Design of Long Optimal Interleavers for Turbo Codes

1
Design of Long Optimal Interleavers for Turbo
Codes

• Nancy B. List
• Advisor Douglas B. Williams

2
Historical Background

• In 1948 Shannon published his famous paper A
  Mathematical Theory of Communication.
• Transmitting at rates lt C reliable
  communication
• Transmitting at rates gt C unreliable
  communication
• Codes exist for which the BER in the received
  sequence can be made arbitrarily small

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What are Turbo Codes?

• Turbo codes are a class of error correcting codes
  codes introduced in 1993 that come closer to
  approaching Shannons limit than any other class
  of error correcting codes.
• Turbo codes achieve their remarkable performance
  with relatively low complexity encoding and
  decoding algorithms.

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Building Blocks of Turbo Codes

• 2 recursive convolutional encoders
• 1 interleaver
• These elements can be connected in parallel or in
  series.

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Parallel Configuration

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Series Configuration

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Recursive Convolutional Encoders

• 3-Delay-State Recursive Convolutional Encoder

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Properties of Recursive Convolutional Encoders

• Infinite impulse response
• Most input sequences are associated with high
  weight parity sequences.

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Decoding Algorithm

• The input bits to the individual encoders are
  decoded separately using a soft-output APP
  decoding algorithm.
• Information about the decoded input bits is
  passed iteratively between the two decoders.
• The decoding algorithm minimizes the bit error
  probability of the input sequence.

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Block Diagram of Decoder

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Interleaver

• The interleavers function is to permute low
  weight code words in one encoder into high weight
  code words for the other encoder.
• Most input sequences are associated with parity
  sequences that are not self-terminating.
• Input sequences with self-terminating parity
  sequences form terminating code words.

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Interleaver

• Input sequences which form terminating code words
  in both encoders form the low weight code words
  of a turbo coding scheme.
• Error patterns corresponding to low weight code
  words are the least likely to be corrected by the
  decoder.
• We have focused our interleaver design method on
  permuting terminating code words into
  non-terminating code words.

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Recursive Convolutional Encoder as a Dynamical
System

• A recursive convolutional encoder can be
  represented as a dynamical system

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Recursive Convolutional Encoder as a Dynamical
System

• In practice,there are a finite number, N, of bits
  to be encoded.
• We can think of these bits as being placed into a
  length-N vector m to be queued into the encoder.

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Recursive Convolutional Encoder as a Dynamical
System

• Alternatively, the encoder can be represented as
  an autonomous dynamical system

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Recursive Convolutional Encoder as a Dynamical
System

• The final state of the encoder after a length-N
  message has been encoded is given by

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Recursive Convolutional Encoder as a Dynamical
System

• The final state of the encoder after a length-N
  message has been encoded is given by
• where

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Impulse Response of Delay States

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Terminating Code Words

• Terminating code words leave the encoder in the
  zero state after all their bits have been
  encoded.
• Thus, the following is satisfied if the input
  sequence in m results in a terminating code word

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Terminating Code Words

• We see that terminating code words constitute the
  dual space to the vector space spanned by the
  delay state impulse response vectors.
• Similarly, we can show that at least
• of the terminating code words cannot be
  interleaved into non-terminating code words.

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Interleaving by Sub-vector

• The relationship between terminating code words
  and the periodic impulse response of the delay
  states leads to our new method of interleaver
  design.
• We call this method interleaving by sub-vector.

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Interleaving by Sub-vector

• As before, if m corresponds to a terminating code
  word, then
• If then

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Interleaving by Sub-vector

• If we are able to permute
• such that
• then

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Interleaving by Sub-vector

• If P interleaves the maximum fraction of length-7
  terminating code words, then it can be shown that
  it interleaves the maximum fraction of length-N
  terminating code words.
• P is optimal in the sense that it
  orthogonalizes the two encoders to the largest
  degree possible.
• In order to reduce the BER of our coding scheme,
  we need to be target certain terminating code
  words by our interleaver.

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Bounds on BER of Turbo Codes

• We use the idea of a uniform random interleaver
  to calculate the probability of a bit error due
  to a low-weight error pattern in a turbo coding
  scheme.
• Def A uniform random interleaver maps a
  weight-d input sequence to one encoder into all
  possible weight-d input sequences for the second
  encoder with equal probability.

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Bounds on BER of Turbo Codes

• The lowest weight-d input sequence corresponding
  to a terminating code word is weight-2.

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Weight-2 Input Sequences

• Error patterns corresponding to terminating code
  words with weight-2 input dominate the BER at low
  SNRs.
• We need to design our interleaver to permute
  these code words into non-terminating code words
  for the other encoder.

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Distance Spectrum Interpretation of BER

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Optimal Long Interleavers

• By interleaving over a sub-vector corresponding
  to more than one period of the delay state
  impulse response, we can
• interleave all low weight code words with
  weight-2 input sequences into non-terminating
  code words, and
• increase the free distance of the coding scheme.

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Interleaving by Sub-vector

• Interleaving by sub-vector effectively permutes
  the columns of a matrix with information bits
  read in row-wise.
• To reduce the probability of error due to the
  terminating code words that are not interleaved,
  we perform random interleaving over the columns
  of this matrix.
• Random column interleaving is the key to being
  able to scale up the interleaver to any length.

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Results

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Results

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Future Work

• The method of interleaving by sub-vector can also
  be used with serially concatenated turbo codes.
• Bit error rates of serially concatenated turbo
  codes are inversely proportional to higher powers
  of the interleaver length
• Using the method of interleaving by sub-vector,
  it should be possible to make the BER decrease as

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Bounds on BER of Serially Concatenated Turbo Codes

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Bounds on BER of Serially Concatenated Turbo Codes

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Bounds on BER of Serially Concatenated Turbo Codes