Non-Linear Codes for Asymmetric Channels, applied to Optical Channels

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Non-Linear Codes for Asymmetric Channels, applied
to Optical Channels

• Miguel Griot

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Outline

• 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
• Characteristics of Trellis Codes
• Design Technique
• Results for 6-user OR-MAC 100-user OR-MAC
• Concatenation with High-Rate Block Codes
• Results for 6-user OR-MAC
• 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

Receiver
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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 thoroughly studied in the
  literature are linear codes.
• We observed in previous slide that, 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. Theoretical
  bounds are found to prove that these codes are
  capacity achieving under ML decoding. There is no
  design technique described for these codes.
• Non-linear Trellis Coded Modulation
• This work introduces a novel design technique for
  non-linear trellis codes with an arbitrary
  density of ones.
• To my knowledge, it is the first work that
  addresses this task.

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Interleaver Division Multiple Access

• One successful approach to uncoordinated multiple
  access is 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
  memory-less Z-Channel.
• Performance obtained for a 6-user OR-MAC using
  IDMA, and for the corresponding Z-Channel were
  the same in my simulations.

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Trellis Codes Characteristics

• Memory given by a state. In the trellis
  representation, for each state, and each possible
  input, an output value and the next state is
  given.
• Generally next state and output given by
  generator polynomials.
• Initial state the all-zero state.
• Zero Termination.
• They are NOT capacity achieving
• We are achieving around 30 of full capacity
  (around 43 of the achievable rate when treating
  other users as noise)
• Low complexity compared to capacity achieving
  codes (Turbo-Codes, LDPC)
• ML decoding Viterbi Decoding

State at time (t1)
0010
State at time t
1100
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Metric for the Z-Channel, for Maximum Likelihood
decoding

• Given a received word, the decoded codeword will
  be the one that maximizes , or given
  equally likely codewords .
• For the Z-Channel, if one codeword has a 1
  in a position where the received word r has a 0,
  then
• Among the possible transmitted codewords (where
  there are no 1-to-0 transitions)
• where is the number of 0-to-1 transitions,
  and is the number of 0-to-0 transitions
• Note that for the possible transmitted
  codewords is actually the number of zeros in the
  received word, which is the same for all possible
  codewords.
• Now, , so the most likely codeword is
  the one that presents the less number of 0-to-1
  transitions.

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Definition of distance for the Z-Channel

• The distance between two codewords measures the
  likelihood that one transmitted codeword will be
  wrongly decoded as the other codeword.
• In the Z-Channel, a transmitted 1 will always
  induce a received 1.
• Define the directional Hamming distance
  as the
  number of ones that have to be added to a
  codeword so that all ones of codeword are
  present in the received word.
• Example
• Now

• A Maximum-Likely (ML) decoder will always decode
  the codeword with larger Hamming weight.

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Definition of distance for the Z-Channel (2)

• For two codewords with different Hamming weight,
  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 .
• Only if is transmitted, an error will be
  produced in the decoder.
• Then, the directional distance of interest is
• which is the larger of both directional Hamming
  distances.
• For two codewords with equal Hamming weight,
  errors can be made in both directions, and both
  directional Hamming distances are equal, and
  equal to the maximum of both.
• In any case, the proper pairwise deign metric is
  
• And the overall objective is to maximize

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Greedy definition of distance

• In a trellis code, the design is made
  branch-wise for each state, and each input, we
  assign the next state, and the output.
• Due to its non-linearity, last definition cannot
  be applied branch-wise.
• It is impossible to tell from one branch, which
  codeword will have more Hamming weight.
• Hence, we have to consider both branch-wise
  directional Hamming distances.
• The safest branch-wise metric would be
• This is the definition used in our design of
  NL-TCM.

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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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Choosing all branches to have at least distance
of 1 between each other

• It would be desirable if possible, that all
  branches had at least distance of 1 between each
  other.
• In the case where all branches have the same
  Hamming weight w then we can have up to
  different branches with a distance of at least 1.
  
