Progress report Miguel Griot Andres I. Vila Casado Wen-Yen Weng Richard Wesel UCLA Graduate School of Engineering - Electrical Engineering Program Communication.

Progress report

Miguel Griot

Andres I. Vila Casado

Wen-Yen Weng

Richard Wesel

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

Communication Systems Laboratory

Results up to last meeting Non-linear trellis codes for OR-MAC (Completed)

Design criteria. BER analytical bounding technique. Results for any number of users.

Parallel concatenated NLTC for OR-MAC Design criteria. BER analytical bounding technique. Results for 6 and 24 users.

Theoretically achievable Sum-rates for more general channels, in particular coherent interference model.

Preliminary results for 6-user optical MAC with coherent interference, using NLTC.

Non-linear trellis codes for OR-MAC Design Criteria:

Extension to Ungerboeck’s rules. We maximize the minimum free distance of

the code, using the proper directional distance definition for the Z-Channel.

BER bounding technique for Z-Channel Transfer function bound technique.

Bit Error Rate Bound for the Z-Channel We use the transfer function bound technique on

[Viterbi ‘71] for linear codes, and extended by [Biglieri ‘90] for non-linear codes, modifying the pairwise error probability measure.

Given two codewords

Replace and the

transfer function bound technique can be readily applied to the NLTC to yield an upper bound to its BER over the Z-Channel.

ˆ,n nX X

ˆ ˆ( , ) ( , )ˆ ˆmax ( , ), ( , )

ˆ ˆ

2 2

n n n nD Dn n n n

D D

n n n ne e

d X X d X Xd X X d X X

P X X P X X

ˆ( , )

ˆ with 2

n nDd X X

n neP X X

Bit Error Rate Bound for the Z-Channel

Product states:

where and denote the state at the encoder and receiver respectively. G denotes a ‘good product-state’ and B denotes a ‘bad product-state’.

Transition matrix:

For each transition in the product-state diagram the branch is labeled as:

es rs

where , is the directional distance in the output,

and , is the Hamming distance in the input.

D e r

H e r

d x x

d u u

Bit Error Bound for the Z-Channel Transfer function:

where:

Then:

Results : 6-user OR-MAC

0.2 0.3 0.4 0.5 0.6 0.7

10-8

10-6

10-4

10-2

100

BE

R

NL-TCM 1/17NL-TCM 1/18

NL-TCM 1/20

NL-TCM 1/20 FPGA

Bound 1/17

Bound 1/18Bound 1/20

Large number of users Main results:

For any number of users, we achieve the same sum-rate with similar performance.

Tight BER analytical bound for Z-Channel provided.

N n SR BER

6 20 0.3 0.439

100 344 0.291 0.4777

300 1000 0.3 0.4901

900 3000 0.3 0.4906

1500 5000 0.3 0.4907

51.1046 1051.2157 1051.2403 1051.2508 10

51.0214 10

Concatenation with Outer Block Code A concatenation of an NLTC with a high rate block code provides

a very low BER, at low cost in terms of rate.

Results: A concatenation of the rate-1/20 NL-TCM code with (255

bytes,247 bytes) Reed-Solomon code has been tested for the 6-user OR-MAC scenario.

This RS-code corrects up to 8 erred bits.

Although we don’t have simulations for the 100-user case, it may be inferred that a similar BER would be achieved.

Block-Code Encoder NL-TC Encoder

Z-Channel

Block-Code Decoder NL-TC Decoder

Rate Sum-rate p BER

0.0484 0.29 0.125 0.4652

102.48 10

Parallel Concatenated NL-TCs Capacity achieving.

Design criteria: An extension of Benedetto’s uniform

interleaver analysis for parallel concatenated non-linear codes has been derived.

This analysis provides a good tool to design the constituent trellis codes.

NL-TC

Interleaver NL-TC

Parallel Concatenated NL-TCs The uniform interleaver analysis proposed by

Benedetto, evaluates the bit error probability of a parallel concatenated scheme averaged over all (equally likely) interleavers of a certain length.

Maximum-likelihood decoding is assumed. However, this analysis doesn’t directly apply to our

codes: It is applied to linear codes, the all-zero codeword is

assumed to be transmitted. The constituent NL-TCM codes are non-linear, hence all the possible codewords need to be considered.

In order to have a better control of the ones density, non-systematic trellis codes are used in our design. Benedetto’s analysis assumes systematic constituent codes.

An extension of the uniform interleaver analysis for non-linear constituent codes has been derived.

Results

6 users

• Parallel concatenationof 8-state, duo-binaryNLTCs. • Sum-rate = 0.6• Block-length = 8192• 12 iterations in message-passing algorithm

General Model for Optical MAC

User 1 User 2 User N

Receiver

1X 2X NX

1 2( 0) ( , , , )NP Y f X X X

1,

( 0) 0,

,m

P Y

if all users transmit a 0

if one and only one user transmits a 1

if m users transmit a 1 and the rest a 0

Model The can be chosen any way, depending on the

actual model to be used. Examples:

Coherent interference:

constant

'm s

m

2 , 0, 2m m

2 , 2mm m

( 2)( )(1 ), 2mm e m

2

1

, 2

0,2 ,

( 1/ 2)

i

mj

mi

i

P e m

U i

threshold

Achievable sum-rates n users with equal ones density p. Joint Decoding

Treating other users as noise – Binary Asymmetric Channel:

0 2

0 1

( ) max 1 1

1, 0

n nn j n kj k

JD p j kj k

n nSR n H p p p p H

j k

0

1

0

1

YiX

p

1 p

1

1

11

0

11

11

11 1

11

nn kk

kk

nn kk

kk

np p

k

np p

k

( ) max 1 1 1BAC pSR n n H p p p H p H

Sum-rate for coherent interence

0 20 40 60 80 100 120 140 160 180 2000.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95Sum-rate: Coherent interference

number of users

Sum

-rat

e

joint decoding

other users noise

We provide an analytical lower bound to the achievable sum-rate for ANY number of users, for both joint decoding and treating other users as noise.

