52 lines
3.1 KiB
TeX
52 lines
3.1 KiB
TeX
\chapter{Introduction}%
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\label{chapter:introduction}
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Channel coding using binary linear codes is a way of enhancing the reliability
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of data by detecting and correcting any errors that may occur during
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its transmission or storage.
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One class of binary linear codes, \ac{LDPC} codes, has become especially
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popular due to being able to reach arbitrarily small probabilities of error
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at code rates up to the capacity of the channel \cite[Sec. II.B.]{mackay_rediscovery},
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while retaining a structure that allows for very efficient decoding.
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While the established decoders for \ac{LDPC} codes, such as \ac{BP} and the
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\textit{min-sum algorithm}, offer good decoding performance, they are suboptimal
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in most cases and exhibit an \textit{error floor} for high \acp{SNR}
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\cite[Sec. 15.3]{ryan_lin_2009}, making them unsuitable for applications
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with extreme reliability requirements.
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Optimization based decoding algorithms are an entirely different way of approaching
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the decoding problem.
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The first introduction of optimization techniques as a way of decoding binary
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linear codes was conducted in Feldman's 2003 Ph.D. thesis and a subsequent paper,
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establishing the field of \ac{LP} decoding \cite{feldman_thesis}, \cite{feldman_paper}.
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There, the \ac{ML} decoding problem is approximated by a \textit{linear program}, i.e.,
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a linear, convex optimization problem, which can subsequently be solved using
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several different algorithms \cite{alp}, \cite{interior_point},
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\cite{original_admm}, \cite{pdd}.
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More recently, novel approaches such as \textit{proximal decoding} have been
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introduced. Proximal decoding is based on a non-convex optimization formulation
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of the \ac{MAP} decoding problem \cite{proximal_paper}.
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The motivation behind applying optimization methods to channel decoding is to
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utilize existing techniques in the broad field of optimization theory, as well
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as to find new decoding methods not suffering from the same disadvantages as
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existing message passing based approaches or exhibiting other desirable properties.
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\Ac{LP} decoding, for example, comes with strong theoretical guarantees
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allowing it to be used as a way of closely approximating \ac{ML} decoding
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\cite[Sec. I]{original_admm},
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and proximal decoding is applicable to non-trivial channel models such
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as \ac{LDPC}-coded massive \ac{MIMO} channels \cite{proximal_paper}.
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This thesis aims to further the analysis of optimization based decoding
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algorithms as well as to verify and complement the considerations present in
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the existing literature.
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Specifically, the proximal decoding algorithm and \ac{LP} decoding using
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the \ac{ADMM} \cite{original_admm} are explored within the context of
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\ac{BPSK} modulated \ac{AWGN} channels.
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Implementations of both decoding methods are produced, and based on simulation
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results from those implementations the algorithms are examined and compared.
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Approaches to determine the optimal value of each parameter are derived and
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the computational and decoding performance of the algorithms is examined.
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An improvement on proximal decoding is suggested, achieving up to 1 dB of gain,
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depending on the parameters chosen and the code considered.
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