Constellation Shaping for Pragmatic Binary Turbo Coded Modulation

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Constellation Shaping for Pragmatic Binary Turbo
Coded Modulation

• Assaf Gurevitz
• Under the supervision of Dr. Danny Raphaeli

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Outline

• Motivation
• Previous work
• Proposed shaping algorithm
• System structure
• Performance comparison
• Summary and conclusions
• Future work

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Motivation

• We would like to apply constellation shaping to a
  turbo code for coding with high spectral
  efficiency.
• It is well known from information theory that the
  capacity of the AWGN is achieved for
• Our goal is performance close to channel
  capacity via a non uniform discrete distribution
  combined with a turbo code.

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Turbo coded modulation techniques
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Previous work

• Before Turbo codes
• Gallager (1968) showed that binary codes can be
  used for assignment of nonequiprobable discrete
  distributions that achieve capacity.
• Forney (1989) used the idea of an infinite
  lattice code to show that the maximal achievable
  shaping gain is that of an infinite sphere, which
  equals to 1.53 dB.
• Calderbank (1990) introduced a shaping technique
  that uses a shaping code to select between
  subconstellations of equal size.
• After Turbo codes
• Wachsmann (1999) combined shaping in the
  framework of multilevel codes. Shaping included a
  multidimensional trellis shaping code.
• Rimoldi (1997) also used a multilevel coding
  scheme by adding together independent encoders.
  The central limit theorem ensures the Gaussian
  distribution of the sum.
• Wei (2002) applied a trellis code used as an
  inner code and a parity check code as an outer
  shaping code

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Capacity gain

• Consider an AWGN channel having discrete inputs c
  with discrete probabilities
  , denoting the noise variance by
  we can express the output pdf function as
• The capacity of the discrete channel is given by
  the maximum of the mutual information
• We consider the power reduction compared to
  equiprobable transmission as the desired capacity
  gain.

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Capacity gain(2)

Optimization for all possible input probabilities
is difficult. Therefore we turn to a sub-optimal
discrete Gaussian distribution, where K is a
normalization factor. The parameter
governs the tradeoff between average power
and entropy H(c). As an example, consider the
transmission of R2.0 and 3.0 bits/dim using a
16-PAM constellation. The constellation points
are c1, .. ,c16 -15,-13,-11,-1,1,1,3,,11,13
,15.
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• It was first shown by R.Gallager (1968) that
  binary codes can be used for mappings of
  nonequiprobable letters that achieve capacity on
  an arbitrary discrete memoryless channel.
• Our Shaping algorithm

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Proposed shaping algorithm
We approximate the discrete Gaussian distribution
by using a binary distribution. Calculating
the capacity gain with this distribution
gives
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Proposed shaping algorithm(2)
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The encoder
In pragmatic binary turbo coded modulation a
single binary turbo code of rate 1/3 is used as
the component code. Its encoder outputs are
suitably multiplexed and punctured to obtain
parity and information bits. The
spectral efficiency is
bits/s/Hz. The aim of the bit interleavers is to
spread as much as possible, after deinterleaving,
the bits associated to the same channel symbol.
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The mapper

• The signal mapper associates each word of m
  encoded bits into one of the M-PAM channel
  symbols.
• Mapping is performed differently with respect to
  the signaling method. In an equiprobable scheme,
  we map m encoded bits into one of the
  symbols using Gray code.
• In a nonequiprobable scheme we apply a table that
  maps m-bit equiprobable input words into
  nonequiprobable M-ary PAM symbols.
• The signal mapper also determines the ordering of
  the bits within the m-tuple defining the symbol.
• We reorder the the m-tuples so that parity bits
  will be associated to lowest LLRs, and
  systematic bits associated to the largest LLRs.
  In this way we obtain better results from the
  iterative process.

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The mapper
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The decoder

• The receiver calculates the log-likelihood
  function for each encoded binary digit. The
  stream of LLRs is then deinterleaved and
  depucntured before passing to the decoder.
• The decoder performs iterated maximum a
  posteriori (MAP) estimation using a turbo
  feedback mechanism. The decoder uses its
  processed output as a priori input for the next
  iteration.

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LLR calculation

• The LLR of the received bits can be expressed as
• For a received signal and using
  Bayes rule we can further write,
• is the set of signal mapper outputs having
  inputs
• We use the bit LLR calculation block in the turbo
  decoder iterations. The extrinsic data, the added
  values for both information and code bits will be
  used by the soft output mapper as a-priori input
  to the next iteration. We can write for the new
  a-priori probabilities

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Performance
We considered the performance in two
cases. Case1 Transmission rate R2.0 bits/dim.
Comparison between the two schemes
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Performance
Case2 Transmission rate R3.0 bits/dim.
Comparison between the two schemes
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Results
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Performance comparison

• Fragouli and Wessel (2001) showed that by careful
  code selection and symbol interleaver design
  they can reach SNR of 0.5 dB from constrained
  capacity (without shaping) for rate R1 and 2
  bits/dim using Turbo TCM.
• Wachsmann (1999) combined shaping with multilevel
  codes and achieved SNR within 1 dB from the
  Shannon limit for rate R2.0 bits/dim.
• Benedetto (2000) used a versatile improved binary
  pragmatic scheme using optimized encoders, and
  reached results within 0.2 dB of the best known
  TCM.
• Wei (2002) defined and applied a new algorithm
  called iterative viterbi decoding algorithm, in
  which a trellis code is used as an inner code and
  a parity check code is used as an outer code.
  Using trellis shaping the performance is 1.25 dB
  from the Shannon limit at a transmission rate of
  3.0 bits/dim.

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Summary and Conclusions

• The high gain was achieved by applying
  nonequiprobable signaling to pragmatic binary
  turbo coded modulation by using nonequiprobable
  signaling.
• We proposed a technique that makes it possible to
  approach the true capacity gain of a finite
  constellation AWGN channel.
• Our nonequiprobable signaling technique is very
  easy to implement and adds a negligible load on
  the turbo decoder.
• We showed for an example of 6 bits/QAM symbol, a
  gain of 0.93 dB out of the available 1.07 dB, and
  transmission within 1.2 dB from the Shannon limit.

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Future work

• Apply our shaping scheme to low density parity
  check codes (LDPC) or any other binary turbo or
  turbo-like code.
• Try to find a shaping scheme suitable for turbo
  TCM which is similar to our binary shaping
  scheme.
• Use the same scheme on multilevel codes.