Interleaver-Division Multiple Access on the OR Channel

1
Interleaver-Division Multiple Access on the OR
Channel
UCLA Electrical Engineering Department-Communicati
on Systems Laboratory

• Miguel Griot, Andres Vila Casado, Wen-Yen Weng,
  Herwin Chan, Juthika Basak, Eli Yablanovitch,
  Ingrid Verbauwhede, Bahram Jalali, Rick Wesel

2
The OR Multiple Access Channel (OR-MAC)
0
User 1
0
User 2
0
Receiver
0
User N

• If all users transmit zeros, receiver sees a zero.

3
The OR Multiple Access Channel (OR-MAC)
1
User 1
0
User 2
1
Receiver
0
User N

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

4
OR-MAC Capacity Region

• 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.

Rate of User 2
Rate of User 1

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

5
Interleaver Division Multiple Access(IDMA)
Interleaver 1
Encoder
User 1
Interleaver 2
User 2
Same Encoder
Interleaver N
Still Same Encoder
User N

• 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.

6
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, its very practical for gigabit decoding
  rates.

7
Single-User Decoding produces a Z-Channel

• N users, all transmitting with the same ones
  density p
• P(X1)p
• P(X0)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

8
Theoretical Sum-Rate Comparison
Optimal ones densities
Users Joint Decoding Others as noise
2 0.293 0.286
6 0.109 0.108
12 0.056 0.056
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What can I achieve with a latency of 35 bits?
Interleaver 1
Trellis Encoder
User 1
Interleaver 2
User 2
Trellis Encoder
Viterbi Decoder
Interleaver N
Trellis Encoder
User N

• 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.

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Trellis Code Parameters
State at time (t1)
Input 0
State at time t
Input 1

• Density of ones p.
• Rate-1/n (1 input bit, n output bits).
• S2v states (represented by v bits).
• 2S branches.

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Controlling the ones density

• There are S/2 of these sub-graphs.
• In this subgraph of four branches, assign a
  Hamming weight of w1 to i branches, and a
  Hamming weight of w to (4-i) branches.

12
Example Hamming Weight Assignments

• 6-user Rate-1/20 Trellis Code
• n20, desired p0.125
• Two branches with weight 2
• Two branches with weight 3
• 100-user rate-1/400 Trellis Code
• n400, desired p0.0069
• One branch with weight 2
• Three branches with weight 3

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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.

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Trellis Code Design

• A trellis branch belongs to many codewords, so we
  dont know which direction is the important one.
• Code design branch metric chooses the smaller of
  the two directional distances

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Ungerboecks Rule

• Maximize distance between all splits and merges.
• All output values have Hamming distance of w or
  w1.
• There are at least v1 branches in an error event.

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Extending Ungerboecks rule

• One can extend Ungerboecks rule into the trellis.

0
1
Maximize split
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Extending Ungerboecks rule

• One can extend Ungerboecks rule into the trellis.

0
0
1
1
0
1
Maximize
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Extending Ungerboecks rule

• One can extend Ungerboecks rule into the trellis.

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
0
1
1
Maximize
0
1
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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
  arent used in previous branches
• Example 100-user OR-MAC,

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Trellis Code Simulations, Bounds, FPGA Results
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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.

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C-Simulation Performance Results 6-user OR-MAC
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Could do Joint Decoding Results for Rate-1/20
p0.125 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
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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
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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
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Optical MAC Coding Architecture
2 Gbps
Reed Solomon (255, 237)
Trellis Code 1/20
int
OR channel
93 Mbps
5 other tx
Correct extra errors
Separate different transmitters
Asychronous Access code
27
Demonstration
FPGA with both Transmitter and Receiver (slide
29)
FPGA Transmitters
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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 todays
  technology. An FPGA implementation has been built
  to prove this fact.
• Turbo coded single-user implementation (week
  supports) 60.

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Future work Capacity achieving codes

• Capacity achieving codes.
• Although they may not be feasible for optical
  speeds, with todays technology, Turbo codes and
  LDPC codes will be feasible in the near future
• Part of my immediate futures 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.

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Non-linear Turbo Like codes

• Serial concatenation CC NL-TCM
• Parallel concatenated NL-TCMs

CC
Interleaver
NL-TCM
NL-TCM
Interleaver
NL-TCM
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Code details

• For all implementations, states were
  used.
• 6-user OR-MAC
• n20 Sum-rate 0.30
• 2 branches with Hw2, 2 with Hw3 (p1/8).
• h3, g2
• n 18 Sum-rate 1/3
• 3 branches with Hw2, 1 with Hw3 (p1/8).
• h2,g2
• n 17 Sum-rate 0.353
• all with Hw2 (p2/170.118).
• h2,g2
• 100-user OR-MAC,
• n 400 w2, i3 (p 0.006875)
• n 360 w2, i2 (p 0.006944)
• for both cases