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@PHDTHESIS{Bcherer:82812,
      author       = {Böcherer, Georg},
      othercontributors = {Mathar, Rudolf},
      title        = {{C}apacity achieving probabilistic shaping for noisy and
                      noiseless channels},
      address      = {Aachen},
      publisher    = {Publikationsserver der RWTH Aachen University},
      reportid     = {RWTH-CONV-143171},
      pages        = {145 S. : graph. Darst.},
      year         = {2012},
      note         = {Aachen, Techn. Hochsch., Diss., 2012},
      abstract     = {In Shannon theory, the key step in calculating the capacity
                      of a communication channel is to determine the channel input
                      distribution that maximizes the mutual information between
                      transmitted and received symbols. For many channels, this so
                      called capacity-achieving distribution is not uniform. In
                      digital communication systems, there is a binary interface
                      that separates the system into a source-related part and a
                      channel-related part. A natural question is how the bit
                      stream at the binary interface can be mapped to channel
                      input symbols in such a way that the resulting distribution
                      is close to capacity-achieving. The topic of this thesis is
                      to answer this question. In this thesis, dyadic
                      distributions are considered. Dyadic distributions can be
                      generated by parsing the bit stream at the binary interface
                      by a prefix-free code. A device that implements this
                      procedure is called a prefix-free matcher. In the first part
                      of this thesis, three types of discrete memoryless channels
                      are considered. First, unconstrained channels, second,
                      channels with input symbols of unequal duration, and third,
                      channels with a constraint on the average symbol cost. Both
                      the noiseless and the noisy case are considered. A new
                      algorithm called geometric Huffman coding is proposed. It is
                      shown that this algorithm efficiently finds the optimal
                      prefix-free matcher for the noiseless case. For the
                      noiseless and the noisy case, it is shown that the generated
                      dyadic distributions are asymptotically capacity-achieving
                      when blocks of symbols are jointly generated and when the
                      blocklength goes to infinity. In the second part of the
                      thesis, noiseless channels with memory are considered. For
                      the general setting, a fundamental relation between the
                      combinatorial and the probabilistic notion of capacity is
                      derived. For noiseless channels that are generated by graphs
                      with finitely many states, it is shown that prefix-free
                      matchers can be used to transmit a binary stream over the
                      channel at a rate as close to capacity as desired. For
                      various examples prefix-free matchers achieve higher rates
                      than previously known techniques. The topic of the last part
                      is how prefix-free matchers can be combined with error
                      correction. For the general class of not necessarily binary
                      systematic block codes, analytic formulas for shaping gain
                      and coding gain are derived. The shaping gain determines the
                      performance loss that results from using a distribution
                      different from the capacity-achieving one, and the coding
                      gain quantifies the loss due to sub-optimal codes. The gains
                      are assessed for three modes of operation, namely when none,
                      some, or all symbols are matched to the capacity-achieving
                      distribution by a prefix-free matcher. It is proven that in
                      the latter case, the optimal shaping gain is asymptotically
                      achieved. The results are evaluated for a binary symmetric
                      channel where the input symbols zero and one are of unequal
                      duration. The numerical results reveal that prefix-free
                      matchers in practice allow to operate ldpc codes with a
                      shaping gain that does not degrade capacity.},
      keywords     = {Kanalkapazität (SWD) / Wahrscheinlichkeitsverteilung (SWD)
                      / Codierung (SWD) / Endlicher Automat (SWD) /
                      Low-Density-Parity-Check-Code (SWD) / Huffman-Code (SWD) /
                      Konvexe Optimierung (SWD)},
      cin          = {613410},
      ddc          = {620},
      cid          = {$I:(DE-82)613410_20140620$},
      shelfmark    = {94A05 * 94A17 * 94A45 * 94A15},
      typ          = {PUB:(DE-HGF)11},
      urn          = {urn:nbn:de:hbz:82-opus-40977},
      url          = {https://publications.rwth-aachen.de/record/82812},
}