Joint Decoding on the OR Channel

Communication System Laboratory

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

Communication Systems Laboratory

Joint Decoding Architecture

Decoding is done by performing belief propagation over the receiver graph

Performs well at very high Sum-Rates

High decoding complexity for a large number of users

Requires either bit synchronism or timing knowledge of all the transmitters

Encoder 2

Encoder 1

Encoder N

Interleaver 1

Interleaver 2

Interleaver N

Same CodeRandomly picked (different with very high probability)

ElementaryMulti-User

Decoder(Threshold)

Interleaver 1

Interleaver N

De-Interl 1

De-Interl N

DecoderDEC-1

DecoderDEC-N

Joint Decoding Results

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.910

-6

10-5

10-4

10-3

10-2

Sum-Rate

BE

R

Joint DecodingJoint Decoding With Block Code

6 users

Turbo Codes for the OR Channel

Communication System Laboratory

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

Communication Systems Laboratory

Parallel Concatenated NL-TCs Increases complexity and latency with respect

to NLTC. 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

OR Channel when treating other users as noise:

Can we provide the same sum-rate and performance for any number of users?

Communication Systems Laboratory

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

Communication Systems Laboratory

Theoretical answer

Theoretically: YES.

2 3 4 5 6 7 8 9 10 11 120

0.2

0.4

0.6

0.8

1

Number of users

Cap

acity

Sum rate comparison

Other users as noise

Joint decodingp

1 = 0.5

ln(2)

Our Experience: NL-TCM NL-TCM: looked like we don’t have a limit in

the number of users. Results for 100 user case:

And we were right in that case.

0.498370.0069440.27781/360

0.494890.0068750.251/400

BERpSum-rateRate 64.54 10

79.45 10

Comparison: Number of output bits n0 & number of ones M vs number of users

Comparison: n0(N) & M(N)

Number of ones is increasing

increasing

Comparison: n0(N) & M(N)

Same number of ones.Ungerboeck’s extension: moving deeper into the trellis.

increasing

Comparison: n0(N) & M(N)

Best code at this point….

All branches different

increasing

Comparison: n0(N) & M(N)

is the best code at this point.

increasing

v=6

N n0 SR BER

100 344 0.291 0.4777

300 1000 0.3 0.4901

900 3000 0.3 0.4906

1500 5000 0.3 0.4907

15000 50000 0.3 0.4908

51.1046 1051.2157 1051.2403 1051.2508 1051.2508 10

We can support any number of users in the OR-MAC with basically same decoding complexity for each user, and practically same performance.

Moreover:

Unused bits(Bunch of zeros)

Moreover: Denote N* the minimum number of users for

which n0 > M.

For every N greater than N* we can use the same encoder and decoder

Design for N* .

Encoder Add Zeros Interleaver

De-InterleaverDelete unused bits

Decoder

Limitation for Non-linear Turbo Codes With 8-state constituent non-linear trellis

codes:

16-state constituent non-linear trellis codes should be used for more than 24 users.

Results

6 users

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

With 16-state constituent NL-TCs

For 50 users:For 100 users:

0.4930 0.4965

Around 50 users should be supported.

