Interleaver-Division Multiple Access on the OR Channel

Interleaver-Division Multiple Access on the OR

Channel

Miguel Griot, Andres Vila Casado, Wen-Yen Weng, Herwin Chan, Juthika Basak, Eli

Yablanovitch, Ingrid Verbauwhede, Bahram Jalali, Rick Wesel

UCLA Electrical Engineering Department-Communication Systems LaboratoryUCLA Electrical Engineering Department-Communication Systems Laboratory

UCLA Electrical Engineering Communication Systems Laboratory 2

The OR Multiple Access Channel (OR-MAC)

If all users transmit zeros, receiver sees a zero.

Receiver

0

0

0

0

User 1

User 2

User N


The OR Multiple Access Channel (OR-MAC)

If any user transmits a one, receiver sees a one.

Receiver

0

1

1

0

User 1

User 2

User N


OR-MAC Capacity Region

All n-tuples with sum-rate ≤1 form the capacity region.

1

1

Can achieve with joint decoding using random codes.

Successive decoding is also optimal if rate-splitting is used at the encoders with jointly optimized ones densities [Grant 01].

Rat

e of

Use

r 2

Rate of User 1


Interleaver Division Multiple Access(IDMA)

Interleaver-Division Multiple Access [Ping 02] allows a single encoder/decoder pair to work for all users.

Introduced for CDMA channels, but easily generalizes to most memoryless MAC channels.

Interleaver 1

Interleaver 2

Interleaver N

EncoderUser 1

User 2

User N

Same Encoder

Still Same Encoder


Ways to use IDMA...

IDMA plus an optimal code (or codes if we have various rates) can achieve optimal sum-rate with joint decoding.

IDMA plus optimal codes and jointly optimized ones densities can achieve the optimal sum-rate with successive decoding.

IDMA plus a short block length code and single-user decoding treating all other users as noise is pretty silly…

No, it’s very practical for gigabit decoding rates.


Single-User Decoding produces a Z-Channel

0

1

0

1

11 (1 )Np

1(1 )Np

YiXp

1 p

N users, all transmitting with the same ones density p: P(X=1)=p P(X=0)=1-p

Focus on a desired user If desired user transmits a 1, a 1 will be received. A 0 will be received only if all users transmit a 0


Theoretical Sum-Rate Comparison

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)

Optimal ones densities:

Users Joint Decoding

Others as noise

2 0.293 0.286

6 0.109 0.108

12 0.056 0.056

To maximize H(Y)Output should Have 50% ones


What can I achieve with a latency of 35 bits?

The required trellis code needs to be nonlinear at the binary level to achieve a very low ones density.

A new definition of distance is required. A design approach is needed.

Interleaver 1

Interleaver 2

Interleaver N

Trellis EncoderUser 1

User 2

User N

Trellis Encoder

Trellis Encoder

Viterbi Decoder


Trellis Code Parameters

Density of ones p. Rate-1/n (1 input bit, n output bits). S=2v states (represented by v bits). 2S branches.

1 2 0, , ,v vX X X 2 0, , ,0vX X

2 0, , ,1vX X

Input 0

Input 1

State at time t:

State at time (t+1):


Controlling the ones density

There are S/2 of these sub-graphs. In this subgraph of four branches, assign a Hamming

weight of w+1 to i branches, and a Hamming weight of w to (4-i) branches.

2 00, , ,vX X

2 01, , ,vX X

2 0, , ,0vX X

2 0, , ,1vX X

Input 0

0

Input 1

1

( )

(4( ))

w floor p n

i round p n w


Example Hamming Weight Assignments

6-user Rate-1/20 Trellis Code n=20, desired p=0.125 Two branches with weight 2 Two branches with weight 3

100-user rate-1/400 Trellis Code n=400, desired p=0.0069 One branch with weight 2 Three branches with weight 3


Directional Distance

Define the directional Hamming distance as the number of ones that have to be added to a codeword so that all ones of the other codeword are present in the received word.

For two codewords with different Hamming weights, if the received word contains all ones from both codewords, the one with larger Hamming weight will be more likely than the codeword with smaller Hamming weight.

1 21 1 2

2 2 1 2 1

111011001001 ( , ) 3

110011 111011 ( , ) 1D

D

rc d c c

c r d c c


Trellis Code Design

A trellis branch belongs to many codewords, so we don’t know which direction is the important one.

Code design branch metric chooses the smaller of the two directional distances:

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


Ungerboeck’s Rule

Maximize distance between all splits and merges.

All output values have Hamming distance of w or w+1.

There are at least v+1 branches in an error event.

