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Abstract
Recent breakthroughs in forward error correction, in the form of low-density parity-check (LDPC) and turbo codes, have seen near Shannon limit performances especially for pointto-
point channels. The construction of capacity-achieving codes in relay channels, for LDPC codes in particular, is currently the subject of intense interest in the research
and development community. This thesis adds to this field, developing methods and supporting theory in designing capacity-achieving LDPC codes for decode-and-forward (DF) schemes in relay channels.
In the first part of the thesis, new theoretical results toward optimizing the achievable rate of DF scheme in half-duplex relay channels under simplified and pragmatic conditions (equal power or equal time allocation) are developed. We derive the closed-form solutions for the optimum parameters (time or power) that maximize the achievable rates of the
DF scheme in the half-duplex relay channel. We also derive the closed-form expression
for the DF achievable rates under these simplified and pragmatic conditions.
The second part of the thesis is dedicated to study the problem of designing several classes
of capacity-achieving LDPC codes in relay channels. First, a new ensemble of LDPC codes,
termed multi-edge-type bilayer-expurgated LDPC (MET-BE-LDPC) codes, is introduced
to closely approach the theoretical limit of the DF scheme in the relay channel. We propose
two design strategies for optimizing MET-BE-LDPC codes; the bilayer approach and
the bilayer approach with intermediate rates. Second, we address the issue of constructing
capacity-achieving distributed LDPC codes in the multiple-access and two-way relay channels,
with broadcast transmissions and time-division multiple accesses. We propose a new methodology to asymptotically optimize the code’s degree distribution when different segments
within the distributed codeword have been transmitted through separate channels
and experienced distinct signal-to-noise ratio in the relay system. Third, we investigate
the use of LDPC codes under the soft-decode-and forward (SDF) scheme in the half-duplex
relay channel. We introduce the concept of a K-layer doping matrix that enables one to
design the rate-compatible (RC) LDPC code with a lower triangular parity-check matrix,
subsequently allowing the additional parity bits to be linearly and systematically encoded
at the relay. We then present the soft-decoding and soft-re-encoding algorithms for the
designed RC-LDPC code so that the relay can forward soft messages to the destination
when the relay fails to decode the source’s messages. Special attention is given to the
detection problem of the SDF scheme. We propose a novel method, which we refer to as
soft fading, to compute the log-likelihood ratio of the received signal at the destination
for the SDF scheme.