Code design for the Binary Asymmetric Channel

Communication System Laboratory

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

Communication Systems Laboratory

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

Simulations

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

Simulations

JD : Joint DecodingOUN: Other Users Noise

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

0.95

Model: m

= + (-)(1-e-(m-2)) for m>=2

number of users

Sum

-rat

eJD = 0.23, = 0.001, = 0.05

JD = 0.16, = 0.0001, = 0.01

OUN = 0.23, = 0.001, = 0.05

OUN = 0.16, = 0.0001, = 0.01

Simulations

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

0.95

Model: m

= ((m-2)) for m>=2

number of users

Sum

-rat

e

JD = 0.23, = 0.8

JD = 0.23, = 0.4

JD = 0.16, = 0.8

JD = 0.16, = 0.8

OUN = 0.23, = , 0.8

OUN = 0.23, = 0.4

OUN = 0.16, = 0.8

OUN = 0.16, = 0.4

Simulations

0 20 40 60 80 100 120 140 160 180 2000.4

0.5

0.6

0.7

0.8

0.9

1

Model: m

= , constant

number of users

Sum

-rat

eJD = 0.3

JD = 0.23

JD = 0.15

OUN = 0.3

OUN = 0.23

OUN = 0.15

Simulations

0 20 40 60 80 100 120 140 160 180 2000.4

0.5

0.6

0.7

0.8

0.9

1

Model: 2 = ,

m = 0 for m>2

number of users

Sum

-rat

eJD = 0.4

JD = 0.3

JD = 0.23

JD = 0.15

OUN = 0.4

OUN = 0.3

OUN = 0.23

OUN = 0.15

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

Lower bounds for Sum-rate (1) Joint Decoding:

n users, with equal ones density p. Using Then:

For the worst case ( constant) the bound is actually very tight.

Note that for the case where

Also note that if (OR channel) , the lower bound becomes 1 for .

1/1 , 1/ 2np

1/ 2,1, ( ) max ( ) max ( ) max ( )

( ) 1 (1 log( ))

m m m mn SR n H f f H

f

'm s

max 1/ 2 max ( ) (max ) ( )m m M m m m m MH H H

1/ 2,1, ( ) max ( ) ( ) ( )M Mn SR n H f f H 0M

1/ 2

Lower bounds for Sum-rate (2) Treating other users as noise:

n users, with equal ones density p. Using Then:

For the worst case ( constant) the bound is again very tight.

Note that if (OR channel) , the lower bound becomes log(2) for .

1/1 , 1/ 2np

21/ 2,1

( )

1 ( )max log(1/ ) log(1/ ) 1 log

( )

log(1/ ) ( ) log(1/ ) (1 )

( ) 1 log( )

BAC

M

M

M

SR n

g

g

H g H

g

'm s

0M 1/ 2

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

Lower bound for Sum-rate For the Binary Asymmetric Channel, there is still a

strictly positive achievable sum-rate for any number of users.

For the Coherent Interference Model, the lower bound for the achievable sum-rate is around 48% (vs. 70% for Z-Channel).

Our target sum-rate for Non-linear trellis codes is 20% (vs. 30% for Z-Channel).

For Parallel Concatenated NL-TCs, our goal will be to achieve a sum-rate of 40%. These codes are under design.

Metric for Z-Channel We use a ‘greedy’ definition of distance (not

the usual Hamming distance). Directional distance between two codewords

(denoted ) is the number of positions at which has a 0 and has a 1.

‘Greedy’ definition of distance:

1 2( , )Dd c c

2c1c

, , min , , ,i j j i D i j D j id c c d c c d c c d c c

Design of NL-TCM for the BAC The metric of the Viterbi decoder for the BAC is:

Where and are the number of 0-to-1 and 1-to-0 transitions from the codeword and the received word , respectively.

The decoded codeword is: The directional distance between two codewords

(denoted ) is the number of positions at which has a 0 and has a 1.

Both directional distances are relevant when computing the probability of error.

A good criteria is maximize the minimum of both directional distances:

This is exactly the same criteria used for NL-TCM codes for the Z-Channel

01 10

1 1| log logn nX Y N N

arg min |nn n n

XX X Y

1 2( , )Dd c c1c 2c

01N 10NnX

nY

, , min , , ,i j j i D i j D j id c c d c c d c c d c c

Design of NL-TCM for the BAC Hence, although the metrics in the Viterbi

decoder are different on the Z-Channel and the BAC, we use the same design technique for both cases.

However, since the achievable rate is lower for the BAC, our target rate will be lower.

We have designed codes for the Coherent Interference Model. Nevertheless, this design technique applies to any model for the 1-to-0 transition probabilies.

Design of NL-TCM for the BAC Results (so far):

6-user MAC 128-state, rate 1/30 NLTC (Sum-rate = 0.2) Coherent interference model.

In order to achieve the same BER than in the OR Channel case:

The number of states had to be increased from 64 to 128 (Increase in complexity).

The sum-rate was decreased from 0.3 to 0.2. Simulations for larger number of users are running. Parallel concatenated NL-TCs are being designed for

this channel.

5

0.283169

0.062156

1.044 10BER