• If , it is possible.
• In the case of branches with different Hamming
  weights, the computation is a little more
  complicated.
• Two different codewords split at some point in
  their trellis paths, and their paths will not
  merge again until at least v1 trellis sections
  after the split.
• In case it is possible to have different output
  values for all branches, then the minimum
  distance of the code is lower-bounded by

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Choosing all branches to have at least distance
of 1 between each other

• In case we need to repeat output values, we can
  allow the following branches to have same output
  value maintaining the bound
• Again consider the sub-graph
• Branches in red (blue) can have same output value
  without affecting the minimum distance.
• Consider two different paths, one traversing
  branch A, and the other traversing branch B at
  some trellis section.
• They traverse at least v branches with different
  output values before that trellis section.
• They traverse at least v branches with different
  output values after that trellis section.
• Their distance is at least 2v.

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Ungerboecks rule

• We have already assigned Hamming weights to the
  branches, and have enumerated all the possible
  output values in order to have different output
  values for all branches (allowing some branches
  to be equal according to previous slide)
• Up to this point we have
• We can further 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

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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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Extending Ungerboecks rule

• The same idea applies for the merges, moving
  backwards in the trellis.
• If we move h trellis sections forward from a
  split (including the split), and g sections
  backwards from a merge (including the merge), the
  new bound becomes
• Now, we have to compute the maximum possible
  values of h and g.

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Extending Ungerboecks rule

• First, lets compute the number of branches that
  need to have maximum distance between each
  otherto cover h sections from a split, and g
  sections backwards from a merge.
• From a splitting point of view
• From the merging point of view
• Each branch, belongs to one group of and one
  group of
• Thus, each branch has to have maximum distance
  with other branches.

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Extending Ungerboecks rule

• Second, we can compute how many branches of
  maximum distance between each other we can have.
  Lets denote this number T.
• For branches with equal Hamming weight w,
• In the general case
• The constraints are
• Each branch has to belong to a group of and
  a group of
• If we choose all constraints
  are satisfied

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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.
• Consider a NL-TCM code with S states, desired
  density p.
• Denote M the sum of all the ones from the outputs
  of all 2S branches.
• Then
• But if then
• It is possible to choose all 2S 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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Performance results

• FPGA implementation
• In order to prove that NL-TCM codes are feasible
  today for optical speeds, a hardware simulation
  engine was built on the Xilinx Virtex2-Pro 2V20
  FPGA.
• Results for the rate-1/20 NL-TCM code are shown
  next.
• Transfer Bound
• Wen-Yen Weng collaborated to this work, with the
  computation a Transfer Function Bound for NL-TCM
  codes.
• It proved to be a very accurate bound, thus
  providing a fast estimation of the performance of
  the NL-TCM codes designed in this work.

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Performance Results 6-user OR-MAC
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Performance Results 6-user OR-MAC
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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.
• The BER for all cases are not as low as required.

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Performance Results 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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Dramatically lowering the BER Concatenation
with Outer Block Code

• Optical systems deliver a very low BER, in our
  work a was required.
• Using only a NL-TCM, the rate would have to be
  very low.
• A better solution is found using the fact that
  when the Viterbi decoding fails, with relatively
  high probability only a small number of bits are
  in error.
• Thus, a high-rate block code that can correct a
  few errors can be attached as an outer code,
  dramatically lowering the BER.

Block-Code Encoder
NL-TCM Encoder
Z-Channel
Block-Code Decoder
NL-TCM Decoder
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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.
• Although we dont have results for the 100-user
  case, it may be inferred that a similar BER would
  be achieved.

Rate Sum-rate p BER
0.0484 0.29 0.125 0.4652
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Conclusions

• I developed 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.
• 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 (1) 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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Non-linear Turbo-like codes

• The NL-TCM will not be a feed-forward encoder.
• The design criteria changes.
• However, the fundamental ideas hold.
• The fact that the RSNL-TCM concatenation
  (hard-decision transmitted from one decoder to
  the other) has such a good BER, makes the serial
  concatenation of CCNL-TCM with soft-decoding
  look promising.

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Future Work(2) More general Channel

• Also, to be more general, I will study the
  Multiple access channel where the 110 case, has
  a positive (although very small) probability.
• Treating other users as noise, one user sees an
  Binary Asymmetric Channel.
• This will be change the metric in the Viterbi
  decoder, the definition of distance used, but
  shouldnt change the design criteria