Uncoordinated Optical Multiple Access using IDMA and Nonlinear TCM

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Uncoordinated Optical Multiple Access using IDMA
and Nonlinear TCM
UCLA Electrical Engineering Department-Communicati
on Systems Laboratory

• PIs Eli Yablanovitch, Rick Wesel, Ingrid
  Verbauwhede, Bahram Jalali, Ming Wu
• Students whose work is discussed here
• Juthika Basak, Herwin Chan, Miguel Griot, Andres
  Vila Casado, Wen-Yen Weng

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Outline of more detailed discussion

• Motivation Optical Channel, Uncoordinated
  Multiple Access.
• Models and Capacity Calculation
• Basic Model the OR Channel
• Treating other users as noise
• Capacity loss vs. complexity reduction.
• The Z channel
• The need for non-linear codes
• Optimal ones density
• Non-linear Trellis Coded Modulation (NL-TCM)
• Definition of distance in the Z-Channel
• Design Technique
• Conclusions
• Future Work

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Motivation Optical Channels, Multiple Access

• Optical Channels
• provide very high data rates, up to tens to
  hundreds of gigabits per second.
• Typically deliver a very low Bit Error Rate
• Wavelength Division (WDMA) or Time Division
  (TDMA) are the most common forms of Multiple
  Access today.
• However, they require considerable coordination.
  
• Objective
• Uncoordinated access to the channel.
• Apply error correcting codes, in order to achieve
  the required BER.
• Maximizing the rate at feasible complexity for
  optical speeds.

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Basic Model The OR Multiple Access Channel
(OR-MAC)

• OR Channel model
• Basic model that can describe the multiple-user
  optical channel with non-coherent combining
• N users transmitting at the same time
• If all users transmit a 0, then a 0 is received
• If even one of them transmits a 1, a 1 is
  received
• 0XX, 1X1

User 1
User 2
Receiver
User N
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OR Channel Theoretical characteristics

• Achievable rate (Capacity)
• The theoretical limits for the MAC, were given by
  Liao and Ahslwede.
• In the case of the OR-MAC, the Theoretical
  Capacity is the triangle of all rate-pairs less
  than the maximum possible sum-rate, which is 1.
• This sum-rate can be theoretically achieved by
• Joint Decoding.
• Sequential decoding (requires coordination).
• Time-Sharing or Wave-length sharing (requires
  coordination).

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Treating other users as noise the Z-Channel

• Joint Decoding and Successive Decoding are fully
  efficient in that one useful bit of information
  is transmitted per time-wavelength slot.
• However, non of these are computationally
  feasible for optical speeds today.
• A practical alternative is to treat all but a
  desired user as noise.
• This alternative, while dramatically reducing the
  decoding complexity, looses up to 30 of full
  capacity, as we will see next.
• When treating other users as noise in an OR-MAC,
  each user sees what is called the Z-Channel.
• My research has been focused on the Z-Channel,
  resulting from the OR-MAC when treating other
  users as noise.

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The 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 it transmits a 1, a 1 will be received.
• If it transmits a 0, a 0 will be received only if
  all other N-1 users transmit a 0

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Maximum achievable sum-rate, when treating other
users as noise.

• Information Theory tells us the optimal ones
  density to transmit for each user.
• When the number of users tends to infinity, the
  optimal ones density tends to
  , which is also the optimal density for joint
  decoding.
• In that case equal probabilities of 1 and 0 is
  perceived at the receiver.
• Note that for a large number of users, the
  optimal ones density becomes very small.
• Surprisingly, the maximum achievable sum-rate is
  always lower-bounded by ln(2)0.6931 and tends to
  ln(2) when the number of users tends to infinity.

