the performance is optimized in terms of BER vs. Eb/No.

0s 1s 2ms 1ms

2mf 0f

system ati c bi t

, 1 - coden m n pari ty bi tsn

1mf

Low-rate turbo code design over AWGN The performance of a code in terms of Eb/No

is driven by the term:

In our case, k = m-1 fixed. Hence, the objective is to maximize the term .

Theorem 1:

Theorem 2: BCH codes satisfy the upper bound with equality. A concatenation with a repetition code maintains the equality.

min 0( / ) ( / )bk n d E NBER e

min /d n1

min 2,

2 1

k

k

dn

n

2 1,k k

Optimal code is linear Optimal structure:

0s 1s 2ms 1ms

(0,1)g (1,1)g( 2,1)mg

1p(r ,1 )

r e pe t i t i o n

2mf 0f

(0, )ng ( 2, )m ng

np(r ,1 )

r e pe t i t i o n

(1, )ng

12 1mn 12 1mr

pari ty bi ts

system ati c bi t

Results:

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.610

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

BE

R

Eb/N

0 [dB]

Performance for different rates and memory elements m

m=3, rate 1/7, N=1024m=3, rate 1/49, N=1024

m=4, rate 1/15, N=1024

m=4, rate 1/127, N=1024

m=4, rate 1/505, N=1024

m=4, rate 1/15, N=8192m=4, rate 1/505, N=8192

Results:

-28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -810

-6

10-5

10-4

10-3

10-2

10-1

BE

R

SNR [dB]

m=3, rate 1/7, N=1024m=3, rate 1/49, N=1024

m=4, rate 1/15, N=1024

m=4, rate 1/127, N=1024

m=4, rate 1/505, N=1024

m=4, rate 1/15, N=8192m=4, rate 1/505, N=8192

Optimal Transmission Strategy for the Broadcast Z Channel

Bike Xie

Miguel Griot

Andres Vila Casado

Richard Wesel

Broadcast Z Channel

X

Y1

Y2

1

0

1

0

1

0

1

2 1 20 1

1 1 2( ; | )R I X Y X2 2 2( ; )R I X Y

X Y1

1

Y2

X2

1p1q2q

2p

2 1

11

The capacity region is the convex hull of the closure of all rate pairs (R1,R2) satisfying

for some probabilities and1q 2q

1 1 2 2 1p q p q

Optimal Transmission Strategy

X Y1

1

Y2

X2

1p1q2q

2p

The optimal transmission strategy is proved to be0

1 1 1(1 ) (1 )1

11

(1 )( 1)Hq

e

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

2 2 1 2 2 1 2( (1 )) ( (1 ))R H q q q H q

1 2 1 1 2 1 1( (1 )) (1 )R q H q q q H

The curve of the capacity region follows from

with the optimal transmission strategy.

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

Communication System

Encoder 2

Encoder 1

OR

OR

OR OR

Decoder 1

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.

•Nonlinear turbo codes that provide a controlled distribution of

ones and zeros are used.

•User 2 with the worse channel decodes message W2 directly.

•User 1 with the better channel has a successive decoding scheme.

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

1 20.15, 0.6

Future Work Gaussian channels with MPSK modulation:

We have proved that the optimal surface of the capacity region can be achieved with independent encoding and group addition.

Nonlinear turbo codes will also be used.