2 00, , ,vX X

2 01, , ,vX X

2 0, , ,0vX X

2 0, , ,1vX X

split

split2 00, , ,vX X

2 01, , ,vX X

2 0, , ,0vX X

2 0, , ,1vX X

merge

merge

min 2 1d w v


Extending Ungerboeck’s rule

One can extend Ungerboeck’s rule into the trellis.

0

1

Maximize split




0

1

Maximize

0

1

0

1


Note that by maximizing the distance between the 8 branches, coming from a split 2 trellis section before, we are maximizing all groups of 4 branches coming from a split in the previous trellis section, and all splits.



0

1Maximize

0

1

0

1


Designing for a very low desired ones density For a low enough desired ones density, all the

branches can be chosen to have maximum distance. The design becomes straight-forward.

It is possible to choose all branches so that there is at most 1 branch that has a 1 in a given position.

Straight-forward design: Assign Hamming weights to branches For each branch, add ones in positions that aren’t

used in previous branches Example: 100-user OR-MAC,

0078.0)2/(10069.0 Sp


Trellis Code Simulations, Bounds, FPGA Results

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

6-user BER 10-5


Results: observations

An error floor can observed for the FPGA rate-1/20 NL-TCM. This is mainly due to the fact that, while

theoretically a 1-to-0 transition means an infinite distance, for implementation constraints those transitions are given a value of 20.

Trace-back depth of 35. Additional coding required to lower BER to

below 10-9.


C-Simulation Performance Results: 6-user OR-MAC

4 5 6 7 8

10-6

10-5

10-4

10-3

users

BE

R

NL-TCM 1/17

NL-TCM 1/18NL-TCM 1/20

6-user BER 10-5


Could do Joint Decoding… Results for Rate-1/20 p=0.125 code.

Number of Users Sum Rate Trellis Code BER

6 0.3 No Errors

8 0.4 2.7e-7

10 0.5 2.0e-5

12 0.6 4.0e-4


C-Simulation Performance Results: NL-TCM only, 100-user OR-MAC

Rate Sum-rate p BER

1/360 0.2778 0.006944 0.49837

1/400 0.25 0.006875 0.49489

64.54 10

79.45 10


Reed-Solomon + NL-TCM : 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. The resulting rate for each user is (247/255).(1/20) The results were obtained using a C program to apply

the RS-code to the FPGA NL-TCM output.

Rate Sum-rate p BER

0.0484 0.29 0.125 0.4652

102.48 10


Optical MAC Coding Architecture

OR channel

5 other tx

Reed Solomon(255, 237)

Trellis Code1/20

int

Correct extra errors

AsychronousAccess code

Separate different transmitters

2 Gbps

93 Mbps


Demonstration

FPGA Transmitters

FPGA with both Transmitter and Receiver (slide 29)


Conclusions

We presented an Interleaver Division Multiple Access system for the OR-MAC.

A relatively low ones density is essential for the OR-MAC channel.

We presented a single-user trellis coded solution.

An arbitrary number of users is supported, maintaining relatively the same efficiency (around 30%).

Joint decoding for six-users increases efficiency to 50-60%.

Although these codes are not capacity achieving,a good part of the capacity is achieved, with a suitable BER for optical needs, and a complexity feasible for optical speeds with today’s technology. An FPGA implementation has been built to prove this fact.

Turbo coded single-user implementation (week supports) 60%.


Future work: Capacity achieving codes

Capacity achieving codes. Although they may not be feasible for optical speeds,

with today’s technology, Turbo codes and LDPC codes will be feasible in the near future

Part of my immediate future’s work will be the design Turbo-Like codes, with an arbitrary ones density.

Most common Turbo-like codes are Parallel concatenation of convolutional codes Serially concatenated convolutional codes.

The convolutional codes will be replaced by properly designed NL-TCMs.


Non-linear Turbo Like codes

Serial concatenation CC + NL-TCM:

Parallel concatenated NL-TCMs:

CC Interleaver NL-TCM

NL-TCM

Interleaver NL-TCM


Code details

For all implementations, states were used. 6-user OR-MAC

n=20 : Sum-rate = 0.30 2 branches with Hw=2, 2 with Hw=3 (p=1/8). h=3, g=2 :

n = 18 : Sum-rate = 1/3 3 branches with Hw=2, 1 with Hw=3 (p=1/8). h=2,g=2 :

n = 17 : Sum-rate = 0.353 all with Hw=2 (p=2/17=0.118). h=2,g=2 :

100-user OR-MAC, n = 400 : w=2, i=3 (p = 0.006875) n = 360 : w=2, i=2 (p = 0.006944) for both cases

62 64

min 2 5 (6 1) 5 12d

min 11d

min 11d

min 14d

Interleaver-Division Multiple Access on the OR Channel

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