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Comparison of capacities
Optimal ones densities
Users Joint Others noise
2 0.293 0.286
6 0.109 0.108
12 0.056 0.056
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The need for non-linear codes

• Linear codes provide equal density of ones and
  zeros in their output (p0.5).
• Most of the codes studied in the literature are
  linear codes.
• For linear codes, the achievable rate tends to
  zero as the number of users increase.
• As the number of users increase, the optimal ones
  density tends to zero.
• Non-linear codes with relatively low density of
  ones are required, to a achieve a good rate.
• Only recently, there has been work on LDPC codes
  with arbitrary density of ones. There is still no
  design technique described for these codes, and
  they cant be decoded at optical speeds today.
• This work introduces a novel design technique for
  non-linear trellis codes with an arbitrary
  density of ones.

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Interleaver Division Multiple Access (IDMA)

• Every user has the same channel code, but each
  users code bits are interleaved by a randomly
  drawn interleaver, with very high probability of
  being unique.
• The receiver is assumed to know the interleaver
  of the desired user.
• With IDMA in the OR-MAC, a receiver should see
  the signal from a desired user, corrupted by a
  memoryless Z-Channel.
• Performance obtained for a 6-user OR-MAC using
  IDMA, and for the corresponding Z-Channel were
  the same in our C simulations.

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Non-linear Trellis Coded Modulation

• Desired density of ones p is given
• Rate of the form 1/n (1 input bit, n output
  bits).
• states (represented by v bits)
• 2S branches
• Feed-forward encoder with 1 input
• Design
• Assign output values to the 2S branches of the
  trellis
• Objective Maximize the minimum distance (greedy
  definition)
• Those outputs have to maintain the desired
  density of ones p.

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Assigning Hamming Weights

• First step assign Hamming weights to the output
  of each branch.
• Using any of the definitions of distance given
  before, codewords with as equal Hamming weight
  between each other lead to better performance.
• In the case of codewords with different Hamming
  weights, the worst-case performance will be
  driven by those codewords with smaller Hamming
  weight.
• Criteria assign as similar Hamming weights to
  the branches as possible, maintaining the density
  of ones as close to the desired density of ones
  as close to the desired p as possible.

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Assigning Hamming Weights

• Consider the following sub-graph
• There are S/2 of these sub-graphs.
• Branches produced by an input bit equal to 0 for
  both states (or 1) go to the same state.
• Define
• In this subgroup of four branches, assign a
  Hamming weight of w1 to i branches, and a
  Hamming weight of w to (4-i) branches.

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Assigning Hamming Weights, Examples

• 6-user OR-MAC, desired density of ones is
  .
• n20 w2, i2
• 2 branches with Hw2, 2 with Hw3 (p1/8).
• n 18 w2, i1
• 3 branches with Hw2, 1 with Hw3 (p1/8).
• n 17 w2, iround(0.5)
• 1 branch with Hw3 and 3 with Hw2 (p0.132)
• all with Hw2 (p2/170.118).
• 100-user OR-MAC,
• n 400 w2, i3 (p 0.006875)
• n 360 w2, i2 (p 0.006944)

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Ungerboecks rule

• We can increase the minimum distance by applying
  Ungerboecks rule maximize the distance between
  all splits and merges.
• Remember that all output values had at least a
  Hamming distance of w.
• For every two different codewords, their paths
  split and merge at least once, and there are at
  least v-1 branches between the split and the
  merge.
• Hence Ungerboecks rule delivers

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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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Performance Results

• 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

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Conclusions

• A novel design technique for non-linear trellis
  codes, that provide a wide range of ones density.
• These codes have been designed for the Z-Channel,
  that arises in the optical multiple access
  channel with IDMA.
• A relatively low ones density is essential for
  the OR-MAC channel, and asymmetric channels in
  general.
• An arbitrary number of users is supported,
  maintaining relatively the same efficiency
  (around 30)
• Although these codes are not capacity achieving,a
  good part of the capacity is achieved, with a
  suitable BER fr optical needs, and a complexity
  feasible for optical speeds with todays
  technology. An FPGA implementation has been built
  to prove this fact.

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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