Lower bound for different This figure shows the lower bounds and the actual sum-rates for

200 users for the worst case ( constant) .

'M s

'm s

JD : Joint DecodingOUN: Other Users Noise

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.4

0.5

0.6

0.7

0.8

0.9

1

Sum-rate lower bound depending on maximum m

Maximum m

sum

rat

e bo

und

/ op

timum

JD optimum

JD sum-rate bound

JD sum-rate for 200 users

OUN optimum OUN sum-rate bound

OUN sum-rate for 200 users

Coherent interference

Progress since last meeting New FPGA demo for 6-user optical multiple

access. Design of NL-TC for optical MAC with

coherent interference, for large number of users.

BER bounding technique for BAC. (Ongoing work) Design of parallel

concatenated NLTC for optical MAC with coherent interference.

Progress: publications & presentations Trellis Codes with Low Ones Density for the OR Multiple Access Channel,

M.Griot, A.Vila Casado, W-Y Weng, H. Chan, J.Basak, E.Yablanovitch, I.Verbauwhede, B. Jalali, R.Wesel. IEEE ISIT, Seattle, 9-14 July 2006.

Presented in IEEE ISIT 2006 by Miguel Griot. Non-linear Turbo Codes for Interleaver-Division Multiple Access on the OR

Channel, M.Griot, A.Vila Casado, R.Wesel. To be presented at IEEE GLOBECOM Technical Conference 2006, Nov. 27 – Dec. 1, San Francisco.

Presentation: Demonstration of Uncoordinated Multiple Access in Optical Communications, H.Chan, A.Vila Casado, J.Basak, M.Griot, W-Y Weng, E.Yablanovitch, I.Verbauwhede, R.Wesel. 2006 43rd Design Automation Conference, July 24-28, San Francisco.

Winner of the 2006 DAC/ISSCC Student Design Contest 1st Prize award, on the Operational System Design category.

Presented by Herwin Chan. Journal Papers under preparation:

Non-linear Trellis Codes for Interleaver-Division Multiple Access on optical channels. (IEEE Trans. Telecommunications)

Includes material presented on ISIT 2006, and NL-TC codes for BAC. Non-linear Turbo Codes for Interleaver-Division Multiple Access on optical

channels. Includes material to be presented on GlobeCom 2006, and PC-NLTC

codes for BAC. (IEEE Trans. Telecommunications) Demonstration of Uncoordinated Multiple Access in Optical

Communications. Includes material presented in DAC Conference 2006.

BER analytical bound

1

ˆ ˆ

ˆ| | |

ˆ ˆ| | |

ˆmin | , |

ˆ ˆ| | | |

ˆ ˆ| | 1

ˆ ˆ| | (1 ) (1 )

ˆ ˆ

i

i

i

n n n n

n n n

n n nY

n n

nn n

i i i iY yi

i i i i i iy

i i i i i iy

n n n

P X X P X X

I P Y X P Y X P Y X

I P Y X P Y X P Y X

P Y X P Y X

P Y X P Y X P y x P y x

x x P y x P y x

x x P y x P y x

P X X P X

ˆ,

(1 ) (1 )n n

Hd X XnX

Results for 6-user MAC 6-user MAC 64-state, rate 1/30 NLTC (Sum-rate = 0.2) Coherent interference model (CI-MAC):

Z-Channel: 0.283169

0.062156

2

1

, 2

0,2 ,

( 1/ 2)

i

mj

mi

i

P e m

U i

threshold

BER bound for 6-user CI-MAC

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4510

-7

10-6

10-5

10-4

10-3

10-2

10-1

BE

RBER and Upper Bound 2 vs. on BAC, = 0.062156

BER Upper Bound 2BER for BACBER for 6-user Opt-MAC

•64-state NL-TC

Model: Coherent interference 128-state NL-TC Sum-rate = 0.2

Users p α β BER6 0.2832 0.0622

32 0.3107 0.0664104 0.3147 0.0677

Results for Optical MAC

33.846 10 56.35 10

26.667 1021.250 10

51.46 1052.71 10

Simulator Features Random data is generated and encoded The signal passes through a parameterizable

channel model Probes are placed at different point of the

receiver to see how the codes react to changes in the channel

Channel Model a and b simulate the degradation of the

transmitted signal due to interference from other transmitters

a – non-coherent combination Probability that a 0 bit turns into 1

b – coherent combination Probability that a 1 bit turns into 0

FPGA Channel Simulator

Hardware transmitter, receiver and channel model simulated on a single FPGA

Effects of changing channel parameters can be evaluated in real time

New Channel codes can be easily tested

FPGA

BER < 10-9

BER < 10-5

ChannelModel

Reed SolomonDecoder(255,237)

Trellis DecoderRate:1/20

transmitter

a b

Measurement Points

FPGA

BER < 10-9

BER < 10-5

ChannelModel

Reed SolomonDecoder(255,237)

Trellis DecoderRate:1/20

transmitter

a b

•Ones density•Channel Errors•One to zero transitions

Non-linear trellis code bit error rate

Total bit error rate

Simulation Interface

Real-time Matlab graphical user interface Real-time control of channel parameters a and b

Channel parameter selection

Real time channel

conditionsBit error rate

measurement at receiver